The nonlinear (p,q)-Schrödinger equation with a general nonlinearity: Existence and concentration

Abstract

We investigate the following class of $(p,q)$-Laplacian problems:$\{\begin{array}{c}-{\epsilon }^p{\Delta }_{p}v-{\epsilon }^q{\Delta }_{q}v+V\left(x\right)\left(\right|v{|}^{p-2}v+\left|v{|}^{q-2}v\right)=f\left(v\right)\text{ in }{R}^N,\\ v\in W^{1,p}\left(R^N\right)\cap W^{1,q}\left(R^N\right),v>0\text{ in }{R}^N,\end{array}$ where $\epsilon >0$ is a small parameter, $N\ge 3$, $1<p<q<N$, ${\Delta }_{s}v:=\mathrm{div}\left(\right|\nabla v{|}^{s-2}\nabla v)$, with $s\in \{p,q\}$, is the s-Laplacian operator, $V:R^N\to R$ is a continuous potential such that ${\inf }_{R^N}V>0$ and $V_0:={\inf }_{\Lambda }V<{\min }_{\partial \Lambda }V$ for some bounded open set $\Lambda \subset R^N$, and $f:R\to R$ is a subcritical Berestycki-Lions type nonlinearity. Using variational arguments, we show the existence of a family of solutions $\left(v_{\epsilon }\right)$ which concentrates around $M:=\{x\in \Lambda :V(x)=V_0\}$ as $\epsilon \to 0$.