Sign-Changing Multi-Peak Solutions for Nonlinear Schrödinger Equations with Compactly Supported Potential

Abstract

In this paper, we aim to prove the existence and concentration of sign-changing H 1(Â? N ) solutions to the following nonlinear Schrodinger equations $$-\varepsilon^2\Delta u_{\varepsilon}+V(x)u_{\varepsilon}=K(x)|u_{\varepsilon}|^{p-1}u_{\varepsilon} $$ with NÂ?3, $1 0. When the potential function V(x) has compact support, $V(x)\not\equiv 0$ and V(x)Â?0, K(x) is permitted to be unbounded under some necessary restrictions, we will show that one sign-changing H 1(Â? N ) solution exists and exhibits concentration profile around local minimum points of the ground energy function $G(\xi)\equiv V^{\theta}(\xi) K^{-\frac{2}{p-1}}(\xi)$ with $\theta=\frac{p+1}{p-1}-\frac{N}{2}$ in the semiclassical limit Â?Â?0.