Strauss’s Radial Compactness and Nonlinear Elliptic Equation Involving a Variable Critical Exponent

Abstract

We study existence of a nontrivial solution of -Δpu(x)+u(x)p-1=u(x)q(x)-1, u(x)≥0, x∈RN, u∈Wrad1,p(RN), under some conditions on q(x), especially, lim inf|x|→∞ q(x)=p. Concerning this problem, we firstly consider compactness and noncompactness for the embedding from Wrad1,p(RN) to Lq(x)(RN). We point out that the decaying speed of q(x) at infinity plays an essential role on the compactness. Secondly, by applying the compactness result, we show the existence of a nontrivial solution of the elliptic equation.