Existence and concentration of solutions for a quasilinear elliptic field equation

Abstract

In this paper, we study the existence and concentration of solution for the following class of quasilinear problem

$$\begin{aligned} -\hbar ^{2} \Delta v+ V(x)v-\hbar ^{p}\Delta _{p} v + W^{\prime }(v)=0\, \,x\in {\mathbb {R}}^{N}, \qquad \qquad \qquad \qquad (P_{\hbar }) \end{aligned}$$

when $\hbar >0$ is small enough, where $v:{\mathbb {R}}^{N}\rightarrow {\mathbb {R}}^{N+1}, 3\le N<p, \hbar >0,$ W is a $C^1$ singular functional that satisfies some technical conditions and the potential V is continuous functions with $\displaystyle \liminf _{\vert x\vert \rightarrow +\infty }V(x)\equiv V_{\infty }>\inf _{x \in {\mathbb {R}}^N}V(x)>0$ . Moreover, we also consider the existence of solution for all $\hbar >0$ when V is a radial function.