Research On The Existence Of Solutions For Two Classes Of Elliptic Equations With Variational Frames

As an important branch of differential equation,nonlinear partial differential equation is the main research field of modern mathematics.In this thesis,we study two classes of elliptic equations with variational frames,under the suitable assumptions,the existence of solutions for the generalized Kadomtsev-Petviashvili equation and the Schr(?)dinger-Bopp-Podolsky equation are proved.The results obtained in this thesis can be used to solve practical nonlinear problems,which is of great theoretical value and practical significance.The specific research works are as follows:The first chapter introduces the background,significance,research status and main innovations of this thesis,and briefly summarizes the research work of this thesis.Chapter two studies the generalized Kadomtsev-Petviashvili equation.By using the Symmetric Mountain Pass Theorem,we get the existence of infinitely many nontrivial solutions in bounded domain without Ambrosetti-Rabinowitz condition.In addition,the existence of ground state solutions for the Kadomtsev-Petviashvili equation in RN is proved by using the method developed by Jeanjea's monotonicity technique.In chapter three,the existence of infinitely many nontrivial solutions for the Schr(?)dinger-Bopp-Podolsky system is studied by means of the Symmetric Mountain Pass Theorem.With the help of the extended Clark Theorem,the existence of multiple non-positive energy solutions and the decay of the solutions are obtained.Under the absence of Ambrosetti-Rabinowitz condition,a family of energy functional is constructed by Jeanjea's monotonicity technique,and the existence of ground state solutions for Schrodinger-Bopp-Podolsky system under suitable conditions is obtained.What's more,we also study the existence of positive solutions for the Schr(?)dingerBopp-Podolsky system as p E(2,6).By constructing a truncation function,we obtain the upper bound of the critical level value,and the boundedness of(PS)sequences as p?(2,4).