Multiplicity results for the fractional laplacian in expanded domains

Abstract

In this paper we establish the multiplicity of nontrivial weak solutions for the problem $(-\Delta {)}^{\alpha }u+u=h(u)$ in ${\Omega }_{\lambda }$,\ $u=0$ on $\partial {\Omega }_{\lambda }$, where ${\Omega }_{\lambda }=\lambda \Omega$, $\Omega$ is a smooth and bounded domain in ${\mathbb{R}}^N,N>2\alpha$, $\lambda$ is a positive parameter, $\alpha \in (0,1)$, $(-\Delta {)}^{\alpha }$ is the fractional Laplacian and the nonlinear term $h(u)$ has a subcritical growth. We use minimax methods, the Ljusternick-Schnirelmann and Morse theories to get multiplicity result depending on the topology of $\Omega$.

References in corpus (1)

• A multiplicity result via Ljusternick-Schnirelmann category and morse theory for a fractional schrödinger equation in ${\mathbb{R}}^N$

Cited by in corpus (1)

• Multiplicity of critical points for the fractional Allen-Cahn energy