Existence And Multiplicity Of Solutions To A Class Of The Dirichlet Problems With P-Laplacian And Weights

In this paper, we study the existence and multiplicity of solutions to a class of the Dirichlet problems with p-Laplacian and weights:Where Ω (?) R^N(N ≥ 3) is a bounded domain with smooth boundary (?)Ω, 1 < p < N, â–³_pu =div(|â–½u|^p-2â–½u) is the p-Laplacian, u^± =max{±u, 0}, u = u⁺—u^-, the weight function a(x) ∈ L^r(Ω), with a(x) > 0, a.e. in Ω, and f satisfies some conditions.The thesis consists of four chapters.In chapter one, we introduce some results on a class of the Dirichlet problems with p-Laplacian and weights:In chapter two, we introduce some basic knowledge of Sobolev spaces W_o^1,p(Ω) and some basic lemmas. In addition, we give some notations.In chapter three, we obtain several results on the existence of solutions to the problem (1.1) with the different values of p, q by using of the minimizing method and the Mountain Pass Lemma.In chapter four, by constructing a vector field and using a descending flow argument as that in T. Bartsch and Liu, we can obtain the existence of sign-changing solutions to the problem (1.1) with 1 < q < p < N.