Qualitative Analysis Of Solutions To Nonlinear High Order And Fractional Order Differential Equations

In this dissertation,we mainly focus on the qualitative analysis of the solutions to a type of higher order differential equation and semi-linear fractional Laplacian equation.Our results mainly concern about the classification of solutions of higher order differential equations,the monotonicity and symmetry properties and the a priori estimate for the solutions of the fractional Laplacian equation.The main results of this dissertation can be summarized as followsThe first chapter is a brief introduction,we introduce the research background of this thesis and the progress in the study of the qualitative analysis of the solutions to nonlinear differential equations.Furthermore,we also clarify the main results of this dissertation and the methods used here to deal with the problems appeared in this dissertationIn the second chapter,we mainly discuss the Liouville type theorem of non-negative solutions to higher order Hardy-Henon equation which is defined in Rn By establishing the equivalence between the differential equation and the integral equation,we can transform the differential equation to its corresponding integral form.Through the method of moving plane in integral forms we can prove the radial symmetry of the solutions.Finally,we use the Pohozaev identity to prove the non-existence of non-negative solutionsIn chapter three,we concentrate on the analysis of the positive solutions to the semi-linear fractional Laplacian equation.We mainly focus on the study of the monotonicity and symmetry properties of the positive solutions to a type of semi-linear fractional Laplacian equation whose domain is strictly convex.Through the establishment of the maximum principles to anti-symmetric functions,we can apply the method of moving planes to nonlocal operators directly in order to prove the monotonicity and symmetry properties of the solutions to a type of fractional order equation which is defined on a strictly convex domainSubsequently,in chapter four,we extend the results of chapter three to more generalized convex domains.Because of the technical difficulty,the method we used in chapter three fails in a more general convex domain which is not strictly convex However,by establishing a more delicate estimate of anti-symmetry functions,we can overcome this obstacle to establish the monotonicity and symmetry properties of the solutions in more general convex domainsIn the last section of this dissertation,we study the uniform estimate of the solutions to a type of semi-linear fractional Laplacian equation which is defined in bounded and smooth domains.We can get an uniform a-priori estimate near the boundary of the domains by using the method of moving planes and the classical potential theory to exclude the possibility of the blowing up phenomenon of the solutions near the boundary of the domain.