Compact Sobolev embeddings and positive solutions to a quasilinear equation with mixed nonlinearities

Abstract

In this paper, we study the existence and multiplicity results of nontrivial positive solutions to the following quasilinear elliptic equation on RN, when N≥2,−Δpu=λK(x)ur−1−V(x)uq−1. Here, K(x),V(x)>0 are suitable potentials, 1<p<r<q<∞, and λ>0 is a parameter. To study this problem, some compact embedding results regarding MVq,p(RN)↪LKr(RN) are proved that unify and extend some recent results of the author [12–14].