Concentration of solutions for non-autonomous double-phase problems with lack of compactness

Abstract

The present paper is devoted to the study of the following double-phase equation (Formula presented.) where N2, 1<p<q<N, q<p with p=NpN-p, :RNR is a continuous non-negative function, (x)=(x), V:RNR is a positive potential satisfying a local minimum condition, V(x)=V(x), and the nonlinearity f:RR is a continuous function with subcritical growth. Under natural assumptions on , V and f, by using penalization methods and Lusternik–Schnirelmann theory we first establish the multiplicity of solutions, and then, we obtain concentration properties of solutions.The present paper is devoted to the study of the following double-phase equation (Formula presented.) where N2, 1<p<q<N, q<p with p=NpN-p, :RNR is a continuous non-negative function, (x)=(x), V:RNR is a positive potential satisfying a local minimum condition, V(x)=V(x), and the nonlinearity f:RR is a continuous function with subcritical growth. Under natural assumptions on , V and f, by using penalization methods and Lusternik–Schnirelmann theory we first establish the multiplicity of solutions, and then, we obtain concentration properties of solutions.