We set

$\displaystyle\Gamma_{j}^{m}:=\\{\Theta\in C(D_{j},\mathbb{R}\times
H^{s}_{r}(\mathbb{R}^{N}))\mid\;$ $\Theta$ is ${\mathbb{G}}$-equivariant;
$\displaystyle\mathcal{I}^{m}(\Theta(0))\leq B_{m}-1;$
$\displaystyle\Theta(\xi)\notin\Omega,\ \mathcal{I}^{m}(\Theta(\xi))\leq
B_{m}-1\ \hbox{for all }\ \xi\in\partial D_{j}\\}$

and

$b_{j}^{m}:=\inf_{\Theta\in\Gamma_{j}^{m}}\sup_{\xi\in
D_{j}}\mathcal{I}^{m}(\Theta(\xi)).$

We notice that for $j=1$ we obtain $\Gamma_{1}^{m}\equiv\Gamma^{m}$ (up to an
even/odd extension from $[0,1]$ to $[-1,1]=D_{1}$) and $b_{1}^{m}\equiv
b_{m}$. So $\Gamma_{j}^{m}$ is a natural extension to build multiple
solutions.

As in the case of $\Gamma^{m}$ and $b_{m}$, we want to prove that
$\Gamma_{j}^{m}\neq\emptyset$ and that, for a fixed $k\in\mathbb{N}$, there
exists an $m_{k}\gg 0$ (possibly equal to $0$) such that, if $m>m_{k}$, then
$b_{j}^{m}<0$ for $j=1\dots k$.

######  Proposition 2.7.5.

  * (i)

For any $\lambda<\lambda_{0}$, $m>0$, $j\in\mathbb{N}$, we have
$\Gamma_{j}^{m}\neq\emptyset$ and $b_{j}^{m}\leq
a_{j}(\lambda)-e^{\lambda}\frac{m}{2}$.

  * (ii)

For any $k\in\mathbb{N}$, set

$m_{k}:=2\inf_{\lambda<\lambda_{0}}\frac{a_{k}(\lambda)}{e^{\lambda}}\geq 0$
(2.7.47)

we have, for any $m>m_{k}$

$b_{j}^{m}<0\quad\hbox{for}\ j=1,2,\dots,k.$

  * (iii)

$m_{k}=0$ for all $k\in\mathbb{N}$ if (g4) holds. That is,

$b_{j}^{m}<0\quad\hbox{for all}\ j\in\mathbb{N}.$

Proof. For (i), the proof is similar to Proposition 2.6.1. We just need to
set, for $\zeta_{\lambda}\in\Gamma_{j}(\lambda)$,

$\psi_{\lambda}(\xi):=\left\\{\begin{aligned}
(\lambda+L(1-2|\xi|),\,0)&&\hbox{ if $|\xi|\in[0,1/2]$,}\\\
\left(\lambda,\,\zeta_{\lambda}\left(\frac{\xi}{|\xi|}(2|\xi|-1)\right)\right)&&\hbox{
if $|\xi|\in(1/2,1]$}\end{aligned}\right.$

and we come up again to the same proof.

For (ii), (iii), we come up with a proof similar to Proposition 2.6.2,
observing in addition that $m_{k}\leq m_{k+1}$ since $a_{k}(\lambda)$ are
increasing in $k$.

By Proposition 2.4.2 and Theorem 2.5.1 $\mathcal{I}^{m}$ satisfies the
$(PSP)_{b}$ condition for $b<0$ and the deformation lemma holds. Let
$m_{k}\geq 0$ be a number given in Proposition 2.7.5. For $m>m_{k}$ we can see
that $b_{j}^{m}<0$ for $j=1,2,\dots,k$ are critical values of
$\mathcal{I}^{m}$. If $b_{j}^{m}$ are different, we directly have multiplicity
of solutions. To deal with the case $b_{j}^{m}=b_{j^{\prime}}^{m}$ for some
$j\not=j^{\prime}$, we need another family of minimax methods, which exploits
the topological information hidden in this equality.

##### Minimax values $c_{j}^{m}$

Let us define minimax families $\Lambda_{j}^{m}$ which allow to find multiple
solutions. We use an idea from [325]. In what follows, we denote by ${\rm
genus}(A)$ the genus of closed symmetric sets $A$ with $0\not\in A$ (see
Appendix A.6).

Define, for each $j\in\mathbb{N}$,

$\displaystyle\Lambda_{j}^{m}:=\\{A=\Theta(\overline{D_{j+l}\setminus Y})$
$\displaystyle\mid$ $\displaystyle l\geq 0,\;\Theta\in\Gamma_{j+l}^{m},$
$\displaystyle Y\subseteq D_{j+l}\setminus\\{0\\}\;\hbox{ is closed, symmetric
in $0$,}$ $\displaystyle\qquad\qquad\qquad\quad\;\hbox{and ${\rm genus}(Y)\leq
l$}\\}$

and

$c_{j}^{m}:=\inf_{A\in\Lambda_{j}^{m}}\sup_{A}\mathcal{I}^{m}.$

In the following lemma, we observe that $\Lambda_{j}^{m}$ includes, in some
way, $\Gamma^{m}_{j}$ and that it inherits the property that the paths
intersect $\partial\Omega$.

######  Lemma 2.7.6.

  * (i)

$\Lambda_{j}^{m}\neq\emptyset$;

  * (ii)

$c_{j}^{m}\leq b_{j}^{m}$;

  * (iii)

for any $A\in\Lambda_{j}^{m}$, we have $A\cap\partial\Omega\neq\emptyset$. As
a consequence, we obtain

$b_{m}=B_{m}=E_{m}\leq c_{j}^{m}.$

Proof. Indeed, we see that, by choosing $l=0$ and $Y=\emptyset$ we have

$\\{A=\Theta(D_{j})\,\mid\,\Theta\in\Gamma_{j}^{m}\\}\subset\Lambda_{j}^{m}$

from which come the first two claims.

Focus on the third claim. Let $A=\Theta(\overline{D_{j+l}\setminus Y})$ and
set $U:=\Theta^{-1}(\Omega)$. By the symmetry in $(\lambda,u)$ of $\Theta$ and
the symmetry in $u$ of $\Omega$ we have that $U$ is symmetric. Moreover, since
$\Theta(0)\in\Omega$, we have that $U\subset D_{j+l}\subset\mathbb{R}^{j+l}$
is a symmetric neighborhood of the origin. By Proposition A.14 we have

${\rm genus}(\partial U)=j+l.$ (2.7.48)

Observe in addition the following chain of inclusions

$\overline{\partial U\setminus Y}=\overline{(\partial U\cap D_{j+l})\setminus
Y}=\overline{(D_{j+l}\setminus Y)\cap\partial
U}\subseteq\overline{D_{j+l}\setminus Y}\cap\overline{\partial
U}=\overline{D_{j+l}\setminus Y}\cap\partial U$

thus

$\Theta\left(\overline{\partial U\setminus
Y}\right)\subseteq\Theta\left(\overline{D_{j+l}\setminus Y}\cap\partial
U\right)\subseteq\Theta\left(\overline{D_{j+l}\setminus
Y}\right)\cap\Theta\left(\partial U\right)=A\cap\Theta\left(\partial
U\right).$

Assume for the moment that it holds

$\Theta(\partial U)\subset\partial\Omega.$ (2.7.49)

Then by the previous computation we have

$\Theta\left(\overline{\partial U\setminus Y}\right)\subseteq
A\cap\partial\Omega.$

Thus, to reach the claim, we need to show that $\overline{\partial U\setminus
Y}\neq\emptyset$. But is an immediate consequence of (2.7.48) and Proposition
A.14 that

${\rm genus}(\overline{\partial U\setminus Y})\geq{\rm genus}(\partial U)-{\rm
genus}(Y)\geq(j+l)-l=j\geq 1$

which directly excludes the possibility that $\overline{\partial U\setminus
Y}$ is empty.

Focus now on (2.7.49); we first observe that, by continuity, we have

$\partial_{D_{j+l}}U=\partial_{D_{j+l}}\big{(}\Theta^{-1}(\Omega)\big{)}\subset\Theta^{-1}(\partial\Omega),$

where $\partial_{D_{j+l}}$ is the boundary with respect to the topology
restricted to $D_{j+l}$, but this is not enough, since $\partial_{D_{j+l}}U$
is generally smaller than $\partial U$ (the boundary made with respect to the
whole space $\mathbb{R}^{j+l}$), which is the one appearing in (2.7.48). Let
thus $\xi\in\partial U$; we need to show that $\Theta(\xi)\in\partial\Omega$.
By definition of $U$, we have
$\Theta(\xi)\in\Theta(\overline{\Theta^{-1}(\Omega)})\subset\overline{\Omega}$;
assume by contradiction $\Theta(\xi)\in\Omega$. We first observe that
$\xi\notin\partial D_{j+l}$, by definition of $\Theta\in\Gamma^{m}_{j+l}$,
thus $\xi$ is in the interior of $D_{j+l}$. We then can find a neighborhood
$N_{1}$ of $\xi$ (with respect to $\mathbb{R}^{j+l}$) contained in $D_{j+l}$,
and a neighborhood $M_{2}$ of $\Theta(\xi)$ contained in $\Omega$; set
$N:=N_{1}\cap\Theta^{-1}(M_{2})$, we have that $N$ is a neighborhood of $\xi$
(with respect to $\mathbb{R}^{j+l}$) contained in $U$, which implies that
$\xi$ is in the interior of $U$, absurd. This concludes the proof of the first
part.

We prove now the consequence. Indeed, for each $A\in\Lambda_{j}^{m}$ we have

$E_{m}=\inf_{\partial\Omega}\mathcal{I}^{m}\leq\inf_{\partial\Omega\cap
A}\mathcal{I}^{m}\leq\sup_{\partial\Omega\cap
A}\mathcal{I}^{m}\leq\sup_{A}\mathcal{I}^{m}$

and thus the claim passing to the infimum over $\Lambda_{j}^{m}$.

Let us now show the main properties of $\Lambda_{j}^{m}$ and $c_{j}^{m}$,
which will actually be the only ones used in the multiplicity result.

######  Proposition 2.7.7.

Let $j\in\mathbb{N}$.

  1. 1.

$\Lambda_{j}^{m}\neq\emptyset$;

  2. 2.

$\Lambda_{j+1}^{m}\subseteq\Lambda_{j}^{m}$, and thus $c_{j}^{m}\leq
c_{j+1}^{m}$;

  3. 3.

let $A\in\Lambda_{j}^{m}$ and $Z\subset\mathbb{R}\times
H^{s}_{r}(\mathbb{R}^{N})$ be ${\mathbb{G}}$-invariant, closed, and such that
$0\notin\overline{P_{2}(Z)}$ and ${\rm genus}(\overline{P_{2}(Z)})\leq i$.
Then $\overline{A\setminus Z}\in\Lambda_{j-i}^{m}$.

Fix now $k\in\mathbb{N}$, and let $m>m_{k}$, where $m_{k}$ has been introduced
in (2.7.47). Then

  * 4.

$c_{j}^{m}<0$ and $\mathcal{I}^{m}$ satisfies $(PSP)_{c_{j}^{m}}$;

  * 5.

if $A\in\Lambda_{j}^{m}$ and $\eta$ is a deformation as in Theorem 2.7.1 for
$b=c_{j}^{m}$, then $\eta(1,A)\in\Lambda_{j}^{m}$.

Proof. Properties $1)$ and $4)$ have already been shown in the Lemma 2.7.6,
while property $2)$ is a consequence of the definition. Let us see properties
$3)$ and $5)$.

  * 3)

Let $A=\Theta(\overline{D_{j+l}\setminus Y})\in\Lambda_{j}^{m}$ and let $Z$ be
${\mathbb{G}}$-invariant, closed and such that $0\notin\overline{P_{2}(Z)}$
and ${\rm genus}(\overline{P_{2}(Z)})\leq i$. Assume it holds

$\displaystyle\overline{A\setminus Z}$
$\displaystyle=\Theta((\overline{D_{j+l}\setminus Y)\setminus\Theta^{-1}(Z)})$
(2.7.50)
$\displaystyle=\Theta(\overline{D_{(j-i)+(l+i)}\setminus(Y\cup\Theta^{-1}(Z))});$

if ${\rm genus}(Y\cup\Theta^{-1}(Z))\leq l+i$ we have the claim. But this is a
direct consequence of the assumptions and Proposition A.14, since

$\displaystyle{\rm genus}(Y\cup\Theta^{-1}(Z))$ $\displaystyle\leq{\rm
genus}(Y)+{\rm genus}(\Theta^{-1}(Z))$ $\displaystyle\leq l+{\rm
genus}(\overline{h(\Theta^{-1}(Z))})$ $\displaystyle=l+{\rm
genus}(\overline{P_{2}(Z)})\leq l+i$

where we have set $h:=P_{2}\circ\Theta$, which is an odd map and thus
admissible for the genus.

Turn now to (2.7.50). Set $B:=D_{j+l}\setminus Y$ and $W:=\Theta^{-1}(Z)$ we
have to prove

$\overline{\Theta(\overline{B})\setminus\Theta(W)}=\Theta(\overline{B\setminus
W}).$

We have

$\overline{\Theta(\overline{B})\setminus\Theta(W)}\subseteq\overline{\Theta(\overline{B}\setminus
W)}\stackrel{{\scriptstyle(i)}}{{\subseteq}}\overline{\Theta(\overline{B\setminus
W})}\stackrel{{\scriptstyle(ii)}}{{=}}\Theta(\overline{B\setminus W})$

and

$\Theta(\overline{B\setminus
W})\stackrel{{\scriptstyle(iii)}}{{\subseteq}}\overline{\Theta(B\setminus
W)}\stackrel{{\scriptstyle(iv)}}{{=}}\overline{\Theta(B)\setminus\Theta(W)}\subseteq\overline{\Theta(\overline{B})\setminus\Theta(W)}$

where

    * (i)

is due to the fact that $W$ is closed;

    * (ii)

$\overline{B\setminus W}\subseteq D_{j+l}$ is compact, thus
$\Theta(\overline{B\setminus W})$ is closed;

    * (iii)

derives from the continuity of $\Theta$;

    * (iv)

is due to the fact that $W$ is a preimage.

  * 5)

Consider $0<\bar{\varepsilon}<1$, $b=c_{j}^{m}\geq B_{m}$ and $\eta$ as in the
deformation lemma, and fix $A=\Theta(\overline{D_{j+l}\setminus
Y})\in\Lambda_{j}^{m}$ with $\Theta\in\Gamma_{j+l}^{m}$. To show that
$\eta(1,A)\in\Lambda_{j}^{m}$ and conclude the proof, it is sufficient to show
that $\tilde{\Theta}:=\eta(1,\Theta)\in\Gamma_{j+l}^{m}$ as well.

    * $\bullet$

$\tilde{\Theta}(-\xi)=\eta(1,\Theta(-\xi))=\eta(1,\Theta_{1}(-\xi),\Theta_{2}(-\xi))=\eta(1,\Theta_{1}(\xi),-\Theta_{2}(\xi))$
and thus

$\displaystyle\big{(}\tilde{\Theta}_{1}(-\xi),\tilde{\Theta}_{2}(-\xi)\big{)}=\Big{(}\eta_{1}\big{(}1,\Theta_{1}(\xi),-\Theta_{2}(\xi)\big{)},\eta_{2}\big{(}1,\Theta_{1}(\xi),-\Theta_{2}(\xi)\big{)}\Big{)}$
$\displaystyle=$
$\displaystyle\Big{(}\eta_{1}\big{(}1,\Theta_{1}(\xi),\Theta_{2}(\xi)\big{)},-\eta_{2}\big{(}1,\Theta_{1}(\xi),\Theta_{2}(\xi)\big{)}\Big{)}=\big{(}\tilde{\Theta}_{1}(\xi),-\tilde{\Theta}_{2}(\xi)\big{)}$

which shows that $\tilde{\Theta}_{1}$ is even and $\tilde{\Theta}_{2}$ is odd.

    * $\bullet$

By Lemma 2.7.6, for $\xi=0$ and $\xi\in\partial D_{j+l}$ we have
$\mathcal{I}^{m}(\Theta(\xi))\leq B_{m}-1=E_{m}-1\leq
c_{j}^{m}-\bar{\varepsilon}$, thus $\Theta(\xi)\in[\mathcal{I}^{m}\leq
c_{j}^{m}-\bar{\varepsilon}]$. Therefore
$\tilde{\Theta}(\xi)=\eta(1,\Theta(\xi))=\Theta(\xi)$ for $\xi=0$ and
$\xi\in\partial D_{j+l}$, and the same properties are satisfied.

####  2.7.3 Multiplicity theorem

Fix $k\in\mathbb{N}^{*}$, and let $\Lambda_{j}^{m}$ and $c_{j}^{m}$ be given
in the previous Section for $j=1\dots k$. Exploiting the properties given in
Proposition 2.7.7, we can find multiple solutions.

######  Theorem 2.7.8.

Fix $k\in\mathbb{N}^{*}$, and assume $m>m_{k}$. We have that

$c_{1}^{m}\leq c_{2}^{m}\leq\dots\leq c_{k}^{m}<0$

are critical values of $\mathcal{I}^{m}$. Moreover

  * (i)

if, for some $q\geq 1$,

$c_{j}^{m}<c_{j+1}^{m}<\dots<c_{j+q}^{m}$

then we have $q+1$ different nonzero critical values, and thus $q+1$ different
(pairs of) nontrivial solutions of (2.1.2);

  * (ii)

if instead, for some $q\geq 1$,

$c_{j}^{m}=c_{j+1}^{m}=\dots=c_{j+q}^{m}\equiv b$ (2.7.51)

then

${\rm genus}(P_{2}(K_{b}^{PSP}))\geq q+1$ (2.7.52)

and thus (by Proposition A.14) $\\#P_{2}(K_{b}^{PSP})=+\infty$, which means
that we have infinite different solutions of (2.1.2).

Summing up, we have at least $k$ different (pairs of) solutions of (2.1.2)
which satisfy the Pohozaev identity (2.1.3).

Proof. It is sufficient to show only the property (2.7.52) on the genus:
indeed by choosing $q=0$ we have that, for each $j$,
$\\#(K_{c_{j}^{m}}^{PSP})\geq 1$ and thus $c_{j}^{m}$ is a nontrivial critical
value.

By the $(PSP)_{b}$ we have that $K_{b}^{PSP}$ is compact, thus
$P_{2}(K_{b}^{PSP})$ is compact; moreover it is symmetric with respect to $0$
and does not contain $0$ (see Corollary 2.4.3).

By Proposition A.14 we can find a (closed, symmetric with respect to origin,
not containing the zero) neighborhood $N$ of $P_{2}(K_{b}^{PSP})$ which
preserves the genus, i.e. ${\rm genus}(N)={\rm genus}(P_{2}(K_{b}^{PSP}))$. We
can easily think $N$ as a projection of a neighborhood $Z$ of $K_{b}^{PSP}$
(i.e. $N=P_{2}(Z)$) satisfying the properties of Proposition 2.7.7.

By Theorem 2.7.1, there exist a sufficiently small $\varepsilon$ and an $\eta$
such that $\eta([\mathcal{I}^{m}\leq b+\varepsilon]\setminus
Z)\subseteq[\mathcal{I}^{m}\leq b-\varepsilon]$. Corresponding to
$\varepsilon$, by definition of $c_{j}^{m}$, there exists an
$A\in\Lambda_{j+q}^{m}$ such that $\sup_{A}\mathcal{I}^{m}<b+\varepsilon$,
that is $A\subseteq[\mathcal{I}^{m}\leq b+\varepsilon]$. Thus, being
$\eta(1,\cdot)$ continuous

$\displaystyle\eta(1,\overline{A\setminus Z})$
$\displaystyle\subseteq\eta(1,\overline{[\mathcal{I}^{m}\leq
b+\varepsilon]\setminus Z})\subseteq\overline{\eta(1,[\mathcal{I}^{m}\leq
b+\varepsilon]\setminus Z)}$
$\displaystyle\subseteq\overline{[\mathcal{I}^{m}\leq
b-\varepsilon]}=[\mathcal{I}^{m}\leq b-\varepsilon],$

and hence

$\sup_{\eta(1,\overline{A\setminus Z})}\mathcal{I}^{m}\leq b-\varepsilon.$
(2.7.53)

On the other hand, assume by contradiction that ${\rm
genus}(P_{2}(K_{b}^{PSP}))\leq q$, i.e. ${\rm genus}(P_{2}(Z))\leq q$. We use
now the properties on $c_{j}^{m}$ and $\Lambda_{j}^{m}$.

Replacing $j$ with $j+q$ and $i$ with $q$ and applying Proposition 2.7.7, we
have $\overline{A\setminus Z}\in\Lambda_{j}^{m}$; by property 5) of
Proposition 2.7.7 we obtain $\eta(1,\overline{A\setminus
Z})\in\Lambda_{j}^{m}$, which implies (by definition of $c_{j}^{m}$)

$\sup_{\eta(1,\overline{A\setminus Z})}\mathcal{I}^{m}\geq c_{j}^{m}=b.$

This is a contradiction with (2.7.53), and thus concludes the proof.

Proof of Theorem 2.1.3. As consequence of Theorem 2.7.8, we derive (i). We
pass to prove (ii). Under condition (g4), we have $m_{k}=0$ for all
$k\in\mathbb{N}$. Thus for any $j\in\mathbb{N}$, $c_{j}^{m}$ is a critical
value of $\mathcal{I}^{m}$ and $c_{j}^{m}\leq b_{j}^{m}<0$. Since $c_{j}^{m}$
is an increasing sequence, we have $c_{j}^{m}\to\bar{c}\leq 0$ as
$j\to\infty$. We need to show that $\bar{c}=0$.

By contradiction we assume $\bar{c}<0$. Then $K^{PSP}_{\bar{c}}$ is compact
and $K^{PSP}_{\bar{c}}\cap(\mathbb{R}\times\\{0\\})=\emptyset$. It follows
that $q={\rm genus}(P_{2}(K^{PSP}_{\bar{c}}))<\infty$. Arguing as in the proof
of Theorem 2.7.8, let $\delta>0$ be such that $q={\rm
genus}(P_{2}(N_{\delta}(K^{PSP}_{\bar{c}})))<\infty$. By Theorem 2.7.1, there
exist $\varepsilon\in(0,1)$ small and $\eta:[0,1]\times\mathbb{R}\times
H^{s}_{r}(\mathbb{R}^{N})\to\mathbb{R}\times H^{s}_{r}(\mathbb{R}^{N})$
satisfying

$\eta(1,[\mathcal{I}^{m}\leq\bar{c}+\varepsilon]\setminus
N_{\delta}(K^{PSP}_{\bar{c}}))\subseteq[\mathcal{I}^{m}\leq\bar{c}-\varepsilon]$
(2.7.54)

and

$\eta(t,\lambda,u)=(\lambda,u)\quad\hbox{if $\,\mathcal{I}^{m}(\lambda,u)\leq
B_{m}-1$}.$ (2.7.55)

We can choose $j\in\mathbb{N}$ sufficiently large such that
$c_{j}^{m}>\bar{c}-\varepsilon$ and take $B\in\Lambda_{j+q}^{m}$ such that
$B\subset[\mathcal{I}^{m}\leq\bar{c}+\varepsilon]$. Then we have

$\overline{B\setminus N_{\delta}(K^{PSP}_{\bar{c}})}\in\Lambda_{j}^{m}.$

From equations (2.7.54), (2.7.55) we derive
$c_{j}^{m}\leq\bar{c}-\varepsilon$, which gives a contradiction.

######  Remark 2.7.9.

We observe that, even if the problem is invariant under translations, the
found solutions are not translations of a same solution since they all are
radially symmetric. Moreover, assuming (g4), since $0>c_{j}^{m}\to 0$ we
easily find a sequence of solutions with distinct energy levels.

###  2.8 $L^{2}$-minimum

In Theorems 2.1.1 and 2.1.2 we find a solution via mountain pass minimax
methods. We remark that this solution is characterized as minimizer of the
functional $\mathcal{L}$ on $\mathcal{S}_{m}$, where
$\mathcal{L}:\,H_{r}^{s}(\mathbb{R}^{N})\to\mathbb{R}$ is defined by

$\mathcal{L}(u):=\frac{1}{2}\|(-\Delta)^{s/2}u\|_{2}^{2}-\int_{\mathbb{R}^{N}}G(u)$

and $\mathcal{S}_{m}$ is the $L^{2}$-sphere in $H_{r}^{S}(\mathbb{R}^{N})$,
i.e.

$\mathcal{S}_{m}:=\\{u\in H_{r}^{s}(\mathbb{R}^{N})\mid\|u\|_{2}^{2}=m\\}.$

Set

$\kappa_{m}:=\inf_{u\in\mathcal{S}_{m}}\mathcal{L}(u).$

######  Proposition 2.8.1.

Assume (g1)–(g3), and let $m\geq m_{0}$, where $m_{0}$ is introduced in
Proposition 2.6.2. We have that the following statements hold.

  * (i)

The Mountain Pass level and the ground state level coincide, i.e.

$\kappa_{m}=b_{m}.$ (2.8.56)

In particular, thanks to Corollary 2.6.3, there exists a ground state of
$\mathcal{L}_{|\mathcal{S}_{m}}$.

  * (ii)

Every ground state of $\mathcal{L}_{|\mathcal{S}_{m}}$ satisfies the Pohozaev
identity (2.1.3) with $\mu$ the associated Lagrange multiplier. Thanks to
(2.8.56), the same conclusion holds for every Mountain Pass solution at level
$b_{m}$.

  * (iii)

Every ground state of $\mathcal{L}_{|\mathcal{S}_{m}}$ has a positive
associated Lagrange multiplier. This means that every ground state of
$\mathcal{L}_{|\mathcal{S}_{m}}$ is a solution of problem (2.1.2).

Moreover, if (g4) holds, then $m_{0}=0$.

Proof. (i) Let $u_{*}$ be the Mountain Pass solution obtained in Corollary
2.6.3, which verifies $\|u_{*}\|_{2}^{2}=m$. Thus,

$\kappa_{m}\leq{\mathcal{L}}(u_{*})=b_{m}<0.$ (2.8.57)

In particular, by (2.8.57) we can find a minimizing sequence
$(u_{n})_{n}\subset\mathcal{S}_{m}$ for $\kappa_{m}$ satisfying
${\mathcal{L}}(u_{n})<0$, and thus we can set

$e^{\lambda_{n}}:=\frac{2}{Nm}\left(s\|(-\Delta)^{s/2}u_{n}\|_{2}^{2}-N\mathcal{L}(u_{n})\right)>0$

so that ${\mathcal{P}}(\lambda_{n},u_{n})=0$, i.e.,
$(\lambda_{n},u_{n})\in\partial\Omega$. At this point Proposition 2.6.2
implies

$\kappa_{m}+o(1)={\mathcal{L}}(u_{n})={\mathcal{I}}^{m}(\lambda_{n},u_{n})\geq
E_{m}=b_{m}.$

Passing to the limit, together with (2.8.57), we have (2.8.56).

(ii) Let $u_{0}$ be a minimizer of ${\mathcal{L}}$ on $\mathcal{S}_{m}$.
Corresponding to $u_{0}$, there exists a Lagrange multiplier
$\mu_{0}\in\mathbb{R}$ such that

$(-\Delta)^{s/2}u_{0}+\mu_{0}u_{0}=g(u_{0}),$

and thus, in particular,

$\|(-\Delta)^{s/2}u_{0}\|_{2}^{2}+\mu_{0}\|u_{0}\|_{2}^{2}-\int_{\mathbb{R}^{N}}g(u_{0})u_{0}\,dx=0.$
(2.8.58)

We show first that $u_{0}$ satisfies the Pohozaev identity. In fact, we
consider the $\mathbb{R}$-action
$\Phi:\,\mathbb{R}\times\mathcal{S}_{m}\to\mathcal{S}_{m}$ defined by

$(\Phi_{\theta}v)(x):=e^{\frac{N}{2}\theta}v(e^{\theta}x),$ (2.8.59)

since $\|\Phi_{\theta}v\|_{2}^{2}=\|v\|_{2}^{2}$. Then we have

${\mathcal{L}}(\Phi_{\theta}u_{0})=\frac{1}{2}e^{2s\theta}\|(-\Delta)^{s/2}u_{0}\|_{2}^{2}-e^{-N\theta}\int_{\mathbb{R}^{N}}G\big{(}e^{\frac{N}{2}\theta}u_{0}\big{)}.$

Since $u_{0}$ is a minimizer, we have
$\frac{d}{d\theta}\bigr{|}_{\theta=0}{\mathcal{L}}(\Phi_{\theta}u_{0})=0$,
that is,

$s\|(-\Delta)^{s/2}u_{0}\|_{2}^{2}+N\int_{\mathbb{R}^{N}}G(u_{0})-\frac{N}{2}\int_{\mathbb{R}^{N}}g(u_{0})u_{0}\,dx=0.$
(2.8.60)

From (2.8.58) and (2.8.60), the Pohozaev identity follows

$\frac{N-2s}{2}\|(-\Delta)^{s/2}u_{0}\|_{2}^{2}+\frac{N}{2}\mu_{0}\|u_{0}\|_{2}^{2}-N\int_{\mathbb{R}^{N}}G(u_{0})=0.$
(2.8.61)

(iii) Finally, from (2.8.57) we have ${\mathcal{L}}(u_{0})=\kappa_{m}<0$, that
is

$\frac{1}{2}\|(-\Delta)^{s/2}u_{0}\|_{2}^{2}-\int_{\mathbb{R}^{N}}G(u_{0})=\kappa_{m}<0,$
(2.8.62)

which joined to (2.8.61) gives $\mu_{0}>0$. This concludes the proof.

######  Remark 2.8.2.

By [275, Theorem 4.1], we have that actually every $L^{2}$-minimum is radially
symmetric (up to a translation). Thus $\kappa_{m}$ coincide with the infimum
made on the $L^{2}$-ball of the whole space $H^{s}(\mathbb{R}^{N})$.

###  2.9 Relation between constrained and unconstrained problems

Let $0<\mu<\mu_{0}$ and $m>0$. By joining the results of Proposition 2.6.2 and
Proposition 4.2.9, we proved the following relation.

$\kappa(m)=\inf_{\mu\in(0,\mu_{0})}\big{(}p(\mu)-\mu\,m\big{)}$ (2.9.63)

where we slightly changed the definition of $L^{2}$-minimum

$\kappa(m):=\inf_{\begin{subarray}{c}u\in H^{s}_{r}(\mathbb{R}^{N})\\\
\frac{1}{2}\|u\|_{2}^{2}=m\end{subarray}}\left(\frac{1}{2}\|(-\Delta)^{s/2}u\|_{2}^{2}-\int_{\mathbb{R}^{N}}G(u)\right)$

and of Pohozaev minimum

$p(\mu):=\inf_{\begin{subarray}{c}u\in
H^{s}_{r}(\mathbb{R}^{N})\setminus\\{0\\}\\\
\|(-\Delta)^{s/2}u\|_{2}^{2}+2^{*}_{s}\left(\frac{\mu}{2}\|u\|_{2}^{2}-\int_{\mathbb{R}^{N}}G(u)\right)=0\end{subarray}}\left(\frac{1}{2}\|(-\Delta)^{s/2}u\|_{2}^{2}-\int_{\mathbb{R}^{N}}G(u)\,+\frac{\mu}{2}\|u\|_{2}^{2}\right);$

we recall that, when $s\in(\frac{1}{2},1)$ or $g\in
C^{\sigma}_{loc}(\mathbb{R})$ for some $\sigma>1-2s$, $p(\mu)$ is actually a
ground state level.

The relation between the unconstrained and the constrained problem is an old-
fashioned problem, which has been deeply investigated in a recent paper by
Jeanjean and Lu [236] in the case $s=1$. We see that equation (2.9.63) gives
an interesting relation between the two energy levels: this relation may be
also reformulated by saying that

$\kappa(m)=-p^{*}(m)$ (2.9.64)

where $p^{*}$ is the _Legendre transform_ of $a$. A relation of this type, but
in a different framework, has been also obtained by Dovetta, Serra and Tilli
in a very recent paper [162]. Here, relying on the convexity of the energy
functions (due to the polynomial shape of $g$), they exploit (2.9.64) in order
to achieve interesting results.

We believe thus that (2.9.64) could give more insights in the study of the
relation between these two problems.

##  3 Chapter Choquard-Hartree-Pekar equations: multiplicity of solutions

In this Chapter we study the following nonlinear _Choquard-Hartree-Pekar
equation_

$-\Delta u+\mu u=(I_{\alpha}*F(u))F^{\prime}(u)\quad\hbox{in}\
\mathbb{R}^{N},$

where $N\geq 3$, $\alpha\in(0,N)$, $I_{\alpha}$ is the Riesz potential, and
$F$ is an almost optimal subcritical nonlinearity. The goal is to prove
existence of infinitely many solutions $u\in H^{1}_{r}(\mathbb{R}^{N})$, by
assuming $F$ odd or even.

We analyze the two cases: $\mu$ is a fixed positive constant or $\mu$ is
unknown and the $L^{2}$-norm of the solution is prescribed, i.e.
$\int_{\mathbb{R}^{N}}u^{2}=m>0$. Since the presence of the nonlocality
prevents to apply the classical approach introduced by Berestycki and Lions in
[51], we implement a new construction of multidimensional odd paths, and we
find a nonlocal counterpart of their multiplicity result. In particular we
extend the existence result in [302], due to Moroz and Van Schaftingen.

This Chapter is mainly based on the paper [116].

###  3.1 Convolution with Riesz potential: a self-interaction

Given a nonlinearity $F\in C^{1}(\mathbb{R},\mathbb{R})$ and set
$f:=F^{\prime}$, we are interested to seek for multiple solutions $u\in
H^{1}_{r}(\mathbb{R}^{N})$ of the nonlocal equation

$-\Delta u+\mu u=(I_{\alpha}*F(u))f(u)\quad\hbox{in}\ \mathbb{R}^{N},$ (3.1.1)

where $N\geq 3$ and $\alpha\in(0,N)$. In literature the semilinear equation
(3.1.1) with nonlocal source has several physical motivations and it is
usually called nonlinear _Choquard_ (or _Hartree_ , or _Pekar_) _equation_.

In 1954 the equation (3.1.1) with $N=3$, $\alpha=2$ and
$F(s)=\frac{1}{2}|s|^{2}$, that is

$-\Delta u+\mu u=\left(\frac{1}{4\pi|x|}*|u|^{2}\right)u\quad\text{in
$\mathbb{R}^{3}$},$ (3.1.2)

was elaborated by Pekar in [313] (see also [260]) to describe the quantum
theory of a polaron at rest, through the use of the Newton potential
$\frac{1}{4\pi|x|}$. The idea of the convolution as a feature of interaction
of a body with itself was exploited also by other authors: in 1976 it was
arisen in the work [264] suggested by Choquard [106] on the modeling of an
electron trapped in its own hole, in a certain approximation to Hartree-Fock
theory of one-component plasma (see also [194, 196, 355]). In 1996 the same
equation was derived by Penrose in his discussion on the self-gravitational
collapse of a quantum mechanical wave-function [314, 315, 316, 299] (see also
[362, 361, 196]) and in that context it is referred as _Schrödinger-Newton
system_ (see (1.3.39)). See also Section 3.1 for a derivation concerning
exotic stars.

If $u$ is a solution of (3.1.2), then we notice that the wave function

$\psi(x,t)=e^{i\mu t}u(x),\quad(x,t)\in\mathbb{R}^{3}\times[0,+\infty)$

is a solitary wave of the time-dependent _Hartree equation_ [216]

$i\psi_{t}=-\Delta\psi-\left(\frac{1}{4\pi|x|}*|\psi|^{2}\right)\psi\quad\text{in
$\mathbb{R}^{3}\times(0,+\infty)$};$ (3.1.3)

thus (3.1.2) represents the stationary nonlinear Hartree equation.

As already pointed out in Chapter 2, the study of standing waves of (3.1.3)
has been pursed in two main directions, which opened two different challenging
research fields.

A first topic regards the search for solutions of (3.1.2) with a prescribed
frequency $\mu$ and free mass, the so-called _unconstrained_ problem. The
second line of investigation of the problem (3.1.3) consists of prescribing
the mass $m>0$ of $u$, thus conserved by $\psi$ in time

$\int_{\mathbb{R}^{3}}|\psi(x,t)|^{2}\,dx=m\quad\forall\,t\in[0,+\infty),$

and letting the frequency $\mu$ to be free. Such problem is usually said
_constrained_.

For the unconstrained problem, the first investigations for existence and
symmetry of the solutions to (3.1.2) go back to the works of Lieb [265] and
Menzala [293], and also to [355, 108, 299] by means of ordinary differential
equations techniques. We mention also the recent papers by Lenzmann [257] and
by Winter and Wei [375] about the nondegeneracy of the unique radial solution
of (3.1.2).

Variational methods were also employed to derive existence and qualitative
results of standing wave solutions for more generic values of $\alpha\in(0,N)$
and of power type nonlinearities $F(t)=\frac{1}{p}|t|^{p}$: in particular
Moroz and Van Schaftingen [300] (see also [304]) considered the special model

$-\Delta u+\mu u=(I_{\alpha}\ast|u|^{p})|u|^{p-2}u\quad\hbox{in}\
\mathbb{R}^{N},$ (3.1.4)

and they proved that (3.1.4) has solutions if

$2^{\\#}_{\alpha}=\frac{N+\alpha}{N}<p<\frac{N+\alpha}{N-2}=2^{*}_{\alpha}.$
(3.1.5)

When dealing with variational (and regular) solutions, they proved that range
(3.1.5) is optimal. Moreover in [300] they showed that all positive ground
states of (3.1.4) are radially symmetric and monotone decreasing about some
point and derived the decay asymptotics at infinity of such ground states (see
[109] for $p\geq 2$, and also [279]). Furthermore, in [206, 205, 332] the
authors study, for some values of $p$ and $\alpha$, least energy nodal
solutions, odd with respect to a hyperplane; see also [109, 128, 378, 372,
384] for other results on sign-changing solutions with various symmetries and
saddle type solutions.

Recently in [302] Moroz and Van Schaftingen considered the problem (3.1.1)
when $F$ is a Berestycki-Lions type function under the following general
assumptions:

  * (F1)

$F\in C^{1}(\mathbb{R},\mathbb{R})$;

  * (F2)

there exists $C>0$ such that, for every $s\in\mathbb{R}$,

$|sf(s)|\leq C\big{(}|s|^{2^{\\#}_{\alpha}}+|s|^{2^{*}_{\alpha}}\big{)};$

  * (F3)

$\lim_{s\to
0}\frac{F(s)}{|s|^{2^{\\#}_{\alpha}}}=0,\quad\lim_{s\to+\infty}\frac{F(s)}{|s|^{2^{*}_{\alpha}}}=0;$

  * (F4)

$F(s)\not\equiv 0$, that is, there exists $s_{0}\in\mathbb{R}$, $s_{0}\neq 0$
such that $F(s_{0})\neq 0$.

In particular they prove the following theorem (see [302, Theorems 1 and 4]).

######  Theorem 3.1.1 ([302]).

We have the following results.

  * •

Assume (F1)–(F4). Then there exists a ground state solution $u\in
H^{1}(\mathbb{R}^{N})$. Moreover $u\in W^{2,q}_{loc}(\mathbb{R}^{N})$ for each
$q\geq 1$ (in particular, $u$ is Hölder continuous);

  * •

Assume (F1)-(F2), $f$ odd and with constant sign on $(0,+\infty)$. Then every
ground state has strict constant sign (strictly positive or negative) and it
is radially symmetric with respect to some point in $\mathbb{R}^{N}$.

The qualitative result contained in Theorem 3.1.1 will be extended in this
thesis to the case $f$ even, see Theorem 4.5.3. The existence of an infinite
number of standing wave solutions to (3.1.2) was instead faced by Lions in
[271] (see also [128]); here the homogeneity of the source plays a crucial
role in order to work on finite dimensional subspaces. Similar ideas have been
applied in [8, 323] in presence of more general sources satisfying Ambrosetti-
Rabinowitz type conditions. We remark that all these multiplicity results deal
with odd power nonlinearities $f$.

To our knowledge it is still an open problem the existence of infinitely many
radially symmetric solutions for the nonlinear Choquard equation (3.1.1) under
the optimal assumptions (F1)–(F4) and symmetric conditions on the nonlocal
source term $(I_{\alpha}*F(u))f(u)$, and this is the aim of this Chapter. We
note that this nonlinear term is odd both if $f$ is even or odd.

Existence of a solution for the nonlinear Choquard equation (3.1.4) under mass
constraint has been obtained by Ye [388]; see also [261] for odd powers-sum
type functions. More recently, Cingolani and Tanaka in [124] obtained
existence of a solution $u\in H^{1}_{r}(\mathbb{R}^{N})$ to

$\left\\{\begin{aligned} -\Delta u&+\mu u=(I_{\alpha}*F(u))f(u)\quad\hbox{in}\
\mathbb{R}^{N},\\\ &\int_{\mathbb{R}^{N}}u^{2}dx=m,\end{aligned}\right.$
(3.1.6)

assuming that $F$ satisfies (F1), (F4) and it is $L^{2}$-subcritical, namely

  * (CF2)

there exists $C>0$ such that, set $2^{m}_{\alpha}=\frac{N+\alpha+2}{N}$, for
every $s\in\mathbb{R}$,

$|sf(s)|\leq C\big{(}|s|^{2^{\\#}_{\alpha}}+|s|^{2^{m}_{\alpha}}\big{)};$

  * (CF3)

$\lim_{s\to
0}\frac{F(s)}{|s|^{2^{\\#}_{\alpha}}}=0,\quad\lim_{s\to+\infty}\frac{F(s)}{|s|^{2^{m}_{\alpha}}}=0.$

The existence result in [124] relies on a Lagrangian formulation of the
problem, in the spirit of Chapter 2.

Multiplicity of radial standing wave solutions to (3.1.3) with prescribed
$L^{2}$-norm has been instead faced again by Lions in [271] (see also [118]
for the planar logarithmic Choquard equation); as regards instead the case of
general nonlinearities $f$, recently Bartsch et al. [37] obtained the
existence of infinitely many solutions of (3.1.6) by assuming that $f$ is an
odd function which satisfies monotonicity and Ambrosetti-Rabinowitz
conditions. We highlight that the restriction on odd functions is not just a
matter of symmetry of the functional, but it is related also to some sign
restriction on the function $f$. The authors in [37] rely on mountain pass and
Concentration-Compactness arguments, together with the use of a stretched
functional, i.e. a functional in an augmented space which takes into
consideration scaling properties and the Pohozaev identity.

It remains open the challenging problem of the existence of infinitely many
solutions for the constrained nonlinear Choquard equation (3.1.6) under
optimal assumptions on the nonlinearity $f$, when monotonicity and Ambrosetti-
Rabinowitz type conditions do not hold or $f$ is not odd.

In the present Chapter we will give an affirmative answer to both the
_unconstrained_ and _constrained_ problems when $F$ satisfies the general
Berestycki-Lions type assumptions (F1)–(F4) and (F1)-(CF2)-(CF3)-(F4)
respectively, together with the symmetric condition

  * (F5)

$F$ is odd or even.

We begin to notice that despite [124], where existence is investigated, to
gain multiplicity the symmetry of the function $F$ plays a crucial role. In
particular, we assume $F$ to be _odd_ or _even_ , which guarantees the
evenness of the energy functional associated to (3.1.1). We emphasize that the
possibility to assume both the symmetries on $F$ is a particular feature of
the nonlocal source: indeed, in the source-local case [51, 224] (see also
Chapter 2), the nonlinear term is usually assumed odd in order to get the
symmetry of the functional. We mention the recent paper [137] where the
existence of a single nonradial solution to (3.1.1) is obtained under the
condition (F5).

We start to analyze the constrained case, which appears, as usual, more
delicate. By virtue of [310], radially symmetric solutions to (3.1.6) can be
characterized as critical points of the $C^{1}$-functional
$\mathcal{L}:H^{1}_{r}(\mathbb{R}^{N})\to\mathbb{R}$

$\mathcal{L}(u):=\frac{1}{2}\int_{\mathbb{R}^{N}}|\nabla
u|^{2}\,dx-\frac{1}{2}\int_{\mathbb{R}^{N}}(I_{\alpha}*F(u))F(u)\,dx,$

constrained on the sphere

$\mathcal{S}_{m}:=\left\\{u\in
H^{1}_{r}(\mathbb{R}^{N})\mid\int_{\mathbb{R}^{N}}u^{2}\,dx=m\right\\}.$

A possible approach to problem (3.1.6) is to minimize $\mathcal{L}$ on the
sphere $\mathcal{S}_{m}$, whenever the functional is here bounded.
Nevertheless, in the spirit of Chapter 2, for the general class of
nonlinearities related to [50, 302], considered in this thesis, we introduce a
Lagrangian formulation of the nonlocal problem (3.1.6), extending a new
approach introduced by Hirata and Tanaka [224] for the local case. We
highlight again the advantage of this method, that can be suitably adapted to
derive multiplicity results of normalized solutions in several different
frameworks.

We recall here briefly the ideas of Chapter 2. Writing
$\mathbb{R}_{+}:=(0,+\infty)$, a solution $(\mu,u)\in\mathbb{R}_{+}\times
H^{1}_{r}(\mathbb{R}^{N})$ of (3.1.6) corresponds to a critical point of the
functional $\mathcal{I}^{m}:\mathbb{R}_{+}\times
H^{1}_{r}(\mathbb{R}^{N})\to\mathbb{R}$ defined by

$\mathcal{I}^{m}(\mu,u):=\frac{1}{2}\int_{\mathbb{R}^{N}}|\nabla
u|^{2}\,dx-\frac{1}{2}\int_{\mathbb{R}^{N}}(I_{\alpha}*F(u))F(u)\,dx+\frac{\mu}{2}\left(\int_{\mathbb{R}^{N}}u^{2}\,dx-m\right).$

We seek for critical points $(\mu,u)\in\mathbb{R}_{+}\times
H^{1}_{r}(\mathbb{R}^{N})$ of $\mathcal{I}^{m}$, namely weak solutions of
$\partial_{u}\mathcal{I}^{m}(\mu,u)=0$ and
$\partial_{\mu}\mathcal{I}^{m}(\mu,u)=0$.

Inspired by the Pohozaev identity, we introduce the Pohozaev functional
$\mathcal{P}:\mathbb{R}_{+}\times H^{1}_{r}(\mathbb{R}^{N})\to\mathbb{R}$ by
setting

$\mathcal{P}(\mu,u):=\frac{N-2}{2}\int_{\mathbb{R}^{N}}|\nabla
u|^{2}\,dx+N\frac{\mu}{2}\int_{\mathbb{R}^{N}}u^{2}\,dx-\frac{N+\alpha}{2}\int_{\mathbb{R}^{N}}(I_{\alpha}*F(u))F(u)\,dx$

and the _Pohozaev set_

$\Omega:=\big{\\{}(\mu,u)\in\mathbb{R}_{+}\times
H^{1}_{r}(\mathbb{R}^{N})\mid\mathcal{P}(\mu,u)>0\big{\\}}\cup\big{\\{}(\mu,0)\mid\mu\in\mathbb{R}_{+}\big{\\}}.$

We note that $\\{(\mu,0)\mid\mu\in\mathbb{R}_{+}\\}\subset{\it int}(\Omega)$
and thus

$\partial\Omega=\big{\\{}(\mu,u)\in\mathbb{R}_{+}\times
H^{1}_{r}(\mathbb{R}^{N})\mid\mathcal{P}(\mu,u)=0,\ u\not\equiv 0\big{\\}},$

where the interior and the boundary are taken with respect to the topology of
$\mathbb{R}_{+}\times H^{1}_{r}(\mathbb{R}^{N})$. Therefore
$(\mu,u)\in\partial\Omega$ if and only if $u\not\equiv 0$ satisfies the
Pohozaev identity $\mathcal{P}(\mu,u)=0$. We recognize a Mountain Pass
structure for the functional $\mathcal{I}^{m}$ in $\mathbb{R}_{+}\times
H_{r}^{1}(\mathbb{R}^{N})$, where the mountain is given by $\partial\Omega$.
We call $\partial\Omega$ a _Pohozaev mountain_ for $\mathcal{I}^{m}$. We
emphasize that under assumptions (F1)-(F2), if $u\in
H^{1}_{r}(\mathbb{R}^{N})$ solves $\partial_{u}\mathcal{I}^{m}(\mu,u)=0$ with
$\mu\in\mathbb{R}_{+}$ fixed, then $\mathcal{P}(\mu,u)=0$.

Using a variant of the Palais-Smale condition [224, 231], which takes into
account the Pohozaev identity, we will prove a deformation theorem which
enables us to apply minimax arguments in the product space
$\mathbb{R}_{+}\times H^{1}_{r}(\mathbb{R}^{N})$. We will prove the existence
of multiple $L^{2}$-normalized solutions detecting minimax structures in such
product space.

We state our main results.

######  Theorem 3.1.2.

Suppose $N\geq 3$, $\alpha\in(0,N)$ and (F1)-(CF2)-(CF3)-(F4)-(F5).

  * (i)

For any $k\in\mathbb{N}$ there exists $m_{k}\geq 0$ such that for every
$m>m_{k}$, the problem (3.1.6) has at least $k$ pairs of nontrivial, distinct,
radially symmetric solutions.

  * (ii)

Assume in addition an $L^{2}$-subcritical growth also at zero, i.e.

    * (CF4)

$\lim_{s\to 0}\frac{|F(s)|}{|s|^{2^{m}_{\alpha}}}=+\infty;$

additionally, if $F$ is odd, assume that there exists $\delta_{0}>0$ such that
$F$ has a constant sign in $(0,\delta_{0}]$ and

$\sup_{s\in(0,\delta_{0}],\,h\in[0,1]}\frac{F(sh)}{F(s)}<+\infty;$ (3.1.7)

for example, this is satisfied if $|F(s)|$ is assumed non-decreasing in
$[0,\delta_{0}]$.

Then $m_{k}=0$ for each $k\in\mathbb{N}$, that is for any $m>0$ the problem
(3.1.6) has countably many pairs of solutions $(\mu_{n},u_{n})_{n}$ satisfying
$\mathcal{L}(u_{n})<0$, $n\in\mathbb{N}$. Moreover we have

$\mathcal{L}(u_{n})\to 0\quad\hbox{as $n\to+\infty$}.$

######  Remark 3.1.3.

We comment condition (3.1.7). Set

$M:=\sup_{s\in(0,\delta_{0}],\,h\in[0,1]}\frac{F(sh)}{F(s)}<+\infty$

we have, when $|F(s)|$ is non-decreasing, $M=1$. As a nontrivial example one
can consider $\beta\in(2^{\\#}_{\alpha},2^{m}_{\alpha})$ and $F$ oscillating
near zero between $|s|^{\beta}$ and $2|s|^{\beta}$, so that $M\leq 2$; for
instance the odd extension of

$F(s):=s^{\beta}\big{(}2+\sin(\tfrac{1}{s})\big{)}\quad\hbox{as $s\to
0^{+}$}.$

If instead $F$ oscillates (not strictly) between $|s|^{\beta_{1}}$ and
$|s|^{\beta_{2}}$, with $2^{\\#}_{\alpha}<\beta_{1}<\beta_{2}<2^{m}_{\alpha}$,
then $M=+\infty$; thus for instance the odd extension of

$F(s):=s^{\beta_{1}}\big{(}1+\sin(\tfrac{1}{s})\big{)}+s^{\beta_{2}}\big{(}1-\sin(\tfrac{1}{s})\big{)}\quad\hbox{as
$s\to 0^{+}$}$

is not covered by (3.1.7).

######  Remark 3.1.4.

We observe that, by substituting $F$ with $-F$, there is no loss of generality
in assuming

$F(s_{0})>0\quad\hbox{for some $s_{0}\neq 0$}$

in (F4) ($s_{0}$ can be chosen positive if, for example, (F5) holds) and

$\lim_{s\to 0^{+}}\frac{F(s)}{|s|^{2^{m}_{\alpha}}}=+\infty$

in (CF4). Thus, for the remaining part of the Chapter, we assume this
positivity on the right-hand side of zero.

A key point of the argument is the construction of multidimensional odd paths.
When $f$ satisfies some Ambrosetti-Rabinowitz condition (i.e., $F$ can be
estimated from below by an homogeneous function $|t|^{p}$), the construction
of such a path classically relies on the equivalence of the $H^{1}$-norm and
the $L^{p}$-norm on finite dimensional subspaces of $H^{1}(\mathbb{R}^{N})$.
When such condition is no longer available, a finer construction is needed: in
the celebrated paper [51] Berestycki and Lions build this path for a local
problem by exploiting an inductive process based on piecewise affine
functions.

In our nonlocal case, in order to prove the existence of multiple solutions
for $m\gg 0$ (point ($i)$ of Theorem 3.1.2), unlike the elaborated approach of
[51] we can obtain the existence of a multidimensional odd path by exploiting
the positivity of the Riesz potential functional. A similar approach can be
implemented to gain existence of infinitely many solutions for any $m>0$ when
$F$ is even (first part of point $(ii)$ of Theorem 3.1.2), since in this case
$F$ can be assumed positive in a neighborhood of the origin. See anyway Remark
3.1.6 below.

A quite delicate issue, instead, comes up when $F$ is odd. Differently from
[224] and Chapter 2, the classical argument given by [51] cannot be applied
directly in the context of nonlinear Choquard equations because of the
presence of a nonlocal source, and we need to implement a new approach to gain
the existence of an admissible odd path.

To this aim we proceed by finding suitable annuli: using characteristic
functions corresponding to the annuli, we construct our multidimensional odd
paths. Here interactions between these characteristic functions produced by
the Riesz potential play a crucial role, in particular the index $\alpha$ is
related to the strength of interaction and the case $\alpha\in(0,1]$ reveals
to be more delicate. To this aim we use sharp estimates for the Riesz
potential obtained by Thim [359].

As a further byproduct of the previous approach we gain the existence of
infinitely many solutions for the unconstrained problem. More precisely,
defined the $C^{1}$-functional
$\mathcal{J}_{\mu}:H^{1}_{r}(\mathbb{R}^{N})\to\mathbb{R}$ by setting

$\mathcal{J}_{\mu}(u):=\frac{1}{2}\int_{\mathbb{R}^{N}}|\nabla
u|^{2}\,dx+\frac{\mu}{2}\int_{\mathbb{R}^{N}}u^{2}\,dx-\frac{1}{2}\int_{\mathbb{R}^{N}}(I_{\alpha}*F(u))F(u)\,dx,$

we establish the following result.

######  Theorem 3.1.5.

Suppose $N\geq 3$, $\alpha\in(0,N)$ and $\mu>0$ fixed. Assume that (F1)–(F5)
hold. Then there exist countably many radial solutions $(u_{n})_{n}$ of the
nonlinear Choquard equation (3.1.1). Moreover we have

$\mathcal{J}_{\mu}(u_{n})\to+\infty\quad\hbox{as $n\to+\infty$}.$

Our multiplicity result is the counterpart of what done in [51] for the local
case with odd nonlinearities and extend the existence result in [302] due to
Moroz and Van Schaftingen.

######  Remark 3.1.6.

We highlight that the easier approach for building a multidimensional path,
based on the positivity of the Riesz kernel (Proposition 1.3.2), cannot
generally be applied to more generally frameworks (also if $F$ is even); for
examples, when dealing with kernels $K=K(x,y)$ which do not makes the
functional

$g\mapsto\int_{\mathbb{R}^{N}}\int_{\mathbb{R}^{N}}K(x,y)g(x)g(y)$

positive (for example, $K(x,y)$ sign-changing). In this case, the approach
here developed, based on suitable annuli, might instead be adapted. This is an
interesting line of research for the future.

The Chapter is organized as follows. In Section 3.2 we focus on the
construction of multidimensional paths, by dealing first with an easier
version based on the positivity of the Riesz potential, and then a refined
version based on some suitable annuli and essential interaction estimates for
non-local terms. Section 3.3 is then dedicated to the study of the asymptotic
behaviour of the mountain pass values, according to variable values of $\mu$.
Afterwards, in Section 3.4, we detect a mountain pass structure, built on the
Pohozaev mountain, for the constrained case, and in Section 3.5 we derive a
Palais-Smale-Pohozaev condition. In Section 3.6 we introduce an augmented
functional which will be used to gain a deformation lemma, and we further
study suitable minimax values defined through the tool of the genus which
allows to prove the main Theorem 3.1.2. Finally in Section 3.7 we deal with
the unconstrained case by proving Theorem 3.1.5.

###  3.2 Multidimensional annuli-shaped paths: even and odd nonlinearities

In this Chapter we briefly denote by $q$ the lower-critical exponent
$2^{\\#}_{\alpha}$ and by $p$ the $L^{2}$-critical exponent $2^{m}_{\alpha}$,
i.e.

$q:=2^{\\#}_{\alpha}=\frac{N+\alpha}{N},\quad
p:=2^{m}_{\alpha}=\frac{N+\alpha+2}{N}.$

Again, to avoid problems with the boundary of $\mathbb{R}_{+}$, we write from
now on (see Section 4.2.2 for a different approach)

$\mu\equiv e^{\lambda}\in(0,+\infty),\quad\lambda\in\mathbb{R}.$

We also set

$\mathcal{D}(u):=\mathcal{D}_{\alpha}(F(u),F(u))=\int_{\mathbb{R}^{N}}(I_{\alpha}*F(u))F(u)\,dx.$

Using Proposition 1.3.1 and (F1)-(F2), we notice that $\mathcal{D}$ is
continuous on $L^{2}(\mathbb{R}^{N})\cap L^{2^{*}}(\mathbb{R}^{N})$, where
$2^{*}=\frac{2N}{N-2}$ is the Sobolev critical exponent, and thus continuous
on $H^{1}_{r}(\mathbb{R}^{N})$; notice that if we assume (CF2), then
$\mathcal{D}$ is continuous also on $L^{2}(\mathbb{R}^{N})\cap
L^{2+\frac{4}{N+\alpha}}(\mathbb{R}^{N})$.

To deal with the unconstrained problem, we further define the
$C^{1}$-functional $\mathcal{J}:\mathbb{R}\times
H^{1}_{r}(\mathbb{R}^{N})\to\mathbb{R}$ by setting

$\mathcal{J}(\lambda,u):=\frac{1}{2}\|\nabla
u\|_{2}^{2}-\frac{1}{2}\mathcal{D}(u)+\frac{e^{\lambda}}{2}\|u\|_{2}^{2},\quad(\lambda,u)\in\mathbb{R}\times
H^{1}_{r}(\mathbb{R}^{N}).$ (3.2.8)

For a fixed $\lambda\in\mathbb{R}$, $u\in H^{1}_{r}(\mathbb{R}^{N})$ is
critical point of $\mathcal{J}(\lambda,\cdot)$ if and only if $u$ solves
(weakly)

$-\Delta u+e^{\lambda}u=(I_{\alpha}*F(u))f(u)\quad\hbox{in}\ \mathbb{R}^{N}.$
(3.2.9)

In this Section we study the geometry of

$u\in H_{r}^{1}(\mathbb{R}^{N})\mapsto\mathcal{J}(\lambda,u)\in\mathbb{R},$

for a fixed $\lambda\in\mathbb{R}$. We introduce a sequence of minimax values
$a_{n}(\lambda)$, $n\in\mathbb{N}^{*}$: these values play important roles to
find multiple solutions for the constrained problem (Theorem 3.1.2) as well as
for the unconstrained problem (Theorem 3.1.5).

For $n\in\mathbb{N}^{*}$ and $\lambda\in\mathbb{R}$ we introduce the set of
paths

$\Gamma_{n}(\lambda):=\big{\\{}\gamma\in
C(D_{n},H^{1}_{r}(\mathbb{R}^{N}))\mid\gamma\hbox{
odd},\,\mathcal{J}(\lambda,\gamma_{|\partial D_{n}})<0\big{\\}}$

and the minimax values

$a_{n}(\lambda):=\inf_{\gamma\in\Gamma_{n}(\lambda)}\sup_{\xi\in
D_{n}}\mathcal{J}(\lambda,\gamma(\xi)).$

For $n\geq 2$ the nonemptiness of $\Gamma_{n}(\lambda)$ has to be checked; for
$n=1$ we refer to [302, claim 1 of Proposition 2.1]. Classically, in the local
framework this fact was proved in [51] by constructing inductively piecewise
affine paths. This construction does not fit the nonlocality interaction given
by the Choquard term, thus we need another approach.

######  Proposition 3.2.1.

Assume (F1)–(F4) and $F(\pm s_{0})\neq 0$. Let $n\in\mathbb{N}^{*}$ and
$\lambda\in\mathbb{R}$. Then $\Gamma_{n}(\lambda)\neq\emptyset$, thus
$a_{n}(\lambda)$ is well defined. Moreover, $a_{n}(\lambda)>0$ and it is
increasing with respect to $\lambda$ and $n$.

Proof. Start observing that the polyhedron

$\Sigma:=\left\\{t=(t_{1},\dots,t_{n})\mid\max_{i=1,\dots,n}|t_{i}|=1\right\\}$

is homeomorphic to $\partial D_{n}$ (we passed from the $L^{2}$ to the
$L^{\infty}$ norm). Let us fix $e_{1},\dots,e_{n}\in
C^{\infty}_{c}(\mathbb{R}^{N})$, each of them between $0$ and $1$, radially
symmetric, equal to one in some annulus $A_{i}$, and such that their supports
are mutually disjoint. Then set $\gamma:\Sigma\to H^{1}_{r}(\mathbb{R}^{N})$
by

$\gamma(t)(x):=s_{0}\sum_{i=1}^{n}t_{i}e_{i}(x)$ (3.2.10)

for every $t=(t_{1},\dots,t_{n})\in\Sigma$ and $x\in\mathbb{R}^{N}$. The map
$\gamma$ is clearly odd and continuous. Moreover every $t\in\Sigma$ has at
least a nontrivial component $|t_{i}|=1$, thus we have
$F(\gamma(t)(x))=F(s_{0}t_{i}e_{i}(x))=F(\pm s_{0})\neq 0$ on $A_{i}$, hence
$F(\gamma(t))\not\equiv 0$. By Proposition 1.3.2 we have

$\mathcal{D}(\gamma(t))>0\quad\hbox{ for each $t\in\Sigma$}.$

Since

$\mathcal{D}\circ\gamma:\Sigma\to\mathbb{R}$

is continuous, and $\Sigma$ is compact, we obtain

$\min_{t\in\Sigma}\mathcal{D}(\gamma(t))=:C>0,$

i.e.

$\mathcal{D}(\gamma(t))\geq C>0\quad\hbox{ for each $t\in\Sigma$}.$

Set moreover $M:=\max_{\Sigma}\|\gamma\|_{H^{1}}^{2}\in\mathbb{R}.$ By
scaling, we obtain

$\displaystyle\mathcal{J}_{\lambda}(\gamma(t)(\cdot/\theta))$
$\displaystyle=\frac{\theta^{N-2}}{2}\|\nabla\gamma(t)\|_{2}^{2}+\frac{\theta^{N}e^{\lambda}}{2}\|\gamma(t)\|_{2}^{2}-\frac{\theta^{N+\alpha}}{2}\mathcal{D}(\gamma(t))$
$\displaystyle\leq\frac{\theta^{N-2}}{2}M+\frac{\theta^{N}e^{\lambda}}{2}M-\frac{\theta^{N+\alpha}}{2}C<0$

for some $\theta=\theta^{*}\gg 0$. Thus we consider
$\tilde{\gamma}:=\gamma(\cdot)(\cdot/\theta^{*}):\partial D_{n}\to
H^{1}_{r}(\mathbb{R}^{N})$. Finally we extend $\tilde{\gamma}$ to $D_{n}$ by

$\tilde{\gamma}(\xi):=|\xi|\gamma\left(\frac{\xi}{|\xi|}\right)$

for every $\xi\in D_{n}\setminus\\{0\\}$, and $\tilde{\gamma}(0):=0$.
Therefore $\tilde{\gamma}\in\Gamma_{n}(\lambda)\neq\emptyset$.

What remains to prove is the monotonicity and positivity of $a_{n}(\lambda)$.
Since $D_{n}\subset D_{n+1}$, we may regard for
$\gamma\in\Gamma_{n+1}(\lambda)$,

$\gamma_{|{D_{n}}}\in\Gamma_{n}(\lambda).$

Thus we have $a_{n}(\lambda)\leq a_{n+1}(\lambda)$. Since
$\mathcal{J}(\lambda,u)$ is monotone in $\lambda$, we also have the
monotonicity with respect to $\lambda$.

The positivity of $a_{1}(\lambda)$ is essentially obtained in [302] (see also
[124]). Thus

$a_{n}(\lambda)\geq a_{1}(\lambda)>0.$

In the proof of Proposition 3.2.1 we hardly relied on the positivity of the
Riesz potential functional given in Proposition 1.3.2, to obtain the existence
of path $\gamma:D_{n}\to H^{1}_{r}(\mathbb{R}^{N})$ and a $C>0$ such that

$\mathcal{D}(\gamma(\xi))\geq C>0\quad\hbox{ for each $\xi\in\partial
D_{n}$}.$ (3.2.11)

Notice moreover that this $\gamma$ satisfies
$\gamma(\theta\xi)=\theta\gamma(\xi)$ for any $\xi\in D_{n}$ and
$\theta\in\mathbb{R}$. Anyway, no good information on the constant $C$
appearing in (3.2.11) are given by this result.

A useful estimate in order to get infinitely many solutions for any $m>0$,
when (CF4) holds, is the one which relates $\mathcal{D}(\theta\gamma)$ to
$F(\theta s_{0})$ (see Lemma 3.2.8 and Section 3.3), that is

$\mathcal{D}(\theta\gamma(\xi))\geq C(F(\pm\theta s_{0}))^{2}\quad\hbox{ for
each $\xi\in\partial D_{n}$ and $\theta\in[0,1]$}$ (3.2.12)

for some uniform $C>0$. When $F$ is positive in a neighborhood of the origin
(which is the case of $F$ even and (CF4)), then one can build a suitable
$\gamma$ which satisfies (3.2.12).

######  Proposition 3.2.2.

Assume (F1)–(F4). Assume moreover that $F$ is positive in some
$[-s_{0},s_{0}]$, $F(\pm s_{0})\neq 0$. Let $n\in\mathbb{N}^{*}$ and
$\lambda\in\mathbb{R}$. Then the path $\gamma\in\Gamma_{n}(\lambda)$ defined
in (3.2.10) satisfies (3.2.12).

Proof. Assume the notation of the proof of Proposition 3.2.1. For each
$t\in\Sigma$, there exists $|t_{k}|=1$, thus, by exploiting that $\theta
s_{0}t_{i}e_{i}\in[-s_{0},s_{0}]$ we obtain

$\displaystyle\mathcal{D}(\theta\gamma(t))$
$\displaystyle=\int_{\mathbb{R}^{N}}\int_{\mathbb{R}^{N}}I_{\alpha}(x-y)F\left(\theta
s_{0}\sum_{i=1}^{n}t_{i}e_{i}(x)\right)F\left(\theta
s_{0}\sum_{j=1}^{n}t_{j}e_{j}(x)\right)$
$\displaystyle=\sum_{i=1}^{n}\sum_{j=1}^{n}\int_{\mathbb{R}^{N}}\int_{\mathbb{R}^{N}}I_{\alpha}(x-y)F\big{(}\theta
s_{0}t_{i}e_{i}(x)\big{)}F\big{(}\theta s_{0}t_{j}e_{j}(x)\big{)}$
$\displaystyle\geq\int_{\mathbb{R}^{N}}\int_{\mathbb{R}^{N}}I_{\alpha}(x-y)F\big{(}\theta
s_{0}t_{k}e_{k}(x)\big{)}F\big{(}\theta s_{0}t_{k}e_{k}(y)\big{)}$
$\displaystyle\geq\big{(}F(\pm\theta
s_{0})\big{)}^{2}\int_{A_{k}}\int_{A_{k}}I_{\alpha}(x-y)$

which is the claim.

When $F$ is odd (and thus it cannot be positive around the origin) it seems
not an easy task to build a $\gamma\in\Gamma_{n}(\lambda)$ satisfying
(3.2.12); indeed some estimate from below on

$\int_{\mathbb{R}^{N}}\int_{\mathbb{R}^{N}}I_{\alpha}(x-y)\frac{F\big{(}\theta\gamma(\xi)(x)\big{)}F\big{(}\theta\gamma(\xi)(y)\big{)}}{\big{(}F(\theta
s_{0})\big{)}^{2}}$

uniform for $\theta\to 0$ seems required; this is related to quotients of the
type $\frac{F(sh)}{F(s)}$ with $s\in(0,s_{0}]$ and $h\in[0,1]$. This is
essentially the meaning of condition (3.1.7).

To deal with this case we need a deep understanding of the Riesz potential on
radial functions. We thus give now a different construction for a
$\gamma\in\Gamma_{n}(\lambda)$: this procedure might be investigated also for
more general Choquard-type equations, where different kernels (possibly sign-
changing) appear.

We start by recalling a result contained in [359, Theorem 1] (see also [294,
Lemma 6.3] and references therein).

######  Theorem 3.2.3 ([359]).

Let $\alpha\in(0,N)$ and $u\in L^{1}(\mathbb{R}^{N})\cap
L^{\infty}(\mathbb{R}^{N})$ be radial. Then $I_{\alpha}*u$ is radial and

$(I_{\alpha}*u)(r)=r^{\alpha}\int_{0}^{\infty}F_{\alpha}\left(\frac{r}{\rho}\right)\left(\frac{\rho}{r}\right)^{\alpha}u(\rho)\,\frac{d\rho}{\rho},$
(3.2.13)

where $F_{\alpha}$ is positive and it satisfies, for some constants $C_{N,0}$,
$C_{N,\infty}$, $C_{N,\alpha}>0$,

$F_{\alpha}(s)\to C_{N,0}\ \hbox{ as $s\to
0$},\quad\frac{F_{\alpha}(s)}{s^{\alpha-N}}\to C_{N,\infty}\ \hbox{ as
$s\to+\infty$}$

and

$\frac{F_{\alpha}(s)}{G_{\alpha}(s)}\to 1\quad\hbox{as}\ s\to 1,$ (3.2.14)

with

$G_{\alpha}(s):=\begin{cases}C_{N,\alpha}&\hbox{if
$\alpha\in(1,N)$},\cr{C_{N,\alpha}}|\log|s-1||&\hbox{if
$\alpha=1$},\cr{C_{N,\alpha}}|s-1|^{\alpha-1}&\hbox{if
$\alpha\in(0,1)$}.\cr\end{cases}$ (3.2.15)

For a proof of Proposition 3.2.1, we prepare some notation and some estimates.
We introduce the annuli

$A(R,h):=\big{\\{}x\in\mathbb{R}^{N}\mid|x|\in[R-h,R+h]\big{\\}},\quad\chi(R,h;\cdot):=\chi_{A(R,h)}$

for any $R\gg h>0$. We have the following key estimates.

######  Lemma 3.2.4.

It results as $h\to 0$

$\iint_{\mathbb{R}^{N}\times\mathbb{R}^{N}}I_{\alpha}(x-y)\chi(1,h;x)\chi(1,h;y)\,dxdy\sim\begin{cases}h^{2}&\hbox{if
$\alpha\in(1,N)$},\cr h^{2}|\log h|&\hbox{if $\alpha=1$},\cr
h^{1+\alpha}&\hbox{if $\alpha\in(0,1)$}.\cr\end{cases}$

Proof. We apply Theorem 3.2.3 to $u(|x|)=\chi(1,h;|x|)$. In particular, by
(3.2.13) we have

$\displaystyle S_{h}$
$\displaystyle:=\iint_{\mathbb{R}^{N}\times\mathbb{R}^{N}}I_{\alpha}(x-y)u(x)u(y)\,dxdy=C\int_{0}^{\infty}(I_{\alpha}*u)(r)u(r)r^{N-1}\,dr$
$\displaystyle=C\int_{0}^{\infty}\int_{0}^{\infty}F_{\alpha}\left(\frac{r}{\rho}\right)\rho^{\alpha-1}r^{N-1}u(\rho)u(r)\,d\rho
dr=C\iint_{[1-h,1+h]^{2}}F_{\alpha}\left(\frac{r}{\rho}\right)\rho^{\alpha-1}r^{N-1}.$

First we note that

$\sup_{\rho,r\in[1-h,1+h]}\left|{\frac{r}{\rho}-1}\right|\to 0\quad\hbox{as}\
h\to 0.$

We consider the following three cases separately:

$\hbox{(i)}\ \alpha\in(1,N),\quad\hbox{(ii)}\ \alpha=1,\quad\hbox{(iii)}\
\alpha\in(0,1).$

(i) When $\alpha\in(1,N)$ we may assume $F(\frac{r}{\rho})\sim
C_{N,\alpha}>0$. Thus

$S_{h}\sim\iint_{[1-h,1+h]^{2}}\rho^{\alpha-1}r^{N-1}\,d\rho dr\sim h^{2}.$

(ii) When $\alpha=1$

$\displaystyle F_{\alpha}\left(\frac{r}{\rho}\right)$ $\displaystyle\sim
G_{1}\left(\frac{r}{\rho}\right)=C_{N,1}\Big{|}\log\Big{|}\frac{r}{\rho}-1\Big{|}\Big{|}$
$\displaystyle\sim\big{|}\log|r-\rho|-\log\rho\big{|}=-\log|r-\rho|+\log\rho.$

Thus

$S_{h}\sim\iint_{[1-h,1+h]^{2}}(-\log|r-\rho|+\log\rho)r^{N-1}\,d\rho dr.$

Set

$\displaystyle A_{h}$
$\displaystyle:=\big{\\{}(\rho,r)|\,|\rho-r|\leq\tfrac{1}{2}h,\,|r-1|\leq\tfrac{1}{2}h\big{\\}},$
$\displaystyle B_{h}$ $\displaystyle:=\big{\\{}(\rho,r)|\,|\rho-r|\leq
2h,\,|r-1|\leq h\big{\\}},$

we have

$A_{h}\subset[1-h,1+h]^{2}\subset B_{h}.$

Hence for some $C$, $C^{\prime}>0$

$C\iint_{A_{h}}(-\log|r-\rho|+\log\rho)r^{N-1}\,d\rho dr\leq S_{h}\leq
C^{\prime}\iint_{B_{h}}(-\log|r-\rho|+\log\rho)r^{N-1}\,d\rho dr.$ (3.2.16)

We compute

$\displaystyle\iint_{B_{h}}(-\log|r-\rho|+\log\rho)r^{N-1}\,d\rho dr$
$\displaystyle\leq\iint_{B_{h}}(-\log|r-\rho|+\log(1+h))(1+h)^{N-1}\,d\rho dr$
$\displaystyle=\iint_{[-2h,2h]\times[1-h,1+h]}(-\log|\tau|+\log(1+h))(1+h)^{N-1}\,d\tau
dr$
$\displaystyle=4h(1+h)^{N-1}\int_{0}^{2h}(-\log\tau)\,d\tau+8h^{2}(1+h)^{N-1}\log(1+h)$
$\displaystyle=4h(1+h)^{N-1}\big{(}-2h\log(2h)+2h\big{)}+8h^{2}(1+h)^{N-1}\log(1+h)$
$\displaystyle\leq C^{\prime\prime}h^{2}|\log h|\quad\hbox{as $h\to 0$}.$

Similarly we have

$\iint_{A_{h}}(-\log|r-\rho|+\log\rho)r^{N-1}\,d\rho dr\geq
C^{\prime\prime\prime}h^{2}|\log h|,$

from which we obtain

$S_{h}\sim h^{2}|\log h|\quad\hbox{as}\ h\to 0.$

(iii) When $\alpha\in(0,1)$

$F_{\alpha}\left(\frac{r}{\rho}\right)\sim
G_{\alpha}\left(\frac{r}{\rho}\right)=C_{N,\alpha}\left|{\frac{r}{\rho}-1}\right|^{\alpha-1}.$

Thus

$S_{h}\sim\iint_{[1-h,1+h]^{2}}\left|{\frac{r}{\rho}-1}\right|^{\alpha-1}\rho^{\alpha-1}r^{N-1}\,d\rho
dr=\iint_{[1-h,1+h]^{2}}\left|{r-\rho}\right|^{\alpha-1}r^{N-1}\,d\rho dr.$

Since

$C\iint_{A_{h}}|r-\rho|^{\alpha-1}(1-h)^{N-1}\,d\rho dr\leq S_{h}\leq
C^{\prime}\iint_{B_{h}}|r-\rho|^{\alpha-1}(1+h)^{N-1}\,d\rho dr,$

we have as in (3.2.16)

$S_{h}\sim h^{1+\alpha}\quad\hbox{as}\ h\to 0.$

This completes the proof.

We show how to use it to build a continuous odd map in
$L^{2}(\mathbb{R}^{N})\cap L^{2^{*}}(\mathbb{R}^{N})$. By a regularization
argument, we will obtain a map in $\Gamma_{n}(\lambda)$.

By scaling, we have

$\displaystyle\iint_{\mathbb{R}^{N}\times\mathbb{R}^{N}}I_{\alpha}(x-y)\chi(R,h;x)\chi(R,h;y)\,dxdy$
$\displaystyle=R^{N+\alpha}\iint_{\mathbb{R}^{N}\times\mathbb{R}^{N}}I_{\alpha}(x-y)\chi\Big{(}1,\frac{h}{R};x\Big{)}\chi\Big{(}1,\frac{h}{R};y\Big{)}\,dxdy$
$\displaystyle\sim\begin{cases}R^{N+\alpha}(\frac{h}{R})^{2}&\hbox{if
$\alpha\in(1,N)$},\cr R^{N+1}(\frac{h}{R})^{2}|\log\frac{h}{R}|&\hbox{if
$\alpha=1$},\cr R^{N+\alpha}(\frac{h}{R})^{1+\alpha}&\hbox{if
$\alpha\in(0,1)$}.\end{cases}$

For $R\geq 2$, we set the thickness of the annuli as

$h_{R}:=\begin{cases}R^{-\frac{N-2+\alpha}{2}}&\hbox{if $\alpha\in(1,N)$},\cr
R^{-\frac{N-1}{2}}(\log R)^{-1/2}&\hbox{if $\alpha=1$},\cr
R^{-\frac{N-1}{1+\alpha}}&\hbox{if $\alpha\in(0,1)$,}\end{cases}$

so that a uniform bound is gained.

######  Corollary 3.2.5.

We have

$\iint_{\mathbb{R}^{N}\times\mathbb{R}^{N}}I_{\alpha}(x-y)\chi(R,h_{R};x)\chi(R,h_{R};y)\,dxdy\in[C_{01},C_{02}]\quad\hbox{for}\
R\geq 2,$ (3.2.17)

where $C_{01}$, $C_{02}>0$ are independent of $R\geq 2$.

Proof. We check (3.2.17) only for $\alpha=1$. We have

$\displaystyle\quad\iint_{\mathbb{R}^{N}\times\mathbb{R}^{N}}I_{\alpha}(x-y)\chi\Big{(}1,\frac{h_{R}}{R};x\Big{)}\chi\Big{(}1,\frac{h_{R}}{R};y\Big{)}\,dxdy$
$\displaystyle\sim
R^{N+1}\Big{(}\frac{h_{R}}{R}\Big{)}^{2}\Big{|}\log\Big{(}\frac{h_{R}}{R}\Big{)}\Big{|}$
$\displaystyle=R^{N+1}\left(\frac{R^{-\frac{N-1}{2}}|\log
R|^{-1/2}}{R}\right)^{2}\left|{\log\left(\frac{R^{-\frac{N-1}{2}}|\log
R|^{-1/2}}{R}\right)}\right|$ $\displaystyle=(\log
R)^{-1}\left|{\log\Big{(}R^{-\frac{N+1}{2}}(\log R)^{-1/2}\Big{)}}\right|$
$\displaystyle=(\log R)^{-1}\left(\frac{N+1}{2}\log R+\frac{1}{2}\log(\log
R)\right)$ $\displaystyle\to\frac{N+1}{2}\quad\text{as}\ R\to\infty.$

whic shows the claim.

Next we compute the interaction effect between $\chi(R^{i},h_{R^{i}};\cdot)$
and $\chi(R^{j},h_{R^{j}};\cdot)$ with $i,j\in\mathbb{N}$, $i\neq j$ and $R\gg
1$.

######  Lemma 3.2.6.

For $i<j$ we have

$\iint_{\mathbb{R}^{N}\times\mathbb{R}^{N}}I_{\alpha}(x-y)\chi(R^{i},h_{R^{i}};x)\chi(R^{j},h_{R^{j}};y)\,dxdy\to
0\quad\hbox{as}\ R\to\infty.$

Proof. Since $\mathop{\rm supp}\chi(R,h_{R};\cdot)=A(R,h_{R})$ we get

$\displaystyle{\rm dist}\big{(}\mathop{\rm
supp}\chi(R^{i},h_{R^{i}};\cdot),\mathop{\rm
supp}\chi(R^{j},h_{R^{j}};\cdot)\big{)}$
$\displaystyle=(R^{j}-h_{R^{j}})-(R^{i}+h_{R^{i}})$
$\displaystyle=R^{j}-O(R^{i}).$

Thus

$\displaystyle I_{R}:=$
$\displaystyle\;\iint_{\mathbb{R}^{N}\times\mathbb{R}^{N}}I_{\alpha}(x-y)\chi(R^{i},h_{R^{i}};x)\chi(R^{j},h_{R^{j}};y)\,dxdy$
$\displaystyle\leq$
$\displaystyle\;C(R^{j}+{O(R^{i})})^{-(N-\alpha)}\|\chi(R^{i},h_{R^{i}};\cdot)\|_{1}\|\chi(R^{j},h_{R^{j}};\cdot)\|_{1}.$

Here

$\|\chi(R,h_{R};\cdot)\|_{1}=\mathop{\rm meas}\nolimits(A(R,h_{R}))\sim
CR^{N-1}h_{R}$

hence

$\displaystyle I_{R}$ $\displaystyle\leq
qC(R^{j}-O(R^{i}))^{-(N-\alpha)}R^{(N-1)i}h_{R^{i}}R^{(N-1)j}h_{R^{j}}$
$\displaystyle\leq C^{\prime}R^{(\alpha-1)j+(N-1)i}h_{R^{i}}h_{R^{j}}.$

When $\alpha\in(1,N)$, we have by the definition of $h_{R}$

$\displaystyle I_{R}$ $\displaystyle\leq
CR^{(\alpha-1)j+(N-1)i}R^{-\frac{1}{2}(N-2+\alpha)(i+j)}$
$\displaystyle=C^{\prime}R^{-\frac{1}{2}(N-\alpha)(j-i)}\to 0\quad\text{as}\
R\to\infty;$

when $\alpha=1$, we obtain

$\displaystyle I_{R}$ $\displaystyle\leq
C^{\prime}R^{(N-1)i}R^{-\frac{1}{2}(N-1)(i+j)}(\log R^{i})^{-\frac{1}{2}}(\log
R^{j})^{-\frac{1}{2}}$
$\displaystyle=C^{\prime}R^{-\frac{1}{2}(N-1)(j-i)}(ij)^{-\frac{1}{2}}(\log
R)^{-1}\to 0\quad\text{as}\ R\to\infty;$

when $\alpha\in(0,1)$,

$\displaystyle I_{R}$ $\displaystyle\leq
C^{\prime}R^{(\alpha-1)j+(N-1)i}R^{{-}\frac{N-1}{{1}+\alpha}(i+j)}$
$\displaystyle=C^{\prime}R^{-\frac{1}{1+\alpha}((N-{\alpha^{2}})j{-}\alpha(N-1)i)}\to
0\quad\text{as}\ R\to\infty.$

This concludes the proof.

We have now the tools to build a refined path $\gamma\in\Gamma_{n}(\lambda)$.

Proof of Proposition 3.2.1 (refined). We construct now a path
$\gamma\in\Gamma_{n}(\lambda)$; this path will moreover satisfy

$\max_{\xi\in D_{n},\,x\in\mathbb{R}^{N}}|\gamma(\xi)(x)|\leq s_{0}.$

Step 1: _Construction of an odd path in $L^{r}$._
For $n\geq 2$ we consider again the polyhedron

$\Sigma=\big{\\{}t=(t_{1},\dots,t_{n})\mid\max_{i=1,\dots,n}|t_{i}|=1\big{\\}}.$

For a large $R\gg 1$, which we will choose later, we define

$\gamma_{R}(t)(x):=\sum_{i=1}^{n}\hbox{sgn}(t_{i})\chi\big{(}R^{i},|t_{i}|h_{R^{i}};x\big{)}:\,\Sigma\to
L^{r}(\mathbb{R}^{N})$

where $r\in[1,+\infty]$. Here we regard $\chi(R^{i},0;x)\equiv 0$, and we
notice that $\gamma_{R}(t)$ is radial for each $t\in\Sigma$. Considered
$s_{0}$, we have

$\displaystyle{\cal D}(s_{0}\gamma_{R}(t))$
$\displaystyle=\sum_{i,j}F(\hbox{sgn}(t_{i})s_{0})F(\hbox{sgn}(t_{j})s_{0})\cdot$
$\displaystyle\quad\cdot\iint_{\mathbb{R}^{N}\times\mathbb{R}^{N}}I_{\alpha}(x-y)\chi(R^{i},|t_{i}|h_{R^{i}};x)\chi(R^{i},|t_{j}|h_{R^{i}};y)\,dxdy.$

We note that

  * (i)

For any $t=(t_{1},\dots,t_{n})\in\Sigma$, there exists at least one $t_{k}$
such that $|t_{k}|=1$.

  * (ii)

By Lemma 3.2.4,

$(F(\pm
s_{0}))^{2}\iint_{\mathbb{R}^{N}\times\mathbb{R}^{N}}I_{\alpha}(x-y)\chi(R^{k},h_{R^{k}};x)\chi(R^{k},h_{R^{k}};y)\,dxdy\geq
C_{0}.$

  * (iii)

By (i) and (ii),

$\sum_{i=1}^{n}(F(\pm
s_{0}))^{2}\iint_{\mathbb{R}^{N}\times\mathbb{R}^{N}}I_{\alpha}(x-y)\chi(R^{i},h_{R^{i}};x)\chi(R^{i},h_{R^{i}};y)\,dxdy\geq
C_{0}.$

  * (iv)

If $i\not=j$, by Lemma 3.2.6,

$\iint_{\mathbb{R}^{N}\times\mathbb{R}^{N}}I_{\alpha}(x-y)\chi(R^{i},h_{R^{i}};x)\chi(R^{j},h_{R^{j}};x)\,dxdy\to
0\quad\hbox{as}\ R\to\infty.$

By (i)–(iv), we have for sufficiently large $R\gg 1$,

${\cal D}(s_{0}\gamma_{R}(t))\geq C>0\quad\hbox{for all}\ t\in\Sigma.$
(3.2.18)

In what follows we fix $R\gg 1$ so that (3.2.18) holds.

Step 2: _Construction of an odd path in $H_{r}^{1}$._
For $0\leq h\ll R$ and $\varepsilon>0$, we set

$\chi_{\varepsilon}(R,h;x):=\begin{cases}1&\text{if}\ x\in A(R,h),\\\
1-\frac{1}{\varepsilon}{\rm dist}(x,A(R,h)),&\text{if}\ {\rm
dist}(x,A(R,h))\in(0,\varepsilon),\\\ 0&\text{otherwise}.\end{cases}$

Here we regard $A(R,0)=\\{x\in\mathbb{R}^{N}|\,|x|=R\\}.$ We note that

$\displaystyle\chi_{\varepsilon}(R,h;\cdot)\in H_{r}^{1}(\mathbb{R}^{N})\
\text{for}\ \varepsilon>0,$
$\displaystyle\chi_{\varepsilon}(R,h;\cdot)\to\chi(R,h;\cdot)\ \text{in}\
L^{r}(\mathbb{R}^{N})\ \text{as}\ \varepsilon\to 0\ \text{for all}\
r\in[1,\infty),$ $\displaystyle\mathop{\rm
supp}\chi_{\varepsilon}(R^{i},h_{R^{i}};\cdot)\cap\mathop{\rm
supp}\chi_{\varepsilon}(R^{j},h_{R^{j}};\cdot)=\emptyset\ \text{for}\ i\not=j\
\text{for $\varepsilon$ small}.$

We set

$\gamma_{\varepsilon,R}(t):=\sum_{i=1}^{n}{\rm
sgn}(t_{i})\chi_{\varepsilon}(R^{i},|t_{i}|h_{R^{i}};\cdot),\quad t\in\Sigma,$
(3.2.19)

$\gamma_{\varepsilon,R}:\Sigma\to H_{r}^{1}(\mathbb{R}^{N})$, continuous. By
(3.2.18) and the continuity of $\mathcal{D}$ on $L^{2}(\mathbb{R}^{N})\cap
L^{2^{*}}(\mathbb{R}^{N})$, we have for $\varepsilon>0$ small

$\mathcal{D}(s_{0}\gamma_{\varepsilon,R}(t))\geq C>0\quad\text{for all}\
t\in\Sigma.$

Since

$\mathcal{J}(\lambda,u(\cdot/\theta))=\frac{1}{2}\theta^{N-2}\|\nabla
u\|_{2}^{2}+\frac{e^{\lambda}}{2}\theta^{N}\|u\|_{2}^{2}-\frac{1}{2}\theta^{N+\alpha}{\cal
D}(u),$

we have for large $\theta\gg 1$

$\mathcal{J}(\lambda,s_{0}\gamma_{\varepsilon,R}(t)(\cdot/\theta))<0\quad\hbox{for
all}\ t\in\Sigma\approx\partial D_{n}.$

Considering $D_{n}=\\{st\mid s\in[0,1],\,t\in\Sigma\\}$ and extending
$s_{0}\gamma_{\varepsilon,{R}}(t)(\cdot/\theta)$ to $D_{n}$ by

$\widetilde{\gamma}(st):=s{s_{0}}\gamma_{\varepsilon,R}(t)(\cdot/\theta),$

finally we obtain a path $\widetilde{\gamma}\in\Gamma_{n}(\lambda)$.

######  Remark 3.2.7.

Even without assuming the positivity of $F$ (see Proposition 3.2.2), we notice
that the construction of an odd map in $L^{r}$ gets easier when $F$ is an even
function. Indeed there is no negative contribution given by the mixed
interactions. We give only an outline of the proof, highlighting that in this
case we do not need to use the fine Theorem 3.2.3 given by [359].

Define for every $i=1,\dots n$ and $s\in[-1,1]$ the annuli

$A_{i}(s):=\big{\\{}x\in\mathbb{R}^{N}\mid|x|\in[2ni-|s|,2ni+|s|]\big{\\}}.$

For every $t=(t_{1},\dots,t_{n})\in\Sigma$ we have that
$A_{1}(t_{1}),\dots,A_{n}(t_{n})$ are disjoint. Moreover, if $t_{i}=0$, then
$\mathop{\rm meas}\nolimits(A_{i}(t_{i}))=0$. Thus we define a continuous, odd
map by

$\gamma(t)(x):=\sum_{i=1}^{n}{\rm
sgn}(t_{i})\chi_{A_{i}(t_{i})}(x):\,\Sigma\to L^{2}(\mathbb{R}^{N})\cap
L^{2^{*}}(\mathbb{R}^{N}).$

Since $F$ is even, we obtain

$\displaystyle\mathcal{D}(s_{0}\gamma(t))$
$\displaystyle=\sum_{i,j}\iint_{A_{i}(t_{i})\times
A_{j}(t_{j})}I_{\alpha}(x-y)F(s_{0}{\rm
sgn}(t_{i})\chi_{A_{i}(t_{i})}(x))F(s_{0}{\rm
sgn}(t_{j})\chi_{A_{j}(t_{j})}(y))\,dxdy$
$\displaystyle=\big{(}F(s_{0})\big{)}^{2}\sum_{i,j}\iint_{A_{i}(t_{i})\times
A_{j}(t_{j})}I_{\alpha}(x-y)\,dxdy\big{(}F(s_{0})\big{)}^{2}\geq C>0,$

where $C$ does not depend on the specific $t$. The regularization to a
$H^{1}_{r}$-path can be done as in the general case (or by mollification), as
well as the extension to $D_{n}$.

We highlight that this construction can be adapted also to the local case, and
thus it gives a simplified construction of a multidimensional path in the
setting of Berestycki and Lions [51].

We are ready now to show that $\gamma_{R,\varepsilon}:\,\Sigma\to
H_{r}^{1}(\mathbb{R}^{N})$, defined in (3.2.19), has the desired property
(3.2.12).

######  Lemma 3.2.8.

Assume (F1)–(F5), and $F>0$ in some $(0,\delta_{0}]$. If $F$ is odd,
additionally assume (3.1.7). Then there exists a constant $A>0$ independent of
$s\in(0,\delta_{0}]$ and $t\in\Sigma$ such that

$\mathcal{D}(s\gamma_{R,\varepsilon}(t))\geq\frac{1}{2}\big{(}F(s)\big{)}^{2}(A+o_{\varepsilon}(1))\quad\hbox{as}\
\varepsilon\to 0;$

here $o_{\varepsilon}(1)$ is a quantity which goes to $0$ as $\varepsilon\to
0$ uniformly in $t\in\Sigma$ and $s\in(0,\delta_{0}]$.

Proof. We prove Lemma 3.2.8 in two steps.

Step 1: For $t\in\Sigma$, set

$a_{ij}(t):=\iint_{\mathbb{R}^{N}\times\mathbb{R}^{N}}I_{\alpha}(x-y)\chi(R^{i},|t_{i}|h_{R^{i}};x)\chi(R^{j},|t_{j}|h_{R^{j}};y)\,dxdy.$

Then for sufficiently large $R>0$, we have

$A:=\inf_{t\in\Sigma}\left(\sum_{i=1}^{n}a_{ii}(t)-\sum_{i\not=j}^{n}a_{ij}(t)\right)>0.$
(3.2.20)

This fact follows from (3.2.17) and Lemma 3.2.6. We fix $R\gg 1$ so that
(3.2.20) holds.

Step 2: _$\mathcal{D}(s\gamma_{R,\varepsilon}(t))\geq\frac{1}{2}F(s)^{2}A$ as
$\varepsilon\to 0$._
We note that for $\varepsilon>0$ small

$\mathop{\rm
supp}\chi_{\varepsilon}(R^{i},|t_{i}|h_{R^{i}};\cdot)\cap\mathop{\rm
supp}\chi_{\varepsilon}(R^{j},|t_{j}|h_{R^{j}};\cdot)=\emptyset\quad\hbox{for}\
i\not=j.$

Thus we have

$\displaystyle\mathcal{D}(s\gamma_{R,\varepsilon}(t))$
$\displaystyle=\sum_{i,j}\iint_{\mathbb{R}^{N}\times\mathbb{R}^{N}}I_{\alpha}(x-y)F(s\,{\rm
sgn}(t_{i})\chi_{\varepsilon}(R^{i},|t_{i}|h_{R^{i}};x))F(s\,{\rm
sgn}(t_{j})\chi_{\varepsilon}(R^{j},|t_{j}|h_{R^{j}};y))$ (3.2.21)
$\displaystyle=:\sum_{i,j}B_{ij}(s,t).$ (3.2.22)

We consider cases $i=j$ and $i\not=j$ separately.

First we focus on the case $i=j$. For both even and odd $F$ we have

$\displaystyle B_{ii}(s,t)$
$\displaystyle=\iint_{\mathbb{R}^{N}\times\mathbb{R}^{N}}I_{\alpha}(x-y)F(s\,{\rm
sgn}(t_{i})\chi_{\varepsilon}(R^{i},|t_{i}|h_{R^{i}};x))F(s\,{\rm
sgn}(t_{i})\chi_{\varepsilon}(R^{j},|t_{i}|h_{R^{i}};y))$
$\displaystyle=\iint_{\mathbb{R}^{N}\times\mathbb{R}^{N}}I_{\alpha}(x-y)F(s\chi_{\varepsilon}(R^{i},|t_{i}|h_{R^{i}};x))F(s\chi_{\varepsilon}(R^{j},|t_{i}|h_{R^{i}};y))$
$\displaystyle\geq\iint_{\mathbb{R}^{N}\times\mathbb{R}^{N}}I_{\alpha}(x-y)F(s\chi(R^{i},|t_{i}|h_{R^{i}};x))F(s\chi(R^{j},|t_{i}|h_{R^{i}};y))$
$\displaystyle=(F(s))^{2}a_{ii}(t),$ (3.2.23)

where we used the positivity of $F$ and the monotonicity of the integral. Next
we consider the case $i\not=j$ for even $F$. Since $F(s)\geq 0$ for
$s\in[-\delta_{0},\delta_{0}]$ we obtain

$B_{ij}(s,t)\geq 0\quad\hbox{for all}\ t\in\Sigma.$ (3.2.24)

Finally we consider the case $i\not=j$ for odd $F$. Since $|F(s)|=F(|s|)$ for
$s\in[-\delta_{0},\delta_{0}]$

$\displaystyle B_{ij}(s,t)$
$\displaystyle=\iint_{\mathbb{R}^{N}\times\mathbb{R}^{N}}I_{\alpha}(x-y)F(s{\,{\rm
sgn}(t_{i})}\chi_{\varepsilon}(R^{i},|t_{i}|h_{R^{i}};x))F(s{\,{\rm
sgn}(t_{j})}\chi_{\varepsilon}(R^{j},|t_{j}|h_{R^{j}};y))$ (3.2.25)
$\displaystyle\geq-\iint_{\mathbb{R}^{N}\times\mathbb{R}^{N}}I_{\alpha}(x-y)F(s\chi_{\varepsilon}(R^{i},|t_{i}|h_{R^{i}};x))F(s\chi_{\varepsilon}(R^{j},|t_{j}|h_{R^{j}};y)).$
(3.2.26)

Setting

$C_{i}(t,\varepsilon):=\big{\\{}x\mid{\rm
dist}(x,A(R^{i},|t_{i}|h_{R^{i}}))\in(0,\varepsilon)\big{\\}}$

we have

$\displaystyle\chi_{\varepsilon}(R^{i},|t_{i}|h_{R^{i}};x)\in(0,1)\quad\hbox{for}\
x\in C_{i}(t_{i},\varepsilon),$
$\displaystyle\chi_{\varepsilon}(R^{i},|t_{i}|h_{R^{i}};x)=\chi(R^{i},|t_{i}|h_{R^{i}};x)\quad\hbox{for}\
x\not\in C_{i}(t_{i},\varepsilon),$ $\displaystyle\mathop{\rm
meas}\nolimits(C_{i}(t_{i},\varepsilon))\to 0\quad\hbox{as}\ \varepsilon\to
0,{\hbox{ uniformly in $t\in\Sigma$}}.$

Thus for $r\in{[1,\infty)}$ and $s\in(0,\delta]$

$\displaystyle\left\|\frac{1}{F(s)}F(s\chi_{\varepsilon}(R^{i},|t_{i}|h_{R^{i}};\cdot))-\chi(R^{i},|t_{i}|h_{R^{i}};\cdot)\right\|_{r}^{r}$
$\displaystyle\leq$
$\displaystyle\int_{C_{i}(t_{i},\varepsilon)}\left|{\frac{1}{F(s)}F(s\chi_{\varepsilon}(R^{i},|t_{i}|h_{R^{i}};x))}\right|^{r}\,dx$
$\displaystyle=$
$\displaystyle\left(\max_{h\in[0,1]}\frac{|F(hs)|}{|F(s)|}\right)^{r}\mathop{\rm
meas}\nolimits(C_{i}(t_{i},\varepsilon))$ $\displaystyle\to 0$
$\displaystyle\quad\hbox{as}\ \varepsilon\to 0\ \hbox{uniformly in}\
t\in\Sigma.$

Here we use the fact that $\max_{h\in[0,1]}\frac{F(sh)}{F(s)}\leq C$, which
follows from the local almost-monotonicity assumption in (CF4). We note that
(3.2) implies, exploiting again (CF4)

$\displaystyle\left|{\frac{1}{(F(s))^{2}}\iint_{\mathbb{R}^{N}\times\mathbb{R}^{N}}I_{\alpha}(x-y)F(s\chi_{\varepsilon}(R^{i},|t_{i}|h_{R^{i}};x))F(s\chi_{\varepsilon}(R^{j},|t_{j}|h_{R^{j}};y))-a_{ij}(t)}\right|$
$\displaystyle\to 0\quad\hbox{as}\ \varepsilon\to 0.$ (3.2.27)

By (3.2.26) and (3.2.27),

$B_{ij}(s,t)\geq-(F(s))^{2}(a_{ij}(t)+o(1))\quad\hbox{as}\ \varepsilon\to 0.$
(3.2.28)

Thus, it follows from (3.2.22)–(3.2.24) and (3.2.28) that

$\displaystyle\mathcal{D}(s\gamma_{R,\varepsilon}(t))$
$\displaystyle\geq(F(s))^{2}\left(\sum_{i=1}^{n}a_{ii}(t)-\sum_{i\not=j}a_{ij}+o(1)\right)$
$\displaystyle\geq\frac{1}{2}(F(s))^{2}(A+o(1))$

This concludes the proof.

###  3.3 Asymptotic analysis of mountain pass values

We end this Section with some key estimates on the asymptotic behaviour of
$a_{n}(\lambda)$ as $\lambda\to\pm\infty$.

######  Proposition 3.3.1.

Assume (F1)–(F4) and let $n\in\mathbb{N}^{*}$.

  * (i)

If (CF3) holds, then
$\lim_{\lambda\to+\infty}\frac{a_{n}(\lambda)}{e^{\lambda}}=+\infty$.

  * (ii)

If (CF4) holds, then
$\lim_{\lambda\to-\infty}\frac{a_{n}(\lambda)}{e^{\lambda}}=0$.

Proof of (i) of Proposition 3.3.1. Recall $q={\frac{N+\alpha}{N}}$,
$p={\frac{N+\alpha+2}{N}}$, and write $\mu=e^{\lambda}$ (and consequently
adapt the notations) for the sake of simplicity.

Since $a_{n}(\mu)\geq a_{1}(\mu)$ for each $n\in\mathbb{N}^{*}$, it is
sufficient to show the claim for $n=1$. By (CF3), for any $\delta>0$ there
exists $C_{\delta}>0$ such that

$|F(s)|\leq\delta|s|^{p}+C_{\delta}|s|^{q}\quad\hbox{for all}\
s\in\mathbb{R}.$

For $v\in H^{1}_{r}(\mathbb{R}^{N})$, setting $u_{s}:=s^{N/2}v(s\cdot)$, we
have

$\displaystyle\mathcal{D}(u_{s})$
$\displaystyle=s^{-N-\alpha}\mathcal{D}(s^{N/2}v)$ $\displaystyle\leq
s^{-N-\alpha}\int_{\mathbb{R}^{N}}\left(I_{\alpha}*(\delta
s^{{\frac{N}{2}}p}|v|^{p}+C_{\delta}s^{{\frac{N}{2}}q}|v|^{q})\right)(\delta
s^{{\frac{N}{2}}p}|v|^{p}+C_{\delta}s^{{\frac{N}{2}}q}|v|^{q})\,dx$
$\displaystyle=s^{2}\int_{\mathbb{R}^{N}}\Big{(}I_{\alpha}*(\delta|v|^{p}+C_{\delta}s^{-1}|v|^{q})\Big{)}(\delta|v|^{p}+C_{\delta}s^{-1}|v|^{q})\,dx$
$\displaystyle=:s^{2}D_{\delta,C_{\delta}s^{-1}}(v),$ (3.3.29)

where we write for $\delta>0$ and $A\geq 0$,

$\displaystyle D_{\delta,A}(v)$
$\displaystyle:=\int_{\mathbb{R}^{N}}\Big{(}I_{\alpha}*(\delta|v|^{p}+A|v|^{q})\Big{)}(\delta|v|^{p}+A|v|^{q})\,dx,$
$\displaystyle\mathcal{J}_{\delta,A}(v)$ $\displaystyle:=\frac{1}{2}\|\nabla
v\|_{2}^{2}+\frac{1}{2}\|v\|_{2}^{2}-\frac{1}{2}D_{\delta,A}(v).$

We also denote by $b(\delta,A)$ the MP value of $\mathcal{J}_{\delta,A}$.
Taking into account the continuity and monotonicity property of $b(\delta,A)$
with respect of each variable $\delta$ and $A$ and observing that
$\mathcal{J}_{\delta,A}$ satisfies the (PS) condition, we have

$b(\delta,A)\to b(\delta,0)\quad\hbox{as}\ A\to 0^{+},$
$b(\delta,0)\to+\infty\quad\hbox{as}\ \delta\to 0^{+}.$

Thus, from (3.3) we have that

$\mathcal{J}(\mu,u_{s})\geq s^{2}\left(\frac{1}{2}\|\nabla
v\|_{2}^{2}+\frac{1}{2}\mu
s^{-2}\|v\|_{2}^{2}-\frac{1}{2}D_{\delta,C_{\delta}s^{-1}}(v)\right).$

Setting $s:=\sqrt{\mu}$, we obtain

$\mathcal{J}(\mu,u_{\sqrt{\mu}})\geq\mu\mathcal{J}_{\delta,C_{\delta}\mu^{-1/2}}(v)$

and thus ${\frac{a_{1}(\mu)}{\mu}}\geq b(\delta,C_{\delta}\mu^{-1/2})$, which
implies

$\liminf_{\mu\to\infty}{\frac{a_{1}(\mu)}{\mu}}\geq\lim_{A\to
0}b(\delta,A)=b(\delta,0).$

Since $\delta>0$ is arbitrary, we gain

$\lim_{\mu\to+\infty}{\frac{a_{1}(\mu)}{\mu}}=+\infty.$

We deal now with the proof of (ii) of Proposition 3.3.1. We highlight that,
when $F$ is even, the proof can be simplified (see [124], Proposition 2.3.6,
and Proposition 3.2.2).

The proof will be based on the key Lemma 3.2.8. We start noticing that, by
(CF4) and Remark 3.1.4, for some $\delta_{0}>0$

$F(s)>0\quad\hbox{for}\ s\in(0,\delta_{0}],$

which implies

  * (i)

when $F$ is even, $F(s)>0$ for all
$s\in[-\delta_{0},\delta_{0}]\setminus\\{0\\}$;

  * (ii)

when $F$ is odd, $F(s)<0$ for all $s\in[-\delta_{0},0)$.

By (CF4), we also note that there exists $L_{s}>0$ with $L_{s}\to+\infty$ as
$s\to 0^{+}$ such that

$F(\sigma)\geq L_{s}\sigma^{p}\quad\hbox{for all}\ \sigma\in[0,s].$ (3.3.30)

Proof of (ii) of Proposition 3.3.1. Let $\gamma_{R,\varepsilon}$ defined in
(3.2.19). For $s_{0}\in(0,\delta_{0}]$ and $\mu>0$, we consider the map

$st\in D_{n}\mapsto
ss_{0}\gamma_{R,\varepsilon}(t)(\cdot/\mu^{-\frac{1}{2}})\in
H_{r}^{1}(\mathbb{R}^{N}).$

We have by Lemma 3.2.8 (since $\varepsilon>0$ is here fixed small, we write
$A$ instead of $A+o_{\varepsilon}(1)$)

$\displaystyle\mu^{-1}\mathcal{J}(\mu,ss_{0}\gamma_{R,\varepsilon}(t)(\cdot/\mu^{-\frac{1}{2}}))$
$\displaystyle=\frac{1}{2}\mu^{-\frac{N}{2}}(ss_{0})^{2}\|\nabla\gamma_{R,\varepsilon}(t)\|_{2}^{2}+\frac{1}{2}\mu^{-\frac{N}{2}}(ss_{0})^{2}\|\gamma_{R,\varepsilon}(t)\|_{2}^{2}-\frac{1}{2}\mu^{-\frac{N}{2}p}\mathcal{D}(ss_{0}\gamma_{R,\varepsilon}(t))$
$\displaystyle\leq\frac{1}{2}\mu^{-\frac{N}{2}}(ss_{0})^{2}\|\gamma_{R,\varepsilon}(t)\|_{H^{1}}^{2}-\frac{1}{4}\mu^{-\frac{N}{2}p}(F(ss_{0}))^{2}A.$

Thus for $\mu$ small

$\mathcal{J}(\mu,s_{0}\gamma_{R,\varepsilon}(t)(\cdot/\mu^{-\frac{1}{2}}))<0\quad\hbox{for}\
t\in\Sigma,$

which implies that $st\mapsto
s_{0}\gamma_{R,\varepsilon}(t)(\cdot/\mu^{-\frac{1}{2}})$ is a path belonging
to $\Gamma_{n}(\mu)$. Moreover by (3.3.30)

$\displaystyle\mu^{-1}a_{n}(\mu)$
$\displaystyle\leq\max_{s\in[0,1],\,t\in\Sigma}\mu^{-1}\mathcal{J}(\mu,ss_{0}\gamma_{R,\varepsilon}(t)(\cdot/\mu^{-\frac{1}{2}}))$
$\displaystyle\leq\max_{s\in[0,1],\,t\in\Sigma}\frac{1}{2}\mu^{-\frac{N}{2}}(ss_{0})^{2}\|\gamma_{R,\varepsilon}(t)\|_{H^{1}}^{2}-\frac{1}{4}\mu^{-\frac{N}{2}p}(F(ss_{0}))^{2}A$
$\displaystyle\leq\max_{s\in[0,1],\,t\in\Sigma}\frac{1}{2}\mu^{-\frac{N}{2}}(ss_{0})^{2}\|\gamma_{R,\varepsilon}(t)\|_{H^{1}}^{2}-\frac{1}{4}L_{s_{0}}(\mu^{-\frac{N}{2}}(ss_{0})^{2})^{p}A$
$\displaystyle\leq C_{s_{0}},$

where

$C_{s_{0}}:=\sup_{\tau\geq
0,\,t\in\Sigma}\left(\frac{1}{2}{\tau}\|\gamma_{R,\varepsilon}(t)\|_{H^{1}}^{2}-\frac{1}{4}L_{s_{0}}A{\tau}^{p}\right)\in\mathbb{R}.$

Thus we have

$\limsup_{\mu\to 0^{+}}\mu^{-1}a_{n}(\mu)\leq C_{s_{0}}.$

Since $C_{s_{0}}\to 0$ as $s_{0}\to 0$, we have (ii) of Proposition 3.3.1.

###  3.4 The Pohozaev mountain

In this Section we start studying the Lagrangian formulation, applying the
previous asymptotic estimates to a Pohozaev geometry. We consider the
functional $\mathcal{I}^{m}:\mathbb{R}\times
H^{1}_{r}(\mathbb{R}^{N})\to\mathbb{R}$ defined by

$\mathcal{I}^{m}(\lambda,u):=\frac{1}{2}\|\nabla
u\|_{2}^{2}-\frac{1}{2}\mathcal{D}(u)+\frac{e^{\lambda}}{2}\bigl{(}\|u\|_{2}^{2}-m\bigr{)},\quad(\lambda,u)\in\mathbb{R}\times
H^{1}_{r}(\mathbb{R}^{N}).$ (3.4.31)

It is immediate that, for any $(\lambda,u)\in\mathbb{R}\times
H^{1}_{r}(\mathbb{R}^{N})$,

$\mathcal{I}^{m}(\lambda,u)=\mathcal{J}(\lambda,u)-\frac{e^{\lambda}}{2}m.$

If (F1)-(F2) hold, by [302, Theorems 2 and 3] we have that each solution $u$
of (3.2.9) belongs to $W^{2,2}_{loc}(\mathbb{R}^{N})$ and it satisfies the
Pohozaev identity

$\frac{N-2}{2}\|\nabla
u\|_{2}^{2}+\frac{N}{2}e^{\lambda}\|u\|_{2}^{2}-\frac{N+\alpha}{2}\,\mathcal{D}(u)=0$
(3.4.32)

or equivalently

$\frac{1}{2^{*}_{\alpha}}\|\nabla
u\|_{2}^{2}+\frac{e^{\lambda}}{2^{\\#}_{\alpha}}\|u\|_{2}^{2}-\mathcal{D}(u)=0$

where $2^{*}_{\alpha}=\frac{N+\alpha}{N-2}$ and
$2^{\\#}_{\alpha}=\frac{N+\alpha}{N}$ are the upper and the lower critical
exponents; again we see that essentially the identity means
$\frac{d}{d\theta}\mathcal{J}(\lambda,u(\cdot/e^{\theta}))_{|\theta=0}=0$.
Inspired by this relation, we also introduce the Pohozaev functional
$\mathcal{P}:\mathbb{R}\times H^{1}_{r}(\mathbb{R}^{N})\to\mathbb{R}$ by
setting

$\mathcal{P}(\lambda,u):=\frac{N-2}{2}\|\nabla
u\|_{2}^{2}-\frac{N+\alpha}{2}\mathcal{D}(u)+\frac{N}{2}e^{\lambda}\|u\|_{2}^{2},\quad(\lambda,u)\in\mathbb{R}\times
H^{1}_{r}(\mathbb{R}^{N}).$ (3.4.33)

We consider the action of ${\mathbb{G}}:=\mathbb{Z}_{2}$ on $\mathbb{R}^{n}$,
$n\in\mathbb{N}^{*}$, and on $\mathbb{R}\times H^{1}_{r}(\mathbb{R}^{N})$,
given by

$(\pm
1,\xi)\in{\mathbb{G}}\times\mathbb{R}^{n}\mapsto\pm\xi\in\mathbb{R}^{n},$
$(\pm 1,\lambda,u)\in{\mathbb{G}}\times\big{(}\mathbb{R}\times
H^{1}_{r}(\mathbb{R}^{N})\big{)}\mapsto(\lambda,\pm u)\in\mathbb{R}\times
H^{1}_{r}(\mathbb{R}^{N}).$

We notice that, under the assumption (F5), $\mathcal{I}^{m}$, $\mathcal{J}$
and $\mathcal{P}$ are invariant under this action, i.e. they are even in $u$:

$\mathcal{I}^{m}(\lambda,-u)=\mathcal{I}^{m}(\lambda,u),\quad\mathcal{J}(\lambda,-u)=\mathcal{J}(\lambda,u),\quad\mathcal{P}(\lambda,-u)=\mathcal{P}(\lambda,u).$

In addition, we observe by the Principle of Symmetric Criticality of Palais
[310] that every critical point of $\mathcal{I}^{m}$ restricted to
$\mathbb{R}\times H^{1}_{r}(\mathbb{R}^{N})$ is actually a critical point of
$\mathcal{I}^{m}$ on the whole $\mathbb{R}\times H^{1}(\mathbb{R}^{N})$.
Finally, we denote by $P_{2}:\mathbb{R}\times H^{1}_{r}(\mathbb{R}^{N})\to
H^{1}_{r}(\mathbb{R}^{N})$ the projection on the second component.

Moreover we consider the _Pohozaev set_

$\Omega:=\big{\\{}(\lambda,u)\in\mathbb{R}\times
H^{1}_{r}(\mathbb{R}^{N})\mid\mathcal{P}(\lambda,u)>0\big{\\}}\cup\big{\\{}(\lambda,0)\mid\lambda\in\mathbb{R}\big{\\}};$

under the assumption (F5), $\Omega$ is symmetric with respect to the axis
$\\{(\lambda,0)\mid\lambda\in\mathbb{R}\\}$, that is,

$(\lambda,u)\in\Omega\implies(\lambda,-u)\in\Omega.$

We start showing the following property, due to the fact that
$\mathcal{D}(u)=o(\|u\|_{H^{1}}^{2})$ as $u\to 0$.

######  Lemma 3.4.1.

We have

$\\{(\lambda,0)\mid\lambda\in\mathbb{R}\\}\subset int(\Omega).$ (3.4.34)

Proof. By

$|F(s)|\lesssim|s|^{q}+|s|^{p}$

where $q=\frac{N+\alpha}{N}$ and $p=\frac{N+\alpha+2}{N}<2^{*}$. Thus

$\|F(u)\|_{\frac{2N}{N+\alpha}}\lesssim\||u|^{q}\|_{\frac{2N}{N+\alpha}}+\||u|^{p}\|_{\frac{2N}{N+\alpha}}=\|u\|_{2}^{q}+\|u\|_{\frac{2Np}{N+\alpha}}^{p}.$

Therefore by Proposition 1.3.1 and Young’s inequality we have

$\displaystyle\int_{\mathbb{R}^{N}}(I_{\alpha}*|F(u)|)|F(u)|\,dx$
$\displaystyle\lesssim\|F(u)\|_{\frac{2N}{N+\alpha}}^{2}\lesssim\left(\|u\|_{2}^{q}+\|u\|_{\frac{2Np}{N+\alpha}}^{p}\right)^{2}$
$\displaystyle\lesssim\|u\|_{2}^{2q}+\|u\|_{\frac{2Np}{N+\alpha}}^{2p}\leq\|u\|_{H^{1}}^{2q}+\|u\|_{H^{1}}^{2p}$

thus

$\mathcal{P}(\lambda,u)\lesssim\|u\|_{H^{1}}^{2}-\|u\|_{H^{1}}^{2q}-\|u\|_{H^{1}}^{2p}>0$

for $\|u\|_{H^{1}}$ small, $u\neq 0$.

By (3.4.34) we detect the _Pohozaev mountain_

$\partial\Omega=\big{\\{}(\lambda,u)\in\mathbb{R}\times
H^{1}_{r}(\mathbb{R}^{N})\mid\mathcal{P}(\lambda,u)=0,\ u\not\equiv
0\big{\\}}.$

We observe that $\partial\Omega\neq\emptyset$, for instance by [302, Theorems
1 and 3].

######  Proposition 3.4.2.

Assume (F1)–(F4) and (F5). We have the following properties.

  * (i)

$\mathcal{J}(\lambda,u)\geq 0$ for all $(\lambda,u)\in\Omega$.

  * (ii)

$\mathcal{J}(\lambda,u)\geq a_{1}(\lambda)>0$ for all
$(\lambda,u)\in\partial\Omega$.

  * (iii)

Assume (CF3). For any $m>0$, we set

$E^{m}:=\inf_{(\lambda,u)\in\partial\Omega}\mathcal{I}^{m}(\lambda,u),\quad\text{
and }\quad
B^{m}:=\inf_{\lambda\in\mathbb{R}}\left(a_{1}(\lambda)-\frac{e^{\lambda}}{2}m\right).$

Then $E^{m}\geq B^{m}>-\infty$. In particular $B^{m}\in\mathbb{R}$ and

$\mathcal{I}^{m}(\lambda,u)\geq B^{m}\quad\hbox{for every}\
(\lambda,u)\in\partial\Omega.$

Proof. We notice that for all $(\lambda,u)\in\Omega$

${\mathcal{J}}(\lambda,u)\geq{\mathcal{J}}(\lambda,u)-\frac{{\mathcal{P}}(\lambda,u)}{N+\alpha}=\frac{\alpha+2}{2(N+\alpha)}\|\nabla
u\|_{2}^{2}+\frac{\alpha}{2(N+\alpha)}e^{\lambda}\|u\|_{2}^{2}\geq 0$

and thus $(i)$ follows. Point $(ii)$ follows from the fact that for each
$\lambda$ the mountain pass level $a_{1}(\lambda)$ coincides with the ground
state energy level (see [301, Section 4.2] and Section 4.3 for details); see
also Remark 3.4.3. Focus on $(iii)$: the fact that $E^{m}\geq B^{m}$ is a
direct consequence of $(ii)$, while the fact that $B^{m}>-\infty$ comes from
Proposition 3.3.1 (i).

######  Remark 3.4.3.

In order to show that $a_{1}(\lambda)>0$, without exploiting the existence
result for the unconstrained problem, we argue as follows (see also [114]).
Let $\gamma\in\Gamma_{1}(\lambda)$; by definition of $\Gamma_{1}(\lambda)$ and
by Proposition 3.4.2 (i) there exists $t^{*}$ such that
$\gamma(t^{*})\in\partial\Omega$ and $\gamma(t^{*})\neq 0$, thus
$\mathcal{P}(\lambda,\gamma(t^{*}))=0$. This means that

$\mathcal{J}(\lambda,\gamma(t^{*}))=\frac{\alpha+2}{2(N+\alpha)}\|\nabla\gamma(t^{*})\|_{2}^{2}+\frac{\alpha\mu}{2(N+\alpha)}\|\gamma(t^{*})\|_{2}^{2}\simeq\|\gamma(t^{*})\|_{H^{1}}^{2}$

thus

$a(\lambda)\gtrsim\inf_{u\in(\partial\Omega)_{\lambda}}\|u\|_{H^{1}}^{2}.$

Since, by (3.4.34), $(\partial\Omega)_{\lambda}$ is far from the line
$(\lambda,0)$, we obtain that the right-hand side is strictly positive, which
is the claim.

From now on we assume (CF3) to give sense to the quantity $B^{m}$. In view of
Proposition 3.4.2 (iii), we set for $m>0$ and $n\in\mathbb{N}^{*}$

$\displaystyle\Gamma_{n}^{m}:=\big{\\{}\Theta\in C(D_{n},\mathbb{R}\times
H^{1}_{r}(\mathbb{R}^{N}))\mid\;$ $\displaystyle\text{$\Theta$ is
${\mathbb{G}}$-equivariant,}\ \mathcal{I}^{m}(\Theta(0))\leq B^{m}-1,$
$\displaystyle\Theta|_{\partial D_{n}}\notin\Omega,\
\mathcal{I}^{m}(\Theta|_{\partial D_{n}})\leq B^{m}-1\big{\\}}$

and

$b_{n}^{m}:=\inf_{\Theta\in\Gamma_{n}^{m}}\sup_{\xi\in
D_{n}}\mathcal{I}(\Theta(\xi));$

we point out that asking $\Theta=(\Theta_{1},\Theta_{2})\in\Gamma^{m}_{n}$ to
be ${\mathbb{G}}$-equivariant means that $\Theta_{1}$ is even and $\Theta_{2}$
is odd, and in particular $\Theta_{2}(0)=0$ which implies
$\Theta(0)\in\Omega$.

######  Proposition 3.4.4.

Assume (F1)-(F2)-(CF3)-(F4)-(F5). We have the following properties.

  * (i)

For any $m>0$ and $n\in\mathbb{N}^{*}$, we have $\Gamma_{n}^{m}\neq\emptyset$
and

$b_{n}^{m}\leq a_{n}(\lambda)-e^{\lambda}\frac{m}{2},$ (3.4.35)

for each $\lambda\in\mathbb{R}$. Moreover, $b_{n}^{m}$ increases with respect
to $n$.

  * (ii)

For any $k\in\mathbb{N}^{*}$ there exists $m_{k}\geq 0$, namely given by

$m_{k}:=2\inf_{\lambda\in\mathbb{R}}\frac{a_{k}(\lambda)}{e^{\lambda}},$
(3.4.36)

such that for $m>m_{k}$

$b_{n}^{m}<0\quad\text{for}\ n=1,2,\dots,k.$

Moreover, $m_{k}$ is increasing with respect to $k$.

  * (iii)

If (CF4) holds, then $m_{k}=0$ for each $k\in\mathbb{N}^{*}$. That is, for
each $m>0$ we have

$b_{n}^{m}<0\quad\text{for all}\ n\in\mathbb{N}^{*}.$

Proof. For given $\lambda\in\mathbb{R}$ and $\zeta\in\Gamma_{n}(\lambda)$, we
will find a $\psi\in\Gamma_{n}^{m}$ such that

$\max_{\xi\in D_{n}}\mathcal{J}(\psi(\xi))\leq\max_{\xi\in
D_{n}}\mathcal{J}(\lambda,\zeta(\xi)),$ (3.4.37)

so that we have

$b_{n}^{m}\leq\max_{\xi\in D_{n}}\mathcal{I}^{m}(\psi(\xi))\leq\max_{\xi\in
D_{n}}\mathcal{J}(\lambda,\zeta(\xi))-{\frac{e^{\lambda}}{2}}m$

and, passing to the infimum over $\Gamma_{n}(\lambda)$, we gain (3.4.35).
To find $\psi\in\Gamma_{n}^{m}$ with (3.4.37), observe that, by definition of
$\Gamma_{n}(\lambda)$ and compactness of $\zeta(\partial D_{n})$, there exists
$C>0$ such that $\mathcal{D}(\zeta(\xi))\geq C>0$ for $\xi\in\partial D_{n}$.
Thus, we have $\mathcal{I}^{m}(\lambda,\zeta(\xi)(\cdot/L))\to-\infty$ and
$\mathcal{P}(\lambda,\zeta(\xi)(\cdot/L))\to-\infty$ as $L\to+\infty$,
uniformly for $\xi\in\partial D_{n}$. Thus, for $L\gg 1$ we obtain, for every
$\xi\in\partial D_{n}$,

$\mathcal{I}^{m}(\lambda,\zeta(\xi)(\cdot/L))\leq B^{m}-1\quad\hbox{ and
}\quad\mathcal{P}(\lambda,\zeta(\xi)(\cdot/L))<0.$ (3.4.38)

We also note that
$\mathcal{I}^{m}(\lambda+L,0)=-{\frac{e^{\lambda+L}}{2}}m\to-\infty$ as
$L\to+\infty$. Thus, for $L\gg 1$, we find that the path
$\psi:D_{n}\to\mathbb{R}\times H^{1}_{r}(\mathbb{R}^{N})$

$\psi(\xi):=\left\\{\begin{aligned} &(\lambda+L(1-2|\xi|),\,0)&\quad\hbox{ if
$|\xi|\in[0,1/2]$,}\\\
&\left(\lambda,\,\zeta\left({\frac{\xi}{|\xi|}}(2|\xi|-1)\right)(\cdot/L)\right)&\quad\hbox{
if $|\xi|\in(1/2,1]$}\end{aligned}\right.$

satisfies $\psi(0)=(\lambda+L,0)\in\mathbb{R}\times\\{0\\}$,
$\mathcal{I}^{m}(\psi(0))\leq B^{m}-1$ and $\mathcal{I}^{m}(\psi(\xi))\leq
B^{m}-1$ for $\xi\in\partial D_{n}$. Thus, by (3.4.38), we obtain
$\psi\in\Gamma^{m}_{n}$ and (3.4.37) holds.
The monotonicity of $b_{n}^{m}$ with respect to $n$ is a consequence of the
definition. Point $(ii)$ follows from (3.4.35) and $(iii)$ follows from
Proposition 3.3.1 (ii).

As a corollary to Proposition 3.4.4, we have the following result.

######  Corollary 3.4.5.

For any $m>0$, we have

$B^{m}=E^{m}=b^{m}_{1},$

i.e. the first minimax value $b_{1}^{m}$ equals the Pohozaev minimum $E^{m}$
on the product space.

Proof. Since any path in $\Gamma^{m}_{n}$ passes through $\partial\Omega$, we
have $b^{m}_{n}\geq E^{m}\geq B^{m}$ for each $n$. On the other hand, passing
to the infimum (3.4.35) we obtain $b_{1}^{m}\leq B^{m}$ and thus the claim.

###  3.5 The Palais-Smale-Pohozaev condition

For every $b\in\mathbb{R}$ we set

$K_{b}^{m}:=\big{\\{}(\lambda,u)\in\mathbb{R}\times
H^{1}_{r}(\mathbb{R}^{N})\mid\mathcal{I}^{m}(\lambda,u)=b,\,\partial_{\lambda}\mathcal{I}^{m}(\lambda,u)=0,\,\partial_{u}\mathcal{I}^{m}(\lambda,u)=0\big{\\}}.$

As already observed, under (F1)-(F2) we have that $\mathcal{P}(\lambda,u)=0$
for each $(\lambda,u)\in K_{b}^{m}$. We notice also that, assuming (F5),
$K_{b}^{m}$ is invariant under the ${\mathbb{G}}$-action, that is

$(\lambda,u)\in K_{b}^{m}\implies(\lambda,-u)\in K_{b}^{m}.$

Under our assumptions on $F$, it seems difficult to verify the standard
Palais-Smale condition for the functional $\mathcal{I}^{m}$. Therefore we
cannot recognize that $K_{b}^{m}$ is compact.

Inspired by [224, 231, 125], we introduce the Palais-Smale-Pohozaev condition,
a weaker compactness condition that takes into account the scaling properties
of $\mathcal{I}^{m}$ through the Pohozaev functional $\mathcal{P}$. Through
this tool we will show that $K_{b}^{m}$ is compact when $b<0$.

######  Definition 3.5.1.

For $b\in\mathbb{R}$, we say that
$(\lambda_{n},u_{n})_{n}\subset\mathbb{R}\times H^{1}_{r}(\mathbb{R}^{N})$ is
a _Palais-Smale-Pohozaev sequence_ for $\mathcal{I}^{m}$ at level $b$ (shortly
a $(PSP)_{b}$ sequence) if

$\mathcal{I}^{m}(\lambda_{n},u_{n})\to b,$ (3.5.39)
$\partial_{\lambda}\mathcal{I}^{m}(\lambda_{n},u_{n})\to 0,$ (3.5.40)
$\|\partial_{u}\mathcal{I}^{m}(\lambda_{n},u_{n})\|_{(H^{1}_{r}(\mathbb{R}^{N}))^{*}}\to
0,$ (3.5.41) $\mathcal{P}(\lambda_{n},u_{n})\to 0.$ (3.5.42)

We say that $\mathcal{I}^{m}$ satisfies the _Palais-Smale-Pohozaev condition_
at level $b$ (shortly the $(PSP)_{b}$ condition) if every $(PSP)_{b}$ sequence
has a strongly convergent subsequence in $\mathbb{R}\times
H^{1}_{r}(\mathbb{R}^{N})$.

We show now the following result.

######  Proposition 3.5.2.

Assume (F1)-(CF2)-(CF3) and let $b<0$. Then $\mathcal{I}^{m}$ satisfies the
$(PSP)_{b}$ condition.

Proof. Let $b<0$ and let $(\lambda_{n},u_{n})\subset\mathbb{R}\times
H^{1}_{r}(\mathbb{R}^{N})$ be a $(PSP)_{b}$ sequence, i.e. satisfying
(3.5.39)–(3.5.42). First we note that by (3.5.40) we obtain

$e^{\lambda_{n}}\big{(}\|u_{n}\|_{2}^{2}-m\big{)}\to 0.$ (3.5.43)

Step 1: _$\lambda_{n}$ is bounded from below and $\|u_{n}\|_{2}^{2}\to m$ as
$n\to+\infty$._
We have by (3.5.42), (3.5.39) and (3.5.43)

$\displaystyle o(1)$ $\displaystyle=\mathcal{P}(\lambda_{n},u_{n})$
$\displaystyle=-\frac{\alpha+2}{2}\|\nabla
u_{n}\|_{2}^{2}+(N+\alpha)\Bigl{(}\mathcal{I}^{m}(\lambda_{n},u_{n})-\frac{e^{\lambda_{n}}}{2}\bigl{(}\|u_{n}\|_{2}^{2}-m\bigr{)}\Bigr{)}+\frac{N}{2}e^{\lambda_{n}}\|u_{n}\|_{2}^{2}$
$\displaystyle=-\frac{\alpha+2}{2}\|\nabla
u_{n}\|_{2}^{2}+(N+\alpha)(b+o(1))+\frac{N}{2}e^{\lambda_{n}}m+o(1).$

Here we used (3.5.43). From the above identity, we derive boundedness of
$\lambda_{n}$ from below, since $b<0$. This result joined to (3.5.43) finally
gives $\|u_{n}\|_{2}^{2}\to m$.

Step 2: _$\lambda_{n}$ and $\|\nabla u_{n}\|_{2}^{2}$ are bounded._
Since, by (3.5.41),
$\varepsilon_{n}:=\|\partial_{u}\mathcal{I}^{m}(\lambda_{n},u_{n})\|_{(H_{r}^{1}(\mathbb{R}^{N}))^{*}}\to
0$, we have

$\|\nabla u_{n}\|_{2}^{2}-\int_{\mathbb{R}^{N}}(I_{\alpha}\ast
F(u_{n}))f(u_{n})u_{n}dx+e^{\lambda_{n}}\|u_{n}\|_{2}^{2}\leq\varepsilon_{n}\|u_{n}\|_{H^{1}}.$
(3.5.44)

We observe that by (CF3) for $\delta>0$ fixed, there exists $C_{\delta}>0$
such that

$|F(s)|\leq\delta|s|^{p}+C_{\delta}|s|^{q}$

where we recall $p=\frac{N+\alpha+2}{N}$ and $q=\frac{N+\alpha}{N}$. Thus

$\|F(u_{n})\|_{\frac{2N}{N+\alpha}}\leq\delta\||u_{n}|^{p}\|_{\frac{2N}{N+\alpha}}+C_{\delta}\||u_{n}|^{q}\|_{\frac{2N}{N+\alpha}}=\delta\|u_{n}\|_{\frac{2Np}{N+\alpha}}^{p}+C_{\delta}\|u_{n}\|_{2}^{q}.$

Therefore by (CF2), Proposition 1.3.1 and Young’s inequality we have

$\displaystyle\int_{\mathbb{R}^{N}}\big{(}I_{\alpha}*|F(u_{n})|\big{)}|f(u_{n})u_{n}|\,dx$
$\displaystyle\leq
C\|F(u_{n})\|_{\frac{2N}{N+\alpha}}\|f(u_{n})u_{n}\|_{\frac{2N}{N+\alpha}}$
$\displaystyle\leq
C\left(\delta\|u_{n}\|_{\frac{2Np}{N+\alpha}}^{p}+C_{\delta}\|u_{n}\|_{2}^{q}\right)\cdot
C^{\prime}\left(\|u_{n}\|_{\frac{2Np}{N+\alpha}}^{p}+\|u_{n}\|_{2}^{q}\right)$
$\displaystyle\leq
CC^{\prime}\delta\|u_{n}\|_{\frac{2Np}{N+\alpha}}^{2p}+CC^{\prime}(\delta+C_{\delta})\left({\frac{\delta}{2}}\|u_{n}\|_{\frac{2Np}{N+\alpha}}^{2p}+{\frac{1}{2\delta}}\|u_{n}\|_{2}^{2q}\right)+CC^{\prime}C_{\delta}\|u_{n}\|_{2}^{2q}$
$\displaystyle\leq
C^{\prime\prime}\delta\|u_{n}\|_{\frac{2Np}{N+\alpha}}^{2p}+C^{\prime\prime}_{\delta}\|u_{n}\|_{2}^{2q}$

and thus, by the Gagliardo-Nirenberg inequality and (3.5.44),

$\displaystyle\|\nabla u_{n}\|_{2}^{2}+e^{\lambda_{n}}\|u_{n}\|_{2}^{2}$
$\displaystyle\leq\int_{\mathbb{R}^{N}}(I_{\alpha}\ast|F(u_{n})|)|f(u_{n})u_{n}|dx+\varepsilon_{n}\|u_{n}\|_{H^{1}}$
$\displaystyle\leq C^{\prime\prime\prime}\delta\|\nabla
u_{n}\|^{2}_{2}\|u_{n}\|_{2}^{2(p-1)}+C_{\delta}^{\prime\prime}\|u_{n}\|_{2}^{\frac{2(N+\alpha)}{N}}+\varepsilon_{n}\|u_{n}\|_{H^{1}}.$

Since by Step 1 $\|u_{n}\|_{2}^{2}=m+o(1)$, we obtain

$\displaystyle(1-C^{\prime\prime\prime}\delta(m+o(1))^{p-1})\|\nabla
u_{n}\|_{2}^{2}+e^{\lambda_{n}}(m+o(1))$ $\displaystyle\leq
C^{\prime\prime}_{\delta}(m+o(1))^{\frac{N+\alpha}{N}}+\varepsilon_{n}(\|\nabla
u_{n}\|_{2}^{2}+m+o(1))^{1/2}.$

For $\delta$ small enough, we have the boundedness of $e^{\lambda_{n}}$ and
$\|\nabla u_{n}\|_{2}$. Hence $\lambda_{n}$ can not go to $+\infty$ and thus
by Step 1 we infer that $\lambda_{n}$ is bounded.

Step 3: _$\lambda_{n}$ and $u_{n}$ strongly converge._
By Steps 1-2, the sequence $(\lambda_{n},u_{n})_{n}$ is bounded in
$\mathbb{R}\times H^{1}_{r}(\mathbb{R}^{N})$ and thus after extracting a
subsequence, denoted in the same way, we may assume that
$\lambda_{n}\to\lambda_{0}$ and $u_{n}\rightharpoonup u_{0}$ weakly in
$H^{1}_{r}(\mathbb{R}^{N})$ for some $(\lambda_{0},u_{0})\in\mathbb{R}\times
H^{1}_{r}(\mathbb{R}^{N})$. Taking into account the assumptions (F1)–(F3) and
the compact embedding of $H^{1}_{r}(\mathbb{R}^{N})$ in
$L^{r}(\mathbb{R}^{N})$ for $r\in(2,2^{*})$, we have by Proposition 1.5.9

$\int_{\mathbb{R}^{N}}(I_{\alpha}\ast
F(u_{n}))f(u_{n})u_{0}\,dx\to\int_{\mathbb{R}^{N}}(I_{\alpha}\ast
F(u_{0}))f(u_{0})u_{0}\,dx$

and

$\int_{\mathbb{R}^{N}}(I_{\alpha}\ast
F(u_{n}))f(u_{n})u_{n}\,dx\to\int_{\mathbb{R}^{N}}(I_{\alpha}\ast
F(u_{0}))f(u_{0})u_{0}\,dx.$

By (3.5.41) we derive that
$\langle\partial_{u}\mathcal{I}^{m}(\lambda_{n},u_{n}),u_{n}\rangle\to 0$ and
$\langle\partial_{u}\mathcal{I}^{m}(\lambda_{n},u_{n}),u_{0}\rangle\to 0$, and
hence $(\nabla
u_{n},\nabla(u_{n}-u_{0}))_{2}+(u_{n},e^{\lambda_{n}}u_{n}-e^{\lambda_{0}}u_{0})_{2}\to
0$. Combining this with $u_{n}\rightharpoonup u_{0}$ and
$\lambda_{n}\to\lambda_{0}$ we get

$\|\nabla u_{n}\|_{2}^{2}+e^{\lambda_{n}}\|u_{n}\|^{2}_{2}\to\|\nabla
u_{0}\|_{2}^{2}+e^{\lambda_{0}}\|u_{0}\|^{2}_{2}$

which implies $u_{n}\to u_{0}$ strongly in $H^{1}_{r}(\mathbb{R}^{N})$.

As a straightforward consequence we obtain the following result.

######  Corollary 3.5.3.

Assume (F1)-(CF2)-(CF3) and let $b<0$. Then
$K_{b}^{m}\cap(\mathbb{R}\times\\{0\\})=\emptyset$ and $K_{b}^{m}$ is compact.

######  Remark 3.5.4.

We emphasize that the $(PSP)_{b}$ condition does not hold at level $b=0$.
Indeed we can consider a $(PSP)_{0}$ unbounded sequence $(\lambda_{n},0)$ with
$\lambda_{n}\to-\infty$.

###  3.6 Genus-shaped critical points

In this Section we essentially follow the lines of Sections 2.4.2–2.7. We give
just an outline, avoiding details and proofs.

####  3.6.1 Augmented functional

We start by achieving a deformation lemma. In order to do this we define

$M:=\mathbb{R}\times\mathbb{R}\times H^{1}_{r}(\mathbb{R}^{N})$

and introduce the augmented functional $\mathcal{H}^{m}:M\to\mathbb{R}$

$\displaystyle\mathcal{H}^{m}(\theta,\lambda,u):=$
$\displaystyle\,\mathcal{I}^{m}(\lambda,u(e^{-\theta}\cdot))$ (3.6.45)
$\displaystyle=$ $\displaystyle\,\frac{e^{(N-2)\theta}}{2}\|\nabla
u\|_{2}^{2}-\frac{e^{(N+\alpha)\theta}}{2}\mathcal{D}(u)+\frac{e^{\lambda}}{2}\bigl{(}e^{N\theta}\|u\|_{2}^{2}-m\bigr{)}$

for all $(\theta,\lambda,u)\in M$, and thus
$\partial_{\theta}\mathcal{H}^{m}(\theta,\lambda,u)=\mathcal{P}(\lambda,u(\cdot/e^{\theta})).$
We point out that, considered the action of ${\mathbb{G}}$ on $M$

${\mathbb{G}}\times M\to M;\,(\pm
1,\theta,\lambda,u)\mapsto(\theta,\lambda,\pm u)$

and assumed (F5), it results that $\mathcal{H}^{m}$ is
${\mathbb{G}}$-invariant. Introducing a metric on $M$ by

${\|(\alpha,\nu,h)\|}_{(\theta,\lambda,u)}^{2}:=\left|\left(\alpha,\nu,\|h(e^{-\theta}\cdot)\|_{H^{1}}\right)\right|^{2}$

for any $(\alpha,\nu,h)\in
T_{(\theta,\lambda,u)}M\equiv\mathbb{R}\times\mathbb{R}\times
H^{1}_{r}(\mathbb{R}^{N})$, we regard $M$ as a Hilbert manifold. We also
denote the dual norm on $T^{*}_{(\theta,\lambda,u)}M$ by
$\|\cdot\|_{(\theta,\lambda,u),*}$, and observe that both
$\|\cdot\|_{(\theta,\lambda,u)}$ and $\|\cdot\|_{(\theta,\lambda,u),*}$
actually depend only on $\theta$.

Denote now $D:=(\partial_{\theta},\partial_{\lambda},\partial_{u})$ the
gradient with respect to all the variables; a direct computation shows that
for any $(\theta,\lambda,u)\in M$

$\displaystyle\|{D\mathcal{H}^{m}(\theta,\lambda,u)\|}_{(\theta,\lambda,u),*}^{2}$
$\displaystyle=|\mathcal{P}(\lambda,u(e^{-\theta}\cdot))|^{2}+|\partial_{\lambda}\mathcal{I}^{m}(\lambda,u(e^{-\theta}\cdot))|^{2}+\|\partial_{u}\mathcal{I}^{m}(\lambda,u(e^{-\theta}\cdot))\|_{(H^{1}_{r}(\mathbb{R}^{N}))^{*}}^{2}.$

We furthermore define the set of critical points of $\mathcal{H}^{m}$ at level
$b$ by

$\displaystyle\tilde{K}_{b}^{m}:=$
$\displaystyle\big{\\{}(\theta,\lambda,u)\in
M\mid\mathcal{H}^{m}(\theta,\lambda,u)=b,\,D\mathcal{H}^{m}(\theta,\lambda,u)=0\big{\\}}$
$\displaystyle=$
$\displaystyle\big{\\{}(\theta,\lambda,u(e^{\theta}\cdot))\mid(\lambda,u)\in
K_{b}^{m},\;\theta\in\mathbb{R}\big{\\}}.$

Finally we introduce the distance between two points as

$\displaystyle{\rm
dist}_{M}\big{(}(\theta_{0},\lambda_{0},h_{0}),(\theta_{1},\lambda_{1},h_{1})\big{)}:=$
$\displaystyle\inf\left\\{\int_{0}^{1}\|\dot{\gamma}(t)\|_{\gamma(t)}\,dt\mid\gamma\in
C^{1}([0,1],M),\;\gamma(0)=(\theta_{0},\lambda_{0},h_{0}),\gamma(1)=(\theta_{1},\lambda_{1},h_{1})\right\\}.$

As a consequence of Proposition 3.5.2 we obtain the following.

######  Proposition 3.6.1.

Assume (F1)-(CF2)-(CF3) and let $b<0$. Then $\mathcal{H}^{m}$ satisfies the
following Palais-Smale-type condition $(\widetilde{PSP})_{b}$: for each
sequence $(\theta_{n},\lambda_{n},u_{n})_{n}\subset M$ such that

$\mathcal{H}^{m}(\theta_{n},\lambda_{n},u_{n})\to b,$
$\|D\mathcal{H}^{m}(\theta_{n},\lambda_{n},u_{n})\|_{(\theta_{n},\lambda_{n},u_{n}),*}\to
0$

as $n\to+\infty$, we have, up to a subsequence,

${\rm dist}_{M}((\theta_{n},\lambda_{n},u_{n}),\tilde{K}_{b}^{m})\to 0.$

####  3.6.2 Deformation theory

We write, for $b\in\mathbb{R}$

$[\mathcal{I}^{m}\leq b]:=\big{\\{}(\lambda,u)\in\mathbb{R}\times
H_{r}^{1}(\mathbb{R}^{N})\mid\mathcal{I}^{m}(\lambda,u)\leq b\big{\\}},$
$[\mathcal{H}^{m}\leq b]_{M}:=\big{\\{}(\theta,\lambda,u)\in
M\mid\mathcal{H}^{m}(\theta,\lambda,u)\leq b\big{\\}}.$

We state the following result.

######  Proposition 3.6.2.

Assume (F1)-(CF2)-(CF3). Let $b<0$, and let $\mathcal{O}$ be a neighborhood of
$K_{b}^{m}$ with respect to the standard distance of $\mathbb{R}\times
H^{1}_{r}(\mathbb{R}^{N})$. Let $\bar{\varepsilon}>0$, then there exist
$\varepsilon\in(0,\bar{\varepsilon})$ and $\eta:[0,1]\times(\mathbb{R}\times
H^{1}_{r}(\mathbb{R}^{N}))\to\mathbb{R}\times H^{1}_{r}(\mathbb{R}^{N})$
continuous such that

  1. 1.

$\eta(0,\cdot,\cdot)=id_{\mathbb{R}\times H^{1}_{r}(\mathbb{R}^{N})}$;

  2. 2.

$\eta$ fixes $[\mathcal{I}^{m}\leq b-\bar{\varepsilon}]$, that is,
$\eta(t,\cdot,\cdot)=id_{[\mathcal{I}^{m}\leq b-\bar{\varepsilon}]}$ for all
$t\in[0,1]$;

  3. 3.

$\mathcal{I}^{m}$ is non-increasing along $\eta$, and in particular
$\mathcal{I}^{m}(\eta(t,\cdot,\cdot))\leq\mathcal{I}^{m}(\cdot,\cdot)$ for all
$t\in[0,1]$;

  4. 4.

if $K_{b}^{m}=\emptyset$, then $\eta(1,[\mathcal{I}^{m}\leq
b+\varepsilon])\subseteq[\mathcal{I}^{m}\leq b-\varepsilon]$;

  5. 5.

if $K_{b}^{m}\not=\emptyset$, then

$\eta(1,[\mathcal{I}^{m}\leq
b+\varepsilon]\setminus\mathcal{O})\subseteq[\mathcal{I}^{m}\leq
b-\varepsilon]$

and

$\eta(1,[\mathcal{I}^{m}\leq b+\varepsilon])\subseteq[\mathcal{I}^{m}\leq
b-\varepsilon]\cup\mathcal{O};$

  6. 6.

if (F5) holds, then $\eta(t,\cdot,\cdot)$ is ${\mathbb{G}}$-equivariant, i.e.
for $\eta=(\eta_{1},\eta_{2})$ we have $\eta_{1}$ even and $\eta_{2}$ odd in
$u$.

To prove this, we work first on the functional $\mathcal{H}$, for which we
obtained a $(\widetilde{PSP})$ condition, which implies that for any $b<0$
there exists $\varepsilon$, $\delta$, $\nu>0$ such that

$\|D\mathcal{H}^{m}(\theta,\lambda,u)\|_{(\theta,\lambda,u),*}\geq\nu$

for $(\theta,\lambda,u)\in M$ satisfying
$\mathcal{H}^{m}(\theta,\lambda,u)\in[b-\varepsilon,b+\varepsilon]$ and ${\rm
dist}_{M}((\theta,\lambda,u),\widetilde{K}_{b}^{m})\geq\delta$.

######  Proposition 3.6.3.

Assume (F1)-(CF2)-(CF3). Let $b<0$, and let $\tilde{\mathcal{O}}$ be a
neighborhood of $\tilde{K}_{b}^{m}$ with respect to ${\rm dist}_{M}$. Let
$\bar{\varepsilon}>0$, then there exist $\varepsilon\in(0,\bar{\varepsilon})$
and $\tilde{\eta}:[0,1]\times M\to M$ continuous such that

  1. 1.

$\tilde{\eta}(0,\cdot,\cdot,\cdot)=id_{M}$;

  2. 2.

$\tilde{\eta}$ fixes $[\mathcal{H}^{m}\leq b-\bar{\varepsilon}]_{M}$, that is
$\tilde{\eta}(t,\cdot,\cdot,\cdot)=id_{[\mathcal{H}^{m}\leq
b-\bar{\varepsilon}]_{M}}$ for all $t\in[0,1]$;

  3. 3.

$\mathcal{H}^{m}$ is non-increasing along $\tilde{\eta}$, and in particular
$\mathcal{H}^{m}(\tilde{\eta}(t,\cdot,\cdot,\cdot))\leq\mathcal{H}^{m}(\cdot,\cdot,\cdot)$
for all $t\in[0,1]$;

  4. 4.

if $\tilde{K}_{b}^{m}=\emptyset$, then $\tilde{\eta}(1,[\mathcal{H}^{m}\leq
b+\varepsilon]_{M})\subseteq[\mathcal{H}^{m}\leq b-\varepsilon]_{M}$;

  5. 5.

if $\tilde{K}_{b}^{m}\neq\emptyset$, then

$\tilde{\eta}(1,[\mathcal{H}^{m}\leq
b+\varepsilon]_{M}\setminus\tilde{\mathcal{O}})\subseteq[\mathcal{H}^{m}\leq
b-\varepsilon]_{M}$

and

$\tilde{\eta}(1,[\mathcal{H}^{m}\leq
b+\varepsilon]_{M})\subseteq[\mathcal{H}^{m}\leq
b-\varepsilon]_{M}\cup\tilde{\mathcal{O}};$

  6. 6.

if (F5) holds, then $\tilde{\eta}(t,\cdot,\cdot)$ is
${\mathbb{G}}$-equivariant, i.e. for
$\tilde{\eta}=(\tilde{\eta}_{0},\tilde{\eta}_{1},\tilde{\eta}_{2})$ we have
$\tilde{\eta}_{0}$, $\tilde{\eta}_{1}$ even and $\tilde{\eta}_{2}$ odd in $u$.

To get Proposition 3.6.2 from Proposition 3.6.3 we introduce

$\pi:(\theta,\lambda,u)\in
M\mapsto(\lambda,u(e^{-\theta}\cdot))\in\mathbb{R}\times
H^{1}_{r}(\mathbb{R}^{N}),$ $\iota:(\lambda,u)\in\mathbb{R}\times
H^{1}_{r}(\mathbb{R}^{N})\mapsto(0,\lambda,u)\in M,$

which are a kind of rescaling projection and immersion satisfying

$\pi\circ\iota=id_{\mathbb{R}\times
H^{1}_{r}(\mathbb{R}^{N})},\quad\pi(\tilde{K}_{b}^{m})=K_{b}^{m},$
$\mathcal{H}^{m}\circ\iota=\mathcal{I}^{m},\quad\mathcal{I}^{m}\circ\pi=\mathcal{H}^{m}.$

For a deformation $\tilde{\eta}$ obtained in Proposition 3.6.3 we thus define

$\eta(t,\lambda,u):=\pi(\tilde{\eta}(t,\iota(\lambda,u))),\quad(t,\lambda,u)\in[0,1]\times(\mathbb{R}\times
H^{1}_{r}(\mathbb{R}^{N})).$ (3.6.46)

####  3.6.3 Multiple critical points

For each $n\in\mathbb{N}^{*}$, define

$\displaystyle\Lambda_{n}^{m}:=\\{A=\Theta(\overline{D_{n+l}\setminus
Y})\mid\;$ $\displaystyle l\in\mathbb{N},\;\Theta\in\Gamma_{n+l}^{m},$
$\displaystyle Y\subseteq D_{n+l}\setminus\\{0\\}\;\hbox{ is closed, symmetric
in $0$}$ $\displaystyle\qquad\qquad\qquad\quad\;\hbox{ and ${\rm genus}(Y)\leq
l$}\\}$

and

$c_{n}^{m}:=\inf_{A\in\Lambda_{n}^{m}}\sup_{A}\mathcal{I}^{m}.$

We notice that
$\left\\{\Theta(D_{n})\right\\}_{\Theta\in\Gamma^{m}_{n}}\subset\Lambda^{m}_{n}$.
In the following lemma, we observe that $\Lambda_{n}^{m}$ and $c_{n}^{m}$
inherit the properties of $\Gamma_{n}^{m}$ and $b_{n}^{m}$, also given by

$A\cap\partial\Omega\not=\emptyset\quad\hbox{for all}\ A\in\Lambda_{1}^{m},$

together with the extra property (v).

######  Proposition 3.6.4.

Assume (F1)-(F2)-(CF3)-(F4). Let $n\in\mathbb{N}^{*}$ and $m>0$. Then

  * (i)

$\Lambda_{n}^{m}\neq\emptyset$.

  * (ii)

$\Lambda_{n+1}^{m}\subseteq\Lambda_{n}^{m}$, and thus $c_{n}^{m}\leq
c_{n+1}^{m}$.

  * (iii)

$c_{n}^{m}\leq b_{n}^{m}$.

  * (iv)

$B^{m}=E^{m}\leq c_{1}^{m}$.

  * (v)

Let $A\in\Lambda_{n}^{m}$ and $Z\subset\mathbb{R}\times
H^{1}_{r}(\mathbb{R}^{N})$ be ${\mathbb{G}}$-invariant, closed, and such that
$0\notin\overline{P_{2}(Z)}$ and ${\rm genus}(\overline{P_{2}(Z)})\leq i<n$.
Then $\overline{A\setminus Z}\in\Lambda_{n-i}^{m}$.

Fix $n\in\mathbb{N}^{*}$ and let $\Lambda_{n}^{m}$ and $c_{n}^{m}$ satisfying
the properties of Proposition 3.6.4. We build now multiple solutions.

######  Proposition 3.6.5.

Assume (F1)-(CF2)-(CF3)-(F4)-(F5). Fix $k\in\mathbb{N}^{*}$ and assume
$m>m_{k}$ (see (3.4.36)). Then

$c_{1}^{m}\leq c_{2}^{m}\leq\dots\leq c_{k}^{m}<0$

are critical values of $\mathcal{I}^{m}$. Moreover

  * (i)

if, for some $q\in\mathbb{N}^{*}$,

$c_{n}^{m}<c_{n+1}^{m}<\dots<c_{n+q}^{m}<0$

then we have $q+1$ different nonzero critical values, and thus $q+1$ different
pairs of nontrivial solutions of (3.1.6);

  * (ii)

if instead, for some $q\in\mathbb{N}^{*}$,

$c_{n}^{m}=c_{n+1}^{m}=\dots=c_{n+q}^{m}=:b<0$ (3.6.47)

then

${\rm genus}(P_{2}(K_{b}^{m}))\geq q+1$ (3.6.48)

and thus $\\#P_{2}(K_{b}^{m})=+\infty$, which means that we have infinite
different solutions of (3.1.6).

Summing up, we have at least $k$ different pairs of nontrivial solutions of
(3.1.6).

###  3.7 The unconstrained problem

In this Section we show how to exploit some of the developed tools also to
obtain infinitely many radial solutions for the _unconstrained problem_
(3.1.1), and give a sketch of the proof of Theorem 3.1.5. Here we assume
(F1)–(F5). We fix $\lambda\in\mathbb{R}$ and write $\mu=e^{\lambda}$; omitting
$\lambda$, we denote
$\mathcal{J}(\cdot):=\mathcal{J}(\lambda,\cdot):H^{1}_{r}(\mathbb{R}^{N})\to\mathbb{R}$,
i.e.

$\mathcal{J}(u):=\frac{1}{2}\|\nabla
u\|_{2}^{2}-\frac{1}{2}\mathcal{D}(u)+\frac{\mu}{2}\|u\|_{2}^{2},\quad u\in
H^{1}_{r}(\mathbb{R}^{N}).$ (3.7.49)

Similarly we write $\mathcal{P}(\cdot):=\mathcal{P}(\lambda,\cdot)$. For every
$b\in\mathbb{R}$ we set

$K_{b}:=\\{u\in
H^{1}_{r}(\mathbb{R}^{N})\mid\mathcal{J}(u)=b,\,\mathcal{J}^{\prime}(u)=0\\}.$

We have the following result.

######  Proposition 3.7.1.

Assume (F1)–(F3) and let $b\in\mathbb{R}$. Then $\mathcal{J}$ satisfies the
Palais-Smale-Pohozaev condition at level $b$ (shortly $(PSP)_{b}$), that is
every sequence $(u_{n})_{n}\subset H^{1}_{r}(\mathbb{R}^{N})$ satisfying

$\mathcal{J}(u_{n})\to b,$ (3.7.50)
$\|\mathcal{J}^{\prime}(u_{n})\|_{(H^{1}_{r}(\mathbb{R}^{N}))^{*}}\to 0,$
(3.7.51) $\mathcal{P}(u_{n})\to 0,$ (3.7.52)

admits a strongly convergent subsequence in $H^{1}_{r}(\mathbb{R}^{N})$. In
particular, $K_{b}$ is compact in $H_{r}^{1}(\mathbb{R}^{N})$.

Proof. First observe that, by (3.7.50) and (3.7.52) we obtain

$\frac{\alpha+2}{2}\|\nabla
u_{n}\|_{2}^{2}+\frac{\alpha}{2}\mu\|u_{n}\|_{2}^{2}=(N+\alpha)b+o(1).$
(3.7.53)

We observe that $b\geq 0$ and the boundedness of $u_{n}$ in
$H_{r}^{1}(\mathbb{R}^{N})$. Thus by (F2)-(F3), $\mathcal{D}^{\prime}(u_{n})$
has a strongly convergent subsequence in $(H_{r}^{1}(\mathbb{R}^{N}))^{*}$ and
by (3.7.51), $u_{n}$ has a strongly convergent subsequence in
$H_{r}^{1}(\mathbb{R}^{N})$. Here we make use of Proposition 1.5.9.

Set $[\mathcal{J}\leq b]:=\\{u\in
H_{r}^{1}(\mathbb{R}^{N})\mid\mathcal{J}_{\lambda}(u)\leq b\\}$. Following the
arguments of Section 3.6 and 3.6.2, we prove the following deformation result
by means of an augmented functional.

######  Proposition 3.7.2.

Assume (F1)–(F3). Let $b\in\mathbb{R}$ and let $\mathcal{O}$ be a neighborhood
of $K_{b}(\lambda)$. Let $\bar{\varepsilon}>0$, then there exist
$\varepsilon\in(0,\bar{\varepsilon})$ and $\eta:[0,1]\times
H^{1}_{r}(\mathbb{R}^{N})\to H^{1}_{r}(\mathbb{R}^{N})$ continuous such that

  1. 1.

$\eta(0,\cdot)=id_{H^{1}_{r}(\mathbb{R}^{N})}$;

  2. 2.

$\eta$ fixes $[\mathcal{J}\leq b-\bar{\varepsilon}]$, that is, $\eta(t,u)=u$
for all $t\in[0,1]$ and $\mathcal{J}(u)\leq b-\bar{\varepsilon}$;

  3. 3.

$\mathcal{J}$ is non-increasing along $\eta$, and in particular
$\mathcal{J}(\eta(t,\cdot))\leq\mathcal{J}(\cdot)$ for all $t\in[0,1]$;

  4. 4.

if $K_{b}=\emptyset$, then $\eta(1,[\mathcal{J}\leq
b+\varepsilon])\subseteq[\mathcal{J}\leq b-\varepsilon]$;

  5. 5.

if $K_{b}\not=\emptyset$, then

$\eta(1,[\mathcal{J}\leq
b+\varepsilon]\setminus\mathcal{O})\subseteq[\mathcal{J}\leq b-\varepsilon]$

and

$\eta(1,[\mathcal{J}\leq b+\varepsilon])\subseteq[\mathcal{J}\leq
b-\varepsilon]\cup\mathcal{O};$

  6. 6.

if (F5) holds, then $\eta(t,\cdot)$ is ${\mathbb{G}}$-equivariant, i.e. it is
odd.

As in Section 3.2, for any $n\in\mathbb{N}^{*}$ we define
$\Gamma_{n}:=\Gamma_{n}(\lambda)$. We note that $\Gamma_{n}\neq\emptyset$ is
shown in Proposition 3.2.1. Now our Theorem 3.1.5 can be obtained through the
arguments given in [325]. Here we just give the definition of another minimax
classes $\Lambda_{n}^{m}$, which ensures the multiplicity of solutions. We set
for $n\in\mathbb{N}^{*}$

$\displaystyle\Lambda_{n}:=\\{A=\Theta(\overline{D_{n+l}\setminus Y})\mid\;$
$\displaystyle l\in\mathbb{N}^{*},\;\Theta\in\Gamma_{n+l},$ $\displaystyle
Y\subseteq D_{n+l}\setminus\\{0\\}\;\hbox{ is closed, symmetric in $0$}$
$\displaystyle\qquad\qquad\qquad\quad\;\hbox{ and ${\rm genus}(Y)\leq l$}\\}$

and

$c_{n}:=\inf_{A\in\Lambda_{n}}\sup_{A}\mathcal{J}.$

Then we have $\\{\gamma(D_{n})|\,\gamma\in\Gamma_{n}\\}\subset\Lambda_{n}$ and
we can also see that

$0<c_{1}\leq c_{2}\leq\cdots\leq c_{n}\leq\cdots.$

Thus we have the following result.

######  Proposition 3.7.3.

Assume (F1)–(F5). Let $n\in\mathbb{N}^{*}$ and $m>0$. Then

  * (i)

$\Lambda_{n}\neq\emptyset$ and $c_{n}\leq c_{n+1}$.

  * (ii)

Let $A\in\Lambda_{n}$ and $Z\subset H^{1}_{r}(\mathbb{R}^{N})$ be
${\mathbb{G}}$-invariant, closed, and such that $0\notin\overline{Z}$ and
${\rm genus}(\overline{Z})\leq i<n$. Then $\overline{A\setminus
Z}\in\Lambda_{n-i}$.

  * (iii)

$c_{n}$ is a critical value of $\mathcal{J}$. Moreover

$c_{n}\to+\infty\quad\hbox{as}\ n\to+\infty.$

In particular, $\mathcal{J}$ has an unbounded sequence of critical values.

Proof. Using Proposition 3.7.2, the proof can be given along the lines in
[325]. See also [125].

Proof of Theorem 3.1.5. Theorem 3.1.5 follows from Proposition 3.7.3.

##  4 Chapter Doubly nonlocal equations: qualitative and quantitative results

This Chapter is dedicated to the study of the following _fractional Choquard
equation_

$(-\Delta)^{s}u+\mu u=\big{(}I_{\alpha}*F(u)\big{)}F^{\prime}(u)\quad\hbox{in
$\mathbb{R}^{N}$}$

where $N\geq 2$, $s\in(0,1)$, $\alpha\in(0,N)$, $\mu>0$ and $F\in
C^{1}(\mathbb{R})$ is a general nonlinearity, in the spirit of Berestycki and
Lions assumptions. After having achieved existence of positive solutions and
ground states, we will focus on the study of some qualitative properties of
these solutions: boundedness, regularity, $L^{1}$-summability, positivity,
radial symmetry and asymptotic decay. We will stress how the interplay between
a fractional framework and a nonlocal nonlinearity, generally nonhomogeneous,
obstructs the application of classical techniques. Some results generalize the
ones presented in [138] and extend [302, 79]; in particular, some new results
are stated also for the limiting case $s=1$. In addition, we will see that the
interaction of the two nonlocalities arises a new critical threshold.

This Chapter is mainly based on the papers: [114] (see also [113]) for Section
4.2, [115] for Sections 4.2, 4.3, 4.4.1–4.4.3, 4.6.1, [112] for Sections 4.3,
4.4.4-4.4.5, 4.5, [197] for Sections 4.6.2–4.6.7, and [117] for Section 4.7.

###  4.1 An example of double nonlocality: collapse of boson stars

In Sections 2.1 and 3.1 we highlighted the importance in physics of the
fractional Laplacian and of the Hartree-type terms. Combinations of the two
arise as well in different frameworks: for example equations of the type

$(-\Delta)^{s}u+\mu u=\big{(}I_{\alpha}*F(u)\big{)}f(u)\quad\hbox{in
$\mathbb{R}^{N}$}$ (4.1.1)

where $N\geq 2$, $s\in(0,1)$, $\alpha\in(0,N)$, $\mu>0$ and $f=F^{\prime}\in
C(\mathbb{R})$, can be found in quantum chemistry [24, 142, 215] (see also
[103] for some orbital stability results): here (4.1.1) appears in the study
of the mean field limit of weakly interacting molecules and in the physics of
multi-particle systems. In particular the equation applies to the study of
graphene [276], where the nonlocal nonlinearity describes the short time
interactions between particles. Doubly nonlocal equations appear also in the
dynamics of populations [85], where small or large values of $s$ better model
specific environments.

One of the main applications anyway arises in the study of exotic stars:
minimization properties related to (4.1.1) play indeed a fundamental role in
the mathematical description of the dynamics of pseudorelativistic boson stars
and their gravitational collapse [169, 222, 256, 257, 258, 269, 192, 193, 194,
195], as well as the evolution of attractive fermionic systems, such as white
dwarf stars [214]. In fact, the study of the ground states to (4.1.1) gives
information on the size of the critical initial conditions for the solutions
of the corresponding pseudorelativistic equation [256], where a critical value
is given by the Chandrasekhar limiting mass. In particular, when
$s=\frac{1}{2}$, $N=3$, $\alpha=2$ and $f(u)=u$, we obtain

$\sqrt{-\Delta}u+\mu u=\left(\frac{1}{4\pi|x|}*u^{2}\right)u\quad\hbox{in
$\mathbb{R}^{3}$}$ (4.1.2)

related to the so called _massless boson stars equation_ [189, 258, 222],
where the pseudorelativistic operator $\sqrt{-\Delta+m}$ collapses to the
square root of the Laplacian. Here $f(t)=|t|^{r-2}t$ with $r=2$ is
$L^{2}$-critical: in this Chapter, when dealing with the mass-constrained
problem, we essentially address the subcritical case $r\in(\frac{5}{3},2)$,
but we believe that this result, together with the developed minimax tools,
can be a first step towards the study of the $L^{2}$-mass critical (and
supercritical) case, since for these problems the minimization approach is
generally not well posed. Moreover, the high generality assumed on the
function $f$ could be useful in the study of different physical problems.

Mathematically, concerning the fractional Schrödinger equation with Hartree
nonlinearity, we mention the papers [138, 139] where D’Avenia, Siciliano and
Squassina considered the case of pure power nonlinearities and obtained
existence and qualitative properties of the solutions. We mention also [103,
202] for some orbital stability results, [104] for a Strichartz estimates
approach, and [129] for the unidimensional case. Other results can be found in
[342, 41, 277] for superlinear nonlinearities, in [219] for some local
perturbation, in [218, 305] for critical equations and in [386] for
concentration phenomena with strictly noncritical and monotone sources.

The existence of $L^{2}$-normalized solutions was investigated when
$F(t)=|t|^{p}$ in [382] (see also [212, 203] for $L^{2}$-supercritical Cauchy
problems by scattering), while in [102] it has been addressed the non-
autonomous unconstrained case. In [141], symmetry and monotonicity of positive
solutions are shown for the fractional Hartree equation for $\mu=0$ and a
critical power nonlinearity, by means of the direct method of moving planes.
Regularity results for a class of doubly nonlocal equations on bounded domains
are obtained in [207].

Some theoretical aspects related to the study of doubly nonlocal equations,
both in the operator and in the source, remain open for general nonlinearities
$F$, in particular when $F$ is not a power function or $F$ is odd.

In the present Chapter we are interested to derive some qualitative properties
of the solutions to (4.1.1), also in these special cases. In particular, after
having stated existence of free and normalized solutions, we will focus our
attention on the study of regularity of solutions (boundedness,
$L^{1}$-summability, Hölder continuity, differentiability), moving then to
positivity and symmetry of ground states, to tackle at the end the asymptotic
behaviour at infinity. The precise statements will be presented throughout the
Chapter: these results generalize some of the ones in [138] from the case of
power functions to general nonlinearities of Berestycki-Lions type; moreover,
we extend some results of [302] to the fractional framework, and some results
of [79] to the Choquard framework.

The achieving of these results requires some technical effort in order to deal
with the two nonlocalities and their interaction, as well as the
nonhomogeneity and the nonregularity of the function $f$. In particular, we
highlight some of the difficulties that arise in this general framework.

In the proof of the positivity, for instance (as well as in the proof of the
existence), the presence of the fractional power of the Laplacian does not
allow to use the fact that every solution satisfies the Pohozaev identity to
conclude that, if $|u|$ is a solution, then it satisfies the Pohozaev
identity; moreover, the conservation of the norm of the gradient does not hold
anymore, i.e. $\|\nabla|u|\|_{2}=\|\nabla u\|_{2}$ is not generally true in
the fractional framework, and an inequality is needed. In addition, when
dealing with $f$ even other information about $u$ are lost through
inequalities; this is not the case when dealing with $f$ odd [302].
Furthermore, the presence of the Choquard term, which scales differently from
the $L^{2}$-norm term, does not allow to implement the classical minimization
argument of [131, 50] (see (5.5.78)), which is useful to deal with the
absolute value of $u$. Similarly, the nonhomogeneity of the nonlinearity $f$
obstructs the minimization approach of [138, 300]. Thus, a new approach is
needed, and it relies on a fiber map which sends solutions to the Pohozaev set
(see Proposition 4.5.5). When dealing with $f$ even, this technique allows to
treat also the case $s=1$, generalizing [302].

Regarding the $L^{1}$-summability, the possibility of including a critical
behaviour in zero (that is, $F(t)\sim t^{2^{\\#}_{\alpha}}$) is not relevant
when dealing with pure power functions [138, 300], since no solution exists in
this case: this growth is instead relevant for general $f$ (for example,
suitable sum of powers). Contrary to the case of noncritical nonlinearities,
when $f$ is critical it is not possible to implement a simple bootstrap
argument to achieve that every solution is in $L^{1}$: a new method is thus
needed, and it is based on a suitable combination of bootstrap argument and
fixed point theorems (see Proposition 4.4.10). The study of this case is new
even for $s=1$, improving [300, 302].

When studying the asymptotic behaviour of solutions, especially when $f$ has a
sublinear growth, the interaction of the two nonlocalities is quite strong,
and new phenomena arise: indeed, contrary to the local case $s=1$ [300], here
the effect of the fractional Laplacian and of the Choquard term give rise to a
new threshold depicting the qualitative profile of ground states at infinity
(see Theorem 4.6.11). From a technical point of view, new difficulties arise
related to the explicit computation of the fractional Laplacian, and to the
computation of concave powers, requiring a more delicate analysis and the
implementation of new inequalities (see Sections 1.2.2 and 1.2.4). This result
is new even for power functions, improving [138].

We refer to the following Sections for the detailed statements of the results.

The Chapter is organized as follows. In the remaining part of the Section we
will briefly give a physical interpretation of equation (4.1.1) in the
framework of gravitational collapses. In Section 4.2 we will deal with
existence of solutions, both for the unconstrained problem and the constrained
one, by highlighting some approach different from the ones developed in
Chapters 2 and 3; some properties related to the energy minimum levels and
existence of positive solutions will be then investigated in Section 4.3.
Section 4.4 will be devoted to the study of regularity of positive solutions,
including boundedness and Hölder regularity; moreover we will gain
$L^{1}$-summability of solutions through a combination of bootstrap and fixed
point maps arguments. Then in Section 4.5 we will exploit these results in
order to gain positivity and radial symmetry of Pohozaev minima, by the
implementation of maximum principles on some fiber maps. Afterwards, we will
investigate in Section 4.6 the asymptotic decay of ground states, focusing
especially on the case of $f$ sublinear, which raises some new phenomenon.
Finally in Section 4.7 we furnish a proof of the Pohozaev identity in the
doubly nonlocal framework, by assuming the solutions merely $C^{1}$.

##### Physical derivation

Here we want to show how equation (4.1.1), in the particular case $N=3$,
$s=\frac{1}{2}$, $\alpha=2$ and $F(u)=\frac{1}{2}u^{2}$, can be derived from a
significant physical framework regarding boson stars. Aim of this Section is
just to give an idea of the process, without any aim of accuracy or rigors. We
refer to [93, 169, 196, 192, 213, 257, 258, 268, 269, 296, 307] (see also
[189, 270, 308, 360] and [193, 214, 256]) for complete expositions on the
topic.

The goal is to show how the equation (4.1.2) is strictly connected with the
self-gravitational collapse of boson stars. Actually a similar derivation
holds also for neutron stars and white dwarfs, with some little complications.

Let us consider thus a group of $n$ _bosons_ (i.e. particles with entire spin,
described by symmetric functions, and which do not respond to the Pauli
exclusion principle). We assume these bosons to form a _boson star_ , i.e.
$n\gg 0$ and we assume most of them (i.e. up to $o(N)$ particles) to be close
one to each other and moving at a fast speed: these particles are at a same
coherent state $\psi$ (and for this reason called _condensate_) and create a
trap for the remaining particles.

Due to the high speed of the particles, we cannot ignore the special
relativistic effect; on the other hand, since the masses are not too big, we
can ignore the effect of general relativity (this is not the case, instead, of
neutron stars). Thus we consider the total relativistic energy

$E^{2}=(pc)^{2}+(mc^{2})^{2}$

summation of the kinetic energy and the energy at rest; here $p$ is the
momentum, $m$ the mass of the boson particle at rest, $c$ the speed of light.
Thus we obtain, in momentum representation,

$E=\sqrt{|\xi|^{2}c^{2}+m^{2}c^{4}};$

passing through a quantization $p\mapsto-i\hbar\nabla$ to the coordinate
representation (and setting $\hbar:=1$) we obtain the _pseudorelativistic_
operator

$E=\sqrt{-c^{2}\Delta+m^{2}c^{4}}.$

We observe that, letting $c\to+\infty$ (that is, the velocities are far from
the one of the light) we obtain the operator $-\frac{1}{2m}\Delta$, that is
the nonrelativistic operator (i.e. the classical Laplacian). We set instead,
from now on, $c:=1$ for the sake of simplicity. Thus

$E=\sqrt{-\Delta+m^{2}}$

which is formally defined through the Fourier symbol
$\mathcal{F}^{-1}\big{(}(|\xi|^{2}+m^{2})^{1/2}\widehat{u}\big{)}$ (see also
Remark 1.2.6).

We consider now the interaction between the particles: this interaction can be
treated classically as a Newton two-bodies interaction, and thus given by the
quantity

$-k\frac{1}{|x_{i}-x_{j}|}$

where $k$ is a coupling constant (proportional to $G$, the gravitational
constant).

Thus we come up with the Hamiltonian of the system

$\mathcal{H}_{n}:=\sum_{i=1}^{n}\sqrt{-\Delta_{i}+m^{2}}-\frac{k}{2}\sum_{i\neq
j}^{n}\frac{1}{|x_{i}-x_{j}|}.$

When dealing with dwarf stars, additional pieces given by the interaction (due
to the Pauli exclusion principle) appear; anyway, for $n\gg 0$, one can ignore
these pieces: this is called the _Hartree approximation of Hartree-Fock
theory_.

Now we are interested in what happens when $n\gg 0$, that is, when the
particles act like a single body, in what is called the _mean field limit_ :

$n\to+\infty,\quad k\to 0,\quad nk=const;$

formally this is given by assuming that the state of motion $\psi_{n}(t)$ can
be factorized at each $t$ – fact that is not generally true, even if one
starts from a factorized state at $t=0$. We highlight that the powers
appearing in the relation $nk=const$ are typical of the boson star framework,
and are indeed different in other frameworks (for instance, in the case of
white dwarfs, we have $n^{3/2}k=const$).

By letting $n\to+\infty$ one can formally prove that, in some precise sense,
the motion of the (single body) boson star converges to the motion of the
following (time-dependent) PDE

$iu_{t}=\sqrt{-\Delta+m^{2}}u-\left(\int_{\mathbb{R}^{3}}\frac{u^{2}(y)}{|x-y|}dy\right)u\quad\hbox{
in $(0,+\infty)\times\mathbb{R}^{3}$}$ (4.1.3)

that is (up to constant)

$iu_{t}=\sqrt{-\Delta+m^{2}}u-(I_{2}*u^{2})u\quad\hbox{ in
$(0,+\infty)\times\mathbb{R}^{3}$.}$

We focus now on this equation, and on the corresponding energy functional

$E(u):=\frac{1}{2}\int_{\mathbb{R}^{3}}|(-\Delta+m)^{1/4}u|^{2}-\frac{1}{2}\int_{\mathbb{R}^{3}}\big{(}I_{2}*u^{2})u^{2}.$

By exploiting the Hardy-Littlewood-Sobolev and the fractional Gagliardo-
Nirenberg inequalities we obtain

$\int_{\mathbb{R}^{3}}\big{(}I_{2}*u^{2})u^{2}\leq\overline{C}\|(-\Delta)^{1/4}u\|_{2}^{2}\|u\|_{2}^{2},$
(4.1.4)

which combined with the trivial inequality $|\xi|^{2}+m^{2}\geq|\xi|^{2}$
gives

$E(u)\geq\frac{1}{2}\|(-\Delta)^{1/4}u\|_{2}^{2}\big{(}1-\overline{C}\|u\|_{2}^{2}\big{)}$

for some $\overline{C}>0$. Set

$\int_{\mathbb{R}^{3}}u^{2}=:M$

the total mass of the boson star (interpreting $u^{2}(x)$ as the density in
$x\in\mathbb{R}^{3}$) we obtain

$E(u)\geq\frac{1}{2}\|(-\Delta)^{1/4}u\|_{2}^{2}\big{(}1-\overline{C}M\big{)};$

from this we see that $E(u)$ could be or be not bounded from below on the
sphere $\\{u\in H^{1/2}(\mathbb{R}^{3})\mid\|u\|_{2}^{2}=M\\}$ depending on
the size of $M$: this is actually a phenomenon related to the _$L^{2}$
-critical growth_ $2=\frac{3+2+2\frac{1}{2}}{3}=2^{m}_{2,\frac{1}{2}}$ in
$N=3$. More precisely, one can prove that there exists a constant $M_{*}$,
related to the best constant of the inequality (4.1.4), such that

$\inf_{\|u\|_{2}^{2}=M}E(u)\;\left\\{\begin{aligned} &\geq 0&\quad\hbox{if
$M<M_{*}$},\\\ &=-\infty&\quad\hbox{if $M>M_{*}$}.\end{aligned}\right.$

As a further consequence, one might study the dynamical properties of
$u(x,t)=e^{it}u(x)$, solution of (4.1.3), showing that

$u=u(x,t)\;\left\\{\begin{aligned} &\hbox{exists for each $t>0$}&\quad\hbox{if
$M<M_{*}$},\\\ &\hbox{explodes in finite time}&\quad\hbox{if
$M>M_{*}$}.\end{aligned}\right.$

This is why $M_{*}$, called _Chandrasekhar mass_ , is related to the _self-
gravitational collapse_ of boson stars (i.e., the collapse due to their own
gravity). One could show that $M_{*}$ is related to a number of particles of
the size of $\sim 10^{38}$, that is, the number of particles that can be
approximately found in a mountain.

As already highlighted, $M_{*}$ is related to the best constant of the
inequality (4.1.4). And one can show that the optimizers $Q$ of this
inequality satisfy the following equation

$\sqrt{-\Delta}Q+\mu Q=\big{(}I_{2}*Q^{2}\big{)}Q\quad\hbox{in
$\mathbb{R}^{3}$}$

for some $\mu>0$ (actually $M_{*}$ equals the $L^{2}$ norm squared of $Q$).
And this is the equation we study.

###  4.2 Different approaches for the existence problem

In this Section we briefly sketch how to get existence of _free_ and
_constrained_ solutions. The techniques are based on the ideas of Chapters 2
and 3. Anyway we present here a different approach to handle the boundary of
$\mathbb{R}_{+}$, instead of considering the change of variable
$\mu=e^{\lambda}$. With this aim, we will give a proof of some details,
referring to Chapters 2-3 for all the other proofs.

This first Section is based on the paper [114] and [115] (see also [113]). For