Ground State Solutions for a Semilinear Elliptic Equation Involving Concave-convex Nonlinearities

Abstract

This work is devoted to the existence and multiplicity properties of the ground state solutions of the semilinear boundary value problem $-Δu=λa(x)u|u|^{q-2}+ b(x)u|u|^{2^∗-2}$ in a bounded domain coupled with Dirichlet boundary condition. Here $2^∗$ is the critical Sobolev exponent, and the term ground state refers to minimizers of the corresponding energy within the set of nontrivial positive solutions. Using the Nehari manifold method we prove that one can find an interval L such that there exist at least two positive solutions of the problem for $λ∈Λ$.