The Positive And Sign-changing Solutions Of Kirchhoff-schrodinger-poisson System

With the continuous development of mathematical research, it has been found that the variational method is used to study the differential equation more easily when solving the physical problem, and the variational method is paid more and more attention. The development of variational methods has gone through two stages. Before 1950s, it was the first stage of the classical variational methods.Since 70s, the second phase of the finite element method has been introduced and developed from solid mechanics and structural mechanics to fluid mechanics and other fields. During the development of the second stage, the researchers have devel-oped the critical point theory, such as the mountain pass theorem and the fountain theorem, and used it to study nonlinear equations, especially the nonlinear elliptic boundary value problems, a lot of meaningful results have been obtained in the corresponding equation, up to now.the Kirchhoff-type equation was first presented by Kirchhoff as an extension of the free vibrations of elastic strings. The Schrodinger-Poisson system has been first.introduced as a physical model describing a charged wave interacting with its own electrostatic field in quantum mechanic. With the continuous development of the above two types of equations, the Kirchhoff-Sclirodinger-Poisson system is studied and some results on multiplicity.In this paper, we obtain the existence of positive solutions, negative solutions,sign-changing solutions and infinitely many sign-changing solutions for two types of the Kirchhoff-Schirodinger-Poisson system by using the variational methods, the mountain pass theorem and a version of the symmetric mountain pass theorem.The dissertation contains three chapters.In chapter 1, we mainly introduce the research status of Kirchhoff-Schrodinger-Poisson system and some basic knowledge and symbols commonly used in this pa-per.In chapter 2, we consider the following Kirchhoff-Schrodinger-Poisson system:where a > O.b ?0,? > 0.1 <q < 2.4 < p < 6. F(?.x.u) = ?f(x)|u|q-2u+g(x)|u|p-2u.Under appropriate conditions, we can use the variational methods to get a positive solution to the problem, which is a local minimizer of the energy function. We can get another different positive solution by using the mountain pass theorem.In chapter 3. we consider the following Kirchhoff-Schrodinger-Poisson system:where a > 0,b ? 0, f(x.u): R3 × R ? R is a nontrival function. We can get the existence of positive and negative solutions by using a sign-changing version of the mountain pass theorem. Under appropriate conditions, we use a version of the symmetric mountain pass theorem to obtain the existence of infinitely many sign-changing solutions to the problem.