Positive solutions of semilinear elliptic problems in unbounded domains with unbounded boundary

Abstract

This paper is concerned with the existence and multiplicity of positive solutions of the equation −Δu+u=up−1, 2<p<2∗=2NN−2, with Dirichlet zero data, in an unbounded smooth domain Ω⊂RN having unbounded boundary. Under the assumptions:(h1)∃τ1,τ2,…,τk∈R+∖{0}, 1⩽k⩽N−2, such that(x1,x2,…,xN)∈Ω⟺(x1,…,xi−1,xi+τi,…,xN)∈Ω,∀i=1,2,…,k,(h2)∃R∈R+∖{0} such that RN∖Ω⊂{(x1,x2,…,xN)∈RN:∑j=k+1Nxj2⩽R2} the existence of at least k+1 solutions is proved.