Study On Some Fourth-Order Elliptic Equations And Related Problems

In this thesis,we study some fourth-order elliptic equations and related problems,including local pointwise inequalities for biharmonic equations with negative exponents,overdetermined problems for biharmonic equations and p-Laplacian equations,and Liouville-type theorems for generalized biharmonic equations and related equations.This thesis is composed of five chapters:In Chapter 1,we summarize the backgrounds and current studies of pointwise inequalities,overdetermined problems,and Liouville-type theorems for fourth-order elliptic equations,and briefly outline the main work of this thesis.In Chapter 2,we study local pointwise inequalities for biharmonic equations with negative exponents.We prove a local pointwise inequality without the growth condition,which answers an open question in[Ngo et al.,Nonlinearity,2018,31(12):5484-5499],that is,whether the growth condition is necessary?The proof method is based on the construction of a new auxiliary function.As an application,we obtain some known pointwise inequalities.In Chapter 3,we study the overdetermined problems for biharmonic equations and p-Laplacian equations.Assume that(M,g)is an n-dimensional complete Riemannian manifold with nonnegative Ricci curvature.Firstly,we study an overdetermined problem for biharmonic equations on Riemannian manifolds.Assume that Ω(?)M M is a smooth bounded domain,we deduce that if the mean curvature H is positive on boundary,then the overdetermined problem has a solution only if Ω is an Euclidean ball and M is isometric to Rn.The proof method is based on Hopf lemma and Reilly’s famous lemma about a domain in Riemannian manifolds is isometric to a ball in Euclidean space.We overcome the restriction that some symmetry tools on Riemannian manifolds are no longer applicable.This conclusion is a generalization of the overdetermined problem for the biharmonic equation in Euclidean space.Secondly,we study an overdetermined problem for p-Laplacian equations on Riemannian manifolds.Assume that Ω (?) M is a bounded and connected domain with boundary (?)Ω of class C2,α.We prove that the solution u of the torsional rigidity problem of the p-Laplacian equation satisfies uv|uv|p-2=-1/nH on boundary if and only if Ω is an Euclidean ball and M is isometric to Rn.The proof method is based on an integral identity.As an application,we prove the overdetermined problem of the p-Laplacian equation,and obtain the Heintze-Karcher inequality and the Soap Bubble theorem.In Chapter 4,we study the Liouville-type theorems for generalized biharmonic equations and related equations.Firstly,under some natural assumptions of potential functions,we prove the mean value inequality for generalized subharmonic functions,and obtain the Liouville-type theorem for generalized harmonic functions.Secondly,we extend the mean value inequality of generalized subharmonic functions to generalized subbiharmonic functions,and obtain the Liouville-type theorem for generalized biharmonic functions.Thirdly,as an application of the mean value inequality of generalized subharmonic functions,we obtain a new proof of the Liouville-type theorem for generalized Schrodinger-type equations that are not dependent on the maximum principle.Finally,we obtain the Liouville-type theorem for generalized Lane-Emden-type equations.In Chapter 5,we briefly summarize this thesis and provide prospects for future study.