Existence Of Solutions Of Non-cooperative Elliptic Systems And Multiplicity

In this paper, we studies existence and multiplicity of solutions for noncoop-erative elliptic systems on the bounded smooth domainΩin RN via variational methods.This thesis consists of four chapters. The first chapter is for a introduction.In chapter two. we investigates the following noncooperative elliptic systems where q>1. Firstly, when nonlinearity f satisfies Ambrosetti-Rabinowitz su-pcrlincar growth condition. we prove that this system has a nontrivial solution. Then, we further prove existence of nontrivial solution under general superlin-car condition. Lastly, we still obtain a existence result for asymptotically linear problem via the new linking concept introduced by Martin Schechter and Kyril Tintarev.In chapter three, we consider the following resonant noncooperative elliptic systems where a.b.d,μεR. h1:h2εL2(Ω).g1·g2:Ω×R→R are Caratheodory functions. Using a generalized Landesman-Lazer condition and infinite linking concept introduced by Benci and Rabinowitz. we obtain two existence results of solutions.In chapter four. we study semilinear elliptic equation and noncooperative elliptic systems near resonance. Firstly, for the following semilinear elliptic equa-tion whereλ∈R, h∈L2(Ω), g:Ω×R→R is a Caratheodory function, we prove this equation has two solutions whenλis closed to any non-principal eigen-valuesλk(k≥2) from above and bellow respectively by using the generalized Landesman-Lazer condition. Then, with regard to the noncooperative elliptic systems given in Chapter three. we establish two multiplicity results whenμapproach any eigenvalue by a similar discussion.