The Existence Of Solution For Three Semilinear Elliptic Equations Without Compactness

In this thesis, by using some approach in variaticnal methods, such as minimax principle, moutain-pass lemma and concentration-compactness principle, we study the existence and multiplity of solutions to three classes of elliptic partial differential equations.In introduction, we recall some background of problems in the thesis.In chapter one, we give some preliminary knowledge.In chapter two, we consider the following semilinear equation with combined nonlinearities on RN By using the moutain-pass lemma, we get a nontrivial solution to the (2.1.1).In chapter three, we consider a semilinear elliptic equation with Hardy critical exponent: By symmetry mountain pass theorem and concentration compactness priciple, we obtain the existence of multiple solutions to (3.1.1).In chapter four, we consider the solution to the semilinear elliptic systems with critical exponent: By concentration compactness principle we can obtain: whenλis small enough, there exists a local minimum to the (4.1.1).