Multiple positive solutions for a Schrödinger-Poisson-Slater equation with critical growth

Abstract

We consider the multiplicity of positive solutions for a Schrödinger-Poisson-Slater equation of the type−△u+(u2⁎1|4πx|)u=μ|u|p−1u+g(x)|u|4u,inR3, where μ>0, 3<p<5, and g∈C(R3,R+). Using Ekeland's variational principle and the well-known arguments of the concentration-compactness principle of Lions (1984) [24,25], when g has one local maximum point, we obtain a positive ground-state solution for all μ>0, while for g with k strict local maximum points, we prove that the equation has at least k distinct positive solutions for μ>0 small. The proof uses a minimization procedure on the Nehari manifold.