Infinitely many solutions for quasilinear systems with critical exponent

Abstract

In this paper, we consider the following quasilinear system: (P)−∑i,j=1NDj(aij(u)Diu)+12∑i,j=1NDsaij(u)DiuDju=au+|u|2∗−2u+α2∗|u|α−2u|v|βinΩ,−∑i,j=1NDj(aij(v)Div)+12∑i,j=1NDsaij(v)DivDjv=bv+|v|2∗−2v+β2∗|u|α|v|β−2vinΩ,u=v=0on∂Ω,where Di=∂∂xi,Dsaij(s)=ddsaij(s), a,b>0 are constants, α,β>1 and α+β=2∗,2∗=2NN−2 is the Sobolev critical exponent for the embedding of H01(Ω) into Lp(Ω),Ω⊂RN is an open bounded domain with smooth boundary. By accurate estimates of the concentration compactness and the asymptotic behavior of the approximation solutions to the subcritical problems, we prove that if N≥7, the problem (P) admits infinitely many solutions.