Multi-bump solutions for Schrödinger equation involving critical growth and potential wells

Abstract

In this paper, we consider the following Schrodinger equation with critical growth $$-\Delta u+(\lambda a(x)-\delta)u=|u|^{2^*-2}u \quad \hbox{ in } \mathbb{R}^N, $$ where $N\geq 5$, $2^*$ is the critical Sobolev exponent, $\delta>0$ is a constant, $a(x)\geq 0$ and its zero set is not empty. We will show that if the zero set of $a(x)$ has several isolated connected components $\Omega_1,\cdots,\Omega_k$ such that the interior of $\Omega_i (i=1, 2, ..., k)$ is not empty and $\partial\Omega_i (i=1, 2, ..., k)$ is smooth, then for any non-empty subset $J\subset \{1,2,\cdots,k\}$ and $\lambda$ sufficiently large, the equation admits a solution which is trapped in a neighborhood of $\bigcup_{j\in J}\Omega_j$. Our strategy to obtain the main results is as follows: By using local mountain pass method combining with penalization of the nonlinearities, we first prove the existence of single-bump solutions which are trapped in the neighborhood of only one isolated component of zero set. Then we construct the multi-bump solution by summing these one-bump solutions as the first approximation solution. The real solution will be obtained by delicate estimates of the error term, this last step is done by using Contraction Image Principle.