# On Hamiltonian systems with critical Sobolev exponents

Angelo Guimarães  Instituto de Ciências Matemáticas e de Computação
Universidade de São Paulo
Caixa Postal 668, CEP 13560-970 - São Carlos - SP - Brazil<EMAIL_ADDRESS>and  Ederson Moreira dos santos  Instituto de Ciências Matemáticas e de
Computação
Universidade de São Paulo
Caixa Postal 668, CEP 13560-970 - São Carlos - SP - Brazil
<EMAIL_ADDRESS>
###### Abstract.

In this paper we consider lower order perturbations of the critical Lane-Emden
system posed on a bounded smooth domain $\Omega\subset\mathbb{R}^{N}$, with
$N\geq 3$, inspired by the classical results of Brezis and Nirenberg [4]. We
solve the problem of finding a positive solution for all dimensions $N\geq 4$.
For the critical dimension $N=3$ we show a new phenomenon, not observed for
scalar problems. Namely, there are parts on the critical hyperbola where
solutions exist for all $1$-homogeneous or subcritical superlinear
perturbations and parts where there are no solutions for some of those
perturbations.

###### Key words and phrases:

Lane-Emden systems; Critical hyperbola; Critical dimension; Positive
solutions.

###### 2010 Mathematics Subject Classification:

35J47, 35J30, 35B33.

E.M. dos Santos has been partially supported by CNPq grant 309006/2019-8. A.
Guimarães has been supported by CAPES (Finance code 001). We thank Professors
Jéssyca L. F. Melo Gurjão and Marcelo Furtado for some enlightening
discussions at the beginning of this project.

##  1\. Introduction

In the memorable paper [4] from 1983, Brezis and Nirenberg considered the
perturbed Lane-Emden equation with critical growth

$-\Delta u=\lambda u^{t}+u^{2^{*}-1}\ \ \text{in}\ \ \Omega,\ \ u>0\ \
\text{in}\ \ \Omega,\ \ u=0\ \ \text{on}\ \ \partial\Omega,$ (1.1)

in a bounded smooth domain $\Omega\subset\mathbb{R}^{N}$, $N\geq 3$, where
$2^{*}=2N/(N-2)$ is the critical Sobolev exponent for the embedding of
$H^{1}_{0}(\Omega)$, with $1\leq t<2^{*}-1$. In particular, they discovered a
surprising difference between the cases $N\geq 4$ and $N=3$, the latter named
as critical dimension. For the particular case with $t=1$, namely for

$-\Delta u=\lambda u+u^{2^{*}-1}\ \ \text{in}\ \ \Omega,\ \ u>0\ \ \text{in}\
\ \Omega,\ \ u=0\ \ \text{on}\ \ \partial\Omega,$ (1.2)

they proved the existence of a solution for every
$0<\lambda<\lambda_{1}(\Omega)$, the optimal interval for existence, for
$N\geq 4$. In contrast, with $N=3$, they showed the existence of
$0<\lambda^{*}<\lambda_{1}(\Omega)$ such that no solution exists for
$0<\lambda<\lambda^{*}$; see [4, Theorem 1.2 and Corollary 1.1]. Here
$\lambda_{1}=\lambda_{1}(\Omega)$ stands for the first eigenvalue of
$(-\Delta,H^{1}_{0}(\Omega))$.

The notion of critical growth for Hamiltonian systems, as independently
introduced by Mitidieri [21] and van der Vorst [25], soon after considered by
several authors, including Clément et al. [5] and Peletier-van der Vorst [24],
is given by the so-called critical hyperbola. In 1998, Hulshof et al. [13]
analyzed the version of (1.2) in the framework of Hamiltonian systems, namely
they considered

$\left\\{\begin{array}[]{lll}-\Delta u=\lambda v+|v|^{p-1}v\text{ in
}\Omega,\\\ -\Delta v=\mu u+|u|^{q-1}u\text{ in }\Omega,\\\ u,v=0\text{ on
}\partial\Omega,\end{array}\right.$

with $N\geq 4$, for $(p,q)$ on the critical hyperbola

$\frac{1}{p+1}+\frac{1}{q+1}=\frac{N-2}{N}.$ (1.3)

Very interesting results were proved in [13, Theorem 2] and we think that
three important problems were left open:

  * a)

What does happen in dimension $N=3$ ?

  * b)

What is the meaning of critical dimension for Hamiltonian elliptic systems ?

  * c)

The investigation of the general $1$-homogenous perturbation of the critical
Lane-Emden system, namely (HS) ahead with $rs=1$, which includes $r=s=1$ as a
particular case.

Item c) deserves some extra comments, since the most accurate 1-homogenous
perturbation to Hamiltonian systems, given below in (HS), is induced by the
hyperbola of points $(r,s)$ such that $rs=1$. Indeed, this hyperbola has been
named as the spectral curve for Hamiltonian systems; see [22, 17, 18] for
linear operators and [11] in the fully nonlinear scenario. In this paper we
address these three questions and present some results observed in the
framework of Hamiltonian systems which are non-existent for scalar problems.
In order to accomplish that, consider the following Hamiltonian system

$\left\\{\begin{array}[]{lll}-\Delta u=\lambda|v|^{r-1}v+|v|^{p-1}v\text{ in
}\Omega,\\\ -\Delta v=\mu|u|^{s-1}u+|u|^{q-1}u\text{ in }\Omega,\\\
u,v=0\text{ on }\partial\Omega,\end{array}\right.$ (HS)

in a bounded smooth domain $\Omega\subset\mathbb{R}^{N}$, $N\geq 3$,
$\lambda>0$ and $\mu>0$. Here $(p,q)$ lies on the critical hyperbola, that is
$p>0$ and $q>0$ satisfy (1.3), and $(r,s)$ is such that

$0<r<p,\ \ 0<s<q,\ \ rs\geq 1.$ (1.4)

Since $\lambda>0$ and $\mu>0$, the critical growth system (HS) can be seen as
a lower order perturbation of the Lane-Emden critical system

$-\Delta u=|v|^{p-1}v,\quad-\Delta v=|u|^{q-1}u\quad\text{in}\ \ \Omega,\quad
u=v=0\quad\partial\Omega,$

as (1.1) is a lower order perturbation of the critical Lane-Emden equation

$-\Delta u=u^{2^{*}-1},\ \ u>0\ \ \text{in}\ \ \Omega,\ \ u=0\ \ \text{on}\ \
\partial\Omega.$

Moreover, condition (1.4) on $(r,s)$ for (HS) corresponds to condition $1\leq
t<2^{*}-1$ for (1.1).

The main result proved in this paper reads as follows.

######  Theorem 1.1.

Let $\lambda>0$, $\mu>0$, assume (1.4) and in case $rs=1$ also suppose that
$\lambda\mu^{r}$ is suitably small. If $N\geq 4$ or, $N=3$ and $p\leq 7/2$ or
$p\geq 8$, then (HS) has a classical positive solution.

The precise condition on the size of $\lambda\mu^{r}$ (for the case with
$rs=1$) is specified at (2.4) and (2.5) ahead. Actually, such condition
appeared before in [20] and corresponds to the hypothesis
$\lambda<\lambda_{1}$ for equation (1.2). Moreover, as proved in [25, Theorem
4.2], if such condition is not verified, then (HS) may have no positive
solution in starshaped domains. Also observe that, in case of $N=3$, $(7/2,8)$
and $(8,7/2)$ are symmetric points on the critical hyperbola (1.3).

We call the attention to the fact that, when $\lambda=\mu$, $r=s$, $p=q$, any
solution of (HS) is such that $u=v$ (see [23, Example 4.3]), which makes (HS)
and (1.1) to be equivalent in this case. With this in mind, for $N=3$, the so-
called critical dimension for (1.1), we prove the existence of solutions for
$(p,q)$ lying on some parts of the critical hyperbola (even if $r=s=1$), which
brings new results when comparing to [13, Theorem 2], where the case $N=3$ is
not considered. Indeed, when setting side by side our results with [13,
Theorem 2], our contribution is threefold: we treat the case $N=3$; for $N=4$
we do not impose $p\neq 2$ or $p\neq 5$; for $N\geq 3$ we consider the natural
1-homogenous ($rs=1$) or superlinear ($rs>1$) perturbations, while [13,
Theorem 2] is restricted to the case with $r=s=1$, $p>1$ and $q>1$. In
particular, for $N\geq 4$, we cover all the points $(p,q)$ on the critical
hyperbola, which includes points with $p<1$ or $q<1$ for $N>4$. Figure 1 ahead
illustrates the existence result given by Theorem 1.1 for $N\geq 4$.

Figure 1. Given any $(p,q)$ on the critical hyperbola, any $(r,s)$ satisfying
(1.4) is
admissible for finding a positive solution to (HS).

Our paper may also serve as motivation for future investigation. In view of
the results in [16, Theorem 1.1], that consider $r=s=1$, it could be
interesting to study blowing up phenomena for system (HS), with $rs=1$, as
$\lambda=\mu\to 0$.

We recall that in the critical dimension $N=3$ it is not possible to prove the
existence of solution for (1.1) in the full range $1\leq t<5$. Indeed, as in
[4, Corollary 2.3], such existence results is proved only for $3<t<5$. This
motivates to introduce the following definition and open problems.

######  Definition 1.2.

For $N=3$, let $(p,q)$ be a point on the critical hyperbola (1.3), $\Omega$ be
a bounded regular domain, and $(r,s)$ satisfying (1.4). We say that $(p,q)$ is
on a _Critical Region_ if, for some $\Omega$ and some $(r,s)$, (HS) has no
positive solution for some $\lambda$ and $\mu$ small. On the other hand,
$(p,q)$ is on a _Noncritical Region_ if for all $\Omega$, all $(r,s)$
satisfying (1.4) and $(\lambda,\mu)$ suitably small, then (HS) has a positive
solution.

Open problems.

  1. (1)

To find the critical region of the critical hyperbola (1.3) for $N=3$.

  2. (2)

A simpler problem, but still challenging, is to find the optimal values
$7/2<p_{*}\leq p^{*}<8$ such that (HS) has no solution for any $p_{*}\leq
p\leq p^{*}$ with $r=s=1$, $\lambda=\mu$ small, with
$\Omega=B(0,1)\subset\mathbb{R}^{3}$.

For this second question, due to the results in [4, Theorem 1.2] and
conditions (4.50) ahead in this paper, we know that $4\leq p_{*}\leq 5\leq
p^{*}\leq 13/2$. Figure 2 illustrates the open problem regarding what should
be critical and noncritical regions of the critical hyperbola for $N=3$.

Figure 2. Critical Hyperbola for $N=3$

Finally, we make a link between critical/noncritical regions of the critical
hyperbola associated to Hamiltonian systems and the critical dimensions for
the biharmonic operator under Navier boundary conditions. We recall that,
according to [26], the dimensions $N=5,6,7$ are named as critical for the
study of

$\Delta^{2}u=\mu u+u^{\frac{N+4}{N-4}}\ \ \text{in}\ \ \Omega,\quad u=\Delta
u=0\ \ \text{on}\ \ \partial\Omega,$ (1.5)

a particular case of (HS) with $\lambda=0$ and $p=1$; see also [12, 1] for the
case with Dirichlet boundary conditions for the biharmonic and polyharmonic
operators, respectively. A first attempt to understand the phenomenon of
critical dimension for Hamiltonian systems was presented in [20]. However, the
asymmetric perturbation in [20] makes the problem more like a nonlinear
version of the biharmonic equation (1.5), as the counterpart of the
$p-$Laplacian version for (1.1). In the case with $\lambda>0$ and $\mu>0$ in
(HS), the natural symmetric perturbation of the critical Lane-Emden system, we
recover that the only critical dimension is $N=3$, as it happens to the scalar
problem (1.1), unveiling the notions of critical and noncritical regions of
the critical hyperbola for $N=3$.

This paper is organized as follows. In Section 2 we present the variational
approach to treat the system, writing (HS) as the fourth order equations (P)
or (P’). We define the energy functionals associated to these equations, show
that they have the mountain pass geometry and present an upper bound for their
mountain pass levels. Section 3 is devoted to localize the range where such
functionals satisfy the $(PS)_{c}$-condition and to the proof of Theorem 1.1.
Finally, we accommodate in Section 4 some technical estimates which are
crucial for the variational treatment.

##  2\. Variational approach and Mountain Pass Geometry

To deal with (HS), following the same approach as in [20], define

$\begin{array}[]{ccc}f_{\lambda}(t)=\lambda|t|^{r-1}t+|t|^{p-1}t,\ \
\overline{F}_{\lambda}(t)=\displaystyle\int_{0}^{t}f_{\lambda}^{-1}(t)dt,\\\
g_{\mu}(t)=\mu|t|^{s-1}t+|t|^{q-1}t,\ \
\overline{G}_{\mu}(t)=\displaystyle\int_{0}^{t}g_{\mu}^{-1}(t)dt,\end{array}$
(2.1)

and rewrite (HS) as one of the fourth-order equations under Navier boundary
conditions

$\left\\{\begin{array}[]{lll}\Delta(f_{\lambda}^{-1}(\Delta
u))=\mu|u|^{s-1}u+|u|^{q-1}u\text{ in }\Omega,\\\ u,\Delta u=0\text{ on
}\partial\Omega,\end{array}\right.$ (P)
$\left\\{\begin{array}[]{lll}\Delta(g_{\mu}^{-1}(\Delta
v))=\lambda|v|^{r-1}v+|v|^{p-1}v\text{ in }\Omega,\\\ v,\Delta v=0\text{ on
}\partial\Omega.\end{array}\right.$ (P’)

Associated with (P) and (P’), we consider the $C^{1}(E_{p},\mathbb{R})$ and
$C^{1}(E_{q},\mathbb{R})$ functionals

$I_{F}(u)=\displaystyle\int_{\Omega}\overline{F}_{\lambda}(\Delta
u)dx-\frac{\mu}{s+1}\displaystyle\int_{\Omega}|u|^{s+1}dx-\frac{1}{q+1}\displaystyle\int_{\Omega}|u|^{q+1}dx,$
(2.2) $I_{G}(u)=\displaystyle\int_{\Omega}\overline{G}_{\mu}(\Delta
v)dx-\frac{\lambda}{r+1}\displaystyle\int_{\Omega}|v|^{r+1}dx-\frac{1}{p+1}\displaystyle\int_{\Omega}|v|^{p+1}dx,$
(2.3)

where $E_{t}:=W^{2,\frac{t+1}{t}}(\Omega)\cap W^{1,\frac{t+1}{t}}_{0}(\Omega)$
is endowed with the norm $\|u\|=|\Delta u|_{\frac{t+1}{t}}$. Throughout in
this paper $|w|_{\theta}$ stand for the $L^{\theta}(\Omega)$-norm of $w$.

The variational treatment of (HS) given by studying (P) or (P’) is usually
called reduction by inversion. This idea has been used by P. L. Lions [19] and
in several other papers, as for example in [7, 6, 15, 2]. Here, since the
functions $f_{\lambda}$ and $g_{\mu}$ are not pure power, and due to the
critical growth nature of (HS), we prove in Section 4.1 some sharp estimates
on $f_{\lambda}$, whose corresponding versions to $g_{\mu}$ also hold. In
order to capture in this inversion the contribution of the term
$\lambda|u|^{r-1}u$, to downsize the Mountain Pass level, we compute at Lemma
4.9 some integrals on rings involving the ground state solutions of the Lane-
Emden critical system on $\mathbb{R}^{N}$, where terms associated to
$u\mapsto\lambda|u|^{r-1}u$ are dominant.

######  Definition 2.1.

We say that $u\in E_{p}$ is a weak solution of (P) iff $I_{F}^{\prime}(u)=0$.
A function $u\in C^{2}(\overline{\Omega})$ such that $f_{\lambda}^{-1}(\Delta
u)\in C^{2}(\Omega)$ is a classical solution of (P) iff satisfies (P)
pointwise. Similarly, we define weak and classical solutions of (P’).
Moreover, $(u,v)$ is a classical solution of (HS), iff $u,v\in
C(\overline{\Omega})\cap C^{2}(\Omega)$ satisfy (HS) pointwise.

######  Lemma 2.2.

If $u$ is a weak solution of (P), then it is a classical solution of (P). The
converse is also true. Moreover, $u$ is a classical solution of (P), iff
$(u,v)$ is a classical solution of (HS), with $v=f_{\lambda}^{-1}(-\Delta u)$.

###### Proof.

We can mimic the proof of [20, Lemma 1], which is based on the arguments in
[9, Section 4] and [14, Section 3]. ∎

Next we show that the functionals $I_{F}$ and $I_{G}$ have the Mountain Pass
geometry and obtain upper bounds for their Mountain Pass levels. For the cases
with $rs=1$ we introduce the conditions

$\displaystyle\lambda^{1/r}\mu\leq\frac{(2|\Omega|)^{\frac{r-p}{r(p+1)}}}{2^{\frac{r+1}{r}}}\mathscr{C}^{\frac{r+1}{r}}_{r,\Omega},$
(2.4)
$\displaystyle\lambda\mu^{1/s}\leq\frac{(2|\Omega|)^{\frac{s-q}{s(q+1)}}}{2^{\frac{s+1}{s}}}\mathscr{C}^{\frac{s+1}{s}}_{s,\Omega},$
(2.5)

on the size of $(\lambda,\mu)$, where

$\mathscr{C}_{r,\Omega}=\inf\\{\|u\|;\ u\in E_{p}\text{ and
}|u|_{\frac{r+1}{r}}=1\\},\ \ \mathscr{C}_{s,\Omega}=\inf\\{\|v\|;\ v\in
E_{q}\text{ and }|v|_{\frac{s+1}{s}}=1\\}.$ (2.6)

######  Remark 2.3.

Conditions (2.4) and (2.5) for the case with $rs=1$ are natural and correspond
to the hypothesis on $\lambda$ and $\mu$ in [13, Theorem 2] to treat (HS) with
$r=s=1$, and to the hypothesis $\lambda<\lambda_{1}$ in [4] to study (1.2).

######  Proposition 2.4.

Let $(p,q)$ and $(r,s)$ be as in (1.3) and (1.4).

  1. -

Then $I_{F}$ has the Mountain Pass geometry with a local minimum at zero,
under the additional condition (2.4) when $rs=1$.

  2. -

Then $I_{G}$ has the Mountain Pass geometry with a local minimum at zero,
under the additional condition (2.5) when $rs=1$.

###### Proof.

Observe that $I_{F}(0)=0$ and, from (4.3),

$\begin{array}[]{ccc}I_{F}(u)\leq\displaystyle\frac{p}{p+1}\|u\|^{\frac{p+1}{p}}-\frac{\mu}{s+1}|u|_{s+1}^{s+1}-\frac{1}{q+1}|u|_{q+1}^{q+1},\
\ \forall\,u\in E_{p}.\end{array}$ (2.7)

Then, $I_{F}(tu)\rightarrow-\infty$ when $t\rightarrow\infty$ and $u\neq 0$.

On the other hand, by Lemma 4.5,

$I_{F}(u)=\displaystyle\int_{|\Delta u|\leq
2\lambda^{\frac{p}{p-r}}}\overline{F}_{\lambda}(\Delta
u)dx+\displaystyle\int_{|\Delta
u|>2\lambda^{\frac{p}{p-r}}}\overline{F}_{\lambda}(\Delta
u)dx-\frac{\mu}{s+1}|u|_{s+1}^{s+1}-\frac{1}{q+1}|u|_{q+1}^{q+1}\vspace{5pt}\\\
\geq\displaystyle\frac{1}{2^{\frac{r+1}{r}}\lambda^{1/r}}\frac{r}{r+1}\int_{|\Delta
u|\leq 2\lambda^{\frac{p}{p-r}}}|\Delta
u|^{\frac{r+1}{r}}dx+\frac{1}{2^{\frac{p+1}{p}}}\frac{p}{p+1}\displaystyle\int_{|\Delta
u|>2\lambda^{\frac{p}{p-r}}}|\Delta u|^{\frac{p+1}{p}}dx\vspace{5pt}\\\
\quad-\frac{\mu}{s+1}|u|_{s+1}^{s+1}-\frac{1}{q+1}|u|_{q+1}^{q+1}.$ (2.8)

By Jensen’s inequality, for a nonnegative measurable function $a$ and
$\alpha>1$,

$\int_{\omega}(a(t))^{\alpha}dt\geq|\omega|^{1-\alpha}\left(\int_{\omega}a(t)dt\right)^{\alpha}.$
(2.9)

Since $0<r<p$, with $\alpha=\frac{r+1}{r}\frac{p}{p+1}>1$, it follows that

$I_{F}(u)\geq\displaystyle\frac{(meas(|\Delta u|\leq
2\lambda^{\frac{p}{p-r}}))^{1-\alpha}}{2^{\frac{r+1}{r}}\lambda^{1/r}}\frac{r}{r+1}\left(\int_{|\Delta
u|\leq 2\lambda^{\frac{p}{p-r}}}|\Delta
u|^{\frac{p+1}{p}}dx\right)^{\alpha}\\\
+\displaystyle\frac{1}{2^{\frac{p+1}{p}}}\frac{p}{p+1}\int_{|\Delta
u|>2\lambda^{\frac{p}{p-r}}}|\Delta
u|^{\frac{p+1}{p}}dx\displaystyle-\frac{\mu}{s+1}|u|_{s+1}^{s+1}-\frac{1}{q+1}|u|_{q+1}^{q+1}\\\
\geq\displaystyle\frac{|\Omega|^{\frac{r-p}{r(p+1)}}}{2^{\frac{r+1}{r}}\lambda^{1/r}}\frac{r}{r+1}\left(\int_{|\Delta
u|\leq 2\lambda^{\frac{p}{p-r}}}|\Delta
u|^{\frac{p+1}{p}}dx\right)^{\alpha}\\\
+\displaystyle\frac{1}{2^{\frac{p+1}{p}}}\frac{p}{p+1}\int_{|\Delta
u|>2\lambda^{\frac{p}{p-r}}}|\Delta
u|^{\frac{p+1}{p}}dx\displaystyle-\frac{\mu}{s+1}|u|_{s+1}^{s+1}-\frac{1}{q+1}|u|_{q+1}^{q+1}.$
(2.10)

For $u\neq 0$ such that

$\displaystyle\frac{1}{2^{\frac{p+1}{p}}}\frac{p}{p+1}\|u\|^{-\frac{p-r}{rp}}\geq\frac{|\Omega|^{\frac{r-p}{r(p+1)}}}{2^{\frac{r+1}{r}}\lambda^{1/r}}\frac{r}{r+1}\
\text{ i.e.}\ \ \
\|u\|\leq\left(\frac{p(r+1)}{(p+1)r}\right)^{\frac{pr}{p-r}}2|\Omega|\lambda^{\frac{p}{p-r}},$
(2.11)

it follows that

$I_{F}(u)\geq\displaystyle\frac{|\Omega|^{\frac{r-p}{r(p+1)}}}{2^{\frac{r+1}{r}}\lambda^{1/r}}\frac{r}{r+1}\left[\left(\int_{|\Delta
u|\leq 2\lambda^{\frac{p}{p-r}}}|\Delta
u|^{\frac{p+1}{p}}dx\right)^{\alpha}+\left(\int_{|\Delta
u|>2\lambda^{\frac{p}{p-r}}}|\Delta
u|^{\frac{p+1}{p}}dx\right)^{\alpha}\right]\\\
\displaystyle-\frac{\mu}{s+1}|u|_{s+1}^{s+1}-\frac{1}{q+1}|u|_{q+1}^{q+1}.$
(2.12)

Since $1-\alpha=-\frac{p-r}{r(p+1)}=-\frac{p}{p+1}\frac{p-r}{pr}$ and
$(a+b)^{\alpha}\leq 2^{\alpha-1}(a^{\alpha}+b^{\alpha})$ for $a,b\geq 0$, we
infer that

$I_{F}(u)\geq\displaystyle\frac{(2|\Omega|)^{\frac{r-p}{r(p+1)}}}{2^{\frac{r+1}{r}}\lambda^{1/r}}\frac{r}{r+1}\|u\|^{\frac{r+1}{r}}-\frac{\mu}{s+1}|u|_{s+1}^{s+1}-\frac{1}{q+1}|u|_{q+1}^{q+1}$
(2.13)

for all $u\in E$ such that
$\|u\|\leq\left(\frac{p(r+1)}{(p+1)r}\right)^{\frac{pr}{p-r}}2|\Omega|\lambda^{\frac{p}{p-r}}$.

If $\frac{r+1}{r}<s+1$, i.e. $\frac{1}{r}<s$, and since $s<q$, it follows that
$I_{F}$ has the Mountain Pass geometry with a local minimum at zero.

On the other hand, if $\frac{r+1}{r}=s+1$, i.e. $\frac{1}{r}=s$, for $u\neq
0$, we infer that

$\begin{array}[]{rcl}(s+1)I_{F}(u)&\geq&\frac{(2|\Omega|)^{\frac{r-p}{r(p+1)}}}{2^{\frac{r+1}{r}}\lambda^{1/r}}\|u\|^{\frac{r+1}{r}}-\mu|u|^{\frac{r+1}{r}}_{\frac{r+1}{r}}-\frac{s+1}{q+1}|u|_{q+1}^{q+1}\vspace{10pt}\\\
&=&\left(\frac{(2|\Omega|)^{\frac{r-p}{r(p+1)}}}{2^{\frac{r+1}{r}}\lambda^{1/r}}-\mu|u|^{\frac{r+1}{r}}_{\frac{r+1}{r}}\|u\|^{-\frac{r+1}{r}}\right)\|u\|^{\frac{r+1}{r}}-\frac{s+1}{q+1}|u|_{q+1}^{q+1}\vspace{10pt}\\\
&\geq&\left(\frac{(2|\Omega|)^{\frac{r-p}{r(p+1)}}}{2^{\frac{r+1}{r}}\lambda^{1/r}}-\mu\frac{1}{\mathscr{C}_{r,\Omega}^{\frac{r+1}{r}}}\right)\|u\|^{\frac{r+1}{r}}-\frac{s+1}{q+1}|u|_{q+1}^{q+1}\end{array}$

and (2.4) gives
$\frac{(2|\Omega|)^{\frac{r-p}{r(p+1)}}}{2^{\frac{r+1}{r}}\lambda^{1/r}}-\mu\frac{1}{\mathscr{C}_{r,\Omega}^{\frac{r+1}{r}}}>0$,
and again $I_{F}$ has the Mountain Pass geometry around zero, since
$\frac{1}{r}=s<q$. ∎

Let $S$ be the Sobolev constant for the embedding $E_{p}\hookrightarrow
L^{q+1}(\Omega)$, namely

$S=\inf_{u\in E_{p},|u|_{q+1}=1}\|u\|.$

######  Proposition 2.5.

Suppose (1.3) and (1.4), $\mu>0,\ \lambda>0$ and in the case $rs=1$ also
assume (2.4). If $N\geq 4$ and $p\leq(N+2)/(N-2)$, or $N=3$ and $p\leq 7/2$,
then the mountain pass level $c_{F}$ of the functional $I_{F}$ is such that
$c_{F}\in(0,\frac{2}{N}S^{\frac{pN}{2(p+1)}})$.

###### Proof.

See Section 4.2. ∎

##  3\. (PS)c condition

When treating (P), the main difficulty is the lack of compactness for the
embedding $E_{p}\hookrightarrow L^{q+1}(\Omega)$. Here we localize the levels
$c$ for which the $(PS)_{c}$ condition holds. Throughout this section (1.3),
(1.4), $\mu,\lambda>0$ are assumed, and the main results is the following.

######  Proposition 3.1.

$I_{F}$ satisfies the $(PS)_{c}$ condition for all
$c<\frac{2}{N}S^{\frac{pN}{2(p+1)}}$.

We split the proof of this proposition in several lemmas.

######  Lemma 3.2.

Every (PS) sequence for $I_{F}$ is bounded

###### Proof.

Let $(u_{n})$ be a (PS) sequence for $I_{F}$. So, using Corollary 4.3 and
Lemma 4.6, there exists $c\in\mathbb{R}$ and a positive sequence
$(\epsilon_{n})$ with $\epsilon_{n}\rightarrow 0$ such that

$\displaystyle\epsilon_{n}\|u_{n}\|+c\geq
I_{F}(u_{n})-\frac{I_{F}^{\prime}(u_{n})u_{n}}{s+1}=\displaystyle\int_{\Omega}\overline{F}_{\lambda}(\Delta
u_{n})-\frac{f_{\lambda}^{-1}(\Delta u_{n})\Delta
u_{n}}{s+1}dx+\frac{(q-s)|u_{n}|_{q+1}^{q+1}}{(s+1)(q+1)}\\\
\geq\displaystyle\int_{|\Delta u_{n}|\geq
2\lambda^{\frac{p}{p-r}}}\overline{F}_{\lambda}(\Delta
u_{n})-\frac{f_{\lambda}^{-1}(\Delta u_{n})\Delta
u_{n}}{s+1}dx\geq\tau\displaystyle\int_{|\Delta u_{n}|\geq
2\lambda^{\frac{p}{p-r}}}|\Delta u_{n}|^{\frac{p+1}{p}}dx\\\
\geq\tau\left(\displaystyle\int_{\Omega}|\Delta
u_{n}|^{\frac{p+1}{p}}dx-\displaystyle\int_{|\Delta u_{n}|\leq
2\lambda^{\frac{p}{p-r}}}|\Delta
u_{n}|^{\frac{p+1}{p}}dx\right)\geq\tau\|u_{n}\|^{\frac{p+1}{p}}-2^{\frac{p+1}{p}}\lambda^{\frac{p+1}{p-r}}|\Omega|,$
(3.1)

which implies the boundedness of $(\|u_{n}\|)$. ∎

To localize the levels where $I_{F}$ satisfies the $(PS)$ condition the
following result, due to P.-L. Lions, is necessary.

######  Lemma 3.3.

Given a bounded sequence $(u_{n})$ in $E_{p}$, there exists a subsequence,
also denoted here by $(u_{n})$, such that:

  * (i)

$u_{n}\rightharpoonup u$ in $E_{p}$.

  * (ii)

$u_{n}\rightarrow u$ a.e. in $\Omega$ and in $L^{\theta}(\Omega)$, for all
$1\leq\theta<q+1$.

  * (iii)

$\left|\Delta
u_{n}\right|^{\frac{p+1}{p}}\stackrel{{\scriptstyle*}}{{\rightharpoonup}}\gamma$
in the sense of measures on $\overline{\Omega}$.

  * (iv)

$\left|u_{n}\right|^{q+1}\stackrel{{\scriptstyle*}}{{\rightharpoonup}}\nu$ in
the sense of measures on $\overline{\Omega}$.

  * (v)

There exist an at most countable index set $J$, a family of points
$\\{x_{j}:j\in J\\}\subset\overline{\Omega}$ and two sequences
$\\{\nu_{j}:j\in J\\},\\{\gamma_{j}:j\in J\\}\subset(0,+\infty)$ such that:

$\displaystyle{\nu=\left|u\right|^{q+1}+\sum_{j\in
J}\nu_{j}\delta_{x_{j}},\,\,\gamma\geq\left|\Delta
u\right|^{\frac{p+1}{p}}+\sum_{j\in J}\gamma_{j}\delta_{x_{j}}},$
$\displaystyle{S\,\nu_{j}^{\frac{p+1}{p}\frac{1}{q+1}}\leq\gamma_{j}\,\,\hbox{for
all}\,\,j\in J,\,\,\hbox{in particular}\,\,\sum_{j\in
J}\nu_{j}^{\frac{p+1}{p}\frac{1}{q+1}}<+\infty}.$

  * (vi)

$\nabla u_{n}\rightharpoonup\nabla u$ in
$\left(W^{1,\frac{p+1}{p}}(\Omega)\right)^{N}$.

  * (vii)

$\nabla u_{n}\rightarrow\nabla u$ a.e. in $\Omega$ and in
$\left(L^{\sigma}(\Omega)\right)^{N}$, for all $1\leq\sigma<\sigma^{*}$, with
$\sigma^{*}>\frac{p+1}{p}$ depending on the critical Sobolev embedding of
$W^{1,\frac{p+1}{p}}(\Omega)$.

###### Proof.

See [19, Lemma I.1] or [10, Lemma 3.3]. ∎

An improvement of the previous lemma is given next.

######  Lemma 3.4.

If $(u_{n})$ is a (PS)-sequence for $I_{F}$, then there exists a subsequence,
for short also denoted by $(u_{n})$, satisfying (i)-(vii) from Lemma 3.3 with
the additional fact that $J$ is at most finite.

###### Proof.

Let $x_{j}\in\overline{\Omega}$ be a point in the singular support of $\mu$
and $\nu$. Let $\zeta\in C^{\infty}_{c}(\mathbb{R}^{N})$ such that
$0\leq\zeta\leq 1$, $\zeta\equiv 1$ in $B(0,1)$ and $supp(\zeta)\subset
B(0,2)$. Moreover for each $\theta>0$ define
$\zeta_{\theta}(x):=\zeta(\frac{x-x_{j}}{\theta})$. So there exists constants
$c_{1}$ and $c_{2}$ independent of $\theta$ such that

$|\nabla\zeta_{\theta}(x)|\leq\frac{c_{1}}{\theta},\
|\Delta\zeta_{\theta}(x)|\leq\frac{c_{2}}{\theta^{2}},\
\forall\,x\in\mathbb{R}^{N}.$

By [10, Lemma 3.4 ], $u_{n}\zeta_{\theta}\in E_{p},\ \forall\,n\in\mathbb{N}\
and\ \theta>0$. Fixing $\theta>0$, since $(\zeta_{\theta}u_{n})$ is bounded in
$E_{p}$, $\langle I_{F}^{\prime}(u_{n}),\zeta_{\theta}u_{n}\rangle=o(1)$ that
is

$\begin{array}[]{lll}o(1)&=\displaystyle\int_{\overline{\Omega}}f_{\lambda}^{-1}(\Delta
u_{n})\Delta
u_{n}\zeta_{\theta}dx-\displaystyle\int_{\overline{\Omega}}|u_{n}|^{q+1}\zeta_{\theta}dx-\mu\displaystyle\int_{\overline{\Omega}}|u_{n}|^{s+1}\zeta_{\theta}dx\\\
&\quad+\displaystyle\int_{\overline{\Omega}}f_{\lambda}^{-1}(\Delta
u_{n})u_{n}\Delta\zeta_{\theta}dx+2\displaystyle\int_{\overline{\Omega}}f_{\lambda}^{-1}(\Delta
u_{n})\nabla u_{n}\nabla\zeta_{\theta}dx.\end{array}$ (3.2)

On the other hand, $\zeta_{\theta}(x)\xrightarrow{\theta\rightarrow
0}\delta_{x_{j}}(x),\ \forall\,x\in\Omega$. So from $u_{n}\rightharpoonup u$
in $E_{p}$ and $E_{p}\subset\subset L^{s+1}(\Omega)$

$\displaystyle\int_{\overline{\Omega}}|u_{n}|^{s+1}\zeta_{\theta}dx\xrightarrow{n\rightarrow\infty}\displaystyle\int_{\overline{\Omega}}|u|^{s+1}\zeta_{\theta}dx\xrightarrow{\theta\rightarrow
0}0.$ (3.3)

Now, by (4.2) and Hölder inequality
$\left(\frac{1}{p+1}+\frac{1}{q+1}+\frac{2}{N}=1\right)$, there exists $C>0$
independent of $n$ and $\theta$ such that

$\displaystyle\left|\int_{\overline{\Omega}}f_{\lambda}^{-1}(\Delta
u_{n})u_{n}\Delta\zeta_{\theta}dx\right|\leq\\!\int_{\Omega}|\Delta
u_{n}|^{1/p}|u_{n}||\Delta\zeta_{\theta}|dx\leq
C\\!\left(\\!\displaystyle\int_{\overline{\Omega}}|u_{n}|^{q+1}\left|\Delta\zeta\left(\frac{x-x_{j}}{\theta}\right)\right|^{\frac{q+1}{2}}\\!dx\right)^{\frac{1}{q+1}}\\!\\!\\!\\!.$
(3.4)

From the definition of the weak* convergence

$\displaystyle\int_{\overline{\Omega}}|u_{n}|^{q+1}\left|\Delta\zeta\left(\frac{x-x_{j}}{\theta}\right)\right|^{\frac{q+1}{2}}dx\xrightarrow{n\rightarrow\infty}\displaystyle\int_{\overline{\Omega}}\left|\Delta\zeta\left(\frac{x-x_{j}}{\theta}\right)\right|^{\frac{q+1}{2}}d\nu,$
(3.5)

and since
$|\Delta\zeta(\frac{x-x_{j}}{\theta})|^{(\frac{q+1}{2})}\xrightarrow{\theta\rightarrow
0}0\ \forall\,x\in\Omega$, by the Lebesgue dominated convergence theorem,

$\int_{\overline{\Omega}}\left|\Delta\zeta\left(\frac{x-x_{j}}{\theta}\right)\right|^{\frac{q+1}{2}}d\nu\xrightarrow{\theta\rightarrow
0}0.$ (3.6)

We also have

$\left|\displaystyle\int_{\overline{\Omega}}f_{\lambda}^{-1}(\Delta
u_{n})\nabla u_{n}\nabla\zeta_{\theta}dx\right|\leq
C\left(\displaystyle\int_{\overline{\Omega}}\left(\frac{1}{\theta}\left|\nabla\zeta\left(\frac{x-x_{j}}{\theta}\right)\right||\nabla
u_{n}|\right)^{\frac{p+1}{p}}dx\right)^{\frac{p}{p+1}}\ \ \text{and}\\\
\displaystyle\int_{\overline{\Omega}}\\!\left(\frac{1}{\theta}\left|\nabla\zeta\left(\frac{x-x_{j}}{\theta}\right)\right||\nabla
u_{n}|\right)^{\frac{p+1}{p}}\\!\\!dx\xrightarrow{n\rightarrow\infty}\displaystyle\int_{\overline{\Omega}}\\!\left(\frac{1}{\theta}\left|\nabla\zeta\left(\frac{x-x_{j}}{\theta}\right)\right||\nabla
u|\right)^{\frac{p+1}{p}}\\!\\!dx=O(\theta^{N-\frac{p+1}{p}}).$ (3.7)

Given $\epsilon>0$ let $M(\epsilon)>0$ be such that
$f_{\lambda}^{-1}(t)t\geq\frac{1}{1+\epsilon}|t|^{\frac{p+1}{p}}$ for all
$|t|\geq M(\epsilon)$. Then define

$A_{n}:=\\{x\in B(x_{j},2\theta)\cap\overline{\Omega};|\Delta u_{n}(x)|\geq
M(\epsilon)\\},B_{n}:=(\overline{\Omega}\cap B(x_{j},2\theta))\backslash
A_{n}.$ (3.8)

Then,

$\displaystyle\int_{\overline{\Omega}}f_{\lambda}^{-1}(\Delta u_{n})\Delta
u_{n}\zeta_{\theta}dx=\displaystyle\int_{A_{n}}f_{\lambda}^{-1}(\Delta
u_{n})\Delta
u_{n}\zeta_{\theta}dx+\displaystyle\int_{B_{n}}f_{\lambda}^{-1}(\Delta
u_{n})\Delta u_{n}\zeta_{\theta}dx\\\
\geq\frac{1}{1+\epsilon}\displaystyle\int_{\overline{\Omega}}|\Delta
u_{n}|^{\frac{p+1}{p}}\zeta_{\theta}dx+\displaystyle\int_{B_{n}}(f_{\lambda}^{-1}(\Delta
u_{n})\Delta u_{n}-\frac{1}{1+\epsilon}|\Delta
u_{n}|^{\frac{p+1}{p}})\zeta_{\theta}dx\\\
=\frac{1}{1+\epsilon}\displaystyle\int_{\overline{\Omega}}|\Delta
u_{n}|^{\frac{p+1}{p}}\zeta_{\theta}dx+\displaystyle\int_{B_{n}}(f_{\lambda}^{-1}(\Delta
u_{n})\Delta u_{n}-\frac{1}{1+\epsilon}|\Delta
u_{n}|^{\frac{p+1}{p}})\zeta_{\theta}dx\longrightarrow\frac{1}{1+\epsilon}\gamma_{j}$
(3.9)

by taking the limit as $n\rightarrow\infty$ and after as $\theta\rightarrow
0$, because

$\lim_{\theta\rightarrow
0}\limsup_{n\rightarrow\infty}\left|\displaystyle\int_{B_{n}}(f_{\lambda}^{-1}(\Delta
u_{n})\Delta u_{n}-\frac{1}{1+\epsilon}|\Delta
u_{n}|^{\frac{p+1}{p}})\zeta_{\theta}dx\right|\\\ \leq\lim_{\theta\rightarrow
0}\limsup_{n\rightarrow\infty}\displaystyle\int_{B_{n}}(f_{\lambda}^{-1}(M)M+\frac{1}{1+\epsilon}M^{\frac{p+1}{p}})\zeta_{\theta}dx=0.$
(3.10)

Then, from all the above estimates ranging from (3.2) to (3.9), we infer that

$0=\lim_{\theta\rightarrow 0}lim_{n\rightarrow\infty}\langle
I_{F}^{\prime}(u_{n}),u_{n}\zeta_{\theta}\rangle\geq\frac{\gamma_{j}}{1+\epsilon}-\nu_{j}$
(3.11)

which implies that $\nu_{j}\geq\frac{\gamma_{j}}{1+\epsilon}$ for all
$\epsilon>0$, and hence $\nu_{j}\geq\gamma_{j}$. In contrast, since $0\leq
f_{\lambda}^{-1}(\Delta u_{n})\Delta u_{n}\leq|\Delta u_{n}|^{\frac{p+1}{p}}$,
it follows from all the above estimates ranging from (3.2) to (3.9) that

$0=\lim_{\theta\rightarrow 0}\lim_{n\rightarrow\infty}\langle
I_{F}^{\prime}(u_{n}),u_{n}\zeta_{\theta}\rangle\leq\gamma_{j}-\nu_{j},$
(3.12)

which implies

$\gamma_{j}=\nu_{j}.$ (3.13)

Then, from Lemma 3.3, $\nu_{j}\geq S\nu_{j}^{\frac{p+1}{p}\frac{1}{q+1}}$ and
so

$\nu_{j}\geq S^{\frac{pN}{2(p+1)}},$ (3.14)

since $\frac{pN}{2(p+1)}=\left(1-\frac{1}{q+1}\frac{p+1}{p}\right)^{-1}$ and
$\gamma_{j}>0$. Combining this with $\sum_{j\in
J}\nu_{j}^{\frac{p+1}{p}\frac{1}{q+1}}<+\infty$, it follows that $J$ is at
most finite. ∎

######  Lemma 3.5.

Given a bounded sequence $(u_{n})$ in $E_{p}$,
$K\subset\subset\Omega\backslash\\{x_{j}:j\in J\\}$ with $\\{x_{j}:j\in J\\}$
from Lemma 3.3, then $u_{n}\rightarrow u$ in $L^{q+1}(K)$, up to a
subsequence.

###### Proof.

See the proof of [10, Lemma 3.6]. ∎

######  Lemma 3.6.

If $(u_{n})$ is a (PS)-sequence for $I_{F}$, and $\\{x_{j}:j\in J\\}$ from
Lemma 3.3, then for every $j\in J$, up to a subsequence,

$\lim_{n\rightarrow\infty}\int_{\overline{\Omega}}\left[\overline{F}_{\lambda}(\Delta
u_{n})-\frac{1}{s+1}f_{\lambda}^{-1}(\Delta u_{n})\Delta
u_{n}\right]dx\geq\frac{ps-1}{(p+1)(s+1)}\gamma_{j}.$ (3.15)

###### Proof.

Consider the even function
$H_{p}(t):=|t|^{-\frac{p+1}{p}}\left(\overline{F}_{\lambda}(t)-\frac{1}{s+1}f_{\lambda}^{-1}(t)t\right)$.
By Lemma 4.1,

$\lim_{t\rightarrow\infty}H_{p}(t)=\lim_{t\rightarrow\infty}\left[\frac{ps-1}{(p+1)(s+1)}\frac{f_{\lambda}^{-1}(t)}{t^{1/p}}-\frac{p-r}{p+1}\frac{\lambda}{r+1}\frac{|f^{-1}_{\lambda}(t)|^{r+1}}{t^{\frac{p+1}{p}}}\right]=\frac{ps-1}{(p+1)(s+1)}.$
(3.16)

Then, given $\epsilon>0$ small, there exists $t_{0}>0$ such that

$H_{p}(t)>c_{\epsilon}>0\ \ \ \ \ \text{for all }|t|>t_{0},$

with $c_{\epsilon}=\frac{ps-1}{(p+1)(s+1)}-\epsilon$, which implies that

$\overline{F}_{\lambda}(t)-\frac{1}{s+1}f_{\lambda}^{-1}(t)t>c_{\epsilon}|t|^{\frac{p+1}{p}}\
\ \ \ \ \text{for all }|t|>t_{0}.$ (3.17)

Let $A_{n}$, $B_{n}$ and $\zeta_{\theta}$ be like in Lemma 3.4, with $A_{n}$
and $B_{n}$ associated with $t_{0}$. So, from (3.17) and Corollary 4.3,

$\displaystyle\int_{\overline{\Omega}}\overline{F}_{\lambda}(\Delta
u_{n})-\frac{1}{s+1}f_{\lambda}^{-1}(\Delta u_{n})\Delta
u_{n}dx\geq\displaystyle c_{\epsilon}\left(\displaystyle\int_{A_{n}}|\Delta
u_{n}|^{\frac{p+1}{p}}\zeta_{\theta}dx+\displaystyle\int_{B_{n}}|\Delta
u_{n}|^{\frac{p+1}{p}}\zeta_{\theta}dx\right)\\\
+\displaystyle\int_{B_{n}}\left[\overline{F}_{\lambda}(\Delta
u_{n})-\frac{1}{s+1}f_{\lambda}^{-1}(\Delta u_{n})\Delta
u_{n}-c_{\epsilon}|\Delta u_{n}|^{\frac{p+1}{p}}\right]\zeta_{\theta}dx\\\
=\displaystyle c_{\epsilon}\displaystyle\int_{\overline{\Omega}}|\Delta
u_{n}|^{\frac{p+1}{p}}\zeta_{\theta}dx+\displaystyle\int_{B_{n}}\left[\overline{F}_{\lambda}(\Delta
u_{n})-\frac{1}{s+1}f_{\lambda}^{-1}(\Delta u_{n})\Delta
u_{n}-c_{\epsilon}|\Delta u_{n}|^{\frac{p+1}{p}}\right]\zeta_{\theta}dx.$
(3.18)

On the other hand,

$\lim_{\theta\to
0}\limsup_{n\rightarrow\infty}\left|\displaystyle\int_{B_{n}}\left[\overline{F}_{\lambda}(\Delta
u_{n})-\frac{1}{s+1}f_{\lambda}^{-1}(\Delta u_{n})\Delta
u_{n}-c_{\epsilon}|\Delta
u_{n}|^{\frac{p+1}{p}}\right]\zeta_{\theta}dx\right|\\\ \leq\lim_{\theta\to
0}\limsup_{n\rightarrow\infty}\displaystyle\int_{\overline{\Omega}}\left[\overline{F}_{\lambda}(t_{0})+\frac{1}{s+1}f_{\lambda}^{-1}(t_{0})t_{0}+c_{\epsilon}|t_{0}|^{\frac{p+1}{p}}\right]\zeta_{\theta}dx=0,$
(3.19)

and, from Lemma 3.3,

$\displaystyle\lim_{\theta\to
0}\limsup_{n\rightarrow\infty}c_{\epsilon}\displaystyle\int_{\overline{\Omega}}|\Delta
u_{n}|^{\frac{p+1}{p}}\zeta_{\theta}dx\geq c_{\epsilon}\gamma_{k}.$ (3.20)

From (3.18), (3.19), (3.20) and the arbitrariness of $\epsilon>0$, we get the
desired inequality. ∎

######  Lemma 3.7.

If $(u_{n})$ is a (PS)-sequence for $I_{F}$,
$K\subset\subset\Omega\backslash\\{x_{j}:j\in J\\}$ with $\\{x_{j}:j\in J\\}$
from Lemma 3.3, then up to a subsequence,

$\displaystyle\int_{K}(f_{\lambda}^{-1}(\Delta u_{n})-f_{\lambda}^{-1}(\Delta
u))(\Delta u_{n}-\Delta u)dx\xrightarrow{n\rightarrow\infty}0.$ (3.21)

###### Proof.

Let $\delta=dist(K,\\{x_{j}:j\in J\\})$. For each $\theta\in(0,\delta)$,
consider $A_{\theta}=\\{x\in\Omega:dist(x,K)<\theta\\}$ and $\xi_{\theta}\in
C_{c}^{\infty}(\Omega)$, $0\leq\xi_{\theta}\leq 1$, $\xi_{\theta}\equiv 1$ on
$A_{\theta/2}$ and $\xi_{\theta}\equiv 0$ on $\Omega\backslash A_{\theta}$.
So, by the monotonicity of $f_{\lambda}^{-1}$,

$0\leq\displaystyle\int_{K}(f_{\lambda}^{-1}(\Delta
u_{n})-f_{\lambda}^{-1}(\Delta u))(\Delta u_{n}-\Delta
u)dx\leq\displaystyle\int_{\Omega}(f_{\lambda}^{-1}(\Delta
u_{n})-f_{\lambda}^{-1}(\Delta u))(\Delta u_{n}-\Delta u)\xi_{\theta}dx\\\
=\displaystyle\int_{\Omega}f_{\lambda}^{-1}(\Delta u_{n})\Delta
u_{n}\xi_{\theta}-f_{\lambda}^{-1}(\Delta u_{n})\Delta
u\xi_{\theta}-f_{\lambda}^{-1}(\Delta u)(\Delta u_{n}-\Delta
u)\xi_{\theta}dx.$ (3.22)

Fixing $\theta>0$, since $I_{F}^{\prime}(u_{n})\rightarrow 0$ and
$(u_{n}\xi_{\theta})$ is bounded in $E$, then $\langle
I_{F}^{\prime}(u_{n}),\xi_{\theta}u\rangle=o(1)$ and $\langle
I_{F}^{\prime}(u_{n}),\xi_{\theta}u_{n}\rangle=o(1)$ that is

$o(1)\\!=\\!\\!\displaystyle\int_{\overline{\Omega}}f_{\lambda}^{-1}(\Delta
u_{n})(\Delta u\xi_{\theta}+u\Delta\xi_{\theta}+2\nabla
u\nabla\xi_{\theta})dx-\\!\\!\displaystyle\int_{\overline{\Omega}}|u_{n}|^{q-1}u_{n}\xi_{\theta}udx-\mu\\!\\!\displaystyle\int_{\overline{\Omega}}|u_{n}|^{s-1}u_{n}\xi_{\theta}udx$
(3.23)
$o(1)\\!=\\!\\!\displaystyle\int_{\overline{\Omega}}f_{\lambda}^{-1}(\Delta
u_{n})(\Delta u_{n}\xi_{\theta}+u_{n}\Delta\xi_{\theta}+2\nabla
u_{n}\nabla\xi_{\theta})dx-\\!\displaystyle\int_{\overline{\Omega}}|u_{n}|^{q+1}\xi_{\theta}dx-\mu\\!\displaystyle\int_{\overline{\Omega}}|u_{n}|^{s+1}\xi_{\theta}dx.$
(3.24)

From (3.22), (3.23), (3.24), Lemma 3.3, Lemma 3.5 and [10, Lemma 2.4], it
follows that

$0\leq\displaystyle\int_{K}(f_{\lambda}^{-1}(\Delta
u_{n})-f_{\lambda}^{-1}(\Delta u))(\Delta u_{n}-\Delta
u)dx\leq\displaystyle\int_{\Omega}f_{\lambda}^{-1}(\Delta
u_{n})\Delta\xi_{\theta}(u_{n}-u)dx\\\
+2\displaystyle\int_{\Omega}f_{\lambda}^{-1}(\Delta
u_{n})\nabla\xi_{\theta}\nabla(u_{n}-u)dx-\displaystyle\int_{\Omega}f_{\lambda}^{-1}(\Delta
u)(\Delta u_{n}-\Delta u)\xi_{\theta}dx+o(1)\\\ \leq\displaystyle
C\left(\int_{A_{\theta}}|u_{n}-u|^{q+1}dx\right)^{\frac{1}{q+1}}+C\left(\int_{\Omega}|\nabla
u_{n}-\nabla u|^{\frac{p+1}{p}}dx\right)^{\frac{p}{p+1}}\\\
-\displaystyle\int_{\Omega}f_{\lambda}^{-1}(\Delta u)(\Delta u_{n}-\Delta
u)\xi_{\theta}dx+o(1)=o(1).\qed$ (3.25)

######  Lemma 3.8.

If $(u_{n})$ is a (PS)-sequence for $I_{F}$, then $\Delta
u_{n}\xrightarrow{n\rightarrow\infty}\Delta u$ a.e. in $\Omega$, up to a
subsequence.

###### Proof.

Let $K\subset\subset\Omega\backslash\\{x_{j}\\}_{j\in J}$. By the inverse of
the Lebesgue dominated convergence theorem, there exists a subsequence of the
integrand in (3.21) that converges a.e. in $K$. Using [8, Lemma 6] with

$X=\mathbb{R},\ \beta_{n}=f_{\lambda}^{-1},\ \beta=f_{\lambda}^{-1},\
\text{and }\xi_{n}=\Delta u_{n}(x)$

we get $\Delta u_{n}\rightarrow\Delta u\ a.e.\ in\ K$. Since $K$ is an
arbitrary compact subset of $\Omega\backslash\\{x_{j}\\}_{j\in J}$, we
conclude that $\Delta u_{n}\rightarrow\Delta u\ a.e.\ in\ \Omega$. ∎

######  Lemma 3.9.

If $(u_{n})$ is a (PS)-sequence for $I_{F}$, then $f_{\lambda}^{-1}(\Delta
u_{n})\rightharpoonup f_{\lambda}^{-1}(\Delta u)\text{ in }L^{p+1}(\Omega)$,
up to a subsequence.

###### Proof.

Since, up to a subsequence,

$\left\\{\begin{array}[]{lll}\Delta
u_{n}\xrightarrow{n\rightarrow\infty}\Delta u\ \ \ \ \text{a.e. in }\Omega,\\\
(\Delta u_{n})\text{ is bounded in }L^{\frac{p+1}{p}}(\Omega),\text{ and}\\\
|f_{\lambda}^{-1}(\Delta u_{n})|\leq|\Delta u_{n}|^{1/p},\end{array}\right.$
(3.26)

we infer that $f_{\lambda}^{-1}(\Delta
u_{n})\xrightarrow{n\rightarrow\infty}f_{\lambda}^{-1}(\Delta u)$ a.e. in
$\Omega$, $(f_{\lambda}^{-1}(\Delta u_{n}))$ is bounded in $L^{p+1}(\Omega)$,
and hence $f_{\lambda}^{-1}(\Delta u_{n})\rightharpoonup
f_{\lambda}^{-1}(\Delta u)\text{ in }L^{p+1}(\Omega)$. ∎

######  Proposition 3.10.

If $(u_{n})$ is a $(PS)$-sequence for $I_{F}$, then there exist a subsequence,
still denoted by $(u_{n})$, such that $u_{n}\rightharpoonup u$ in $E_{p}$ and
$u$ is a weak solution of (P).

###### Proof.

First, since $(u_{n})$ is a $(PS)$-sequence to $I_{F}$, $\langle
I_{F}^{\prime}(u_{n}),w\rangle\rightarrow 0$, for all $w\in E_{p}$. On the
other hand, up to a subsequence,

$\left\\{\begin{array}[]{lll}f_{\lambda}^{-1}(\Delta u_{n})\rightharpoonup
f_{\lambda}^{-1}(\Delta u)\text{ in }L^{p+1}(\Omega),\\\
|u_{n}|^{q-1}u_{n}\rightharpoonup|u|^{q-1}u\text{ in
}L^{\frac{q+1}{q}}(\Omega)\text{ and}\\\
|u_{n}|^{s-1}u_{n}\rightarrow|u|^{s-1}u\text{ in
}L^{\frac{s+1}{s}}(\Omega).\end{array}\right.$ (3.27)

Thus, for all $w\in E_{p}$, $\langle
I_{F}^{\prime}(u_{n}),w\rangle\rightarrow\langle I_{F}^{\prime}(u),w\rangle$.
Then $\langle I_{F}^{\prime}(u),w\rangle=0$, for all $w\in E_{p}$, that is,
$u$ is a weak solution of (P). ∎

###### Proof of Proposition 3.1.

Let $(u_{n})$ be a $(PS)_{c}$ sequence for $I_{F}$ with
$c<\frac{2}{N}S^{\frac{pN}{2(p+1)}}$. By contradiction, suppose that
$J\neq\emptyset$. We can suppose the assertions of Lemma 3.4, with
$\gamma_{j}=\nu_{j}$, $\nu_{j}\geq S^{\frac{pN}{2(p+1)}}$, for all $j\in J$,
by (3.13) and (3.14). Since $(u_{n})$ is bounded in $E_{p}$, $\langle
I_{F}^{\prime}(u_{n}),u_{n}\rangle=o(1)$, and Lemmas 3.3 and 3.6, we infer
that

$c=\lim_{n\rightarrow\infty}I_{F}(u_{n})-\frac{1}{s+1}\langle
I_{F}^{\prime}(u_{n}),u_{n}\rangle\\\
=\lim_{n\rightarrow\infty}\displaystyle\int_{\overline{\Omega}}\overline{F}_{\lambda}(\Delta
u_{n})-\frac{1}{s+1}f_{\lambda}^{-1}(\Delta u_{n})\Delta
u_{n}dx+\left(\frac{1}{s+1}-\frac{1}{q+1}\right)\displaystyle\int_{\overline{\Omega}}|u_{n}|^{q+1}dx\\\
\geq\displaystyle\left(\frac{ps-1}{(p+1)(s+1)}+\frac{1}{s+1}-\frac{1}{q+1}\right)\nu_{j}=\frac{2}{N}\nu_{j}\geq\frac{2}{N}S^{\frac{pN}{2(p+1)}}$
(3.28)

for every $j\in J$, which is a contradiction. Hence, $J=\emptyset$.

Then, from Lemmas 3.3 and 3.5, since $L^{q+1}(\Omega)$ is uniformly convex,
$u_{n}\rightarrow u$ in $L^{q+1}(\Omega)$.

Let $v_{n}=u_{n}-u$, thus $v_{n}\rightharpoonup 0$ in $E_{p}$, $\Delta
v_{n}\rightarrow 0\ a.e.$ in $\Omega$ and $v_{n}\rightarrow 0$ in
$L^{q+1}(\Omega)$. Since

$|(a+b)f_{\lambda}^{-1}(a+b)-af_{\lambda}^{-1}(a)|\leq|a+b|^{\frac{p+1}{p}}+|a|^{\frac{p+1}{p}}\leq
2^{p}(|b|^{\frac{p+1}{p}}+|a|^{\frac{p+1}{p}}),\ \ \forall\,a,b\in\mathbb{R},$
(3.29)

from [3, Theorem 2], with $j(t)=tf_{\lambda}^{-1}(t)$,

$\int_{\Omega}f_{\lambda}^{-1}(\Delta u_{n})\Delta
u_{n}dx=\int_{\Omega}f_{\lambda}^{-1}(\Delta u+\Delta v_{n})(\Delta u+\Delta
v_{n})dx\\\ =\int_{\Omega}f_{\lambda}^{-1}(\Delta u)\Delta
u+f_{\lambda}^{-1}(\Delta v_{n})\Delta v_{n}dx+o(1).$ (3.30)

Since $u$ is weak solution of (P),

$o(1)=\langle
I_{F}^{\prime}(u_{n}),u_{n}\rangle=\displaystyle\int_{\Omega}f_{\lambda}^{-1}(\Delta
u_{n})\Delta u_{n}-|u_{n}|^{q+1}-\mu|u_{n}|^{s+1}dx\\\ =\displaystyle\langle
I_{F}^{\prime}(u),u\rangle+\int_{\Omega}f_{\lambda}^{-1}(\Delta v_{n})\Delta
v_{n}dx+o(1)=\int_{\Omega}f_{\lambda}^{-1}(\Delta v_{n})\Delta v_{n}dx+o(1),$
(3.31)

that is, $f_{\lambda}^{-1}(\Delta v_{n})\Delta v_{n}\rightarrow 0$ in
$L^{1}(\Omega)$. Then, Lemma 4.4 and Jensen’s inequality (2.9) lead to

$0\leftarrow\displaystyle\int_{\Omega}f_{\lambda}^{-1}(\Delta v_{n})\Delta
v_{n}dx\geq\frac{1}{2^{1/p}}\int_{|\Delta v_{n}|\geq
2\lambda^{\frac{p}{p-r}}}|\Delta
v_{n}|^{\frac{p+1}{p}}dx+\frac{1}{(2\lambda)^{1/r}}\int_{|\Delta v_{n}|\leq
2\lambda^{\frac{p}{p-r}}}|\Delta v_{n}|^{\frac{r+1}{r}}dx\vspace{5pt}\\\
\geq\frac{1}{2^{1/p}}\int_{|\Delta v_{n}|\geq 2\lambda^{\frac{p}{p-r}}}|\Delta
v_{n}|^{\frac{p+1}{p}}dx+\frac{1}{(2\lambda)^{1/r}}|\Omega|^{1-\alpha}\left(\int_{|\Delta
v_{n}|\leq 2\lambda^{\frac{p}{p-r}}}|\Delta
v_{n}|^{\frac{p+1}{p}}dx\right)^{\alpha},$ (3.32)

with $\alpha=\frac{p}{p+1}\frac{r+1}{r}$. Therefore $|\Delta
v_{n}|^{\frac{p+1}{p}}\to 0$ in $L^{1}(\Omega)$, that is, $u_{n}\to u$ in
$E_{p}$. ∎

At this point we have all the tools at hand to prove our main result.

###### Proof of Theorem 1.1.

Suppose, without loss of generality, that $p\leq q$. The case $q\leq p$ can be
handled similarly, by using $I_{G}$ instead of $I_{F}$. By Propositions 2.4,
2.5, 3.1 and Lemma 2.2, the existence of a classical solution is a direct
consequence of the Mountain Pass theorem.

Next, we prove that any Mountain Pass solution is signed. Let $u$ be a
Mountain Pass solution of (P). So, by Lemma 2.2, $u\in
C^{2}(\overline{\Omega})$ and $u=0$ on $\partial\Omega$. Then, by the
classical strong maximum principle for second-order elliptic operators, it is
enough to show that $\Delta u$ does not change sign in $\Omega$. By
contradiction, suppose that $\Delta u$ changes sign in $\Omega$, and let
$\omega$ be the solution of

$\left\\{\begin{array}[]{crl}-\Delta\omega=|\Delta u|&\text{in }\Omega,\\\
\omega=0&\text{on }\partial\Omega.\end{array}\right.$ (3.33)

By the Strong Maximum Principle, $\omega>|u|$ in $\Omega$ and we infer that

$c_{F}\leq\max_{t\geq 0}I_{F}(t\omega)=\max_{t\geq
0}\left\\{\int_{\Omega}\overline{F}_{\lambda}(t\Delta\omega)dx-\frac{\mu}{s+1}t^{s+1}\int_{\Omega}|\omega|^{s+1}dx-\frac{t^{q+1}}{q+1}\int_{\Omega}|\omega|^{q+1}dx\right\\}\\\
<\max_{t\geq 0}\left\\{\int_{\Omega}\overline{F}_{\lambda}(t\Delta
u)dx-\frac{\mu}{s+1}t^{s+1}\int_{\Omega}|u|^{s+1}dx-\frac{t^{q+1}}{q+1}\int_{\Omega}|u|^{q+1}dx\right\\}=\max_{t\geq
0}I_{F}(tu)=c_{F},$ (3.34)

which is a contradiction. Hence $\Delta u$ does not change sign in $\Omega$,
and therefore, up to multiplication by $-1$, $u>0$ and $-\Delta u>0$ in
$\Omega$. Finally, by Lemma 2.2, with $v=f^{-1}(-\Delta u)$, $(u,v)$ is a
positive classical solution of (HS). ∎

##  4\. Appendix

###  4.1. Some technical properties of the auxiliary functions

Ahead in this section, where (1.4) is assumed, some properties of the
functions $f_{\lambda}^{-1},\ \overline{F}_{\lambda},\ g_{\mu}^{-1}$ and
$\overline{G}_{\mu}$, as defined in (2.1), are given. Indeed, we can consider
$f_{\lambda}^{-1}$ and $\overline{F}_{\lambda}$ and infer the respective
properties for the others. We start by showing some useful inequalities.
Observe that

$t^{p}<f_{\lambda}(t)\text{ and }\lambda t^{r}<f_{\lambda}(t)\ \ \text{for}\ \
t>0$ (4.1)

and writing $\tau=f_{\lambda}(t)$ we get

$f_{\lambda}^{-1}(\tau)<\tau^{1/p}\text{ and
}f_{\lambda}^{-1}(\tau)<\frac{\tau^{1/r}}{\lambda^{1/r}}\ \ \text{for}\ \
t>0.$ (4.2)

So,

$\overline{F}_{\lambda}(\tau)\leq\frac{p}{p+1}|\tau|^{\frac{p+1}{p}}\text{ and
}\overline{F}_{\lambda}(\tau)\leq\frac{r}{r+1}\frac{1}{\lambda^{1/r}}|\tau|^{\frac{r+1}{r}}\
\ \forall\,\tau\in\mathbb{R}.$ (4.3)

The next lemmas are used to obtain the geometric condition and upper bounds
for the critical level of the Mountain Pass Theorem for the functionals
$I_{F}$ and $I_{G}$.

######  Lemma 4.1.

$\overline{F}_{\lambda}(t)=\frac{p}{p+1}f^{-1}_{\lambda}(t)t-\frac{p-r}{p+1}\frac{\lambda}{r+1}|f^{-1}_{\lambda}(t)|^{r+1}$
for all $t\in\mathbb{R}$

###### Proof.

Set
$M(t):=\frac{p}{p+1}f_{\lambda}^{-1}(t)t-\frac{p-r}{p+1}\frac{\lambda}{r+1}|f^{-1}_{\lambda}(t)|^{r+1}-\overline{F}_{\lambda}(t)$.
Then $M(0)=0$, $M$ is even, and for $t>0$

$M^{\prime}(t)=\displaystyle\frac{1}{p+1}\left(-f_{\lambda}^{-1}(t)+\frac{pt}{\lambda
r[f_{\lambda}^{-1}(t)]^{r-1}+p[f_{\lambda}^{-1}(t)]^{p-1}}-\frac{\lambda(p-r)\left[f^{-1}_{\lambda}(t)\right]^{r}}{\lambda
r[f_{\lambda}^{-1}(t)]^{r-1}+p[f_{\lambda}^{-1}(t)]^{p-1}}\right)\\\
\displaystyle=\frac{1}{p+1}\left(\frac{\lambda(p-r)\left[f^{-1}_{\lambda}(t)\right]^{r}}{\lambda
r[f_{\lambda}^{-1}(t)]^{r-1}+p[f_{\lambda}^{-1}(t)]^{p-1}}-\frac{\lambda(p-r)\left[f^{-1}_{\lambda}(t)\right]^{r}}{\lambda
r[f_{\lambda}^{-1}(t)]^{r-1}+p[f_{\lambda}^{-1}(t)]^{p-1}}\right)=0,$ (4.4)

which implies the desired identity. ∎

######  Lemma 4.2.

$\overline{F}_{\lambda}(t)=\lambda\frac{r}{r+1}|f^{-1}_{\lambda}(t)|^{r+1}+\frac{p}{p+1}|f^{-1}_{\lambda}(t)|^{p+1}$
for all $t\in\mathbb{R}$.

###### Proof.

The argument follows as in the proof of the last Lemma, observing that for all
$t>0$

$\frac{d}{dt}\left(\lambda\frac{r}{r+1}|f^{-1}_{\lambda}(t)|^{r+1}+\frac{p}{p+1}|f^{-1}_{\lambda}(t)|^{p+1}\right)=\frac{\lambda
r\left[f^{-1}_{\lambda}(t)\right]^{r}+p\left[f^{-1}_{\lambda}(t)\right]^{p}}{\lambda
r\left[f^{-1}_{\lambda}(t)\right]^{r-1}+p\left[f^{-1}_{\lambda}(t)\right]^{p-1}}=f_{\lambda}^{-1}(t),$
(4.5)

which coincides with $\frac{d}{dt}\overline{F}_{\lambda}(t)$. ∎

######  Corollary 4.3.

$\displaystyle\overline{F}_{\lambda}(t)\geq\frac{f_{\lambda}^{-1}(t)t}{s+1}$,
for all $t\in\mathbb{R}.$

###### Proof.

Since
$f^{-1}_{\lambda}(t)t=f^{-1}_{\lambda}(t)f_{\lambda}(f_{\lambda}^{-1}(t))=\lambda|f_{\lambda}^{-1}(t)|^{r+1}+|f_{\lambda}^{-1}(t)|^{p+1}$,
from Lemma 4.2 and (1.4), for all $t\in\mathbb{R}$,

$\overline{F}_{\lambda}(t)-\frac{f_{\lambda}^{-1}(t)t}{s+1}=\lambda\left(\frac{r}{r+1}-\frac{1}{s+1}\right)|f^{-1}_{\lambda}(t)|^{r+1}+\left(\frac{p}{p+1}-\frac{1}{s+1}\right)|f^{-1}_{\lambda}(t)|^{p+1}\geq
0.\qed$

######  Lemma 4.4.

$f_{\lambda}^{-1}(t)\geq\left\\{\begin{array}[]{cl}\displaystyle\left(\frac{t}{2\lambda}\right)^{1/r},&\forall\
0<t\leq 2\lambda^{\frac{p}{p-r}},\\\
\displaystyle\left(\frac{t}{2}\right)^{1/p},&\forall\ t\geq
2\lambda^{\frac{p}{p-r}}.\end{array}\right.$

###### Proof.

Observe that
$2\lambda^{\frac{p}{p-r}}=\lambda(\lambda^{\frac{1}{p-r}})^{r}+(\lambda^{\frac{1}{p-r}})^{p}=f_{\lambda}(\lambda^{\frac{1}{p-r}})$
and write $z=f_{\lambda}^{-1}(t)$.

If $t\leq 2\lambda^{\frac{p}{p-r}}$, applying $f_{\lambda}^{-1}$ to this
inequality, one gets $z\leq
f^{-1}_{\lambda}(2\lambda^{\frac{p}{p-r}})=\lambda^{\frac{1}{p-r}}$, that is,
$z^{p}\leq\lambda z^{r}$, and so $t=z^{p}+\lambda z^{r}\leq 2\lambda z^{r}$,
which implies $\left(\frac{t}{2\lambda}\right)^{1/r}\leq f_{\lambda}^{-1}(t).$

If $t\geq 2\lambda^{\frac{p}{p-r}}$, then $z\geq\lambda^{\frac{1}{p-r}}$ and
$2z^{p}\geq\lambda z^{r}+z^{p}=t$, which implies
$f_{\lambda}^{-1}(t)\geq\left(t/2\right)^{1/p}$, as desired. ∎

######  Lemma 4.5.

$\overline{F}_{\lambda}(t)\geq\left\\{\begin{array}[]{ll}\displaystyle\frac{r}{r+1}\lambda^{-1/r}\left(\frac{t}{2}\right)^{\frac{r+1}{r}}+\frac{p}{p+1}\left(\frac{t}{2\lambda}\right)^{\frac{p+1}{r}},&\forall\
|t|\leq 2\lambda^{\frac{p}{p-r}},\vspace{5pt}\\\
\displaystyle\frac{r}{r+1}\lambda\left(\frac{t}{2}\right)^{\frac{r+1}{p}}+\frac{p}{p+1}\left(\frac{t}{2}\right)^{\frac{p+1}{p}},&\forall\
|t|\geq 2\lambda^{\frac{p}{p-r}}.\end{array}\right.$

###### Proof.

It is a straightforward consequence of Lemmas 4.2 and 4.4. ∎

######  Lemma 4.6.

For $\tau=\frac{ps-1}{2^{\frac{p+1}{p}}(p+1)(s+1)}$,

$\overline{F}_{\lambda}(t)-\frac{f_{\lambda}^{-1}(t)t}{s+1}\geq\tau|t|^{\frac{p+1}{p}}\
\ \forall\,|t|\geq 2\lambda^{\frac{p}{p-r}}.$ (4.6)

###### Proof.

By Lemma 4.4 and the proof of Corollary 4.3, for all $|t|\geq
2\lambda^{\frac{p}{p-r}},$

$\overline{F}_{\lambda}(t)-\frac{f_{\lambda}^{-1}(t)t}{s+1}=\lambda\left(\frac{r}{r+1}-\frac{1}{s+1}\right)|f^{-1}_{\lambda}(t)|^{r+1}+\left(\frac{p}{p+1}-\frac{1}{s+1}\right)|f^{-1}_{\lambda}(t)|^{p+1}\\\
\geq\frac{ps-1}{(p+1)(s+1)}|f^{-1}_{\lambda}(t)|^{p+1}\geq\frac{ps-1}{(p+1)(s+1)}\left(\frac{|t|}{2}\right)^{\frac{p+1}{p}}.\qed$
(4.7)

######  Lemma 4.7.

For all $\alpha,\ \beta\in\mathbb{R}$, there exists $\theta\in(0,\ 1)$ such
that

$\displaystyle 0\leq f^{-1}_{\lambda}(\alpha+\beta)(\alpha+\beta)\leq
f^{-1}_{\lambda}(\alpha)\alpha+\frac{r+1}{r}|\beta|\,|f^{-1}_{\lambda}(\alpha+\theta\beta)|.$
(4.8)

###### Proof.

Consider the function $m(t)=f_{\lambda}^{-1}(t)t$. Then, $m$ is even,
$m^{\prime}(0)=0$ and

$m^{\prime}(t)=\displaystyle
f_{\lambda}^{-1}(t)+\frac{tf_{\lambda}^{-1}(t)}{\lambda
r\left[f_{\lambda}^{-1}(t)\right]^{r}+p\left[f_{\lambda}^{-1}(t)\right]^{p}},\
\ \text{for $t>0$}.$

Hence,

$0<m^{\prime}(t)<f_{\lambda}^{-1}(t)+\frac{tf_{\lambda}^{-1}(t)}{\lambda
r\left[f_{\lambda}^{-1}(t)\right]^{r}+r\left[f_{\lambda}^{-1}(t)\right]^{p}}=\frac{r+1}{r}f_{\lambda}^{-1}(t)\
\ \forall\ t>0,$

which implies that $|m^{\prime}(t)|\leq\frac{r+1}{r}|f_{\lambda}^{-1}(t)|$ for
all $t\in\mathbb{R}.$ By the mean value theorem, there exists $\theta\in(0,1)$
such that

$0\leq
f^{-1}_{\lambda}(\alpha+\beta)(\alpha+\beta)=f^{-1}_{\lambda}(\alpha)\alpha+m^{\prime}(\alpha+\theta\beta)\beta\leq
f^{-1}_{\lambda}(\alpha)\alpha+\frac{r+1}{r}|\beta|\,|f^{-1}_{\lambda}(\alpha+\theta\beta)|.\qed$

######  Lemma 4.8.

$\displaystyle\lim_{t\to\infty}\frac{t^{\frac{p+1}{p}}-f_{\lambda}^{-1}(t)t}{t^{\frac{r+1}{p}}}=\frac{\lambda}{p}.$

In particular, given $0<c<\frac{\lambda}{p}$, there exists $t_{0}>0$ such that

$0\leq
f_{\lambda}^{-1}(t)t\leq|t|^{\frac{p+1}{p}}-c\,|t|^{\frac{r+1}{p}},\quad\forall\,|t|\geq
t_{0}.$ (4.9)

###### Proof.

For $t>0$, writing $t=f_{\lambda}(\tau)$,

$\displaystyle\frac{t^{\frac{p+1}{p}}-f_{\lambda}^{-1}(t)t}{t^{\frac{r+1}{p}}}=\frac{(\lambda\tau^{r}+\tau^{p})^{\frac{p+1}{p}}-\lambda\tau^{r+1}-\tau^{p+1}}{\left(\lambda\tau^{r}+\tau^{p}\right)^{\frac{r+1}{p}}}=\frac{(\lambda\tau^{r-\frac{p(r+1)}{p+1}}+\tau^{p-\frac{p(r+1)}{p+1}})^{\frac{p+1}{p}}-\lambda-\tau^{p-r}}{\tau^{\frac{(r-p)(r+1)}{p}}\left(\lambda+\tau^{p-r}\right)^{\frac{r+1}{p}}}\\\
=\frac{(\lambda+\tau^{p-r})[\tau^{\frac{r-p}{p}}(\lambda+\tau^{p-r})^{\frac{1}{p}}-1]}{\tau^{\frac{(r-p)(r+1)}{p}}\left(\lambda+\tau^{p-r}\right)^{\frac{r+1}{p}}}=\frac{\tau^{\frac{r-p}{p}}(\lambda+\tau^{p-r})^{\frac{1}{p}}-1}{\tau^{\frac{(r-p)(r+1)}{p}}\left(\lambda+\tau^{p-r}\right)^{\frac{r+1}{p}-1}}.$
(4.10)

Then, with $y=\tau^{r-p}$, $y\overset{t\to\infty}{\longrightarrow}0^{+}$, and
it follows that

$\displaystyle\lim_{t\to\infty}\frac{t^{\frac{p+1}{p}}-f_{\lambda}^{-1}(t)t}{t^{\frac{r+1}{p}}}=\displaystyle\lim_{y\to
0^{+}}\frac{y^{1/p}(\lambda+y^{-1})^{1/p}-1}{y^{\frac{r+1}{p}}(\lambda+y^{-1})^{\frac{r+1}{p}-1}}=\displaystyle\lim_{y\to
0^{+}}\frac{(\lambda y+1)^{1/p}-1}{y(\lambda y+1)^{\frac{r+1}{p}-1}},$ (4.11)

and applying the L’Hôpital rule,

$\displaystyle\lim_{t\to\infty}\frac{t^{\frac{p+1}{p}}-f_{\lambda}^{-1}(t)t}{t^{\frac{r+1}{p}}}=\lim_{y\to
0^{+}}\frac{\frac{\lambda}{p}(\lambda y+1)^{1/p-1}}{(\lambda
y+1)^{\frac{r+1}{p}-1}+\left(\frac{r+1}{p}-1\right)\lambda y(\lambda
y+1)^{\frac{r+1}{p}-2}}=\frac{\lambda}{p}.\qed$

###  4.2. Upper bound for the Mountain Pass level

Let $(p,q)$ be on the critical hyperbola (1.3) and $(\varphi,\psi)$ be a
positive radial solution of the problem

$-\Delta\varphi=|\psi|^{p-1}\psi,\ \ -\Delta\psi=|\varphi|^{q-1}\varphi,\ \
\text{in}\ \ \mathbb{R}^{N},$ (4.12)

whose qualitative and quantitative properties are described in [15].

We recall that $(\varphi,\psi)$ has de following decay at infinity:

$\begin{array}[]{lllll}a)&\frac{2}{N-2}<p<\frac{N}{N-2},&\displaystyle\lim_{t\rightarrow\infty}t^{p(N-2)-2}\varphi(t)=b&\text{and}&\displaystyle\lim_{t\rightarrow\infty}t^{N-2}\psi(t)=c,\\\
\vspace{2pt}\par
b)&p=\frac{N}{N-2},&\displaystyle\lim_{t\rightarrow\infty}\dfrac{t^{N-2}}{\log{t}}\varphi(t)=b&\text{and}&\displaystyle\lim_{t\rightarrow\infty}t^{N-2}\psi(t)=c,\\\
\vspace{2pt}\par
c)&\frac{N}{N-2}<p<\frac{N^{2}+2N-4}{N^{2}-4N+4},&\displaystyle\lim_{t\rightarrow\infty}t^{N-2}\varphi(t)=b&\text{and}&\displaystyle\lim_{t\rightarrow\infty}t^{N-2}\psi(t)=c,\\\
\vspace{2pt}\par
d)&p=\frac{N^{2}+2N-4}{N^{2}-4N+4},&\displaystyle\lim_{t\rightarrow\infty}t^{N-2}\varphi(t)=b&\text{and}&\displaystyle\lim_{t\rightarrow\infty}\dfrac{t^{N-2}}{\log{t}}\psi(t)=c,\\\
\vspace{2pt}\par
e)&\frac{N^{2}+2N-4}{N^{2}-4N+4}<p,&\displaystyle\lim_{t\rightarrow\infty}t^{N-2}\varphi(t)=b&\text{and}&\displaystyle\lim_{t\rightarrow\infty}t^{q(N-2)-2}\psi(t)=c,\end{array}$
(4.13)

where $b>0$ and $c>0$ are constants and $t=|x|$. Fix $a\in\Omega$. Let
$\xi_{a}\in{\it C}_{c}^{\infty}(\mathbb{R}^{N})$ be a function such that
$0\leq\xi_{a}(x)\leq 1$ for all $x\in\mathbb{R}^{N}$, $\xi_{a}\equiv 1$ in
$B(a,\rho/2)$, $\xi_{a}\equiv 0$ in $B(a,\rho)^{c}$ and
$B(a,\rho)\subset\subset\Omega$, $\rho>0$.

######  Lemma 4.9.

Suppose (1.4) and let
$U_{\delta,a}:=\delta^{\frac{-N}{q+1}}\xi_{a}(x)\varphi\left(\frac{x-a}{\delta}\right)$,
where $\varphi$ is defined by (4.12) and
$V_{\delta,a}=|U_{\delta,a}|^{-1}_{q+1}U_{\delta,a}$. Then, for everery
$t\in[m,\overline{m}]$, with $m>0$ and $m,\,\overline{m}$ independent of
$\delta$, it holds:

$\bullet$ if $N=3$ with $p\leq 7/2$, or $N\geq 4$ with $p\leq\frac{N+2}{N-2}$,
then

$\int_{\Omega}f^{-1}_{\lambda}(t\Delta V_{\delta,a}(x)t\Delta
V_{\delta,a}(x))dx<t^{\frac{p+1}{p}}S,\ \ \ \ \text{for $\delta>0$ suitably
small},$ (4.14)

$\bullet$ if $N=3$ with $7/2<p<11$, or $N\geq 4$ with
$\frac{N+2}{N-2}<p\leq\frac{N^{2}+2N-4}{N^{2}-4N+4}$, then

$\\!\\!\\!\int_{\Omega}f^{-1}_{\lambda}(t\Delta V_{\delta,a}(x)t\Delta
V_{\delta,a}(x))dx\\!<\\!\left\\{\begin{array}[]{lll}t^{\frac{p+1}{p}}S+c_{1}t^{\frac{r+1}{r}}\delta^{\frac{N(r+1)}{r(p+1)}}-c_{2}t^{\frac{r+1}{p}}\delta^{\frac{N}{q+1}\frac{N}{N-2}},&\text{if
}\,r<\frac{2}{N-2},\\\
t^{\frac{p+1}{p}}S+c_{1}t^{\frac{r+1}{r}}\delta^{\frac{N(r+1)}{r(p+1)}}-c_{2}\delta^{\frac{N}{q+1}\frac{N}{N-2}}|\log(\delta)|,&\text{if
}\,r=\frac{2}{N-2},\\\
t^{\frac{p+1}{p}}S+c_{1}t^{\frac{r+1}{r}}\delta^{\frac{N(r+1)}{r(p+1)}}-c_{2}t^{\frac{r+1}{p}}\delta^{\frac{N(p-r)}{p+1}},&\text{if
}\,r>\frac{2}{N-2},\end{array}\right.$ (4.15)

$\bullet$ if $\frac{N^{2}+2N-4}{N^{2}-4N+4}<p$, then

$\\!\\!\\!\int_{\Omega}f^{-1}_{\lambda}(t\Delta V_{\delta,a}(x)t\Delta
V_{\delta,a}(x))dx\\!<\\!\left\\{\begin{array}[]{lll}t^{\frac{p+1}{p}}S+c_{1}t^{\frac{r+1}{r}}\delta^{\frac{N(r+1)}{r(p+1)}}-c_{2}\lambda\delta^{\frac{Nq}{q+1}}|\log(\delta)|,&\\!\text{if
}r+1=\frac{p+1}{q+1},\\\
t^{\frac{p+1}{p}}S+c_{1}t^{\frac{r+1}{r}}\delta^{\frac{N(r+1)}{r(p+1)}}-c_{2}\lambda\delta^{\frac{Nq}{q+1}},&\\!\text{if
}r+1<\frac{p+1}{q+1},\\\
t^{\frac{p+1}{p}}S+c_{1}t^{\frac{r+1}{r}}\delta^{\frac{N(r+1)}{r(p+1)}}-c_{2}\lambda\delta^{\frac{N(p-r)}{p+1}},&\\!\text{if
}r+1>\frac{p+1}{q+1}.\end{array}\right.$ (4.16)

###### Proof.

First, observe that

$\Delta V_{\delta}=\gamma_{\delta,a}(x)+\sigma_{\delta,a}(x),$

where

$\gamma_{\delta,a}(x):=|U_{\delta}|^{-1}_{q+1}\delta^{\frac{-N}{q+1}}\delta^{-2}\Delta\varphi\left(\frac{x-a}{\delta}\right)$

and

$\sigma_{\delta,a}(x):=|U_{\delta}|^{-1}_{q+1}\delta^{\frac{-N}{q+1}}\left(2\delta^{-1}\nabla\xi_{a}(x)\nabla\varphi\left(\frac{x-a}{\delta}\right)+\varphi\left(\frac{x-a}{\delta}\right)\Delta\xi_{a}(x)\right).$

So,

$\displaystyle\int_{\Omega}f^{-1}_{\lambda}(t\Delta V_{\delta,a}(x))t\Delta
V_{\delta,a}(x)dx=\\\
\displaystyle\int_{B(a,\rho/2)}f^{-1}_{\lambda}(t\xi_{a}(x)\gamma_{\delta,a}(x)+t\sigma_{\delta,a}(x))(t\xi_{a}(x)\gamma_{\delta,a}(x)+t\sigma_{\delta,a}(x))dx\\\
+\displaystyle\int_{\Omega\backslash
B(a,\rho/2)}f^{-1}_{\lambda}(t\xi_{a}(x)\gamma_{\delta,a}(x)+t\sigma_{\delta,a}(x))(t\xi_{a}(x)\gamma_{\delta,a}(x)+t\sigma_{\delta,a}(x))dx,$
(4.17)

and since $supp\left(\Delta V_{\delta,a}(x)\right)\subset\overline{B}(a,\rho)$
and $supp(t\sigma_{\delta,a}(x))\subset\overline{R(a,\rho/2,\rho)}$, where
$R(a,\rho/2,\rho):=B(a,\rho)\setminus\overline{B}(a,\rho/2)$, one has

$\displaystyle\int_{\Omega}f^{-1}_{\lambda}(t\Delta V_{\delta,a}(x))t\Delta
V_{\delta,a}(x)dx=\displaystyle\int_{B(a,\rho/2)}f^{-1}_{\lambda}(t\gamma_{\delta,a}(x))t\gamma_{\delta,a}(x)dx\\\
+\displaystyle\int_{R(a,\rho/2,\rho)}f^{-1}_{\lambda}(t\xi_{a}(x)\gamma_{\delta,a}(x)+t\sigma_{\delta,a}(x))(t\xi_{a}(x)\gamma_{\delta,a}(x)+t\sigma_{\delta,a}(x))dx.$
(4.18)

Now we split the estimate in two steps, which correspond to the principal part

$h_{\delta,a}:=\int_{B(a,\rho/2)}f^{-1}_{\lambda}\left(t\gamma_{\delta,a}(x)\right)t\gamma_{\delta,a}(x),$
(4.19)

and to the residual part

$j_{\delta,a}:=\displaystyle\int_{R(a,\rho/2,\rho)}f^{-1}_{\lambda}(t\xi_{a}(x)\gamma_{\delta,a}(x)+t\sigma_{\delta,a}(x))(t\xi_{a}(x)\gamma_{\delta,a}(x)+t\sigma_{\delta,a}(x))dx.$
(4.20)

Step 1. Estimate of (4.19)

Using (4.2) and the asymptotic behavior of $\Delta\varphi$ as in [15, Theorem
2] and [10, Lemma 6.2], one gets

$\begin{array}[]{ccc}\displaystyle
h_{\delta,a}=\int_{\mathbb{R}^{N}}\left|t\gamma_{\delta,a}(x)\right|^{\frac{p+1}{p}}dx+\int_{B(a,\rho/2)}f^{-1}_{\lambda}\left(t\gamma_{\delta,a}(x)\right)t\gamma_{\delta,a}(x)dx-\int_{\mathbb{R}^{N}}\left|t\gamma_{\delta,a}(x)\right|^{\frac{p+1}{p}}dx\\\
\displaystyle=t^{\frac{p+1}{p}}S+\int_{B(a,\rho/2)}f^{-1}_{\lambda}\left(t\gamma_{\delta,a}(x)\right)t\gamma_{\delta,a}(x)-|t\gamma_{\delta,a}(x)|^{\frac{p+1}{p}}dx-\int_{\mathbb{R}^{N}\setminus
B(a,\rho/2)}\left|t\gamma_{\delta,a}(x)\right|^{\frac{p+1}{p}}dx.\end{array}$
(4.21)

The behavior of the last term is already known by [10], namely

$-\int_{\mathbb{R}^{N}\setminus
B(a,\rho/2)}\left|t\gamma_{\delta,a}(x)\right|^{\frac{p+1}{p}}dx\leq\left\\{\begin{array}[]{lll}-t^{\frac{p+1}{p}}C\delta^{\frac{N(p+1)}{q+1}},&\text{
if }q>\frac{N}{N-2},\\\
-t^{\frac{p+1}{p}}C|\log(\delta)|^{p+1}\delta^{\frac{N(p+1)}{q+1}},&\text{ if
}q=\frac{N}{N-2},\\\ -t^{\frac{p+1}{p}}C\delta^{qN},&\text{ if
}q<\frac{N}{N-2}.\end{array}\right.$ (4.22)

Next, we estimate

$i_{\delta,a}:=\int_{B(a,\rho/2)}f^{-1}_{\lambda}\left(t\gamma_{\delta,a}(x)\right)t\gamma_{\delta,a}(x)-|t\gamma_{\delta,a}(x)|^{\frac{p+1}{p}}dx.$
(4.23)

We consider three parts of the ball $B(a,\rho/2)$, namely the ball
$B(a,\delta)$ and the rings $R(a,\delta,\delta^{M})$ and
$R(a,\delta^{M},\rho/2)$, where the number $M<1$ will be defined ahead. This
splitting involving rings is key argument to capture the contribution of the
term $\lambda|u|^{r-1}r$ to downsize the Mountain Pass level.

Step 1.1. By the behavior of $\Delta\varphi(x)$ it is known that there exists
(for $\delta$ sufficiently small) $c>0$ such that if $|x-a|<\delta$ then
$c<t|\Delta\varphi(\frac{x-a}{\delta})|$, so (4.9) and
$\frac{N+2(q+1)}{q+1}=\frac{pN}{p+1}$, can be used to infer that

$\displaystyle\int_{B(a,\delta)}f^{-1}_{\lambda}\left(t\gamma_{\delta,a}(x)\right)t\gamma_{\delta,a}(x)-|t\gamma_{\delta,a}(x)|^{\frac{p+1}{p}}dx\displaystyle\leq-c\lambda\int_{B(a,\delta)}|t\gamma_{\delta,a}(x)|^{\frac{r+1}{p}}dx\\\
\displaystyle\displaystyle\leq-c^{\prime}\lambda\int_{B(a,\delta)}\delta^{-N\frac{r+1}{p+1}}dx=-C\lambda\delta^{N\frac{p-r}{p+1}}.$
(4.24)

Step 1.2. Now focus the attention on the $R(a,\delta,\delta^{M})$-term. In
this ring, since $1<\frac{|x-a|}{\delta}<\delta^{M-1}$, with $M<1$ to be
defined, the asymptotic decay of $\gamma_{\delta,a}(x)$ present in [10, Lemma
6.2] can be used, and three cases have to be analized.

Case 1: $q>\frac{N}{N-2}$.
In this case it is known that

$\delta^{\frac{-pN}{p+1}}\left|\Delta\varphi\left(\frac{x-a}{\delta}\right)\right|\geq
c|x-a|^{-p(N-2)}\delta^{p(N-2)-\frac{pN}{p+1}},$

the last term is grater than a constant if $|x-a|\leq\delta^{M}$ with

$(1-M)p(N-2)-\frac{pN}{p+1}=0\iff
M=\frac{p(N-2)-2}{(N-2)(p+1)}=\frac{N}{N-2}\frac{1}{q+1},$

observing that $0<M<1$, and $\delta^{M}<\rho/2$ since $\delta\to 0$. Applying
Lemma 4.8, it follows that

$\int_{R(a,\delta,\delta^{M})}f^{-1}_{\lambda}\left(t\gamma_{\delta,a}(x)\right)t\gamma_{\delta,a}(x)-|t\gamma_{\delta,a}(x)|^{\frac{p+1}{p}}dx\leq-c\lambda\int_{R(a,\delta,\delta^{M})}\left|t\gamma_{\delta,a}(x)\right|^{\frac{r+1}{p}}dx\\\
\leq\displaystyle-c\lambda\int_{R(a,\delta,\delta^{M})}|x-a|^{-(r+1)(N-2)}\delta^{N\frac{r+1}{q+1}}dx\displaystyle=-c\lambda\delta^{N\frac{r+1}{q+1}}\int_{\delta}^{\delta^{M}}y^{1-r(N-2)}dy,$
(4.25)

and hence

$\displaystyle\int_{R(a,\delta,\delta^{M})}f^{-1}_{\lambda}\left(t\gamma_{\delta,a}(x)\right)t\gamma_{\delta,a}(x)-|t\gamma_{\delta,a}(x)|^{\frac{p+1}{p}}dx\leq-\left\\{\begin{array}[]{lll}c\lambda\delta^{\frac{N}{q+1}\frac{N}{N-2}}|\log(\delta)|,&\text{if
}r=\frac{2}{N-2},\\\ \displaystyle
c\lambda\frac{\delta^{\frac{N}{q+1}\frac{N}{N-2}}-\delta^{N\frac{p-r}{p+1}}}{2-r(N-2)},&\text{if
}r\neq\frac{2}{N-2}.\end{array}\right.$

Case 2: $q=\frac{N}{N-2}$.
In this case it is known that

$\delta^{\frac{-pN}{p+1}}\left|\Delta\varphi\left(\frac{x-a}{\delta}\right)\right|\geq
c\left(\log\left(\frac{|x-a|}{\delta}\right)+1\right)^{p}|x-a|^{-p(N-2)}\delta^{p(N-2)-\frac{pN}{p+1}},$

so one can use $M=\frac{N}{N-2}\frac{1}{q+1}$ and proceed as in Case 1, to
obtain the same estimate, which could be even better.

Case 3: $q<\frac{N}{N-2}$.
In this case it is known that

$\delta^{\frac{-pN}{p+1}}\left|\Delta\varphi\left(\frac{x-a}{\delta}\right)\right|\geq
c|x-a|^{-\frac{p(q+1)N}{p+1}}\delta^{\frac{pqN}{p+1}},$

and this is grater than a constant if $|x-a|\leq\delta^{M}$ with

$\frac{Mp(q+1)N}{p+1}=\frac{pqN}{p+1}\iff M=\frac{q}{q+1}.$

So Lemma 4.8 can be applied, leading that

$\int_{R(a,\delta,\delta^{M})}f^{-1}_{\lambda}\left(t\gamma_{\delta,a}(x)\right)t\gamma_{\delta,a}(x)-|t\gamma_{\delta,a}(x)|^{\frac{p+1}{p}}dx\leq-c\lambda\int_{R(a,\delta,\delta^{M})}\left|t\gamma_{\delta,a}(x)\right|^{\frac{r+1}{p}}dx\\\
\leq\displaystyle-c\lambda\int_{R(a,\delta,\delta^{M})}|x-a|^{-\frac{(r+1)(q+1)}{p+1}N}\delta^{\frac{r+1}{p+1}qN}dx\displaystyle=-c\lambda\delta^{\frac{r+1}{p+1}qN}\int_{\delta}^{\delta^{M}}y^{N-1-\frac{(q+1)(r+1)}{p+1}N}dy$
(4.26)

and so

$\displaystyle\int_{R(a,\delta,\delta^{M})}f^{-1}_{\lambda}\left(t\gamma_{\delta,a}(x)\right)t\gamma_{\delta,a}(x)-|t\gamma_{\delta,a}(x)|^{\frac{p+1}{p}}dx\leq-\left\\{\begin{array}[]{lll}c\lambda\delta^{\frac{Nq}{q+1}}|\log(\delta)|,&\text{if
}r+1=\frac{p+1}{q+1},\\\ \displaystyle
c\lambda\frac{\delta^{\frac{Nq}{q+1}}-\delta^{N\frac{p-r}{p+1}}}{N-N\frac{(q+1)(r+1)}{p+1}},&\text{if
}r+1\neq\frac{p+1}{q+1}.\end{array}\right.$

Step 1.3. Finally we estimate the $R(a,\delta^{M},\rho/2)$-term. In this ring,
$\left|\frac{x-a}{\delta}\right|>1$, and the asymptotic behavior of
$\gamma_{\delta,a}(x)$ present in [10, Lemma 6.2] can be used one more time,
but in this case $\gamma_{\delta,a}$ becomes small, and then
$f^{-1}_{\lambda}\left(t\gamma_{\delta,a}(x)\right)t\gamma_{\delta,a}(x)-|t\gamma_{\delta,a}(x)|^{\frac{p+1}{p}}\leq-
c_{\lambda}|t\gamma_{\delta,a}(x)|^{\frac{p+1}{p}}$, and one can proceed as in
(4.22).

At this point, from Steps 1.1, 1.2, and 1.3, we can write the estimates for
$i_{\delta,a}$ defined in (4.23). But before doing this, note that in all the
three cases of Step 1.2, the term $\delta^{\frac{N(p-r)}{p+1}}$ (dominant term
in Step 1.1) appear, so it does not need to be repeated.

$i_{\delta,a}\leq\left\\{\begin{array}[]{rl}-c\lambda\delta^{\frac{N}{q+1}\frac{N}{N-2}}|\log(\delta)|-C\delta^{\frac{N(p+1)}{q+1}},&\text{if
}q>\frac{N}{N-2},\,r=\frac{2}{N-2},\\\ -\displaystyle
c\lambda\frac{\delta^{\frac{N}{q+1}\frac{N}{N-2}}-\delta^{N\frac{p-r}{p+1}}}{2-r(N-2)}-C\delta^{\frac{N(p+1)}{q+1}},&\text{if
}q>\frac{N}{N-2},\,r\neq\frac{2}{N-2},\\\
-c\lambda\delta^{\frac{N}{q+1}\frac{N}{N-2}}|\log(\delta)|-C|\log(\delta)|^{p+1}\delta^{\frac{N(p+1)}{q+1}},&\text{if
}q=\frac{N}{N-2},\,r=\frac{2}{N-2},\\\
-c\lambda\frac{\delta^{\frac{N}{q+1}\frac{N}{N-2}}-\delta^{N\frac{p-r}{p+1}}}{2-r(N-2)}-C|\log(\delta)|^{p+1}\delta^{\frac{N(p+1)}{q+1}},&\text{if
}q=\frac{N}{N-2},\,r\neq\frac{2}{N-2},\\\
-c\lambda\delta^{\frac{Nq}{q+1}}|\log(\delta)|-C\delta^{qN},&\text{if
}q<\frac{N}{N-2},\,r+1=\frac{p+1}{q+1},\\\ -\displaystyle
c\lambda\frac{\delta^{\frac{Nq}{q+1}}-\delta^{N\frac{p-r}{p+1}}}{N-N\frac{(q+1)(r+1)}{p+1}}-C\delta^{qN},&\text{if
}q<\frac{N}{N-2},\,\,r+1\neq\frac{p+1}{q+1}.\par\end{array}\right.$ (4.27)

Now we summarize all the calculation made in Step 1. To estimate
$h_{\delta,a}$, from (4.21), we must deal with (4.22) and (4.27). Observe that
the majorante in (4.22) also appears in the second terms in (4.27). Then, the
estimate for (4.21) follows from the comparison among the powers of $\delta$
in (4.27).

For $q>\frac{N}{N-2}$, the terms to be compared are

$-C\delta^{\frac{N(p+1)}{q+1}}\text{ and
}-\left\\{\begin{array}[]{lll}c\lambda\delta^{\frac{N}{q+1}\frac{N}{N-2}}|\log(\delta)|,&\text{if
}r=\frac{2}{N-2},\\\ \displaystyle
c\lambda\frac{\delta^{\frac{N}{q+1}\frac{N}{N-2}}-\delta^{N\frac{p-r}{p+1}}}{2-r(N-2)},&\text{if
}r\neq\frac{2}{N-2},\end{array}\right.$ (4.28)

and the first part is always weaker. Of course
$N\frac{p-r}{p+1}>\frac{N}{q+1}\frac{N}{N-2}$ if $r<\frac{2}{N-2}$, and
$\frac{N(p+1)}{q+1}>\frac{N}{q+1}\frac{N}{N-2}$ (since $p>\frac{2}{N-2}$
always), so in this case the dominant term is
$-c\lambda\delta^{\frac{N}{q+1}\frac{N}{N-2}}$. The same analysis shows that
if $r=\frac{2}{N-2}$ the dominant term is
$c\lambda\delta^{\frac{N}{q+1}\frac{N}{N-2}}|\log(\delta)|$. If
$r>\frac{2}{N-2}$, then $N\frac{p-r}{p+1}<\frac{N}{q+1}\frac{N}{N-2}$ and the
dominant term is $-c\delta^{N\frac{p-r}{p+1}}$.

When $q=\frac{N}{N-2}$ the analysis done before gives that the dominant term
is

$-\left\\{\begin{array}[]{lll}c\lambda\delta^{\frac{N}{q+1}\frac{N}{N-2}}|\log(\delta)|,&\text{if
}r=\frac{2}{N-2},\\\ \displaystyle
c\lambda\delta^{\frac{N}{q+1}\frac{N}{N-2}},&\text{if }r<\frac{2}{N-2},\\\
-t^{\frac{p+1}{p}}C|\log(\delta)|^{p+1}\delta^{\frac{N(p+1)}{q+1}}-C\lambda\delta^{N\frac{p-r}{p+1}}&\text{if
}r>\frac{2}{N-2}.\end{array}\right.$

Finaly, if $q<\frac{N}{N-2}$ the terms that we have to compare are

$-t^{\frac{p+1}{p}}C\delta^{qN}\text{ and
}-\left\\{\begin{array}[]{lll}c\lambda\delta^{\frac{Nq}{q+1}}|\log(\delta)|,&\text{if
}r+1=\frac{p+1}{q+1},\\\ \displaystyle
c\lambda\frac{\delta^{\frac{Nq}{q+1}}-\delta^{N\frac{p-r}{p+1}}}{N-N\frac{(q+1)(r+1)}{p+1}},&\text{if
}r+1\neq\frac{p+1}{q+1}.\end{array}\right.$

If $r+1<\frac{p+1}{q+1}$, it is easy to see that
$\frac{Nq}{q+1}<N\frac{p-r}{p+1}$ and surely $\frac{Nq}{q+1}<Nq,$ so
$-c\lambda\delta^{\frac{Nq}{q+1}}$ is the dominant term. The same computation
ensures that the dominant term is
$-c\lambda\delta^{\frac{Nq}{q+1}}|\log(\delta)|,$ if $r+1=\frac{p+1}{q+1}$.
Finally, if $r+1>\frac{p+1}{q+1}$ an analogous computation show that the term
$-C\lambda\delta^{N\frac{p-r}{p+1}}$ is the dominant. Then, putting it all
together,

$h_{\delta,a}\leq
t^{\frac{p+1}{p}}S-\left\\{\begin{array}[]{rl}c\lambda\delta^{\frac{N}{q+1}\frac{N}{N-2}}|\log(\delta)|,&\text{if
}q\geq\frac{N}{N-2},\,r=\frac{2}{N-2},\\\
c\lambda\delta^{\frac{N}{q+1}\frac{N}{N-2}},&\text{if
}q\geq\frac{N}{N-2},\,r<\frac{2}{N-2},\\\
c\lambda\delta^{\frac{N(p-r)}{p+1}},&\text{if
}q\geq\frac{N}{N-2},\,r>\frac{2}{N-2},\\\
\lambda\delta^{\frac{Nq}{q+1}}|\log(\delta)|,&\text{if
}q<\frac{N}{N-2},\,r+1=\frac{p+1}{q+1},\\\
c\lambda\delta^{\frac{Nq}{q+1}},&\text{if
}q<\frac{N}{N-2},\,\,r+1<\frac{p+1}{q+1},\\\
c\lambda\delta^{\frac{N(p-r)}{p+1}},&\text{if
}q<\frac{N}{N-2},\,\,r+1>\frac{p+1}{q+1}.\end{array}\right.$ (4.29)

Step 2 Estimate of the residual part (4.20).

Here $\left|\frac{x-a}{\delta}\right|\geq\frac{\rho}{2\delta}\to\infty$,
uniformly with respect to $x\in R(a,\rho/2,\rho)$, as $\delta\to 0$. Then the
asymptotic behavior of $\gamma_{\delta,a}$ and $\sigma_{\delta,a}$ present in
[10, Lemma 6.2] reads

$\sigma_{\delta,a}(x)\leq\left\\{\begin{array}[]{ll}c|U_{\delta}|^{-1}_{q+1}\delta^{\frac{N}{p+1}}(|x-a|^{-N+1}+|x-a|^{-N+2}),&\text{
if }p>\frac{N}{N-2},\\\
c|U_{\delta}|^{-1}_{q+1}\delta^{\frac{N}{p+1}}|\log(\delta)|(|\log|x-a|+1|)(|x-a|^{-N+1}(1+|x-a|)),&\text{
if }p=\frac{N}{N-2},\\\
c|U_{\delta}|^{-1}_{q+1}\delta^{\frac{pN}{q+1}}(|x-a|^{-p(N-2)+2}+|x-a|^{-p(N-2)+1}),&\text{
if }p<\frac{N}{N-2},\end{array}\right.$ (4.30)

and,

$\gamma_{\delta,a}\leq\left\\{\begin{array}[]{ll}c|x-a|^{-p(N-2)}\delta^{\frac{pN}{q+1}},&\text{
if }q>\frac{N}{N-2}\\\
c(\left|\log\left(|x-a|\right)\right|^{p+1}+1)^{\frac{p}{p+1}}|x-a|^{-p(N-2)}|\log\delta|^{p}\delta^{\frac{pN}{q+1}},&\text{
if }q=\frac{N}{N-2}\\\
c|x-a|^{\frac{p(q+1)N}{p+1}}\delta^{\frac{pqN}{p+1}},&\text{ if
}q<\frac{N}{N-2}.\end{array}\right.$ (4.31)

From this, it follows that $|j_{\delta,a}|$ is bounded from above by

$|R(a,\rho/2,\rho)|\left\\{\begin{array}[]{lll}f^{-1}_{\lambda}\left(ct\left(\delta^{\frac{pN}{q+1}}+\delta^{\frac{pN}{q+1}}\right)\right)ct\left(\delta^{\frac{pN}{q+1}}+\delta^{\frac{pN}{q+1}}\right),&\text{if
}p<\frac{N}{N-2},\\\
f^{-1}_{\lambda}\left(ct\left(\delta^{\frac{pN}{q+1}}+\delta^{\frac{N}{p+1}}|\log(\delta)|\right)\right)ct\left(\delta^{\frac{pN}{q+1}}+\delta^{\frac{N}{p+1}}|\log(\delta)|\right),&\text{if
}p=\frac{N}{N-2},\\\
f^{-1}_{\lambda}\left(ct\left(\delta^{\frac{pN}{q+1}}+\delta^{\frac{N}{p+1}}\right)\right)ct\left(\delta^{\frac{pN}{q+1}}+\delta^{\frac{N}{p+1}}\right),&\text{if
}p,q>\frac{N}{N-2},\\\
f^{-1}_{\lambda}\left(ct\left(|\log\delta|^{p}\delta^{\frac{pN}{q+1}}+\delta^{\frac{N}{p+1}}\right)\right)ct\left(|\log\delta|^{p}\delta^{\frac{pN}{q+1}}+\delta^{\frac{N}{p+1}}\right),&\text{if
}q=\frac{N}{N-2},\\\
f^{-1}_{\lambda}\left(ct\left(\delta^{\frac{pqN}{p+1}}+\delta^{\frac{N}{p+1}}\right)\right)ct\left(\delta^{\frac{pqN}{p+1}}+\delta^{\frac{N}{p+1}}\right),&\text{if
}q<\frac{N}{N-2},\end{array}\right.$ (4.32)
$\leq|R(a,\rho/2,\rho)|\left\\{\begin{array}[]{lll}f^{-1}_{\lambda}\left(tc\delta^{\frac{pN}{q+1}}\right)tc\delta^{\frac{pN}{q+1}},&\text{
if }p<\frac{N}{N-2},\\\
f^{-1}_{\lambda}\left(tc\delta^{\frac{N}{p+1}}|\log(\delta)|\right)tc\delta^{\frac{N}{p+1}}|\log(\delta)|,&\text{
if }p=\frac{N}{N-2},\\\
f^{-1}_{\lambda}\left(tc\delta^{\frac{N}{p+1}}\right)tc\delta^{\frac{N}{p+1}},&\text{
if }p,q>\frac{N}{N-2},\\\
f^{-1}_{\lambda}\left(tc\delta^{\frac{N}{p+1}}\right)tc\delta^{\frac{N}{p+1}},&\text{
if }q=\frac{N}{N-2},\\\
f^{-1}_{\lambda}\left(tc\delta^{\frac{N}{p+1}}\right)tc\delta^{\frac{N}{p+1}},&\text{
if }q<\frac{N}{N-2},\end{array}\right.$ (4.33)
$=|R(a,\rho/2,\rho)|\left\\{\begin{array}[]{rll}f^{-1}_{\lambda}\left(tc\delta^{\frac{pN}{q+1}}\right)tc\delta^{\frac{pN}{q+1}},&\text{
if }p<\frac{N}{N-2},\\\
f^{-1}_{\lambda}\left(tc\delta^{\frac{N}{p+1}}|\log(\delta)|\right)tc\delta^{\frac{N}{p+1}}|\log(\delta)|,&\text{
if }p=\frac{N}{N-2},\\\
f^{-1}_{\lambda}\left(tc\delta^{\frac{N}{p+1}}\right)tc\delta^{\frac{N}{p+1}},&\text{
if }p>\frac{N}{N-2},\end{array}\right.$ (4.34)

and using the asymptotic behavior of $f_{\lambda}^{-1}$, one concludes that

$|j_{\delta,a}|\leq\left\\{\begin{array}[]{rll}c_{\lambda}\left(t\delta^{\frac{pN}{q+1}}\right)^{\frac{r+1}{r}},&\text{
if }p<\frac{N}{N-2},\\\
c_{\lambda}\left(t\delta^{\frac{N}{p+1}}|\log(\delta)|\right)^{\frac{r+1}{r}},&\text{
if }p=\frac{N}{N-2},\\\
c_{\lambda}\left(t\delta^{\frac{N}{p+1}}\right)^{\frac{r+1}{r}},&\text{ if
}p>\frac{N}{N-2}.\end{array}\right.$ (4.35)

Step 3: Comparison of the residual parts in (4.29) and (4.35).

Step 3.1: $p<\frac{N}{N-2}$. First, observe that this implies
$q>\frac{N}{N-2}$. To obtain (4.14), from the comparison (4.28) to obtain
(4.29), it is enough to verify that
$\frac{N(p+1)}{q+1}<\frac{pN}{q+1}\frac{r+1}{r}$, which is equivalent to
$p+1<p\frac{r+1}{r}$, that is $r<p$, which is always the case. Hence, the
Lemma is proved in this case.

Step 3.2: $p=\frac{N}{N-2}$. Again this implies $q>\frac{N}{N-2}$, and the
procedure to obtain (4.14) is identical to that from Step 3.1.

Step 3.3: $\frac{N}{N-2}<p\leq\frac{N+2}{N-2}$.

If $r<\frac{2}{N-2}$, to obtain (4.14), from (4.29) and (4.35), one must
decide when $\frac{N^{2}}{(q+1)(N-2)}<{\frac{N}{p+1}\frac{r+1}{r}}$, that is

$\frac{N(p+1)}{(q+1)(N-2)}<\frac{r+1}{r}\iff\frac{p(N-2)-2}{N-2}-1<\frac{1}{r}\iff
r<\frac{1}{p-\frac{N}{N-2}},$

and this is always true for $N\geq 4$. If $N=3$, these conditions read

$3<p\leq 5,\quad r<2,\quad\text{and}\quad r<\frac{1}{p-3}$

which are satisfied with the extra condition $p\leq 7/2$.

If $r=\frac{2}{N-2}$, then $\frac{r+1}{r}=\frac{N}{2}$, to obtain (4.14), from
(4.29) and (4.35), one must decide when

$\frac{p+1}{q+1}\leq\frac{N-2}{2}\iff\frac{p(N-2)-2}{N}\leq\frac{N-2}{2}\iff
p\leq\frac{N}{2}+\frac{2}{N-2},$

and (remember $p\leq\frac{N+2}{N-2}$) this is always true for $N\geq 4$, and
with $N=3$ these conditions read $p\leq 7/2$.

If $r>\frac{2}{N-2}$, to obtain (4.14), from (4.29) and (4.35), one must
decide when $\frac{N(p-r)}{p+1}<\frac{N}{p+1}\frac{r+1}{r}$, that is

$p-r<\frac{r+1}{r}\iff 0<r^{2}+(1-p)r+1.$ (4.36)

Remember that $p\leq\frac{N+2}{N-2}$. Then, for $N>4$, (4.36) is true because
such second order polynomial has no real roots. For $N=4$ and
$p<\frac{N+2}{N-2}$, again (4.36) has no real root and (4.36) is verified. For
$N=4$ and $p=\frac{N+2}{N-2}$, such polynomial has $1$ as real root and
$r>1=2/(N-2)$, hence (4.36) is verified. With $N=3$, the largest real root of
such polynomial is less or equal to $2$ for $p\leq 7/2$, hence (4.36) is
verified because $r>2=2/(N-2)$. This finishes the proof of the lemma. ∎

######  Remark 4.10.

The estimate of $\int_{\Omega}f^{-1}_{\lambda}(t\Delta V_{\delta,a}(x))t\Delta
V_{\delta,a}(x)dx$ from Lemma 4.9 deserves some comments.

When evaluating $\int_{\Omega}f^{-1}_{\lambda}(t\Delta V_{\delta,a}(x))t\Delta
V_{\delta,a}(x)dx$, the leading term comes from
$\int_{B(0,\rho/2)}f^{-1}_{\lambda}\left(t\gamma_{\delta,a}(x)\right)t\gamma_{\delta,a}(x)dx$,
which carries by itself a (negative) remainder that has to be compared with
the residual term $\int_{R(0,\rho/2,\rho)}f_{\lambda}^{-1}(t\Delta
V_{\delta,a}(x))t\Delta V_{\delta,a}(x)dx$. With $N\geq 4$ or $N=3$ and $p\leq
7/2$ the remainder $\int_{R(0,\rho/2,\rho)}f_{\lambda}^{-1}(t\Delta
V_{\delta,a}(x))t\Delta V_{\delta,a}(x)dx$ is smaller than the negative part
of the remainder term in $\int_{B(0,\rho/2)}f^{-1}_{\lambda}(t\Delta
V_{\delta,a}(x))t\Delta V_{\delta,a}(x)dx$, which brings down the functional
when comparing it to the problem without the perturbation $\lambda u^{r}$, and
this plays an important role in the results in this paper. At this point, it
is important to compare the estimates (4.14), (4.15) and (4.16) with [10, Eq.
(6.4)].

We are almost prepared for the proof of Proposition 2.5. Going on this
direction, observe that if (1.3) and (1.4) are satisfied, then

$\lim_{t\to\infty}I_{F}(tV_{\delta,a})=-\infty$ (4.37)

and the $\max_{t\geq 0}I(tV_{\delta,a})$ is achieved at some $t_{\delta}>0$,
thus,

$0=I_{F}^{\prime}(t_{\delta}V_{\delta,a})=\displaystyle\int_{\Omega}f_{\lambda}^{-1}(t_{\delta}\Delta
V_{\delta,a})\Delta V_{\delta,a}\,dx-
t_{\delta}^{s}|V_{\delta,a}|^{s+1}_{s+1}-t_{\delta}^{q},$ (4.38)

from where we infer that

$t_{\delta}^{q+1}=\displaystyle\int_{\Omega}f_{\lambda}^{-1}(t_{\delta}\Delta
V_{\delta,a})t_{\delta}\Delta V_{\delta,a}\,dx-
t_{\delta}^{s+1}|V_{\delta,a}|^{s+1}_{s+1}.$ (4.39)

######  Lemma 4.11.

Suppose (1.3) and (1.4). Then $t_{\delta}$, as $\delta\rightarrow 0$, is
bounded form below and above.

###### Proof.

Suppose by contradiction that $t_{\delta}\overset{\delta\to
0}{\longrightarrow}0$. Define $A_{\delta}:=\\{x\in\Omega;|t_{\delta}\Delta
V_{\delta,a}(x)|<1+\lambda\\}$ and $B_{\delta}=\Omega\backslash A_{\delta}$,
small $\delta$ give us

$Ct_{\delta}^{1/r}\leq\displaystyle\frac{t_{\delta}^{1/p}}{(1+\lambda)^{1/p}}\int_{B_{\delta}}|\Delta
V_{\delta,a}(x)|^{\frac{p+1}{p}}dx+\frac{t_{\delta}^{1/r}}{(1+\lambda)^{1/r}}\int_{A_{\delta}}|\Delta
V_{\delta,a}(x)|^{\frac{r+1}{r}}dx\\\
\displaystyle\leq\int_{\Omega}f_{\lambda}^{-1}(t_{\delta}\Delta
V_{\delta,a}(x))\Delta
V_{\delta,a}(x)dx=t_{\delta}^{s}\,o(\delta)+t_{\delta}^{q}\,O(1).$ (4.40)

which is a contradiction by the fact that $1/r\leq s$ and $1/r<q$, so
$t_{\delta}\nrightarrow 0$. Observe that in the case $rs=1$ the $o(\delta)$
produces the contradiction.

Now observe that by (4.38) and the estimates present in [20]

$t_{\delta}^{q}\leq
t_{\delta}^{\frac{1}{p}}S+t_{\delta}^{\frac{1}{p}}o(\delta)\Longrightarrow
t_{\delta}^{\frac{pq-1}{p}}\leq S+o(\delta)$ (4.41)

that is, $t_{\delta}\leq k<\infty$ for all $\delta$ sufficiently small. ∎

###### Proof of Proposition 2.5.

Now, using Lemma 4.1 and identity (4.39)

$\max_{t\geq
0}I_{F}(tV_{\delta,a})=I_{F}(t_{\delta}V_{\delta,a})=\int_{\Omega}\overline{F}_{\lambda}(t_{\delta}\Delta
V_{\delta,a})dx-\frac{t_{\delta}^{q+1}}{q+1}-\frac{\mu}{s+1}t_{\delta}^{s+1}|V_{\delta,a}|^{s+1}_{s+1}\\\
=\frac{p}{p+1}\int_{\Omega}f_{\lambda}^{-1}(t_{\delta}\Delta
V_{\delta,a})t_{\delta}\Delta
V_{\delta,a}dx-\lambda\frac{p-r}{p+1}\frac{\left|f_{\lambda}^{-1}(t_{\delta}\Delta
V_{\delta,a})\right|^{r+1}_{r+1}}{r+1}-\frac{t_{\delta}^{q+1}}{q+1}-\frac{\mu
t_{\delta}^{s+1}|V_{\delta,a}|^{s+1}_{s+1}}{s+1}\\\
=\frac{2}{N}\int_{\Omega}f_{\lambda}^{-1}(t_{\delta}\Delta
V_{\delta,a})t_{\delta}\Delta
V_{\delta,a}dx-\lambda\frac{p-r}{p+1}\frac{\left|f_{\lambda}^{-1}(t_{\delta}\Delta
V_{\delta,a})\right|^{r+1}_{r+1}}{r+1}-\frac{\mu(q-s)t_{\delta}^{s+1}|V_{\delta,a}|^{s+1}_{s+1}}{(q+1)(s+1)}.$
(4.42)

By (4.38) and Lemma 4.9 one gets

$t_{\delta}^{q}\leq
t_{\delta}^{1/p}S-Ct_{\delta}^{1/p}\delta^{\frac{N(p+1)}{q+1}}-t_{\delta}^{s}\mu|V_{\delta,a}|^{s+1}_{s+1}\quad\Rightarrow\quad
t_{\delta}<S^{\frac{p}{pq-1}}.$ (4.43)

Combining this with (4.42) and Lemma 4.9, we infer that

$\max_{t\geq
0}I_{F}(tV_{\delta,a})<\displaystyle\frac{2}{N}t_{\delta}^{\frac{p+1}{p}}S-\frac{\mu(q-s)}{(q+1)(s+1)}t_{\delta}^{s+1}|V_{\delta,a}|^{s+1}_{s+1}<\frac{2}{N}S^{\frac{pN}{2(p+1)}},$
(4.44)

which concludes the proof. ∎

######  Remark 4.12.

For $N=3$, we mention that the estimates from (4.15) and (4.16) can be used to
prove the existence of a positive solution to (HS) for the pairs $(p,q)$ on
the critical hyperbola (1.3) that are not included in Theorem 1.1, namely with
$7/2<p<8$, and for some (not all) $(r,s)$ as in (1.4). This remark is linked
to the condition $3<t<5$ in [4, Corollary 2.3] to prove the existence of a
solution to (1.1).

By (4.38) and Lemma 4.9 one gets

$t_{\delta}^{q}\leq
t_{\delta}^{1/p}S+c_{1}t_{\delta}^{\frac{r+1}{r}}\delta^{\frac{3(r+1)}{r(p+1)}}+i_{\delta,a}-t_{\delta}^{s}\mu|V_{\delta,a}|^{s+1}_{s+1}.$
(4.45)

Combining this with (4.42) and Lemma 4.9, we infer that

$\max_{t\geq
0}I_{F}(tV_{\delta,a})\leq\displaystyle\frac{2}{3}t_{\delta}^{\frac{p+1}{p}}S+c_{1}t_{\delta}^{\frac{r+1}{r}}\delta^{\frac{3(r+1)}{r(p+1)}}+i_{\delta,a}-\frac{\mu(q-s)}{(q+1)(s+1)}t_{\delta}^{s+1}|V_{\delta,a}|^{s+1}_{s+1},$
(4.46)

and this is smaller than $\frac{2}{3}S^{\frac{3p}{2(p+1)}}$ if, and only if,

$C\delta^{\frac{3(r+1)}{r(p+1)}}\leq
c_{1}\mu|V_{\delta,a}|^{s+1}_{s+1}-i_{\delta,a}=c_{1}\mu|V_{\delta,a}|^{s+1}_{s+1}+|i_{\delta,a}|.$
(4.47)

To get (4.47) it is sufficient to verify, as $\delta\rightarrow 0$, that

$c\delta^{\frac{3(r+1)}{r(p+1)}}<|i_{\delta,a}|,$ (4.48)

or

$c\delta^{\frac{3(r+1)}{r(p+1)}}<c_{1}\mu|V_{\delta,a}|^{s+1}_{s+1}=\left\\{\begin{array}[]{ll}C\mu\delta^{\frac{3(s+1)}{p+1}},&\text{
if }s<2,\\\ C\mu\delta^{\frac{9}{p+1}}|\log\delta|,&\text{ if }s=2,\\\
C\mu\delta^{3-\frac{3(s+1)}{q+1}},&\text{ if }s>2,\end{array}\right.$ (4.49)

where the behavior of $|V_{\delta,a}|_{s+1}$ in (4.49) is given in [20, eq.
(36)].

To obatin (4.48), we keep all the calculation from Step 3.3 for the case with
$7/2<p\leq(N+2)/(N-2)=5$. Then, we execute similar estimates for $5<p<8$.
Therefore, for $7/2<p<8$, using the residual terms in the first three lines
(that is $q\geq\frac{N}{N-2}=3$) of (4.29), which is a refinement of (4.27),
one can see that (4.48) holds, iff

$\begin{array}[]{rl}r<\frac{1}{p-3}&\text{ if }r<2,\\\ p\leq\frac{7}{2}&\text{
if }r=2,\\\ 0<r^{2}+(1-p)r+1&\text{ if }r>2,\end{array}$ (4.50)

otherwise, the term of (4.35) is dominant.

Let us now consider (4.49). For $s<2$ the inequality is equivalent to
$\frac{r+1}{r}>s+1$, that is $rs<1$, which is a contradiction with condition
(1.4). For $s=2$ the inequality is equivalent to $\frac{r+1}{r}\geq 3$, that
is $r\leq 1/2$, and this together with (1.4), gives $r=1/2$. Finally, for
$s>2$ the inequality is true if

$\frac{3(r+1)}{r(p+1)}>3-\frac{3(s+1)}{q+1}\quad\text{that is,}\quad
s+1>q+1-\frac{r+1}{r}\frac{q+1}{p+1}=\frac{q+1}{p+1}(p-\frac{1}{r}).$ (4.51)

Therefore, given any $(p,q)$ on the critical hyperbola (1.3) with $N=3$,
$7/2<p<8$, $(r,s)$ as in (1.4) with the one of the extra conditions (4.50),
$(r,s)=(1/2,2)$ or (4.51), the mountain pass level of $I_{F}$ is in the range
of compactness and the mountain pass theorem ensures the existence of a
solution.

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