Multiple positive solutions of semilinear elliptic equations with weight function in exterior domains

Abstract

In this paper, we are concerned with the multiplicity of solutions for semilinear elliptic problems with weight functions in exterior domains. We prove that, if the decay of the weight function at spatial infinity is sufficiently slow, then the equation admits at least three solutions and two of them escape away to the spatial infinity as the decay rate of the weight function tends to $0$. The result is proved via the variational method combined with the Ljusternik--Schnirelman-type multiplicity theorem.