# On a Perturbed Critical p-Kirchhoff-Type Problem

G. N. Cunha  Instituto de Matemática e Estatística, Universidade Federal de
Goiás, Goiânia GO74001-970, Brazil<EMAIL_ADDRESS>,  F. Faraci
Department of Mathematics and Computer Sciences, University of Catania, 95125
Catania, Italy<EMAIL_ADDRESS>and  K. Silva  Instituto de Matemática e
Estatística, Universidade Federal de Goiás, Goiânia GO74001-970, Brazil
<EMAIL_ADDRESS>
###### Abstract.

In this paper we deal with a stationary non-degenerate $p-$Kirchhoff type
problem with critical non-linearity and a subcritical parametrized
perturbation. We work on bounded domains of the Euclidean space, without any
restriction on the dimension or on $p>0$. Variational methods will be utilized
in combination with an analysis of the fibering functions of the energy
functional, in order to obtain ground state solutions, as well as Mountain
Pass solutions, depending on the values of the parameter. A local analysis of
the energy functional will allow us to obtain non-trivial solutions even
beyond the extremal parameter.

###### Contents

  1. 1 Introduction
  2. 2 Abstract Results
  3. 3 Existence Results
    1. 3.1 Global Minimizers
    2. 3.2 Local Minimizers
    3. 3.3 Mountain Pass Solutions
  4. 4 Non-Existence Results

_Mathematics Subject Classification (2010)_ : 35J20, 35B33.

_Key words and phrases_ : Critical Nonlinearity, Extremal Parameter, Fibering
Maps, Kirchhoff Term, Subcritical Perturbation, Variational Methods.

##  1\. Introduction

In this paper we deal with the following stationary Kirchhoff type problem:

$\left\\{\begin{array}[]{lr}-M\left(\displaystyle{\int_{\Omega}|\nabla
u|^{p}}dx\right)\Delta_{p}u=|u|^{p^{\star}-2}u+\lambda f(x,u)&in\ \ \
\Omega\\\ u=0&on\ \partial\Omega,\end{array}\right.$ (1)

where $1<p<+\infty$, $\Omega$ is a bounded domain in ${\mathbb{R}}^{N}$ with
smooth boundary, $p^{\star}=\frac{pN}{N-p}$, $M:[0,+\infty[\mapsto[0,+\infty[$
is a continuous function with $\hat{M}(t):=\displaystyle{\int_{0}^{t}M(s)ds}$,
$f:\Omega\times\mathbb{R}\mapsto\mathbb{R}$ is a Carathéodory function with
primitive $F(x,t)=\displaystyle{\int_{0}^{t}f(x,s)ds}$ for each $x\in\Omega$,
$t\in{\mathbb{R}}$.

According to the survey [18], Kirchhoff problems arise from the study of the
transverse oscillations of a stretched string. The original equation is

$\rho
hu_{tt}-\left\\{\rho_{0}+\frac{Eh}{2L}\int_{0}^{L}|u_{x}|^{2}dx\right\\}u_{xx}+\delta
u_{x}+f(x,u)=0,$ (2)

where $u=u(t,x)$ is the lateral displacement at the time $t$ and at the space
coordinate $x$, $E$ the Young modulus, $\rho$ the mass density, $h$ the cross
section area, $L$ the length of the string, $\rho_{0}$ the initial axial
tension, $\delta$ the resistance modulus, and $f$ is the external force. When
$\delta=f=0$, (2) was introduced by Kirchhoff in [13]. Further details and a
study of the physical phenomena described by Kirchhoff’s classical theory can
be found in [19]. In the last years, the existence and multiplicity of
solutions for Kirchhoff problems with a critical non-linearity have received
considerable attention. As a matter of fact, the main difficulty in dealing
with such problems is the lack of compactness of the Sobolev embedding
$W_{0}^{1,p}(\Omega)\subset L^{p^{\star}}(\Omega)$, which prevents the
application of standard variational methods.

The existence and multiplicity of solutions of Kirchhoff type equations with
critical exponents have been investigated by using different techniques, such
as truncation and variational methods, the Nehari manifold approach, the
Ljusternik-Schnirelmann category theory, genus theory (see for instance
[3]–[8] and the references therein).

In the present paper, we apply an idea introduced in [6], which was inspired
by the fibering method in [17] and the notion of extremal parameters described
in [12], to analize the topological changes occured on the energy functional
as the parameter $\lambda$ varies. With such a technique, we do not need to
consider the second order derivative of the fiber function, except for the
non-existence results. More specifically we employ the Second Lions’s
Concentration Compactness principle to obtain Mountain Pass type solutions
(see [1], [7],[10],[11],[15],[16],[20]), but also to establish the sequential
weak lower semi-continuity of the energy functional, with a proof inspired by
[14], which is, in turn (as far as we know), the only work so far where this
approach has been utilized. Also, in the cases beyond the extremal parameter,
we prove it is still possible to obtain non-trivial solutions, as long as we
minimize the energy functional locally, as in [6].

We are going to look for solutions of problem (1) in the Sobolev space
$W^{1,p}_{0}(\Omega)$. This linear space is endowed with the norm

$\|u\|:=\left(\int_{\Omega}|\nabla u|^{p}dx\right)^{\frac{1}{p}}$

and continuously embedded into $L^{p^{\star}}(\Omega)$ with embedding constant

$\displaystyle{S=\sup_{u\in
W_{0}^{1,p}(\Omega)\setminus\\{0\\}}\frac{\|u\|_{p^{\star}}^{p^{\star}}}{\|u\|^{p^{\star}}}}.$
(3)

A weak solution for problem (1) is a critical point of the energy functional
$\Phi_{\lambda}:W_{0}^{1,p}(\Omega)\mapsto{\mathbb{R}}$ given by

$\displaystyle{\Phi_{\lambda}(u)=\frac{1}{p}\hat{M}(\|u\|^{p})-\frac{1}{p^{\star}}\|u\|^{p^{\star}}_{p^{\star}}-\lambda\int_{\Omega}F(x,u(x))dx.}$

In order to control the behavior of the fibers at $0$ and $+\infty$, as well
as to establish the coercivity of the energy functional, we need the following
hypotheses on the non-local term $M$:

  1. $(\rho_{1})$:

$\displaystyle{\lim_{t\to 0^{+}}M(t)>0}$;

  2. $(\rho_{2})$:

$\displaystyle{\lim_{t\to+\infty}\frac{M(t)}{t^{\frac{r-p}{p}}}>0}$ for some
$r>p^{\star}$.

The sequential weak lower semi-continuity of the energy functional, on the
other hand, is associated with the following conditions:

  1. $(\beta_{1})$:

$\displaystyle{\inf_{t>0}\frac{\hat{M}(t)}{t^{\frac{p^{\star}}{p}}}\geq
S\frac{p}{p^{\star}}}$;

  2. $(\beta_{2})$:

$\displaystyle{\hat{M}(t+s)\geq\hat{M}(t)+\hat{M}(s)}$ for all $t>0$ and
$s>0$.

Notice that conditions $\eqref{beta1}$ and $\eqref{beta2}$ imply that
$\hat{M}$ is strictly increasing.

The existence of a Mountain Pass solution comes mainly from the next
condition. This hypothesis (which is stronger than ($\beta_{1}$)) is also
related to the non-existence results:

  1. $(\gamma_{1})$:

$\displaystyle{\inf_{t>0}\frac{M(t)}{t^{\frac{p^{\star}}{p}-1}}>S}$

An example of a function satisfying conditions ($\rho_{1}$), ($\rho_{2}$),
($\beta_{1}$), ($\beta_{2}$), and ($\gamma_{1}$), is

$M(t):=a+bt^{\alpha-1},$ (4)

for suitable values of $\alpha>1$ and $a,b>0$:

  * (i)

Assume $\alpha>\frac{N}{N-p}$; then

$\displaystyle{\inf_{t>0}\frac{\hat{M}(t)}{t^{\frac{p^{\star}}{p}}}}\geq\frac{p}{p^{\star}}S$

if, and only if,

$a^{\frac{N(\alpha-1)-\alpha
p}{p}}b\geq\left[\frac{p}{p^{\star}}S\frac{N(\alpha-1)-p\alpha}{N(\alpha-1)-p\alpha+p}\right]^{\frac{(N-p)(\alpha-1)}{p}}\alpha\frac{p}{N(\alpha-1)-p\alpha}.$

  * (ii)

Also, for $\alpha>\frac{N}{N-p}$, the following assertion holds true:

$\displaystyle{\inf_{t>0}\frac{M(t)}{t^{\frac{p^{\star}}{p}-1}}}>S$

if, and only if,

$a^{\frac{N(\alpha-1)-\alpha
p}{p}}b>\left[S\frac{N(\alpha-1)-p\alpha}{N(\alpha-1)-p\alpha+p}\right]^{\frac{(N-p)(\alpha-1)}{p}}\frac{p}{N(\alpha-1)-p\alpha}.$

  * (iii)

For $\alpha>1$, $\displaystyle{\lim_{t\to 0}M(t)>0}$.

  * (iv)

For any $r>p^{\star}$ such that $\alpha p>r$, there holds

$\displaystyle{\lim_{t\to+\infty}\frac{M(t)}{t^{\frac{r-p}{p}}}>0}.$

  * (v)

For $\alpha>1$, the inequality

$\hat{M}(t+s)\geq\hat{M}(t)+\hat{M}(s)\ \ \ \ \forall\ t,s\ \in[0,+\infty[$

holds true.

  * (vi)

For $\alpha\in(1,\frac{N}{N-p}]$, there holds

$\displaystyle{\inf_{t>0}\frac{\hat{M}(t)}{t^{\frac{p^{\star}}{p}}}}=\left\\{\begin{array}[]{lr}0&{\rm
if}\ \ \alpha<\frac{N}{N-p}\\\ \frac{b}{\alpha}&{\rm if}\ \
\alpha=\frac{N}{N-p}.\end{array}\right.$

Some comparison with related literature is in order. For
$S=S_{N}^{-\frac{p^{\star}}{p}}$ in (3), and $\alpha=p=2$ in (4), we obtain
the problem studied in the work [6]; in that paper, the conditions
corresponding to ($\beta_{1}$) and ($\gamma_{1}$) would be, respectively,

$a^{\frac{N-4}{2}}b\geq\frac{(N-4)^{\frac{N-4}{2}}}{N^{\frac{N-2}{2}}S_{N}^{\frac{N}{2}}}4$

and

$a^{\frac{N-4}{2}}b>\frac{(N-4)^{\frac{N-4}{2}}}{(N-2)^{\frac{N-2}{2}}S_{N}^{\frac{N}{2}}}2.$

In the present paper we also achieve an improvement with respect to [5]
concerning the semicontinuity property: in that paper, the condition
corresponding to ($\beta_{1}$), i.e.,

$\displaystyle{\inf_{l>0}\frac{\hat{M}(l)}{l^{\frac{p^{\star}}{p}}}\geq
c_{p}}$

where

$c_{p}=\left\\{\begin{array}[]{lcl}\left(2^{p-1}-1\right)^{\frac{p^{\star}}{p}}\frac{p}{p^{\star}}S_{N}^{-\frac{P^{\star}}{p}}&if&p\geq
2,\\\
2^{2p^{\star}-1-\frac{p^{\star}}{p}}\frac{p}{p^{\star}}S_{N}^{-\frac{p^{\star}}{p}}&if&1<p<2,\end{array}\right.$

is more restrictive than $(\beta_{1})$ since for $p\neq 2$ there holds

$c_{p}>\frac{p}{p^{\star}}S_{N}^{-\frac{p^{\star}}{p}}.$

The following hypotheses on the perturbation $f$ will be used throughout this
work.

  1. $(f_{1})$:

There exist $c_{1},c_{2}>0$ and $q\in(p,p^{\star})$ such that $|f(x,t)|\leq
c_{1}+c_{2}|t|^{q-1}$ for $t\in{\mathbb{R}}$, and a.e. in $\Omega$;

  2. $(f_{2})$:

$\displaystyle{\lim_{t\to 0}}\frac{f(x,t)}{|t|^{p-1}}=0$ uniformly on
$x\in\Omega$;

  3. $(f_{3})$:

$f(x,t)>0$ for every $t>0$, a.e in $\Omega$; and $f(x,t)<0$ for every $t<0$,
a.e in $\Omega$. Moreover there exists $\mu>0$ such that $f(x,t)\geq\mu>0$ for
a.a. $x\in\Omega$ and every $t\in I$, being $I$ an open interval of
$(0,+\infty)$.

An example of a function satisfying the conditions ($f_{1}$), ($f_{2}$), and
($f_{3}$), is

$\begin{array}[]{rcl}f(x,t)=|t|^{q-2}t&\forall&t\in{\mathbb{R}},\end{array}$

where $q\in(p,p^{\star})$ is fixed.

Now we introduce the main results of this paper, which we prove in the next
sections.

The first result justifies the necessity for the parameterized perturbation.

######  Theorem 1.1.

Under condition ($\gamma_{1}$) there exists a number $\lambda_{1}^{\star}>0$
such that for all $-\infty<\lambda<\lambda_{1}^{\star}$ problem (1) possesses
only the trivial solution.

The following existence result is the main goal of this paper.

######  Theorem 1.2.

Assume conditions ($\rho_{1}$), ($\rho_{2}$), ($\beta_{1}$), ($\beta_{2}$),
($f_{1}$), ($f_{2}$), and ($f_{3}$). Then, there exists
$\lambda_{0}^{\star}\geq 0$ such that

  * (i)

if $\lambda>\lambda_{0}^{\star}$, then the energy functional $\Phi_{\lambda}$
has a global minimizer $u_{\lambda}$ such that
$I_{\lambda}=\Phi_{\lambda}(u_{\lambda})<0$ ( in particular that
$u_{\lambda}\neq 0$);

  * (ii)

if $\lambda=\lambda_{0}^{\star}$, then the energy functional $\Phi_{\lambda}$
has a global minimizer $u_{\lambda_{0}^{\star}}$ such that
$I_{\lambda_{0}^{\star}}=0$; if the inequality in condition ($\beta_{1}$) is
strict, then $u_{\lambda_{0}^{\star}}\neq 0$;

  * (iii)

if $\lambda<\lambda_{0}^{\star}$, then for all $u\in
W_{0}^{1,p}(\Omega)\backslash\\{0\\}$ there holds $\Phi_{\lambda}(u)>0$.
Therefore, $u_{\lambda}=0$ is the only global minimizer of $\Phi_{\lambda}.$

The next result shows that although we may not find non-trivial global
minimizers for the energy functional for $\lambda<\lambda_{0}^{\star}$, we may
still find non-trivial local minimizers, as long as the parameter $\lambda$ is
close enough to $\lambda_{0}^{\star}$.

######  Theorem 1.3.

Assume conditions ($\rho_{1}$), ($\rho_{2}$), ($\beta_{1}$), ($\beta_{2}$),
($f_{1}$), ($f_{2}$), and ($f_{3}$). If the inequality in condition
($\beta_{1}$) is strict, then there exists $\epsilon>0$ small enough so that
for each $\lambda\in(\lambda_{0}^{\star}-\epsilon,\lambda_{0}^{\star})$ the
energy functional $\Phi_{\lambda}$ possesses a local minimizer with positive
energy.

The next result states the existence of a mountain pass type solution.

######  Theorem 1.4.

Assume conditions ($\rho_{1}$), ($\rho_{2}$), ($\beta_{2}$), ($\gamma_{1}$),
($f_{1}$), ($f_{2}$), and ($f_{3}$). Then, there exists $\epsilon>0$ small
enough such that for each $\lambda>\lambda_{0}^{\star}-\epsilon$, problem (1)
has a solution of mountain pass type.

##  2\. Abstract Results

Now we proceed to describe the abstract results which allow us to deduce our
main theorems, stated in Section 1.

For each $u\in W_{0}^{1,p}(\Omega)\backslash\\{0\\}$ and $\lambda\geq 0$, we
define the fiber function $\psi_{\lambda,u}:[0,+\infty[\mapsto{\mathbb{R}}$ by
$\psi_{\lambda,u}(t):=\Phi_{\lambda}(tu)$.

######  Lemma 2.1.

Assume conditions ($f_{1}$) and ($f_{2}$). Then, the following assertions hold
true:

  * (i):

Under condition ($\rho_{1}$), there exist
$\epsilon_{1}=\epsilon_{1}(\lambda,u)>0$ and
$\epsilon_{2}=\epsilon_{2}(\lambda,u)>0$ such that $\psi_{\lambda,u}(t)>0\
\forall t\in(0,\epsilon_{1})$, and $\psi^{\prime}_{\lambda,u}(t)>0\ \forall
t\in(0,\epsilon_{2})$;

  * (ii):

Under condition ($\rho_{2}$), there holds
$\displaystyle{\lim_{t\to\infty}\psi_{\lambda,u}(t)}=+\infty$ and
$\displaystyle{\lim_{t\to\infty}\psi^{\prime}_{\lambda,u}(t)=+\infty}$.

###### Proof.

We will prove the claim for $\psi_{\lambda,u}$. In fact, write

$\psi_{\lambda,u}(t)=t^{p}\|u\|^{p}\left[\frac{1}{p}\frac{\hat{M}(t^{p}\|u\|^{p})}{t^{p}\|u\|^{p}}-\frac{t^{p^{\star}-p}}{p^{\star}}\frac{\|u\|_{p^{\star}}^{p^{\star}}}{\|u\|^{p}}-\frac{\lambda}{\|u\|^{p}}\int_{\Omega}\frac{F(x,tu(x))}{t^{p}}dx\right].$

Note that by conditions ($f_{1}$) and ($f_{2}$), for each $\varepsilon>0$
there exists $c>0$ such that

$|F(x,t)|\leq\varepsilon|t|^{p}+c|t|^{q}\ \hbox{for all $t\in{\mathbb{R}}$,
a.e. in $\Omega$. }$

Therefore,

$\lim_{t\to 0}\int_{\Omega}\frac{F(x,tu(x))}{t^{p}}dx=0.$

By assumption ($\rho_{1}$) and De l’Hospital rule

$\displaystyle{\lim_{t\to
0}\frac{\hat{M}(t^{p}\|u\|^{p})}{t^{p}\|u\|^{p}}>0},$

and the first conclusion in $(i)$ follows.

By $(\rho_{2})$ and continuity of $M$, there exists positive constants
$c_{1},c_{2}$ such that

$\hat{M}(t)\geq c_{1}t^{\frac{r}{p}}-c_{2}\ \hbox{for all $t\geq 0$}.$

Thus, from $(f_{1})$, and possibly different constants $c_{i}$,

$\displaystyle\psi_{\lambda,u}(t)$
$\displaystyle=\frac{1}{p}{\hat{M}(t^{p}\|u\|^{p})}-\frac{t^{p^{\star}}}{p^{\star}}{\|u\|_{p^{\star}}^{p^{\star}}}-{\lambda}\int_{\Omega}{F(x,tu(x))}dx$
$\displaystyle\geq\frac{1}{p}c_{1}t^{r}\|u\|^{r}-\frac{t^{p^{\star}}}{p^{\star}}{\|u\|_{p^{\star}}^{p^{\star}}}-c_{3}t^{q}\|u\|_{q}^{q}-c_{2}$

and the first claim in $(ii)$ holds. ∎

For each $u\in W_{0}^{1,p}(\Omega)\backslash\\{0\\}$, consider now the
following system:

$\left\\{\begin{array}[]{l}\psi_{\lambda,u}(t)=0\\\
\psi^{\prime}_{\lambda,u}(t)=0\\\
\psi_{\lambda,u}(t)=\inf_{s>0}\psi_{\lambda,u}(s).\end{array}\right.$ (5)

######  Lemma 2.2.

Assume conditions ($f_{1}$), ($f_{2}$), ($\rho_{1}$), ($\rho_{2}$) and
($\beta_{1}$). Then, system (5) has a solution $(\lambda_{0}(u),t_{0}(u))$ for
each $u\in W_{0}^{1,p}(\Omega)\backslash\\{0\\}$. Furthermore, the solution is
unique with respect to $\lambda$.

###### Proof.

We proceed by showing first that for each $u\in
W_{0}^{1,p}(\Omega)\backslash\\{0\\}$, the set

$\Lambda_{u}:=\\{\lambda\geq 0:\ \ \inf_{t\geq 0}\psi_{\lambda,u}(t)\geq 0\\}$

is non empty. Fix $\bar{\lambda}>0$. By Lemma 2.1, there exist
$\epsilon_{1},\delta_{1}>0$ such that $\psi_{\bar{\lambda},u}>0$ in
$(0,\epsilon_{1})\cup(\delta_{1},+\infty)$.

For $\lambda\leq\bar{\lambda}$, since
$\psi_{\lambda,u}\geq\psi_{\bar{\lambda},u}$, the fiber $\psi_{\lambda,u}$ is
positive over $(0,\epsilon_{1})\cup(\delta_{1},+\infty)$.

Take a positive monotone sequence $\\{\lambda_{k}\\}_{k\geq 1}$ converging to
zero. The sequence of continuous real functions
$\\{\psi_{\lambda_{k},u}\\}_{k\geq 1}$ converges uniformly to $\psi_{0,u}$
over the compact interval $[\epsilon_{1},\delta_{1}]$. We observe that
$\inf_{[\epsilon_{1},\delta_{1}]}\psi_{0,u}>0$ as it follows by condition
($\beta_{1}$) and by the fact that the constant $S$ in (3) is not attained.
Therefore, there exists a $k_{0}$ such that the fiber $\psi_{\lambda_{k},u}$
is positive over the interval $[\epsilon_{1},\delta_{1}]$ for all $k>k_{0}$.
For $k$ large enough such that $\lambda_{k}<\bar{\lambda}$, we get that
$\lambda_{k}\in\Lambda_{u}$. Also, $\Lambda_{u}$ is bounded from above, since
for fixed $t_{0}>0$, $\lim_{\lambda\to\infty}\psi_{\lambda,u}(t_{0})=-\infty$.

Now we define the canditate $\lambda_{0}(u):=\sup\Lambda_{u}$. For each
positive $\epsilon\leq\epsilon_{0}$ (where $\epsilon_{0}$ is fixed), we denote
by $t_{0}(\epsilon)$, the first critical point of
$\psi_{\lambda_{0}(u)+\epsilon,u}$ such that

$\psi_{\lambda_{0}(u)+\epsilon,u}(t_{0}(\epsilon))<0\ .$ (6)

Since the function $\epsilon\mapsto\psi_{\lambda_{0}(u)+\epsilon,u}(t)$ is
decreasing, the map $\epsilon\mapsto t_{0}(\epsilon)$ is bounded from below by
the first root of the fiber $\psi_{\lambda_{0}(u)+\epsilon_{0},u}$. Define the
candidate $t_{0}(u):=\liminf_{\epsilon\to 0}t_{0}(\epsilon)$. By taking the
liminf on inequality (6) as $\epsilon$ goes to zero, we get
$\psi_{\lambda_{0}(u),u}(t_{0}(u))\leq 0$. Since $\lambda_{0}(u)$ belongs to
$\Lambda_{u}$ as well, there holds $\psi_{\lambda_{0}(u),u}(t)\geq 0$ for all
$t>0$, in particular for $t=t_{0}(u)$. Therefore,
$\psi_{\lambda_{0}(u),u}(t_{0}(u))=0$

Let us now prove the uniqueness.

Assume that the ordered pairs
$(\lambda_{0}(u),t_{0}(u))$,$(\lambda^{{}^{\prime}}_{0}(u),t^{{}^{\prime}}_{0}(u))$
are both solutions of system (5). Assume without loss of generality that
$\lambda_{0}^{{}^{\prime}}(u)\geq\lambda_{0}(u)$. Then,
$0\leq\psi_{\lambda_{0}^{{}^{\prime}}(u),u}(t_{0}(u))\leq\psi_{\lambda_{0}(u),u}(t_{0}(u))=0$,
which implies $\lambda_{0}^{{}^{\prime}}(u)=\lambda_{0}(u)$. The proof is
concluded. ∎

######  Corollary 2.1.

For all $u\in W_{0}^{1,p}(\Omega)\backslash\\{0\\}$ and all $k\geq 0$, there
holds $\lambda_{0}(ku)=\lambda_{0}(u)$.

###### Proof.

For $u\in W_{0}^{1,p}(\Omega)\backslash\\{0\\}$ and $k\geq 0$, there holds
$ku\in W_{0}^{1,p}(\Omega)\backslash\\{0\\}$. System (5) possesses a solution
$(\lambda_{0}(ku),t_{0}(ku))$, where the first coordinate is unique:

$\left\\{\begin{array}[]{l}\psi_{\lambda_{0}(ku),ku}(t_{0}(ku))=0\\\
\psi^{{}^{\prime}}_{\lambda_{0}(ku),ku}(t_{0}(ku))=0\\\
\psi_{\lambda_{0}(ku),ku}(t_{0}(ku))=\inf_{l>0}\psi_{\lambda_{0}(ku),ku}(l).\end{array}\right.$
(7)

System (7) may be rewritten as

$\left\\{\begin{array}[]{l}\psi_{\lambda_{0}(ku),u}(kt_{0}(ku))=0\\\
\psi^{{}^{\prime}}_{\lambda_{0}(ku),u}(kt_{0}(ku))=0\\\
\psi_{\lambda_{0}(ku),u}(kt_{0}(ku))=\inf_{l>0}\psi_{\lambda_{0}(ku),u}(kl)=\inf_{l>0}\psi_{\lambda_{0}(ku),u}(l).\end{array}\right.$

By uniqueness, we conclude that

$\lambda_{0}(ku)=\lambda_{0}(u).$

∎

Now, we define an extremal parameter which will play an important role in our
analysis:

$\lambda^{\star}_{0}:=\inf_{u\in
W_{0}^{1,p}(\Omega)\backslash\\{0\\}}\lambda_{0}(u).$ (8)

######  Lemma 2.3.

Assume conditions ($f_{1}$), ($f_{2}$), and ($f_{3}$). The following
assertions hold true.

  * (i):

Under condition ($\beta_{1}$), there holds $\lambda_{0}^{\star}\geq 0$.

  * (ii):

If the inequality in condition ($\beta_{1}$) is strict, then
$\lambda_{0}^{\star}>0$ and viceversa.

###### Proof.

$(i)$ Let us perform a proof by contradiction. Assume there exists a sequence
$\\{u_{k}\\}_{k\geq 1}\in W_{0}^{1,p}(\Omega)\backslash\\{0\\}$ such that (we
use the notation $\lambda_{k}:=\lambda_{0}(u_{k})$)

$-\infty\leq\lim_{k\to\infty}\lambda_{k}=\lambda_{0}^{\star}<0.$ (9)

We may assume by Corollary (2.1) that $\|u_{k}\|=1$ for all $k\geq 1$. By the
definition of $\lambda_{0}(u_{k})$, there exists a sequence
$\\{t_{k}\\}_{k\geq 1}$ of positive numbers (which is bounded, according to
Lemma (2.1) item $(ii)$ ) such that

$\frac{1}{p}\hat{M}(t_{k}^{p}\|u_{k}\|^{p})-\frac{1}{p^{\star}}t_{k}^{p^{\star}}\|u_{k}\|^{p^{\star}}_{p^{\star}}=\lambda_{k}\int_{\Omega}F(x,t_{k}u_{k})dx\
\ \forall\ k\geq 1.$

By the Sobolev embbeding,

$\frac{1}{p}\hat{M}(t_{k}^{p})-\frac{1}{p^{\star}}t_{k}^{p^{\star}}S\leq\lambda_{k}\int_{\Omega}F(x,t_{k}u_{k})dx\
\ \forall\ k\geq 1.$ (10)

The contradiction follows from (10) and (9). Therefore, there holds
$\lambda_{0}^{\star}\geq 0$.

$(ii)$ Let $L>S\frac{p}{p^{\star}}$ such that
$\inf_{t>0}\frac{\hat{M}(t)}{t^{\frac{p^{\star}}{p}}}\geq L.$ Arguing as in
the proof of item $(i)$, assume by contradiction that
$\displaystyle{\lim_{k\to+\infty}\lambda_{k}=\lambda_{0}^{\star}=0}$, where
$\lambda_{k}=\lambda_{0}(u_{k})$ with $\|u_{k}\|=1$. Thus, there exists a
sequence $\\{t_{k}\\}_{k\geq 1}$ of positive numbers such that

$\left(L-S\frac{p}{p^{\star}}\right)t_{k}^{p^{\star}}\leq{\hat{M}(t_{k}^{p}})-S\frac{p}{p^{\star}}t_{k}^{p^{\star}}\leq\lambda_{k}p\int_{\Omega}{F(x,t_{k}u_{k})}dx.$

The right hand side tends to zero since $\lambda_{k}\to 0$ and
$\left\\{\int_{\Omega}F(x,t_{k}u_{k})dx\right\\}$ is bounded due to the growth
of $F$, and the fact that $\\{t_{k}\\}$ and $\\{u_{k}\\}$ are bounded. This
implies that $\displaystyle\lim_{k\to+\infty}t_{k}=0$. Dividing the previous
inequality by $t_{k}^{p}$, we get

$\frac{\hat{M}(t_{k}^{p})}{t_{k}^{p}}-S\frac{p}{p^{\star}}t_{k}^{p^{\star}-p}\leq{\lambda_{k}}p\int_{\Omega}\frac{F(x,t_{k}u_{k})}{t_{k}^{p}}dx,$

which contradicts assumption ($\rho_{1}$) when passing to the limit as
$k\to\infty$. Therefore, $\lambda_{0}^{\star}>0$.

Let us prove the viceversa. Assume that condition ($\beta_{1}$) holds with
equality. We will prove that $\lambda_{0}^{\star}=0$. Without loss of
generality we may assume that $0\in\Omega$. Fix a ball of radius $r>0$ such
that $B_{2r}(0)\subset\Omega$ and let $\varphi$ a function in
$C^{\infty}_{0}(B_{2r}(0))$ such that $\varphi(x)=1$ in $B_{r}(0)$,
$0\leq\varphi\leq 1$ and $|\nabla\varphi|\leq 2$.

Put

$v_{\varepsilon}(x)=\frac{\varphi(x)}{\left(\varepsilon+|x|^{\frac{p}{p-1}}\right)^{\frac{N-p}{p}}}\qquad\hbox{and}\qquad
u_{\varepsilon}(x)=\frac{v_{\varepsilon}(x)}{\|v_{\varepsilon}\|}.$

By [9] (see also [2]), there exists a constant $K=K(N,p)$ such that

$\displaystyle\|v_{\varepsilon}\|^{p}=K\varepsilon^{-\frac{N-p}{p}}+O(1);\qquad\displaystyle\|v_{\varepsilon}\|_{p^{\star}}^{p}=KS^{\frac{p}{p^{\star}}}\varepsilon^{-\frac{N-p}{p}}+O(\varepsilon);$
$\displaystyle\|u_{\varepsilon}\|=1;\qquad\displaystyle\|u_{\varepsilon}\|_{p^{\star}}^{p^{\star}}=S+O(\varepsilon^{\frac{N}{p}}).$

We deduce in particular that

$\displaystyle\|v_{\varepsilon}\|=K^{\frac{1}{p}}\varepsilon^{-\frac{N-p}{p^{2}}}+O(\varepsilon^{\frac{(N-p)(p-1)}{p^{2}}}).$

Let $t_{0}>0$ such that

$\inf_{t>0}\frac{\hat{M}(t)}{t^{\frac{p^{\star}}{p}}}=\frac{\hat{M}(t_{0}^{p})}{t_{0}^{p^{\star}}}=S\frac{p}{p^{\star}}.$

Then,

$\displaystyle\psi_{\lambda,u_{\varepsilon}}(t_{0})$ $\displaystyle=$
$\displaystyle\frac{1}{p}\hat{M}(t_{0}^{p})-\frac{1}{p^{\star}}t_{0}^{p^{\star}}\|u_{\varepsilon}\|^{p^{\star}}_{p^{\star}}-\lambda\int_{\Omega}F(x,t_{0}u_{\varepsilon})dx$
$\displaystyle=$
$\displaystyle-\frac{1}{p^{\star}}t_{0}^{p^{\star}}O(\varepsilon^{\frac{N}{p}})-\lambda\int_{\Omega}F(x,t_{0}u_{\varepsilon})dx.$

Let us estimate $\displaystyle\int_{\Omega}F(x,t_{0}u_{\varepsilon})dx$ from
below. By assumption ($f_{3}$), one has that $f(x,s)\geq\mu\chi_{I}(s)$ (being
$\chi_{I}$ the characteristic function of the interval $I$), so there exist
$\alpha,\beta>0$ such that
$F(x,s)\geq\tilde{F}(s):=\mu\int_{0}^{s}\chi_{I}(t)dt\geq\beta$ for every
$s\geq\alpha$. Following Corollary 2.1 of [2] and using the positivity and
monotonicity of $F$,

$\displaystyle\int_{\Omega}F(x,t_{0}u_{\varepsilon})dx$
$\displaystyle\geq\int_{|x|\leq r}F(x,t_{0}u_{\varepsilon})dx\geq\int_{|x|\leq
r}F\left(x,\frac{t_{0}}{\|v_{\varepsilon}\|(\varepsilon+|x|^{\frac{p}{p-1}})^{\frac{N-p}{p}}}\right)dx$
$\displaystyle\geq\int_{|x|\leq
r}\tilde{F}\left(\frac{t_{0}}{\|v_{\varepsilon}\|(\varepsilon+|x|^{\frac{p}{p-1}})^{\frac{N-p}{p}}}\right)dx$
$\displaystyle=c_{1}\varepsilon^{\frac{N(p-1)}{p}}\int_{0}^{r\varepsilon^{-\frac{p-1}{p}}}\tilde{F}\left(\frac{t_{0}}{\|v_{\varepsilon}\|}\left(\frac{\varepsilon^{-1}}{1+s^{\frac{p}{p-1}}}\right)^{\frac{N-p}{p}}\right)s^{N-1}ds$

One has that

$\tilde{F}\left(\frac{t_{0}}{\|v_{\varepsilon}\|}\left(\frac{\varepsilon^{-1}}{1+s^{\frac{p}{p-1}}}\right)^{\frac{N-p}{p}}\right)\geq\beta\hbox{
if $s$ is such that}\
\frac{t_{0}}{\|v_{\varepsilon}\|}\left(\frac{\varepsilon^{-1}}{1+s^{\frac{p}{p-1}}}\right)^{\frac{N-p}{p}}\geq\alpha.$
(11)

Notice that the second inequality of (11) is equivalent to

$\frac{t_{0}\varepsilon^{-\frac{(N-p)(p-1)}{p^{2}}}}{(K^{\frac{1}{p}}+O(\varepsilon^{\frac{N-p}{p}}))(1+s^{\frac{p}{p-1}})^{\frac{N-p}{p}}}\geq\alpha.$

Now, fix $c_{2}<\frac{t_{0}}{\alpha}$. If $s\leq
c_{2}\varepsilon^{-\frac{(p-1)^{2}}{p^{2}}}$, for $\varepsilon$ small enough,
the above inequality holds true. This implies that for an eventually smaller
$r$,

$\displaystyle\int_{\Omega}F(x,t_{0}u_{\varepsilon})dx\geq
c_{3}\varepsilon^{\frac{N(p-1)}{p}}\int_{0}^{r\varepsilon^{-\frac{(p-1)^{2}}{p^{2}}}}\beta
s^{N-1}ds=c_{4}\varepsilon^{\frac{N(p-1)}{p^{2}}},$

for some positive constant $c_{4}$. Hence,

$\psi_{\lambda,u_{\varepsilon}}(t_{0})\leq-\frac{1}{p^{\star}}t_{0}^{p^{\star}}O(\varepsilon^{\frac{N}{p}})-\lambda
c_{4}\varepsilon^{\frac{N(p-1)}{p^{2}}}=\varepsilon^{\frac{N(p-1)}{p^{2}}}\left(O(\varepsilon^{\frac{N}{p^{2}}})-\lambda
c_{4}\right)<0,$

for small $\varepsilon>0$. We deduce that
$\lambda_{0}(u_{\varepsilon})<\lambda$ and because of the arbitrariness of
$\lambda,$ $\lambda_{0}^{\star}=0$. ∎

Now we prove a continuity result, which will be useful for proving the
existence of a minimizer of our problem. The proof we present here was
inspired by [14, Theorem 2.1].

######  Lemma 2.4.

Assume conditions ($\beta_{1}$) and ($\beta_{2}$). Then, for all
$\\{u_{k}\\}_{k\geq 1}\subset W_{0}^{1,p}(\Omega)$ such that
$u_{k}\rightharpoonup u\in W_{0}^{1,p}(\Omega)$, and $\\{\lambda_{k}\\}_{k\geq
1}\subset{\mathbb{R}}$ such that
$\lambda_{k}\rightarrow\lambda\in{\mathbb{R}}$,

$\Phi_{\lambda}(u)\leq\liminf_{k\to\infty}\Phi_{\lambda_{k}}(u_{k}).$

###### Proof.

Let $\\{u_{k}\\}_{k\geq 1}\subset W_{0}^{1,p}(\Omega)$ be such that
$u_{k}\rightharpoonup u\in W_{0}^{1,p}(\Omega)$. Let’s call
$\liminf_{k\to\infty}\Phi_{\lambda_{k}}(u_{k})=L$. By the second
Concentration-Compactness Lemma of Lions, there exist an at most countable
index set $J$, a set of points $\\{x_{j}\\}_{j\in J}\subset\overline{\Omega}$
and two families of positive numbers $\\{\eta_{j}\\}_{j\in J}$,
$\\{\nu_{j}\\}_{j\in J}$ such that

$\displaystyle|\nabla u_{k}|^{p}$ $\displaystyle\rightharpoonup
d\eta\geq|\nabla u|^{p}+\sum_{j\in J}\mu_{j}\delta_{x_{j}},$
$\displaystyle|u_{k}|^{p^{*}}$ $\displaystyle\rightharpoonup
d\nu=|u|^{p^{*}}+\sum_{j\in J}\nu_{j}\delta_{x_{j}},$

(weak star convergence in the sense of measures), where $\delta_{x_{j}}$ is
the Dirac mass concentrated at $x_{j}$ and such that

$S^{-\frac{p}{p^{\star}}}\nu_{j}^{\frac{p}{p^{*}}}\leq\mu_{j}\qquad\mbox{for
every $j\in J$}.$

Thus, we deduce

$\displaystyle L$ $\displaystyle=$
$\displaystyle\frac{1}{p}\hat{M}\left(\liminf_{k\to\infty}\int_{\Omega}|\nabla
u_{k}|^{p}dx\right)-\frac{1}{p^{\star}}\left(\liminf_{k\to\infty}\int_{\Omega}|u_{k}|^{p^{\star}}dx\right)$
$\displaystyle-\liminf_{k\to\infty}\lambda_{k}\int_{\Omega}F(x,u_{k}(x))dx$
$\displaystyle\overset{\eqref{beta2}}{\geq}$
$\displaystyle\frac{1}{p}\hat{M}\left(\int_{\Omega}|\nabla u|^{p}dx+\sum_{j\in
J}\mu_{j}\right)-\frac{1}{p^{\star}}\left(\int_{\Omega}|u|^{p^{\star}}dx+\sum_{j\in
J}\nu_{j}\right)-\lambda\int_{\Omega}F(x,u(x))dx$
$\displaystyle\overset{\eqref{beta2}}{\geq}$
$\displaystyle\Phi_{\lambda}(u)+\frac{1}{p}\sum_{j\in
J}\hat{M}\left(\mu_{j}\right)-\frac{1}{p^{\star}}\sum_{j\in J}\nu_{j}$
$\displaystyle\overset{\eqref{beta1}}{\geq}$
$\displaystyle\Phi_{\lambda}(u)+\frac{S}{p^{\star}}\sum_{j\in
J}\mu_{j}^{\frac{p^{\star}}{p}}-\frac{1}{p^{\star}}\sum_{j\in J}\nu_{j}$
$\displaystyle=$ $\displaystyle\Phi_{\lambda}(u).$

∎

######  Corollary 2.2.

Assume that condition ($\beta_{2}$) holds true and the inequality in condition
($\beta_{1}$) is strict. Let $\\{u_{k}\\}_{k\geq 1}\subset
W_{0}^{1,p}(\Omega)$ be a sequence such that $u_{k}\rightharpoonup u\in
W_{0}^{1,p}(\Omega)$ and
$\displaystyle{\Phi_{\lambda}(u)=\lim_{k\to\infty}\Phi_{\lambda_{k}}(u_{k})}$.
Then, $u_{k}\rightarrow u\in W_{0}^{1,p}(\Omega)$.

###### Proof.

Arguing as in the proof of Lemma (2.4) we deduce that inequality in (2) is
strict, that is $L>\Phi_{\lambda}(u)$ which contradicts our assumption. Hence
$J$ must be empty, that is

$\lim_{k\to\infty}\int_{\Omega}|u_{k}|^{p^{\star}}dx=\int_{\Omega}|u|^{p^{\star}}dx$

and the uniform convexity of $L^{p^{\star}}(\Omega)$ implies that $u_{k}\to
u\mbox{ in }L^{p^{\star}}(\Omega).$ By the fact that
$\Phi_{\lambda_{k}}(u_{k})\to\Phi_{\lambda}(u)$, it follows that
$\hat{M}(\|u_{k}\|^{p})\to\hat{M}(\|u\|^{p})$ which ensures our claim because
of the strict monotonicity of $\hat{M}$. ∎

##  3\. Existence Results

In this section, we study the existence of global and local minimizers for the
energy functional, as well as Mountain Pass solutions. The existence of
minimizers will be guaranteed by conditions ($\beta_{1}$) and ($\beta_{2}$),
while the existence of a Mountain Pass solution will be achieved under
assumption ($\gamma_{1}$). Throughout the sequel we will always assume
($\rho_{1}$) and ($\rho_{2}$).

###  3.1. Global Minimizers

First we look for global minimizers. Consider the problem

$I_{\lambda}:=\inf\left\\{\Phi_{\lambda}(u)\ :\ u\in
W_{0}^{1,p}(\Omega)\right\\}$ (13)

######  Proposition 3.1.

The infimum in problem (13) is attained by some $u_{\lambda}\in
W_{0}^{1,p}(\Omega)$, under conditions ($\beta_{1}$) and ($\beta_{2}$). If
$\lambda>\lambda_{0}^{\star}$, then $I_{\lambda}<0$ and $u_{\lambda}\neq 0$.
If $\lambda<\lambda_{0}^{\star}$, then $I_{\lambda}=0$ and $u_{\lambda}=0$.

###### Proof.

Condition ($\rho_{2}$), combined with the following inequality, gives us the
coercivity of the energy functional:

$\begin{array}[]{lcl}\Phi_{\lambda}(u)&\geq&\displaystyle{\frac{1}{p}\hat{M}(\|u\|^{p})-\frac{S}{p^{\star}}\|u\|^{p^{\star}}-\lambda\int_{\Omega}F(x,u(x))dx}\\\
&=&\displaystyle{\|u\|^{r}\left[\frac{1}{p}\frac{\hat{M}(\|u\|^{p})}{\left(\|u\|^{p}\right)^{\frac{r}{p}}}-\frac{S}{p^{\star}}\|u\|^{p^{\star}-r}-\lambda\int_{\Omega}\frac{F(x,u(x))}{\|u\|^{r}}dx\right]}.\end{array}$

Conditions ($\beta_{1}$) and ($\beta_{2}$) give us the sequential weak lower
semi continuity of the energy functional, as proven in Lemma (2.4). Therefore,
$I_{\lambda}$ is reached. In order to analyse the sign of $I_{\lambda}$, we
resort to system (5) and Definition (8).

If $0\leq\lambda<\lambda_{0}^{\star}$, then for all $u\in
W_{0}^{1,p}(\Omega)\backslash\\{0\\}$, $\lambda<\lambda_{0}(u)$ and so
$0=\psi_{\lambda_{0}(u),u}(t_{0}(u))\leq\psi_{\lambda_{0}(u),u}(1)<\psi_{\lambda,u}(1)=\Phi_{\lambda}(u)$.
Since $\Phi_{\lambda}(0)=0$, we are done.

If $\lambda>\lambda_{0}^{\star}$, there exists $u\in
W_{0}^{1,p}(\Omega)\backslash\\{0\\}$ such that $\lambda>\lambda_{0}(u)$. And
thus
$0=\psi_{\lambda_{0}(u),u}(t_{0}(u))>\psi_{\lambda,u}(t_{0}(u))=\Phi_{\lambda}(t_{0}(u)u)$.
∎

######  Proposition 3.2.

Assume condition ($\beta_{1}$) with strict inequality. Then, there exists
$u_{\lambda_{0}^{\star}}\in W_{0}^{1,p}(\Omega)\setminus\\{0\\}$ such that
$I_{\lambda_{0}^{\star}}=\Phi_{\lambda_{0}^{\star}}(u_{\lambda_{0}^{\star}})=0$.

###### Proof.

Fix a sequence $\lambda_{k}\downarrow\lambda_{0}^{\star}$. From Proposition
3.1, for each $k$ we can find $u_{k}\in W_{0}^{1,p}(\Omega)\backslash\\{0\\}$
such that $I_{\lambda_{k}}=\Phi_{\lambda_{k}}(u_{k})<0$. Since
$\lambda_{k}\downarrow\lambda_{0}^{\star}$ and assumptions ($f_{1}$),($f_{2}$)
hold true, the coercivity of $\Phi_{\lambda_{0}^{\star}}$ tells us that
$\\{u_{k}\\}$ is bounded, and therefore we may assume that
$u_{k}\rightharpoonup u_{\lambda_{0}^{\star}}$. From Lemma 2.4 we obtain

$\Phi_{\lambda_{0}^{\star}}(u_{\lambda_{0}^{\star}})\leq\liminf_{k\to\infty}\Phi_{\lambda_{k}}(u_{k})\leq
0.$

Since for all $u\in W_{0}^{1,p}(\Omega)\backslash\\{0\\}$ one has
$\lambda_{0}^{\star}\leq\lambda_{0}(u)$, there holds
$0=\psi_{\lambda_{0}(u),u}(t_{0}(u))\leq\psi_{\lambda_{0}(u),u}(1)\leq\psi_{\lambda_{0}^{\star},u}(1)=\Phi_{\lambda_{0}^{\star}}(u)$.
Therefore,
$I_{\lambda_{0}^{\star}}=\Phi_{\lambda_{0}^{\star}}(u_{\lambda_{0}^{\star}})=0$.

Let us prove that $u_{\lambda_{0}^{\star}}\neq 0$.

Assume the contrary. Let $L>S\frac{p}{p^{\star}}$ such that
$\inf_{t>0}\frac{\hat{M}(t)}{t^{\frac{p^{\star}}{p}}}\geq L.$ Thus,

$\left(L-S\frac{p}{p^{\star}}\right)\|u_{k}\|^{p^{\star}}\leq{\hat{M}(\|u_{k}\|^{p}})-\frac{p}{p^{\star}}\|u_{k}\|_{p^{\star}}^{p^{\star}}\leq\lambda_{k}p\int_{\Omega}{F(x,u_{k})}dx.$

The right hand side tends to zero by the growth of $F$, and by using that
$u_{k}\to 0$ in $L^{q}(\Omega)$ for $q<p^{\star}$. Hence, $u_{k}\to 0$ in
$W_{0}^{1,p}(\Omega)$. Dividing the previous inequality by $\|u_{k}\|^{p}$, we
get

$\frac{\hat{M}(\|u_{k}\|^{p})}{\|u_{k}\|^{p}}-S\frac{p}{p^{\star}}\|u_{k}\|^{p^{\star}-p}\leq\frac{\lambda_{k}}{\|u_{k}\|^{p}}p\int_{\Omega}{F(x,u_{k})}dx.$

And since the right hand side is still tending to zero, we get the desired
contradiction, by ($\rho_{1}$). ∎

Proof of Theorem 1.2. The existence of a global minimizer for $\Phi_{\lambda}$
follows by the coercivity of the energy functional and the lower
semicontinuity property given by Lemma 2.4. The rest of the proof is a
consequence of Propositions 3.1 and 3.2.∎

###  3.2. Local Minimizers

We have already proved that the energy functional possesses a global minimizer
$u_{\lambda}\in W_{0}^{1,p}(\Omega)$, regardless of $\lambda$. However, when
$0\leq\lambda<\lambda_{0}^{\star}$ Proposition (3.1) tells us that
$u_{\lambda}=0$, while Proposition (3.2) states the existence of a non-trivial
solution for $\lambda=\lambda_{0}^{\star}$. Therefore, for
$0\leq\lambda<\lambda_{0}^{\star}$ we tackle a different minimization problem.
Let $\delta>0$ and define

$I_{\lambda}^{\delta}:=\inf_{u\in K_{\delta}}\Phi_{\lambda}(u),$ (14)

where

$K_{\delta}:=\left\\{u\in W_{0}^{1,p}(\Omega)\ |\ d(u,K)\leq\delta\right\\}$

and

$K:=\left\\{u\in W_{0}^{1,p}(\Omega)\backslash\\{0\\}\ |\
\Phi_{\lambda_{0}^{\star}}(u)=0\right\\}.$

######  Remark 3.1.

Notice that by Proposition (3.2) there holds $K\neq\emptyset$, as long as the
inequality in condition ($\beta_{1}$) is strict.

We are going to prove that problem (14) has a solution $u_{\lambda}^{\delta}$
which, for $0\leq\lambda<\lambda_{0}^{\star}$ close enough to
$\lambda_{0}^{\star}$, does not belong to $\partial K_{\delta}$ and for
$\delta$ small enough is non trivial.

The proof of this fact is based on the following lemmas.

######  Lemma 3.1.

Assume conditions ($f_{1}$), and ($f_{2}$) and ($\beta_{1}$) with strict
inequality. Then, there exists $\bar{\delta}$ such that for
$0<\delta<\bar{\delta}$, $0\notin K_{\delta}$.

###### Proof.

Assume by contradiction that there exists $\delta_{n}\to 0$ with
$d(0,K)\leq\delta_{n}$ for all $n\geq 1$. Therefore, $d(0,K)=0$, so
$0\in\bar{K}$. This implies the existence of $\\{u_{n}\\}_{n\geq 1}\subset K$,
$u_{n}\rightarrow 0$ in $W_{0}^{1,p}(\Omega)$. Thus,

$\begin{array}[]{c}\displaystyle{0=\|u_{n}\|^{p}\left[\frac{1}{p}\frac{\hat{M}(\|u_{n}\|^{p})}{\|u_{n}\|^{p}}-\frac{1}{p^{\star}}\frac{\|u_{n}\|_{p^{\star}}^{p^{\star}}}{\|u_{n}\|^{p}}-\lambda_{0}^{\star}\int_{\Omega}\frac{F(x,u_{n}(x))}{\|u_{n}\|^{p}}dx\right]}\
\geq\\\
\displaystyle{\|u_{n}\|^{p}\left[\frac{1}{p}\frac{\hat{M}(\|u_{n}\|^{p})}{\|u_{n}\|^{p}}-\frac{1}{p^{\star}}S\|u_{n}\|^{p^{\star}-p}-\lambda_{0}^{\star}\int_{\Omega}\frac{F(x,u_{n}(x))}{\|u_{n}\|^{p}}dx\right]}\end{array}$

and the latter is positive as it follows by ($\rho_{1}$), ($f_{1}$), and
($f_{2}$), leading to a contradiction. ∎

######  Lemma 3.2.

Assume conditions ($f_{1}$), ($f_{2}$), ($\beta_{2}$) and ($\beta_{1}$) with
strict inequality. Then, there exists $\bar{\delta}$ such that for
$0<\delta<\bar{\delta}$,

$\inf_{u\in\partial K_{\delta}}\Phi_{\lambda_{0}^{\star}}(u)>0.$

###### Proof.

Suppose by contradiction that for $\delta$ small enough there exists
$\\{u_{k}\\}_{k\geq 1}\subset\partial K_{\delta}$ such that
$\lim_{k\to\infty}\Phi_{\lambda_{0}^{\star}}(u_{k})=0$. Then, by the
coercivity of the energy functional, $u_{k}\rightharpoonup w$ in
$W_{0}^{1,p}(\Omega)$ up to a subsequence. Also, by the weak lower semi-
continuity of $\Phi_{\lambda_{0}^{\star}}$ it follows that
$\Phi_{\lambda_{0}^{\star}}(w)=0$, which means that $w\in K\cup\\{0\\}$ and
that $u_{k}\rightarrow w$ strongly in $W_{0}^{1,p}(\Omega)$ by Corollary
(2.2), i.e. $w\in\partial K_{\delta}$. Subsequently, by Lemma (3.1), $w\neq
0$, and we reach the desired contradiction: $w\in K$ along with
$d(w,K)=\delta>0$. ∎

######  Lemma 3.3.

Assume conditions ($f_{1}$), ($f_{2}$), ($\beta_{2}$), and ($\beta_{1}$) with
strict inequality. Then, the set $K$ is compact in $W_{0}^{1,p}(\Omega)$ with
the strong topology.

###### Proof.

Let $\\{v_{n}\\}_{n\geq 1}\subset K$. The definition of $K$ and the coercivity
of the energy functional imply that the sequence $\\{v_{n}\\}_{n\geq 1}\subset
K$ is bounded in $W_{0}^{1,p}(\Omega)$. And consequently $v_{n}\rightharpoonup
v$ in $W_{0}^{1,p}(\Omega)$ up to a subsequence. By the sequential weak lower
semi-continuity of the energy functional we obtain that
$0\leq\Phi_{\lambda_{0}^{\star}}(v)\leq\liminf_{n\to\infty}\Phi_{\lambda_{0}^{\star}}(v_{n})=0$,
which implies that $v\in K\cup\\{0\\}$. Therefore, we obtain that
$v_{n}\rightarrow v$ in $W_{0}^{1,p}(\Omega)$ by Corollary (2.2). Since
$K\subset K_{\delta}$ for all $\delta>0$, Lemma (3.1) finishes the proof. ∎

######  Lemma 3.4.

Assume conditions ($f_{1}$), ($f_{2}$), ($\beta_{2}$) and condition
($\beta_{1}$) with strict inequality. Then, the set $K_{\delta}$ is
sequentially weakly closed in $W_{0}^{1,p}(\Omega)$.

###### Proof.

Let $\\{u_{n}\\}_{n\geq 1}\subset K_{\delta}$ be such that
$u_{n}\rightharpoonup u_{0}$ in $W_{0}^{1,p}(\Omega)$. Since
$d(u_{n},K)\leq\delta$ for all $n\geq 1$, we will obtain that
$d(u_{0},K)\leq\delta$ if we show that $d(u_{0},K)\leq\liminf_{n}d(u_{n},K)$.

To accomplish so, assume there exists a positive constant $c$ such that

$d(u_{0},K)>c>\liminf_{n\to\infty}d(u_{n},K).$

By Lemma (3.3), $K$ is a compact subset of
$\left(W_{0}^{1,p}(\Omega),\|\cdot\|\right)$. Therefore, for each $n$, there
exists $v_{n}\in K$ such that $d(u_{n},K)=\|u_{n}-v_{n}\|$. Up to a
subsequence, $v_{n}\rightarrow v_{0}\in K$. The following chain of
inequalities leads us to a contradiction:

$\begin{array}[]{rrl}\|u_{0}-v_{0}\|&\leq&\liminf_{n\to\infty}\|u_{n}-v_{0}\|\\\
&\leq&\liminf_{n\to\infty}(\|u_{n}-v_{n}\|+\|v_{n}-v_{0}\|)\\\
&=&\liminf_{n\to\infty}(d(u_{n},K)+\|v_{n}-v_{0}\|)\\\
&<&c+\lim_{n\to\infty}\|v_{n}-v_{0}\|\\\ &<&d(u_{0},K)\end{array}$

∎

Proof of Theorem 1.3. Let $\delta>0$ small enough be such that $0\notin
K_{\delta}$ (see Lemma (3.1). By (14), for $\lambda\leq\lambda_{0}^{\star}$
there exits a sequence $\\{u_{n}\\}_{n\geq 1}\subset K_{\delta}$ such that

$\lim_{n\to\infty}\Phi_{\lambda}(u_{n})=I_{\lambda}^{\delta}.$

Since the functional $\Phi_{\lambda}$ is coercive, $\\{u_{n}\\}_{n\geq 1}$
must be bounded on $W_{0}^{1,p}(\Omega)$. And therefore up to a subsequence

$u_{n}\rightharpoonup u_{\lambda}^{\delta}\in W_{0}^{1,p}(\Omega).$

By Lemma (3.4), $u_{\lambda}^{\delta}\in K_{\delta}$ and so

$I_{\lambda}^{\delta}:=\inf_{u\in
K_{\delta}}\Phi_{\lambda}(u)=\Phi_{\lambda}(u_{\lambda}^{\delta}).$

Let us show that if we take the parameter $\lambda\leq\lambda_{0}^{\star}$
close enough to $\lambda_{0}^{\star}$, $u_{\lambda}^{\delta}$ does not belong
to $\partial K_{\delta}$. Otherwise, there exists a sequence of positive
numbers $\\{\lambda_{k}\\}_{k\geq 1}$, with
$\lambda_{k}\leq\lambda_{0}^{\star}$, and
$\lim_{k\to\infty}\lambda_{k}=\lambda_{0}^{\star}$, such that
$u_{\lambda_{k}}^{\delta}\in\partial K_{\delta}$ for each $k$. Since $\partial
K_{\delta}\subset K_{\delta}$, for all $k\geq 1$,

$I_{\lambda_{k}}^{\delta}=\inf_{u\in\partial
K_{\delta}}\Phi_{\lambda_{k}}(u).$

Fix any $u_{0}\in K$. From the considerations above, we obtain that for all
$k\geq 1$,

$\inf_{u\in\partial
K_{\delta}}\Phi_{\lambda_{0}^{\star}}(u)\leq\inf_{u\in\partial
K_{\delta}}\Phi_{\lambda_{k}}(u)=\inf_{u\in
K_{\delta}}\Phi_{\lambda_{k}}(u)\leq\inf_{u\in
K}\Phi_{\lambda_{k}}(u)\leq\Phi_{\lambda_{k}}(u_{0}).$

Since
$\Phi_{\lambda_{k}}(u_{0})\rightarrow\Phi_{\lambda_{0}^{\star}}(u_{0})=0$, and
$\Phi_{\lambda_{0}^{\star}}\geq 0$ we obtain that

$\inf_{u\in\partial K_{\delta}}\Phi_{\lambda_{0}^{\star}}(u)=0,$

which contradicts Lemma (3.2). Thus, $u_{\lambda}^{\delta}\notin\partial
K_{\delta}$ and so it is a local minimizer of $\Phi_{\lambda}$.

Also, by Lemma (3.1) $u_{\lambda}^{\delta}\neq 0$, and by Remark (LABEL:eitaa)
there holds $I_{\lambda}^{\delta}>0$. ∎

###  3.3. Mountain Pass Solutions

In this section we prove the Palais-Smale property for the energy functional
and our existence result will be a consequence of the Mountain Pass theorem.

First, we establish the Palais-Smale condition. We follow the proof of [5,
Theorem 1.3 ].

######  Lemma 3.5.

Assume condition ($\gamma_{1}$). Then, for any $\lambda\geq 0$ the energy
functional $\Phi_{\lambda}$ satisfies the Palais-Smale property.

###### Proof.

Let $\\{u_{k}\\}_{k\geq 1}$ be a Palais Smale sequence at a level $c$, i.e. a
sequence satisfying the conditions

$\left\\{\begin{array}[]{lll}\displaystyle{\lim_{k\to\infty}\Phi_{\lambda}(u_{k})}&=&c\\\
\displaystyle{\lim_{k\to\infty}\Phi^{{}^{\prime}}_{\lambda}(u_{k})}&=&0.\end{array}\right.$

Since

$\displaystyle{\inf_{t>0}\frac{M(t)}{t^{\frac{p^{\star}}{p}-1}}>S},$

we may assume there exists a $L>S$ such that

$\begin{array}[]{lr}M(t)>Lt^{\frac{p^{\star}}{p}-1}&\forall t\geq
0.\end{array}$ (15)

Then,

$\begin{array}[]{lr}\hat{M}(t)\geq
L\frac{p}{p^{\star}}t^{\frac{p^{\star}}{p}}&\forall t\geq 0.\end{array}$

In order to prove the coercivity of the energy functional, we may argue as in
the proof of Proposition (3.1). The sequence $\\{u_{k}\\}_{k\geq 1}$ is
bounded in $W_{0}^{1,p}(\Omega)$. And, up to subsequences, the following holds
true.

$\left\\{\begin{array}[]{lccr}u_{k}\rightharpoonup
u&in&W_{0}^{1,p}(\Omega)&\\\ u_{k}\rightarrow
u&in&L^{q}(\Omega),&q\in[1,+\infty)\\\ u_{k}\rightarrow u&a.e\
on&\Omega.&\end{array}\right.$

By the Concentration-Compactness Lemma of Lions there exist an at most
countable index set $J$, a set of points $\\{x_{j}\\}_{j\in
J}\subset\overline{\Omega}$ and two families of positive numbers
$\\{\eta_{j}\\}_{j\in J}$, $\\{\nu_{j}\\}_{j\in J}$ such that

$\displaystyle|\nabla u_{k}|^{p}$ $\displaystyle\rightharpoonup
d\eta\geq|\nabla u|^{p}+\sum_{j\in J}\mu_{j}\delta_{x_{j}},$
$\displaystyle|u_{k}|^{p^{*}}$ $\displaystyle\rightharpoonup
d\nu=|u|^{p^{*}}+\sum_{j\in J}\nu_{j}\delta_{x_{j}},$

(weak star convergence in the sense of measures), where $\delta_{x_{j}}$ is
the Dirac mass concentrated at $x_{j}$ and such that

$S^{-\frac{p}{p^{\star}}}\nu_{j}^{\frac{p}{p^{*}}}\leq\mu_{j}\qquad\mbox{for
every $j\in J$}.$

We will prove that $J$ is empty. Assume by contradiction that there exists an
index $j_{0}\in J$. And, for $\epsilon>0$ define the following smooth function
on $\Omega$.

$\phi_{\epsilon}(x)=\left\\{\begin{array}[]{crc}1,&x\in&B(x_{0},\epsilon)\\\
0,&x\in&\Omega\backslash B(x_{0},2\epsilon)\end{array}\right.$

such that $|\nabla\phi_{\epsilon}(x)|\leq\frac{2}{\epsilon}.$ For each
$\epsilon>0$, $\\{u_{k}\phi_{\epsilon}\\}_{k\geq 1}$ is bounded in
$W_{0}^{1,p}(\Omega)$. Therefore,

$\lim_{k\to+\infty}\Phi_{\epsilon}^{{}^{\prime}}(u_{k})(u_{k}\phi_{\epsilon})=0.$

And thus

$\begin{array}[]{lll}o(1)&=&M(\|u_{k}\|^{p})\displaystyle{\int_{\Omega}|\nabla
u_{k}|^{p-2}\nabla u_{k}(\nabla
u_{k}\phi_{\epsilon})dx-\int_{\Omega}|u_{k}|^{p^{\star}}\phi_{\epsilon}dx}-\lambda\displaystyle{\int_{\Omega}f(x,u_{k})u_{k}\phi_{\epsilon}dx}\\\
&=&M(\|u_{k}\|^{p})\displaystyle{\left[\int_{\Omega}|\nabla
u_{k}|^{p}\phi_{\epsilon}+u_{k}|\nabla u_{k}|^{p-2}\nabla
u_{k}\nabla\phi_{\epsilon}dx\right]-\int_{\Omega}|u_{k}|^{p^{\star}}\phi_{\epsilon}dx}-\lambda\displaystyle{\int_{\Omega}f(x,u_{k})u_{k}\phi_{\epsilon}dx}\end{array}$
(16)

On the other hand, by applying the Lebesgue Dominated Convergence Theorem, we
may prove that

$\displaystyle{\lim_{\epsilon\to 0}\lim_{k\to\infty}\int_{\Omega}u_{k}|\nabla
u_{k}|^{p-2}\nabla u_{k}\nabla\phi_{\epsilon}dx}=0,$
$\begin{array}[]{lcl}\displaystyle{\lim_{\epsilon\to
0}\lim_{k\to\infty}\int_{\Omega}|u_{k}|^{p^{\star}}\phi_{\epsilon}dx}=\displaystyle{\lim_{\epsilon\to
0}\int_{B(x_{0},2\epsilon)}|u_{k}|^{p^{\star}}\phi_{\epsilon}dx+\nu_{j_{0}}}=\nu_{j_{0}},\par\end{array}$

and, by condition ($f_{1}$), that

$\lim_{\epsilon\to
0}\lim_{k\to+\infty}\displaystyle{\int_{\Omega}f(x,u_{k})u_{k}\phi_{\epsilon}dx}=0.$

Since $M(\|u_{k}\|^{p})$ is bounded in ${\mathbb{R}}$, we deduce that

$\displaystyle{\lim_{\epsilon\to
0}\lim_{k\to\infty}M(\|u_{k}\|^{p})\int_{\Omega}u_{k}|\nabla
u_{k}|^{p-2}\nabla u_{k}\nabla\phi_{\epsilon}dx}=0.$

Furthermore, we have that

$\begin{array}[]{lcl}\displaystyle{\lim_{k\to\infty}M(\|u_{k}\|^{p})\int_{\Omega}|\nabla
u_{k}|^{p}\phi_{\epsilon}dx}&\geq&\displaystyle{\lim_{k\to\infty}\left[M\left(\int_{\Omega}|\nabla
u_{k}|^{p}dx\right)\int_{B(x_{0},2\epsilon)}|\nabla
u_{k}|^{p}\phi_{\epsilon}dx\right]}\\\
&\geq&\displaystyle{\lim_{k\to\infty}\left[L\left(\int_{B(x_{0},2\epsilon)}|\nabla
u_{k}|^{p}\phi_{\epsilon}dx\right)^{\frac{p^{\star}}{p}-1}\int_{B(x_{0},2\epsilon)}|\nabla
u_{k}|^{p}\phi_{\epsilon}dx\right]}\\\
&\geq&L\left[\displaystyle{\int_{B(x_{0},2\epsilon)}|\nabla
u|^{p}\phi_{\epsilon}dx+\mu_{j_{0}}}\right]^{\frac{p^{\star}}{p}}.\end{array}$

The above outcomes, combined together give us that

$\begin{array}[]{lcl}0\geq
L\mu_{j_{0}}^{\frac{p^{\star}}{p}}-\nu_{j_{0}}=\left(L-S\right)\mu_{j_{0}}^{\frac{p^{\star}}{p}}\geq
0,\end{array}$

which means that $\mu_{j_{0}}=0$ and, subsequently that $\nu_{j_{0}}=0$, a
contradiction. Thus, $J=\emptyset$ and so

$\displaystyle{\lim_{k\to\infty}\int_{\Omega}|u_{k}|^{p^{\star}}dx=\int_{\Omega}|u|^{p^{\star}}dx},$
(17)

which implies that $\\{u_{k}\\}$ converges to $u\in L^{p^{\star}}$ strongly.

Let us finally prove that $u_{k}\to u$ strongly in $W_{0}^{1,p}(\Omega)$. We
already know by hypothesis that

$\begin{array}[]{lcl}\displaystyle
0={\lim_{k\to\infty}\Phi^{{}^{\prime}}_{\lambda}(u_{k})(u_{k}-u)}&=&\displaystyle{\lim_{k\to\infty}\left[M(\|u_{k}\|^{p})\displaystyle{\int_{\Omega}|\nabla
u_{k}|^{p-2}\nabla
u_{k}\nabla(u_{k}-u)}dx-\int_{\Omega}|u_{k}|^{p^{\star}-2}u_{k}(u_{k}-u)dx\right.}\\\
&&\left.-\lambda\displaystyle{\int_{\Omega}f(x,u_{k})(u_{k}-u)dx}\right],\end{array}$

and, by using condition ($f_{1}$) and (17) we obtain that

$\displaystyle{\lim_{k\to+\infty}\int_{\Omega}f(x,u_{k})(u_{k}-u)dx=0},$

$\displaystyle{\lim_{k\to\infty}\int_{\Omega}|u_{k}|^{p^{\star}-2}u_{k}(u_{k}-u)dx}=0.$

Thus,

$\displaystyle{\lim_{k\to\infty}M(\|u_{k}\|^{p})\int_{\Omega}|\nabla
u_{k}|^{p-2}\nabla u_{k}\nabla(u_{k}-u)dx}=0.$

If $\displaystyle{\lim_{k\to\infty}M(\|u_{k}\|^{p})}=0$, then from (15) one
has that $\displaystyle{\lim_{k\to\infty}\|u_{k}\|=0}$, which means that
$u_{k}\rightarrow 0$ strongly in $W_{0}^{1,p}(\Omega)$. Otherwise,
$\displaystyle{\limsup_{k\to\infty}M(\|u_{k}\|^{p})}>0$, that implies

$\displaystyle{\lim_{k\to\infty}\int_{\Omega}|\nabla u_{k}|^{p-2}\nabla
u_{k}\nabla(u_{k}-u)dx=0}.$

Since $\\{u_{k}\\}_{k\geq 1}$ converges weakly to $u$, we know that

$\displaystyle{\lim_{k\to\infty}\int_{\Omega}|\nabla u|^{p-2}\nabla
u\nabla(u_{k}-u)dx=0}.$

Then, we deduce that

$\displaystyle{\lim_{k\to\infty}\int_{\Omega}\left(|\nabla u_{k}|^{p-2}\nabla
u_{k}-|\nabla u|^{p-2}\nabla u\right)\nabla(u_{k}-u)dx=0},$

and as a consequence, we obtain that $u_{k}\rightarrow u$ in
$W_{0}^{1,p}(\Omega)$. ∎

Let us point out now that the energy functional complies with the mountain
pass geometry. Notice that when $\lambda\geq\lambda_{0}^{\star}$ this is
obvious since $0$ is a strict local minimizer (see Lemma 3.6) and
$u_{\lambda}$ is a global minimizer with energy level less or equal than zero.
Thus, the interesting case is when
$\lambda_{0}^{\star}-\epsilon<\lambda<\lambda_{0}^{\star}$.

######  Lemma 3.6.

Under condition ($\gamma_{1}$) the following statements hold true.

  * (i)

For $R>0$ small enough there exists $\sigma=\sigma(R)>0$ such that
$\Phi_{\lambda}(u)\geq\sigma$ for all $u\in W_{0}^{1,p}(\Omega)$ with
$\|u\|=R$;

  * (ii)

For $0<R<\|u_{\lambda_{0}^{*}}\|$ in item $(i)$, there exists $\epsilon>0$
small enough such that $\Phi_{\lambda}(u_{\lambda_{0}^{*}})<\sigma$ for all
$\lambda>\lambda_{0}^{\star}-\epsilon$.

###### Proof.

$(i)$ By ($\rho_{1}$) there exists $C>0$ and $\delta>0$ such that if
$\|u\|<\delta$ then

$\hat{M}(\|u\|^{p})>C\|u\|^{p}.$

Also, from ($f_{2}$) we deduce that for fixed $\epsilon>0$ we may choose
$\delta>0$ in such a way that
$\displaystyle{\int_{\Omega}F(x,u)dx}\leq\epsilon\|u\|^{p}$ for
$\|u\|<\delta$.

Consequently, the following inequality holds true:

$\Phi_{\lambda}(u)\geq\left(\frac{C}{p}-\lambda\epsilon\right)\|u\|^{p}-\frac{1}{p^{\star}}\|u\|_{p^{\star}}^{p^{\star}}\geq\|u\|^{p}\left[\left(\frac{C}{p}-\lambda\epsilon\right)-\frac{S}{p^{\star}}\|u\|^{p^{\star}-p}\right].$

Therefore, if we take $\epsilon>0$ such that $\frac{C}{p}-\lambda\epsilon>0$,
and take $R<\delta=\delta(\epsilon)$ small enough so that, for all $u\in
W_{0}^{1,p}(\Omega)$ such that $\|u\|=R$, there holds
$\frac{S}{p^{\star}}\|u\|^{p^{\star}-p}<\frac{C}{p}-\lambda\epsilon$, we are
led into the desired conclusion.

$(ii)$ From Proposition (3.1),
$\Phi_{\lambda_{0}^{\star}}(u_{\lambda_{0}^{\star}})=0$. Therefore, the
monotonicity of the function
$\lambda\mapsto\Phi_{\lambda}(u_{\lambda_{0}^{\star}})$ ensures that
$\Phi_{\lambda}(u_{\lambda_{0}^{\star}})\leq 0$ for all
$\lambda\geq\lambda_{0}^{\star}$, and that
$0<\Phi_{\lambda}(u_{\lambda_{0}^{\star}})<\sigma$ for
$\lambda_{0}^{\star}-\epsilon<\lambda<\lambda_{0}^{\star}$ with $\epsilon>0$
small enough. ∎

Proof of Theorem 1.4. Define

$c_{\lambda}=\inf_{g\in\Gamma}\ \max_{0\leq t\leq
1}\Phi_{\lambda_{0}^{\star}}(g(t)),$

where

$\Gamma:=\left\\{g\in C\left([0,1];W_{0}^{1,p}(\Omega)\right)\ |\
g(0)=0,g(1)=u_{\lambda_{0}^{\star}}\right\\}.$

By Lemmas 3.5, 3.6 and the Mountain Pass Theorem it follows that for each
$\lambda>\lambda_{0}^{\star}-\epsilon$, the set

$K_{c_{\lambda}}=\left\\{u\in W_{0}^{1,p}(\Omega)\ |\
\Phi_{\lambda}(u)=c_{\lambda},\ \Phi^{{}^{\prime}}_{\lambda}(u)=0\right\\}$

is not empty. ∎

##  4\. Non-Existence Results

In this section we establish a non-existence result under condition
($\gamma_{1}$), which is a stronger hypothesis than $\eqref{beta1}$. We also
assume that $M$ is of class $C^{1}({\mathbb{R}})$, and that $f(x,\cdot)\in
C^{1}({\mathbb{R}})$ for all $x\in\Omega$. We recall that hypotheses
($\rho_{1}$) and ($\rho_{2}$) hold true.

For each $u\in W_{0}^{1,p}(\Omega)\setminus\\{0\\}$ consider the following
system:

$\left\\{\begin{array}[]{l}\psi^{{}^{\prime}}_{\lambda,u}(t)=0\\\
\psi^{{}^{\prime\prime}}_{\lambda,u}(t)=0\\\
\psi^{{}^{\prime}}_{\lambda,u}(t)=\inf_{s>0}\psi^{{}^{\prime}}_{\lambda,u}(s).\end{array}\right.$
(18)

######  Lemma 4.1.

Assume conditions ($f_{1}$), ($f_{2}$) and ($\gamma_{1}$) . Then, for each
$u\in W_{0}^{1,p}(\Omega)\backslash\\{0\\}$ system (18) possesses a solution
$(\lambda_{1}(u),t_{1}(u))$ which is unique with respect to $\lambda$.

###### Proof.

The proof is similar to Lemma (2.2). ∎

######  Lemma 4.2.

Assume conditions ($f_{1}$), ($f_{2}$) and ($\gamma_{1}$). Then, for all $u\in
W_{0}^{1,p}(\Omega)\backslash\\{0\\}$ there holds
$\lambda_{1}(u)<\lambda_{0}(u)$.

###### Proof.

Let us perform a proof by contradiction. Suppose there exists a $u\in
W_{0}^{1,p}(\Omega)\backslash\\{0\\}$ such that
$\lambda_{0}(u)\leq\lambda_{1}(u)$. Then,
$\psi^{{}^{\prime}}_{\lambda_{0}(u),u}(t)\geq\psi^{{}^{\prime}}_{\lambda_{1}(u),u}(t)\geq
0$ for all $t\geq 0$. Therefore, $\psi_{\lambda_{0}(u),u}$ is non-decreasing
over $[0,+\infty)$. However, by Lemma 2.1 we know that
$\psi_{\lambda_{0}(u),u}(t)>0$ for $t>0$ small enough,
$\psi_{\lambda_{0}(u),u}(t_{0})=0$ for some $t_{0}\in(0,+\infty)$, and that
$\lim_{t\to+\infty}\psi_{\lambda_{0}(u),u}(t)=+\infty$. The proof is complete.
∎

Let us define the extremal parameter

$\lambda^{\star}_{1}:=\inf_{u\in
W_{0}^{1,p}(\Omega)\backslash\\{0\\}}\lambda_{1}(u).$

Notice that by Lemma 4.2, $\lambda_{1}^{\star}\leq\lambda_{0}^{\star}$.

######  Lemma 4.3.

Assume conditions ($f_{1}$), ($f_{2}$) and ($\beta_{1}$) with strict
inequality. Then, $\lambda_{0}^{\star}=\lambda_{0}(u_{\lambda_{0}^{\star}})$,
where $u_{\lambda_{0}^{\star}}$ is as in Proposition (3.2).

###### Proof.

Let $u:=u_{\lambda_{0}^{\star}}$. From the definition of $\lambda_{0}(u)$ and
$\lambda_{0}^{\star}$,

$\begin{array}[]{lrlccl}0=&I_{\lambda_{0}^{\star}}&=&\Phi_{\lambda_{0}^{\star}}(u)&\geq&\Phi_{\lambda_{0}(u)}(u)\\\
&&=&\psi_{\lambda_{0}(u),u}(1)&\geq&\psi_{\lambda_{0}(u),u}(t_{0}(u))\\\
&&=&0.&&\end{array}$

Therefore, $\Phi_{\lambda_{0}^{\star}}(u)=\Phi_{\lambda_{0}(u)}(u)$ or

$\begin{array}[]{c}\displaystyle{\frac{1}{p}\hat{M}(\|u\|^{p})-\frac{1}{p^{\star}}\|u\|^{p^{\star}}_{p^{\star}}-\lambda_{0}^{\star}\int_{\Omega}F(x,u(x))dx=\frac{1}{p}\hat{M}(\|u\|^{p})-\frac{1}{p^{\star}}\|u\|^{p^{\star}}_{p^{\star}}-\lambda_{0}(u)\int_{\Omega}F(x,u(x))dx}.\end{array}$

Therefore, $\lambda_{0}^{\star}=\lambda_{0}(u_{\lambda_{0}^{\star}})$. ∎

######  Lemma 4.4.

Assume conditions ($f_{1}$), ($f_{2}$) and ($\gamma_{1}$). Then,
$\lambda_{1}^{\star}<\lambda_{0}^{\star}$.

###### Proof.

From Lemma (4.2) we obtain the desired inequality:

$\lambda_{1}^{\star}\leq\lambda_{1}(u_{\lambda_{0}^{\star}})<\lambda_{0}(u_{\lambda_{0}^{\star}})=\lambda_{0}^{\star}.$

∎

We are ready to prove our non existence result.

Proof of Theorem 1.1 Assume $\lambda<\lambda_{1}^{\star}$. Then, for all $u\in
W_{0}^{1,p}\backslash\\{0\\}$, $\lambda<\lambda_{1}(u)$. Therefore,
$\psi^{{}^{\prime}}_{\lambda,u}(t)>\psi^{{}^{\prime}}_{\lambda_{1}(u),u}(t)\geq\psi^{{}^{\prime}}_{\lambda_{1}(u),u}(t_{1}(u))=0$
for all positive $t$, and therefore the energy functional has no non-zero
critical points. ∎

Acknowledgments G. N. Cunha has been supported by FAPEG, Fundação de Amparo à
Pesquisa do Estado de Goiás. F. Faraci has been supported by Università degli
Studi di Catania, PIACERI 2020-2022, Linea di intervento 2, Progetto ”MAFANE”
and by the Gruppo Nazionale per l’Analisi Matematica, la Probabilità e le loro
Applicazioni (GNAMPA) of the Istituto Nazionale di Alta Matematica (INdAM). K.
Silva has been supported by CNPq-Grant 308501/2021-7.

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# Irrationality of the general smooth quartic $3$-fold using intermediate
Jacobians

Benson Farb Supported in part by National Science Foundation Grant No.
DMS-181772 and the Eckhardt Faculty Fund.

###### Abstract

We prove that the intermediate Jacobian of the Klein quartic $3$-fold $X$ is
not isomorphic, as a principally polarized abelian variety, to a product of
Jacobians of curves. As corollaries we deduce (using a criterion of Clemens-
Griffiths) that $X$, as well as the general smooth quartic $3$-fold, is
irrational. These corollaries were known: Iskovskih-Manin [IM] proved that
every smooth quartic $3$-fold is irrational. However, the method of proof here
is different than that of [IM] and is significantly simpler.

##  1 Introduction

A smooth quartic $3$-fold is a smooth, degree $4$ hypersurface $Y$ in complex
projective space $\mathbb{P}^{4}$. For such a $Y$ there is a Hodge
decomposition

$H^{3}(Y;\mathbb{C})=H^{2,1}(Y)\oplus H^{1,2}(Y)$

and an attached intermediate Jacobian

$\operatorname{J}(Y):=\frac{H^{1,2}(Y)^{*}}{\displaystyle
i(H_{3}(Y;\mathbb{Z}))}$

where the embedding $i:H_{3}(Y;\mathbb{Z})\to H^{1,2}(Y)^{*}$ is defined by
sending $\alpha\in H_{3}(Y;\mathbb{Z})$ to the linear functional
$\omega\mapsto\int_{\alpha}\omega$. The complex torus $\operatorname{J}(Y)$ is
a $30$-dimensional abelian variety. It has a principal polarization defined by
the Hermitian form

$Q(\alpha,\beta):=2i\int_{Y}\alpha\wedge\bar{\beta}.$

The Klein quartic $3$-fold $X$ is the smooth, degree $4$ hypersurface

$X:=\\{[x_{0}:\cdots:x_{4}]:x_{0}^{3}x_{1}+x_{1}^{3}x_{2}+x_{2}^{3}x_{3}+x_{3}^{3}x_{4}+x_{4}^{3}x_{0}=0\\}\subset\mathbb{P}^{4}.$

$X$ admits a non-obvious faithful action of
$\mathbb{Z}/61\mathbb{Z}\rtimes\mathbb{Z}/5\mathbb{Z}$ by automorphisms; see
§2. We will use these symmetries to prove the following.

######  Theorem 1.1 (Intermediate Jacobian).

The intermediate Jacobian $\operatorname{J}(X)$ of the Klein quartic $3$-fold
$X$ is not isomorphic, as a principally polarized abelian variety, to a
product of Jacobians of smooth curves.

A short argument using resolution of singularities (Corollary 3.26 of [CG])
gives the Clemens-Griffiths criterion : if $Y$ is rational then
$\operatorname{J}(Y)$ is isomorphic as a principally polarized abelian variety
(henceforth p.p.a.v.) to a product of Jacobians of smooth curves. Theorem 1.1
thus implies:

######  Corollary 1.2 (Irrationality of Klein).

The Klein quartic $3$-fold is irrational: it is not birational to
$\mathbb{P}^{3}$.

The intermediate Jacobian determines a period mapping
$\operatorname{J}:{\mathcal{M}}_{4,3}\to{\mathcal{A}}_{30}$ from the moduli
space of smooth quartic $3$-folds to the moduli space of $30$-dimensional
principally polarized abelian varieties. $\operatorname{J}$ is a holomorphic
map between quasiprojective varieties. Since the target ${\mathcal{A}}_{30}$
is the quotient of a bounded symmetric domain by an arithmetic lattice,
Theorem 3.10 of Borel [Bo] gives that $\operatorname{J}$ is in fact a
morphism. Let ${\mathcal{P}}\subset{\mathcal{A}}_{30}$ denote the subvariety
consisting of products of Jacobians of smooth curves. Then
$\operatorname{J}^{-1}({\mathcal{P}})$ is a subvariety of
${\mathcal{M}}_{4,3}$. Theorem 1.1 implies that the inclusion
$\operatorname{J}^{-1}({\mathcal{P}})\subset{\mathcal{M}}_{4,3}$ is strict.
The irreducibility of ${\mathcal{M}}_{4,3}$ then gives:

######  Corollary 1.3 (Irrationality is general).

The general smooth quartic $3$-fold is irrational.111In other words, there is
a subvariety $V\subsetneq{\mathcal{M}}_{4,3}$ such that each
$X\in{\mathcal{M}}_{4,3}\setminus V$ is irrational.

Context. Corollaries 1.2 and 1.3 are not new. Iskovskih-Manin [IM] proved in
1971 that any smooth quartic $3$-fold $X$ is irrational. In contrast, Segre
had constructed in [Se] (see also §9 of [IM]) examples of such $X$ that are
unirational: there is a dominant rational map $\mathbb{P}^{3}\dashrightarrow
X$. Iskovskih-Manin prove their theorem by developing the “method of maximal
singularities” to prove that any birational map $X\dashrightarrow X$ has
finite order, and noting that this is of course not true for $\mathbb{P}^{3}$.
This initiated the modern theory of birational superrigidity; see, e.g.
Cheltsov [Ch] for a survey and details. More recently, Colliot-Thélène-Pirutka
[CP], building on a method of Voisin using the Chow group of $0$-cycles,
proved that the very general smooth quartic $3$-fold is not stably rational.

Around the same time as Iskovskih-Manin, Clemens-Griffiths [CG] used their
criterion mentioned above to prove that any smooth cubic $3$-fold $Y$ is
irrational, even though any such $Y$ is unirational. The bulk of their proof
is showing that $\operatorname{J}(Y)$ is not a product of Jacobians of curves.

Intermediate Jacobians have been used (via the Clemens-Griffiths criterion) to
prove irrationality for many $3$-folds, but not (as far as we can tell) for
smooth quartic $3$-folds; see Beauville’s survey [B1], in particular the table
on page $6$. The proof of Theorem 1.1 uses the symmetry of $X$ in a crucial
way, and follows an idea of Beauville (see [B1, B2], and also Zarhin [Z]) to
whom we owe an intellectual debt. It may be worth noting that the proofs of
all of the results in this paper use technology available already in 1972.

Acknowledgements. I thank Nick Addington and Jeff Achter for useful
discussions, and Ronno Das and János Kollár for corrections on an earlier
version of this paper. I am also extremely grateful to Curt McMullen, whose
many useful comments on an earlier version of this paper greatly improved its
exposition.

##  2 Proof of Theorem 1.1

In this note we always work in the category of principally polarized abelian
varieties. The polarization is crucial for the proofs that follow. For any
p.p.a.v $A$, denote by $\operatorname{Aut}(A)$ the group of automorphisms of
$A$ respecting the polarization; in particular $\operatorname{Aut}(A)$ is
finite (see, e.g. [BL], Corollary 5.1.9). Without the polarization this is no
longer true: consider the automorphism of
$A:=\mathbb{C}^{2}/\mathbb{Z}[i]^{2}$ induced by $(z,w)\mapsto(2z+w,z+w)$,
which is an infinite order algebraic automorphism of $A$.

Recall that the Jacobian ${\rm Jac}(C)$ of a smooth, projective curve $C$ is a
p.p.a.v., with polarization induced by the intersection pairing on
$H_{1}(C;\mathbb{Z})$. We will need the following.

######  Lemma 2.1.

Let $C$ be any smooth, projective curve of genus $g\geq 2$ and let ${\rm
Jac}(C)$ denote its Jacobian. Assume that the biholomorphic automorphism group
$\operatorname{Aut}(C)$ has odd order. Then for any
$G\subset\operatorname{Aut}({\rm Jac}(C))$ the following hold.

  1. 1.

Any cyclic subgroup of $G$ has order at most $4g+2$.

  2. 2.

If $g\geq 4$ and if $G$ is metacyclic (meaning that $G$ has a cyclic normal
subgroup $N\lhd G$ such that $G/N$ is cyclic) then $|G|\leq 9(g-1)$.

###### Proof.

For any smooth projective curve $C$ of genus $g\geq 2$ the natural map
$\rho:\operatorname{Aut}(C)\to\operatorname{Aut}({\rm Jac}(C))$ is injective;
see, e.g. [FM], Theorem 6.8. The classical Torelli theorem gives that $\rho$
is surjective if $C$ is hyperelliptic, and otherwise $[\operatorname{Aut}({\rm
Jac}(C)):\rho(\operatorname{Aut}(C))]=2$, the remaining automorphism being the
standard involution that every p.p.a.v has. Since $|G|$ is assumed to be odd,
there is a subgroup $\tilde{G}\subset\operatorname{Aut}(C)$ such that
$\rho:\tilde{G}\to G$ is an isomorphism. Both parts of the lemma now follow
from the corresponding statements for subgroups of $\operatorname{Aut}(C)$;
see e.g. Theorem 7.5 of [FM] (which is classical) and Proposition 4.2 of
[Sch], a result of Schweizer. ∎

###### Proof of Theorem 1.1.

Let $X$ be the Klein quartic $3$-fold, and let $\zeta:=e^{2\pi
i/(3^{5}+1)}=e^{2\pi i/244}$. The group
$G:=\mathbb{Z}/61\mathbb{Z}\rtimes\mathbb{Z}/5\mathbb{Z}$ acts on $X$ by
automorphisms via the maps

$\begin{array}[]{l}\phi([x_{0}:x_{1}:x_{2}:x_{3}:x_{4}]):=[\zeta
x_{0}:\zeta^{-3}x_{1}:\zeta^{9}x_{2}:\zeta^{-27}x_{3}:\zeta^{81}x_{4}]\\\ \\\
\psi([x_{0}:x_{1}:x_{2}:x_{3}:x_{4}]):=[x_{1}:x_{2}:x_{3}:x_{4}:x_{0}]\end{array}$

of order $61$ and $5$, respectively 222The somewhat surprisingly large order
automorphism $\phi$ is based on Klein, and as far as we can tell was first
written down by Z. Zheng in [Zh], Lemma 3.2.; in fact
$G\cong\operatorname{Aut}(X)$ (see [GLMV], Theorem B), but we will not need
this. For any smooth, degree $d\geq 3$ hypersurface in $\mathbb{P}^{n},n>1$,
the action of $\operatorname{Aut}(X)$ on $H^{3}(X;\mathbb{Z})$ is faithful
(see, e.g., Chap.1, Cor. 3.18 of [H]). Since in addition
$\operatorname{Aut}(X)$ preserves the Hodge decomposition of
$H^{3}(X;\mathbb{C})$, it follows that $\operatorname{Aut}(X)$, hence $G$,
acts faithfully on $\operatorname{J}(X)$ by p.p.a.v automorphisms.

Suppose that $X$ is rational. The Clemens-Griffiths criterion gives an
isomorphism of p.p.a.v.:

$A:=\operatorname{J}(X)\cong A_{1}^{n_{1}}\times\cdots\times A_{r}^{n_{r}}$
(2.1)

where each $A_{i}:={\rm Jac}(C_{i})$ is the Jacobian of a smooth, projective
curve $C_{i}$ and where $A_{i}\not\cong A_{j}$ if $i\neq j$. They also show
(Corollary 3.23 of [CG]) that each $A_{i}$ is irreducible 333A p.p.a.v $A$ is
irreducible if any morphism $A^{\prime}\to A$ of p.p.a.v is $0$ or an
isomorphism., and that the decomposition of any p.p.a.v into a product of
p.p.a.v as in (2.1) is unique.

Now, $G$ acts on $A$ as p.p.a.v. automorphisms. The uniqueness of the
decomposition (2.1) implies that each $A_{i}^{n_{i}}$ is $G$-invariant. Note
that

$30=\dim(A)=\sum_{i=1}^{r}n_{i}\dim(A_{i}).$ (2.2)

Since each $A_{i}$ is irreducible, the action of $G$ on $A_{i}^{n_{i}}$ gives
a representation

$G\to\operatorname{Aut}(A_{i}^{n_{i}})\cong\operatorname{Aut}(A_{i})^{n_{i}}\rtimes
S_{n_{i}}$

whose composition with the projection to $S_{n_{i}}$ records the permutation
of the direct factors of $A_{i}^{n_{i}}$.

Since the $G$-action on $A$ is faithful and $\mathbb{Z}/61\mathbb{Z}$ has
prime order, there exists some $i$ (after re-labeling assume $i=1$) so that
$\mathbb{Z}/61\mathbb{Z}$ acts faithfully on $A_{1}^{n_{1}}$. By the orbit-
stabilizer theorem, the orbit of any direct factor $A_{1}$ of $A_{1}^{n_{1}}$
under the prime order subgroup $\mathbb{Z}/61\mathbb{Z}\subset G$ has $1$ or
$61$ elements; but the latter is impossible by (2.2) since $\dim(A_{1})\geq
1$. Thus $\mathbb{Z}/61\mathbb{Z}$ leaves each individual direct factor
$A_{1}$ invariant.

Fix such a direct factor $B\cong A_{1}$ on which $\mathbb{Z}/61\mathbb{Z}$
acts faithfully (such a factor must exist since $\mathbb{Z}/61\mathbb{Z}$ acts
faithfully on $A_{1}^{n_{1}}$, as noted above). Recall that $B\cong
A_{1}\cong{\rm Jac}(C_{1})$ for some smooth projective curve $C_{1}$ of genus
$g\geq 1$. Note that in fact $g\geq 2$ since otherwise $\dim(B)=1$ and so
$A_{1}$ does not admit a p.p.a.v. automorphism of order $>6$. Thus Lemma
2.1(1) applies, giving

$61\leq 4\cdot{\rm genus}(C_{1})+2=4\dim(B)+2$

and so $\dim(A_{1})=\dim(B)={\rm genus}(C_{1})\geq 15$. Again by the orbit-
stabilizer theorem, the orbit of $B$ in the set of direct factors of
$A_{1}^{n_{1}}$ under the prime order subgroup $\mathbb{Z}/5\mathbb{Z}\subset
G$ has $1$ or $5$ elements. Since $\dim(B)={\rm genus}(C_{1})\geq 15$ and
$n_{1}\cdot{\rm genus}(C_{1})\leq 30$, the latter is not possible; that is,
$B$ is $\mathbb{Z}/5\mathbb{Z}$-invariant, and so $G$-invariant.

Now, the definition of $\phi$ and $\psi$ above give that
$G\cong\mathbb{Z}/61\mathbb{Z}\rtimes\mathbb{Z}/5\mathbb{Z}$ is a nontrivial
semidirect product; that is, $G$ is not a direct product. For any homomorphism
$\mu:C\rtimes D\to E$ of a nontrivial semidirect product of finite simple
groups (e.g. cyclic groups of prime order) to any group, if $\mu$ is not
faithful on $D$ then it is not faithful on $C$ (and indeed $\mu$ is trivial in
this case). Since the $\mathbb{Z}/61\mathbb{Z}$-action on $B$ is faithful, it
follows that the $\mathbb{Z}/5\mathbb{Z}$ action on $B$ is faithful. From this
it follows that the $G$-action on $B$ is faithful (consider the kernel $K$ of
the $G$-action, and note that $K\cap\mathbb{Z}/61\mathbb{Z}=0$ and so
$K<\mathbb{Z}/5\mathbb{Z}$, so that $K$ is trivial).

Note that

$|G|=61\cdot 5=305>261=9\cdot(30-1)>9({\rm genus}(C_{1})-1).$ (2.3)

Since ${\rm genus}(C_{1})\geq 15\geq 4$ and since $G$ is metacyclic, Lemma
2.1(2) applies. Its conclusion contradicts (2.3). Thus $X$ is not rational.

∎

######  Remark 2.2.

One might hope to replace the use of Lemma 2.1(2) by something simpler, such
as the Hurwitz bound $|\operatorname{Aut}(C)|\leq 84(g-1)$. However, a quick
check of the numerology shows that this is not enough to obtain a
contradiction.

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Dept. of Mathematics
University of Chicago
E-mail<EMAIL_ADDRESS>