On The Existence Of Multiple Solution Of The Semilinear Polyharmonic Elliptic Equations And Quasilinear Elliptic Equations

In this paper, we mainly study the existence of multiple solutions for semilinear polyharmonic elliptic:equations involving critical Sobolev exponent and the existence of a nontrivial solution and nodal soliton solutions for generalized quasilinear elliptic equations.The thesis consists of four chapters:In chapter one,we summarize the background of the related problems and state the main results of the present thesis. We also give some preliminary results and notations used in the whole thesis.In chapter two, we study the existence of multiple solutions for the following semilinear polyharmonic elliptic equation involving critical Sobolev exponent: where Ω is a bounded domain in RN with N≥2k+1,1<q<2,λ>0,f, g are continuous functions on Ω which are somewhere positive but which may change sign on Ω. k*=2N/N-2k is the critical Sobolev exponent. By extracting the Palais-Smale sequence in the Nehari manifold, the existence of multiple nontrivial solutions to this equation is verified.In chapter three, we are concerned with the existence of nontrivial positive solutions for the following generalized quasilinear elliptic equation where1<p<N, g(t):Râ†'R+is a C1nondecreasing function with respect to|t|, g(0)=1; h:Râ†'R is a differential function, the potential V(x):RNâ†'R is positive in RN. By introducing a new change of variable replacement, we prove the existence of a nontrivial positive solution via the Mountain pass theorem. We generalize the main results of Yaotian Shen and Youjun Wang in [92] from the case p=2to the cases1<p<N.In chapter four, we are concerned with the existence of node soliton solutions for the following generalized quasilinear elliptic equation where1<p<N, g(t):Râ†'R+is an even differential function and g’(s)≥0, Vs≥0; h:Râ†'R is an odd differential function, the potential V(x):RNâ†'R is positive in RN. By using a new change of variables and minimization argument, we obtain a sign-changing minimizer with k nodes of a minimization problem. We generalize the main results of Y. Deng, S. Peng and J. Wang in [29] from a quasilinear Schrodinger equation to a generalized quasilinear elliptic equation.