Existence Of Positive Solution For A P-Laplace Equation With Critical Nonlinearities

In this Master's degree thesis we consider a p-Laplace equation with dual critical nonlinearities: where△pu= div(|▽u|p-2▽u),0∈Ω(?) RN(N≥3) is a bounded domain with smooth boundary,βis a positive parameter. By using variational methods, we obtain the existence of positive solution for the equation.The organization of this thesis is as the following:In chapter 1, we introduce variational methods and consider the background, and the recent development of the problem.In chapter 2, we collect some basic materials in partial differential equation and some important theorems which will be used in the next chapter such as Sobolev space, embedding theorem, maximum theorem, mountain pass theorem and so on.In chapter 3, we use variational method to solve the problem. Via mountain pass theorem and a test function to verify local PS conditions, we obtain that the problem (P1) has at least one positive solution u∈W01,p(Ω) whenλand q satisfyλ> -λ1 and max-In chapter 4, we prove Lemma 3.2.1, it is important in the paper, because it helps us to overcome the difficulties which lie in the proof of critical nonlinearities' compactness, and it enables us to solve the problem.