On a Critical Exponential p & N Equation Type: Existence and Concentration of Changing Solutions

Abstract

In this paper we study a class of quasilinear equation with exponential critical growth. More precisely, we show existence of a family of nodal solutions, i.e, sign-changing solutions for the problem

Such nodal solutions concentrate on the minimum points set of the potential V, changes sign exactly once in ${\mathbb {R}}^{N}$ and have exponential decay at infinity. Here we use variational methods and Del Pino and Felmer’s technique (Del Pino and Felmer 1996) in order to overcome the lack of compactness.

Appendix

In this appendix we show existence of nodal solution for the problem

where r is the constant that appears in the hypothesis $(f_5)$ and $V_{\infty }$ is a positive constant. We have associated to the problem $(P_r)$ the functional

$$\begin{aligned} I_r(u)=\dfrac{1}{p}\displaystyle \int \limits _{\Omega }\left[ k_2|\nabla u|^p+V_{\infty } k_4|u|^p \right] dx+\dfrac{1}{N}\displaystyle \int \limits _{\Omega }\left[ |\nabla u|^N+ V_{\infty } |u|^N \right] dx- \dfrac{1}{r}\displaystyle \int \limits _{\Omega }|u|^r dx \end{aligned}$$

and the set

$$\begin{aligned} {\mathcal {N}}^\pm _r=\{u\in W^{1,N}_0(\Omega )|~u^\pm \ne 0~\text {and}~I_r'(u)u^\pm =0\} \end{aligned}$$

Then, we can prove that there exists $w_r\in {\mathcal {N}}_r^\pm$ such that

$$\begin{aligned} I_r(w_r)=c_r:=\inf \limits _{{\mathcal {N}}^\pm _r}I_r&\text {and}&I'_r(w_r)=0. \end{aligned}$$

(5.2)

Lemma 5.5

For each $u\in W^{1,N}_0(\Omega )$ such that $u^\pm \ne 0$, there exists a unique pair $(t,s) \in (0,+\infty )\times (0,+\infty )$, such that $t u^++su^-\in {\mathcal {N}}^\pm _r$.

Proof

Note that if $u^{\pm }\in W^{1,N}_0(\Omega )\backslash \{0\}$ and $\gamma >0$, we have

$$\begin{aligned} \frac{I_r(\gamma u^{\pm })}{\gamma ^r}= & {} \dfrac{\gamma ^{p-r}}{p}\displaystyle \int \limits _{\Omega }\left[ k_2|\nabla u^{\pm }|^p+V_\infty k_4|u^{\pm }|^p \right] dx\\&+\dfrac{\gamma ^{N-r}}{N}\displaystyle \int \limits _{\Omega }\left[ |\nabla u^{\pm }|^N+ V_\infty |u^{\pm }|^N \right] dx- \dfrac{1}{r}\displaystyle \int \limits _{\Omega }|u^{\pm }|^r dx. \end{aligned}$$

Then,

$$\begin{aligned} \lim \limits _{\gamma \rightarrow 0} \dfrac{I_r( \gamma u^{\pm })}{\gamma ^r}=+\infty ~~\text {and}~~\lim \limits _{\gamma \rightarrow +\infty } \dfrac{I_r(\gamma u^{\pm })}{\gamma ^r}=- \dfrac{1}{r}\displaystyle \int \limits _{\Omega }|u^{\pm }|^r dx<0. \end{aligned}$$

Consequently, there exists $t,~s \in (0,+\infty )$ such that

$$\begin{aligned} I_r(t u^+):=\sup \limits _{\gamma \ge 0} I_r (\gamma u^+)~~ \text {and}~~ I_r(s u^-):=\sup \limits _{\gamma \ge 0} I_r (\gamma u^-). \end{aligned}$$

This implies that

$$\begin{aligned} I'_r(tu^++su^-)(tu^++su^-)=I'_r(tu^+)tu^++I'_r(su^-)su^-=0 \end{aligned}$$

In order to show the unicity of t and s, consider $f(s)=s^r$ and note that $\dfrac{f(t)}{t^N}$ is nondecreasing in $t>0$. $\square$

Lemma 5.6

The following properties hold:

• (i) There exists a constant $\rho _r>0$ such that $\left[ ~\displaystyle \int \limits _{\Omega }|\nabla u^\pm |^N dx\right] ^{1/N}\ge \rho _r$, for all $u\in {\mathcal {N}}^\pm _r$;

• (ii) There exists a constant $C_r>0$ such that $I_r(u)\ge C_r \displaystyle \int \limits _{\Omega }|\nabla u|^N dx$, for all $u\in {\mathcal {N}}^\pm _r$.

Proof

Using that $I'_r(u)u^\pm =0$ and by Sobolev embeddings there exists a constant $C>0$ such that

$$\begin{aligned} \begin{array}{llll} \displaystyle \int \limits _{\Omega }|\nabla u^\pm |^N dx&{}\le &{} \displaystyle \int \limits _{\Omega }\left[ k_2 |\nabla u^\pm |^p+V_\infty k_4|u^\pm |^p\right] dx+\displaystyle \int \limits _{\Omega }\left[ |\nabla u^\pm |^N+V_\infty |u^\pm |^N\right] dx\vspace{0,2cm}\\ {} &{}=&{} \displaystyle \int \limits _{\Omega } |u^\pm |^r dx \le C \left[ ~\displaystyle \int \limits _{\Omega }|\nabla u^\pm |^N dx\right] ^{r/N}. \end{array} \end{aligned}$$

Since $r>N$, the item (i) follows.

To verify the second assertion observe that

$$\begin{aligned} I_r(u)= & {} I_r(u)-\dfrac{1}{r}I'_r(u) u\\= & {} \left( \dfrac{1}{p}-\dfrac{1}{r} \right) \displaystyle \int \limits _{\Omega }\left[ k_2 |\nabla u|^p+V_\infty k_4|u|^p\right] dx\\&+\left( \dfrac{1}{N}-\dfrac{1}{r} \right) \displaystyle \int \limits _{\Omega }\left[ |\nabla u|^N+V_\infty |u|^N\right] dx \\\ge & {} \left( \dfrac{1}{N}-\dfrac{1}{r} \right) \displaystyle \int \limits _{\Omega }|\nabla u|^N dx. \end{aligned}$$

$\square$

Proposition 5.7

There exists $w_r\in W_0^{1,N}(\Omega )$ such that $w_r$ is a solution of $(P_r)$ and $I_r(w_r)=\inf \nolimits _{{\mathcal {N}}^{\pm }_{r}}I_r$.

Proof

Let $(u_n)$ be a minimizing sequence for $I_r$ in ${\mathcal {N}}^\pm _r$, i.e, a sequence $\{u_n\}\subset {\mathcal {N}}^\pm _r$ such that $I_r(u_n)= c_r+o_n(1)$. Note that, by Lemma 5.6, $(u_n)$ is a bounded sequence in $W^{1,N}_0(\Omega )$. Then there exists $u_r\in W^{1,N}_0(\Omega )$ such that, up to a subsequence, $u_n\rightharpoonup u_r$ in $W^{1,N}_0(\Omega )$. Arguing as in Lemma 2.3 in Castro et al. (1997), it is possible to show that $v\mapsto v^\pm$ is a continuous function of $W_0^{1,N}(\Omega )$ into itself, from which it follows that $u_n^\pm \rightharpoonup u_r^\pm$ in $W_0^{1,N}(\Omega )$. Moreover, by Sobolev embeddings,

$$\begin{aligned} \left\{ \begin{array}{lllllll} u^\pm _n\rightarrow u_r^\pm ~~\text {strongly in}~~L^s(\Omega )~~\text {for any}~~~1\le s< +\infty ,\\ u^\pm _n(x)\rightarrow u_r^\pm (x)~~\text {for a.e}~~x\in \Omega . \end{array}\right. \end{aligned}$$

(5.3)

First we are going to show that $u^\pm _r\ne 0$. In fact, if $u^\pm _r \equiv 0$ then, using the Lemma 5.6 and that $I'_r(u_n)u^\pm _n=0$ for all $n\in {\mathbb {N}}$, we have the following contradiction

$$\begin{aligned} \rho ^N_r\le \displaystyle \int \limits _{\Omega }|\nabla u^\pm _n|^N dx \le \displaystyle \int \limits _{\Omega }| u^\pm _n|^r dx =o_n(1). \end{aligned}$$

It follows from Fatou’s Lemma that

$$\begin{aligned} \displaystyle \int \limits _{\Omega }|\nabla u^\pm _r|^p dx\le \liminf \limits _{n\rightarrow \infty } \displaystyle \int \limits _{\Omega }|\nabla u^\pm _n|^p dx~~\text {and}~~ \displaystyle \int \limits _{\Omega }|\nabla u^\pm _r|^N dx\le \liminf \limits _{n\rightarrow \infty } \displaystyle \int \limits _{\Omega }|\nabla u^\pm _n|^N dx. \end{aligned}$$

Therefore, using compact embedding and Lemma 5.5 we get

$$\begin{aligned} \begin{array}{llllll} c_r\le I_r(t u^{+}_{r} + s u^{-}_{r}) \le \liminf \limits _{n\rightarrow \infty } [~I_r(t u^+_n)+I_r(s u^-_n)~] \le \liminf \limits _{n\rightarrow \infty } [~I_r(u^+_n)+I_r(u^-_n)~]\vspace{0,2cm}\\ ~~~~=\liminf \limits _{n\rightarrow \infty } I_r(u_n)+o_n(1)=c_r. \end{array} \end{aligned}$$

Considering $w_r= t u^+_r + s u^-_r$, we obtain $I_r(w_r)=c_r$ and using a Deformation Lemma (Figueiredo and Nunes 2020, Proof of Theorem 1.1), we conclude that $I'_r(w_r)=0.$ $\square$