Solutions Of Nonlinear Elliptic Equations With Nonlocal Term

This thesis is devoted to the study of existence of solutions of two classes of equations involving nonlocal term. Firstly, we study the existence of solutions of Kirchhoff-type elliptic equations, including of existence of ground state solutions, multiplicity of solutions, negative energy solutions, sign-changing solutions. Secondly, we consider the existence of solutions of the Schr?dinger-Poisson equations. The concrete results and methods is as follows.The thesis consists of six parts:In Chapter 1, introduction. We introduce the physical background and academic meaning of two classes of equations with nonlocal term in the thesis. And we briefly give the development of the related topics in the domestic and overseas at the present time. Finally, we simply present the mainly content of the thesis.In Chapter 2, we consider the existence of some kinds of solutions of Kirchhoff-type elliptic equations involving critical Sobolev exponents in bounded domain of N-dimensional euclidean space( N ?4). When N ?4, for q ?2 and q ?(2, 4), we overcome some difficulties to obtain the boundedness of(PS) sequences through the Nehari manifold and Lions' first concentration compactness lemma, and get local compactness of(PS) sequences by using Lions' second concentration compactness lemma. Then we prove the existence of ground state solutions bu of Kirchhoff-type elliptic equations. Moreover, if b ?0, we disscuss that the relation of Kirchhoff-type elliptic equations and Brezis-Nirenberg problem. And we admit that when b tends to zero, the ground state solution bu of Kirchhoff-type elliptic equations converges to 0u, which is a ground state solutions of Brezis-Nirenberg problem. For q ?(1, 2), we gain that the equation possesses infinitely many negative energy solutions via a cut-off technique and Krasnoselskii's genus theory. When N ?5, under some assumptions on a and b, we prove that there is a positive solution to the equations.In Chapter 3, we deal with the existence of sign-changing solutions of the Kirchhoff-type elliptic equations with pure power nonlinearity. Making use of variational methods, invariant set of descent flow and Krasnoselskii's genus theory, for p ?(2, 6), we turn out that there are infinitely many sign-changing solutions to the Kirchhoff-type elliptic equations. Our result expand the known results which is valid for only p ?(4, 6)In Chapter 4, we disscuss the existence of sign-changing solutions to the Kirchhoff-type elliptic equations involving a general nonlinearity. Through some weak assumptions of the nonlinearity term and using a skill to cut-off energy functional and the sign-changing version Nehari manifold, we assert that there are a least energy sign-changing solution and a signed ground state solution to the Kirchhoff-type elliptic equations and the energy of this sign-changing solution is strictly more than energy of ground state solution.In Chapter 5, we study the other class of ellitpc equations with nonlocal term, that is, Schr?dinger-Poisson equations with critical Sobolev exponents. Firstly, by Lax-Milgram theorem, we know that Schr?dinger-Poisson equations can be reduced into one equation involving a nonlocal term. Then we have the energy functional by the variational methods. Secondly, when p ?1, we constrain the energy functional on Nehari menifod and obtain the boundedness of(PS) sequences. By using the concentration compactness lemm, we show the local compactness of(PS) sequences. Finally, we discover the constraint is a natural constraint about the energy functional. Therefore, we see that the Schr?dinger-Poisson equations possesses a ground state solution. While p ?(0,1), using a cut-off skill to obtain a new energy functional, we show that the Schr?dinger-Poisson equations possesses infinitely many negative energy solutions by Krasnoselskii's genus theory.In Chapter 6, we simply summarize the thesis, as well hope that some results can be optimal and some assumptions can be weaken.