The Existence And Regularity Of Multiple Solutions For Infinitely Degenerate Elliptic Boundary Value Problems

In this paper, we study multiplicity and regularity of solutions for infinitely degen-erate elliptic equations. This article is divided into five chapters. In the first chapter, we introduce the background and basic knowledge about infinitely degenerate elliptic equations. In the second and third chapter, we research the basic inequalities on in-finitely degenerate system of vector fields and the estimates of eigenvalues for infinitely degenerate elliptic operators, which are important in the study of infinitely degenerate elliptic equations. In Chapter4and5, we study existence and multiplicity, boundedness and regularity of solutions for infinitely degenerate elliptic equations. The details are as follows:In Chapter1, we recall the finitely degenerate elliptic operator and equation. Then, we state some relate knowledge and the research status about the infinitely degenerate elliptic operator and equation. At last, we introduce the main work of this paper and preliminaries.In Chapter2, we review Logarithmic Sobolev inequality and Poincare inequality on a class of infinitely degenerate system of vector fields. At the same time, we establish the Hardy’s inequality on some infinitely degenerate system of vector fields: Here HX,01(Ω) is a weighted Sobolev space induced by the system of vector fields X.In Chapter3, we study the estimates of eigenvalues for infinitely degenerate ellip-tic operators. If the infinitely degenerate vector fields X=(X1,…,Xm) verify the logarithmic regularity estimates: where A=(e2+|(?)|2)1/2=<(?)>, s>1,△x=∑j=1mXj2. Then the lower bounds of Dirichlet eigenvalues for the infinitely degenerate elliptic operator-△X is where s is the order of pseudo-differential operator in the logarithmic regularity estimates.Next, we study the behavior of solutions for infinitely degenerate elliptic equations, such as existence and multiplicity, boundedness and regularity. Specifically, we consider following infinitely degenerate elliptic equations: (1) Hardy potential case. where Vn(x) satisfies Hardy’s inequality, a, b∈R.(2) Sublinear symmetrical perturbation case. where g(x, u)=-g(x,-u),|g(x,u)|≤c(1+|u|q-1),1<q<2, a, b∈R.(3) Free perturbation case. where f(x)∈L2(Ω), a,b∈R.(4) Sign-changing coefficient case. where a(x)∈C(Ω) is sign-changing.In Chapter4, making use of the Minimax Method, Symmetrical Mountain Pass Theorem, Ekeland’s Variational Principle, Perturbation Theorem, Nehari Manifold De-composition, we gain some results of existence and multiplicity for the above infinitely degenerate elliptic equations.In Chapter5, applying the iterative technique and the regularity lift theorems of degenerate equations, we obtain the boundedness and regularity of weak solutions for the above infinitely degenerate elliptic equations.