A different approach to ground state solutions for [formula omitted]-Laplacian system with critical exponent

Abstract

In this paper, we give a different method from Ao and Zou (2019) and Chen and Zou (2012) to consider the following nonlinear Schrödinger system with one critical exponent and one subcritical exponent: −Δpu+μ|u|p−2u=|u|q−2u+αλ|u|α−2u|v|βinRN,−Δpv+ν|v|p−2v=|v|p∗−2v+βλ|u|α|v|β−2vinRN,where N≥max{3,p},μ,ν,λ>0,α≥1,β≥1,2≤p<q<p∗andα+β=p,p∗=Np∕(N−p). By using variational methods, we prove that there exists μ0∈(0,1), such that when 0<μ≤μ0, the above system has a positive ground state solution; when μ>μ0, there exists λμ,ν∈μ−μ0ααpνββp,(μα)αp(νβ)βp such that if λ>λμ,ν, the above system has a positive ground state solution, if λ<λμ,ν, the above system has no ground state solution.