Existence And Multiplicity Of Positive Solutions For Some Elliptic Boundary Value Problems

In this thesis, by using the variational method, Nehari method and some analysis techniques, we study the existence and multiplicity of positive solutions for some elliptic boundary value problems.Firstly, we consider the following semilinear elliptic problem with critical exponent where Ω(?)RNN(N≥3) is an open bounded domain with smooth boundary, l<g<2*,λ>0 and 2*=2N/N-2 is the critical Sobolev exponent. The coef-ficient function is nonzero and nonnegative, and g E C(Q) is a positive function. We obtain the existence and multiplicity of positive solutions for problem (0.1) via the variational methods and Nehari method.Secondly, we study the following singular Neumann boundary value problem where Ω(?)RN(N≥3) is a bounded domain with smooth boundary,λ> 0,0< γ<1<p≤2*- 1. The coefficient function is nonzero and nonnegative. P is a nonzero and nonnegative function and satisfies When 1< p< 2*-1, we obtain two solutions of problem (0.2) by Nehari method. By using a minimax method and some analysis techniques, we obtain the existence of solutions for problem (0.2) with p= 2*-1.Next, we consider the following Kirchhoff-type problem with singularity where Ω(?)R3 is a bounded domain, a, b, λ,μ> 0,0<γ< 1. The existence and multiplicity of positive solutions for problem (0.3) are obtained by the Nehari method and a minimax method.Finally, by using a minimax method and some analysis techniques, we study the uniqueness of solutions for the following singular Kirchhoff-type equation where Ω(?) RN (N≥3) is a bounded domain,0<γ<1,λ≥0,0<p≤2*-1, a, b≥0 with a+b> 0. The coefficient with f(x)> 0 for almost every x ∈Ω.