The Existence Of A Class Of P-laplacian Equations With Multiple,

At present, the practical problems in physics, biochemistry, medicine, controlling theory and other disciplines, may be resolved through the Partical differential equation. People had paid more attention to the study of them and got many important results, which made the theory of Partical differential equation more perfect. Based on the previous research, we prove the existence and multiplicity of solutions to a special elliptic equation by variational methods including Mountain Pass Lemma and uniformly convex functional that generalized the notion of uniformly convex norm. Our results generalize the previous results to some extent. In this paper, firstly, we introduce the relevant notions and properties of uniformly convex functional and the (S+) condition, which are the basis to prove the existence and multiplicity of solutions to equations of p-Laplacian type.Secondly, we study the Dirichlet problem of p-Laplacian: and prove the existence of weak solutions. We prove the existence of the solution to the problem which some conditions in [16] are weakened through the functional version of the mountain pass theorem. Especially, when p=2, we get the existence of the negetive solution under suitable conditions.Finally, this paper makes further exploration of the existence of infinitely many solutions by the "Z2-symmetric" version of the mountain pass theorem. Then we get more general conlusions.