Multiplicity results for the fractional laplacian in expanded domains

Abstract

In this paper we establish the multiplicity of nontrivial weak solutions for the problem $(-Δ)^α u +u= h(u)$ in $Ω_λ$,\ $u=0$ on $\partialΩ_λ$, where $Ω_λ=λΩ$, $Ω$ is a smooth and bounded domain in $\mathbb{R}^N, N>2α$, $λ$ is a positive parameter, $α\in (0,1)$, $(-Δ)^α$ 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 $Ω$.