Multiple sign-changing solutions for nonlinear Schrödinger equations with potential well

Abstract

We study the existence of sign-changing solutions for the following nonlinear Schrödinger equation $-\mathrm{\Delta }u+V_{\lambda }\left(x\right)u=f(x,u),x\in {\mathbb{R}}^N,$ where the potential $V_{\lambda }\left(x\right)$ has a potential well with bottom independent of the parameter $\lambda >0$. We show that as $\lambda \to +\mathrm{\infty }$ more and more sign-changing solutions of the nonlinear Schrödinger equation exist. The solutions lie in $H^1\left({\mathbb{R}}^N\right)$ and are localized near the bottom of the potential well, but not near local minima of the potential.

COMMUNICATED BY:

• Dumitru Motreanu

Keywords:

• Sign-changing solutions
• potential well
• asymptotic behavior

MSC(2010):

• 35J20
• 35B40
• 58E05

Disclosure statement

No potential conflict of interest was reported by the authors.