Existence Of Solutions To Several Partial Differential Equations Based On Fibering Method

In this paper,we introduce a new method-fibering method,which can solve some partial differential equation problems effectively.Based on this method,we consider the existence of solutions for three kinds of partial differential equations.This paper is divided into four chapters.In the first chapter,the fibering method is introduced in detail,and its theory is proved.The use of this method is illustrated by a simple example.From this,we can see that the fibering method is very suitable for the case that the nonlinear term of partial differential equation is polynomial form.In the second chapter,we consider a kind of Kirchhoff equation with critical Sobolev exponent:(?)where,?(?)R4 is a bounded region with smooth boundary (?)?,a,b,?,?are positive parameters.Combining with Nehari manifold,it is proved that if ??(0,a?1),? ?(bS2,+?),then the problem(0.4)has at least one pair of nontrivial solutions;if ??(0,a?1),??(0,bS2),then the problem(0.4)has no nontrivial solutions.In the third chapter,we consider the existence of solutions for a class of semilinear elliptic boundary value problem:(?)where,? is a bounded region with a smooth boundary in RN,?>0,a,b,:??R is a smooth function,and a(x)>0,b(x)(?)0.Combining with Sobolev embedding theoremLthe conclusion of this chapter is as follows:supposel<q,p<N+2/N-2.If ??(0,?1),then there is a solution to the problem(0.5);If ?=?1 and ?? b(x)|(?)1|p+1dx<0.the conclusion holds still.Here,(?)1 represents the first eigenfunction corresponding to the first eigenvalue ?1.In the fourth chapter,we consider the existence of nontrivial solutions for a class of Semilinear biharmonic equations with critical exponent:(?)where,?(?) RN is a bounded smooth region in RN,N? 5;vis the unit outward normal to (?)? ?>0;p=2N/N-4 is the critical Sobolev constant o fH02?Lp(?).Combining with Brezis-Lieb lemma,it is proved that ?(?)RN is a bounded smooth region,p=2N/N-4,N? 8.if ? ?(0,?1],then the problem(0.6)has at least a nontrivial solutions.