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 $\Omega$.