Concentration behavior of solutions for quasilinear elliptic equations with steep potential well

Abstract

In this paper, we study the following quasilinear elliptic equation:

$$\begin{aligned} -\triangle u+\lambda V(x)u-[\triangle (1+u^{2})^{\frac{1}{2}}]\frac{u}{2(1+u^{2})^{\frac{1}{2}}}=|u|^{p-2}u,\,\,x\in {{\mathbb {R}}}^{N}, \end{aligned}$$

where $N\ge 3$, $\lambda >0$, $12-4\sqrt{6}<p<2^{*}$, $V\in C({{\mathbb {R}}}^{N},{{\mathbb {R}}})$ and $V^{-1}(0)$ has nonempty interior. At first, we prove the existence of a nontrivial solution $u_{\lambda }$ via variational method. Then, the concentration behavior of $u_{\lambda }$ is also explored on the set $V^{-1}(0)$ as $\lambda \rightarrow \infty$.