The Existence And Properties Of Positive Solutions For Several Classes Of Fractional Partial Differential Equations And Systems

Fractional partial differential equations are widely used in physics,finance,materials science and other fields,the developments of the related theories can promote the progress of multiple interdisciplinary subjects.In the study of partial differential equations,positive solutions or positive radial symmetric solutions have attracted much attention because of the strong physical backgrounds.In this dissertation,we focus on the existence and properties of positive solutions for several classes of fractional partial differential equations and systems.The main methods we used are the principle of symmetric criticality,mountain pass lemma,minimax method,Ekeland’s variational principle and constrained minimization method.The full text is divided into seven chapters.In the first two chapters,we introduce the research backgrounds and some preliminaries.In the next four chapters,we give the detailed research contents.In the last chapter,we summarize the main conclusions and innovation points of the dissertation,then we propose some questions for further study.The following is a brief introduction from chapter three to six:In chapter three,we investigate the symmetry properties of positive solutions to a fractional system with multiple isolated singularities.If the system has two isolated singularities,then the positive solutions are axis-symmetric about the line going through the given singularities.Further,if the system has finitely many isolated singularities,then the positive solutions are cylindrical symmetric about the hyperplane through the given singularities.For the critical fractional system with single singularity,Y.M.Li and J.G.Bao(Nonlinear Anal.2020,191:111636)proved that the positive solutions are radially symmetric,by the extension formulations for fractional Laplacian operator and the method of moving sphere.In this chapter,under a wider range of indicators,we obtain the symmetry properties of positive solutions by applying a direct method of moving plane for fractional system.In chapter four,we study the existence and concentration of positive ground state solutions for a fractional system with non-constant potential functions and saturable nonlinearity.First,by applying the mountain pass lemma with Cerami condition in a symmetric space,the constrained minimization method and the method of moving plane,we get the existence of positive radial symmetric ground state solutions(mountain pass type solutions)for a fractional system with constant potential functions.Then combined with some energy comparison and the concentration compactness principle,we obtain the existence of positive ground state solutions for the system with non-constant potential functions.Moreover,the concentration phenomenon of the positive ground state solutions is also studied.Our results generalize the results about classical system of R.Lehrer et al.(Nonlinear Anal.2020,197:111841)under some more relaxed indicator conditions.In chapter five,we investigate the existence of positive normalized ground state solutions for a fractional Kirchhoff equation with Sobolev critical exponent.The positive radial symmetric ground state solutions(minimizers)are obtained by applying the constrained minimization method,Pohozaev identity and Ekeland’s variational principle.Our results generalize the results of L.Jeanjean et al.(arXiv:2008.12084.2020)where the authors studied the existence of positive normalized solutions for a classical Schrodinger equation.In addition,a more concise and universal method is given to prove that the L2-norm of the weak limit for the minimizing sequence is non-splitting.In chapter six,we study the existence of positive normalized solutions or positive radial symmetric normalized solutions for a fractional Kirchhoff system.In L2-critical case,the nonnegative solutions(minimizers)are obtained by constrained minimization method when the potential functions are trapping potentials and the coefficient of the coupling term is small.Under some further restrictions on the exponents of the coupling term and some growth condition on potential functions,we obtain the decay estimate and H?lder continuity of the nonnegative solutions.Further,we get the positivity of the minimizers.In L2supercritical case,by the principle of symmetric criticality and minimax method,we obtain the positive radial symmetric solutions(mountain pass type solutions)when the coefficient of the coupling term is large.