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 onq(x), especially,lim inf|x|→∞ q(x)=p. Concerning this problem, we firstly consider compactness and noncompactness for the embedding fromWrad1,p(RN)toLq(x)(RN). We point out that… Click to show full 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 onq(x), especially,liminf|x|→∞q(x)=p. Concerning this problem, we firstly consider compactness and noncompactness for the embedding fromWrad1,p(RN)toLq(x)(RN). We point out that the decaying speed ofq(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.
