Researches On The Existence Of Ground State Solutions For Two Classes Of Elliptic Partial Differential Equations With Nonlocal Terms

We consider a class of fractional Schr?dinger equations with a general nonlinearity and a class of p-Kirchhoff type equations,and obtain the ground state solutions of these two classes of elliptic partial differential equations under certain assumptions.In chapter one,we summarize the relevant background and research status of these two classes of equations,and briefly explain the structure of this thesis as well as some related notations and conceptions.In chapter two,we obtain the ground state solutions for a class of fractional Schr?dinger equations with a general nonlinearity by applying the variational method.Firstly,according to the classical extension method,we transform this kind of nonlocal problem into a local problem defined in the upper half space of higher dimension,and by studying the solution of local equation of “extended space”,we obtain the solution of corresponding nonlocal problem.Secondly,we construct a functional of the local equation of higher dimension,and apply the General Minimax principle to its composite functional,then by studying the geometric structure of the composite functional,we construct a bounded Palais-Smale(PS)sequence with an extra property.Finally,by using the truncation technique combined with the energy analysis,and through the standard concentration-compactness argument,we prove the existence results.In chapter three,we also mainly apply the variational method to obtain the ground state solutions for a class of p-Kirchhoff type equations.We use the General Minimax principle and the Pohozeav identity to deal with such problems.Firstly,we construct a composite functional which is related to the functional of original problem and verify that the composite functional possesses the Mountain-pass geometry.Secondly,by studying the geometric structure of the composite functional,we construct a bounded(PS)sequence that satisfies an extra property,and then through the standard concentration-compactness argument,we can get the existence results.In chapter four,we conclude and prospect the whole thesis.