Existence Of Solutions To Biharmonic Equations With Hardy-Sobolev Exponents On Bounded Domains

Partial differential equation is an important branch in mathematics which relates closely to geometry and physics,etc.The existence of solution for elliptic equations plays an important role in the theoretical research of partial differential equations,and provides theoretical foundation for physics and geometry.In recent years,the existence of solutions to elliptic equations involving biharmonic operators has also been extensively studied,which has both important scientific significance and research value for the theory of elastic thin plates in physics and conformal geometry related to the Paneitz-Branson operator.In this thesis,we investigate the existence of the solutions to the following biharmonic equation involving the Hardy-Sobolev exponent and Dirichlet boundary condition (?) where ?(?)RN is a bounded smooth domain with 0,?>0,2?q?2*,2?p?2#(s),2*?(?)and 2#(s)=(?)are the Sobolev critical exponent and Hardy-Sobolev critical exponent respectively,(?)is the outer normal derivative at (?)?.The thesis is organized as follows.Chapter 1,we introduce background and significance of this thesis,and state the main results.We also present the basic knowledge and preliminary theorems in the end of this section.Chapter 2,we show the asymptotic estimates of the extremal functions of Hardy-Sobolev inequality related to biharmonic operator.Furthermore,we calculate some useful estimates to the extremal functions.Chapter 3,We mainly discuss the nonexistence,existence of weak solutions to(P?)with critical Hardy-Sobolev exponent.Firstly,we employ Rellich-Pohozaev identity to prove the nonexistence of nontrivial solution on star-shaped domain for?>0,q?2*;or ?<0,2?q?2*;or ?=0,2?q<2*,provided ? is stictly star-shaped;or ???1,q=2,provided u? 0 and ?=BR.Secondly,in order to overcome the difficulty of lack of compactness for critical Hardy-Sobolev exponent,we establish global compactness and obtain the(PS)c condition.And then we prove existence of at least one solution to(P?)with ?>0,2<q<2*or 0<?<?1,q?2 by Mountain-Pass theorem.Finally,we discuss the multiple sign-changing solutions to(P?)with q=2,p=2#(s)by Linking theorem.Chapter 4,we show the existence of infinite-many solutions to(P?)in the subcritical case ?>0,2?q<2*,2?p<2#(s).