Multiplicity Of Nontrivial Solutions For Two Kinds Of Elliptic Type Equations With Logarithmic Nonlinearity

Multiplicity of nontrivial solutions for two kinds of elliptic type equations with loga-rithmic nonlinearity were studied by variational method on bounded domain.Firstly,we considered the multiplicity of a kind of p-Laplacian type equation with sign-changing loga-rithmic nonlinearity,and got that the equation had at least two nontrivial solutions.Next we studied the multiplicity of nontrivial solutions of a class of p-Kirchhoff equation,got that the equation had at least two nontrivial solutions.The main theoretical bases are variational method,Mountain pass lemma and Nehari manifold.Firstly,we consider the multiplicity of nontrivial solutions for the following p-Laplacian type equation with sign-changing type equation where ?p is the p-Laplacian operator,? is a smooth bounded domain in RN,f,g:?? R,p?(1,N).The main result is as follows.Theorem 2.1.1.Suppose that f,g ?(?)are sign-changing on ?,and satisfies that then the equation(P1)has at least two nontrivial solutions,where |?|N is the measure of ?,Secondly,we investigate the multiplicity of nontrivial solutions for the following p-Kirchhoff type equation-(a+b)??|(?)u|pdx)?pu=?f(x)|u|q-2u+g(x)|u|r-2u+|u|p-2ulong|u|,x??u=0,x ?(?)?where a,b are positive constants,?>0 is a parameter,? is a smooth bounded domain in RN,f,g?c(?),1<q<p<2p<r<p*.The main results are as follows.Theorem 3.1.1.Suppose that there exist a nonempty domain ?1(?)? satisfying g(x)>0 for x ? ?,then exist AO>0 such that for ??(0,?0),the equation(P2)has at least two nontrivial solutions.The structure of this paper is as follows.In chapter 1,the essential ideology of variational methods and the recent development of P-Laplacian type problems are introduced.Moreover,the research work and main results of this paper are stated.In chapter 2,the preliminaries to prove the multiplicity of nontrivial solutions for p-Laplacian type equation with sign-changing logarithmic nonlinearities on a bounded domain are presented.More importantly,the proof of the main result is given.In chapter 3,the preliminaries to prove the multiplicity of nontrivial solutions for p-Kirchhoff with sign-changing logarithmic nonlinearities on a bounded domain are presented.More importantly,the proofs of the main results are given.