Multiplicity results for the fractional laplacian in expanded domains

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In this paper we establish the multiplicity of nontrivial weak solutions for the problem (-Δ)α u +u= h(u) in Ωλ,u=0 on ∂Ωλ, where Ωλ=λΩ, Ω is a smooth and bounded domain in Rⁿ, N>2α, λ is a positive parameter, α∈ (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.