Multiplicity Of Solutions For Elliptic Equations With Logarithmic Nonlinearity

The multiple solutions of p-Laplacian equation with logarithmic nonlinearity and existence of infinitely many solutions of biharmonic equation involving logarithmic non-linearity in bounded domain are investigated by using variational methodsFirstly,by decomposition the Nehari manifold of the energy functional combined with the logarithmic Sobolev inequality and minimizing sequence method,we consider the multiple solutions of p-Laplacian equation Dirichlet boundary-value problem with sign-changing weight function and logarithmic nonlinearity in bounded domain.where ? is a smooth bounded domain of RN,p??1,N?,p-Laplacian operator ?pu:=div?|?u|p-2?u?,??C?????.The main result is as follows.Theorem 1,Let ? change sign in ???,satisfying where |?|N is the volume of ?,the constant??t?as the usual ?-function.Then?P₁?possesses at least two nontrivial solutionsSecondly,by using variational methods combined with the fountain theorem,we consider the existence of infinitely many solutions of biharmonic equations Dirichlet boundary-value problem with logarithmic nonlinearity in bounded domain.where ?2 denotes the biharmonic operator,? is a bounded domain in RN?N?3?with smooth boundary and b,d ?R are two constants.The main results is as followsTheorem 2.The problem?P₂?possesses infinitely many solutions {uk}_k=1^+? and there is a positive constant C such that ||uk|| L2 ?CkN/2.Moreover7,problem?P₂?has a ground state solutionThe structure of this paper is as follows.In the first chapter,the basic theory of variational methods and the recent research work of partial differential equations with logarithmic nonlinearity by using variational methods are introduced,we are mainly introduced the related research of equations with p-Laplacian operator and biharmonic operator.Then the main contents and conclusions of this paper are statedIn the second chapter,the necessary knowledge for proving the multiplicity of nontrivial solutions of equation?P₁?is stated and the proof process of the main results is given.In the third chapter,the necessary knowledge for proving the existence of infinitely many solutions of equation?P₂?is stated and the proof process of the main results is given.