Two Types of Solutions to a Class of (p,q)-Laplacian Systems with Critical Sobolev Exponents in RN

Abstract

We focus on the following elliptic system with critical Sobolev exponents:  -div⁡∇up-2∇u+m(x)up-2u=λup⁎-2u+(1/η)Gu(u,v),  x∈RN; -div⁡∇vq-2∇v+n(x)vq-2v=μvq⁎-2v+(1/η)Gv(u,v),  x∈RN; u(x)>0,v(x)>0,  x∈RN, where μ,λ>0,1<p≤q<N, either η∈(1,p) or η∈(q,p⁎), and critical Sobolev exponents p⁎=pN/(N-p) and q⁎=qN/(N-q). Conditions on potential functions m(x),n(x) lead to no compact embedding. Relying on concentration-compactness principle, mountain pass lemma, and genus theory, the existence of solutions to the elliptic system with η∈(q,p⁎) or η∈(1,p) will be established.