On The Global Structure Of Solutions For The Prescribed Mean Curvature Problems

This dissertation focuses on the global structure of solutions for the prescribed mean curvature problems.The main content can be divided into three parts:studying the global structure of radial positive and sign-changing solutions for the prescribed mean curvature problems in a ball,the exact number of positive solutions and the extremal solutions of the Dirichlet problem for the one-dimensional Minkowski-curvature equation;investigating the spectrum structure for eigenvalue problems involving mean curvature operators in Euclidean and Minkowski spaces;discussing the global structure of radial positive solutions of nonlinear elliptic systems with Neumann boundary conditions and the exact number of positive solutions of the Neumann problem for the one-dimensional Minkowski-curvature equation.Our main results are as follows.1.By applying the unilateral global bifurcation theory and some preliminary results on the superior limit of a sequence of connected components,we establish the global structure of radial positive solutions of Dirichlet problem(?)(?)on the behavior of nonlinear term f(?,s)near s = 0.We prove the existence of connect-ed component of radial positive solutions for the prescribed mean curvature problems in an annular domain via global bifurcation techniques.And then,we use the component obtained in an annular domain to construct the desired component of radial positive so-lutions for the prescribed mean curvature problem in a ball.It is worth remarking that the study of global behavior of the radial positive solutions is very useful for computing the numerical solutions.These results extend,improve and unify the corresponding ones of Bereanu,Jebelean and Torres[J.Funct.Anal.2013]and Coelho,Corsato and Rivetti[Topol.Methods Nonlinear Anal.2014].In addition,by applying the unilateral glob-al bifurcation theory and some preliminary results on the superior limit of a sequence of connected components,we also establish the global structure of radial sign-changing solutions of the above Dirichlet problem.Our results partially improve the main ones of Capietto,Dambrosio and Zanolin[Ann.Mat.Pura Appl.(4)2001].2.We are concerned with quasilinear Dirichlet problemwhere ? ?(-?,0)U(0,?)is a constant.We show that any nontrivial solution u of the above problem has only finite many of simple zeros in[0,1],all of humps of u are same,and the first hump is symmetric around the middle point of its domain.We also describe the global structure of the set of nontrivial solutions of the above problem.Some results of this dissertation complement the corresponding ones of Cano-Casanova,Lopez-Gomez and Takimoto[J.Differential Equations,2012],who only established the existence of a component of positive solution of the above problem as ?>0.3.We start with the study of the exact multiplicity and bifurcation diagrams of positive solutions of the quasilinear two-point boundary value problemThe arguments are based upon a time map method.Next,if nonlinearity f(x,s)is decreasing with respect to s,we construct the well-ordered lower and upper solutions and obtain extremal solutions of Dirichlet problem for one-dimensional Minkowski-curvature equationby using the monotone iterative technique.4.We consider radial positive solutions of elliptic systems of the formwhere essentially ?,? are assumed to be radially nondecreasing weights and f,g are nondecreasing in each component.We show the existence of at least one nondecreasing nontrivial radial solutions by using bifurcation techniques.Our results is sharp and is a complement for one of the main results in Bonheure,Serra and Tilli[J.Funct.Anal.2013].Motivated by the existence of multiple positive solutions for a class of semipositone Neumann two point boundary value problem,and then,we study the existence of multiple positive solutions of the Neumann problem for the one-dimensional Minkowski-curvature equationby using the quadrature technique,and it is the first time to obtain the existence of mul-tiple positive solutions and the exact number of positive solutions of the above problem.