# Existence, nonexistence, and asymptotic behavior of solutions for
$\displaystyle N$-Laplacian equations involving critical exponential growth in
the whole $\displaystyle\mathbb{R}^{N}$

Anderson L. A. de Araujo  Universidade Federal de Viçosa, Departamento de
Matemática
Av. Peter Henry Rolfs, s/n, Viçosa, MG, Brazil, CEP 36570-900.
E-mail<EMAIL_ADDRESS>Luiz F. O. Faria  Universidade Federal de Juiz
de Fora, ICE, Departamento de Matemática
Rua José Lourenço Kelmer, s/n Juiz de Fora, MG, Brazil, CEP 36036-900
E-mail<EMAIL_ADDRESS>
###### Abstract

In this paper, we are interested in studying the existence or non-existence of
solutions for a class of elliptic problems involving the $\displaystyle
N$-Laplacian operator in the whole space. The nonlinearity considered involves
critical Trudinger-Moser growth. Our approach is non-variational, and in this
way we can address a wide range of problems not yet contained in the
literature. Even $\displaystyle W^{1,N}(\mathbb{R}^{N})\hookrightarrow
L^{\infty}(\mathbb{R}^{N})$ failing, we establish
$\displaystyle\|u_{\lambda}\|_{L^{\infty}(\mathbb{R}^{N})}\leq
C\|u\|_{W^{1,N}(\mathbb{R}^{N})}^{\Theta}$ (for some $\displaystyle\Theta>0$),
when $\displaystyle u$ is a solution. To conclude, we explore some asymptotic
properties.

Keywords: Dirichlet problem; Galerkin approximation; Critical growth; Moser-
Trudinger inequality; $\displaystyle N$-Laplacian.
_2020 MSC_ : 35A01; 35A25; 35B09; 35B20; 35B33; 35B40; 35J92.

##  1 Introduction

Let $\displaystyle W_{0}^{1,p}(\Omega)$,
$\displaystyle\Omega\subseteq\mathbb{R}^{N}$, $\displaystyle N\geq 2$, be the
Sobolev space endowed with the norm

$\displaystyle\|u\|_{W^{1,p}(\Omega)}=\left(\int_{\Omega}(|\nabla
u|^{p}+|u|^{p})dx\right)^{\frac{1}{p}}.$

If $\displaystyle p<N$, the critical growth $\displaystyle p^{*}=Np/(N-p)$
means that $\displaystyle\|u\|_{W_{0}^{1,p}(\Omega)}\hookrightarrow
L^{q}(\Omega)$, $\displaystyle N\leq p\leq p^{*}$. The case $\displaystyle
p=N$ is a borderline case in the sense of Sobolev embeddings. As is well-
known, one has $\displaystyle\|u\|_{W_{0}^{1,N}(\Omega)}\hookrightarrow
L^{q}(\Omega)$, $\displaystyle q\geq N$, but a function in $\displaystyle
W_{0}^{1,N}(\Omega)$ may have a local singularity and this causes the failure
of the embedding $\displaystyle W^{1,N}(\Omega)\not\subset
L^{\infty}(\Omega)$. If $\displaystyle\Omega$ is a bounded domain, another
kind of maximal growth was proved by Yudovič [38], Pohožaev [35] and Trudinger
[36]. They proved, in an independent way, that

$u\in
W^{1,N}_{0}(\Omega)\Longrightarrow\int_{\Omega}e^{|u|^{N^{\prime}}}dx<\infty,$
(1)

where $\displaystyle N^{\prime}=\frac{N}{N-1}$. Moreover, for any higher
growth, the corresponding integral can be infinite for a suitable choice of
$\displaystyle u$. After that, Moser [30] improved this assertion, showing
that if $\displaystyle u\in W_{0}^{1,N}(\Omega)$, then

$\sup_{\|\nabla u\|_{L^{N}(\Omega)}\leq
1}\displaystyle\int_{\Omega}e^{\alpha|u|^{N^{\prime}}}dx\left\\{\begin{array}[]{ll}\leq
c|\Omega|,&\mbox{ if }\alpha\leq\alpha_{N},\\\ =\infty,&\mbox{ if
}\alpha>\alpha_{N},\end{array}\right.$ (2)

where $\displaystyle\alpha_{N}=N\omega_{N-1}^{1/(N-1)}$, $\displaystyle c$ is
a constant which depends on $\displaystyle N$, and $\displaystyle\omega_{N-1}$
is the measure of the unit sphere in $\displaystyle\mathbb{R}^{N}$. Inequality
$\displaystyle\eqref{TMz}$ is now called Moser-Trudinger inequality and the
term $\displaystyle e^{\alpha_{N}|u|^{N^{\prime}}}$ is known as critical
Moser-Trudinger growth.

This well-known Moser-Trundinger inequality has been generalized in many ways.
In the case $\displaystyle\Omega=\mathbb{R}^{N}$, B. Ruf when $\displaystyle
N=2$ in [33], and Y.X. Li and B. Ruf in [25] for $\displaystyle N>2$, proved
the following assertion

$\sup_{\|u\|_{W^{1,N}(\mathbb{R}^{N})}\leq
1}\displaystyle\int_{\mathbb{R}^{N}}\phi_{N}(\alpha|u|^{\frac{N}{N-1}})dx\left\\{\begin{array}[]{ll}\leq
C(\alpha,N)^{N^{\prime}},&\mbox{ if }\alpha\leq\alpha_{N},\\\ =\infty,&\mbox{
if }\alpha>\alpha_{N},\end{array}\right.$ (3)

where $\displaystyle\alpha_{N}>0$ is a constant and

$\phi_{N}(t)=e^{t}-\sum_{j=0}^{N-2}\frac{t^{j}}{j!}=\sum_{j=N-1}^{\infty}\frac{t^{j}}{j!},\,\,t\geq
0.$ (4)

######  Remark 1.

Notice that $\displaystyle\phi_{N}(t)$ is a increasing function.

In this paper, we consider the existence, nonexistence, and asymptotic
behavior of positive solutions for a class of $\displaystyle N$-Laplacian
elliptic equations related to the critical growth (3) in the whole space
$\displaystyle\mathbb{R}^{N}$. More precisely, we are concerned with the
following problem:

$-\Delta_{N}u+|u|^{N-2}u=\lambda a(x)|u|^{q-2}u+f(u)\,\,\mbox{ in
}\,\,\mathbb{R}^{N},$ (5)

where $\displaystyle\lambda>0$, $\displaystyle 2\leq N$, $\displaystyle
1<q<N$, $\displaystyle a$ is a positive function such that $\displaystyle a\in
L^{\frac{N}{N-q}}(\mathbb{R}^{N})\cap L^{\infty}(\mathbb{R}^{N})$,
$\displaystyle f:\mathbb{R}\to\mathbb{R}$ is a continuous function satisfying
the growth condition

$0\leq f(t)t\leq
a_{1}|t|^{p}\phi_{N}(\alpha|t|^{\frac{N}{N-1}}),\,\,t\in\mathbb{R},$ (6)

where $\displaystyle N<p<+\infty$, and $\displaystyle\alpha>0$.

In this setting, we mean by a solution of problem $\displaystyle\eqref{P}$ any
function $\displaystyle u\in W^{1,N}(\mathbb{R}^{N})\cap
C_{loc}^{1}(\mathbb{R}^{N})$, such that

$\displaystyle\displaystyle\int_{\mathbb{R}^{N}}\left(|\nabla u|^{N-2}\nabla
u\nabla\phi+|u|^{N-2}u\phi-\lambda a(x)|u|^{q-2}u\phi-f(u)\phi\right)dx=0,$

for all $\displaystyle\phi\in C_{0}^{\infty}(\mathbb{R}^{N})$.

Besides existence, this paper is regarding explore some asymptotic properties
of the obtained solutions. If $\displaystyle u_{\lambda}$ is a solution of
(5), we are interested in investigating the following properties:

$u_{\lambda}(x)\to 0\mbox{ as }|x|\to\infty,$ (7)
$\|u_{\lambda}\|_{W^{1,N}(\mathbb{R}^{N})}\rightarrow 0\mbox{ as }\lambda\to
0^{+},$ (8)

and

$\|u_{\lambda}\|_{L^{\infty}(\mathbb{R}^{N})}\rightarrow 0\mbox{ as
}\lambda\to 0^{+}.$ (9)

Furthermore, concerning to problem (5), we scrutinize the nonexistence of
solution for $\displaystyle\lambda$ large enough.

The class of equations in (5) appears as a model for several problems in the
fields of electromagnetism, astronomy, and fluid dynamics. They can be used to
accurately describe the behavior of electric, gravitational, and fluid
potentials. Problems of the form (5) are important in several applications to
the study of evolution equations of $\displaystyle N$-laplacian type that
appear in non-Newtonian fluids, turbulent flows in porus media and other
contexts.

The case $\displaystyle N=2$ is also related to the stationary problem
associated with the following initial value Schrödinger equation

$i\partial_{t}u+\Delta u+\mu g(u)=0,\;\;\mbox{ in
}\mathbb{R}_{t}\times\mathbb{R}^{2}_{x},$ (10)

with data

$u_{0}:=u(0,\cdot)\in W^{1,2}(\mathbb{R}^{2}),$ (11)

where $\displaystyle u:=u(t,x)$ is a complex-valued function of
$\displaystyle(t,x)\in\mathbb{R}\times\mathbb{R}^{2}$, and

$g(u):=u(e^{4\pi|u|^{2}}-1).$ (12)

Schrödinger equations involving exponential nonlinearities have several
applications, such as the self-trapped beans in plasma [23]. In [7], the
author considered Schrödinger equation with decreasing exponential
nonlinearity. The stationary problem associated to (10) is given by

$\Delta\phi-\phi+g(\phi)=0,\;\;\phi\in W^{1,2}(\mathbb{R}^{2}).$ (13)

If $\displaystyle\phi\in W^{1,2}(\mathbb{R}^{2})$ is a solution to (13), then
$\displaystyle e^{it}\phi$ is a solution to (10) called a soliton or a
standing wave. Equations of type (13) arise in various other contexts of
physics such that, the classical approximation in statistical mechanics,
constructive field theory, false vacuum in cosmology, nonlinear optics, laser
propagations, etc. They are also called nonlinear Euclidean scalar field
equations, see [3, 8, 16, 18].

Recently, there has been considerable interest in the study of existence
results for problems of the form

$\left\\{\begin{array}[]{lcc}-\Delta_{N}u+V(x)|u|^{N-2}u=g(x,u)&\textup{in}&\mathbb{R}^{N},\\\
u>0&\textup{on}&\mathbb{R}^{N},\end{array}\right.$ (14)

where $\displaystyle g(x,u)$ is continuous and behaves like
$\displaystyle\phi_{N}(\alpha|u|^{\frac{N}{N-1}})$ as
$\displaystyle|u|\rightarrow+\infty$, we would like to mention [11, 12, 29, 4,
13].

In general, the potential $\displaystyle V:\mathbb{R}^{N}\to\mathbb{R}$ is
bounded away from zero, i.e.

$\displaystyle V(x)\geq c>0\,\,\,x\in\mathbb{R}^{N}.$

If $\displaystyle V$ is large at infinity in some suitable sense, then the
loss of compactness due to the unboundedness of the domain
$\displaystyle\mathbb{R}^{N}$ can be overcome and vanishing phenomena can be
ruled out. So, a natural framework for the function space setting of problem
(14) is given by the subspace $\displaystyle E$ of $\displaystyle
W^{1,N}\left(\mathbb{R}^{N}\right)$ defined as

$\displaystyle E:=\left\\{u\in
W^{1,N}\left(\mathbb{R}^{N}\right):\int_{\mathbb{R}^{N}}V(x)|u|^{N}dx<+\infty\right\\}$

endowed with the norm

$\displaystyle\|u\|_{E}:=\left(\int_{\mathbb{R}^{N}}\left(|\nabla
u|^{N}+V(x)|u|^{N}\right)dx\right)^{\frac{1}{N}}\forall u\in E.$

Under appropriate assumptions on the potential $\displaystyle V$, the
embedding

$E\hookrightarrow L^{p}\left(\mathbb{R}^{N}\right)$ (15)

turns out to be compact. For instance, if

$V^{-1}\in L^{\frac{1}{N-1}}\left(\mathbb{R}^{N}\right),$ (16)

then the embedding (15) is compact for any $\displaystyle p\geq 1$ (see e.g.
[37, Lemma 2.4]), while assuming the weaker condition

$V^{-1}\in L^{1}\left(\mathbb{R}^{N}\right)$ (17)

the embedding (15) becomes compact only for $\displaystyle p\geq N$ (see [9],
see also [14] for other compactness results).

The authors of [12, 37, 13, 24], considering a potential $\displaystyle V$
satisfying (16) or (17), obtained existence results for equations of the form
(14) and even more general equations. Their proofs rely, crucially, on the
compact embeddings (15), given by (16) and (17) (in particular, on the compact
embedding of $\displaystyle E$ into $\displaystyle
L^{N}\left(\mathbb{R}^{N}\right)$).

Most papers treat problem (14) employing variational methods. Then, usually,
it is assumed that $\displaystyle g$ has subcritical or critical growth and
sometimes asking

$g(s)\geq c|s|^{p-1},\mbox{ for each }s\geq 0\mbox{ where }c>0\mbox{ and }p>N$
(18)

are constants, see [11, 2].

Another common assumption on $\displaystyle g$ is the so-called Ambrosetti-
Rabinowitz condition, that is,

$\displaystyle\exists R>0\mbox{ and }\theta>N\mbox{ such that }0<\theta
G(x,s)\leq sg(x,s)\,\,\forall|s|\geq R\mbox{ and }x\in\mathbb{R}^{N},$

where $\displaystyle G(s)=\int_{0}^{s}g(t)dt$, see [11, 12, 29, 24].

Even when the Ambrosetti-Rabinowitz is dropped, it is usually to be assumed
some additional condition to obtain compactness of the Palais-Samle sequences
or Cerami sequences. See, for instance, [2, 24], where the authors assume,
respectively, the set of conditions

$\begin{array}[]{c}\left.\begin{array}[]{r}\exists t_{0}>0\mbox{ and
}M>0\mbox{ such that }0<G(x,s)\leq Mg(x,s),\\\ \forall|s|\geq t_{0}\mbox{ and
}x\in\Omega;\\\ 0<NG(x,s)\leq sg(x,s)\,\,\forall|s|\geq 0\mbox{ and
}x\in\Omega,\end{array}\right\\}\end{array}$ (19)

and

$\left.\begin{array}[]{r}H(x,t)\leqslant H(x,s)\mbox{ for all }0<t<s,\forall
x\in\mathbb{R}^{N},\\\ \mbox{where
}H(x,u)=uf(x,u)-NF(x,u).\end{array}\right\\}$ (20)

The case when the potential $\displaystyle V$ is constant, i.e. $\displaystyle
V(x)=c$ for any $\displaystyle x\in\mathbb{R}^{N}$, is much less developed. In
such a case, only a few existence results are known, and we should like to
mention [29] (and references therein), where the authors proved existence
results by means Variational approach. In [11], the author consider
$\displaystyle V(x)=1$ and $\displaystyle g(x,t)=\lambda
h(x)|t|^{q-2}t+f(t),\,x\in\mathbb{R}^{N}$, as in (6), with $\displaystyle f\in
C^{1}(\mathbb{R},\mathbb{R})$ satisfying the Ambrosetti-Rabinowitz condition.

In the following, we state our main results.

######  Theorem 1.1.

Suppose that $\displaystyle f:\mathbb{R}\to\mathbb{R}$ is a continuous
function satisfying assumption $\displaystyle\eqref{growth}$. Then there
exists $\displaystyle\lambda^{*}>0$ such that for every
$\displaystyle\lambda\in(0,\lambda^{*})$ problem $\displaystyle\eqref{P}$
admits a positive solution $\displaystyle u_{\lambda}\in
W^{1,N}(\mathbb{R}^{N})\cap C^{1}_{loc}(\mathbb{R}^{N})$. Furthermore,

$\displaystyle\|u_{\lambda}\|_{W^{1,N}(\mathbb{R}^{N})}\rightarrow 0,$

as $\displaystyle\lambda\rightarrow 0^{+}$.

######  Proposition 1.

Suppose that $\displaystyle f(t)=|t|^{p-1}\phi_{N}(\alpha|t|^{\frac{N}{N-1}})$
and let

$\displaystyle\lambda^{*}=\sup\\{\lambda>0;\eqref{P}\mbox{ has a solution
}u_{\lambda}\mbox{ in }W^{1,N}(\mathbb{R}^{N})\\}.$

Then $\displaystyle\lambda^{*}<\infty$.

######  Theorem 1.2.

Suppose that $\displaystyle f:\mathbb{R}\to\mathbb{R}$ is a continuous
function satisfying assumption $\displaystyle\eqref{growth}$. Then any
solution $\displaystyle u_{\lambda}$,
$\displaystyle\lambda\in(0,\lambda^{*})$, given by Theorem 1.1, satisfies
$\displaystyle u_{\lambda}(x)\to 0$ as $\displaystyle|x|\to\infty$.

######  Corollary 1.

Any solution $\displaystyle u_{\lambda}$,
$\displaystyle\lambda\in(0,\lambda^{*})$, given by Theorem 1.1, satisfies

$\displaystyle\|u_{\lambda}\|_{L^{\infty}(\mathbb{R}^{N})}\rightarrow 0,$

as $\displaystyle\lambda\rightarrow 0^{+}$.

Notice that in this paper we don not impose any extra hypotheses on
$\displaystyle f$ beyond (6). To compare our condition on the nonlinearity of
problem (5) with we have found in the literature, we present two examples that
follow.

######  Example 1.

As the first example of a function satisfying (6), we have

$f(t):=h(t)\phi_{N}(\alpha|t|^{\frac{N}{N-1}})$ (21)

where

$\displaystyle h(t)=|t|^{p-2}t\sin^{2}{(t)}.$

Now, define

$g(x,t):=\lambda a(x)t^{q-1}+h(t)\phi_{N}(\alpha|t|^{\frac{N}{N-1}}),$ (22)

where $\displaystyle a\in L^{\frac{N}{N-q}}(\mathbb{R}^{N})\cap
L^{\infty}(\mathbb{R}^{N})$ is a positive function. Notice that (22) satisfies
neither (18) nor Ambrosetti-Rabinowitz condition nor (20). First, let us
verify that $\displaystyle g$ does not satisfy (18). Let $\displaystyle c>0$
be a positive fixed constant. Taking the sequence $\displaystyle t_{k}=2k\pi$,
with $\displaystyle k$ a positive integer, $\displaystyle q<N$ and
$\displaystyle a\in L^{\infty}(\mathbb{R}^{N})$, there holds $\displaystyle
g(x,t_{k})=\lambda a(x)t_{k}^{q-1}<ct_{k}^{p-1}$, for $\displaystyle k$ large
enough.

Now, let us define

$\displaystyle\begin{array}[]{lll}H(x,t)&=&t[\lambda
a(x)t^{q-1}+|t|^{p-2}t\sin^{2}{(t)}\phi_{N}(\alpha|t|^{\frac{N}{N-1}})]\\\
&&-N\int_{0}^{t}(\lambda
a(x)s^{q-1}+|s|^{p-2}s\sin^{2}{(s)}\phi_{N}(\alpha|s|^{\frac{N}{N-1}}))ds.\end{array}$

Still considering the sequence $\displaystyle t_{k}=2k\pi$, with
$\displaystyle k$ a positive integer, $\displaystyle H$ satisfies

$\displaystyle H(x,t_{k})=\lambda
a(x)t_{k}^{q}\left(1-\frac{N}{q}\right)-N\int_{0}^{t_{k}}|s|^{p-2}s\sin^{2}{(s)}\phi_{N}(\alpha|s|^{\frac{N}{N-1}})ds<0.$

Notice that $\displaystyle H(x,t_{k})>H(x,t_{k+1})$. Therefore, (22) satisfies
neither Ambrosetti-Rabinowitz condition nor (19) nor (20).

######  Example 2.

The second example of a function satisfying (6) we would like to mention is

$f(t):=|t|^{p-2}t[(\sin{(t)})_{+}]\phi_{N}(\alpha|t|^{\frac{N}{N-1}}),\,\mbox{
where }z_{+}=\max\\{z,0\\}.$ (23)

Notice that $\displaystyle f$ is a continuous function but it is not
derivable. Indeed, since

$\displaystyle\begin{array}[]{lll}f^{\prime}_{-}(\pi)&=&\lim_{l\to
0^{-}}\frac{f(\pi+l)}{l}=\lim_{l\to
0^{-}}\frac{|\pi+l|^{p-2}(\pi+l)\sin{(\pi+l)}\phi_{N}(\alpha|\pi+l|^{\frac{N}{N-1}})}{l}\\\
&=&-\lim_{l\to
0^{-}}|\pi+l|^{p-2}(\pi+l)\frac{\sin{(l)}}{l}\phi_{N}(\alpha|\pi+l|^{\frac{N}{N-1}})\\\
&=&-\pi^{p-1}\phi_{N}(\alpha\pi^{\frac{N}{N-1}})<0,\end{array}$

and

$\displaystyle f^{\prime}_{+}(\pi)=\lim_{l\to 0^{+}}\frac{f(\pi+l)}{l}=0,$

so $\displaystyle f^{\prime}(\pi)$ does not exists. In particular,
$\displaystyle f$ does not belong to $\displaystyle
C^{1}(\mathbb{R},\mathbb{R})$, as it imposed in [11] to obtain the existence
result. Furthermore, if $\displaystyle N\geq 2$, (23) satisfies neither
Ambrosetti-Rabinowitz condition nor (20). Indeed, let us define

$\displaystyle\begin{array}[]{lll}\tilde{H}(t)&=&t(|t|^{p-2}t[(\sin{(t)})_{+}]\phi_{N}(\alpha|t|^{\frac{N}{N-1}}))\\\
&&-N\int_{0}^{t}|s|^{p-2}s[(\sin{(s)})_{+}]\phi_{N}(\alpha|s|^{\frac{N}{N-1}})ds.\end{array}$

Considering the sequence $\displaystyle t_{k}=2k\pi$, with $\displaystyle k$
being a positive integer, $\displaystyle\tilde{H}$ satisfies

$\displaystyle\tilde{H}(t_{k})=-N\int_{0}^{t_{k}}|s|^{p-2}s[(\sin{(s)})_{+}]\phi_{N}(\alpha|s|^{\frac{N}{N-1}})ds<0.$

Notice that $\displaystyle\tilde{H}(t_{k})>\tilde{H}(t_{k+1})$. Therefore,
(23) does not satisfies neither Ambrosetti-Rabinowitz condition nor (19) nor
(20).

By observing these simple examples above, we can see that our results are not
included in the previous literature. Therefore, it brings novelty in the study
of such equations in the field.

The solution of (5) is obtained as the limit of auxiliarily problems in
bounded domains. The technique combines the Galerkin method, comparison
principle of lower and upper solutions, and regularity scheme. As in [10], we
also consider a special class of normed spaces of finite dimension. However,
to clarify this approach we show Lemma 2.3, which is a result of independent
interest. Such a lemma, together Brouwer’s fixed point theorem, allows us, in
the scheme of the Galerkin method, to work in general Banach spaces of finite
dimensions with general norms.

Due to the presence of the critical term
$\displaystyle\phi_{N}(\alpha|u|^{\frac{N}{N-1}})$ and since we are in the
whole space $\displaystyle\mathbb{R}^{N}$, we had to overcome several
difficulties. One of them is a suitable improvement needed in the estimates of
the approximating functions of $\displaystyle f$ by comparing it with [10,
Lemma 2.2.] (see Lemma 3.2 bellow). Another delicate point in our approach is
the regularity needed to ensure that the solution, obtained in the limit, does
not vanish. In the bounded domain, we obtain regularity up to the boundary for
the auxiliary problems considering approximate functions in the sense of
Strauss [34]. For the domain-wide solution case, a kind of a priori estimate
in the sup norm is presented, see Proposition 4 (and the consequences of that
outcome).

Now we proceed to introduce the organization of the rest of the paper. Section
2 presents comparison principles, some preliminaries results that are useful
through the text and an important result labeled as Lemma 2.3. Section 3 is
concerning approximating functions that enable us to obtain regularity up to
the boundary to the approximating solutions. In Section 4, we study a class of
auxiliary problems in bounded domains. Section 5 is devoted to the proof of
our main results.

##  2 Preliminaries

In this section, we state some preliminaries results which will be necessary
throughout the paper.

Let $\displaystyle u\in W^{1,N}_{0}(D)$. In what follows, let us denote by
$\displaystyle\tilde{u}$ the canonical extension of $\displaystyle u$ by
$\displaystyle 0$ outside $\displaystyle D$, that is,

$\tilde{u}(x)=\left\\{\begin{array}[]{ll}u(x)&\mbox{ if }x\in D,\\\ 0&\mbox{
if }x\in\mathbb{R}^{N}\setminus D.\end{array}\right.$ (24)

It is well known that $\displaystyle u\in W_{0}^{1,N}(D)$ implies
$\displaystyle\tilde{u}\in W^{1,N}(\mathbb{R}^{N})$ (see e.g. [5, Proposition
9.18]).

The next technical lemma was proved in [1, Lemma 2.3].

######  Lemma 2.1.

Let $\displaystyle\alpha>0$ and $\displaystyle r>1$. Then, for every
$\displaystyle\beta>r$, there exists a constant $\displaystyle C:=C(\beta)$
such that

$\displaystyle\left(\phi_{N}(\alpha|u|^{\frac{N}{N-1}})\right)^{r}\leq
C\phi_{N}(\beta\alpha|u|^{\frac{N}{N-1}}),$

where $\displaystyle\phi_{N}$ is given in (4).

A complete proof of the next result can be found in [15, Lemma 3].

######  Lemma 2.2.

Let $\displaystyle 1<q<N$. For any constants $\displaystyle b>0$, the problem

$\left\\{\begin{array}[]{lll}-\Delta_{N}u+|u|^{N-2}u=b|u|^{q-2}u&\mbox{in}&D,\\\
u>0&\mbox{in}&D,\\\ u=0&\mbox{on}&\partial D,\end{array}\right.$ (25)

admits a solution $\displaystyle u_{b}\in C^{1}_{0}(\overline{D})$ satisfying
$\displaystyle\partial u_{b}/\partial\nu<0$ on $\displaystyle\partial D$.

We conclude this section by presenting a lemma, which is a consequence of
Brouwer’s Fixed Point Theorem. However, our statement is a subtle (but very
useful) generalization by comparing it with the literature. In particular,
this result allows us to work in general Banach spaces, with freedom in
choosing the norm (see the proof of Lemma 4.2). We will adopt
$\displaystyle|x|_{2}=\sqrt{\left\langle x,x\right\rangle}$ to denote the
usual euclidean norm in $\displaystyle\mathbb{R}^{d}$ and
$\displaystyle\|x\|_{d}$ to denote a general norm in
$\displaystyle\mathbb{R}^{d}$. The proof of next lemma follows some ideas like
in Kesavan [21], where the result for the particular case
$\displaystyle\|x\|_{d}:=|x|_{2}$ is presented.

######  Lemma 2.3.

Let $\displaystyle
F:(\mathbb{R}^{d},\|\cdot\|_{d})\rightarrow(\mathbb{R}^{d},\|\cdot\|_{d})$ be
a continuous function such that $\displaystyle\left\langle
F(\xi),\xi\right\rangle\geq 0$ for every $\displaystyle\xi\in\mathbb{R}^{d}$
with $\displaystyle\|\xi\|_{d}=\varrho$ for some $\displaystyle\varrho>0$, and
$\displaystyle\langle\cdot,\cdot\rangle^{1/2}=|\cdot|_{2}$. Then, there exists
$\displaystyle z_{0}$ in the closed ball
$\displaystyle\overline{B}^{d}_{\varrho}(0):=\\{z\in\mathbb{R}^{d};\|z\|_{d}\leq\varrho\\}$
such that $\displaystyle F(z_{0})=0$.

###### Proof.

Firstly, there exists $\displaystyle c(d)>0$ such that

$\|x\|_{d}\leq c(d)|x|_{2},\,\,\ \forall x\in\mathbb{R}^{d}.$ (26)

Suppose, $\displaystyle F(x)\neq 0$ for all $\displaystyle
x\in\overline{B}^{d}_{\varrho}(0)$. Define

$\displaystyle
g:(\mathbb{R}^{d},\|\cdot\|_{d})\rightarrow(\mathbb{R}^{d},\|\cdot\|_{d})$

by

$\displaystyle g(x)=-\frac{\varrho}{\|F(x)\|_{d}}F(x)$

which maps $\displaystyle\overline{B}^{d}_{\varrho}(0)$ into itself and is
continuous. Hence it has a fixed point $\displaystyle x_{0}$, by Brouwer’s
Fixed Point Theorem. Since $\displaystyle x_{0}=g(x_{0})$, we have
$\displaystyle\|x_{0}\|_{d}=\|g(x_{0})\|_{d}=\varrho>0$. But then by (26)

$\displaystyle 0<\varrho^{2}=\|x_{0}\|^{2}_{d}\leq
c(d)^{2}|x_{0}|^{2}_{2}=c(d)^{2}\left\langle
x_{0},x_{0}\right\rangle=c(d)^{2}\left\langle x_{0},g(x_{0})\right\rangle$
$\displaystyle=-c(d)^{2}\frac{\varrho}{\|F(x_{0})\|_{d}}\left\langle
x_{0},F(x_{0})\right\rangle\leq 0,$

by assumptions, which is a contradiction. ∎

###  2.1 Comparison principle

In this ection, we assume that $\displaystyle D$ is a bounded domain in
$\displaystyle\mathbb{R}^{N}$ with $\displaystyle C^{2}$ boundary
$\displaystyle\partial D$. In the following, we present a couple of comparison
principles for a subsolution and a supersolution of the problem

$\left\\{\begin{array}[]{ll}-\Delta_{N}u+|u|^{N-2}u=g(u)&\mbox{ in }D,\\\
u=0&\mbox{ on }\partial D,\end{array}\right.$ (27)

where $\displaystyle g:\mathbb{R}\rightarrow\mathbb{R}$ is a continuous
function.

We say that $\displaystyle u_{1}\in W^{1,N}({D})$ is a subsolution of problem
(27) if $\displaystyle u_{1}\leq 0$ on $\displaystyle\partial D$ and

$\displaystyle\displaystyle\int_{D}(|\nabla u_{1}|^{N-2}\nabla
u_{1}\nabla\varphi+|u_{1}|^{N-2}u_{1}\varphi)dx\leq\int_{D}g(u_{1})\varphi dx$

for all $\displaystyle\varphi\in W^{1,N}_{0}({D})$ with
$\displaystyle\varphi\geq 0$ in $\displaystyle{D}$ provided the integral
$\displaystyle\int_{D}g(u_{1})\varphi dx$ exists. We say that $\displaystyle
u_{2}\in W^{1,N}({D})$ is a supersolution of (27) if the reversed inequalities
are satisfied with $\displaystyle u_{2}$ in place of $\displaystyle u_{1}$ for
all $\displaystyle\varphi\in W^{1,N}_{0}({D})$ with $\displaystyle\varphi\geq
0$ in $\displaystyle{D}$.

The next comparison results are particular cases of the ones achieved in [15,
Theorem 3, Theorem 5].

######  Proposition 2.

Let $\displaystyle g:\mathbb{R}\rightarrow\mathbb{R}$ be a continuous function
such that $\displaystyle g(t)/t^{N-1}$ is decreasing for $\displaystyle t>0$.
Assume that $\displaystyle u_{1}$ and $\displaystyle u_{2}$ are a positive
subsolution and a positive supersolution of problem (27), respectively. If
$\displaystyle u_{2}(x)>u_{1}(x)=0$ for all $\displaystyle x\in\partial D$,
$\displaystyle u_{i}\in C^{1,\alpha}(\overline{D})$ with some
$\displaystyle\alpha\in(0,1)$, $\displaystyle\Delta_{N}u_{i}\in
L^{\infty}(D)$, for $\displaystyle i,j=1,2$, then $\displaystyle u_{2}\geq
u_{1}$ in $\displaystyle D$.

Whenever $\displaystyle u_{1}$ and $\displaystyle u_{2}$ satisfy the
homogeneous Dirichlet boundary condition we can state the following result.

######  Proposition 3.

Let $\displaystyle g:\mathbb{R}\rightarrow\mathbb{R}$ be a continuous function
such that $\displaystyle g(t)/t^{N-1}$ is decreasing for $\displaystyle t>0$.
Assume that $\displaystyle u_{1},u_{2}\in C_{0}^{1,\alpha}(\overline{D})$ with
some $\displaystyle\alpha\in(0,1)$, are a positive subsolution and a positive
supersolution of problem (27), respectively. If
$\displaystyle\Delta_{N}u_{i}\in L^{\infty}(D)$, for $\displaystyle i,j=1,2$,
$\displaystyle u_{1}/u_{2}\in L^{\infty}(D)$ and $\displaystyle u_{2}/u_{1}\in
L^{\infty}(D)$, then $\displaystyle u_{2}\geq u_{1}$ in $\displaystyle{D}$.

##  3 Approximating functions

To prove Theorem 1.1, we approximate $\displaystyle f$ by Lipschitz functions
$\displaystyle f_{k}:\mathbb{R}\to\mathbb{R}$ defined by

$f_{k}(s)=\displaystyle\left\\{\begin{array}[]{lcc}-k[G(-k-\frac{1}{k})-G(-k)],&\textup{if}&s\leq-k,\\\
-k[G(s-\frac{1}{k})-G(s)],&\textup{if}&-k\leq s\leq-\frac{1}{k},\\\
k^{2}s[G(-\frac{2}{k})-G(-\frac{1}{k})],&\textup{if}&-\frac{1}{k}\leq s\leq
0,\\\ k^{2}s[G(\frac{2}{k})-G(\frac{1}{k})],&\textup{if}&0\leq
s\leq\frac{1}{k},\\\ k[G(s+\frac{1}{k})-G(s)],&\textup{if}&\frac{1}{k}\leq
s\leq k,\\\ k[G(k+\frac{1}{k})-G(k)],&\textup{if}&s\geq k,\\\
\end{array}\right.$ (28)

where $\displaystyle G(s)=\int_{0}^{s}f(\xi)d\xi$.

The following approximation result was proved in [34].

######  Lemma 3.1.

Let $\displaystyle f:\mathbb{R}\to\mathbb{R}$ be a continuous function such
that $\displaystyle sf(s)\geq 0$ for every $\displaystyle s\in\mathbb{R}$.
Then there exists a sequence $\displaystyle f_{k}:\mathbb{R}\to\mathbb{R}$ of
continuous functions satisfying

(i) $\displaystyle sf_{k}(s)\geq 0$ for every $\displaystyle s\in\mathbb{R}$;

(ii) $\displaystyle\forall\,k\in\mathbb{N}$ $\displaystyle\exists\,c_{k}>0$
such that $\displaystyle|f_{k}(\xi)-f_{k}(\eta)|\leq c_{k}|\xi-\eta|$ for
every $\displaystyle\xi,\eta\in\mathbb{R}$;

(iii) $\displaystyle f_{k}$ converges uniformly to $\displaystyle f$ in
bounded subsets of $\displaystyle\mathbb{R}$.

The sequence $\displaystyle f_{k}$ of the previous lemma has some additional
properties presented below. Here, we present a suitable improvement in the
estimates of the approximating functions $\displaystyle f_{k}$ by comparing it
with [10, Lemma 2.2.].

######  Lemma 3.2.

Let $\displaystyle f:\mathbb{R}\to\mathbb{R}$ be a continuous function
satisfying (6) for every $\displaystyle s\in\mathbb{R}$. Then the sequence
$\displaystyle f_{k}$ of Lemma 3.1 satisfies

(i) $\displaystyle\forall\,k\in\mathbb{N}$, $\displaystyle 0\leq sf_{k}(s)\leq
C_{1}|s|^{p}\phi_{N}(2^{\frac{N}{N-1}}\alpha\,|s|^{\frac{N}{N-1}})$ for every
$\displaystyle|s|\geq\frac{1}{k}$;

(ii) $\displaystyle\forall\,k\in\mathbb{N}$, $\displaystyle 0\leq
sf_{k}(s)\leq C_{2}\frac{1}{k^{p-2}}|s|^{2}$ for every
$\displaystyle|s|\leq\frac{1}{k}$,

where $\displaystyle C_{1}$ and $\displaystyle C_{2}$ are positive constants
independent of $\displaystyle k$.

###### Proof.

Everywhere in this proof, the constant $\displaystyle a_{1}$ is the one of
(6).

First step. Suppose that $\displaystyle-k\leq s\leq-\frac{1}{k}$.

By the mean value theorem, there exists
$\displaystyle\eta\in(s-\frac{1}{k},s)$ such that

$\displaystyle
f_{k}(s)=-k[G(s-\frac{1}{k})-G(s)]=-kG^{\prime}(\eta)(s-\frac{1}{k}-s)=f(\eta)$

and

$\displaystyle sf_{k}(s)=sf(\eta).$

Since $\displaystyle s-\frac{1}{k}<\eta<s<0$ and $\displaystyle f(\eta)<0$, we
have $\displaystyle sf(\eta)\leq\eta f(\eta)$. Therefore, by using Remark 1,
we obtain

$\displaystyle\begin{array}[]{rcl}sf_{k}(s)\leq\eta
f(\eta)&\leq&a_{1}|\eta|^{p}\phi_{N}(\alpha\,|\eta|^{\frac{N}{N-1}})\\\
&\leq&a_{1}|s-\frac{1}{k}|^{p}\phi_{N}(\alpha\,|s-\frac{1}{k}|^{\frac{N}{N-1}})\\\
&\leq&a_{1}(|s|+\frac{1}{k})^{p}\phi_{N}(\alpha\,(|s|+\frac{1}{k})^{\frac{N}{N-1}})\\\
&\leq&a_{1}(2|s|)^{p}\phi_{N}(\alpha\,(2|s|)^{\frac{N}{N-1}})\\\
&=&a_{1}2^{p}|s|^{p}\phi_{N}(2^{\frac{N}{N-1}}\alpha\,|s|^{\frac{N}{N-1}}).\end{array}$

Second step. Assume $\displaystyle\frac{1}{k}\leq s\leq k$.

By the mean value theorem, there exists
$\displaystyle\eta\in(s,s+\frac{1}{k})$ such that

$\displaystyle
f_{k}(s)=k[G(s+\frac{1}{k})-G(s)]=kG^{\prime}(\eta)(s+\frac{1}{k}-s)=f(\eta)$

and

$\displaystyle sf_{k}(s)=sf(\eta).$

Since $\displaystyle 0<s<\eta<s+\frac{1}{k}$ and $\displaystyle f(\eta)>0$, we
have $\displaystyle sf(\eta)\leq\eta f(\eta)$. Therefore,

$\displaystyle\begin{array}[]{rcl}sf_{k}(s)\leq\eta
f(\eta)&\leq&a_{1}|\eta|^{p}\phi_{N}(\alpha\,|\eta|^{\frac{N}{N-1}})\\\
&\leq&a_{1}|s+\frac{1}{k}|^{p}\phi_{N}(\alpha\,|s+\frac{1}{k}|^{\frac{N}{N-1}})\\\
&\leq&a_{1}(2|s|)^{p}\phi_{N}(\alpha\,(2|s|)^{\frac{N}{N-1}})\\\
&=&a_{1}2^{p}|s|^{p}\phi_{N}(2^{\frac{N}{N-1}}\alpha\,|s|^{\frac{N}{N-1}}).\end{array}$

Third step. Suppose that $\displaystyle|s|\geq k$, then

$f_{k}(s)=\displaystyle\left\\{\begin{array}[]{lcc}-k[G(-k-\frac{1}{k})-G(-k)],&\textup{if}&s\leq-k\\\
k[G(k+\frac{1}{k})-G(k)],&\textup{if}&s\geq k.\\\ \end{array}\right.$ (29)

If $\displaystyle s\leq-k$, by the mean value theorem, there exists
$\displaystyle\eta\in(-k-\frac{1}{k},-k)$ such that

$\displaystyle
f_{k}(s)=k[G(-k-\frac{1}{k})-G(-k)]=-kG^{\prime}(\eta)(-k-\frac{1}{k}-(-k))=f(\eta)$

and

$\displaystyle sf_{k}(s)=sf(\eta).$

Since $\displaystyle-k-\frac{1}{k}<\eta<-k<0$ and $\displaystyle
k<|\eta|<k+\frac{1}{k}$, we conclude that

$\begin{array}[]{rcl}sf_{k}(s)=\frac{s}{\eta}\eta
f(\eta)&\leq&\frac{|s|}{|\eta|}a_{1}|\eta|^{p}\phi_{N}(\alpha\,|\eta|^{\frac{N}{N-1}})=a_{1}|s||\eta|^{{p}-1}\phi_{N}(\alpha\,|\eta|^{\frac{N}{N-1}})\\\
&\leq&a_{1}|s|(k+\frac{1}{k})^{{p}-1}\phi_{N}(\alpha\,(k+\frac{1}{k})^{\frac{N}{N-1}})\\\
&\leq&a_{1}|s|(|s|+\frac{1}{k})^{{p}-1}\phi_{N}(\alpha\,(|s|+\frac{1}{k})^{\frac{N}{N-1}})\\\
&\leq&a_{1}|s|(2|s|)^{{p}-1}\phi_{N}(\alpha\,(2|s|)^{\frac{N}{N-1}})\\\
&\leq&a_{1}2^{{p}}|s|^{p}\phi_{N}(2^{\frac{N}{N-1}}\alpha\,|s|^{\frac{N}{N-1}}).\end{array}$
(30)

If $\displaystyle s\geq k$, by the mean value theorem, there exists
$\displaystyle\eta\in(k,k+\frac{1}{k})$ such that

$\displaystyle
f_{k}(s)=k[G(k+\frac{1}{k})-G(k)]=kG^{\prime}(\eta)(k+\frac{1}{k}-k)=f(\eta).$

By computations similar to conclude (30) one has

$\displaystyle sf_{k}(s)=sf(\eta)=\frac{s}{\eta}\eta
f(\eta)\leq\frac{|s|}{|\eta|}a_{1}|\eta|^{p}\phi_{N}(\alpha\,|\eta|^{\frac{N}{N-1}})\leq
a_{1}2^{{p}}|s|^{p}\phi_{N}(2^{\frac{N}{N-1}}\alpha\,|s|^{\frac{N}{N-1}}).$

Fourth step. Assume $\displaystyle-\frac{1}{k}\leq s\leq\frac{1}{k}$, then

$f_{k}(s)=\displaystyle\left\\{\begin{array}[]{lcc}k^{2}s[G(-\frac{2}{k})-G(-\frac{1}{k})],&\textup{if}&-\frac{1}{k}\leq
s\leq 0\\\ k^{2}s[G(\frac{2}{k})-G(\frac{1}{k})],&\textup{if}&0\leq
s\geq\frac{1}{k}.\\\ \end{array}\right.$ (31)

If $\displaystyle-\frac{1}{k}\leq s\leq 0$, by the mean value theorem, there
exists $\displaystyle\eta\in(-\frac{2}{k},-\frac{1}{k})$ such that

$\displaystyle
f_{k}(s)=k^{2}s[G(-\frac{2}{k})-G(-\frac{1}{k})]=k^{2}sG^{\prime}(\eta)(-\frac{2}{k}-(-\frac{1}{k}))=-ksf(\eta).$

Therefore

$\begin{array}[]{lll}sf_{k}(s)&=&-ks^{2}f(\eta)=-k\frac{s^{2}}{\eta}\eta
f(\eta)\leq k\frac{s^{2}}{|\eta|}\eta f(\eta)\\\
&\leq&a_{1}k|s|^{2}|\eta|^{{p}-1}\phi_{N}(\alpha\,|\eta|^{\frac{N}{N-1}})\leq
a_{1}k|s|^{2}(\frac{2}{k})^{{p}-1}\phi_{N}(\alpha\,|\eta|^{\frac{N}{N-1}})\\\
&\leq&a_{1}2^{{p}-1}\frac{|s|^{2}}{k^{p-2}}\phi\left(\alpha\,\left(\frac{2}{k}\right)^{\frac{N}{N-1}}\right)\\\
&\leq&a_{1}2^{{p}-1}\exp(2^{\frac{N}{N-1}}\alpha)\frac{1}{{k^{p-2}}}|s|^{2}.\end{array}$
(32)

If $\displaystyle 0\leq s\leq\frac{1}{k}$, by the mean value theorem, there
exists $\displaystyle\eta\in(\frac{1}{k},\frac{2}{k})$ such that

$\displaystyle
f_{k}(s)=k^{2}s[G(\frac{2}{k})-G(\frac{1}{k})]=k^{2}sG^{\prime}(\eta)(\frac{2}{k}-\frac{1}{k})=ksf(\eta).$

By similar computations to conclude (32) one obtains

$\displaystyle sf_{k}(s)=ks^{2}f(\eta)=k\frac{s^{2}}{|\eta|}\eta f(\eta)\leq
a_{1}2^{{p}-1}\exp(2^{\frac{N}{N-1}}\alpha)\frac{1}{{k^{p-2}}}|s|^{2}.$

The proof of the lemma follows by taking $\displaystyle C_{1}=a_{1}2^{p}$ ad
$\displaystyle C_{2}=a_{1}2^{{p}-1}\exp(2^{\frac{N}{N-1}}\alpha)$, where
$\displaystyle a_{1}$ is given in (6). ∎

##  4 Solution on bounded domains

From now on, we assume that $\displaystyle D$ is a bounded domain in
$\displaystyle\mathbb{R}^{N}$ with $\displaystyle C^{2}$ boundary
$\displaystyle\partial D$. For $\displaystyle r\geq 1$, we denote by
$\displaystyle\|u\|_{L^{r}(D)}$ the usual norm on the space $\displaystyle
L^{r}(D)$. We endow $\displaystyle W_{0}^{1,N}(D)$ with the norm
$\displaystyle\|u\|_{W^{1,N}(D)}^{N}=\|\nabla
u\|_{L^{N}(D)}^{N}+\|u\|_{L^{N}(D)}^{N}$.

In this section, we focus on the existence of a positive solution for the
problem:

$\displaystyle\left\\{\begin{array}[]{lll}-\Delta_{N}u+|u|^{N-2}u=\lambda
a(x)|u|^{q-2}u+f(u)&\mbox{in}&D,\\\ u>0&\mbox{in}&D,\\\
u(x)=0&\mbox{on}&\partial D.\end{array}\right.$ $None$

Here $\displaystyle\lambda>0$ is a parameter, $\displaystyle 1<q<N$,
$\displaystyle f:[0,\infty)\rightarrow\mathbb{R}$ is a continuous function
satisfying (6).

We say that $\displaystyle u\in W_{0}^{1,N}(D)$ is a solution of
$\displaystyle(PD)$ if $\displaystyle u(x)>0$ in $\displaystyle D$ and

$\displaystyle\displaystyle\displaystyle\int_{D}|\nabla u|^{N-2}\nabla
u\nabla\phi dx+\int_{D}|u|^{N-2}u\phi dx=\lambda\int_{D}a(x)|u|^{q-2}u\phi
dx+\int_{D}f(u)\phi dx,$

for all $\displaystyle\phi\in W_{0}^{1,N}(D)$.

The existence of a solution for problem $\displaystyle(PD)$ is stated below.

######  Theorem 4.1.

Suppose that $\displaystyle f:[0,\infty)\to\mathbb{R}$ is a continuous
function satisfying (6). Then there exists $\displaystyle\lambda^{*}>0$ such
that for every $\displaystyle\lambda\in(0,\lambda^{*})$ problem
$\displaystyle(PD)$ admits a solution $\displaystyle u_{\lambda}\in
W^{1,N}_{0}(D)$ such that $\displaystyle\partial u_{\lambda}/\partial\nu<0$ on
$\displaystyle\partial D$, where $\displaystyle\nu$ stands for the outer
normal to $\displaystyle\partial D$.

###  4.1 Approximate equation

In the proof of Theorem 4.1 we utilize the following auxiliary problem:

$\displaystyle\left\\{\begin{array}[]{lll}-\Delta_{N}u+|u|^{N-2}u=\lambda
a(x)|u|^{q-2}u+f_{n}(u)+\frac{\varphi}{n}&\mbox{in}&D\\\ u>0&\mbox{in}&D\\\
u(x)=0&\mbox{on}&\partial D,\end{array}\right.$ $None$

with $\displaystyle n>0$ a integer number, $\displaystyle\varphi$ is a fixed
positive function such that $\displaystyle\varphi\in
L^{\infty}(\mathbb{R}^{N})\cap L^{N^{\prime}}(\mathbb{R}^{N})$ and
$\displaystyle f_{n}$ is given by Lemma 3.1 and Lemma 3.2.

######  Lemma 4.2.

There exists $\displaystyle\lambda^{*}>0$ and $\displaystyle
n^{*}\in\mathbb{N}$ such that $\displaystyle(PD_{n})$ has a solution
$\displaystyle u_{n}\in C^{1}_{0}(\overline{D})$ such that
$\displaystyle\partial u_{n}/\partial\nu<0$ on $\displaystyle\partial D$ for
every $\displaystyle\lambda\in(0,\lambda^{*})$ and $\displaystyle n\geq
n^{*}$. Furthermore,

$\displaystyle\|u_{n}\|_{W^{1,N}(D)}\leq\varrho,$

where $\displaystyle\varrho$ does not depend on $\displaystyle n$.

###### Proof.

Let $\displaystyle\mathcal{B}=\\{w_{1},w_{2},\dots,w_{n},\dots\\}$ be a
Schauder basis (see [17, 5] for details) for the Banach space
$\displaystyle(W_{0}^{1,N}(D),\|\cdot\|_{W^{1,N}(D)})$. For each positive
integer $\displaystyle m$, let

$\displaystyle W_{m}=[w_{1},w_{2},\dots,w_{m}]$

be the $\displaystyle m$-dimensional subspace of $\displaystyle
W^{1,N}_{0}(D)$ generated by $\displaystyle\\{w_{1},w_{2},\dots,w_{m}\\}$ with
norm induced from $\displaystyle W^{1,N}_{0}(D)$. Let
$\displaystyle\xi=(\xi_{1},\ldots,\xi_{m})\in\mathbb{R}^{m}$, notice that

$|\xi|_{m}:=\|\sum_{j=1}^{m}\xi_{j}w_{j}\|_{W^{1,N}(D)}$ (33)

defines a norm in $\displaystyle\mathbb{R}^{m}$ (see [10] for the details).

By using the above notation, we can identify the spaces
$\displaystyle(W_{m},\|\cdot\|_{W^{1,N}(D)})$ and
$\displaystyle(\mathbb{R}^{m},|\cdot|_{m})$ by the isometric linear
transformation

$u=\sum_{j=1}^{m}\xi_{j}w_{j}\in
W_{m}\mapsto\xi=(\xi_{1},\ldots,\xi_{m})\in\mathbb{R}^{m}.$ (34)

Now, define the function $\displaystyle F:\mathbb{R}^{m}\to\mathbb{R}^{m}$
such that

$\displaystyle F(\xi)=(F_{1}(\xi),F_{2}(\xi),\dots,F_{m}(\xi)),$

where $\displaystyle\xi=(\xi_{1},\xi_{2},...,\xi_{m})\in\mathbb{R}^{m}$,

$\displaystyle\begin{array}[]{lll}F_{j}(\xi)&=&\int_{D}|\nabla u|^{N-2}\nabla
u\nabla
w_{j}dx+\int_{D}|u|^{N-2}uw_{j}dx-\lambda\int_{D}a(x)(u_{+})^{q-1}w_{j}dx\\\
&&-\int_{D}f_{n}(u_{+})w_{j}dx-\frac{1}{n}\int_{D}\varphi w_{j}dx,\end{array}$

$\displaystyle j=1,2,\dots,m$, and $\displaystyle
u=\sum_{i=1}^{m}\xi_{i}w_{i}\in W_{m}$. Therefore,

$\begin{array}[]{lll}\left\langle
F(\xi),\xi\right\rangle&=&\displaystyle\int_{D}|\nabla
u|^{N}dx+\int_{D}|u|^{N}dx-\lambda\int_{D}a(x)(u_{+})^{q}dx\\\
&&-\int_{D}f_{n}(u_{+})u_{+}dx-\frac{1}{n}\int_{D}\varphi udx,\end{array}$
(35)

where $\displaystyle u_{+}=\max\\{u,0\\}$, $\displaystyle u_{-}=u_{+}-u$.

Given $\displaystyle u\in W_{m}$ we define

$\displaystyle D^{+}_{n}=\\{x\in D:|u(x)|\geq\frac{1}{n}\\}$

and

$\displaystyle D^{-}_{n}=\\{x\in D:|u(x)|<\frac{1}{n}\\}.$

Thus, we can write (35) as

$\displaystyle\left\langle F(\xi),\xi\right\rangle=\left\langle
F(\xi),\xi\right\rangle_{P}+\left\langle F(\xi),\xi\right\rangle_{N},$

where

$\displaystyle\begin{array}[]{lll}\left\langle
F(\xi),\xi\right\rangle_{P}&=&\displaystyle\int_{D^{+}_{n}}|\nabla
u|^{N}dx+\int_{D^{+}_{n}}|u|^{N}dx-\lambda\int_{D^{+}_{n}}a(x)(u_{+})^{q}dx\\\
&&-\int_{D^{+}_{n}}f_{n}(u_{+})u_{+}dx-\frac{1}{n}\int_{D^{+}_{n}}\varphi
udx\end{array}$

and

$\displaystyle\begin{array}[]{lll}\left\langle
F(\xi),\xi\right\rangle_{N}&=&\displaystyle\int_{D^{-}_{n}}|\nabla
u|^{N}dx+\int_{D^{-}_{n}}|u|^{N}dx-\lambda\int_{D^{-}_{n}}a(x)(u_{+})^{q}dx\\\
&&-\int_{D^{-}_{n}}f_{n}(u_{+})u_{+}dx-\frac{1}{n}\int_{D^{-}_{n}}\varphi
udx.\end{array}$

Step 1. In what follows, $\displaystyle C$ denotes a generic real constant.
Since the embedding $\displaystyle W^{1,N}(\mathbb{R}^{N})\subset
L^{\tau}(\mathbb{R}^{N})$ is continuous for all $\displaystyle\tau\geq N$ (see
[5, Corollary 9.11]), we have

$\int_{D^{+}_{n}}|a(x)|(u_{+})^{q}dx\leq
C\|a\|_{L^{N/(N-q)}(\mathbb{R}^{N})}\|\tilde{u}\|^{q}_{W^{1,N}(\mathbb{R}^{N})}\leq\frac{K_{1}}{2}\|\tilde{u}\|^{q}_{W^{1,N}(\mathbb{R}^{N})}.$
(36)

By virtue of Lemmas 2.1 and 3.2, we get

$\begin{array}[]{lll}\displaystyle\int_{D^{+}_{n}}f_{n}(u_{+})u_{+}dx&\leq&\displaystyle
C_{1}\int_{D^{+}_{n}}|u_{+}|^{p}\phi_{N}(2^{\frac{N}{N-1}}\alpha|u_{+}|^{\frac{N}{N-1}})dx\\\
&\leq&\left(\displaystyle\int_{\mathbb{R}^{N}}|\tilde{u}|^{Np}dx\right)^{\frac{1}{N}}\left(\int_{\mathbb{R}^{N}}(\phi_{N}(2^{\frac{N}{N-1}}\alpha|\tilde{u}|^{\frac{N}{N-1}}))^{N^{\prime}}dx\right)^{\frac{1}{N^{\prime}}}\\\
&\leq&K_{2}\|\tilde{u}\|_{W^{1,N}(\mathbb{R}^{N})}^{p}\left(\int_{\mathbb{R}^{N}}(\phi_{N}(N2^{\frac{2N}{N-1}}\alpha|\tilde{u}|^{\frac{N}{N-1}}))dx\right)^{\frac{1}{N^{\prime}}}\\\
&\leq&K_{2}C(\alpha,N)\|\tilde{u}\|_{W^{1,N}(\mathbb{R}^{N})}^{p},\end{array}$
(37)

for $\displaystyle\|\tilde{u}\|_{W^{1,N}(\mathbb{R}^{N})}$ small enough, where
$\displaystyle\tilde{u}$ is given in (24). Indeed, if

$\|\tilde{u}\|_{W^{1,N}(\mathbb{R}^{N})}\leq\frac{1}{4}\left(\frac{\alpha_{N}}{N\alpha}\right)^{(N-1)/N},$
(38)

then (3) guarantees the following estimate

$\begin{array}[]{rll}\left(\int_{\mathbb{R}^{N}}(\phi_{N}(N2^{\frac{2N}{N-1}}\alpha|\tilde{u}|^{\frac{N}{N-1}}))dx\right)^{\frac{1}{N^{\prime}}}&=&\\\
\left(\int_{\mathbb{R}^{N}}(\phi_{N}(N2^{\frac{2N}{N-1}}\alpha\|\tilde{u}\|^{{\frac{N}{N-1}}}_{W^{1,N}(\mathbb{R}^{N})}|\frac{\tilde{u}}{\|\tilde{u}\|_{W^{1,N}(\mathbb{R}^{N})}}|^{\frac{N}{N-1}}))dx\right)^{\frac{1}{N^{\prime}}}&\leq&C(\alpha,N).\end{array}$
(39)

Now, since $\displaystyle\varphi\in L^{N^{\prime}}(\mathbb{R}^{N})$, we have

$\begin{array}[]{rcl}\displaystyle\int_{D^{+}_{n}}\varphi
udx&\leq&\displaystyle\|\varphi\|_{L^{N^{\prime}}(\mathbb{R}^{N})}\|\tilde{u}\|_{L^{N}(\mathbb{R}^{N})}\\\
&\leq&\displaystyle\frac{K_{3}}{2}\|\tilde{u}\|_{W^{1,N}(\mathbb{R}^{N})}.\end{array}$
(40)

It follows from (36), (37) and (40) that

$\begin{array}[]{rcl}\left\langle
F(\xi),\xi\right\rangle_{P}&\geq&\displaystyle\int_{D^{+}_{n}}|\nabla
u|^{N}dx+\int_{D^{+}_{n}}|u|^{N}dx-\lambda\frac{K_{1}}{2}\|\tilde{u}\|^{q}_{W^{1,N}(\mathbb{R}^{N})}\\\
&&-\displaystyle
K_{2}C(\alpha,N)\|\tilde{u}\|^{p}_{W^{1,N}(\mathbb{R}^{N})}-\frac{K_{3}}{2n}\|\tilde{u}\|_{W^{1,N}(\mathbb{R}^{N})}.\end{array}$
(41)

######  Remark 2.

Notice that the constants $\displaystyle K_{1}$, $\displaystyle K_{2}$ and
$\displaystyle K_{3}$ do depend on neither $\displaystyle|D|=\int_{D}dx$ nor
$\displaystyle n$.

Step 2. In a similarly way, we obtain

$\int_{D^{-}_{n}}|a(x)|(u_{+})^{q}dx\leq\frac{K_{1}}{2}\|\tilde{u}\|^{q}_{W^{1,N}(\mathbb{R}^{N})}.$
(42)

By virtue of Lemma 3.2 (ii) we obtain

$\begin{array}[]{lll}\int_{D^{-}_{n}}f_{n}(u_{+})u_{+}dx&\leq&C_{2}\frac{1}{n^{p-2}}\int_{D^{-}_{n}}|u_{+}|^{2}dx\\\
&\leq&C_{2}\frac{1}{n^{p-2}}\left(\int_{D^{-}_{n}}dx\right)^{\frac{N-2}{N}}\left(\int_{\mathbb{R}^{N}}|\tilde{u}|^{N}dx\right)^{\frac{2}{N}}\\\
&\leq&C_{2}|D|^{(N-2)/N}\frac{1}{n^{p-2}}\|\tilde{u}\|_{W^{1,N}(\mathbb{R}^{N})}^{2}.\end{array}$
(43)

We also have

$\begin{array}[]{rcl}\displaystyle\int_{D^{-}_{n}}\varphi
udx&\leq&\displaystyle\frac{K_{3}}{2}\|\tilde{u}\|_{W^{1,N}(\mathbb{R}^{N})}.\end{array}$
(44)

It follows from (42), (43) and (44) that

$\begin{array}[]{lll}\left\langle
F(\xi),\xi\right\rangle_{N}&\geq&\displaystyle\int_{D^{-}_{n}}|\nabla
u|^{N}dx+\int_{D^{-}_{n}}|u|^{N}dx-\lambda\frac{K_{1}}{2}\|\tilde{u}\|_{W^{1,N}(\mathbb{R}^{N})}^{q}\\\
&&-\frac{C_{2}|D|^{(N-2)/N}}{n^{p-2}}\|\tilde{u}\|^{2}_{W^{1,N}(\mathbb{R}^{N})}-\frac{K_{3}}{2n}\|\tilde{u}\|_{W^{1,N}(\mathbb{R}^{N})}.\end{array}$
(45)

Using that
$\displaystyle\|u\|^{N}_{W^{1,N}(D)}=\|\tilde{u}\|^{N}_{W^{1,N}(\mathbb{R}^{N})}=\|\nabla\tilde{u}\|^{N}_{L^{N}(\mathbb{R}^{N})}+\|\tilde{u}\|^{N}_{L^{N}(\mathbb{R}^{N})}$,
inequalities (41) and (45) imply

$\begin{array}[]{rcl}\displaystyle\left\langle
F(\xi),\xi\right\rangle&\geq&\displaystyle\|\tilde{u}\|^{N}_{W^{1,N}(\mathbb{R}^{N})}-\lambda
K_{1}\|\tilde{u}\|^{q}_{W^{1,N}(\mathbb{R}^{N})}-\displaystyle
K_{2}C(\alpha,N)\|\tilde{u}\|^{p}_{W^{1,N}(\mathbb{R}^{N})}\\\
&&-\frac{C_{2}|D|^{(N-2)/N}}{n^{p-2}}\|\tilde{u}\|^{2}_{W^{1,N}(\mathbb{R}^{N})}-\frac{K_{3}}{n}\|\tilde{u}\|_{W^{1,N}(\mathbb{R}^{N})}.\end{array}$
(46)

Now, let
$\displaystyle|\xi|_{m}=\|\tilde{u}\|_{W^{1,N}(\mathbb{R}^{N})}=\varrho$ for
some $\displaystyle\varrho>0$ to be chosen later. Thus, we have

$\displaystyle\begin{array}[]{rcl}\displaystyle\left\langle
F(\xi),\xi\right\rangle&\geq&\displaystyle\varrho^{N}-\lambda
K_{1}\varrho^{q}-\displaystyle
K_{2}C(\alpha,N)\varrho^{p}-\frac{C_{2}|D|^{(N-2)/N}}{n^{p-2}}\varrho^{2}-\frac{K_{3}}{n}\varrho.\end{array}$

If $\displaystyle\varrho$ is such that

$\displaystyle\varrho\leq\frac{1}{(2K_{2}C(\alpha,N))^{\frac{1}{p-N}}},$

then

$\displaystyle\varrho^{N}-K_{2}C(\alpha,N)\varrho^{p}\geq\frac{\varrho^{N}}{2}.$

Thus, by choosing

$\varrho:=\min\left\\{\frac{1}{(2K_{2}C(\alpha,N))^{\frac{1}{p-N}}},\frac{1}{4}\left(\frac{\alpha_{N}}{N\alpha}\right)^{(N-1)/N}\right\\},$
(47)

we obtain

$\displaystyle\begin{array}[]{rcl}\displaystyle\left\langle
F(\xi),\xi\right\rangle&\geq&\displaystyle\frac{\varrho^{N}}{2}-\lambda
K_{1}\varrho^{q}-\frac{C_{2}|D|^{(N-2)/N}}{n^{p-2}}\varrho^{2}-\frac{K_{3}}{n}\varrho.\end{array}$

Now, define $\displaystyle\varsigma:=\frac{\varrho^{N}}{2}-\lambda
K_{1}\varrho^{q}$. If we choose

$\displaystyle\lambda^{*}:=\frac{\varrho^{N-q}}{4K_{1}}>0,$

then $\displaystyle\varsigma>\frac{\varrho^{N}}{4}$ for all $\displaystyle
0<\lambda<\lambda^{*}$. Now, we choose $\displaystyle n^{*}\in\mathbb{N}$ such
that

$\displaystyle\frac{C_{2}|D|^{(N-2)/N}}{n^{p-2}}\varrho^{2}+\frac{K_{3}}{n}\varrho<\frac{\varsigma}{2},$

for every $\displaystyle n\geq n^{*}$. Notice that $\displaystyle n^{*}$
depends on the domain $\displaystyle D$. Since
$\displaystyle\xi\in\mathbb{R}^{m}$ is such that
$\displaystyle|\xi|_{m}=\varrho$, then for $\displaystyle\lambda<\lambda^{*}$
and $\displaystyle n\geq n^{*}$ we obtain

$\left\langle F(\xi),\xi\right\rangle\geq\frac{\varsigma}{2}>0.$ (48)

Since $\displaystyle f_{n}$ is a Lipschitz function (for each $\displaystyle
n\in\mathbb{N}$), it easy to see that $\displaystyle
F:\mathbb{R}^{m}\rightarrow\mathbb{R}^{m}$ is a continuous function. Thus, for
each $\displaystyle\lambda<\lambda^{*}$ and $\displaystyle n>n^{*}$ fixed,
Lemma 2.3 ensure the existence of $\displaystyle y\in\mathbb{R}^{m}$ with
$\displaystyle|y|_{m}\leq\varrho$ and such that $\displaystyle F(y)=0$. In
other words, there exists $\displaystyle u_{m}\in W_{m}$ verifying

$\|u_{m}\|_{W^{1,N}(D)}\leq\varrho,$ (49)

and such that

$\begin{array}[]{l}\int_{D}|\nabla u_{m}|^{N-2}\nabla u_{m}\nabla
wdx+\int_{D}|u_{m}|^{N-2}u_{m}wdx=\\\
\lambda\int_{D}a(x)(u_{m+})^{q-1}wdx+\int_{D}f_{n}(u_{m+})wdx+\frac{1}{n}\int_{D}\varphi
wdx,\end{array}$ (50)

for all $\displaystyle w\in W_{m}$.

######  Remark 3.

It is important to mention that $\displaystyle\varrho$, given in (47), does
not depend on the domain $\displaystyle D$, $\displaystyle m$ nor
$\displaystyle n$.

Since $\displaystyle W_{m}\subset W_{0}^{1,N}(D)$
$\displaystyle\forall\,m\in\mathbb{N}$ and $\displaystyle\varrho$ does not
depend on $\displaystyle m$, then $\displaystyle(u_{m})_{m\in\mathbb{N}}$ is a
bounded sequence in $\displaystyle W_{0}^{1,N}(D)$. Therefore, for some
subsequence, there exists $\displaystyle u\in W_{0}^{1,N}(D)$ such that

$u_{m}\rightharpoonup u\,\,\,\mbox{weakly in}\,\,\,W_{0}^{1,N}(D),$ (51)
$u_{m}\to u\,\,\,\mbox{in}\,\,\,L^{s}(D)\,\,\,s\geq N,$ (52) $u_{m}\to
u,\,\,\,\mbox{a.e. in}\,\,\,D.$ (53)

Thus,

$\|u\|_{W^{1,N}(D)}\leq\liminf_{m\to\infty}\|u_{m}\|_{W^{1,N}(D)}\leq\varrho.$
(54)

We claim that

$u_{m}\rightarrow u\mbox{ in }W^{1,N}_{0}(D).$ (55)

Indeed, using the fact that
$\displaystyle\mathcal{B}=\\{w_{1},w_{2},\dots,w_{n},\dots\\}$ is a Schauder
basis of $\displaystyle W^{1,N}_{0}(D)$, for every $\displaystyle u\in
W^{1,N}_{0}(D)$ there exists a unique sequence
$\displaystyle(\alpha_{n})_{n\geq 1}$ in $\displaystyle\mathbb{R}$ such that
$\displaystyle u=\sum_{j=1}^{\infty}\alpha_{j}w_{j}$, so that

$\psi_{m}:=\sum_{j=1}^{m}\alpha_{j}w_{j}\rightarrow u\,\,\mbox{ in
}W_{0}^{1,N}(D)\,\,\mbox{ as }m\rightarrow\infty.$ (56)

Using $\displaystyle w=(u_{m}-\psi_{m})\in W_{m}$ as test function in (50), we
obtain

$\begin{array}[]{l}\int_{D}|\nabla u_{m}|^{N-2}\nabla
u_{m}\nabla(u_{m}-\psi_{m})dx+\int_{D}|u_{m}|^{N-2}u_{m}(u_{m}-\psi_{m})dx=\\\
\lambda\int_{D}a(x)(u_{m+})^{q-1}(u_{m}-\psi_{m})dx+\int_{D}f_{n}(u_{m+})(u_{m}-\psi_{m})dx\\\
+\frac{1}{n}\int_{D}\varphi(u_{m}-\psi_{m})dx.\end{array}$ (57)

From (51), (52) and (56), it is easy to see that

$\begin{array}[]{l}\int_{D}(|u_{m}|^{N-1}+|\lambda
a(x)(u_{m+})^{q-1}|+\frac{1}{n}|\varphi|)|u_{m}-\psi_{m}|dx\leq\\\
(\|u_{n}\|_{L^{N}(D)}^{N-1}+\lambda\|a\|_{L^{N/(N-q)}(D)}\|u_{n}\|^{q-1}_{L^{N}(D)}+\frac{1}{n}\|\varphi\|_{L^{N^{\prime}}(D)})\|u_{m}-\psi_{m}\|_{L^{N}(D)}.\end{array}$
(58)

By continuity of $\displaystyle f_{n}$ and (53) we obtain

$f_{n}(u_{m_{+}})^{N^{\prime}}\to f_{n}(u_{+})^{N^{\prime}}\,\,\,\mbox{ a.e.
in}\,\,\,D.$

By Lemma 3.1, (49), and by using Hölder inequality we obtain

$\begin{array}[]{lll}\displaystyle\int_{D}f_{n}(u_{m+})^{N^{\prime}}dx&\leq&\displaystyle
c_{n}^{N^{\prime}}\int_{D}|u_{m}|^{N^{\prime}}dx\\\
&\leq&c_{n}^{N^{\prime}}C|D|^{N/(N-N^{\prime})}\|u_{m}\|^{N^{\prime}}_{W^{1,N}(D)}\\\
&\leq&c_{n}^{N^{\prime}}C|D|^{N/(N-N^{\prime})}\varrho^{N^{\prime}}.\end{array}$
(59)

Hence, [19, Theorem 13.44] leads to

$f_{n}(u_{m_{+}})\to f_{n}(u_{+})\,\,\,\mbox{ weakly in
}\,\,\,L^{N^{\prime}}(D).$ (60)

Applying (52), (56) and (60), we conclude that

$\lim_{m\rightarrow\infty}\displaystyle\int_{D}f_{n}(u_{m+})(u_{m}-\psi_{m})dx=0.$
(61)

By (49) and (56), we obtain

$\lim_{m\rightarrow\infty}\displaystyle\int_{D}|\nabla u_{m}|^{N-2}\nabla
u_{m}\nabla(u-\psi_{m})dx=0.$ (62)

By (57), (58) (61) and (62), we obtain

$\lim_{m\rightarrow\infty}\displaystyle\int_{D}|\nabla u_{m}|^{N-2}\nabla
u_{m}\nabla(u_{m}-u)dx=0.$ (63)

Now it is sufficient to apply the $\displaystyle(S_{+})-$ property of
$\displaystyle-\Delta_{N}$ (see, e.g., [31, Proposition 3.5.]) to obtain (55).

Now, for every $\displaystyle m\geq k$ we obtain

$\begin{array}[]{l}\int_{D}|\nabla u_{m}|^{N-2}\nabla u_{m}\nabla
w_{k}dx+\int_{D}|u_{m}|^{N-2}u_{m}w_{k}dx=\\\
\lambda\int_{D}a(x)(u_{m+})^{q-1}w_{k}dx+\int_{D}f_{n}(u_{m+})w_{k}dx+\frac{1}{n}\int_{D}\varphi
w_{k}dx,\end{array}$ (64)

for all $\displaystyle w_{k}\in W_{k}$.

It follows from (55) and (60) that

$\begin{array}[]{l}\int_{D}|\nabla u|^{N-2}\nabla u\nabla
w_{k}dx+\int_{D}|u|^{N-2}uw_{k}dx=\\\
\lambda\int_{D}a(x)(u_{+})^{q-1}w_{k}dx+\int_{D}f_{n}(u_{+})w_{k}dx+\frac{1}{n}\int_{D}\varphi
w_{k}dx,\end{array}$ (65)

for all $\displaystyle w_{k}\in W_{k}$. Since
$\displaystyle[W_{k}]_{k\in\mathbb{N}}$ is dense in $\displaystyle
W_{0}^{1,N}(D)$ we conclude that

$\begin{array}[]{l}\int_{D}|\nabla u|^{N-2}\nabla u\nabla
wdx+\int_{D}|u|^{N-2}uwdx=\\\
\lambda\int_{D}a(x)(u_{+})^{q-1}wdx+\int_{D}f_{n}(u_{+})wdx+\frac{1}{n}\int_{D}\varphi
wdx,\end{array}$ (66)

for all $\displaystyle w\in W_{0}^{1,N}(D)$. Furthermore, $\displaystyle u\geq
0$ in $\displaystyle D$. In fact, since $\displaystyle u_{-}\in
W_{0}^{1,N}(D)$ then from (66) we obtain

$\displaystyle\begin{array}[]{lll}-\|u_{-}\|^{N}_{W_{0}^{1,N}(D)}&=&\int_{D}|\nabla
u|^{N-2}\nabla u\nabla u_{-}dx+\int_{D}|u|^{N-2}uu_{-}dx\\\
&=&\lambda\int_{D}a(x)(u_{+})^{q-1}u_{-}dx+\int_{D}f_{n}(u_{+})u_{-}dx+\frac{1}{n}\int_{D}\varphi
u_{-}dx\\\ &\geq&0.\end{array}$

Then $\displaystyle u_{-}\equiv 0$ a.e. in $\displaystyle D$, whence
$\displaystyle u\geq 0$ a.e. in $\displaystyle D$. Moreover, $\displaystyle
u\not\equiv 0$ is valid due to $\displaystyle\frac{\varphi}{n}>0$ in
$\displaystyle D$. By applying the strong maximum principle in [32, Theorem
5.4.1] we obtain $\displaystyle u>0$ in $\displaystyle D$, and [32, Theorem
5.5.1] ensure that $\displaystyle\partial u/\partial\nu<0$ on
$\displaystyle\partial D$ holds. By Lemma 3.1 and [22, Theorem 7.1] we
conclude that $\displaystyle u\in L^{\infty}(D)$. Thus, [26, Theorem 1] and
[27, p. 320] ensure the regularity up to the boundary $\displaystyle u\in
C^{1,\beta}(\overline{D})$, for some $\displaystyle\beta\in(0,1)$.

Therefore, we conclude that proof of the lemma by taking $\displaystyle
u_{n}=u$.

∎

###  4.2  Proof of Theorem 4.1

First we show that $\displaystyle(PD)$ has a positive solution. For each
$\displaystyle n\in\mathbb{N}$, $\displaystyle n>n^{*}$, by Lemma 4.2,
equation $\displaystyle(PD_{n})$ has a solution $\displaystyle u_{n}\in
W^{1,N}_{0}(D)\cap C^{1,\beta}(\overline{D})$. Thus

$\begin{array}[]{l}\displaystyle\int_{D}|\nabla u_{n}|^{N-2}\nabla u_{n}\nabla
wdx+\int_{D}|u_{n}|^{N-2}u_{n}wdx=\\\
\displaystyle\lambda\int_{D}a(x)u_{n}^{q}wdx+\int_{D}f_{n}(u_{n})wdx+\frac{1}{n}\int_{D}\varphi
wdx,\end{array}$ (67)

for all $\displaystyle w\in W_{0}^{1,N}(D)$.

By (54) we have that

$\|u_{n}\|_{W^{1,N}(D)}\leq\varrho,\,\,\forall\,n\in\mathbb{N},$ (68)

and $\displaystyle\varrho$ does not depend on $\displaystyle n$. Thus, along a
subsequence again relabeled as $\displaystyle u_{n}$, there exists
$\displaystyle u\in W^{1,N}_{0}(D)$ such that

$u_{n}\rightharpoonup u\,\,\,\mbox{weakly
in}\,\,\,W^{1,N}_{0}(D),\,\,\mbox{as}\,\,n\rightarrow\infty.$ (69)

Thus,

$\|u\|_{W^{1,N}(D)}\leq\liminf_{n\to\infty}\|u_{n}\|_{W^{1,N}(D)}\leq\varrho.$
(70)

We claim that

$u_{n}\rightarrow u\,\,\,W^{1,N}_{0}(D),\,\,\mbox{as}\,\,n\rightarrow\infty.$
(71)

In fact, the proof of (71) follows in a similarly way as we did in the
previous section.

First, notice that Lemma 3.1 and (68) imply

$\begin{array}[]{lll}\displaystyle\int_{D}f_{n}(u_{n})^{N^{\prime}}dx&\leq&\displaystyle
c_{n}^{N^{\prime}}\int_{D}|u_{n}|^{N^{\prime}}dx\leq
c_{n}^{N^{\prime}}C\varrho^{N^{\prime}}.\end{array}$ (72)

Moreover,

$\displaystyle u_{n}\rightarrow u\,\,\mbox{a.e. in}\,\,D,$

and by the uniform convergence of Lemma 3.1 ($\displaystyle iii$) we have

$f_{n}(u_{n}(\cdot))\rightarrow f(u(\cdot))\,\,\mbox{a.e. in}\,\,D.$ (73)

Hence, [19, Theorem 13.44] leads to

$f_{n}(u_{n})\to f(u)\,\,\,\mbox{ weakly in }\,\,\,L^{N^{\prime}}(D).$ (74)

On the other hand, taking $\displaystyle w=(u_{n}-u)$ as a test function in
(67), we get

$\begin{array}[]{ll}&\displaystyle\int_{D}|\nabla u_{n}|^{N-2}\nabla
u_{n}\nabla(u_{n}-u)dx\\\
\leq&-\int_{D}|u_{n}|^{N-2}u_{n}(u_{n}-u)dx+\lambda\int_{D}a(x)u_{n}^{q-1}(u_{n}-u)dx\displaystyle\\\
&+\int_{D}f_{n}(u_{n})(u_{n}-u)dx+\frac{1}{n}\int_{D}\varphi(u_{n}-u)dx\\\
\leq&\|u_{n}\|^{N-1}_{L^{N}(D)}\|u_{n}-u\|_{L^{N}(D)}\\\
&+\lambda\|a\|_{L^{N/(N-q)}(D)}\|u_{n}\|^{q-1}_{L^{N}(D)}\|u_{n}-u\|_{L^{N}(D)}\\\
&+\|f_{n}(u_{n})\|_{L^{N^{\prime}}(D)}\|u_{n}-u\|_{L^{N}(D)}\\\
&+\frac{1}{n}\|\varphi\|_{L^{N^{\prime}}(D)}\|u_{n}-u\|_{L^{N}(D)}\rightarrow
0\,\,\mbox{ as }n\rightarrow\infty.\end{array}$ (75)

And then, $\displaystyle\limsup_{n\rightarrow\infty}\int_{D}|\nabla
u_{n}|^{N-2}\nabla u_{n}\nabla(u_{n}-u)dx\leq 0$. Therefore, (71) follows by
$\displaystyle(S_{+})$ property.

By using (71) and (74), we can pass to the limit in (67) to obtain

$\begin{array}[]{l}\displaystyle\int_{D}|\nabla u|^{N-2}\nabla u\nabla
wdx+\int_{D}|u|^{N-2}uwdx=\displaystyle\lambda\int_{D}a(x)u^{q-1}wdx+\int_{D}f(u)wdx,\end{array}$
(76)

for all $\displaystyle w\in W_{0}^{1,N}(D)$. Thus, $\displaystyle u$ is a
solution of $\displaystyle(PD)$.

Now, fix a positive constant $\displaystyle\lambda$ such that

$\lambda<\lambda^{*}=\frac{\varrho^{N-q}}{4K_{1}}.$ (77)

Since $\displaystyle a>0$ is a continuous function, define

$\displaystyle a_{D}=\inf_{D}a(x).$

Then, according to Lemma 2.2, there exists a positive solution $\displaystyle
u_{0}$ of

$\displaystyle\left\\{\begin{array}[]{lll}-\Delta_{N}u+|u|^{N-2}u=\lambda
a_{D}u^{q-1}&\mbox{in}&D,\\\ u>0&\mbox{in}&D,\\\ u=0&\mbox{on}&\partial
D.\end{array}\right.$

Let $\displaystyle u_{n}$ be a positive solution of problem
$\displaystyle(PD_{n})$ obtained by Lemma 4.2. We observe that $\displaystyle
u_{0}/u_{n},u_{n}/u_{0}\in L^{\infty}(D)$ because $\displaystyle u_{0}$ and
$\displaystyle u_{n}$ are positive functions belonging to $\displaystyle
C^{1,\beta}_{0}(\overline{D})$ and satisfying $\displaystyle\partial
u_{n}/\partial\nu<0$, $\displaystyle\partial u_{0}/\partial\nu<0$ on
$\displaystyle\partial D$. Notice that

$\lambda a(x)t^{q-1}+f_{n}(t)+\frac{\varphi}{n}\geq\lambda a_{D}t^{q-1}=g(t).$

Hence, $\displaystyle u_{1}=u_{0}$ and $\displaystyle u_{2}=u_{n}$ are a
positive subsolution and a positive supersolution of problem (25),
respectively. Thus, by Proposition 3 we see that $\displaystyle u_{n}\geq
u_{0}$ in $\displaystyle D$ for every $\displaystyle n>n^{*}$. Therefore, by
passing to the limit we obtain

$\displaystyle u\geq u_{0}\mbox{ a.e. in }D.$

Thus, we conclude that $\displaystyle u$ is a positive solution of problem
$\displaystyle(PD)$.

From now, the solution we just found will be labeled as $\displaystyle
u_{\lambda}$ with explicit dependence on $\displaystyle\lambda$. In what
follows, we will deduce that $\displaystyle\|u_{\lambda}\|_{W^{1,N}(D)}\to 0$
as $\displaystyle\lambda\to 0$. Fix the pair
$\displaystyle(\lambda,u_{\lambda})$, where $\displaystyle\lambda$
$\displaystyle\in(0,\lambda^{*})$ and $\displaystyle u_{\lambda}$ is the
corresponding solution of problem $\displaystyle(PD)$. By using $\displaystyle
w=u_{\lambda}$ as a test function in (76), we obtain

$\begin{array}[]{lll}\displaystyle\int_{\mathbb{R}^{N}}|\nabla\tilde{u}_{\lambda}|^{N}dx+\int_{\mathbb{R}^{N}}\tilde{u}_{\lambda}^{N}dx&=&\displaystyle\int_{D}|\nabla
u_{\lambda}|^{N}dx+\int_{D}u_{\lambda}^{N}dx\\\
&=&\displaystyle\lambda\int_{D}a(x)u_{\lambda}^{q}dx+\int_{D}f(u_{\lambda})u_{\lambda}dx\\\
&=&\displaystyle\lambda\int_{\mathbb{R}^{N}}a(x)\tilde{u}_{\lambda}^{q}dx+\int_{\mathbb{R}^{N}}f(\tilde{u}_{\lambda})\tilde{u}_{\lambda}dx\\\
&\leq&\lambda
K_{1}\|\tilde{u}_{\lambda}\|^{q}_{W^{1,N}(\mathbb{R}^{N})}+K_{2}C(\alpha,N)\|\tilde{u}_{\lambda}\|^{p}_{W^{1,N}(\mathbb{R}^{N})},\end{array}$
(78)

where $\displaystyle K_{1}$, $\displaystyle K_{2}$ are given in (36), (37),
respectively.

Since $\displaystyle\tilde{u}_{\lambda}\neq 0$, from (78), we have the
following estimate

$\begin{array}[]{lll}\|\tilde{u}_{\lambda}\|^{N-q}_{W^{1,N}(\mathbb{R}^{N})}(1-K_{2}C(\alpha,N)\|\tilde{u}_{\lambda}\|^{p-N}_{W^{1,N}(\mathbb{R}^{N})})&\leq&\lambda
K_{1}.\end{array}$ (79)

By combining (47) and (68), we obtain

$\displaystyle\|\tilde{u}_{\lambda}\|^{p-N}_{W^{1,N}(\mathbb{R}^{N})}\leq\frac{1}{2K_{2}C(\alpha,N)}.$

Thus,

$\begin{array}[]{lll}\|u_{\lambda}\|_{W^{1,N}(D)}=\|\tilde{u}_{\lambda}\|_{W^{1,N}(\mathbb{R}^{N})}&\leq&(2\lambda
K_{1})^{1/(N-q)}\to 0\quad as\quad\lambda\to 0.\end{array}$ (80)

Thus, the proof of the theorem is complete.

##  5 Proof of the Main Theorem

###  5.1 A priori estimates

In this subsection, for convenience, when necessary, we will omit the notation
$\displaystyle\tilde{u}$. In order to prove Theorem 1.1 and 1.2, it will be
needed a couple of estimates proven in the following results. It is important
to mention that much less is known about the results of regularity for the
$\displaystyle L_{p}$ operator, as can be seen in [20]. This becomes an
obstacle to obtain uniform estimates in the sense of Hölder norm.

Fix $\displaystyle u\in W^{1,N}(D)$ any positive solution of
$\displaystyle(PD)$ given by Theorem 4.1. Here, we will borrow some ideas from
[15]. Define $\displaystyle u_{M}:=\min\\{u,M\\}$ for $\displaystyle M>0$.
Choose $\displaystyle\overline{p}^{*}$ satisfying $\displaystyle
2N^{2}<\overline{p}^{*}$. For $\displaystyle R^{\prime}>R>0$, we take a smooth
function $\displaystyle\eta_{R,R^{\prime}}$ such that $\displaystyle
0\leq\eta_{R,R^{\prime}}\leq 1$,
$\displaystyle\|\eta_{R,R^{\prime}}^{\prime}\|_{\infty}\leq 2/(R^{\prime}-R)$,
$\displaystyle\eta_{R,R^{\prime}}(t)=1$ if $\displaystyle t\leq R$ and
$\displaystyle\eta_{R,R^{\prime}}=0$ if $\displaystyle t\geq R^{\prime}$.

######  Lemma 5.1.

Let $\displaystyle x_{0}\in\mathbb{R}^{N}$, $\displaystyle M>0$,
$\displaystyle R^{\prime}>R>0$, such that $\displaystyle
B(x_{0},R^{\prime})\subset D$, $\displaystyle\gamma_{1}=\frac{N}{N-q}>1$ and
$\displaystyle\gamma_{1}^{\prime}=\frac{N}{q}$ such that $\displaystyle
1/\gamma_{1}+1/\gamma_{1}^{\prime}=1$. Denote
$\displaystyle\eta(x):=\eta_{R,R^{\prime}}(|x-x_{0}|)$. Assume that
$\displaystyle 2N\leq\tilde{p}$ (in particular
$\displaystyle\gamma_{1}^{\prime}\leq\tilde{p}$) and $\displaystyle u\in
W^{1,N}(D)$ a solution of ($\displaystyle PD$), in particular, $\displaystyle
u\in L^{\tilde{p}(N+\beta)}(B(x_{0},R^{\prime}))$ with $\displaystyle\beta\geq
0$. Then it holds:

$\displaystyle\displaystyle\int_{B(x_{0},R^{\prime})}f(u)uu_{M}^{\beta}\eta^{N}dx\,\leq
C(\varrho)\|u\|^{\beta}_{L^{\tilde{p}(N+\beta)}(B(x_{0},R^{\prime}))}B_{R^{\prime}}$
(82)
$\displaystyle\displaystyle\int_{B(x_{0},R^{\prime})}a(x)u^{q}u_{M}^{\beta}\eta^{N}dx\,\leq\|a\|_{L^{\gamma_{1}}(B(x_{0},R^{\prime}))}\|u\|_{L^{\tilde{p}(N+\beta)}(B(x_{0},R^{\prime}))}^{q+\beta}B_{R^{\prime}}$

where $\displaystyle B_{R^{\prime}}:=(1+|B(0,R^{\prime})|)$ and
$\displaystyle|B(0,R^{\prime})|$ denotes the Lebesgue measure of the ball
$\displaystyle B(0,R^{\prime})$.

###### Proof.

According to Hölder’s inequality, we easily show (82). So, we will prove only
(82). By Young’s inequality, Lemmas 2.1, 3.2, and inequality (38), we obtain

$\displaystyle\begin{array}[]{l}\int_{B(x_{0},R^{\prime})}f(u)uu_{M}^{\beta}\eta^{N}dx\\\
\leq
C\int_{B(x_{0},R^{\prime})}|u|^{p}u_{M}^{\beta}\phi_{N}(\alpha|u|^{\frac{N}{N-1}})dx\\\
\leq
C\left(\int_{B(x_{0},R^{\prime})}|u|^{Np}u_{M}^{N\beta}dx\right)^{1/N}\left(\int_{B(x_{0},R^{\prime})}[\phi_{N}(\alpha|u|^{\frac{N}{N-1}})]^{N^{\prime}}dx\right)^{1/N^{\prime}}\\\
\leq
C\left(\int_{B(x_{0},R^{\prime})}|u|^{2Np}dx\right)^{1/2N}\left(\int_{B(x_{0},R^{\prime})}u_{M}^{2N\beta}dx\right)^{1/2N}C(\alpha,N)\\\
\leq
C\|u\|^{p}_{L^{2Np}(B(x_{0},R^{\prime}))}\|u\|^{\beta}_{L^{2N\beta}(B(x_{0},R^{\prime}))}\\\
\leq C(\varrho)\|u\|^{\beta}_{L^{2N\beta}(B(x_{0},R^{\prime}))}\,,\end{array}$

because $\displaystyle\|u\|_{L^{4p}(B(x_{0},R^{\prime}))}\leq
C\|\tilde{u}\|_{W^{1,N}(\mathbb{R}^{N})}\leq C\varrho$.

Since $\displaystyle\tilde{p}(N+\beta)>2N\beta$, by Hölder’s inequality we
obtain

$\displaystyle\left(\int_{B(x_{0},R^{\prime})}u^{2N\beta}dx\right)^{1/2N}\leq\|u\|^{\beta}_{L^{\tilde{p}(N+\beta)}(B(x_{0},R^{\prime}))}|B(x_{0},R^{\prime})|^{\frac{\tilde{p}(N+\beta)-2N\beta}{2N\tilde{p}(N+\beta)}},$

that conclude the inequality (82). ∎

######  Lemma 5.2.

Let $\displaystyle x_{0}\in\mathbb{R}^{N}$, $\displaystyle R^{\prime}>R>0$,
such that $\displaystyle B(x_{0},R^{\prime})\subset D$,
$\displaystyle\gamma_{1}=\frac{N}{N-q}>1$ and
$\displaystyle\gamma_{1}^{\prime}=\frac{N}{q}$. Assume that $\displaystyle
2N\leq\tilde{p}$. As $\displaystyle u\in
L^{\tilde{p}(N+\beta)}(B(x_{0},R^{\prime}))$ with $\displaystyle\beta\geq 0$,
then

$\displaystyle\displaystyle\|u\|_{L^{\frac{\overline{p}^{*}}{N}(N+\beta)}(B(x_{0},R))}^{N+\beta}$
$\displaystyle\displaystyle\leq
2^{N}(N+\beta)^{N}C_{*}^{N}B_{R^{\prime}}(C_{R^{\prime}}+D_{R,R^{\prime}})\max\\{1,\|u\|_{L^{\tilde{p}(N+\beta)}(B(x_{0},R^{\prime}))}\\}^{N+\beta}$

holds with

$\displaystyle\begin{array}[]{l}B_{R^{\prime}}:=1+|B(0,R^{\prime})|,\\\
C_{R^{\prime}}:=C(\varrho)+\lambda^{*}\|a\|_{L^{\gamma_{1}}(B(x_{0},R^{\prime}))},\\\
D_{R,R^{\prime}}:=\frac{N^{N}2^{2N-1}+2^{N-1}}{(R^{\prime}-R)^{N}},\end{array}$

where $\displaystyle C_{*}$ is the positive constant embedding from
$\displaystyle W^{1,N}(\mathbb{R}^{N})$ to $\displaystyle
L^{\overline{p}^{*}}(\mathbb{R}^{N})$.

###### Proof.

Taking $\displaystyle uu_{M}^{\beta}\eta^{N}\in
W^{1,N}_{0}(B(x_{0},R^{\prime}))$ (for $\displaystyle M>0$) as test function
in (76), where $\displaystyle\eta(x)=\eta_{R,R^{\prime}}(|x-x_{0}|)$, and by
Lemma 5.1, we obtain

$\displaystyle\displaystyle
C(\varrho)\|u\|^{\beta}_{L^{\tilde{p}(N+\beta)}(B(x_{0},R^{\prime}))}B_{R^{\prime}}$
$\displaystyle\displaystyle+\lambda\|a\|_{L^{\gamma_{1}}(B(x_{0},R^{\prime}))}\|u\|_{L^{\tilde{p}(N+\beta)}(B(x_{0},R^{\prime}))}^{q+\beta}B_{R^{\prime}}$
$\displaystyle\displaystyle\geq$
$\displaystyle\displaystyle\int_{B(x_{0},R^{\prime})}|\nabla
u|^{N}u_{M}^{\beta}\eta^{N}dx\,+\int_{B(x_{0},R^{\prime})}u_{M}^{N+\beta}\eta^{N}dx\,$
$\displaystyle\displaystyle-\frac{2N}{R^{\prime}-R}\int_{B(x_{0},R^{\prime})}|\nabla
u|^{N-1}u_{M}^{\beta}u\eta^{N-1}dx,$

where we use $\displaystyle|\nabla\eta|\leq 2/(R^{\prime}-R)$. From Young’s
inequality and Hölder’s inequality, we obtain

$\displaystyle\displaystyle\frac{2N}{R^{\prime}-R}\int_{B(x_{0},R^{\prime})}|\nabla
u|^{N-1}u_{M}^{\beta}u\eta^{N-1}dx\,$ (85) $\displaystyle\displaystyle\leq$
$\displaystyle\displaystyle\frac{1}{2}\int_{B(x_{0},R^{\prime})}|\nabla
u|^{N}u_{M}^{\beta}\eta^{N}dx\,+\frac{2^{N}N^{N}2^{N-1}}{(R^{\prime}-R)^{N}}\int_{B(x_{0},R^{\prime})}u^{N+\beta}dx\,$
$\displaystyle\displaystyle\leq$
$\displaystyle\displaystyle\frac{1}{2}\int_{B(x_{0},R^{\prime})}|\nabla
u|^{N}u_{M}^{\beta}\eta^{N}dx\,+\frac{N^{N}2^{2N-1}}{(R^{\prime}-R)^{N}}\|u\|_{L^{\tilde{p}(N+\beta)}(B(x_{0},R^{\prime}))}^{N+\beta}B_{R^{\prime}}.$

Thus, from (5.1) and (85) we have

$\displaystyle\displaystyle
B_{R^{\prime}}\left(C_{R^{\prime}}+\frac{N^{N}2^{2N-1}}{(R^{\prime}-R)^{N}}\right)\max\\{1,\|u\|_{L^{\tilde{p}(N+\beta)}(B(x_{0},R^{\prime}))}\\}^{N+\beta}$
(86) $\displaystyle\displaystyle\geq$
$\displaystyle\displaystyle\frac{1}{2}\int_{B(x_{0},R^{\prime})}|\nabla
u|^{N}u_{M}^{\beta}\eta^{N}dx\,+\int_{B(x_{0},R^{\prime})}u_{M}^{N+\beta}\eta^{N}dx\,.$

Moreover, by using

$\displaystyle\displaystyle\|\nabla(u_{M}^{1+\beta/N}\eta)\|_{L^{N}(\mathbb{R}^{N})}^{N}\leq
2^{N-1}\left\\{\|\eta\nabla(u_{M}^{1+\beta/N})\|_{L^{N}(\mathbb{R}^{N})}^{N}+\|u_{M}^{1+\beta/N}\nabla\eta\|_{L^{N}(\mathbb{R}^{N})}^{N}\right\\}$
$\displaystyle\displaystyle\leq$ $\displaystyle\displaystyle
2^{N-1}\left(1+\frac{\beta}{N}\right)^{N}\int_{B(x_{0},R^{\prime})}|\nabla
u|^{N}u_{M}^{\beta}\eta^{N}dx\,+\frac{2^{2N-1}}{(R^{\prime}-R)^{N}}\int_{B(x_{0},R^{\prime})}u_{M}^{N+\beta}dx\,$

and Hölder’s inequality, due to the embedding from $\displaystyle
W^{1,N}(\mathbb{R}^{N})$ to $\displaystyle
L^{\overline{p}^{*}}(\mathbb{R}^{N})$, we have

$\begin{array}[]{l}\displaystyle\frac{1}{2}\int_{B(x_{0},R^{\prime})}|\nabla
u|^{N}u_{M}^{\beta}\eta^{N}dx\,+\int_{B(x_{0},R^{\prime})}u_{M}^{N+\beta}\eta^{N}dx\\\
\\\ \geq
2^{-N}N^{N}(N+\beta)^{-N}\left\\{\|\nabla(u_{M}^{1+\beta/N}\eta)\|_{L^{N}(\mathbb{R}^{N})}^{N}+\|u_{M}^{1+\beta/N}\eta\|_{L^{N}(\mathbb{R}^{N})}^{N}\right\\}\\\
\\\
\qquad-\frac{2^{N-1}N^{N}}{(N+\beta)^{N}(R^{\prime}-R)^{N}}\int_{B(x_{0},R^{\prime})}u_{M}^{N+\beta}dx\\\
\\\ \geq
2^{-N}N^{N}(N+\beta)^{-N}\|u_{M}^{1+\beta/N}\eta\|_{W^{1,N}(\mathbb{R}^{N})}^{N}\\\
\\\
\qquad-\frac{2^{N-1}}{(R^{\prime}-R)^{N}}\|u\|_{L^{\tilde{p}(N+\beta)}(B(x_{0},R^{\prime}))}^{N+\beta}(1+|B(0,R^{\prime})|)\\\
\\\ \geq
2^{-N}N^{N}(N+\beta)^{-N}C_{*}^{-N}\|u_{M}^{1+\beta/N}\eta\|_{L^{\overline{p}^{*}}(\mathbb{R}^{N})}^{N}\\\
\qquad-\frac{2^{N-1}}{(R^{\prime}-R)^{N}}\|u\|_{L^{\tilde{p}(N+\beta)}(B(x_{0},R^{\prime}))}^{N+\beta}B_{R^{\prime}}\\\
\\\ \geq
2^{-N}N^{N}(N+\beta)^{-N}C_{*}^{-N}\|u_{M}\|_{L^{\overline{p}^{*}(N+\beta)/N}(B(x_{0},R))}^{N+\beta}\\\
\\\
\qquad-\frac{2^{N-1}}{(R^{\prime}-R)^{N}}\|u\|_{L^{\tilde{p}(N+\beta)}(B(x_{0},R^{\prime}))}^{N+\beta}B_{R^{\prime}}.\end{array}$
(87)

Consequently, it follows from (86) and (87) that

$\displaystyle\displaystyle
2^{-N}N^{N}(N+\beta)^{-N}C_{*}^{-N}\|u_{M}\|_{L^{\overline{p}^{*}(N+\beta)/N}(B(x_{0},R))}^{N+\beta}$
(88) $\displaystyle\displaystyle\leq$ $\displaystyle\displaystyle
B_{R^{\prime}}(C_{R^{\prime}}+D_{R,R^{\prime}})\max\\{1,\|u\|_{L^{\tilde{p}(N+\beta)}(B(x_{0},R^{\prime}))}\\}^{N+\beta}.$

The conclusion follows by applying Fatou’s lemma and letting $\displaystyle
M\to\infty$ in (88). ∎

######  Proposition 4.

Assume the assumptions of Lemma 5.2. Let us suppose $\displaystyle x_{0}\in D$
and $\displaystyle a$ a positive function such that $\displaystyle a\in
L^{\frac{N}{N-q}}(\mathbb{R}^{N})$, $\displaystyle f$ satisfying (6),
$\displaystyle R_{*}>0$ such that $\displaystyle B(x_{0},2R_{*})\subset D$. If
$\displaystyle u\in W^{1,N}(D)$ is a solution of ($\displaystyle PD$), then
$\displaystyle u\in L^{\infty}(D)$. Furthermore,
$\displaystyle\|u\|_{W^{1,N}(D)}\leq C$ implies
$\displaystyle\|u\|_{L^{\infty}(D)}\leq\tilde{C}$.

###### Proof.

Since $\displaystyle 2N<\overline{p}^{*}/N$, we can choose
$\displaystyle\tilde{p}$ such that

$\displaystyle 2N<\tilde{p}<\frac{\overline{p}^{*}}{N}.$

Let $\displaystyle R_{*}>0$ satisfying $\displaystyle B(x_{0},2R_{*})\subset
D$. Put

$\displaystyle A:=\lambda^{*}\|a\|_{L^{\gamma_{1}}(\mathbb{R}^{N})}.$

Define sequences $\displaystyle\\{\beta_{n}\\}$,
$\displaystyle\\{R_{n}^{\prime}\\}$ and $\displaystyle\\{R_{n}\\}$ by

$\displaystyle\displaystyle\beta_{0}:=\frac{\overline{p}^{*}}{\tilde{p}}-N>0,\quad\tilde{p}(N+\beta_{n+1})=\frac{\overline{p}^{*}}{N}(N+\beta_{n}),$
$\displaystyle\displaystyle R_{n}^{\prime}:=(1+2^{-n})R_{*},\quad
R_{n}:=R_{n+1}^{\prime}.$

Since $\displaystyle u\in W^{1,N}(D)$, by using the embedding of
$\displaystyle W^{1,N}(D)$ to $\displaystyle L^{\overline{p}^{*}}(D)$, we see
that $\displaystyle u\in
L^{\overline{p}^{*}}(D)=L^{\tilde{p}(N+\beta_{0})}(D)$.

Let $\displaystyle x_{0}\in D$ fixed. Lemma 5.2 guarantees that if
$\displaystyle u\in L^{\tilde{p}(N+\beta_{n})}(B(x_{0},R_{n}^{\prime}))$, then
$\displaystyle u\in
L^{\frac{\overline{p}^{*}}{N}(N+\beta_{n})}(B(x_{0},R_{n}))=L^{\tilde{p}(N+\beta_{n+1})}(B(x_{0},R_{n+1}^{\prime}))$.
Notice that

$\displaystyle\begin{array}[]{ll}B_{R^{\prime}_{n}}&\leq(1+|B(0,2R_{*})|)=:B_{0},\\\
C_{R_{n}^{\prime}}&\leq C(\varrho)+A+1=:C_{0},\\\
D_{R_{n},R_{n}^{\prime}}&=\frac{N^{N}2^{2N-1}+2^{N-1}}{R_{*}^{N}}2^{N(n+1)}=C^{\prime}2^{N(n+1)}=:D_{n}\end{array}$

for any $\displaystyle n\geq 0$ with $\displaystyle C^{\prime}$ independent of
$\displaystyle n$. By setting

$\displaystyle
b_{n}:=\max\\{1,\|u\|_{L^{\tilde{p}(N+\beta_{n})(B(x_{0},R_{n}^{\prime}))}}\\},$

and by Lemma 5.2 we obtain

$b_{n+1}\leq
C^{\frac{1}{N+\beta_{n}}}(N+\beta_{n})^{\frac{N}{N+\beta_{n}}}(C_{0}+D_{n})^{\frac{1}{N+\beta_{n}}}b_{n}$
(89)

for every $\displaystyle n\geq 0$ with $\displaystyle\
C:=2^{N}(C_{*}+1)^{N}B_{0}$. Put $\displaystyle
P:=\tilde{p}N/\overline{p}^{*}<1$. Then, $\displaystyle
N+\beta_{n+1}=(N+\beta_{n})/P$,
$\displaystyle\beta_{n+1}>\beta_{n}/P>\beta_{0}(1/P)^{n+1}\to\infty$ as
$\displaystyle n\to\infty$. Moreover, we see that

$\displaystyle\displaystyle S_{1}$
$\displaystyle\displaystyle:=\sum_{n=0}^{\infty}\frac{1}{N+\beta_{n}}=\frac{1}{N+\beta_{0}}\sum_{n=0}^{\infty}P^{n}=\frac{1}{(N+\beta_{0})(1-P)}<\infty,$
$\displaystyle\displaystyle S_{2}$
$\displaystyle\displaystyle:=\mbox{ln}\prod_{n=0}^{\infty}(N+\beta_{n})^{\frac{N}{N+\beta_{n}}}=\frac{N}{N+\beta_{0}}\sum_{n=0}^{\infty}P^{n}\left(\mbox{ln}(N+\beta_{0})+n\mbox{ln}P^{-1}\right)<\infty$

and

$\displaystyle\displaystyle S_{3}$
$\displaystyle\displaystyle:=\mbox{ln}\prod_{n=0}^{\infty}(C_{0}+D_{n})^{\frac{1}{N+\beta_{n}}}=\sum_{n=0}^{\infty}\frac{P^{n}}{N+\beta_{0}}\mbox{ln}(C_{0}+D_{n})$
$\displaystyle\displaystyle\leq\sum_{n=0}^{\infty}\frac{P^{n}}{N+\beta_{0}}N(n+1)\mbox{ln}(C_{0}+C^{\prime})2<\infty.$

As a result, by iteration in (89) and
$\displaystyle\tilde{p}(N+\beta_{0})=\overline{p}^{*}$, we obtain

$\displaystyle\|u\|_{L^{\frac{\overline{p}^{*}}{N}(N+\beta_{n})}(B(x_{0},R_{*}))}\leq
b_{n}\leq
C^{S_{1}}e^{S_{2}}e^{S_{3}}\max\\{1,\|u\|_{L^{\overline{p}^{*}}(B(x_{0},2R_{*}))}\\}$

for every $\displaystyle n\geq 1$. Letting $\displaystyle n\to\infty$, this
ensures that

$\|u\|_{L^{\infty}(B(x_{0},R_{*}))}\leq
C^{S_{1}}e^{S_{2}}e^{S_{3}}\max\\{1,\|u\|_{L^{\overline{p}^{*}}(B(x_{0},2R_{*}))}\\}.$
(90)

By using the embedding of $\displaystyle W^{1,N}(D)$ to $\displaystyle
L^{\overline{p}^{*}}(D)$, (90) yields that

$\begin{array}[]{ll}\|u\|_{L^{\infty}(B(x_{0},R_{*}))}&\leq
C^{S_{1}}e^{S_{2}}e^{S_{3}}\max\\{1,\|u\|_{L^{\overline{p}^{*}}(D)}\\}\\\
&\leq
C^{S_{1}}e^{S_{2}}e^{S_{3}}\max\\{1,C_{*}\|u\|_{W^{1,N}(D)}\\}\end{array}$
(91)

whence $\displaystyle u$ is bounded in $\displaystyle D$ because
$\displaystyle x_{0}\in D$ is arbitrary and the constant $\displaystyle
C^{S_{1}}e^{pS_{2}}e^{S_{3}}$ is independent of $\displaystyle x_{0}$. ∎

###  5.2 Proof of Theorem 1.1

In this section, we denote $\displaystyle B_{n}:=B_{n}(0)$ the open ball
centered at the origin with radius $\displaystyle n$. Throughout this section,
we will consider $\displaystyle\lambda\in(0,\lambda^{*})$ fixed. The space
$\displaystyle W^{1,N}(B_{n})$ is endowed with the norm

$\displaystyle\|u\|_{N,n}^{N}:=\int_{B_{n}}\left(|\nabla
u|^{N}+|u|^{N}\right)dx\,.$

By applying Theorem 4.1 with $\displaystyle D=B_{n}$ ($\displaystyle
n\in\mathbb{N}$), we obtain a solution $\displaystyle u_{n}\in
W_{0}^{1,N}(B_{n})$ of the problem

$\displaystyle\left\\{\begin{array}[]{lll}-\Delta_{N}u+|u|^{N-2}u=\lambda
a(x)|u|^{q-2}u+f(u)&\mbox{in}&B_{n}\\\ u>0&\mbox{in}&B_{n}\\\
u(x)=0&\mbox{on}&\partial B_{n}.\end{array}\right.$ $None$

Again, (70) and (80) shows the boundedness of $\displaystyle\\{u_{n}\\}_{n>0}$
in $\displaystyle W_{0}^{1,N}(D)$. That is,

$\|u_{n}\|_{N,n}\ \leq\tilde{\varrho}\quad\mbox{for all}\quad n\in\mathbb{N},$
(92)

where

$\displaystyle\tilde{\varrho}:=\min\left\\{(2\lambda
C_{1})^{1/(N-q)},\varrho\right\\}$

is independent of $\displaystyle B_{n}$.

If $\displaystyle n\geq m+1$, notice that

$\begin{array}[]{lll}\int_{B_{m+1}}(|\nabla u_{n}|^{N-2}\nabla
u_{n}\nabla\varphi+|u_{n}|^{N-2}u_{n}\varphi-\lambda
a(x)|u_{n}|^{q-2}u_{n}\varphi-f(u_{n})\varphi)dx=0,\end{array}$ (93)

for all $\displaystyle\varphi\in C^{\infty}_{0}(B_{m+1})$. By (92) we obtain

$\|u_{n}\|_{N,m+1}\leq\|u_{n}\|_{N,n}\leq\tilde{\varrho}.$ (94)

Therefore, there exists $\displaystyle u_{\lambda}\in W^{1,N}(B_{m+1})$ such
that

$\displaystyle\displaystyle u_{n}$ $\displaystyle\displaystyle\rightharpoonup$
$\displaystyle\displaystyle
u_{\lambda}\quad\,\,\,\mbox{in}\,\,W^{1,N}(B_{m+1}),$ (95)
$\displaystyle\displaystyle u_{n}$ $\displaystyle\displaystyle\to$
$\displaystyle\displaystyle u_{\lambda}\quad{\rm in}\ L^{s}(B_{m+1}),\,s\geq
N,$ (96) $\displaystyle\displaystyle u_{n}(x)$
$\displaystyle\displaystyle\rightarrow$ $\displaystyle\displaystyle
u_{\lambda}(x)\quad\,\,\,\mbox{ a.e. }x\in B_{m+1},$ (97)

as $\displaystyle n\to\infty$.

By inequality (91) in Proposition 4 and (94), we infer that

$\begin{array}[]{ll}\|u_{n}\|_{L^{\infty}(B_{m+1})}&\leq
C^{S_{1}}e^{S_{2}}e^{S_{3}}\max\\{1,C_{*}\tilde{\varrho}\\}=:\Theta.\end{array}$
(98)

Now we use regularity result up to the boundary due to Lieberman [27, Theorem
1.7], to conclude from (98) that

$\|u_{n}\|_{C^{1,\beta}(\overline{B_{m}})}\leq\vartheta,$ (99)

where $\displaystyle\beta\in(0,1)$ and $\displaystyle\vartheta$ is independent
of $\displaystyle n$. Thus, using (99) and Arzelà-Ascoli theorem, we conclude
that

$u_{\lambda}\in C^{1,\alpha}(\overline{B_{m}})\mbox{ for some
}\alpha\in(0,\beta).$ (100)

We also have

$\begin{array}[]{l}f(u_{n})\to f(u_{\lambda})\,\,\,\mbox{ weakly in
}\,\,\,L^{N^{\prime}}(B_{m+1}),\end{array}$ (101)

as $\displaystyle n\to\infty$. Indeed, notice that

$\begin{array}[]{rll}\int_{B_{m+1}}|f(u_{n})|^{N^{\prime}}dx\leq\int_{B_{m+1}}(u_{n}^{(p-1)}\phi_{N}(2^{\frac{N}{N-1}}\alpha|u_{n}|^{\frac{N}{N-1}}))^{N^{\prime}}dx&\leq&\\\
\Theta^{(p-1)N^{\prime}}\int_{B_{m+1}}(\phi_{N}(N2^{\frac{2N}{N-1}}\alpha\|u_{n}\|^{{\frac{N}{N-1}}}_{W^{1,N}(B_{n}}|\frac{u_{n}}{\|u_{n}\|_{W^{1,N}(B_{n})}}|^{\frac{N}{N-1}}))dx&\leq&\\\
\Theta^{(p-1)N^{\prime}}C(\alpha,N)^{N^{\prime}}&<&\infty.\end{array}$ (102)

Since

$\displaystyle u_{n}\rightarrow u_{\lambda}\,\,\mbox{a.e. in}\,\,B_{m+1},$

by the continuity of $\displaystyle f$ we have

$f(u_{n}(\cdot))\rightarrow f(u_{\lambda}(\cdot))\,\,\mbox{a.e.
in}\,\,B_{m+1}.$ (103)

Hence, [19, Theorem 13.44] leads to

$f(u_{n})\to f(u_{\lambda})\,\,\,\mbox{ weakly in
}\,\,\,L^{N^{\prime}}(B_{m+1}).$ (104)

Let us show that $\displaystyle u_{n}$ converges to $\displaystyle
u_{\lambda}$ strongly in $\displaystyle W^{1,N}(B_{m})$. To this end, fix
$\displaystyle l\in\mathbb{N}$ and choose a smooth function
$\displaystyle\psi_{l}$ satisfying $\displaystyle 0\leq\psi_{l}\leq 1$,
$\displaystyle\psi_{l}(r)=1$ if $\displaystyle r\leq m$ and
$\displaystyle\psi_{l}(r)=0$ if $\displaystyle r\geq m+1/l$. Setting
$\displaystyle\eta_{l}(x):=\psi_{l}(|x|)$, we note that
$\displaystyle(u_{n}-u_{\lambda})\eta_{l}\in W_{0}^{1,N}(B_{m+1})\subset
W_{0}^{1,N}(B_{n})$ for any $\displaystyle n\geq m+1$. Denote

$\displaystyle V_{n}=\int_{B_{m}}|\nabla u_{n}|^{N-2}\nabla u_{n}(\nabla
u_{n}-\nabla u_{\lambda})\,dx.$

Using $\displaystyle(u_{n}-u_{\lambda})\eta_{l}$ as test function in
$\displaystyle\eqref{z2}$ and invoking the growth condition (6), we obtain

$\displaystyle\displaystyle V_{n}$ $\displaystyle\displaystyle=$
$\displaystyle\displaystyle\int_{|x|<m+1/l}\left(\lambda
a(x)u_{n}^{q-1}+f(u_{n})-u_{n}^{N-1}\right)(u_{n}-u_{\lambda})\eta_{l}\,dx$
$\displaystyle\displaystyle-\int_{m\leq|x|<m+1/l}|\nabla u_{n}|^{N-2}\nabla
u_{n}\nabla(u_{n}-u_{\lambda})\eta_{l}dx$
$\displaystyle\displaystyle-\int_{m\leq|x|<m+1/l}|\nabla u_{n}|^{N-2}\nabla
u_{n}\nabla\eta_{l}(u_{n}-u_{\lambda})dx$ $\displaystyle\displaystyle\leq$
$\displaystyle\displaystyle\int_{B_{m+1}}(\lambda
a(x)u_{n}^{q-1}+u_{n}^{p-1}\phi_{N}(2^{\frac{N}{N-1}}\alpha|u_{n}|^{\frac{N}{N-1}})+u_{n}^{N-1})|u_{n}-u_{\lambda}|\,dx$
$\displaystyle\displaystyle+\int_{m\leq|x|<m+1/l}|\nabla u_{n}|^{N-1}|\nabla
u_{\lambda}|\,dx$
$\displaystyle\displaystyle+d_{l}\int_{m\leq|x|<m+1/l}|\nabla
u_{n}|^{N-1}|u_{n}-u_{\lambda}|\,dx$ $\displaystyle\displaystyle\equiv$
$\displaystyle\displaystyle I_{n}^{1}+I_{n}^{2}+I_{n}^{3},$

where $\displaystyle d_{l}:=\sup_{|x|<m+1/l}|\nabla\eta_{l}(x)|$.

By Hölder’s inequality and (92) we have

$\displaystyle\displaystyle I_{n}^{1}$ $\displaystyle\displaystyle\leq$
$\displaystyle\displaystyle\|u_{n}-u_{\lambda}\|_{L^{N}(B_{m+1})}\left\\{\lambda\|a\|_{L^{N/(N-q)}(B_{m+1})}\|u_{n}\|^{q}_{W^{1,N}(B_{m}+1)}\right.$
$\displaystyle\displaystyle\left.+\vartheta^{p-1}C(\alpha,N)+\|u_{n}\|^{N-1}_{W^{1,N}(B_{m}+1)}\right\\}$
$\displaystyle\displaystyle\leq$
$\displaystyle\displaystyle\overline{C}\|u_{n}-u_{\lambda}\|_{L^{N}(B_{m+1})},$

where $\displaystyle\overline{C}$ is a positive constant independent of
$\displaystyle u_{n}$, $\displaystyle n$, $\displaystyle m$ and $\displaystyle
l$.

By Hölder’s inequality, the following estimates follow:

$\displaystyle\displaystyle I_{n}^{2}$
$\displaystyle\displaystyle\leq\|u_{n}\|^{N-1}_{W^{1,N}(B_{m+1})}\left(\int_{m\leq|x|<m+1/l}|\nabla
u_{\lambda}|^{N}\,dx\right)^{1/N},$

$\displaystyle\displaystyle I_{n}^{3}$ $\displaystyle\displaystyle\leq
d_{l}\|u_{n}-u_{\lambda}\|_{L^{N}(B_{m+1})}\|\nabla
u_{n}\|_{L^{N}(B_{m+1})}^{N-1}.$

Thereby, from (92) and (96) we derive

$\displaystyle\displaystyle\limsup_{n\to\infty}V_{n}\leq\tilde{\varrho}^{N-1}\left(\int_{m\leq|x|<m+1/l}|\nabla
u_{\lambda}|^{N}\,dx\right)^{1/N},$

for all $\displaystyle l\in\mathbb{N}$. Thus, letting $\displaystyle
l\to\infty$, we obtain

$\limsup_{n\to\infty}V_{n}\leq 0.$ (105)

As known from (95), $\displaystyle u_{n}$ weakly converges to $\displaystyle
u_{\lambda}$ in $\displaystyle W^{1,N}(B_{m})$, so we may write

$\begin{array}[]{lll}V_{n}+o(1)&=&\int_{B_{m}}\left(|\nabla u_{n}|^{N-2}\nabla
u_{n}-|\nabla u_{\lambda}|^{N-2}\nabla u_{\lambda}\right)(\nabla u_{n}-\nabla
u_{\lambda})\,dx\\\ &\geq&\left(\|\nabla u_{n}\|_{L^{N}(B_{m})}^{N-1}-\|\nabla
u_{\lambda}\|_{L^{N}(B_{m})}^{N-1}\right)\left(\|\nabla
u_{n}\|_{L^{N}(B_{m})}-\|\nabla u_{\lambda}\|_{L^{N}(B_{m})}\right)\\\ \\\
&\geq&0.\end{array}$ (106)

By (105) and (106), we obtain $\displaystyle\lim_{n\to\infty}V_{n}=0$,
$\displaystyle\lim_{n\to\infty}\|\nabla u_{n}\|_{L^{N}(B_{m})}=\|\nabla
u_{\lambda}\|_{L^{N}(B_{m})}$. This implies that $\displaystyle u_{n}$
converges to $\displaystyle u_{\lambda}$ strongly in $\displaystyle
W^{1,N}(B_{m})$ because the spaces $\displaystyle W^{1,N}(B_{m})$ is uniformly
convex.

Now, since $\displaystyle u_{n}>0$ in $\displaystyle B_{m}$, we infer that
$\displaystyle u_{\lambda}$ is a nonnegative solution of the problem

$\displaystyle-\Delta_{N}u_{\lambda}+u^{N-1}_{\lambda}=\lambda
a(x)u_{\lambda}^{q-1}+f(u_{\lambda})\quad{\rm in}\ B_{m},\quad u_{\lambda}\geq
0\quad{\rm on}\ \partial B_{m}.$

By using (24) and since $\displaystyle C_{0}^{\infty}(\mathbb{R}^{N})$ is
dense in $\displaystyle W^{1,N}(\mathbb{R}^{N})$, by a diagonal argument,
there exist a relabeled subsequence of $\displaystyle\\{\tilde{u}_{n}\\}$ and
a function $\displaystyle u_{\lambda}\in W^{1,N}(\mathbb{R}^{N})$ such that

$\displaystyle\tilde{u}_{n}\rightharpoonup
u_{\lambda}\,\,\,\mbox{in}\,\,\,W^{1,N}(\mathbb{R}^{N}),$
$\displaystyle\tilde{u}_{n}(x)\rightarrow u_{\lambda}(x)\,\,\,\mbox{for a.e
}x\in\mathbb{R}^{N}.$

These convergence properties, and some iteration process, ensure that
$\displaystyle u_{\lambda}$ is a solution of problem $\displaystyle\eqref{P}$,
that belongs to $\displaystyle C^{1}_{loc}(\mathbb{R}^{N})$ (see (100)).

The next step in the proof is to show that $\displaystyle u_{\lambda}$ does
not vanish in $\displaystyle\mathbb{R}^{N}$. Indeed, Lemma 2.2 provides a
solution $\displaystyle u_{\lambda,m}$ of the problem

$\displaystyle\left\\{\begin{array}[]{lll}-\Delta_{N}u+|u|^{N-2}u=\lambda
a_{m}|u|^{q-2}u&\mbox{in}&B_{m}\\\ u>0&\mbox{in}&B_{m}\\\
u(x)=0&\mbox{on}&\partial B_{m},\end{array}\right.$

where

$\displaystyle a_{m}=\inf_{B_{m}}a(x).$

Since $\displaystyle\lambda a(x)t^{q-1}+f(t)\geq\lambda a_{m}t^{q-1}$ for all
$\displaystyle x\in\mathbb{R}^{N}$. We are thus in a position to apply
Proposition 2 to the functions $\displaystyle u_{\lambda,m}$ and
$\displaystyle\tilde{u}_{n}$ with $\displaystyle n>m$, in place of
$\displaystyle u_{1}=u_{\lambda,m}$ and $\displaystyle u_{2}=\tilde{u}_{n}$,
respectively, which renders $\displaystyle\tilde{u}_{n}\geq u_{\lambda,m}$ in
$\displaystyle B_{m}$ for every $\displaystyle n>m$. This enables us to deduce
that $\displaystyle u_{\lambda}\geq u_{\lambda,m}$ in $\displaystyle B_{m}$,
so $\displaystyle u_{\lambda}(x)>0$ for every $\displaystyle
x\in\mathbb{R}^{N}$, because $\displaystyle m$ was arbitrary chosen.

Furthermore, since $\displaystyle\tilde{u}_{n}$ weakly converges to
$\displaystyle u_{\lambda}$ in $\displaystyle W^{1,N}(\mathbb{R}^{N})$, by
(94) we have

$\|u_{\lambda}\|_{W^{1,N}(\mathbb{R}^{N})}\leq\liminf_{n\rightarrow\infty}\|\tilde{u}_{n}\|_{W^{1,N}(\mathbb{R}^{N})}=\liminf_{n\rightarrow\infty}\|\tilde{u}_{n}\|_{N,n}\leq\tilde{\varrho},$
(107)

and then, we obtain $\displaystyle u_{\lambda}\in W^{1,N}(\mathbb{R}^{N})$,
and

$\|u_{\lambda}\|_{W^{1,N}(\mathbb{R}^{N})}\rightarrow 0,\,\mbox{ as
}\lambda\rightarrow 0.$ (108)

Thus, the proof of Theorem 1.1 is complete.

###  5.3 Proof of Proposition 1

In this proof, we borrowed some ideas from [6]. For the sake of contradiction,
suppose that $\displaystyle\lambda^{*}=\infty$. Thus, there is a sequence
$\displaystyle\lambda_{n}\to\infty$ and corresponding solutions $\displaystyle
u_{\lambda_{n}}>0$ in $\displaystyle\mathbb{R}^{N}$ given by Theorem 1.1. In
what follows, define $\displaystyle a_{R}=\inf_{B_{R}(0)}a(r)$, for some
$\displaystyle R>0$ fixed. Define also

$\displaystyle\mathcal{P}(x,t)=\lambda
a(x)t^{q-1}+t^{p-1}\phi_{N}(\alpha|t|^{\frac{N}{N-1}})$

and

$\displaystyle P_{1}(t)=\Lambda
t^{q-1}+t^{p-1}\phi_{N}(\alpha|t|^{\frac{N}{N-1}}),$

where $\displaystyle\Lambda=\lambda\ a_{R}$.

We claim that there exists a constant $\displaystyle C_{\Lambda}>0$ such that

$P(x,t)\geq C_{\Lambda}t^{N-1}\mbox{ for every }\ t>0.$

Indeed, define the function $\displaystyle Q(t)=P_{1}(t)t^{-(N-1)}$. Then,
since $\displaystyle q<N<p$, $\displaystyle Q(t)\to\infty$ as $\displaystyle
t\to 0^{+}$ and as $\displaystyle t\to\infty$. The minimum value is
$\displaystyle Q(t_{1})=C_{\Lambda}$, where $\displaystyle t_{1}>0$ is the
unique solution of

$\displaystyle\phi_{N}(\alpha
t^{\frac{N}{N-1}})(p-N)t^{p-q}+\alpha\frac{N}{N-1}t^{\frac{(N-1)(p-q+1)+1}{N-1}}\phi_{N-1}(\alpha
t^{\frac{N}{N-1}})=\Lambda(N-q).$

Indeed, let us define

$\displaystyle H(t):=\phi_{N}(\alpha
t^{\frac{N}{N-1}})(p-N)t^{p-q}+\alpha\frac{N}{N-1}t^{\frac{(N-1)(p-q+1)+1}{N-1}}\phi_{N-1}(\alpha
t^{\frac{N}{N-1}}).$

Note that $\displaystyle H(0)=0$ and $\displaystyle H(t)\to\infty$ as
$\displaystyle t\to\infty$. Since

$\displaystyle\begin{array}[]{rcl}H^{\prime}(t)&=&(p-N)(p-q)t^{p-q-1}\phi_{N}(\alpha
t^{\frac{N}{N-1}})\\\
&+&\frac{\alpha\,N(p-N)}{N-1}t^{p-q+\frac{1}{N}}\phi_{N-1}(\alpha
t^{\frac{N}{N-1}})\\\
&+&\frac{N\alpha}{N-1}\left[\frac{(N-1)(p-q+1)+1}{N-1}\right]t^{\frac{(N-1)(p-q+1)+1}{N-1}-1}\phi_{N-1}(\alpha
t^{\frac{N}{N-1}})\\\
&+&\frac{\alpha^{2}N^{2}}{(N-1)^{2}}t^{\frac{(N-1)(p-q+1)+1}{N-1}+\frac{1}{N-1}}\phi_{N-2}(\alpha
t^{\frac{N}{N-1}})>0,\,\,\,\forall t\geq 0,\end{array}$

hence $\displaystyle H$ is increasing. Therefore, there is a unique point
$\displaystyle t_{1}>0$ such that $\displaystyle H(t_{1})=\Lambda(N-q)$.
Consequently, by the arguments before, $\displaystyle t_{1}>0$ is the unique
point of minimum of $\displaystyle Q$, with $\displaystyle
Q(t_{1})=C_{\Lambda}$.

Denote by $\displaystyle\sigma_{1}>0$ and $\displaystyle\varphi_{1}>0$,
respectively, the principal eigenvalue and corresponding eigenfunction of the
eigenvalue problem

$\displaystyle\left\\{\begin{array}[]{lll}-\Delta_{N}(\varphi_{1})=\sigma_{1}|\varphi_{1}|^{N-2}\varphi_{1}&\mbox{in}&B_{R}(0)\\\
\varphi_{1}=0&\mbox{on}&\partial B_{R}(0).\end{array}\right.$

Since $\displaystyle C_{\Lambda}$ increases as $\displaystyle\lambda_{n}$
increases, for $\displaystyle\delta>0$ small enough, there is
$\displaystyle\lambda_{0}$ in this sequence such that the corresponding
constant satisfies $\displaystyle C_{\Lambda_{0}}\geq\sigma_{1}+\delta+1$.
Hence the solution $\displaystyle u_{\lambda_{0}}$ of (5) associated to
$\displaystyle\lambda_{0}$ satisfies $\displaystyle u_{\lambda_{0}}>0$ in
$\displaystyle\mathbb{R}^{N}$ and

$\displaystyle\left\\{\begin{array}[]{lll}-\Delta_{N}u_{\lambda_{0}}\geq(C_{\Lambda_{0}}-1)|u_{\lambda_{0}}|^{N-2}u_{\lambda_{0}}\geq(\sigma_{1}+\delta)|u_{\lambda_{0}}|^{N-2}u_{\lambda_{0}}&\mbox{in}&B_{R}(0)\\\
u_{\lambda_{0}}\geq 0&\mbox{on}&\partial B_{R}(0).\end{array}\right.$

Otherwise, taking $\displaystyle\varepsilon>0$ small enough we obtain
$\displaystyle\varepsilon\varphi_{1}<u_{\lambda_{0}}$ in $\displaystyle
B_{R}(0)$ and

$\displaystyle\left\\{\begin{array}[]{lll}-\Delta_{N}(\varepsilon\varphi_{1})=\sigma_{1}|\varepsilon\varphi_{1}|^{N-2}(\varepsilon\varphi_{1})\leq(\sigma_{1}+\delta)|\varepsilon\varphi_{1}|^{N-2}(\varepsilon\varphi_{1})&\mbox{in}&B_{R}(0)\\\
\varphi_{1}=0&\mbox{on}&\partial B_{R}(0).\end{array}\right.$

By the method of subsolution and supersolution, see [6, Theorem 2.1], there is
a solution $\displaystyle\varepsilon\varphi_{1}\leq\omega\leq u_{\lambda_{0}}$
in $\displaystyle B_{R}(0)$ of

$\displaystyle\left\\{\begin{array}[]{lll}-\Delta_{N}\omega=(\sigma_{1}+\delta)|\omega|^{N-2}\omega&\mbox{in}&B_{R}(0)\\\
\omega=0&\mbox{on}&\partial B_{R}(0).\end{array}\right.$

Hence there is a contradiction to the fact that $\displaystyle\sigma_{1}$ is
isolated (see [28]). Therefore, $\displaystyle\lambda^{*}<\infty$.

###  5.4 Proof of Theorem 1.2

By using the same arguments as in the proof of Theorem 1.1, with
$\displaystyle D=B_{n}$, and considering (107), we can pass to the limit
$\displaystyle n\to+\infty$ in (91) to obtain

$\begin{array}[]{c}\|u_{\lambda}\|_{L^{\infty}(\mathbb{R}^{N})}\leq
C^{S_{1}}e^{S_{2}}e^{S_{3}}\max\\{1,C_{*}\tilde{\varrho}\\}.\end{array}$ (109)

Since $\displaystyle u_{\lambda}$ is bounded in $\displaystyle\mathbb{R}^{N}$,
we put $\displaystyle M_{0}:=\|u_{\lambda}\|_{L^{\infty}(\mathbb{R}^{N})}$.
Then, by Lemma 5.1, we have

$\displaystyle\displaystyle\int_{B(x_{0},R^{\prime})}f(u_{\lambda})u_{\lambda}(u_{\lambda})_{M}^{\beta}\eta^{N}dx\,\leq
C(\varrho)\|u_{\lambda}\|^{\beta}_{L^{\tilde{p}(N+\beta)}(B(x_{0},R^{\prime}))}B_{R^{\prime}}$
(110)
$\displaystyle\displaystyle\int_{B(x_{0},R^{\prime})}a(x)u_{\lambda}^{q}(u_{\lambda})_{M}^{\beta}\eta^{N}dx\,\leq\|a\|_{L^{\gamma_{1}}(B(x_{0},R^{\prime}))}M_{0}^{q}\|u_{\lambda}\|_{L^{\tilde{p}(N+\beta)}(B(x_{0},R^{\prime}))}^{\beta}B_{R^{\prime}}.$
(111) By Hölder inequality, we obtain
$\displaystyle\displaystyle\|u_{\lambda}\|_{L^{\tilde{p}(N+\beta)}(B(x_{0},R^{\prime}))}^{N+\beta}\leq
M_{0}^{N}\|u_{\lambda}\|_{L^{\tilde{p}(N+\beta)}(B(x_{0},R^{\prime}))}^{\beta}B_{R^{\prime}}.$
(112)

Now, fix $\displaystyle x_{0}\in\mathbb{R}^{N}$. It follows from the argument
as in the proof of Lemma 5.2 with (110), (111) and (112) that

$\displaystyle\displaystyle\|u_{\lambda}\|_{L^{\frac{\overline{p}^{*}}{N}(N+\beta)}(B(x_{0},R))}^{N+\beta}$
$\displaystyle\displaystyle\leq
2^{N}(N+\beta)^{N}C_{*}^{N}B_{R^{\prime}}(C_{R^{\prime}}+D_{R,R^{\prime}})(M_{0}+1)^{N}\|u_{\lambda}\|_{L^{\tilde{p}(N+\beta)}(B(x_{0},R^{\prime}))}^{\beta}$

provided $\displaystyle u_{\lambda}\in
L^{\tilde{p}(N+\beta)}(B(x_{0},R^{\prime}))$. Choose
$\displaystyle\gamma_{1}$, $\displaystyle\tilde{p}$ and define sequences
$\displaystyle\\{\beta_{n}\\}$, $\displaystyle\\{R_{n}^{\prime}\\}$ and
$\displaystyle\\{R_{n}\\}$ as in the proof of Proposition 4. Set

$\displaystyle
V_{n}:=\|u_{\lambda}\|_{L^{\tilde{p}(N+\beta_{n})}(B(x_{0},R^{\prime}_{n}))}^{\beta_{n}}.$

The remainder of the proof follows with the same arguments as in in the proof
of [15, Theorem 2] to conclude that

$V_{n}^{\frac{N+\beta_{n-1}}{\beta_{n}}}\leq
C(N+\beta_{n-1})^{N}(C_{0}+D_{n-1})V_{n-1}$ (114)

with $\displaystyle C:=2^{N}C_{*}^{N}B_{0}(M_{0}+1)^{N}$. Recall that

$\displaystyle\beta_{n}+N=P^{-1}(N+\beta_{n-1})\quad{\rm
and}\quad\frac{N}{N+\beta_{0}}=P.$

Define

$\displaystyle
Q_{n}:=\prod_{k=2}^{n+1}\left(1+\frac{P^{k}}{1-P^{k}}\right)=\prod_{k=2}^{n+1}\left(1-P^{k}\right)^{-1}\quad{\rm
and}\quad W_{n}:=(C_{0}+D_{n}).$

Then, the inequality (114) leads to

$\displaystyle\displaystyle\mbox{ln}V_{n}$
$\displaystyle\displaystyle\leq\frac{\beta_{n}}{N+\beta_{n-1}}\left(\mbox{ln}V_{n-1}+\mbox{ln}[C(N+\beta_{n-1})^{N}]+\mbox{ln}W_{n-1}\right)$
$\displaystyle\displaystyle=P^{-1}\left(1-P^{n+1}\right)\left(\mbox{ln}V_{n-1}+N\mbox{ln}[CP^{-n+1}(N+\beta_{0})]+\mbox{ln}W_{n-1}\right)$
$\displaystyle\displaystyle\leq
P^{-1}\left(1-P^{n+1}\right)\mbox{ln}V_{n-1}+NP^{-1}\mbox{ln}[(C+1)P^{-n+1}(N+\beta_{0})]+P^{-1}\mbox{ln}W_{n-1}$
$\displaystyle\displaystyle\leq
P^{-n}\left(\prod_{k=1}^{n}\left(1-P^{k+1}\right)\right)\mbox{ln}V_{0}+N\sum_{k=1}^{n}P^{-k}\mbox{ln}[(C+1)P^{-n+k}(N+\beta_{0})]$
$\displaystyle\displaystyle\qquad+\sum_{k=1}^{n}P^{-k}\mbox{ln}W_{n-k}$
$\displaystyle\displaystyle=P^{-n}Q_{n}^{-1}\mbox{ln}V_{0}+N\sum_{k=1}^{n}P^{-k}\mbox{ln}[(C+1)P^{-n+k}(N+\beta_{0})]+\sum_{k=1}^{n}P^{-k}\mbox{ln}W_{n-k}$

for every $\displaystyle n$ because of
$\displaystyle\mbox{ln}[(C+1)P^{-n+1}(N+\beta_{0})]>0$ and
$\displaystyle\mbox{ln}W_{n}>0$ for all $\displaystyle n$. Therefore, we have

$\displaystyle\displaystyle\mbox{ln}\|u_{\lambda}\|_{L^{\tilde{p}(N+\beta_{n})}(B(x_{0},R^{\prime}_{n}))}=\frac{\mbox{ln}V_{n}}{\alpha_{n}}=\frac{P^{n}\mbox{ln}V_{n}}{N+\beta_{0}-NP^{n}}$
$\displaystyle\displaystyle\leq\frac{Q_{n}^{-1}\mbox{ln}V_{0}}{N+\beta_{0}-NP^{n}}+\frac{\sum_{l=0}^{n-1}P^{l}\mbox{ln}[(C+1)P^{-l}(N+\beta_{0})]}{N+\beta_{0}-NP^{n}}+\frac{\sum_{l=0}^{n-1}P^{l}\mbox{ln}W_{l}}{N+\beta_{0}-NP^{n}}.$
(115)

Here, taking a sufficiently large positive constant $\displaystyle C^{\prime}$
independent of $\displaystyle n$, we see that

$\displaystyle\sum_{l=0}^{n-1}P^{l}\mbox{ln}[(C+1)P^{-l}(N+\beta_{0})]\leq
C^{\prime}\sum_{l=0}^{\infty}P^{l}(l+1)=:S_{1}<\infty$

and

$\displaystyle\begin{array}[]{ll}\sum_{l=0}^{n-1}P^{l}\mbox{ln}W_{l}&\leq
C^{\prime}\sum_{l=0}^{n-1}P^{l}(l+1)\leq
C^{\prime}\sum_{l=0}^{\infty}P^{l}(l+1)=:S_{2}<\infty.\end{array}$

Next, we shall show that $\displaystyle\\{Q_{n}\\}$ is a convergent sequence.
It is easily see that $\displaystyle\\{Q_{n}\\}$ is increasing. Moreover,
setting $\displaystyle d_{k}:=\mbox{ln}\left(1+\frac{P^{k}}{1-P^{k}}\right)$,
we see that

$\displaystyle\begin{array}[]{ll}\lim_{k\to\infty}\frac{d_{k+1}}{d_{k}}&=\lim_{k\to\infty}\frac{\mbox{ln}(1-P^{k+1})}{\mbox{ln}(1-P^{k})}=\lim_{k\to\infty}\frac{1-P^{k}}{1-P^{k+1}}\,P=P<1\end{array}$

by L’Hospital’s rule. This implies that

$\displaystyle\mbox{ln}Q_{n}=\sum_{k=2}^{n+1}\mbox{ln}\left(1+\frac{P^{k}}{1-P^{k}}\right)\leq\sum_{k=1}^{\infty}\mbox{ln}\left(1+\frac{P^{k}}{1-P^{k}}\right)<\infty.$

Therefore, $\displaystyle\\{Q_{n}\\}$ is bounded from above, whence
$\displaystyle\\{Q_{n}\\}$ converges and

$\displaystyle 1<\frac{1}{1-P^{2}}=Q_{1}\leq
Q_{\infty}:=\lim_{n\to\infty}Q_{n}<\infty$

holds. Consequently, letting $\displaystyle n\to\infty$ in (115), we have

$\|u_{\lambda}\|_{L^{\infty}(B(x_{0},R_{*}))}\leq(NS_{1}S_{2})^{\frac{1}{N+\beta_{0}}}\|u_{\lambda}\|_{L^{\overline{p}^{*}}(B(x_{0},2R_{*}))}^{\frac{\beta_{0}}{(N+\beta_{0})Q_{\infty}}}.$
(116)

This yields our conclusion since
$\displaystyle\|u_{\lambda}\|_{L^{\overline{p}^{*}}(B(x_{0},2R_{*}))}\to 0$ as
$\displaystyle|x_{0}|\to\infty$, $\displaystyle\beta_{0}>0$ and the constant
$\displaystyle NS_{1}S_{2}$ is independent of $\displaystyle x_{0}$.

###  5.5 Proof of Corollary 1

Notice that by (109),
$\displaystyle\|u_{\lambda}\|_{L^{\infty}(\mathbb{R}^{N})}$ is uniformly
bounded in the variable $\displaystyle\lambda$. Hence, the constant
$\displaystyle C$ in (114) is uniformly bounded in the variable
$\displaystyle\lambda$.

By (116), we obtain

$\displaystyle\|u_{\lambda}\|_{L^{\infty}(B(x_{0},R_{*}))}\leq(NS_{1}S_{2})^{\frac{1}{N+\beta_{0}}}\|u_{\lambda}\|_{L^{\overline{p}^{*}}(\mathbb{R}^{N})}^{\frac{\beta_{0}}{(N+\beta_{0})Q_{\infty}}}\leq\tilde{C}\|u_{\lambda}\|_{W^{1,N}(\mathbb{R}^{N})}^{\frac{\beta_{0}}{(N+\beta_{0})Q_{\infty}}},$

with $\displaystyle\tilde{C}$ independent of $\displaystyle\lambda$. Since
$\displaystyle x_{0}$ is arbitrary, on has

$\displaystyle\|u_{\lambda}\|_{L^{\infty}(\mathbb{R}^{N})}\leq\tilde{C}\|u_{\lambda}\|_{W^{1,N}(\mathbb{R}^{N})}^{\frac{\beta_{0}}{(N+\beta_{0})Q_{\infty}}}.$

Thus, (108) ensure that

$\displaystyle\|u_{\lambda}\|_{L^{\infty}(\mathbb{R}^{N})}\rightarrow 0,$

as $\displaystyle\lambda\rightarrow 0$.

### Acknowledgements

L. F.O Faria was partially supported by FAPEMIG CEX APQ 02374/17. A.L.A. de
Araujo was partially supported by FAPEMIG FORTIS-10254/2014 and CNPQ.

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