Existence of Multi-peak Solutions for a Class of Quasilinear Problems in Orlicz-Sobolev Spaces

Abstract

The aim of this work is to establish the existence of multi-peak solutions for the following class of quasilinear problems

$$ - \mbox{div} \bigl(\epsilon^{2}\phi\bigl(\epsilon|\nabla u|\bigr)\nabla u \bigr) + V(x)\phi\bigl(\vert u\vert\bigr)u = f(u)\quad\mbox{in } \mathbb{R}^{N}, $$

where $\epsilon$ is a positive parameter, $N\geq2$, $V$, $f$ are continuous functions satisfying some technical conditions and $\phi$ is a $C^{1}$-function.

Appendix: New Properties Involving Orlicz-Sobolev Spaces

In this appendix, we will prove some results which were used in the present paper. Our first result is associated with an important property involving Orlicz-Sobolev spaces, which is well known for Sobolev spaces. Here, we follows the same steps found in [16, Theorem 3.2] (or [1, Theorem 8.35]), however our proof can be applied for unbounded domains.

Proposition 5.1

There exists $M^{*}>0$, which is independent of $\epsilon$, such that

$$\begin{aligned} \Vert u\Vert_{\varPhi_{*}, \varOmega_{\epsilon, i}}\leq M^{*} \Vert u \Vert_{\widetilde{X}_{\epsilon, i}} \quad\textit{for all } u \in\widetilde{X}_{\epsilon, i}. \end{aligned}$$

Proof

In what follows, we define $\upsilon(t) = (\varPhi_{*}(t) )^{1- \frac{1}{N}}$. Firstly, notice that

$$\begin{aligned} \biggl|\frac{d}{dt}\upsilon(t) \biggr|\leq\frac{N-1}{N}\widetilde{ \varPhi} ^{-1} \bigl(\upsilon(t)^{\frac{N}{N-1}} \bigr) \quad\mbox{for all } t>0. \end{aligned}$$

(5.1)

For each $u \in\widetilde{X}_{\epsilon, i}\cap L^{\infty}( \varOmega_{\epsilon, i})$ and $k>0$, the function $\nu:= \upsilon \circ ( \frac{|u|}{k} ) \in W^{1,1}(\varOmega_{\epsilon, j})$ and

$$ \frac{\partial\nu(x)}{\partial x_{j}} = \upsilon^{\prime} \biggl( \frac{|u|}{k}(x) \biggr)\frac{\operatorname{sgn}u(x)}{k}\frac{\partial u (x)}{ \partial x_{j}}. $$

By [1, Theorem 4.12], once $\varOmega_{{\epsilon,j}}$ verifies the uniform cone condition for all $\epsilon>0$, we know that the constant associated with the embedding $W^{1,1}(\varOmega_{\epsilon , j}) \hookrightarrow L^{\frac{N}{N-1}}(\varOmega_{\epsilon, j})$ does not depend on $\epsilon$, that is, there exists a positive constant $C$, which is independent of $u$ and $\epsilon$, such that

$$ \Vert\nu\Vert_{L^{\frac{N}{N-1}}(\varOmega_{\epsilon, j})}\leq C \Biggl(\sum_{j=1}^{N} \biggl\Vert \frac{\partial\nu}{\partial x_{j}} \biggr\Vert _{ L^{1}(\varOmega_{\epsilon, j})} + \Vert\nu \Vert_{ L^{1}(\varOmega_{\epsilon, j})} \Biggr), $$

or equivalently,

$$\begin{aligned} \biggl[ \int_{\varOmega_{\epsilon, i}}\varPhi_{*} \biggl(\frac{|u|}{k} \biggr)dx \biggr]^{1- \frac{1}{N}}\leq\frac{C}{k}\sum _{j=1}^{N} \int_{\varOmega_{\epsilon, i}} \biggl|\upsilon^{\prime} \biggl(\frac{|u|}{k} \biggr)\frac{\partial u }{\partial x_{j}} \biggr|dx + C \int_{\varOmega_{\epsilon, i}} \biggl| \upsilon \biggl( \frac{|u|}{k} \biggr) \biggr| dx. \end{aligned}$$

Setting $k = \Vert u\Vert_{\varPhi_{*}, \varOmega_{\epsilon, i}}$, the Holder’s inequality together with (5.1) yields

$$\begin{aligned} 1\leq\frac{2C}{k}\frac{N-1}{N}\sum _{j=1}^{N} \biggl\Vert \widetilde{\varPhi} ^{-1} \biggl(\varPhi_{*} \biggl(\frac{|u|}{k} \biggr) \biggr) \biggr\Vert _{\widetilde{\varPhi}, \varOmega_{\epsilon, i}} \biggl\Vert \frac{ \partial u }{\partial x_{j}} \biggr\Vert _{\varPhi, \varOmega_{\epsilon, i}} + C \int_{\varOmega_{\epsilon, i}} \biggl| \upsilon \biggl( \frac{|u|}{k} \biggr) \biggr| dx. \end{aligned}$$

(5.2)

Now, a direct computation leads to

$$ \int_{\varOmega_{\epsilon, i}} \biggl|\upsilon \biggl( \frac{|u|}{k} \biggr)\biggr| dx \leq\frac{2}{k}\frac{N-1}{N} \biggl\Vert \widetilde{\varPhi}^{-1} \biggl( \varPhi_{*}\biggl(\frac{|u|}{k}\biggr) \biggr) \biggr\Vert _{\widetilde{\varPhi}, \varOmega_{\epsilon, i}}\Vert u \Vert_{\varPhi, \widetilde{\varOmega}_{\epsilon, i}}. $$

Since

$$ \biggl\Vert \widetilde{\varPhi}^{-1} \biggl(\varPhi_{*} \biggl( \frac{|u|}{k} \biggr) \biggr) \biggr\Vert _{\widetilde{\varPhi}, \varOmega_{\epsilon, i}}\leq1, $$

we get,

$$\begin{aligned} \int_{\varOmega_{\epsilon, i}} \biggl|\upsilon \biggl( \frac{|u|}{k} \biggr)\biggr| dx \leq& \frac{2}{k}\frac{N-1}{N}\Vert u \Vert_{\varPhi, \widetilde{\varOmega}_{\epsilon, i}}. \end{aligned}$$

(5.3)

From (5.2)–(5.3),

$$\begin{aligned} 1 \leq& \frac{2C}{k}\frac{N-1}{N} \Vert\nabla u \Vert_{\varPhi, \widetilde{\varOmega}_{\epsilon, i}}+ \frac{2}{k} \frac{N-1}{N}\Vert u \Vert_{\varPhi, \widetilde{\varOmega}_{\epsilon, i}}. \end{aligned}$$

Hence, there exists $M_{*}>0$, independent of $\epsilon$ such that

$$ \Vert u\Vert_{\varPhi_{*}, \varOmega_{\epsilon, i}}\leq M^{*} \Vert u \Vert_{\widetilde{X}_{\epsilon, i}} \quad \mbox{for all } u \in\widetilde{X}_{\epsilon, i}\cap L^{\infty}( \varOmega_{\epsilon, i}), $$

obtaining the desired result. □

As a byproduct of the above proof, we have the following corollary

Corollary 5.1

Let $(y_{n,i})$ the sequence obtained in (4.6). There is $C>0$, which is independent of $\rho$ and $n \in\mathbb{N}$, such that

$$\begin{aligned} \int_{\mathbb{R}^{N}\backslash\bigcup_{j = 1}^{p}B_{\rho}(y_{n, i})} \varPhi_{*}(|v|)dx \leq C \Vert v \Vert_{W^{1, \varPhi}(\mathbb{R}^{N}\backslash\bigcup_{j = 1}^{p}B_{ \rho}(y_{n, i}))}, \end{aligned}$$

for all $v \in W^{1,\varPhi}(\mathbb{R}^{N}\backslash\bigcup_{j = 1}^{p}B _{\rho}(y_{n, i}))$.

Proof

The corollary follows by repeating the same steps used in the proof Proposition 5.1. The main point that we would like to point out is the fact that the constant associated with the embedding

$$ W^{1,1}\biggl(\mathbb{R}^{N}\backslash\bigcup_{j = 1}^{p}B_{\rho}(y_{n, i}) \biggr) \hookrightarrow L^{\frac{N}{N-1}}\biggl(\mathbb{R}^{N}\backslash \bigcup_{j = 1} ^{p}B_{\rho}(y_{n, i})\biggr) $$

is also independent of $\rho$ and $n \in\mathbb{N}$, because $\varTheta_{\rho,n,i}=\mathbb{R}^{N}\backslash\bigcup_{j = 1}^{p}B_{ \rho}(y_{n, i})$ verifies the uniform cone condition for all $\rho>0$ and $n \in\mathbb{N}$. □

The next result is also well known for Sobolev spaces, however for Orlicz-Sobolev spaces we do not know any reference. Here, we adapt some arguments found in [6].

Proposition 5.2

Let $\varrho>0$ and $\epsilon_{n}\in(0, +\infty)$ with $\epsilon _{n}\rightarrow0$. Let $v_{n, i}\subset\widetilde{X}_{\epsilon_{n}}$, $i$ be a sequence and a constant $C_{0}>0$ such that

$$ \Vert v_{n, i}\Vert_{\widetilde{X}_{\epsilon_{n}, i}} \leq C_{0}\quad \textit{and}\quad\lim_{n\rightarrow+\infty}\sup_{y \in\mathbb{R} ^{N}} \int_{B_{\varrho}(y)\cap\varOmega_{\epsilon_{n}, i}}\varPhi\bigl(|v_{n, i}|\bigr)dx = 0. $$

Then,

$$ \lim_{n\rightarrow+\infty} \int_{\varOmega_{\epsilon_{n}, i}}B\bigl(|v_{n, i}|\bigr)dx = 0, $$

for any N-function $B$ verifying $\Delta_{2}$-condition,

$$ \lim_{t\rightarrow0}\frac{B(t)}{\varPhi(t)} = 0\quad\textit{and}\quad \lim _{|t|\rightarrow+\infty}\frac{B(t)}{\varPhi_{*}(t)} = 0. $$

Proof

Firstly, note that given $\eta>0$ there exists $\kappa>0$ such that

$$ B\bigl(|v_{n, i}|\bigr)\leq\eta\varPhi_{*}\bigl(|v_{n, i}|\bigr),\quad \mbox{for } |v _{n, i}|\geq\kappa. $$

As $(\|v_{n, i}\|_{\widetilde{X}_{\epsilon_{n}, i}})$ is bounded in ℝ, we have

$$ \int_{\varOmega_{\epsilon_{n}, i}}B\bigl(|v_{n, i}|\bigr)dx \leq\eta C + \int_{\varOmega_{\epsilon_{n}, i}\cap[|v_{n, i}|\leq\kappa]}B\bigl(|v_{n, i}|\bigr)dx $$

which implies

$$\begin{aligned} \limsup_{n \to+ \infty} \int_{\varOmega_{\epsilon_{n}, i}}B\bigl(|v_{n, i}|\bigr)dx \leq\eta C + \limsup _{n \to+ \infty} \int_{\varOmega_{\epsilon_{n}, i}\cap[|v_{n, i}|\leq\kappa]}B\bigl(|v_{n, i}|\bigr)dx. \end{aligned}$$

(5.4)

We will show that

$$\begin{aligned} \limsup_{n \to+ \infty} \int_{\varOmega_{\epsilon_{n}, i}\cap[|v_{n, i}|\leq\kappa]}B\bigl(|v_{n, i}|\bigr)dx = 0. \end{aligned}$$

(5.5)

For this purpose, we consider for each $\zeta>0$ enough small, the function $\chi_{\zeta} \in C_{0}^{1}(\mathbb{R})$ given by

$$\begin{aligned} \chi_{\zeta}(s) =\left \{ \textstyle\begin{array}[c]{lll} 1, &\quad \mbox{if } |s|\le\kappa- \zeta, \\ a_{1}(s), &\quad \mbox{if } -(\kappa+ \zeta) \leq s\leq-(\kappa- \zeta), \\ a_{2}(s), &\quad \mbox{if } \kappa- \zeta\leq s\leq\kappa+ \zeta, \\ 0, &\quad \mbox{if } |s|\geq\kappa+ \zeta, \end{array}\displaystyle \right . \end{aligned}$$

where $a_{1}, a_{2}\in C^{1} (\mathbb{R};[0, 1] )$, $a_{1}$ is nondecreasing and $a_{2}$ is nonincreasing. Next, let us define the auxiliary function

$$ u_{n, i}(x) = \chi_{\zeta}\bigl(\bigl|v_{n, i}(x)\bigr| \bigr)v_{n, i}(x). $$

Notice that

$$ \int_{\varOmega_{\epsilon_{n}, i}}B\bigl(|u_{n, i}|\bigr)dx \geq \int_{\varOmega_{\epsilon_{n}, i}\cap[|v_{n, i}|\leq\kappa- \zeta]}B\bigl(|v _{n, i}|\bigr)dx. $$

(5.6)

Thereby, (5.5) follows by showing the limit below

$$ \limsup_{n \to+ \infty} \int_{\varOmega_{\epsilon_{n}, i}}B\bigl(|u_{n, i}|\bigr)dx = 0. $$

(5.7)

In fact, gathering the above limit with (5.6), we derive that

$$\begin{aligned} \limsup_{n \to+ \infty} \int_{\varOmega_{\epsilon_{n}, i}\cap[|v_{n, i}|\leq\kappa- \zeta]}B\bigl(|v _{n, i}|\bigr)dx = 0. \end{aligned}$$

Since

$$ \int_{\varOmega_{\epsilon_{n}, i}\cap[\kappa- \zeta\leq|v_{n, i}| \leq\kappa]}B\bigl(|v_{n, i}|\bigr)dx = o_{n}(1) $$

and

$$\begin{aligned} \int_{\varOmega_{\epsilon_{n}, i}\cap[|v_{n, i}|\leq\kappa]}B\bigl(|v_{n, i}|\bigr)dx =& \int_{\varOmega_{\epsilon_{n}, i}\cap[\kappa- \zeta\leq|v_{n, i}| \leq\kappa]}B\bigl(|v_{n, i}|\bigr)dx \\ &{} + \int_{\varOmega_{\epsilon_{n}, i}\cap[|v_{n, i}|\leq\kappa- \zeta]}B\bigl(|v _{n, i}|\bigr)dx, \end{aligned}$$

we deduce that

$$ \limsup_{n \to+ \infty} \int_{\varOmega_{\epsilon_{n}, i}\cap[|v_{n, i}|\leq\kappa]}B\bigl(|v_{n, i}|\bigr)dx=0, $$

showing (5.5). Now, by (5.4) and (5.5),

$$\begin{aligned} \limsup_{n \to+ \infty} \int_{\varOmega_{\epsilon_{n}, i}}B\bigl(|v_{n, i}|\bigr)dx \leq\eta C. \end{aligned}$$

By using that $\eta$ is arbitrary, it follows that

$$\begin{aligned} \limsup_{n \to+ \infty} \int_{\varOmega_{\epsilon_{n}, i}}B\bigl(|v_{n, i}|\bigr)dx = 0, \end{aligned}$$

proving the proposition. Now, we observe that (5.7) follows by repeating the same approach explored in [6, Theorem 3.1]. □