Sign-changing solutions for asymptotically linear Schr\"odinger equation in bounded domains

Abstract

In this article we study the Schrodinger equation $$ -\Delta u=f(x,u),\quad x\in\Omega, \quad u\in H_0^1(\Omega), $$ where $\Omega$ is a bounded domain in $\mathbb{R}^N$ and f(x,u) is asymptotically linear at infinity with respect to u. Inspired by the works of Salvatore [14] on sign-changing solutions, in which $f(x,u)$ is asymptotically linear at zero with respect to $u$, we prove, via the constraint variational method and the quantitative deformation lemma, that the equation possesses one sign-changing solution with exactly two nodal domains.

Keywords

• Schrodinger equation
• sign-changing solutions
• asymptotically linear