Research On Some Elliptic Equations And Systems Involving Critical Exponent

In this thesis, we mainly study some elliptic equations and elliptic systems with critical exponent by means of variational methods.The thesis consists of four chapters:In Chapter One, we summarize the background of the related problems and state the main results of the present thesis. Moreover, we also introduce some preliminary knowledge and notations.In Chapter Two, we consider the following singular elliptic equation with Dirichlet boundary condition which involves the Caffarelli-Kohn-Nirenberg in-equalities. On the one hand, we show that the existence and nonexistence results of sign-changing solutions for the above problem by the Ljusternik-Schnirelaman theory and a Pohozaev-type identity; On the other hand, by means of variational method and Nehari manifold, we obtain least energy sign-changing solutions in some ranges of the parameters μ and λ, in particular, our result generalizes the existence results of sign-changing solutions to lower dimensions5and6.In Chapter Three, we study the following nonlinear Schrodinger system with Sobolev critical exponent where N>5,λ1,λ2>0,β≠0,2<α, r<2*,2*:=2N/N-2. By virtue of the Mountain Pass lemma, Ekeland variational principle and Nehari mainfold, we show that this critical system has a positive radial solution for β∈(0,+∞)∪[-1,0).In Chapter Four, we devote to the following elliptic system with Sobolev critical exponent where α,β>1, α+β=2*:=2N/N-2(N≥3) and Ω=RN or Ω is a smooth bounded domain in RN. On the one hand, we obtain a uniqueness result on the least energy solutions and show that a manifold of a type of positive solutions non-degenerate for the above system with Ω=RN in some ranges of α,β, N; On the other hand, when Ω is a smooth bounded domain in RN, we establish a global compactness result and by means of this global compactness result, we extend a classical result of Cor on in [30] on the existence of positive solutions of single equation with critical exponent on domains with nontrivial topology to the above elliptic system.