Existence of solution for a class of problem in whole $\mathbb{R}^N$ without the Ambrosetti-Rabinowitz condition

TL;DR

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Contribution

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Abstract

In this paper we study the existence of solution for a class of elliptic problem in whole $\mathbb{R}^N$ without the well known Ambrosetti-Rabinowitz condition. Here, we do not assume any monotonicity condition on $f(s)/s$ for $s>0$.

Equations

$$I(u)=\frac{1}{2}{\int }_{\Omega }|\nabla u{|}^{2}dx-{\int }_{\Omega }F(u)dx,\forall u\in H_0^1(\Omega ),$$

$$Z(s)\le DZ(t),\text{mbox}for0\le s\le t,$$

$$V(x)<V_{\infty }=\underset{|x|\to +\infty }{\lim }V(x).$$

$$V(x+y)=V(x),\forall x\in {\mathbb{R}}^N\text{mbox}and\forall y\in {\mathbb{Z}}^N.$$

$$V(x)\to +\infty \text{mbox}as|x|\to +\infty .$$

$$V(x)=1+\lambda W(x),$$

$$\mathcal{H}(s)=sh(s)-2H(s)\ge {\sigma }_o{s}^2,\text{mbox}forall|s|\ge s_o.$$

$$0<\frac{h({\rm Y})}{f({\rm Y})}=\max \left\{\frac{h(t)}{f(t)}:t\in [0,{\rm Y}]\right\}\text{mbox}and\frac{h({\rm Y})}{f({\rm Y})}=\min \left\{\frac{h(t)}{f(t)}:t\in [{\rm Y},+\infty )\right\}.$$

$$|G(s)|\le c|s|+\frac{c}{p}|s{|}^p,\forall s\in \mathbb{R}.$$

$$\underset{|s|\to +\infty }{\lim }\frac{g(s)}{|s{|}^{p-2}s}=1,\text{mbox}forsomep>2.$$

$$|g(s)s-\theta G(s)|\le \frac{(\theta -2)V_o|s{|}^2}{4\theta },\text{mbox}forall|s|\le s_o.$$

$$E=\left\{u\in D^{1,2}({\mathbb{R}}^N);{\int }_{{\mathbb{R}}^N}(|\nabla u{|}^2+V(x)|u{|}^2)dx<+\infty \right\}$$

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

$$J(u)=\frac{1}{2}\Vert u{\Vert }^{2}dx-{\int }_{{\mathbb{R}}^N}G(u)dx.$$

$$J^{\prime }(u)v={\int }_{{\mathbb{R}}^N}(\nabla u\nabla v+uv)dx-{\int }_{{\mathbb{R}}^N}g(u)vdx,\forall u,v\in E.$$

$$c=\underset{\gamma \in \Gamma }{\inf }J(\gamma (t))>0$$

Full text

Claudianor Alves was partially supported by CNPq/Brazil Proc. 304804/2017-7 ; Marco A.S. Souto was partially supported by CNPq/Brazil Proc. 306082/2017-9

1. Introduction

In the last decades a lot of authors have studied the existence of solution for elliptic problems of the form

[TABLE]

where $\Omega\subset\mathbb{R}^{N}$ is a smooth bounded domain and $f$ is a continuous function with subcritical growth satisfying some technical conditions. In general, the main conditions on $f$ are the following:

[TABLE]

There is $q\in(2,2^{*})$, where $2^{*}=\frac{2N}{N-2}$, if $N\geq 3$ or $q\in(2,+\infty)$ if $N=1,2$ such that

[TABLE]

(Ambrosetti-Rabinowitz condition) There are $\theta>2$ and $M>0$ such that

[TABLE]

where $F(s)=\displaystyle\int_{0}^{s}f(t)\,dt$.

By using the Mountain Pass Theorem due to Ambrosetti and Rabinowitz [2], it is possible to prove that the energy functional associated with $(P)$, given by

[TABLE]

satisfies the well known $(PS)$ condition. Here, assumption $(f_{3})$ permits to prove, of a very easy way, that all $(PS)$ sequences are bounded. However, in the last years, we have observed that in a lot papers, the authors have used a more weak condition that $(f_{3})$, more precisely, instead of $(f_{3})$ the following conditions are assumed:

[TABLE]

and

[TABLE]

The condition $(f_{5})$ is sometimes replaced by

[TABLE]

The literature is large for problems without Ambrosseti-Rabinowitz condition, we would like to cite the papers by Costa and Magalhães [6], Jeanjean and Tanaka [11], Liu and Wang [15], Miyagaki and Souto [16], Schechter and Zou [19], Struwe and Tarantello [20], Wang and Wei [21], Zhou [23] and their references. In general, the main tool used in the above mentioned papers is Mountain Pass Theorem with Cerami’s condition found in Bartolo, Benci and Fortunato [5].

Here, we would like point out that in the seminal paper [6], Costa and Magalhães established the existence of solution for $(P)$ without Ambrosseti-Rabinowitz condition by supposing, among others conditions,

[TABLE]

with $\mu>\frac{N}{2}(q-2)$ and $q$ as in $(f_{2})$. Since that $\Omega$ is a bounded domain, the authors were able to show that $(f_{2})$ and $(f_{6})$ combine to give the boundedness of $(PS)$ sequences for the energy functional associated with $(P)$, which is a key point to prove $(PS)$ condition. In that paper, an interesting point is that the authors did not assume any monotonicity condition on $\frac{f(s)}{s}$ or $\mathcal{F}(s)=f(s)s-2F(s)$ for $s>0$.

The existence of solution for elliptic problem in whole $\mathbb{R}^{N}$ like

[TABLE]

without Ambrosseti-Rabinowitz condition also have been considered in some papers, see for example, Jeanjean [10], Liu [14] and their references. In [10], Jeanjean has proved a very interesting Abstract Theorem that permits to work with a large class of problem without Ambrosetti-Rabinowitz condition in bounded or unbounded domain. In that paper the author assumes $V=1$ and that there is $D\in[1,+\infty)$ such that

[TABLE]

where $Z(s)=\frac{1}{2}f(s)s-F(s)$ for $s\geq 0$. In [14], Liu used essentially the same arguments explored in [16], [19] and [20] by supposing that $V$ is $\mathbb{Z}^{N}$-periodic or

[TABLE]

Related to the function $f$, Liu also assumed a condition like (1.1).

Motivated by the above references, the present paper is concerned with the existence of solution for $(P_{1})$ without Ambrosseti-Rabinowitz condition. Here, $f:\mathbb{R}\to\mathbb{R}$ is continuous function that satisfies $(f_{1})$ and the following conditions:

There are $p\in\left(2,\frac{2N+4}{N}\right)$ and $C>0$ such that

[TABLE]

For each $s_{o}>0$ there is a $\sigma_{o}>0$ such that

[TABLE]

Here, it is important to recall that $(f_{8})$ is weaker than $(f_{3})$.

Related to the $V$, we assume that it is continuous and belongs to one of the following classes:

Class 1:   $V$ is a periodic function

• $(V_{1})$

$V$ is $\mathbb{Z}^{N}$ periodic continuous function, that is,

[TABLE]

• $(V_{2})$

$\displaystyle\inf_{x\in\mathbb{R}^{N}}V(x)=V_{o}>0$.

Class 2:   $V$ is coercive :

• $(V_{3})$

[TABLE]

Class 3:   $V$ is a Barstch & Wang type potential, that is,

• $(V_{4})$

$V$ is a continuous function of the form

[TABLE]

where $\lambda>0$ and $W$ is a nonnegative function.

Our first result is related to the case where the Ambrosetti-Rabinowiz condition only holds at infinity.

Theorem 1.1**.**

Assume $(f_{1})-(f_{3})$ and $(f_{5})^{\prime}$. Suppose that $V$ belongs to Class 1,2 or 3, then, $(P_{1})$ has a ground state solution. When $V$ belongs to Class 3, the existence of solution holds for large $\lambda$.

The Theorem 1.1 is a crucial step to prove our second result that establishes the existence of solution for $(P_{1})$ without Ambrosetti-Rabinowiz condition and it has the following statement

Theorem 1.2**.**

Assume $(f_{1}),(f_{7})-(f_{8})$ and that $V$ belongs to Class 1,2 or 3. Then, $(P_{1})$ has a ground state solution. When $V$ belongs to Class 3, the existence of solution holds for large $\lambda$.

As a consequence of Theorem 1.2, we can consider a more general class of problems like

[TABLE]

where $\beta>0$ and $V,f$ satisfy the conditions of Theorem 1.2. Related to the $h$, we assume that it is continuous and satisfies:

• $(h_{1})$

$\displaystyle{\lim_{s\to 0}\frac{h(s)}{s}=0}$ ;

• $(h_{2})$

There are $C>0$ and $r\in(2,2^{*})$ such that $|h(s)|\leq C(1+|s|^{r-1})$ for all $s\in\mathbb{R}$;

• $(h_{3})$

For each $s_{o}>0$, there is a $\sigma_{o}>0$ such that

[TABLE]

• $(h_{4})$

There is $\Upsilon>0$ such that

[TABLE]

We would like point out that $(h_{4})$ holds, for example, if $h(s)/f(s)$ is a nondecreasing function for $s>0$.

Our third theorem is the following

Theorem 1.3**.**

Assume $(f_{1}),(f_{7})-(f_{8})$, $(h_{1})-(h_{4})$ and that $V$ belongs to Class 1,2 or 3. Then, setting $\beta^{*}=\frac{f(\Upsilon)}{h(\Upsilon)}$, there is $\Upsilon^{*}>0$ such that $(P_{2})$ has a nontrivial solution for all $\beta\in[0,\beta^{*})$ and $\Upsilon\in[\Upsilon^{*},+\infty)$. As in Theorem 1, the existence of solution for Class 3 holds when $\lambda$ is large.

The plan of the paper is as follows: Section 2 deals with Theorem 1.1, while Sections 3 and 4 are devoted to the proofs of Theorems 1.2 and 1.3 respectively. In Section 5, we point out that if $\displaystyle{\frac{h(s)}{f(s)}\to+\infty}$, we also have an existence result replacing $(h_{3})$ by a weaker assumption, namely:   For each $s_{o}>0$, there is a $\sigma_{o}>0$ such that

[TABLE]

2. Proof of Theorem 1.1

In this section, we will show the existence of solution for a class of auxiliary problem, which proves Theorem 1.1 and will be used in the proof of Theorem 1.2. Specifically, we will study the existence of solution for the following class of problem

[TABLE]

where $V$ belongs to Class 1,2 or 3, and $g:\mathbb{R}\to\mathbb{R}$ is a continuous function verifying:

• $(g_{1})$

$\displaystyle{\lim_{s\to 0}\frac{g(s)}{s}=0}$ ;

• $(g_{2})$

There are $C>0$ and $p\in\left(2,\frac{2N}{N-2}\right)$ such that $|g(s)|\leq C(1+|s|^{p-1})$ for all $s\in\mathbb{R}$ ;

• $(g_{3})$

There are $\theta>2$ and $M>0$ such that $sg(s)-\theta G(s)>0$ for all $|s|\geq M$ ;

• $(g_{4})$

$\mathcal{G}(s)=sg(s)-2G(s)>0$ for all $0<|s|\leq M$.

Observe that $(f_{1})-(f_{3})$ and $(f_{5})^{\prime}$ imply $(g_{1})-(g_{4})$.

In the sequel, we mention some facts that involve conditions $(g_{1})-(g_{4})$. It is easy to check that $(g_{1})-(g_{2})$ imply in the inequality below

[TABLE]

The condition $(g_{3})$ is the usual Ambrosetti-Rabinowitz at infinity, however we would like point out that we are not assuming this condition near the origin, which is usually assumed in whole $\mathbb{R}$, when we are considering elliptic problem in whole $\mathbb{R}^{N}$. Moreover, we do not assume any monotonicity condition on $\displaystyle{\frac{g(s)}{s}}$, which is in general assumed in a lot of papers without the Ambrosetti-Rabinowitz condition. Finally, we recall that $(g_{3})$ is satisfied if

[TABLE]

From $(g_{1})$, there exists $s_{o}>0$ such that

[TABLE]

After these remarks, we are able to prove the existence of solution for $(AP)$.

2.1. The variational approach

In this subsection, related to the function $V$, we assume only condition $(V_{2})$. From now on, we set

[TABLE]

endowed with the norm

[TABLE]

and functional $J:E\to\mathbb{R}$ by

[TABLE]

From $(g_{1})-(g_{3})$, $J$ is well defined in $E$ and $J\in C^{1}(E,\mathbb{R})$ with

[TABLE]

Moreover, it is very easy to check that $J$ also satisfies the mountain pass geometry. In what follows, we denote by $c>0$ the mountain pass level associated with $J$, that is,

[TABLE]

where

[TABLE]

Associated with $c$, we have a Cerami sequence $(v_{n})\subset E$, that is,

[TABLE]

Proposition 2.1**.**

The sequence $(v_{n})$ is bounded in $E$.

Proof.

As $(v_{n})$ is a Cerami sequence, we have

[TABLE]

For each $n$, we set

[TABLE]

where $s_{o}$ was fixed in (2.3). From $(g_{4})$,

[TABLE]

where $\sigma_{o}=\inf\{\mathcal{G}(s):|s|\in[s_{o},M]\}>0$ and $|\mathcal{A}_{n}|$ denotes the Lebesgue’s measure of $\mathcal{A}_{n}$. This inequality implies that $(|\mathcal{A}_{n}|)$ is a bounded sequence. In the sequel, we consider the sets

[TABLE]

and

[TABLE]

From (2.3),

[TABLE]

and by $(g_{3})$,

[TABLE]

On the other hand, $(g_{3})$ gives

[TABLE]

for some $M_{0}>0$. The above analysis ensure that

[TABLE]

Since $(|\mathcal{A}_{n}|)$ is a bounded sequence, it follows that $(v_{n})$ is bounded in $E$.

∎

2.2. Existence of solution for $(AP)$: The periodic case

In this section we assume that $V$ verifies $(V_{1})-(V_{2})$. By Subsection 2.1, we know that there is a bounded Cerami sequence $(v_{n})$ associated with the mountain pass level $c$, that is,

[TABLE]

From boundedness of $(v_{n})$, we deduce that $(v_{n})$ is a $(PS)_{c}$ sequence, that is

[TABLE]

Moreover, we can assume that for some subsequence, there is $u\in E$ such that

[TABLE]

[TABLE]

and

[TABLE]

Since $J$ is invariant by translation, by Lions [13], we can assume that $v\not=0$. Hence, $v$ is a nontrivial critical point for $J$, and so, $v$ is a nontrivial solution for $(AP)$. The reader can see more details of how we can use [13] in [1], [8], [9] and [12].

2.3. The existence of solution for $(AP)$ : The Coercive case

In this case, it is well known that the following compact embedding holds

[TABLE]

This compact embedding together with the boundedness of $(PS)$ sequences permit to prove that $J$ verifies the well known (PS) condition, and so, the mountain pass level $c$ is a positive critical value for $(AP)$. This prove that $(AP)$ has a nontrivial solution.

We would like to point out that the same conclusion holds if $(V_{3})$ is replaced by

[TABLE]

where $|A|$ denotes the Lebesgue’s measure of a mensurable set $A\subset\mathbb{R}^{N}$. The last condition also implies in the compact embedding (2.6).

Finally, we would like point out that if $V$ is radially symmetric, the compactness (2.6) also holds in $H_{rad}^{1}(\mathbb{R}^{N})$, for more details see Willem [22, Corollary 1.26]. The existence of solution follows as in [22, Section 1.6], where the above compact embedding (2.6) and the Principle of symmetric criticality due to Palais apply an important role in the arguments.

2.4. Existence of solution for (AP): $V$ is a Barstch & Wang type potential

The approach explored in the previous subsection can be also used to study the existence of solution when the potential $V$ is of the form

[TABLE]

where $\lambda>0$ and $W:\mathbb{R}^{N}\to\mathbb{R}$ is a nonnegative continuous function that satisfies

[TABLE]

Arguing as [4], there is $\alpha>0$, which is independent of $\lambda$, such that

[TABLE]

Moreover, it is possible to prove that there are $R>0$ independent of $\lambda>0$, and $\lambda^{*}>0$ such that

[TABLE]

From (2.7)-(2.8),

[TABLE]

showing that $v$ is nontrivial. Hence $(AP)$ has a nontrivial solution for $\lambda$ large enough.

Before concluding this section, the reader is invited to see that the results showed in this section prove the Theorem 1.1.

3. Proof of Theorem 1.2

In this section we deal with the proof of Theorem 1.2, and the results obtained in Section 2 will be crucial in our approach.

In what follows, we consider the auxiliary problem

[TABLE]

where $\displaystyle{q=\frac{2N+4}{N}}$ and $\varepsilon>0$.

It is easy to check that $g_{\varepsilon}(s)=f(s)+\varepsilon|s|^{q-2}u$ satisfies $(g_{1})-(g_{4})$ for $\theta=q$. Hence, from the previous sections, for fixed $\varepsilon>0$, there is a nontrivial solution $u_{\varepsilon}$ of $(AP)_{\varepsilon}$ such that

[TABLE]

where

[TABLE]

and $c_{\varepsilon}$ is the mountain pass level associated with $J_{\varepsilon}$, that is,

[TABLE]

Since $J_{\varepsilon}(v)\leq J_{0}(v)$ for all $v\in E$, we have $c_{\varepsilon}\leq c_{0}$ for all $\varepsilon>0$. Furthermore, it is possible to prove that there is $\alpha>0$, which is independent of $\varepsilon\in[0,1]$, such that

[TABLE]

As an immediate consequence of the above inequality, we have

[TABLE]

The next result establishes an important estimate for $\{u_{\varepsilon}\}$ in $E$, which is a key point in our approach.

Proposition 3.1**.**

There is $\varepsilon_{0}>0$ such that $\{u_{\varepsilon}\}$ is bounded in $E$ for all $\varepsilon\in[0,\varepsilon_{0}]$.

Proof.

For each $\varepsilon>0$, we have $\mathcal{G}_{\varepsilon}\geq\mathcal{F}$. Thus, by $(f_{8})$,

[TABLE]

Arguing as (2.3), there exists $s_{o}>0$, which is independent of $\varepsilon>0$ small, such that

[TABLE]

On the other hand, from $(f_{7})$, for each $\varepsilon>0$, there is $C_{\varepsilon}>0$ verifying

[TABLE]

So, setting

[TABLE]

and using (3.3) and (3.4), we get

[TABLE]

that is,

[TABLE]

Recalling that

[TABLE]

and fixing $\varepsilon_{0}=\frac{S}{8}(\frac{\sigma}{c_{0}})^{\frac{2}{N}}$, we derive that

[TABLE]

This proves the boundedness of $\{u_{\varepsilon}\}$ for $\varepsilon\in(0,\varepsilon_{0}]$ in $E$.

∎

In the sequel, for each $n\in\mathbb{N}$, we denote by $u_{n}$ the solution of $(AP)_{\frac{1}{n}}$, that is,

[TABLE]

By Proposition 3.1, the sequence $(u_{n})$ is bounded in $E$, hence for some subsequence, there is $u\in E$ such that

[TABLE]

[TABLE]

and

[TABLE]

From this, for each $\phi\in E$,

[TABLE]

Then taking the limit of $n\to+\infty$, we find

[TABLE]

showing that $u$ is a solution of $(P_{1})$. Our goal is proving that $u$ is a nontrivial solution. Have this in mind, firstly we prove the result below

Lemma 3.2**.**

$u_{n}\not\to 0$* in $L^{p}(\mathbb{R}^{N})$, where $p$ was given in $(f_{7})$.*

Proof.

Since $u_{n}$ is a solution of $(AP)_{\frac{1}{n}}$, it follows that

[TABLE]

By $(f_{1})$ and $(f_{7})$,

[TABLE]

for some positive constant $C$. Consequently,

[TABLE]

for some positive constant $C$. Using the fact that $(u_{n})$ is bounded in $E$, we have that $(u_{n})$ is also bounded in $L^{q}(\mathbb{R}^{N})$. Thus, supposing by contradiction that $u_{n}\to 0$ in $L^{p}(\mathbb{R}^{N})$, we obtain

[TABLE]

which contradicts (3.2). ∎

Conclusion of the proof of Theorem 1.2

Now, we can argue as in Subsections 2.2, 2.3 and 2.4 to deduce that $u\not=0$. For example, in the periodic case, the Lions’ result together with Lemma 3.2 ensures that we can assume that $u\not=0$. For the others cases, we argue exactly as in Section 2.

4. Proof of Theorem 1.3

In what follows, we consider the function $h_{\Upsilon}:\mathbb{R}\to\mathbb{R}$ given by

[TABLE]

and $f_{\Upsilon}:\mathbb{R}\to\mathbb{R}$ by

[TABLE]

Using the above notations, our intention is proving the existence of a nontrivial solution $u_{\beta}$ with $\|u_{\beta}\|_{\infty}<\Upsilon$ for the following auxiliary problem

[TABLE]

Hereafter, we fix $\beta^{*}=\frac{f(\Upsilon)}{h(\Upsilon)}$ and $\beta\in[0,\beta^{*}]$. A simple computation shows that $f_{\Upsilon}$ satisfies $(f_{1}),(f_{7})-(f_{8})$, more precisely,

• $(i)$

$\displaystyle{\lim_{s\to 0}\frac{f_{\Upsilon}(s)}{s}=0}$ ;

• $(ii)$

There are $C>0$ and $p\in\left(2,\frac{2N+4}{N}\right)$ such that $|f_{\Upsilon}(s)|\leq C(1+|s|^{p-1})$ for all $s\in\mathbb{R}$;

• $(iii)$

For each $s_{o}>0$, there is $\sigma>0$ such that

[TABLE]

where $F_{\Upsilon}(s)=\int_{0}^{s}f_{\Upsilon}(t)\,dt$.

Indeed, we will check only $(ii)-(iii)$, because $(i)$ is immediate. By $(h_{4})$,

[TABLE]

leading to

[TABLE]

and

[TABLE]

On the other hand, if $s\in[0,\Upsilon]$, we have

[TABLE]

Now, if $s\geq\Upsilon$,

[TABLE]

and so, by (4.1),

[TABLE]

As a consequence of the above analysis,

[TABLE]

From this, we can apply Theorem 1.2 to get a solution $u_{\beta}\in E$ of $(AP)_{\beta,\Upsilon}$. Our next step is proving that there is $\Upsilon^{*}>0$ such that $\|u_{\beta}\|_{\infty}\leq\Upsilon^{*}$ for $\Upsilon\geq\Upsilon^{*}$, because this estimate permits to conclude that $u_{\beta}$ is a solution of $(P_{2})$ for $\Upsilon\geq\Upsilon^{*}$, which shows Theorem 1.3. However, the existence of $\Upsilon^{*}$ follows from the following fact: If $I_{\Upsilon}:E\to\mathbb{R}$ denotes the energy functional associated with $(AP)_{\beta,\Upsilon}$, that is,

[TABLE]

we have

[TABLE]

Hence,

[TABLE]

where $c_{0}$ denotes the mountain pass level associated with the functional $J_{0}$ ( see Section 3). Since $I^{\prime}_{\Upsilon}(u_{\beta})=0$, we can argue as in Section 3 to show that there is $k>0$, which is independent of $\Upsilon$ and $\beta$, such that

[TABLE]

Recalling that by $(ii)$,

[TABLE]

where the constant $C$ does not depend on $\beta$ and $\Upsilon$, the bootstrap argument found [18, Proposition 2.15] ensures that there is $K_{*}>0$, which is independent of $\beta$ and $\Upsilon$ such that

[TABLE]

From this, if $\Upsilon\geq\Upsilon^{*}=K$, it follows that $u_{\beta}$ is a solution for $(P_{2})$, finishing the proof of Theorem 1.3.

5. Final comments

In this section we would like point out that if $\displaystyle{\frac{h(s)}{f(s)}\to+\infty}$ as $s\to+\infty$, we can replace $(h_{3})$ by

• $(h^{\prime}_{3})$

For each $s_{o}>0$, there is a $\sigma_{o}>0$ such that

[TABLE]

Indeed, using this assumption in (4.2), we get

[TABLE]

If $\displaystyle{\frac{h(s)}{f(s)}\to+\infty}$ as $s\to+\infty$, we derive that $\beta^{*}\to 0$ as $\Upsilon^{*}\to+\infty$. Hence,

[TABLE]

for $\Upsilon^{*}$ large enough. From this, Theorem 1.3 still holds for $\Upsilon^{*}$ large enough.