Abstract
In Chap. 5, we consider the boundary value problems for a class of Monge–Ampère equations. First we prove that any solution on the ball is radially symmetric by the moving plane argument. Then we show that there exists a critical radius such that, if the radius of a ball is smaller than this critical value, then there exists a solution, and vice versa. Using a comparison between domains we prove that this phenomenon occurs for every domain. By using the Lyapunov–Schmidt reduction method we get the local structure of the solutions near a degenerate point; by Leray–Schauder degree theory, a priori estimates, and using bifurcation theory we get the global structure.
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Zhang, Z. (2013). Solutions of a Class of Monge–Ampère Equations. In: Variational, Topological, and Partial Order Methods with Their Applications. Developments in Mathematics, vol 29. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-30709-6_5
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DOI: https://doi.org/10.1007/978-3-642-30709-6_5
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