Existence And Uniqueness Of Positive Solutions For Two Classes Of Singular Elliptic Equations

Since the singular term,the well-known critical points theory can not be applied directly,for example the Mountain Pass Lemma.In this dissertation,we investigate the existence and uniqueness of positive solutions for two classes of singular elliptic equations by exploiting the global extremum of energy functional and the constrained extremum on Nehari manifold.Firstly,we study the uniqueness of positive solutions for a class of resonant singular p-Laplacian equation#12 where ?(?)RN is a non-empty bounded region with smooth boundary(?)?,?pu=div(|?u|p-2?u),1<p<N,f?Lp*/p*+?-1(?)is a non-zero and non-negative function,O<?<1.By adopting the variational method,we obtain a global minimum point corresponding to the energy functional of the equation.After that,we prove this global minimum point is indeed the positive solution of the equation combined with the definition of the solution.Furthermore,some analysis techniques are utilized to prove the uniqueness of the positive solution.We extend this result and improve the recent results on the singular p-Laplacian equations.Then,we consider the existence of positive solutions for a class of strongly singular linear Kirchhoff problem#12 where ?(?)RN(N?3)is a non-empty bounded region with smooth boundary(?)?,a,b?0 and a+b>0,m>0,?>1,f f?L1(?)is a non-zero and non-negative function,g?L?(?).Due to the strong singularity,we first establish constraint sets for the solution by using the Nehari manifold.The minimum point of the energy functional corresponding to the problem is defined on the constraint set.Then,we use the variational method and some analysis techniques to prove the existence of the minimum point.Moreover,we prove that it is indeed a positive solution of the problem.We extend the recent results on strongly singular Kirchhoff problems.Finally,we make a brief summary of the existence and uniqueness of the positive solutions for two classes of singular elliptic equations studied in this dissertation,and discuss about the future work.