Research On The Solutions Of Two Types Of Elliptic Equations

The variational method is based on the theory of critical point,and the differential equation boundary value problem is transformed into a variational problem to prove the existence,multiplicity,and approximate solution of the solution.In this paper,we use the variational method to study two kinds of elliptic differential equations with strong physi-cal background,namely the Schrodinger Poisson equation with critical exponent and the Kirchhoff equation with p-Laplacian operator.In the second chapter,we study the following Schrodinger Poisson equations with critical exponentswhere ??R3 is a bounded smooth domain,r ?(0,1),and f,g ?C(?)are nonnegative and nontrivial.By using the Nehari manifold,Ekeland's variational principle and the compact-ness principle to get the constant M4,Then for 0<?<M4,u? ? N?+,this equation at least has one positive solution.In the third chapter,we study the following Kirchhoff equations with p-Laplace operatorwhere ??R3 is a bounded smooth domain,?p = div(|?u|p-2?u)is the p-Laplacian with 1<p<N.There are some parameters,they are a,>0,a + b>0,??0,0<s<1 and 0<r ? p*-1.For any x??,such that f ? Lp*+s-1/p*(?)with f(x)>0,and p*=N-p/Np Using variational method,The convergence of the Legege control theorem and the first eigenvalue of the operator,this equation possesses a unique positive solution.