The Existence Of Solutions Of Boundary Value Problems For Some Classes Of Nonlinear Equations Involving Gradient Terms

The boundary value problems of differential equations are often used to describe the models of practical problems,which have important applications in the fields of mathematics,physics,engineering and so on.In all kinds of equation problems,the boundary value problems of second-order differential equations play an important role.From the view point of mechanics,since this kind of equation describes the fact of Newton's principle of determinacy,which is a basic physical law to describe the motion of objects,the related problems appear in various scientific and engineering models and have always been widely concerned.In the case of “conservative” in which the nonlinear term is independent of gradient,people have established a variety of methods,A large part of them is the application of variational method.The physical correspondence of variational method is “least action principle”,which is a natural law widely followed by motion.For the problem of nonlinear term with gradient term,it is generally “nonconservative” and the well-developed variational method can not be directly applied.The existing methods mainly focus on the topological degree method and the upper-lower solution methods.In this thesis,the existence and multiplicity of solutions of several kinds of differential equations with gradient or derivative terms are established by using some nonlinear analysis methods,such as variational method,fixed point theory and Nehari manifold method,and the characterization of the sign information of the solutions is given.These results will supply the known results in related fields and are expected to be applied to other nonlinear problems.There are three classes of problems in this thesis.They are Dirichlet boundary problem of elliptic equation involving gradient term,problem of radial solution of elliptic equation with mixed boundary value and problem of periodic solution for second order ordinary differential equation with derivative term.The first aspect of this thesis is concerned with the existence of solution for nonlinear elliptic equation with gradient term.It is assumed that the nonlinearity is continuous and related to the gradient term.It is also assumed that the nonlinearity is local Lipschitz continuous,asymptotically linear at the origin and the infinity and two asymptotic slopes are located at two sides of the first eigenvalue of the operator.Under this condition,we establish the existence of at least one positive solution and one negative solution.In addition,we consider the existence of solutions for the superlinear case.The assumptions of this problem are uniformly superlinear condition,subcritical growth condition,uniformly monotone condition and local Lipschitz condition.Under these conditions,we prove that there exists at least one positive solution and one negative solution.The second aspect of this thesis is the existence of solution to the mixed boundary value problem of elliptic equation related with gradient term in annular domain.The nonlinear term is continuously differentiable.In the case of asymptotically linear,we establish the existence of nontrivial radial solution under some non-resonant conditions.When the asymptotic slopes of the nonlinear term at origin and infinity are located at two sides of the first eigenvalue of the operator,we prove that there are at least two nontrivial radial solutions,one of which is positive and the other is negative.In addition to the asymptotically linear problem,we also establish the results for the superlinear case,in which the nonlinearity is assumed to be superlinear at origin and infinity and satisfies the local Lipschitz condition.Under these conditions,we establish the existence of nontrivial radial solution.We also obtain the existence of at least one positive solution and one negative solution when only continuous condition are imposed.The third aspect of this thesis is the existence of periodic solutions for second order differential equation with derivative term.The nonlinearity is assumed to be continuous,periodic with respect to time,odd with respect to the second component,and even with respect to the third component.It is also assumed that the superlinear growth condition and the local Lipschitz condition are satisfied.We establish the existence of periodic solution for the period small enough,and give the characterization of sign-changing information of the periodic solution.The thesis consists of six chapters as follows.The first chapter is the introduction,which introduces the practical application background and the main results of this thesis.The second chapter is the preliminaries,which introduce the basic concepts and main lemmas used in this thesis.The main parts of this thesis are organized from Chapter3 to Chapter 6.Chapter 3 studies the boundary value problem of the second order elliptic equation with gradient term.Under the assumption that the nonlinear term is asymptotically linear,we establish the existence and multiplicity of solution.Chapter 4is devoted to the existence of the solution of superlinear problem involving gradient term without classical Ambrosetti-Rabinowitz condition.In Chapter 5,we study the radial solution of the mixed boundary value problem of the second order elliptic equation.In the superlinear and asymptotically linear case,we obtain the existence and multiplicity of solution,respectively.In Chapter 6,we study the periodic solution of ordinary differential equation with derivative term.By using the critical point theory,we obtain the existence of periodic solution.