# The method of the energy function and applications

Claudianor O. Alves  ,  Tiago L. C. Coelho  and  João R. Santos Júnior∗
Unidade Acadêmica de Matemática - UAMAT
Universidade Federal de Campina Grande
CEP: 58109-970, Campina Grande - Pb, Brazil<EMAIL_ADDRESS>
Programa de Doutorado em Matemática
Instituto de Ciências Exatas e Naturais
Universidade Federal do Pará
Avenida Augusto corrêa 01, 66075-110, Belém, PA, Brazil
<EMAIL_ADDRESS>
Faculdade de Matemática
Instituto de Ciências Exatas e Naturais
Universidade Federal do Pará
Avenida Augusto corrêa 01, 66075-110, Belém, PA, Brazil<EMAIL_ADDRESS>
###### Abstract.

In this work, we establish a new method to find critical points of
differentiable functionals defined in Banach spaces which belong to a suitable
class ($\mathcal{J}$) of functionals. Once given a functional $J$ in the class
($\mathcal{J}$), the central idea of the referred method consists in defining
a real function $\zeta$ of a real variable, called energy function, which is
naturally associated to $J$ in the sense that the existence of real critical
points for $\zeta$ guarantees the existence of critical points for the
functional $J$. As a consequence, we are able to solve some variational
elliptic problems, whose associated energy functional belongs to
($\mathcal{J}$) and provide a version of the mountain pass theorem for
functionals in the class ($\mathcal{J}$) that allows us to obtain mountain
pass solutions without the so-called Ambrosetti-Rabinowitz condition.

###### Key words and phrases:

Energy function method, mountain pass theorem, nonlinear elliptic equations.

###### 2010 Mathematics Subject Classification:

35J20, 35J25, 35Q74.

C. O. Alves was partially supported by CNPq/Brazil 307045/2021-8 and Projeto
Universal FAPESQ-PB 3031/2021. J. R. Santos Júnior is the corresponding author
and he was partially supported by CNPq 313766/2021-5, Brazil.

###### Contents

  1. 1 Introduction
  2. 2 The Method (MEF)
    1. 2.1 Application to a semilinear problem
  3. 3 A version of the Mountain Pass Theorem
  4. 4 Some applications
    1. 4.1 A general nonlinear problem
    2. 4.2 A Berestycki-Lions type problem
    3. 4.3 An anisotropic equation

##  1\. Introduction

Over the years, many methods have been developed to attack different classes
of partial differential equations. Even though some methods are renowned for
their range of applications, it is of course expected that each one of them
has its advantages and disadvantages. In fact, the experience shows us that
depending on the level of complexity or specificity of a partial differential
equation certain approaches are more appropriated and effective than others.
In this context, it is crucial to have a diversified theoretical framework
which allows that different techniques complement each other in order to
provide a more complete understanding of the problems.

In this work, we have established an alternative method that allows us to
complement some classical results of the literature, involving differential
equations. In fact, this referred method, which we will call Method of the
Energy Function (MEF), has proved to be effective to treat some relevant
classes of variational elliptic partial differential problems, for which, some
progress is made in the present article, see Section 4. For instance, by using
the energy function approach, we obtain a version of the Mountain Pass Theorem
suitable for problems that do not require the Ambrosetti-Rabinowitz growth
condition, see Theorem 3.1.

In order to give a more clear idea about the subject we are going to treat,
let us consider a functional $J:E\rightarrow\mathbb{R}$ in the form
$J=\Psi-\Phi$, where $E$ is a reflexive Banach space and $\Psi,\Phi\in
C^{1}(E,\mathbb{R})$. When $J$ is of class $(\mathcal{J})$ in $E$, that is,
$\Phi$ and $\Psi$ satisfy certain conditions to be stated in the next section,
it is possible to relate $J$ with a real $C^{1}$-function
$\zeta:[0,\infty)\rightarrow\mathbb{R}$, which we will call the energy
function.

The main feature on the approach we are about to describe is that it relates
the existence of critical points for $\zeta$ with the existence of critical
points for the functional $J$. More precisely, under appropriated conditions,
we can “change” the effort of searching for critical points of functionals
defined in infinite-dimensional Banach spaces for the task of finding critical
points of a one variable real function, which is certainly more treatable,
since it does not require more than basic tools of the classical differential
Calculus to be executed.

The idea of trying to relate the energy functional of a certain elliptic
problem to a real function was inspired by the work of Arcoya, Santos Júnior
and Suárez [3], which studied a problem involving a degenerate Kirchhoff
operator of the form

(1.1) $\left\\{\begin{array}[]{lll}-m(\|u\|)\Delta u=f(u)&\mbox{in}&\Omega,\\\
\hskip 56.9055ptu=0&\mbox{on}&\partial\Omega,\end{array}\right.$

where $\Omega\subset\mathbb{R}^{N}$ ($N\geq 1$) is a smooth bounded domain,
$m:[0,\infty)\rightarrow\mathbb{R}$ and $f:\mathbb{R}\rightarrow\mathbb{R}$
are continuous functions satisfying appropriated conditions. The authors in
[3] defined a function $\alpha:[0,\infty)\rightarrow\mathbb{R}$ given by

(1.2) $\alpha(r)=\max_{\|u\|^{2}\leq r\atop u\in
H_{0}^{1}(\Omega)}\int_{\Omega}F^{\ast}(u)dx,$

where

$F^{\ast}(t)=\int_{0}^{t}f^{\ast}(s)ds,\,\,\mbox{with}\,\,f^{\ast}(t)=\left\\{\begin{array}[]{lll}f(0)&\mbox{if}&t<0,\\\
f(t)&\mbox{if}&0\leq t<s_{\ast},\\\ 0&\mbox{if}&s_{\ast}<t,\end{array}\right.$

and showed that $\alpha$ is differentiable in $(0,\infty)$ with

$\alpha^{\prime}(r)=\frac{1}{2r}\max_{u\in\mathcal{S}_{r}}\int_{\Omega}f^{\ast}(u)udx,$

where

$\mathcal{S}_{r}:=\left\\{u\in H_{0}^{1}(\Omega);\|u\|^{2}\leq
r\,\,\mbox{and}\,\,\int_{\Omega}F^{\ast}(u)dx=\alpha(r)\right\\}.$

It is important to point out that it is not obvious or intuitive that
$\alpha$, defined as in (1.2), is a differentiable, or even continuous, map
regarding to $r$, mainly because the function which attains the maximum in
(1.2) cannot be unique. In fact, in the particular situation considered in
[3], in order to prove the differentiability of $\alpha$, the authors were
inspired by arguments first used in Szulkin and Weth [27]. Another important
detail on [3] that should be highlighted is that, the map $\alpha$ is only
used to show that the weak solution found to the problem (1.1) is nontrivial
and no relation was established between critical points of $\alpha$, or even
of any other real function involving $\alpha$, and critical points of the
energy functional associated to the problem (1.1).

The paper is divided as follows: In Section 2, we introduce the energy
function $\zeta:[0,\infty)\rightarrow\mathbb{R}$ (see (2.1)) in an abstract
context and give general conditions under which $\zeta$ is
$C^{1}((0,\infty),\mathbb{R})$ and its critical points have a strong connexion
with a certain class of critical points of $J$, see Theorem 2.8. Furthermore,
we provide an application of the method (MEF) to a class of semilinear
problems which complements results obtained in [25]. In Section 3, we apply
the method (MEF) to get a new version of the Mountain Pass Theorem, see
Theorem 3.1. Some interesting facts about Theorem 3.1 are the following: 1) It
provides a new characterization of the mountain pass level as a “maxmin” type
constant. In fact, at least in our best knowledge, such a characterization has
not previously appeared in the literature; 2) It allows to treat problems
involving nonlinearities satisfying conditions weaker than Ambrosetti-
Rabinowitz condition [2], which have not been required in papers that address
the matter, see for instance [1, 12, 15, 13, 18, 19, 20, 21, 22, 24, 23] and
references there in. Finally, in Section 4, in order to illustrate the
applicability of Theorem 3.1, we revisite the studies made in Willem and Zou
[28], Schechter and Zou [26], Miyagaki and Souto [23], Azzollini and Pomponio
[4] and Barile and Figueiredo [5].

##  2\. The Method (MEF)

Throughout this article, $E$ denotes a reflexive Banach space. We say that a
functional $J$ belongs to class $(\mathcal{J})$ in $E$, if
$J:E\rightarrow\mathbb{R}$, $J=\Psi-\Phi$ and $\Psi,\Phi\in
C^{1}(E,\mathbb{R})$ satisfy the following hypotheses:

  * $(\Psi_{1})$

$\Psi$ is convex, coercive and weakly lower semicontinuous;

  * $(\Psi_{2})$

$\tilde{\mathcal{A}}_{r}=\\{u\in E;\,\Psi(u)\leq r\\}$ for all $r\geq 0$,
$\tilde{\mathcal{A}}_{0}=\\{0\\}$ and
$0\in\mbox{int}(\tilde{\mathcal{A}}_{r})$ for each $r>0$;

  * $(\Psi_{3})$

$\Psi^{\prime}(u)u\neq 0$ for all $u\in\partial\tilde{\mathcal{A}}_{r}$, where
$\partial\tilde{\mathcal{A}}_{r}=\\{u\in E;\,\Psi(u)=r\\}$;

  * $(\Psi_{4})$

For each $u\not=0$ and $r>0$, there exists a unique $t_{u}(r)>0$ such that
$t_{u}(r)u\in\partial\tilde{\mathcal{A}}_{r}$;

  * $(\Psi_{5})$

If $u_{n}\rightharpoonup u_{0}$ in $E$ and $\Psi(u_{n})\to\Psi(u_{0})$, then
$u_{n}\to u_{0}$ in $E$;

  * $(\Phi_{1})$

$\Phi$ and $u\mapsto\Phi^{\prime}(u)u/\Psi^{\prime}(u)u$ are weakly upper
semicontinuous;

  * $(\Phi_{2})$

If $\Phi^{\prime}(u)=0$, then $\Phi(u)\leq 0$;

  * $(\Phi_{3})$

There exists a sequence $\\{u_{n}\\}\subset E$ such that $u_{n}\rightarrow 0$
and $\Phi(u_{n})>0$, for all $n\in\mathbb{N}$.

Some consequences can be concluded from previous assumptions:

  * $(1)$

The coercivity in $(\Psi_{1})$ ensures that the boundaries
$\partial\tilde{\mathcal{A}}_{r}$ are nonempty, and it also implies the
boundedness of the sets $\tilde{\mathcal{A}}_{r}$.

  * $(2)$

From $(\Psi_{2})$, it follows that $\Psi(u)=0$ if, and only if, $u=0$ and,
moreover, $\Psi(u)>0$ if $u\neq 0$.

  * $(3)$

Combining information in (2), the coercivity in $(\Psi_{1})$ and $(\Psi_{3})$,
we conclude that $\Psi^{\prime}(u)u>0$ for all $u\neq 0$.

######  Proposition 2.1.

Suppose that $\Psi$ satisfies $(\Psi_{1})$,
$0\in\mbox{int}(\tilde{\mathcal{A}}_{r})$ for each $r>0$, and $\Phi$ is weakly
upper semicontinuous and satisfies $(\Phi_{2})-(\Phi_{3})$. Then, for each
$r>0$, $\Phi$ is bounded from above on $\tilde{\mathcal{A}}_{r}$ and its
maximum is attained at the boundary. In other words, there exists
$u_{r}\in\partial\tilde{\mathcal{A}}_{r}$ such that

$\Phi(u_{r})=\max_{u\in\tilde{\mathcal{A}}_{r}}\Phi(u)=\max_{u\in\partial\tilde{\mathcal{A}}_{r}}\Phi(u).$

###### Proof.

Since $\Psi$ satisfies $(\Psi_{1})$, we have that $\tilde{\mathcal{A}}_{r}$ is
convex, closed and bounded, and so, $\tilde{\mathcal{A}}_{r}$ is weakly closed
and bounded, from where it follows that $\tilde{\mathcal{A}}_{r}$ is weakly
compact. On the other hand, as $\Phi$ is weakly upper semicontinuous, it is
bounded from above on $\tilde{\mathcal{A}}_{r}$ and there exists
$u_{r}\in\tilde{\mathcal{A}}_{r}$ such that

$\Phi(u_{r})=\max_{u\in\tilde{\mathcal{A}}_{r}}\Phi(u).$

Now, we are going to show that $u_{r}\in\partial\tilde{\mathcal{A}}_{r}$.
Arguing by contradiction, assume that $u_{r}$ belongs to the interior of
$\tilde{\mathcal{A}}_{r}$, which is open, because $\Psi$ is continuous and

$\mathcal{A}_{r}:=\mbox{int}(\tilde{\mathcal{A}}_{r})=\\{u\in
E;\,\Psi(u)<r\\}.$

In this case, $u_{r}$ is a critical point of $\Phi$, that is,
$\Phi^{\prime}(u_{r})=0$. Using ($\Phi_{3}$) together with the fact that
$0\in\mbox{int}(\tilde{\mathcal{A}}_{r})$ for each $r>0$ , there exists
$n_{0}\in\mathbb{N}$ such that $u_{n_{0}}\in\tilde{\mathcal{A}}_{r}$ and

$0<\Phi(u_{n_{0}})\leq\max_{u\in\tilde{\mathcal{A}}_{r}}\Phi(u)=\Phi(u_{r}).$

However, from ($\Phi_{2}$), we also have that $\Phi(u_{r})\leq 0$, which is a
contradiction, proving that $u_{r}\in\partial\mathcal{A}_{r}$.

∎

######  Remark 2.2.

Let us consider the set

$\mathcal{G}_{r}:=\left\\{u\in\tilde{\mathcal{A}_{r}};\hskip
2.84544pt\Phi(u)=\max_{v\in\tilde{\mathcal{A}_{r}}}\Phi(v)\right\\}.$

It is an immediate consequence of the proof of Proposition 2.1 that, if $\Psi$
satisfies $(\Psi_{1})$ and $\Phi$ is weakly upper semicontinuous, then the set
$\mathcal{G}_{r}$ is nonempty. So, it makes sense to consider the hypothesis

  * $(\Phi_{2}^{\prime})$

For each $r>0$, there exists $c_{r}>0$, such that
$c_{r}\Phi(u)\leq\Phi^{\prime}(u)u$ for all $u\in\mathcal{G}_{r}$.

Note that, Proposition 2.1 (as well as all subsequent results that make use of
it) still holds true if we replace hypothesis $(\Phi_{2})$ by
$(\Phi_{2}^{\prime})$. It is important to point out that $(\Phi_{2})$ and
$(\Phi_{2}^{\prime})$ does not imply each other.

From previous remark, the set $\mathcal{G}_{r}$ is well-defined. Moreover, by
Proposition 2.1, one has
$\mathcal{G}_{r}\subset\partial\tilde{\mathcal{A}_{r}}$.

In the following, under the assumptions of Proposition 2.1, it is possible to
introduce a real function of real variable which is naturally related to the
functional $J$, in a sense that we will clarify later on (see Theorem 2.8).
This function plays a crucial role in the next results to be presented in this
manuscript.

Suppose that the assumptions in Proposition 2.1 hold. We call energy function
to the function $\zeta:[0,\infty)\rightarrow\mathbb{R}$ defined by

(2.1) $\zeta(r):=r-\varphi(r),$

where
$\varphi(r):=\displaystyle\max_{u\in\partial\tilde{\mathcal{A}}_{r}}\Phi(u)$
if $r>0$ and $\varphi(0)=\Phi(0)$.

Observe that the energy function and the functional $J$ are related by the
following equality

(2.2)
$\zeta(r)=r-\max_{u\in\partial\tilde{\mathcal{A}_{r}}}\Phi(u)=\min_{u\in\partial\tilde{\mathcal{A}}_{r}}J(u)=J(u_{r})\
\mbox{for all $r>0$ and $\zeta(0)=J(0)$}.$

In what follows, our main goal is to show that we can change the task of
finding critical points of $J$ for the simpler task of finding critical points
for $\zeta$. In order to establish this relation, we will study the
differentiability of function $\varphi$.

######  Proposition 2.3.

Suppose that $\Psi$ satisfies $(\Psi_{1})$, $(\Psi_{2})$ and $(\Psi_{4})$, and
$\Phi$ is weakly upper semicontinuous and satisfies $(\Phi_{2})-(\Phi_{3})$.
Then, function $\varphi:[0,\infty)\to\mathbb{R}$ is continuous.

###### Proof.

It is enough to take an arbitrary sequence $\\{r_{n}\\}\subset(0,\infty)$ with
$r_{n}\rightarrow r_{0}\geq 0$. Before that, observe that for each
$n\in\mathbb{N}$, we know from Proposition 2.1 that, there exists a function
$u_{n}\in\partial\tilde{\mathcal{A}_{r_{n}}}$ such that

$\Phi(u_{n})=\max_{u\in\partial\tilde{\mathcal{A}}_{r_{n}}}\Phi(u)=\max_{u\in\tilde{\mathcal{A}}_{r_{n}}}\Phi(u).$

We will show that $\\{u_{n}\\}$ is bounded. In fact, since $\\{r_{n}\\}$ is
bounded, there exists $R>0$ such that $\Psi(u_{n})=r_{n}\leq R$ for all
$n\in\mathbb{N}$, showing that $\\{u_{n}\\}\subset\tilde{\mathcal{A}}_{R}$,
which is a bounded set by $(\Psi_{1})$. Hence, up to a subsequence,

(2.3) $u_{n}\rightharpoonup u_{0}\,\,\mbox{in}\,\,E.$

Next, we will consider two cases, namely $r_{0}=0$ and $r_{0}>0$.

Case 1: $r_{0}=0$.

In this case, by $(\Psi_{2})$, the lower weak semicontinuity of $\Psi$ and
(2.3), one has

$0\leq\Psi(u_{0})\leq\liminf_{n\rightarrow\infty}\Psi(u_{n})=\liminf_{n\rightarrow\infty}r_{n}=0.$

Therefore, $\Psi(u_{0})=0$ and, again, by $(\Psi_{2})$, $u_{0}=0$.

Finally, from $(\Phi_{3})$, there exists $\\{v_{n}\\}\subset E$ with $v_{n}\to
0$ and $\Phi(v_{n})>0$. Now, from $(\Psi_{2})$, if necessary, we can
renumerate $\\{v_{n}\\}$ in such a way that
$v_{n}\in\tilde{\mathcal{A}}_{r_{n}}$. Thus, since $\Phi$ is continuous and
weakly upper semicontinuous, we get

$\Phi(0)=\lim_{n\to\infty}\Phi(v_{n})\leq\liminf_{n\rightarrow\infty}\Phi(u_{n})\leq\limsup_{n\rightarrow\infty}\Phi(u_{n})\leq\Phi(0),$

showing that

$\varphi(r_{n})=\Phi(u_{n})\to\Phi(0)=\varphi(0).$

Case 2: $r_{0}>0$.

Since $\Psi$ is weakly lower semicontinuous,

$\Psi(u_{0})\leq\liminf_{n\rightarrow\infty}\Psi(u_{n})=\liminf_{n\rightarrow\infty}r_{n}=r_{0},$

which means that $u_{0}\in\tilde{\mathcal{A}}_{r_{0}}$. Thus,

(2.4) $\Phi(u_{0})\leq\varphi(r_{0}).$

On the other hand, consider $u_{\ast}\in\partial\tilde{\mathcal{A}}_{r_{0}}$
such that

$\varphi(r_{0})=\Phi(u_{\ast}).$

Gathering $(\Psi_{4})$ and coercivity of $\Psi$, there exists a bounded
sequence $\\{t_{n}\\}\subset(0,+\infty)$ such that
$t_{n}u_{\ast}\in\partial\tilde{\mathcal{A}}_{r_{n}}$. Then

$\Phi\left(t_{n}u_{\ast}\right)\leq\Phi(u_{n}),\quad\forall n\in\mathbb{N}.$

######  Claim 2.4.

$t_{n}\rightarrow 1$ as $n\to\infty$.

Assume that for some subsequence of $\\{t_{n}\\}$, still denoted by itself, we
have $t_{n}\to t_{0}$. Since $\Psi$ is continuous, we must have

$\Psi(t_{0}u_{\ast})=\lim_{n\rightarrow\infty}\Psi(t_{n}u_{\ast})=\lim_{n\rightarrow\infty}r_{n}=r_{0}=\Psi(u_{\ast}),$

from where it follows that
$t_{0}u_{\ast}\in\partial\tilde{\mathcal{A}}_{r_{0}}$. As
$u_{\ast}\in\partial\tilde{\mathcal{A}}_{r_{0}}$, $(\Psi_{4})$ yields that
$t_{0}=1$ and the Claim is proved.

The claim above permits to conclude that

$\varphi(r_{0})=\Phi\left(u_{\ast}\right)=\limsup_{n\rightarrow\infty}\Phi\left(t_{n}u_{\ast}\right)\leq\limsup_{n\rightarrow\infty}\Phi(u_{n}).$

From $(\Phi_{1})$,

(2.5)
$\varphi(r_{0})\leq\limsup_{n\rightarrow\infty}\Phi(u_{n})\leq\Phi(u_{0}).$

Employing (2.4) and (2.5), one finds

$\varphi(r_{0})=\Phi(u_{0}).$

Again, by $(\Psi_{4})$ and coercivity of $\Psi$, there exists a bounded
sequence $\\{s_{n}\\}\subset(0,+\infty)$ such that
$s_{n}u_{0}\in\partial\tilde{\mathcal{A}}_{r_{n}}$ and $s_{n}\to 1$. Hence,

$\Phi(u_{0})=\liminf_{n\rightarrow\infty}\Phi(s_{n}u_{0})\leq\liminf_{n\rightarrow\infty}\Phi(u_{n}).$

On the other hand, as $\Phi$ is weakly upper semicontinuous,

$\limsup_{n\rightarrow\infty}\Phi(u_{n})\leq\Phi(u_{0}).$

The last two inequalities give

$\lim_{n\rightarrow\infty}\Phi(u_{n})=\Phi(u_{0}).$

Therefore,

$\lim_{n\rightarrow\infty}\varphi(r_{n})=\lim_{n\rightarrow\infty}\Phi(u_{n})=\Phi(u_{0})=\varphi(r_{0}).$

∎

######  Proposition 2.5.

Suppose that $\Psi$ and $\Phi$ satisfy $(\Psi_{1})-(\Psi_{3})$ and
$(\Phi_{1})-(\Phi_{3})$ respectively. Then, the functional
$u\mapsto\frac{\Phi^{\prime}(u)u}{\Psi^{\prime}(u)u}$ attains the maximum in
the set

$\mathcal{G}_{r}=\left\\{u\in\tilde{\mathcal{A}_{r}};\,\Phi(u)=\varphi(r)\right\\}.$

###### Proof.

By $(\Psi_{1})$, $\mathcal{G}_{r}$ is a bounded set. Now, we will prove that
$\mathcal{G}_{r}$ is weakly closed. Since
$\tilde{\mathcal{A}_{r}}\subset\overline{B}_{R}(0)$ for some $R>0$,
$\overline{B}_{R}(0)$ is weakly closed and $\overline{B}_{R}(0)$ is metrizable
by the weak topology, it is enough to prove that if
$\\{u_{n}\\}\subset\mathcal{G}_{r}$ and $u_{n}\rightharpoonup u_{0}$ in $E$,
then $u_{0}\in\mathcal{G}_{r}$. From $(\Psi_{1})$,

(2.6) $\Psi(u_{0})\leq\liminf_{n\rightarrow\infty}\Psi(u_{n})=r,$

which means that $u_{0}\in\tilde{\mathcal{A}}_{r}$ and

$\Phi(u_{0})\leq\varphi(r).$

On the other hand, since $\Phi$ is weakly upper semicontinuous and
$\\{u_{n}\\}\subset\mathcal{G}_{r}$, it follows that

(2.7) $\varphi(r)=\limsup_{n\rightarrow\infty}\Phi(u_{n})\leq\Phi(u_{0}).$

So, from (2.6), (2.7) and Proposition 2.1, we get $u_{0}\in\mathcal{G}_{r}$.
Therefore, $\mathcal{G}_{r}$ is weakly closed and bounded and, consequently,
weakly compact. Furthermore, by $(\Psi_{3})$ and $(\Phi_{1})$, the function
$u\mapsto\Phi^{\prime}(u)u/\Psi^{\prime}(u)u$ is well defined and is weakly
upper semicontinuous, then the maximum of this functional is attained in
$\mathcal{G}_{r}$. ∎

We say that a function $u_{r}\in\mathcal{G}_{r}$ is an energy maximum type
point if

$\frac{\Phi^{\prime}(u_{r})u_{r}}{\Psi^{\prime}(u_{r})u_{r}}=\max_{u\in\mathcal{G}_{r}}\frac{\Phi^{\prime}(u)u}{\Psi^{\prime}(u)u}.$

Observe that an energy maximum type point
$u_{r}\in\partial\tilde{\mathcal{A}}_{r}$ maximizes at the same time $\Phi$ in
$\tilde{\mathcal{A}}_{r}$ and
$u\mapsto\frac{\Phi^{\prime}(u)u}{\Psi^{\prime}(u)u}$ in $\mathcal{G}_{r}$,
and, for sure, it is nontrivial when $r>0$.

Before proving the differentiability of function $\varphi$, we need the
following lemma about the differentiability of mapping $r\mapsto t_{u}(r)$,
for each $u\neq 0$ fixed.

######  Lemma 2.6.

Suppose that $\Psi$ is coercive and satisfies $(\Psi_{2})-(\Psi_{4})$. Then,
for each $u\neq 0$ fixed, the application $r\mapsto t_{u}(r)$, given in
$(\Psi_{4})$, is $C^{1}((0,\infty),\mathbb{R})$ with

(2.8) $t_{u}^{\prime}(r)=\frac{1}{\Psi^{\prime}(t_{u}(r)u)u},\ \mbox{for all
$r>0$}.$

###### Proof.

In what follows, we define $h:[0,\infty)\to\mathbb{R}$ by $h_{u}(t)=\Psi(tu)$.
From the coercivity of $\Psi$ and $(\Psi_{2})-(\Psi_{3})$, we infer that
$h_{u}(0)=0$, $\displaystyle\lim_{t\to\infty}h_{u}(t)=\infty$ and $h_{u}$ is
increasing. Thus, there exists the inverse $h_{u}^{-1}$ and from $(\Psi_{4})$

$t_{u}(r)=h_{u}^{-1}(r),\ \mbox{for all $r>0$}.$

The result follows now from the chain’s rule. ∎

After that previous analysis, we are ready to prove that $\varphi$ is $C^{1}$
in $(0,\infty)$.

######  Proposition 2.7.

Suppose that $J=\Psi-\Phi$ belongs to class $(\mathcal{J})$ in $E$. Then,
$\varphi\in C^{1}((0,\infty),\mathbb{R})$ and

$\varphi^{\prime}(r)=\max_{u\in\mathcal{G}_{r}}\frac{\Phi^{\prime}(u)u}{\Psi^{\prime}(u)u}.$

###### Proof.

Fix $r_{0}>0$ and take $u_{r_{0}}\in\mathcal{G}_{r_{0}}$ and
$u_{r_{0}+r}\in\mathcal{G}_{r_{0}+r}$, with $r>0$. Thereby,

$\varphi(r_{0}+r)-\varphi(r_{0})=\Phi(u_{r_{0}+r})-\Phi(u_{r_{0}}).$

By $(\Psi_{4})$, there exists a unique $t_{u_{r_{0}}}(r_{0}+r)$ such that
$t_{u_{r_{0}}}(r_{0}+r)u_{r_{0}}\in\partial\tilde{\mathcal{A}}_{r_{0}+r}$. So,

$\Phi\left(t_{u_{r_{0}}}(r_{0}+r)u_{r_{0}}\right)-\Phi(u_{r_{0}})\leq\varphi(r_{0}+r)-\varphi(r_{0}).$

Since $t_{u_{r_{0}}}(r_{0})=1$, it follows from Lemma 2.6 and Mean Value
Theorem that there exists a constant $0<c_{r}<1$ such that

$\Phi\left(t_{u_{r_{0}}}(r_{0}+r)u_{r_{0}}\right)-\Phi(u_{r_{0}})=r\frac{\Phi^{\prime}\left(t_{u_{r_{0}}}(r_{0}+c_{r}r)u_{r_{0}}\right)u_{r_{0}}}{\Psi^{\prime}(t_{u_{r_{0}}}(r_{0}+c_{r}r)u_{r_{0}})u_{r_{0}}}.$

Consequently,

(2.9)
$r\frac{\Phi^{\prime}\left(t_{u_{r_{0}}}(r_{0}+c_{r}r)u_{r_{0}}\right)u_{r_{0}}}{\Psi^{\prime}(t_{u_{r_{0}}}(r_{0}+c_{r}r)u_{r_{0}})u_{r_{0}}}\leq\varphi(r_{0}+r)-\varphi(r_{0}).$

On the other hand, there exists a unique $t_{u_{r_{0}+r}}(r_{0})$ such that
$t_{u_{r_{0}+r}}(r_{0})u_{r_{0}+r}\in\partial\tilde{\mathcal{A}}_{r_{0}}$.
Then

$\varphi(r_{0}+r)-\varphi(r_{0})\leq\Phi(u_{r_{0}+r})-\Phi\left(t_{u_{r_{0}+r}}(r_{0})u_{r_{0}+r}\right).$

Again, as $t_{u_{r_{0}+r}}(r_{0}+r)=1$, it follows from Lemma 2.6 and Mean
Value Theorem that there exists a constant $0<d_{r}<1$ such that

(2.10) $\varphi(r_{0}+r)-\varphi(r_{0})\leq
r\frac{\Phi^{\prime}\left(t_{u_{r_{0}+r}}(r_{0}+d_{r}r)u_{r_{0}+r}\right)u_{r_{0}+r}}{\Psi^{\prime}\left(t_{u_{r_{0}+r}}(r_{0}+d_{r}r)u_{r_{0}+r}\right)u_{r_{0}+r}}.$

From (2.9) and (2.10),

(2.11)
$r\frac{\Phi^{\prime}\left(t_{u_{r_{0}}}(r_{0}+c_{r}r)u_{r_{0}}\right)u_{r_{0}}}{\Psi^{\prime}(t_{u_{r_{0}}}(r_{0}+c_{r}r)u_{r_{0}})u_{r_{0}}}\leq\varphi(r_{0}+r)-\varphi(r_{0})\leq
r\frac{\Phi^{\prime}\left(t_{u_{r_{0}+r}}(r_{0}+d_{r}r)u_{r_{0}+r}\right)u_{r_{0}+r}}{\Psi^{\prime}\left(t_{u_{r_{0}+r}}(r_{0}+d_{r}r)u_{r_{0}+r}\right)u_{r_{0}+r}}.$

Since $\Psi,\Phi$ are $C^{1}$ functionals, $r\mapsto t_{u_{r_{0}}}(r)$ is
continuous (see Lemma 2.6) and $t_{u_{r_{0}}}(r_{0})=1$, it is clear that

$\frac{\Phi^{\prime}\left(t_{u_{r_{0}}}(r_{0}+c_{r}r)u_{r_{0}}\right)u_{r_{0}}}{\Psi^{\prime}(t_{u_{r_{0}}}(r_{0}+c_{r}r)u_{r_{0}})u_{r_{0}}}\to\frac{\Phi^{\prime}(u_{r_{0}})u_{r_{0}}}{\Psi^{\prime}(u_{r_{0}})u_{r_{0}}},\quad\mbox{as}\quad
r\to 0.$

In particular, if we choose $u_{r_{0}}\in\mathcal{G}_{r_{0}}$ as a maximum
energy type point, we have

(2.12)
$\frac{\Phi^{\prime}\left(t_{u_{r_{0}}}(r_{0}+c_{r}r)u_{r_{0}}\right)u_{r_{0}}}{\Psi^{\prime}(t_{u_{r_{0}}}(r_{0}+c_{r}r)u_{r_{0}})u_{r_{0}}}\to\max_{u\in\mathcal{G}_{r_{0}}}\frac{\Phi^{\prime}(u)u}{\Psi^{\prime}(u)u},\quad\mbox{as}\quad
r\to 0.$

The behaviour of the right side term in inequality (2.11), as $r$ tends to
zero, is much more delicate to be analyzed, since $u_{r_{0}+r}$ is depending
on $r$. To do that, we need to prove some previous claims.

Claim 1: For each sequence $r_{n}\to 0^{+}$, we have that
$u_{r_{0}+r_{n}}\rightharpoonup u_{\ast}$, for some maximum energy type point
$u_{\ast}\in\mathcal{G}_{r_{0}}$.

From $(\Psi_{4})$ and Proposition 2.1, for each
$u\in\partial\tilde{\mathcal{A}}_{r_{0}}$, one has
$t_{u}(r_{0}+r_{n})u\in\partial\tilde{\mathcal{A}}_{r_{0}+r_{n}}$ with

(2.13) $\Phi(t_{u}(r_{0}+r_{n})u)\leq\Phi(u_{r_{0}+r_{n}})$

Moreover, by $(\Psi_{1})$, the sequence $\\{u_{r_{0}+r_{n}}\\}$ is bounded,
and so, up to a subsequence, there exists $u_{\ast}\in E$ such that

(2.14) $u_{r_{0}+r_{n}}\rightharpoonup u_{\ast}.$

From Lemma 2.6, $(\Phi_{1})$, (2.13) and (2.14),

$\Phi(t_{u}(r_{0})u)\leq\Phi(u_{\ast}).$

Since $u\in\partial\tilde{\mathcal{A}}_{r_{0}}$, it follows from $(\Psi_{4})$
that $t_{u}(r_{0})=1$. Thus

(2.15) $\Phi(u)\leq\Phi(u_{\ast}),\ \mbox{for all
$u\in\partial\tilde{\mathcal{A}}_{r_{0}}$}.$

On the other hand, from $\Psi(u_{r_{0}+r_{n}})=r_{0}+r_{n}$, (2.14) and
$(\Psi_{1})$,

(2.16) $\Psi(u_{\ast})\leq r_{0}.$

It is a consequence of (2.15), (2.16) and Proposition 2.1 that
$u_{\ast}\in\partial\tilde{\mathcal{A}}_{r_{0}}$ (in particular $u_{\ast}$ is
nontrivial) and

$\Phi(u_{\ast})=\max_{u\in\tilde{\mathcal{A}}_{r_{0}}}\Phi(u),$

and so,

(2.17) $u_{\ast}\in\mathcal{G}_{r_{0}}.$

Since $u\mapsto\Phi^{\prime}(u)u/\Psi^{\prime}(u)u$ is also weakly upper
semicontinuous, by replacing $\Phi$ for
$u\mapsto\Phi^{\prime}(u)u/\Psi^{\prime}(u)u$ and repeating the same previous
argument, we conclude that

$\frac{\Phi^{\prime}(u)u}{\Psi^{\prime}(u)u}\leq\frac{\Phi^{\prime}(u_{\ast})u_{\ast}}{\Psi^{\prime}(u_{\ast})u_{\ast}},\
\mbox{for all $u\in\mathcal{G}_{r_{0}}$}.$

The proof of the claim follows now from (2.14), (2.17) and last inequality.

Claim 2: $t_{u_{r_{0}+r_{n}}}(r_{0}+d_{r_{n}}r_{n})\to 1$ as $n\to\infty$.

From $(\Psi_{4})$,

(2.18)
$\Psi(t_{u_{r_{0}+r_{n}}}(r_{0}+d_{r_{n}}r_{n})u_{r_{0}+r_{n}})=r_{0}+d_{r_{n}}r_{n}.$

Moreover, by $(\Psi_{1})$, $\\{u_{r_{0}+r_{n}}\\}$ is bounded for $n$ large
enough. Since $\Psi(u_{r_{0}+r_{n}})=r_{0}+r_{n}$, $\\{u_{r_{0}+r_{n}}\\}$ is
far away from the origin of $E$ and, consequently,
$\\{t_{u_{r_{0}+r_{n}}}(r_{0}+d_{r_{n}}r_{n})\\}$ is bounded in $\mathbb{R}$.
Therefore, up to a subsequence,

(2.19) $t_{u_{r_{0}+r_{n}}}(r_{0}+d_{r_{n}}r_{n})\to t_{\ast}$

for some $t_{\ast}\geq 0$. From $(\Psi_{1})$, $(\Psi_{2})$, Claim 1, (2.18)
and (2.19),

$\Psi(t_{\ast}u_{\ast})\leq\lim_{n\rightarrow\infty}\Psi(t_{u_{r_{0}+r_{n}}}(r_{0}+d_{r_{n}}r_{n})u_{r_{0}+r_{n}})=\lim_{n\rightarrow\infty}(r_{0}+d_{r_{n}}r_{n})=r_{0}=\Psi(u_{\ast}).$

Showing that $t_{\ast}u_{\ast}\in\tilde{\mathcal{A}}_{r_{0}}$. On the other
hand, since $h_{u_{\ast}}(t)=\Psi(tu_{\ast})$ is increasing (see proof of
Lemma 2.6), we conclude that $t_{\ast}\leq 1$. Next, we are going to prove
that $(\Psi_{5})$ implies $t_{\ast}=1$. In fact, it is a consequence of Claim
1, $(\Psi_{5})$ and

$\Psi(u_{r_{0}+r_{n}})=r_{0}+r_{n}\to r_{0}=\Psi(u_{\ast}),$

that

(2.20) $u_{r_{0}+r}\to u_{\ast}\ \mbox{in $E$}.$

From (2.18), (2.19), (2.20) and the continuity of $\Psi$, we get

$\Psi(t_{\ast}u_{\ast})=r_{0}.$

Since $u_{\ast}\in\partial\tilde{\mathcal{A}}_{r_{0}}$, it follows from
$(\Psi_{4})$ that $t_{\ast}=1$, and the claim is proved.

Hence, from Claims 1 and 2,

(2.21)
$\limsup_{n\to\infty}\frac{\Phi^{\prime}\left(t_{u_{r_{0}+r_{n}}}(r_{0}+d_{r_{n}}r_{n})u_{r_{0}+r_{n}}\right)u_{r_{0}+r_{n}}}{\Psi^{\prime}\left(t_{u_{r_{0}+r_{n}}}(r_{0}+d_{r_{n}}r_{n})u_{r_{0}+r_{n}}\right)u_{r_{0}+r_{n}}}\leq\frac{\Phi^{\prime}(u_{\ast})u_{\ast}}{\Psi^{\prime}(u_{\ast})u_{\ast}}=\max_{u\in\mathcal{G}_{r_{0}}}\frac{\Phi^{\prime}(u)u}{\Psi^{\prime}(u)u}.$

Since $r_{n}\to 0^{+}$ is an arbitrary sequence, it follows from (2.11),
(2.12) and (2.21) that $\varphi$ is differentiable, and

$\varphi^{\prime}(r_{0})=\max_{u\in\mathcal{G}_{r_{0}}}\frac{\Phi^{\prime}(u)u}{\Psi^{\prime}(u)u}.$

The continuity of $\varphi^{\prime}$ is also a straightforward consequence of
$(\Psi_{5})$. ∎

A careful analysis about the proof of Proposition 2.7 shows us that if
$(\Psi_{5})$ is not required, we can still guarantee the existence of the
right hand derivative of $\varphi$, more specifically, we can proof that

$\varphi^{\prime}_{+}(r_{0})=\max_{u\in\mathcal{G}_{r_{0}}}\frac{\Phi^{\prime}(u)u}{\Psi^{\prime}(u)u},\
\forall\ r_{0}>0,$

and

$\limsup_{r\to
0^{-}}\frac{\varphi(r_{0}+r)-\varphi(r_{0})}{r}\leq\max_{u\in\mathcal{G}_{r_{0}}}\frac{\Phi^{\prime}(u)u}{\Psi^{\prime}(u)u},\
\forall\ r_{0}>0.$

######  Theorem 2.8.

Let $E$ be a reflexive Banach space and $J$ be a functional of class
$(\mathcal{J})$ in $E$. A maximum energy type point $u_{r}\in\mathcal{G}_{r}$
is a critical point of the functional $J$, for some $r>0$ if, and only if, $r$
is a critical point of the energy function $\zeta$.

###### Proof.

Suppose that $u_{r}$ is a maximum energy type point. By Proposition 2.1, one
has that $u_{r}$ is a maximum point of $\Phi$ restricted to
$\partial\tilde{\mathcal{A}}_{r}$. From $(\Psi_{3})$ and Lagrange multipliers
theorem (see, for example, [17, Proposition 14.3]), there exists
$\lambda_{r}\in\mathbb{R}$ such that

$\Phi^{\prime}(u_{r})v=\lambda_{r}\Psi^{\prime}(u_{r})v,\hskip
2.84544pt\mbox{for all}\hskip 2.84544ptv\in E.$

As $(\Psi_{3})$ occurs, by choosing $v=u_{r}$, we obtain

(2.22)
$\lambda_{r}=\frac{\Phi^{\prime}(u_{r})u_{r}}{\Psi^{\prime}(u_{r})u_{r}}.$

On the other hand, from (2.2), the same function $u_{r}$ is also the minimum
of the functional $J$ restricted to $\partial\tilde{\mathcal{A}}_{r}$. So,
again, from $(\Psi_{3})$ and Lagrange multipliers theorem, there exists
$\gamma_{r}\in\mathbb{R}$ such that

(2.23) $J^{\prime}(u_{r})v=\gamma_{r}\Psi^{\prime}(u_{r})v,\,\,\mbox{for
all}\,\,v\in E,$

or equivalently,

$\Psi^{\prime}(u_{r})v-\Phi^{\prime}(u_{r})v=\gamma_{r}\Psi^{\prime}(u_{r})v,\,\,\mbox{for
all}\,\,v\in E.$

By choosing $v=u_{r}$ and dividing the expression by
$\Psi^{\prime}(u_{r})u_{r}$, we arrive at

$1-\frac{\Phi^{\prime}(u_{r})u_{r}}{\Psi^{\prime}(u_{r})u_{r}}=\gamma_{r}.$

Finally, employing (2.22), we obtain the equality

(2.24) $\gamma_{r}=1-\lambda_{r}.$

By Proposition 2.7 and definition of $\zeta$,

$\zeta^{\prime}(r)=1-\frac{\Phi^{\prime}(u_{r})u_{r}}{\Psi^{\prime}(u_{r})u_{r}}.$

In other words,

(2.25)
$\zeta^{\prime}(r)=\frac{J^{\prime}(u_{r})u_{r}}{\Psi^{\prime}(u_{r})u_{r}}.$

Therefore, from equality above, if $u_{r}$ is a critical point of $J$, then
$\zeta^{\prime}(r)=0$. Now suppose that $r$ is a critical point of $\zeta$,
that is, $\zeta^{\prime}(r)=0$. Then

$\frac{\Phi^{\prime}(u_{r})u_{r}}{\Psi^{\prime}(u_{r})u_{r}}=1.$

Using (2.22), one gets

$\lambda_{r}=1.$

Then, by (2.24),

$\gamma_{r}=0$

which combines with (2.23) to give

$J^{\prime}(u_{r})v=0,\,\mbox{for all}\,v\in E,$

showing that $u_{r}$ is a critical point for $J$. ∎

######  Remark 2.9.

A careful analysis of previous proof, more specifically, of equality (2.25),
gives us a sufficient condition, in order a point in the Nehari set
$\mathcal{N}=\\{u\in E\backslash\\{0\\}:J^{\prime}(u)u=0\\}$ be a critical
point of $J\in(\mathcal{J})$. In fact, if $u\in\mathcal{N}$ is a maximum
energy type point in $\mathcal{G}_{r}$, then, by (2.25), we have
$\zeta^{\prime}(r)=0$. Now, from Theorem 2.8, $u$ is a critical point of $J$.

###  2.1. Application to a semilinear problem

Let us consider the following class of semilinear problems

(2.26) $\left\\{\begin{array}[]{lll}\displaystyle-\Delta
u=f(x,u)&\mbox{in}&\Omega,\\\ \hskip
19.91684ptu=0&\mbox{on}&\partial\Omega,\end{array}\right.$

where $\Omega\subset\mathbb{R}^{N}(N\geq 2)$ is a smooth bounded domain and
$f:\Omega\times\mathbb{R}\rightarrow\mathbb{R}$ is a Carathéodory function.

The classical result of Hammerstein [14] proves that if $f$ satisfies a linear
growth condition, and

$\limsup_{|t|\to\infty}2F(x,t)/t^{2}<\lambda_{1},\ \mbox{uniformly in $x$},$

then problem (2.26) has a solution, where $\lambda_{1}$ is the first
eigenvalue of laplacian operator with Dirichlet boundary condition. Some years
later, in 1986, Mawhin, Ward and Willem [25] improve Hammerstein’s result by
allowing the previous upper limit to “touch” the first eigenvalue of
laplacian, except on a set of positive measure, that is, the authors consider
that there exists $\theta\in L^{\infty}(\Omega)$ such that

(2.27) $\limsup_{|t|\to\infty}2F(x,t)/t^{2}\leq\theta(x)\leq\lambda_{1},\
\mbox{uniformly in $x$},$

with $\theta(x)<\lambda_{1}$ on subset of $\Omega$ of positive measure. The
main goal of this section is to complement the works previously mentioned, by
using the method (MEF). To be more precise, we are going to show that if $f$
is nonnegative in $(0,\infty)$ and positive for small values of $t>0$, then
hypothesis (2.27) can be relaxed.

In this section, for each $\beta\in L^{s}(\Omega)$ with $s>N/2$,
$\lambda_{1}(\beta)$ stands for the first eigenvalue of

(2.28) $\left\\{\begin{array}[]{lll}\displaystyle-\Delta
u=\lambda\beta(x)u&\mbox{in}&\Omega,\\\ \hskip
19.91684ptu=0&\mbox{on}&\partial\Omega.\end{array}\right.$

Function $f$ satisfies the following assumptions:

  * $(f_{1})$

There exists $C>0$ such that

$|f(x,t)|\leq C\left(1+|t|^{p-1}\right),\,\mbox{for
all}\,t\in(0,\infty)\,\mbox{and a.e. in \,$\Omega$},$

with $1<p<2^{\ast}$ if $N\geq 3$ and $p>1$ if $N=2$;

  * $(f_{2})$

$f$ is nonnegative in $(0,\infty)$ and positive in $(0,\varepsilon)$, for some
$\varepsilon>0$;

  * $(f_{3})$

There exist functions $\alpha_{\ast}$, $\alpha^{\ast}$ and $\eta$ such that

$\displaystyle\left\\{\begin{array}[]{lll}\alpha^{\ast},\eta\in
L^{\infty}(\Omega)\ \mbox{and
$\lambda_{1}(\alpha_{\ast})<1\leq\lambda_{1}(\eta)$}&\mbox{if}&f(x,0)=0,\\\
\hskip 19.91684pt\eta\in L^{\infty}(\Omega)\mbox{ and
$1\leq\lambda_{1}(\eta)$}&\mbox{if}&f(x,0)>0,\end{array}\right.$

with

$\alpha_{\ast}(x)=\liminf_{t\rightarrow 0^{+}}2F(x,t)/t^{2},\
\alpha^{\ast}(x)=\limsup_{t\rightarrow 0^{+}}2F(x,t)/t^{2},\ \mbox{uniformly
in $x$},$

and

$\eta(x)=\limsup_{t\rightarrow\infty}2F(x,t)/t^{2},\ \mbox{uniformly in $x$}.$

Some considerations are now necessary:

  1. (1)

Observe that, from $(f_{2})$, the primitive
$F^{\ast}(t)=\int_{0}^{t}f^{\ast}(x,s)\,ds$ is positive and nondecreasing for
all $t>0$. Thus, $0\leq\alpha_{\ast}\leq\alpha^{\ast}$ and $\eta\geq 0$.
Consequently, $(f_{3})$ implies that $\alpha_{\ast}\in L^{\infty}(\Omega)$
and, by [9], there exist positive first eigenvalues
$\lambda_{1}(\alpha_{\ast})$ and $\lambda_{1}(\eta)$.

  2. (2)

If $f(x,0)>0$, then the generalized L’Hospital rule implies that
$\alpha_{\ast}=\alpha^{\ast}=\infty$. In fact, in this case

$\infty=\liminf_{t\rightarrow 0^{+}}f(x,t)/t\leq\liminf_{t\rightarrow
0^{+}}2F(x,t)/t^{2}=\alpha_{\ast}(x).$

  3. (3)

It is important to point out that inequality $1\leq\lambda_{1}(\eta)$ in
$(f_{3})$ is considerably weaker than (2.27). In fact, it is a consequence of
Proposition 1.12A in [9] that (2.27) implies $1<\lambda_{1}(\eta)$. On the
other hand, there exist functions $\eta$ which satisfy $(f_{3})$ and are
greater than $\lambda_{1}$ in some subset $\Omega_{0}\subset\Omega$ with
positive measure (and so, they are not covered by hypothesis (2.27) in [25]).
In effect, for each $k\in\mathbb{N}$, let us consider the set of positive
measure

$\Omega_{k}=\\{x\in\Omega:dist(x,\partial\Omega)<1/k\\}$

and

$\eta_{k}(x)=\frac{\lambda_{1}}{2}\chi_{\Omega\backslash\Omega_{k}(x)}+\frac{3\lambda_{1}}{2}\chi_{\Omega_{k}}(x),$

where $\chi_{A}$ denotes the characteristic function of a measurable set
$A\subset\Omega$. It is clear that $\eta_{k}\in L^{\infty}(\Omega)$ and

(2.30) $\eta_{k}\to\eta_{\infty}=\frac{\lambda_{1}}{2}\ \mbox{in
$L^{N/2}(\Omega)$}$

with

(2.31) $\lambda_{1}(\eta_{\infty})=2>1.$

It follows from (2.30), (2.31) and [9], that

(2.32) $\lambda_{1}(\eta_{k})\to 2.$

Thus, from (2.32) there exists $k_{\ast}\in\mathbb{N}$ such that, defining
$\eta_{\ast}:=\eta_{k_{\ast}}$, we have

(2.33) $\lambda_{1}(\eta_{\ast})>1.$

Now, consider for example the function

(2.34) $f(x,t)=\left\\{\begin{array}[]{lll}-f(x,-t),&\mbox{if}&t<0,\\\
\eta_{\ast}(x)t^{2}&\mbox{if}&0\leq t<1/4,\\\
\eta_{\ast}(x)\frac{t^{3}}{(3/16)+t^{2}}&\mbox{if}&t\geq
1/4.\end{array}\right.$

We have that $f$ satisfies $(f_{1})-(f_{3})$ with
$\alpha_{\ast}=\alpha^{\ast}=0$ and $\eta=\eta_{\ast}$ (see (2.33)). Moreover,

$\eta(x)=\frac{3\lambda_{1}}{2}>\lambda_{1}\ \mbox{in $\Omega_{k_{\ast}}$}.$

In order to find a solution for (2.26), we will first prove the existence of a
solution to the auxiliary problem

(2.35) $\left\\{\begin{array}[]{lll}\displaystyle-\Delta
u=f^{\ast}(x,u)&\mbox{in}&\Omega,\\\ \hskip
19.91684ptu=0&\mbox{on}&\partial\Omega,\end{array}\right.$

where

(2.36) $f^{\ast}(x,t)=\left\\{\begin{array}[]{lll}f(x,0),&\mbox{if}&t<0,\\\
f(x,t),&\mbox{if}&t\geq 0.\end{array}\right.$

For sure, $f^{\ast}$ satisfies $(f_{1})-(f_{3})$.

The functional $J$ associated with the problem (2.35) is given by

$J(u)=\Psi(u)-\Phi(u),$

where

$\Psi(u)=\frac{1}{2}\|u\|^{2}\hskip 14.22636pt\mbox{and}\hskip
14.22636pt\Phi(u)=\int_{\Omega}F^{\ast}(x,u)dx.$

Now, we are going to show that $J$ belongs to class $(\mathcal{J})$ in
$H_{0}^{1}(\Omega)$. In fact, by norm properties, hypothesis $(\Psi_{1})$ is
clearly satisfied. Moreover, by considering the change of variable
$r=\frac{1}{2}\rho^{2}$, we get $\tilde{\mathcal{A}}_{r}=B_{\rho}$ and
$\partial\tilde{\mathcal{A}}_{r}=S_{\rho}$, where $B_{\rho}$ and $S_{\rho}$
denote, respectively, the closed ball and the sphere of radius $\rho$ and
centered at the origin of $H_{0}^{1}(\Omega)$. Thus, the hypotheses
$(\Psi_{2})-(\Psi_{3})$ are certainly satisfied, $(\Psi_{4})$ holds with
$t_{u}(r)=r/\|u\|$, and $(\Psi_{5})$ follows from the uniform convexity of
$H_{0}^{1}(\Omega)$.

In the next lemmas, we prove that $\Phi$ verifies $(\Phi_{1})-(\Phi_{3})$.

######  Lemma 2.10.

The functional $\Phi$ satisfies $(\Phi_{1})$.

###### Proof.

Suppose that

$u_{n}\rightharpoonup u\hskip 5.69046pt\mbox{in}\hskip
5.69046ptH_{0}^{1}(\Omega).$

By compact embeddings, up to a subsequence, one has

$u_{n}\rightarrow u\hskip 5.69046pt\mbox{in}\hskip 5.69046ptL^{p}(\Omega),$
$u_{n}(x)\to u(x)\quad\mbox{a.e. in}\quad\Omega,$

and there exists $g\in L^{p}(\Omega)$ such that

$|u_{n}|\leq g(x),\quad\mbox{a.e. in}\quad\Omega.$

Therefore,

(2.37) $F^{\ast}(x,u_{n}(x))\rightarrow F^{\ast}(x,u(x))\hskip
5.69046pt\mbox{a.e.}\hskip 2.84544pt\mbox{in}\hskip 5.69046pt\Omega,$

and by $(f_{1})$,

$|F^{\ast}(x,u_{n}(x))|\leq
C\left(|u_{n}(x)|+\frac{1}{p}|u_{n}(x)|^{p}\right)\leq
C\left(g(x)+\frac{1}{p}g(x)^{p}\right)\quad\mbox{a.e. in}\quad\Omega.$

Hence, by Lebesgue’s dominated convergence theorem

$\Phi(u_{n})=\int_{\Omega}F^{\ast}(x,u_{n})dx\rightarrow\int_{\Omega}F^{\ast}(x,u)dx=\Phi(u),$

showing that $\Phi$ is weakly continuous. Since $u\mapsto\Phi^{\prime}(u)u$ is
also continuous and nonnegative, $\Psi^{\prime}(u)u=\|u\|^{2}$ is positive and
weakly lower semicontinuous, we conclude that
$u\mapsto\Phi^{\prime}(u)u/\Psi^{\prime}(u)u$ is weakly upper semicontinuous.

∎

######  Lemma 2.11.

The functional $\Phi$ satisfies $(\Phi_{2})$.

###### Proof.

Suppose that $\Phi^{\prime}(u)=0$, that is,

$\int_{\Omega}f^{\ast}(x,u)vdx=0,\hskip 2.84544pt\mbox{for all}\hskip
2.84544ptv\in H_{0}^{1}(\Omega).$

By [7, Corollary 4.24], $f^{\ast}(x,u)=0$ a.e. in $\Omega$, and so, by
($f_{2}$), we must have $u\leq 0$. From this, $F^{\ast}(x,u)\leq 0$ and $\Phi$
verifies ($\Phi_{2}$).

∎

######  Lemma 2.12.

The functional $\Phi$ satisfies $(\Phi_{3})$.

###### Proof.

Since $F^{\ast}(x,t)$ is positive in $\Omega\times(0,\infty)$, it is enough to
choose a positive function $v\in H_{0}^{1}(\Omega)$ and define $u_{n}=(1/n)v$.
It is clear that $u_{n}\to 0$ in $H_{0}^{1}(\Omega)$ and $\Phi(u_{n})>0$ for
all $n\in\mathbb{N}$.

∎

The next result ensures that any solution for (2.35) is also a solution for
(2.26).

######  Lemma 2.13.

Let $r=\frac{1}{2}\rho^{2}$. If

$u_{0}\in\mathcal{G}_{r}=\left\\{u\in B_{\rho};\hskip
2.84544pt\Phi(u)=\max_{u\in S_{\rho}}\int_{\Omega}F^{\ast}(x,u)dx\right\\},$

then $u_{0}\geq 0$ a.e. in $\Omega$.

###### Proof.

Supposing by contraction that $u_{0}\not\equiv|u_{0}|$, then we must have
$|[u_{0}<0]>0$. Since

$\int_{[u_{0}<0]}F^{\ast}(x,u_{0})dx<\int_{[u_{0}<0]}F^{\ast}(x,|u_{0}|)dx$

and

$\int_{[u_{0}\geq 0]}F^{\ast}(x,u_{0})dx=\int_{[u_{0}\geq
0]}F^{\ast}(x,|u_{0}|)dx,$

it follows that

$\Phi(u_{0})=\int_{\Omega}F^{\ast}(x,u_{0})dx<\int_{\Omega}F^{\ast}(x,|u_{0}|)dx=\Phi(|u_{0}|),$

with $|u_{0}|\in S_{\rho}$. Since $u_{0}$ maximizes $\Phi$ on $S_{\rho}$, we
get a contradiction. Therefore, $u_{0}=|u_{0}|$. ∎

######  Theorem 2.14.

Under hypotheses $(f_{1})-(f_{3})$, problem (2.26) has a nonnegative and
nontrivial weak solution.

###### Proof.

Since $J$ is of class $(\mathcal{J})$ in $H_{0}^{1}(\Omega)$, the energy
function $\zeta:[0,\infty)\rightarrow\mathbb{R}$ is given by

$\zeta(r)=r-\max_{u\in\partial\tilde{\mathcal{A}}_{r}}\int_{\Omega}F^{\ast}(x,u)dx.$

Let $\sigma(\rho)=\zeta((1/2)\rho^{2})$. Then,

$\sigma(\rho)=\frac{1}{2}\rho^{2}-\max_{u\in
S_{1}}\int_{\Omega}F^{\ast}(x,\rho u)dx,$

and, by Proposition 2.7, it is $C^{1}$ in $(0,+\infty)$. Moreover, since
$\sigma^{\prime}(\rho)=\zeta^{\prime}((1/2)\rho^{2})\rho$, it follows that if
$\rho>0$ is a critical point of $\sigma$, then $r=(1/2)\rho^{2}$ is a critical
point of $\zeta$.

In order to apply Theorem 2.8, we will show that $\zeta$ has a critical point.
Have this in mind, let $u_{\ast}$ be a positive function with
$\|u_{\ast}\|=1$. Then

$\frac{\sigma(\rho)}{\rho^{2}}=\frac{1}{2}-\max_{u\in
S_{1}}\int_{\Omega}\frac{F^{\ast}(x,\rho
u)}{\rho^{2}}dx\leq\frac{1}{2}-\int_{\Omega}\frac{F^{\ast}(x,\rho
u_{\ast})}{(\rho u_{\ast})^{2}}u_{\ast}^{2}dx.$

Employing $(f_{3})$ and Fatou’s Lemma, one gets

(2.38) $\limsup_{\rho\rightarrow
0^{+}}\frac{\sigma(\rho)}{\rho^{2}}\leq\frac{1}{2}-\liminf_{\rho\rightarrow
0^{+}}\int_{\Omega}\frac{F^{\ast}(x,\rho u_{\ast})}{(\rho
u_{\ast})^{2}}u_{\ast}^{2}dx\leq\frac{1}{2}-\frac{1}{2}\int_{\Omega}\alpha_{\ast}(x)u_{\ast}^{2}dx.$

Now, we have two cases to consider. If $f(x,0)>0$, then $\alpha_{\ast}=\infty$
and (2.38) implies

$\limsup_{\rho\rightarrow 0^{+}}\frac{\sigma(\rho)}{\rho^{2}}=-\infty.$

By other side, if $f(x,0)=0$ then $\alpha_{\ast}\in L^{\infty}(\Omega)$ and,
in this case, again by $(f_{3})$ and replacing $u_{\ast}$ by the positive
eigenfunction $e_{1}$ of $\lambda_{1}(\alpha_{\ast})$ such that $\|e_{1}\|=1$,
we get from (2.38)

$\limsup_{\rho\rightarrow
0^{+}}\frac{\sigma(\rho)}{\rho^{2}}\leq\frac{1}{2}\left(1-\frac{1}{\lambda_{1}(\alpha_{\ast})}\right)<0.$

In any case, $\sigma$ is negative for $\rho$ small enough. As $\sigma(0)=0$,
our next step is to prove that for each $\varepsilon>0$ there exists
$\tilde{\rho}>0$ such that $\sigma(\rho)>-\varepsilon$ if $\rho>\tilde{\rho}$,
because this is enough to conclude the existence of a critical point for
$\sigma$. For that, let us consider a sequence
$\\{\rho_{n}\\}\subset(0,\infty)$ with $\rho_{n}\rightarrow\infty$,
$\\{v_{n}\\}\subset H_{0}^{1}(\Omega)$, $\|v_{n}\|=1$ and
$\\{\rho_{n}v_{n}\\}\subset\mathcal{G}_{\rho_{n}}$. Thus

$\frac{\sigma(\rho_{n})}{\rho_{n}^{2}}=\frac{1}{2}-\int_{[v_{n}>0]}\frac{F^{\ast}(x,\rho_{n}v_{n})}{(\rho_{n}v_{n})^{2}}v_{n}^{2}dx,$

that is,

(2.39)
$\frac{\sigma(\rho_{n})}{\rho_{n}^{2}}=\frac{1}{2}-\int_{\Omega}\frac{F^{\ast}(x,\rho_{n}v_{n})}{(\rho_{n}v_{n})^{2}}v_{n}^{2}\chi_{[v_{n}>0]}dx.$

Since $\\{v_{n}\\}$ is a bounded sequence with $v_{n}\geq 0$ (see Lemma 2.13),
for some subsequence, we derive that there exists $v_{0}\in
H_{0}^{1}(\Omega)$, with $v_{0}\geq 0$ and $\|v_{0}\|\leq 1$, such that

$v_{n}\rightharpoonup v_{0}\hskip 2.84544pt\hskip 2.84544pt\mbox{in}\hskip
2.84544pt\hskip 2.84544ptH_{0}^{1}(\Omega),$ $v_{n}\rightarrow v_{0}\hskip
2.84544pt\hskip 2.84544pt\mbox{in}\hskip 2.84544pt\hskip
2.84544ptL^{2}(\Omega),$ $v_{n}(x)\rightarrow v_{0}(x)\hskip 2.84544pt\hskip
2.84544pt\mbox{a.e.}\hskip 2.84544pt\hskip 2.84544pt\mbox{in}\hskip
2.84544pt\hskip 2.84544pt\Omega$

and there exists $h\in L^{2}(\Omega)$ such that

$|v_{n}|\leq h(x)\hskip 2.84544pt\hskip 2.84544pt\mbox{a.e.}\hskip
2.84544pt\hskip 2.84544pt\mbox{in}\hskip 2.84544pt\hskip 2.84544pt\Omega.$

Since $F^{\ast}$ is nonnegative, conditions $(f_{1})$ and $(f_{3})$ yield that
there exists $K>0$ such that

(2.40) $\left|\frac{F^{\ast}(x,t)}{t^{2}}\right|\leq K,\quad\forall
t\in\mathbb{R}\quad\mbox{and a.e. in }\quad\Omega.$

From (2.40),

(2.41)
$\left|\frac{F^{\ast}(x,\rho_{n}v_{n}(x))}{(\rho_{n}v_{n}(x))^{2}}v_{n}^{2}(x)\chi_{[v_{n}>0]}(x)\right|\leq|h(x)|^{2}K\hskip
2.84544pt\hskip 2.84544pt\mbox{a.e.}\hskip 2.84544pt\hskip
2.84544pt\mbox{in}\hskip 2.84544pt\hskip 2.84544pt\Omega.$

If $v_{0}=0$, the compactness Sobolev embedding together with (2.40) gives

$\liminf_{n\rightarrow\infty}\frac{\sigma(\rho_{n})}{\rho_{n}^{2}}=\frac{1}{2}>0.$

Since $F$ is positive, for the case $v_{0}\not=0$, observe that,

$\liminf_{n\rightarrow\infty}\frac{\sigma(\rho_{n})}{\rho_{n}^{2}}\geq\frac{1}{2}-\limsup_{n\rightarrow\infty}\int_{[v_{0}>0]}\left[\frac{F^{\ast}(x,\rho_{n}v_{n})}{(\rho_{n}v_{n})^{2}}\right]\chi_{[v_{n}>0]}(x)v_{n}^{2}dx$

Since

$\chi_{[v_{n}>0]}(x)\to 1\ \mbox{on $[v_{0}>0]$},$

we employ (2.41) together with the version of Fatou’s lemma for the upper
limit to conclude that

$\liminf_{n\rightarrow\infty}\frac{\sigma(\rho_{n})}{\rho_{n}^{2}}\geq\frac{1}{2}-\frac{1}{2}\int_{\Omega}\eta(x)v_{0}^{2}dx.$

Thus, by $(f_{3}),$

(2.42)
$\liminf_{n\rightarrow\infty}\frac{\sigma(\rho_{n})}{\rho_{n}^{2}}\geq\frac{1}{2}\left(1-\frac{1}{\lambda_{1}(\eta)}\right)\|v_{0}\|^{2}\geq
0.$

Therefore, $\sigma$ has a critical point $\rho_{\ast}>0$. Consequently,
$r_{\ast}=(1/2)\rho_{\ast}^{2}$ is a critical point of $\zeta$ and, by Theorem
2.8, there exists a maximum energy type point
$u_{r_{\ast}}\in\mathcal{G}_{r_{\ast}}$ which is a nontrivial and nonnegative
solution of problem (2.35). Since $f$ and $f^{\ast}$ coincide for nonnegative
values, we conclude that $u_{r_{\ast}}$ is also a solution of problem (2.26).
∎

##  3\. A version of the Mountain Pass Theorem

In this section, we will provide a version of the mountain pass theorem by
using the Method (MEF).

######  Theorem 3.1.

Let $E$ be a reflexive Banach space and $J$ be a functional of class
$(\mathcal{J})$ in $E$. Suppose that there exist $\alpha,\rho>0$ and $w\in E$,
with $w\in\mathcal{G}_{R}$, such that

  * $(H_{1})$

$J(u)\geq\alpha>J(0)$ for all $u\in\partial\tilde{\mathcal{A}}_{\rho}$;

  * $(H_{2})$

$J(w)<\alpha,$ with $R>\rho$.

Then there holds the equality below

(3.1)
$c_{\ast}=\max_{r\in[0,R]}\min_{u\in\partial\tilde{\mathcal{A}}_{r}}J(u)\leq\inf_{\gamma\in\Gamma}\max_{t\in[0,1]}J(\gamma(t))=c,$

where $\Gamma=\\{\gamma\in C([0,1],E);\gamma(0)=0,\gamma(1)=w\\}$. Moreover,
$c_{\ast}>\max\\{J(0),J(w)\\}$ is a critical value of $J$.

###### Proof.

Since $J$ is a functional of class $(\mathcal{J})$, the energy function

$\zeta(r)=\min_{u\in\partial\tilde{\mathcal{A}}_{r}}J(u)$

is well-defined. By assumption $(H_{1})$,

$\zeta(\rho)=\min_{u\in\partial\tilde{\mathcal{A}}_{\rho}}J(u)>J(0)=\zeta(0).$

On the other hand, from $(H_{2})$,

$\zeta(R)=\min_{u\in\partial\tilde{\mathcal{A}}_{R}}J(u)\leq
J(w)<\min_{u\in\partial\tilde{\mathcal{A}}_{\rho}}J(u)=\zeta(\rho).$

So, due to the geometry of $\zeta$, there exists $r_{\ast}\in(0,R)$ such that

$\zeta(r_{\ast})=\max_{r\in(0,R)}\zeta(r)=\max_{r\in[0,R]}\zeta(r)\geq\zeta(\rho)>\max\\{J(0),J(w)\\}\quad\mbox{and}\quad\zeta^{\prime}(r_{\ast})=0$

Hence, by Theorem 2.8, the number $c_{\ast}$ given below

$\max\\{J(0),J(w)\\}<c_{\ast}=\max_{r\in(0,R)}\zeta(r)=\max_{r\in[0,R]}\zeta(r)=\max_{r\in[0,R]}\min_{u\in\partial\tilde{\mathcal{A}}_{r}}J(u).$

is a critical value of $J$.

Next, we will prove that

$c_{\ast}=\max_{r\in[0,R]}\min_{u\in\partial\tilde{\mathcal{A}}_{r}}J(u)\leq\inf_{\gamma\in\Gamma}\max_{t\in[0,1]}J(\gamma(t))=c.$

Consider a path $\gamma\in\Gamma$ and the critical point $r_{\ast}\in(0,R)$
given above. Since $\gamma(0)=0\in\tilde{\mathcal{A}}_{r_{\ast}}$,
$\gamma(1)=w\not\in\tilde{\mathcal{A}}_{r_{\ast}}$ (otherwise, we would have
$J(w)\geq c_{\ast}$) and $\gamma([0,1])$ is connected, we have that
$\gamma([0,1])\cap\partial\tilde{\mathcal{A}}_{r_{\ast}}$ is a nonempty set.
So,

$\max_{r\in[0,R]}\zeta(r)=\zeta(r_{\ast})=\min_{u\in\partial\tilde{\mathcal{A}}_{r_{\ast}}}J(u)\leq\max_{t\in[0,1]}J(\gamma(t)),\,\mbox{for
all}\,\gamma\in\Gamma,$

from where it follows that

$\max_{r\in[0,R]}\zeta(r)\leq\inf_{\gamma\in\Gamma}\max_{t\in[0,1]}J(\gamma(t))=c,$

or equivalently,

(3.2)
$c_{\ast}=\max_{r\in[0,R]}\min_{u\in\partial\tilde{\mathcal{A}}_{r}}J(u)\leq\inf_{\gamma\in\Gamma}\max_{t\in[0,1]}J(\gamma(t))=c.$

∎

Next result gives a sufficient condition for the equality $c=c_{\ast}$.

######  Corollary 3.2.

Let $E$ be a reflexive Banach space and $J$ be a functional of class
$(\mathcal{J})$ in $E$ satisfying assumptions $(H_{1})-(H_{2})$ given in
Theorem 3.1. If the mountain pass level $c$ coincides with the infimum of $J$
on the Nehari manifold $\mathcal{N}$ associated to $J$, that is,

$c=\inf_{u\in\mathcal{N}}J(u),\ \mbox{where}\ \mathcal{N}=\\{u\in
E\backslash\\{0\\}:J^{\prime}(u)u=0\\},$

then $c_{\ast}=c=\inf_{u\in\mathcal{N}}J(u)$.

###### Proof.

Since $c_{\ast}$ is a critical value of $J$, from Theorem 3.1, there exists
$u_{\ast}\in E$ such that $J(u_{\ast})=c_{\ast}>J(0)$. Therefore,
$u_{\ast}\neq 0$, and $u_{\ast}\in\mathcal{N}$. Showing that
$c_{\ast}\geq\inf_{u\in\mathcal{N}}J(u)$. The result now follows from
inequality (3.1). ∎

Situations where $c$ coincides with $\inf_{u\in\mathcal{N}}J(u)$ are well
known in the literature and cover a large class of concrete problems in
applications. In these particular cases, Corollary 3.2 provides a new
characterization for the mountain pass level constant $c$, then it can be seen
as a maxmin level.

Another possible advantage to have a “mountain pass theorem” for maxmin
critical levels is in the determination of ground state critical points for
$J$, that is, a critical point $w\in\mathcal{N}$ satisfying
$J(w)=\inf_{u\in\mathcal{N}}J(u)$. Next result gives a step in this direction.

######  Corollary 3.3.

Let $E$ be a reflexive Banach space and $J$ be a functional of class
$(\mathcal{J})$ in $E$ satisfying assumptions $(H_{1})-(H_{2})$ given in
Theorem 3.1. If,

(3.3) $J(u)\geq J(tu),\ \mbox{for all}\ t>0\ \mbox{and}\ u\in\mathcal{N},$

then $J$ has a ground state critical point.

###### Proof.

As we have seen in the proof of previous corollary,

(3.4) $c_{\ast}\geq\inf_{u\in\mathcal{N}}J(u).$

To conclude the proof, it is enough to prove the opposite inequality. From
Theorem 3.1, there exists a critical point $u_{\ast}$ for $J$ satisfying
$J(u_{\ast})=c_{\ast}=\zeta(r_{\ast})$. Let $u\in\mathcal{N}$. It follows from
$(\Psi_{4})$ that there exists $t_{u}(r_{\ast})>0$ such that
$t_{u}(r_{\ast})u\in\partial\tilde{\mathcal{A}}_{r_{\ast}}$. Thus, by (3.3),

(3.5) $J(u)\geq J(t_{u}(r_{\ast})u)\geq J(u_{\ast})=c_{\ast}.$

where the last inequality in (3.5) is a consequence of
$J(u_{\ast})=\min_{u\in\partial\tilde{\mathcal{A}}_{r_{\ast}}}J(u)$. Since
$u\in\mathcal{N}$ is arbitrary in (3.5), we conclude that

(3.6) $\inf_{u\in\mathcal{N}}J(u)\geq c_{\ast}.$

The result follows now from (3.4) and (3.6).

∎

##  4\. Some applications

###  4.1. A general nonlinear problem

In this section we are going to investigate the following problem

(4.1) $\left\\{\begin{array}[]{lll}\displaystyle-\Delta
u=f(x,u)&\mbox{in}&\Omega,\\\ \hskip
19.91684ptu=0&\mbox{on}&\partial\Omega,\end{array}\right.$

where $\Omega\subset\mathbb{R}^{N}(N\geq 2)$ is a smooth bounded domain and
$f:\Omega\times\mathbb{R}\to\mathbb{R}$ is a Carathéodory function satisfying
the growth condition $(f_{1})$, introduced in Subsection 2.1.

In [2], Ambrosetti and Rabinowitz proved existence of nontrivial solution for
(4.1) by considering the following conditions on $f$:

  * $(F_{2})$

$\displaystyle\lim_{t\to 0}\frac{f(x,t)}{|t|}=0$;

  * $(AR)$

there are constants $\nu>2$ and $r\geq 0$ such that

$0<\nu F(x,t)\leq tf(x,t),|t|\geq r,\ F(x,t)=\int_{0}^{t}f(x,s)\,ds.$

Condition (AR) is very important to provide some compactness for Palais-Smale
sequences, which is a crucial point in the Mountain Pass Theorem presented in
[2]. However, despite its technical relevance, hypothesis (AR) is quite
restrictive and in the last years, many authors have concentrate efforts in
the search for weaker conditions that still guarantee the existence of
nontrivial solution for (4.1). Observe that (AR) implies

(4.2) $F(x,t)\geq c_{3}|t|^{\nu}-c_{4},\ x\in\Omega,\ |t|\geq r.$

In 1994, Costa and Magalhães [8] studied (4.1) under the following assumptions
on $f$:

  * $(F_{1})_{q}$

$\displaystyle\limsup_{|t|\rightarrow\infty}\frac{2F(x,t)}{|t|^{q}}\leq
b<\infty$ uniformly for a.e. $x$ in $\Omega$;

  * $(F_{1})_{\mu}$

$\displaystyle\liminf_{|t|\rightarrow\infty}\frac{H(x,t)}{|t|^{\mu}}\geq a>0$
uniformly for a.e. $x$ in $\Omega$;

  * $(F_{3})$

$\displaystyle\limsup_{t\rightarrow
0}\frac{2F(x,t)}{t^{2}}\leq\alpha<\lambda_{1}<\beta\leq\liminf_{|t|\rightarrow\infty}\frac{2F(x,t)}{t^{2}}$
uniformly for a.e. $x$ in $\Omega$,

where $H(x,t)=f(x,s)s-2F(x,t)$, $\mu>(N/2)(q-2)$ and, in practice, $q$ is
smaller than $p$ in $(f_{1})$. Under hypotheses $(F_{1})_{q}$, $(F_{1})_{\mu}$
and $(F_{3})$, the authors were able to prove the existence of nontrivial
solution to (4.1). Since $\mu>2$ in (AR), it follows from (4.2), that
hypothesis (AR) implies $(F_{1})_{\mu}$ (for instance, when $q\leq 2+(4/N)$)
and the last inequality in $(F_{3})$.

In 2003, Willem and Zou [28] also considered a more general version of (4.1)
and were able to prove existence of nontrivial solution by replacing (AR). In
fact, instead (AR), the authors considered the condition

  * $(WZ)$

$tf(x,t)\geq 0,\ t\in\mathbb{R}\ \mbox{and}\ tf(x,t)\geq c_{0}|t|^{\nu},\
|t|\geq r$,

where $c_{0}>$, $r\geq 0$ and $\nu>2$. Hypothesis (WZ) is certainly weaker
than (AR), but it still implies (4.2).

In 2004, Schechter and Zou [26] obtained an existence result for (4.1), under
conditions $(f_{1}),(F_{2})$ and

  * $(SZ1)$

$\displaystyle\lim_{t\to\infty}\frac{F(x,t)}{t^{2}}=\infty$ uniformly in $x$;

  * $(SZ2)$

$\nu F(x,t)-tf(x,t)\leq C(t^{2}+1)$, $|t|\geq r$, or $H(x,s)$ is convex in
$s$,

for some constants $\nu>2$ and $r\geq 0$. Hypothesis (SZ1) is much more weak
than (AR) and is known in the literature as superquadraticity’s condition. As
observed in [23], the first part of (SZ2) is equivalent to (AR).

In 2008, Miyagaki and Souto [23] solved (4.1) by assuming $f(x,0)=0$,
$(f_{1})$, $(F_{2})$, $(SZ1)$ and the following monotonicity’s condition:
There exists $t_{0}>0$ such that

  * $(MS)$

$t\mapsto f(x,t)/t$ is increasing in $(t_{0},\infty)$ and decreasing in
$(-\infty,-t_{0})$, for all $x\in\Omega$.

Note that (MS) is weaker than the second part of (SZ2).

In the present work, we are going to treat problem (4.1) by supposing that $f$
satisfies conditions $(f_{1})$ and $(f_{2})$, in Subsection 2.1, and

  1. $(f_{3}^{\prime})$

$\displaystyle\left\\{\begin{array}[]{lll}\eta\in L^{s}(\Omega)\ \mbox{with
$s>N/2$, $\alpha\in L^{\infty}(\Omega)$ and $\lambda_{1}(\eta)\leq
1<\lambda_{1}(\alpha)$}&\mbox{if}&\eta\not\equiv\infty,\\\ \hskip
19.91684pt\alpha\in L^{\infty}(\Omega)\mbox{ and
$1\leq\lambda_{1}(\alpha)$}&\mbox{if}&\eta\equiv\infty,\end{array}\right.$

with

(4.4) $\alpha(x)=\limsup_{t\rightarrow 0^{+}}2F(x,t)/t^{2}\ \mbox{and}\
\eta(x)=\liminf_{t\rightarrow\infty}2F(x,t)/t^{2},\ \mbox{uniformly in $x$}.$

It is a consequence of $(f_{3}^{\prime})$ that $f(x,0)=0$.

In order to compare our assumptions with other hypotheses previously
mentioned, observe that $(f_{3}^{\prime})$ is weaker than $(F_{3})$ in [8] or
$(F_{2})$ and $(SZ1)$ in [26]. Moreover, $(f_{2})$ (or even
$(f_{2}^{\prime})$) and $(f_{3}^{\prime})$ together cover functions which do
not satisfy (WZ) in [28] and $(MS)$ in [23]. Let us consider, for instance,
the Caratheódory function $f:\Omega\times\mathbb{R}\to\mathbb{R}$, given by

$f(x,t)=a(x)\log(t+1)(2+\cos t)t\ \mbox{if $t>0$, and $f(x,t)=0$ if $t\leq
0$.}$

for some $a\in L^{\infty}(\Omega)$ with $a>0$. We have that $\alpha(x)=0$ and
$\eta(x)=\infty$, therefore, $f$ satisfies $(f_{1})$, $(f_{2})$ and
$(f_{3}^{\prime})$, but it does not satisfy $(WZ)$ and $(MS)$. Our main
theorem in this section is the following.

######  Theorem 4.1.

Under hypotheses $(f_{1})$, $(f_{2})$ and $(f_{3}^{\prime})$, problem (4.1)
has a nonnegative nontrivial solution.

###### Proof.

As in Subsection 2.1, we consider the auxiliary problem

(4.5) $\left\\{\begin{array}[]{lll}\displaystyle-\Delta
u=f^{\ast}(x,u)&\mbox{in}&\Omega,\\\ \hskip
19.91684ptu=0&\mbox{on}&\partial\Omega,\end{array}\right.$

where

$f^{\ast}(x,t)=\left\\{\begin{array}[]{lll}0,&\mbox{if}&t<0,\\\
f(x,t),&\mbox{if}&t\geq 0.\end{array}\right.$

Again, due to $(f_{1})-(f_{2})$, as in Subsection 2.1, $F^{\ast}$ is positive
and nondecreasing for all $t>0$, functional $J$ associated with the problem
(4.5) is well defined and is of class $(\mathcal{J})$ in $H_{0}^{1}(\Omega)$,
where

$J(u)=\Psi(u)-\Phi(u),$

with

$\Psi(u)=\frac{1}{2}\|u\|^{2}\hskip 14.22636pt\mbox{and}\hskip
14.22636pt\Phi(u)=\int_{\Omega}F^{\ast}(x,u)dx.$

It remains us to prove that $J$ satisfies $(H_{1})$ and $(H_{2})$ given in
Theorem 3.1. Indeed, let us consider a sequence
$\\{\rho_{n}\\}\subset(0,\infty)$ with $\rho_{n}\rightarrow 0$ and
$\\{v_{n}\\}\subset H_{0}^{1}(\Omega)$ with $\|v_{n}\|=1$ and
$\\{\rho_{n}v_{n}\\}\subset\mathcal{G}_{\rho_{n}}$. Since $(f_{2})$ holds,
Lemma 2.13 is still valid here. Thus

(4.6)
$\frac{\sigma(\rho_{n})}{\rho_{n}^{2}}=\frac{1}{2}-\int_{\Omega}\frac{F^{\ast}(x,\rho_{n}v_{n})}{(\rho_{n}v_{n})^{2}}v_{n}^{2}\chi_{[v_{n}>0]}dx,$

and there exists $v_{0}\in H_{0}^{1}(\Omega)$, with $v_{0}\geq 0$ and
$\|v_{0}\|\leq 1$, such that

(4.7) $v_{n}\rightharpoonup v_{0}\hskip 2.84544pt\hskip
2.84544pt\mbox{in}\hskip 2.84544pt\hskip 2.84544ptH_{0}^{1}(\Omega),\
v_{n}\rightarrow v_{0}\hskip 2.84544pt\hskip 2.84544pt\mbox{in}\hskip
2.84544pt\hskip 2.84544ptL^{2}(\Omega)\ \mbox{and}\ v_{n}(x)\rightarrow
v_{0}(x)\hskip 2.84544pt\hskip 2.84544pt\mbox{a.e.}\hskip 2.84544pt\hskip
2.84544pt\mbox{in}\hskip 2.84544pt\hskip 2.84544pt\Omega,$

and there exists $h\in L^{2}(\Omega)$ such that

(4.8) $|v_{n}|\leq h(x)\hskip 2.84544pt\hskip 2.84544pt\mbox{a.e.}\hskip
2.84544pt\hskip 2.84544pt\mbox{in}\hskip 2.84544pt\hskip 2.84544pt\Omega.$

Fixing $R>0$, we have

(4.9)
$\frac{\sigma(\rho_{n})}{\rho_{n}^{2}}=\frac{1}{2}-\int_{[\rho_{n}v_{n}\leq
R]}\frac{F^{\ast}(x,\rho_{n}v_{n})}{(\rho_{n}v_{n})^{2}}v_{n}^{2}\chi_{[v_{n}>0]}dx-\frac{1}{\rho^{2}_{n}}\int_{[\rho_{n}v_{n}>R]}F^{\ast}(x,\rho_{n}v_{n})dx,$

It follows from $(f_{1})$ that there exists $q\in(2,2^{\ast})$ such that
$|F^{\ast}(x,t)|\leq C|t|^{q}$ for all $|t|\geq R$ and some $C>0$. So, from
$\rho_{n}\to 0$, (4.7) and (4.8), we conclude that

(4.10)
$\frac{1}{\rho^{2}_{n}}\int_{[\rho_{n}v_{n}>R]}F^{\ast}(x,\rho_{n}v_{n})dx\to
0.$

On the other hand, since $F^{\ast}$ is nonnegative, conditions $(f_{1})$ and
$(f_{3}^{\prime})$ yield that there exists $K>0$ such that

(4.11) $\left|\frac{F^{\ast}(x,t)}{t^{2}}\right|\leq K,\quad\forall\ |t|\leq
R\quad\mbox{and a.e. in }\quad\Omega.$

From (4.11) and (4.8),

(4.12)
$\left|\frac{F^{\ast}(x,\rho_{n}v_{n}(x))}{(\rho_{n}v_{n}(x))^{2}}v_{n}^{2}(x)\chi_{[v_{n}>0]}(x)\right|\leq|h(x)|^{2}K\hskip
2.84544pt\hskip 2.84544pt\mbox{a.e.}\hskip 2.84544pt\hskip
2.84544pt\mbox{in}\hskip 2.84544pt\hskip 2.84544pt[\rho_{n}v_{n}(x)\leq R].$

If $v_{0}=0$, the compactness Sobolev embeeding together with (4.9) and (4.12)
gives

$\liminf_{n\rightarrow\infty}\frac{\sigma(\rho_{n})}{\rho_{n}^{2}}=\frac{1}{2}>0.$

For the case $v_{0}\not=0$, we employ (4.12) together with versions for the
upper limit of Fatou’s lemma and the convergence

$\chi_{[v_{n}>0]\cap[\rho_{n}v_{n}\leq R]}(x)\to 1,\ \mbox{a.e. in
$[v_{0}>0]$}$

to get

(4.13)
$\liminf_{n\rightarrow\infty}\frac{\sigma(\rho_{n})}{\rho_{n}^{2}}\geq\frac{1}{2}-\frac{1}{2}\int_{\Omega}\alpha(x)v_{0}^{2}dx.$

Thus, by $(f_{3}^{\prime})$

$\liminf_{n\rightarrow\infty}\frac{\sigma(\rho_{n})}{\rho_{n}^{2}}\geq\frac{1}{2}\left(1-\frac{1}{\lambda_{1}(\alpha)}\right)\|v_{0}\|^{2}>0.$

Therefore, there exists $\rho$ small enough such that
$\displaystyle\min_{u\in\mathcal{S}_{\rho}}J(u)=\sigma(\rho)>0$. Showing that
$(H_{1})$ holds.

Now let $e_{1}\in S_{1}$ be the positive eigenfunction associated to
$\lambda_{1}(\eta)$. Then,

$\frac{J(re_{1})}{r^{2}}=\frac{1}{2}-\int_{\Omega}\frac{F^{\ast}(x,re_{1})}{(re_{1})^{2}}e_{1}^{2}dx.$

Since $F^{\ast}$ is a positive function when $t\in(0,\infty)$, the Fatou’s
Lemma gives

$\limsup_{r\rightarrow\infty}\frac{J(re_{1})}{r^{2}}=\frac{1}{2}-\liminf_{r\rightarrow\infty}\int_{\Omega}\frac{F^{\ast}(x,re_{1})}{(re_{1})^{2}}e_{1}^{2}dx\leq\frac{1}{2}-\frac{1}{2}\int_{\Omega}\eta(x)e_{1}^{2}dx.$

If $\eta\equiv\infty$, it follows from previous inequality that

$\limsup_{r\rightarrow\infty}\frac{J(re_{1})}{r^{2}}=-\infty.$

By other side, if $\eta\not\equiv\infty$, by $(f_{3}^{\prime})$, we conclude
that

$\limsup_{r\rightarrow\infty}\frac{J(re_{1})}{r^{2}}\leq\frac{1}{2}\left(1-\frac{1}{\lambda_{1}(\eta)}\right)\leq
0.$

In any case, given $\underline{\alpha}>0$ with $\underline{\alpha}<\alpha$,
there exists $R>0$ sufficiently large such that

$J(Re_{1})\leq\underline{\alpha}<\alpha,$

proving $(H_{2})$. Thereby, by Theorem 3.1, $J$ attains a critical value in

$c_{\ast}=\max_{r\in[0,R]}\min_{\|u\|=r}J(u).$

In other words, there exists $r_{\ast}\in(0,R)$ such that $u_{r_{\ast}}$ is a
nontrivial solution to the problem (4.5). Since $f$ and $f^{\ast}$ coincide
for nonnegative values, we conclude that $u_{r_{\ast}}$ is also a solution to
the problem (4.15).

∎

###  4.2. A Berestycki-Lions type problem

In 1983, Berestycki and Lions [6] were the first to solve the problem

(4.14) $\left\\{\begin{array}[]{lll}\displaystyle-\Delta u=g(u)\hskip
2.84544pt\hskip 2.84544pt\mbox{in}\hskip 2.84544pt\hskip
2.84544pt\mathbb{R}^{N},\\\ \hskip 11.38092ptu\in
H^{1}(\mathbb{R}^{N}),\end{array}\right.$

without using the Ambrosseti and Rabinowitz condition. However, the method
used in [6] is not valid for non-autonomous problems, even though the function
$g(x,u)$ is radially symmetric with respect to $x$.

In 2009, Azzollini and Pomponio [4], motivated by approach explored in
Jeanjean [16], considered the problem above with $g(x,t)=f(t)-V(x)t$, which
can be rewritten of the form

(4.15) $\left\\{\begin{array}[]{lll}\displaystyle-\Delta u+V(x)u=f(u)\hskip
2.84544pt\hskip 2.84544pt\mbox{in}\hskip 2.84544pt\hskip
2.84544pt\mathbb{R}^{N},\\\ \hskip 11.38092ptu\in
H^{1}(\mathbb{R}^{N}),\end{array}\right.$

assuming that $N\geq 1$ and $V\in C^{1}(\mathbb{R}^{N},\mathbb{R})$ satisfying
the following conditions:

  * $(V_{1}^{\prime})$

$V(x)=V(|x|)$;

  * $(V_{2}^{\prime})$

$V(x)\geq 0$ for all $x\in\mathbb{R}^{N}$ and the inequality is strict
somewhere;

  * $(V_{3}^{\prime})$

$\|(\nabla V(\cdot)|\cdot)^{+}\|_{N/2}<2S$, where $\displaystyle
S=\inf_{u\in{D}^{1,2}(\mathbb{R}^{N})\setminus\\{0\\}}\frac{|\nabla
u|_{2}^{2}}{|u|_{2^{\ast}}^{2}}$;

  * $(V_{4}^{\prime})$

$\displaystyle\lim_{|x|\rightarrow\infty}V(x)=0$.

In the present section, we intend to use Theorem 3.1 to complement the main
result found in [4], in the sense we will only consider the following
conditions on potential $V$.

  * $(V_{1})$

$V(x)=V(|x|)$;

  * $(V_{2})$

There exist $V_{0},V_{\infty}>0$ such that $V_{0}\leq V(x)\leq
V_{\infty},\,\forall\,x\in\mathbb{R}^{N}$.

By changing $(V_{2}^{\prime})$–$(V_{4}^{\prime})$ for $(V_{2})$, it is
possible to obtain functions that do not satisfy the conditions of [4], but
that satisfies our hypotheses, see for example the function

$V(x)=\frac{|x|+k_{1}}{|x|+k_{2}},\hskip 2.84544pt\hskip
2.84544pt\mbox{with}\hskip 2.84544pt\hskip 2.84544pt0<k_{1}<k_{2}.$

With respect to $f\in C(\mathbb{R},\mathbb{R})$, let us consider the following
assumptions:

  * $(f_{1})$

$\lim_{t\rightarrow 0}\frac{f(t)}{t}=0\hskip 14.22636pt\mbox{and}\hskip
14.22636pt\limsup_{|t|\rightarrow\infty}\frac{|f(t)|}{|t|^{p-1}}<\infty,\,\,\mbox{where}\,\,2<p<2^{\ast};$

  * $(f_{2})$

There exists $t_{0}>0$ such that $F(t_{0})-\frac{V_{\infty}}{2}t_{0}^{2}>0$;

  * $(f_{3})$

$f(t)>0$ for all $t>0$.

In the sequel, we will restrict the functional $J$ to the subspace
$H^{1}_{rad}(\mathbb{R}^{N})\subset H^{1}(\mathbb{R}^{N})$, because by
Strauss’ Lemma [17, Theorem 1.2, Chapter 6] we have the compact embedding

(4.16) $H^{1}_{rad}(\mathbb{R}^{N})\hookrightarrow
L^{p}(\mathbb{R}^{N}),\,\,\mbox{with}\,\,2<p<2^{\ast}.$

As in the previous section, let us consider the auxiliary problem

(4.17) $\left\\{\begin{array}[]{lll}\displaystyle-\Delta
u+V(x)u=f^{\ast}(u)&\mbox{in}&\mathbb{R}^{N},\\\ \hskip 19.91684ptu\in
H^{1}_{rad}(\mathbb{R}^{N}),\end{array}\right.$

where

$f^{\ast}(t)=\left\\{\begin{array}[]{lll}0,&\mbox{if}&t<0,\\\
f(t),&\mbox{if}&t\geq 0.\end{array}\right.$

A simple computation gives that $f^{\ast}$ satisfies the same conditions as
$f$.

Due to $(f_{3})$, $F^{\ast}$ does not change sign in $(0,\infty)$. Then, from
$(f_{2})$, $F^{\ast}$ is positive and increasing for all $t>0$. The functional
$J:H^{1}_{rad}(\mathbb{R}^{N})\rightarrow\mathbb{R}$ is given by

$J(u)=\Psi(u)-\Phi(u),$

where

$\Psi(u)=\frac{1}{2}\int_{\mathbb{R}^{N}}\left(|\nabla
u|^{2}+V(x)|u|^{2}\right)dx\hskip 14.22636pt\mbox{and}\hskip
14.22636pt\Phi(u)=\int_{\mathbb{R}^{N}}F^{\ast}(u)dx.$

Hereafter, we will denote by $\|\,\,\,\,\|_{*}$ the following norm in
$H^{1}(\mathbb{R}^{N})$

$\|u\|_{*}=\left(\int_{\mathbb{R}^{N}}(|\nabla
u|^{2}+V(x)|u|^{2})\,dx\right)^{\frac{1}{2}}.$

By condition $(V_{2})$ it is easy to check that the norm above is equivalent
the usual norm in $H^{1}(\mathbb{R}^{N})$.

Next, we will show that $J$ belongs to class $(\mathcal{J})$.

######  Lemma 4.2.

The functional $\Phi$ satisfies $(\Phi_{1})$.

###### Proof.

By $(f_{1})$, given $\varepsilon>0$, there exists a constant
$C=C(\varepsilon)>0$ such that

$|f^{\ast}(t)|\leq\varepsilon|t|+C|t|^{p-1},\quad\forall t\in\mathbb{R}$

From this inequality,

(4.18)
$|F^{\ast}(t)|\leq\frac{\varepsilon}{2}|t|^{2}+\frac{C}{p}|t|^{p},\quad\forall
t\in\mathbb{R},$

where $2<p<2^{\ast}$. Setting the functions
$P,Q:\mathbb{R}\rightarrow\mathbb{R}$ by

$P(t)=F^{\ast}(t)\hskip 2.84544pt\hskip 2.84544pt\hskip
2.84544pt\mbox{and}\hskip 2.84544pt\hskip 2.84544pt\hskip
2.84544ptQ(t)=t^{2}+|t|^{2^{\ast}},$

it follows that

(4.19) $\lim_{|t|\rightarrow\infty}\frac{P(t)}{Q(t)}=0$

and

(4.20) $\lim_{t\rightarrow 0}\frac{P(t)}{Q(t)}=0.$

Finally, assuming that $u_{n}\rightharpoonup u_{0}$ in
$H^{1}_{rad}(\mathbb{R}^{N})$, we derive that

(4.21) $\sup_{n\in\mathbb{N}}\int_{\mathbb{R}^{N}}|Q(u_{n}(x))|dx<\infty.$

Therefore, (4.19), (4.20) and (4.21) permit to apply [6, Theorem A.I] to get

$\Phi(u_{n})=\int_{\mathbb{R}^{N}}F^{\ast}(u_{n})dx\rightarrow\int_{\mathbb{R}^{N}}F^{\ast}(u_{0})dx=\Phi(u_{0}),$

proving that $\Phi$ is weakly continuous. It is possible to obtain the same
result for $u\mapsto\Phi^{\prime}(u)u$.

∎

For the other hypotheses of $\Phi$, the procedure is similar to the previous
application. Moreover, it is also true that, fixed $r>0$, if

$u_{0}\in\mathcal{G}_{r}=\left\\{u\in\tilde{\mathcal{A}}_{r};\hskip
2.84544pt\Phi(u)=\max_{u\in\partial\tilde{\mathcal{A}}_{r}}\int_{\mathbb{R}^{N}}F^{\ast}(u)dx\right\\},$

then $u_{0}\geq 0$ a.e. in $\mathbb{R}^{N}$.

######  Theorem 4.3.

Under hypotheses $(V_{1})$–$(V_{2})$ and $(f_{1})$–$(f_{3})$, problem (4.15)
has a nontrivial weak solution.

###### Proof.

In what follows, we will prove that $J$ satisfies $(H_{1})$ and $(H_{2})$ in
Theorem 3.1. Indeed, by (4.18),

$J(u)\geq\|u\|_{\ast}^{2}-\frac{\varepsilon}{2}|u|_{2}^{2}-\frac{C}{p}|u|_{p}^{p}.$

Fixing $\varepsilon>0$ small and using continuous Sobolev embeddings, we
arrive at

$J(u)\geq C_{1}\|u\|_{\ast}^{2}-C_{2}\|u\|_{\ast}^{p}.$

As $p>2$, for $\rho$ small enough, there exists $\alpha>0$ such that

$J(u)\geq\alpha,\hskip 2.84544pt\mbox{for all}\hskip 2.84544ptu\in
H^{1}_{rad}(\mathbb{R}^{N})\hskip 2.84544pt\hskip 2.84544pt\mbox{with}\hskip
2.84544pt\hskip 2.84544pt\|u\|_{\ast}=\rho,$

proving $(H_{1})$.

On the other hand, using $(f_{2})$, note that there exists $\varphi\in
C_{0}^{\infty}(\mathbb{R}^{N})$ such that

(4.22)
$\int_{\mathbb{R}^{N}}F(\varphi)dx-\frac{V_{\infty}}{2}|\varphi|_{2}^{2}>0.$

By taking $\displaystyle w(t,x)=\varphi\left(\frac{x}{t}\right)$ with $t>0$,
we obtain

$\int_{\mathbb{R}^{N}}|\nabla
w|^{2}dx=\frac{1}{t^{2}}\int_{\mathbb{R}^{N}}\left|\nabla\varphi\left(\frac{x}{t}\right)\right|^{2}dx=t^{N-2}\int_{\mathbb{R}^{N}}|\nabla\varphi(x)|^{2}dx$

and

$\int_{\mathbb{R}^{N}}V(x)\left|\varphi\left(\frac{x}{t}\right)\right|^{2}dx=t^{N}\int_{\mathbb{R}^{N}}V(xt)|\varphi(x)|^{2}dx.$

Fixing $R_{t}=\|w\|_{\ast}$, one gets

$R_{t}^{2}=t^{N-2}\int_{\mathbb{R}^{N}}|\nabla\varphi(x)|^{2}dx+t^{N}\int_{\mathbb{R}^{N}}V(xt)|\varphi(x)|^{2}dx.$

Consequently,

$J(w)\leq\frac{t}{2}^{N-2}\int_{\mathbb{R}^{N}}|\nabla\varphi|^{2}dx+\frac{t}{2}^{N}\int_{\mathbb{R}^{N}}V_{\infty}|\varphi|^{2}dx-t^{N}\int_{\mathbb{R}^{N}}F(\varphi)dx,$

that is,

$J(w)\leq\frac{t}{2}^{N-2}\int_{\mathbb{R}^{N}}|\nabla\varphi|^{2}dx-t^{N}\left(\int_{\mathbb{R}^{N}}F(\varphi)dx-\frac{V_{\infty}}{2}\int_{\mathbb{R}^{N}}|\varphi|^{2}dx\right).$

Thus, by (4.22), $J(w)<0$ for $t$ large enough. Note that
$R_{t}\rightarrow\infty$ when $t\rightarrow\infty$, because we have the
inequality below

$t^{N-2}|\nabla\varphi|_{2}^{2}+V_{0}t^{N}|\varphi|_{2}^{2}\leq R_{t}^{2}\leq
t^{N-2}|\nabla\varphi|_{2}^{2}+V_{\infty}t^{N}|\varphi|_{2}^{2}.$

From this, $(H_{2})$ also occurs and we can use the Theorem 3.1 to conclude
that $J$ attains a critical value in

$c_{\ast}=\max_{r\in[0,R]}\min_{\|u\|_{\ast}=r}J(u).$

In other words, there exists $r_{\ast}\in(0,R)$ such that $u_{r_{\ast}}$ is a
nontrivial critical point of $J$ in $H_{rad}^{1}(\mathbb{R}^{N})$. However, we
can apply the Palais Criticality Principle [28] to guarantee that
$u_{r_{\ast}}$ is also a critical point of $J$ in $H^{1}(\mathbb{R}^{N})$ and,
since that $f$ and $f^{\ast}$ coincide for nonnegative values, we deduce that
$u_{r_{\ast}}$ is a solution to the problem (4.15).

∎

###  4.3. An anisotropic equation

Consider the problem

(4.23)
$\left\\{\begin{array}[]{lll}\displaystyle-\sum_{i=1}^{N}\partial_{i}(|\partial_{i}u|^{p_{i}-2}\partial_{i}u)=f(u)&\mbox{in}&\Omega,\\\
\hskip 96.73918ptu=0&\mbox{on}&\partial\Omega,\end{array}\right.$

where $N\geq 2$, $p_{i}$’s are ordered , that is, $1<p_{1}\leq
p_{2}\leq...\leq p_{N}$ and $f:\mathbb{R}\rightarrow\mathbb{R}$ is a
continuous function with $f(0)=0$. In 2009, Di Castro and Montefusco [10]
solved the problem (4.23) considering $f(u)=\lambda|u|^{q-2}u$, with
$\lambda>0$ and $p_{1}<q<p_{N}$. Note that the function chosen by the authors
already satisfies the Ambrosseti and Rabinowitz condition. So, our goal is to
solve (4.23) using a more general function.

In our case, $f$ satisfies the following conditions:

  * $(f_{1})$

There exists $C>0$ such that

$|f(t)|\leq C\left(1+|t|^{q-1}\right)\,\mbox{for
all}\,t\in(0,\infty),\,\mbox{with}\,\,p_{1}<q<p_{N};$

  * $(f_{2})$

$0\leq\limsup_{t\rightarrow
0^{+}}\frac{F(t)}{|t|^{p_{N}}}<\frac{\xi}{\Theta^{p_{N}}}\,\,\mbox{and}\,\,\,\,\mathcal{S}\leq\liminf_{t\rightarrow\infty}\frac{F(t)}{|t|^{p_{N}}},$

where $F(x,t)=\int_{0}^{t}f(x,s)ds$ and $\mathcal{S},\Theta$ and $\xi$ are
given in (4.24), (4.26) and (4.25) respectively.

  * $(f_{3})$

$f(t)>0$ for all $t>0$.

As function space we will consider the space $W_{0}^{1,\vec{p}}(\Omega)$
endowed with the norm

$\|u\|_{\vec{p}}=\sum_{i=1}^{N}\|\partial_{i}u\|_{p_{i}},$

where $\vec{p}=(p_{1},p_{2},\ldots,p_{N})$.

Recall that $W_{0}^{1,\vec{p}}(\Omega)$ is a reflexive Banach space which is
compactly embedded in $L^{s}(\Omega)$ for all $s\in[1,p_{\infty})$ (see [11]),
where $p_{\infty}=\max\\{p_{N},\bar{p}^{\ast}\\}$, $\bar{p}=\displaystyle
N/\left(\displaystyle\sum^{N}_{i=1}\frac{1}{p_{i}}\right)$ is the harmonic
mean of the $p_{i}$’s and

$p^{\ast}:=\displaystyle\frac{N}{\left(\displaystyle\sum^{N}_{i=1}\frac{1}{p_{i}}\right)-1}=\displaystyle\frac{N\overline{p}}{N-\overline{p}}.$

Moreover, we fix $\Theta,\xi>0$ satisfying

(4.24) $|u|_{p_{N}}\leq\Theta\|u\|_{\vec{p}},\quad\forall u\in
W_{0}^{1,\vec{p}}(\Omega)$

and

(4.25)
$\xi\|u\|^{p_{N}}_{\vec{p}}\leq\frac{1}{p_{N}}\sum_{i=1}^{N}\|\partial_{i}u\|^{p_{N}}_{p_{i}},\quad\forall
u\in W_{0}^{1,\vec{p}}(\Omega).$

The existence of such a constant $\Theta$ is guaranteed in [11, Theorem 1].

Hereafter, we assume that $p_{N}<\bar{p}^{\ast}$, and so, we have the compact
embedding $W_{0}^{1,\vec{p}}(\Omega)\hookrightarrow L^{p_{N}}(\Omega)$.
Moreover, we will denote by $\mathcal{S}>0$ the following constant

(4.26) $\mathcal{S}=\min_{u\in
W_{0}^{1,\vec{p}}(\Omega)}\left\\{\sum_{i=1}^{N}\|\partial_{i}u\|^{p_{i}}_{p_{i}}\;:\;\|u^{+}\|_{p_{N}}=1\right\\}.$

######  Theorem 4.4.

Under hypotheses $(f_{1})-(f_{3})$, problem (4.23) has a nontrivial weak
solution.

###### Proof.

We will first prove the existence of a solution to the auxiliary problem

(4.27)
$\left\\{\begin{array}[]{lll}\displaystyle-\sum_{i=1}^{N}\partial_{i}(|\partial_{i}u|^{p_{i}-2}\partial_{i}u)=f^{\ast}(u)&\mbox{in}&\Omega,\\\
\hskip 96.73918ptu=0&\mbox{on}&\partial\Omega,\end{array}\right.$

where $f^{\ast}$ is defined by

$f^{\ast}(t)=\left\\{\begin{array}[]{lll}0,&\mbox{if}&t<0,\\\
f(t),&\mbox{if}&t\geq 0.\end{array}\right.$

Due to $(f_{3})$, $F^{\ast}$ does not change sign in $(0,\infty)$. Then, from
$(f_{2})$, $F^{\ast}$ is positive and increasing for all $t>0$. The functional
$J:W_{0}^{1,\vec{p}}(\Omega)\rightarrow\mathbb{R}$ associated with the problem
(4.27) is given by

$J(u)=\sum_{i=1}^{N}\frac{1}{p_{i}}\int_{\Omega}|\partial_{i}u|^{p_{i}}dx-\int_{\Omega}F^{\ast}(u)dx.$

In the sequel

$\Psi(u)=\sum_{i=1}^{N}\frac{1}{p_{i}}\int_{\Omega}|\partial_{i}u|^{p_{i}}dx\quad\mbox{and}\quad\Phi(u)=\int_{\Omega}F^{\ast}(u)dx.$

######  Lemma 4.5.

Hypothesis $(\Psi_{5})$ is satisfied in $W_{0}^{1,\vec{p}}(\Omega)$.

###### Proof.

Let $(u_{n})\in W_{0}^{1,\vec{p}}(\Omega)$ with $u_{n}\rightharpoonup u$ and
$\displaystyle\lim_{n\to+\infty}\Psi(u_{n})=\Psi(u)$. Then, by the weak
convergence,

$\sum_{i=1}^{N}\frac{1}{p_{i}}\int_{\Omega}|\partial_{i}u|^{p_{i}}dx=\limsup_{n\to+\infty}\sum_{i=1}^{N}\frac{1}{p_{i}}\int_{\Omega}|\partial_{i}u_{n}|^{p_{i}}dx\geq\liminf_{n\to+\infty}\sum_{i=1}^{N}\frac{1}{p_{i}}\int_{\Omega}|\partial_{i}u_{n}|^{p_{i}}dx\geq\sum_{i=1}^{N}\frac{1}{p_{i}}\int_{\Omega}|\partial_{i}u|^{p_{i}}dx,$

hence, for a some subsequence, one has

$\sum_{i=1}^{N}\frac{1}{p_{i}}\int_{\Omega}|\partial_{i}u|^{p_{i}}dx=\sum_{i=1}^{N}\frac{1}{p_{i}}\limsup_{n\to+\infty}\int_{\Omega}|\partial_{i}u_{n}|^{p_{i}}dx\geq\sum_{i=1}^{N}\frac{1}{p_{i}}\liminf_{n\to+\infty}\int_{\Omega}|\partial_{i}u_{n}|^{p_{i}}dx\geq\sum_{i=1}^{N}\frac{1}{p_{i}}\int_{\Omega}|\partial_{i}u|^{p_{i}}dx,$

If there is $i\in\\{1,...,N\\}$ such that

$\liminf_{n\to+\infty}\int_{\Omega}|\partial_{i}u_{n}|^{p_{i}}dx>\int_{\Omega}|\partial_{i}u|^{p_{i}}dx,$

or

$\limsup_{n\to+\infty}\int_{\Omega}|\partial_{i}u_{n}|^{p_{i}}dx>\liminf_{n\to+\infty}\int_{\Omega}|\partial_{i}u|^{p_{i}}dx,$

we get a contradiction. From this,

$\limsup_{n\to+\infty}\int_{\Omega}|\partial_{i}u_{n}|^{p_{i}}dx=\liminf_{n\to+\infty}\int_{\Omega}|\partial_{i}u_{n}|^{p_{i}}dx=\sum_{i=1}^{N}\frac{1}{p_{i}}\int_{\Omega}|\partial_{i}u|^{p_{i}}dx,\quad\forall
i\in\\{1,...N\\}.$

Using the fact that $L^{p_{i}}(\Omega)$ is uniformly convex and that
$\partial_{i}u_{n}\rightharpoonup\partial_{i}u$ in $L^{p_{i}}(\Omega)$, we
derive that

$\partial_{i}u_{n}\to\partial_{i}u\quad\mbox{in}\quad
L^{p_{i}}(\Omega),\quad\forall i\in\\{1,...N\\},$

showing that

$u_{n}\to u\quad\mbox{in}\quad W_{0}^{1,\vec{p}}(\Omega),$

finishing the proof.

∎

Similar to what was done in the previous sections, it is possible to prove
that $J$ is of class $(\mathcal{J})$ and that the functions $u_{r}$ are
nonnegative. It remains us to prove that $J$ satisfies $(H_{1})$ and $(H_{2})$
in Theorem 3.1. Indeed, by $(f_{1})-(f_{2})$, given
$\epsilon\in(0,\frac{\xi}{\Theta^{p_{N}}})$ there exists $C>0$ such that

$F^{\ast}(t)\leq\epsilon|t|^{p_{N}}+C|t|^{\overline{p}^{*}},\quad\forall
t\in\mathbb{R}.$
$J(u)=\sum_{i=1}^{N}\frac{1}{p_{i}}\int_{\Omega}|\partial_{i}u|^{p_{i}}dx-\int_{\Omega}F^{\ast}(u)dx\geq\frac{1}{p_{N}}\sum_{i=1}^{N}\int_{\Omega}|\partial_{i}u|^{p_{i}}dx-\epsilon\Theta^{p_{N}}\|u\|_{\vec{p}}^{p_{N}}\,dx-C\|u\|_{\vec{p}}^{\overline{p}^{*}},$

For $\|u\|_{\vec{p}}\leq 1$, we get

$J(u)\geq\frac{1}{p_{N}}\sum_{i=1}^{N}\|\partial_{i}u\|^{p_{N}}_{p_{i}}-\epsilon\Theta^{p_{N}}\|u\|_{\vec{p}}^{p_{N}}\,dx-C\|u\|_{\vec{p}}^{\overline{p}^{*}}\geq\xi\|u\|_{\vec{p}}^{p_{N}}-\epsilon\Theta^{p_{N}}\|u\|_{\vec{p}}^{p_{N}}\,dx-C\|u\|_{\vec{p}}^{\overline{p}^{*}},$

where $\xi$ was given in (4.25). From $(f_{2})$,

$J(u)\geq C_{1}\|u\|_{\vec{p}}^{p_{N}}-C\|u\|_{\vec{p}}^{\overline{p}^{*}}.$

As $\overline{p}^{*}>p_{N}$, for $\rho$ small enough, there exists $\alpha>0$
such that

$J(u)\geq\alpha,\hskip 2.84544pt\mbox{for all}\hskip 2.84544ptu\in
W_{0}^{1,\vec{p}}(\Omega)\hskip 2.84544pt\hskip 2.84544pt\mbox{with}\hskip
2.84544pt\hskip 2.84544pt\|u\|_{\ast}=\rho,$

proving $(H_{1})$.

Now let $v\in W_{0}^{1,\vec{p}}(\Omega)$ such that

$\mathcal{S}=\sum_{i=1}^{N}\|\partial_{i}u\|^{p_{i}}_{p_{i}}\quad\mbox{and}\quad\|v^{+}\|_{p_{N}}=1$

and $r>0$ such that $rv\in\partial\tilde{\mathcal{A}}_{R}$ . Then,
$r\to+\infty$ if, and only if $R\to+\infty$. Thus,

$\frac{J(rv)}{r^{p_{N}}}=\sum_{i=1}^{N}\frac{r^{p_{i}}}{r^{p_{N}}p_{i}}\int_{\Omega}|\partial_{i}v|^{p_{i}}dx-\int_{[v>0]}\frac{F^{\ast}(rv)}{|rv|^{p_{N}}}|v|^{p_{N}}dx.$

Thus, for $r>1$,

$\limsup_{r\to+\infty}\frac{J(rv)}{r^{p_{N}}}\leq\sum_{i=1}^{N}\int_{\Omega}|\partial_{i}v|^{p_{i}}dx-\mathcal{S}\int_{[v>0]}|v|^{p_{N}}dx=\mathcal{S}-\mathcal{S}\leq
0.$

So, given $\underline{\alpha}>0$ with $\underline{\alpha}<\alpha$, there
exists $r>0$ sufficiently large such that

$J(rv)\leq\underline{\alpha}<\alpha,$

proving $(H_{2})$.

Therefore, by Theorem 3.1, there exists $r_{\ast}\in(0,R)$ such that
$u_{r_{\ast}}$ is a nontrivial solution to the problem (4.27). Since $f$ and
$f^{\ast}$ coincide for nonnegative values, we conclude that $u_{r_{\ast}}$ is
also a solution to the problem (4.23).

∎

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