Existence Of Multiple Solutions And Ground State Solutions Of Nonlinear Schrodinger Systems With Critical Exponents And Perturbed Terms

In this paper,we consider the following coupled Schrodinger system:Here Ω(?) RN(N≥3) is a smooth bounded domain,2*=2N/N-2 is the Sobolev critical exponent. λ1,λ2>-λ1(Ω),μ1,μ2>0,β> 0 is the coupling constant. Whereλ1(Ω) is the first eigenvalue of -Ω,and f(x),g(x)≠ 0. We show that when f(x),g(x) satisfy some condition this system have a bounded solution and a ground state solution. Our main method is to divide the Nehari manifold into three parts by restricting some condition, by restricting f(x) and g(x) we can make the degenerate part only contain (0,0), thus we get two non-degenerate parts of the Nehari manifold, then we will find solutions in two non-degenerate parts respectively and these solutions must be non-trivial.It is easy to decide which solution has minimizer energy by comparing their size. The main difficult lies in the embedding H01(Ω)â†'L2- (Ω)is noncompact. For this, we solved this problem by proving its contradiction. For this system, we show that under the condition of f{x),g(x)> 0, when f(x)> 0,g(x)> 0 its ground state solution has the least energy,when f(x)=0, g(x)=0 its ground state solution has the most energy. That is the ground state solution of this system has the least energy when both of the external terms are nonzero,and has the most energy when it doesn’t have external terms.