Ground state solutions of a Klein–Gordon–Maxwell system for two scalar fields

摘要

In this article, we consider a Klein–Gordon–Maxwell system for two scalar fields: −Δu+m12−(ω1+ϕ)2u=|u|2+|v|2p2−1u,−Δv+m22−(ω2+ϕ)2v=|u|2+|v|2p2−1v,−Δϕ+(u2+v2)ϕ=−ω1u2−ω2v2.$$ left{begin{array}{cc}hfill -Delta u+left[{m}_1^2-{left({omega}_1+phi right)}^2right]u& ={left({left|uright|}^2+{left|vright|}^2right)}^{frac{p}{2}-1}u,hfill {}hfill -Delta v+left[{m}_2^2-{left({omega}_2+phi right)}^2right]v& ={left({left|uright|}^2+{left|vright|}^2right)}^{frac{p}{2}-1}v,hfill {}hfill -Delta phi +left({u}^2+{v}^2right)phi & =-{omega}_1{u}^2-{omega}_2{v}^2.hfill end{array}right. $$ We show that there exists a ground state solution under the condition 2