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Abstract: We study the boundary value problems for Monge–Ampère equations: in , , . First we prove that any solution on the ball is radially symmetric by the argument of moving plane. Then we show there exists a critical radius such that if the radius of a ball is smaller than this critical value there exists a solution, and vice versa. Using the comparison between domains we can prove that this phenomenon occurs for every domain. Finally we consider an equivalent problem with a parameter in Ω, , . By using Lyapunov–Schmidt reduction method we get the local structure of the solutions near a degenerate point; by Leray–Schauder degree theory, a priori estimate and bifurcation theory we get the global structure. [Copyright &y& Elsevier]

In this paper, we use a bifurcation approach, Morse theory to study asymptotically linear and superlinear elliptic boundary value problems, and obtain new results on the exact multiplicity of solutions and sign-changing solutions to asymptotically linear and superlinear problems and the properties of the solutions. [Copyright &y& Elsevier]