On the Monge-Ampère equation with boundary blow-up: existence, uniqueness and asymptotics. (English) Zbl 1148.35022
The authors study the Monge-Ampére equation
\[ \det D^2u= b(x)f(u) \]
in a smooth, strictly convex, bounded domain \(\Omega\) in \(\mathbb R^N\), \(N\geq 2\), with the boundary condition
\[ \lim_{x\to \partial \Omega} u(x)=\infty, \]
where \(f\in C[0,\infty)\cap C^\infty(0,\infty)\) is positive, increasing and such that \(f(0)=0\), and \(b\in C^{\infty}(\bar\Omega)\) is positive in \(\Omega\). Existence of a strictly convex blow-up solution in \(C^\infty(\Omega)\) is guaranteed if there exists a positive and non-decreasing function \(f_0\) on \([0,\infty)\) such that \(f_0(u)\leq f(u)\) for every \(u>0\), \(f_0^{1/N}\) is locally Lipschitz continuous on \([0,\infty)\) and satisfies the Keller-Osserman condition, and, finally, \(f_0^{-1/N}\) is convex in \((0,\infty)\). In particular, this generalizes the result by A. C. Lazer and P. J. McKenna [J. Math. Anal. Appl. 197, No. 2, 341–362 (1996; Zbl 0856.35042)] where existence was proven if \(f(u)=u^q\) with \(q>N\). Moreover the authors provide asymptotic estimates of the behaviour of such solutions near \(\partial \Omega\) and prove uniqueness when the variation of \(f\) at \(\infty\) is regular of index \(q>N\).
\[ \det D^2u= b(x)f(u) \]
in a smooth, strictly convex, bounded domain \(\Omega\) in \(\mathbb R^N\), \(N\geq 2\), with the boundary condition
\[ \lim_{x\to \partial \Omega} u(x)=\infty, \]
where \(f\in C[0,\infty)\cap C^\infty(0,\infty)\) is positive, increasing and such that \(f(0)=0\), and \(b\in C^{\infty}(\bar\Omega)\) is positive in \(\Omega\). Existence of a strictly convex blow-up solution in \(C^\infty(\Omega)\) is guaranteed if there exists a positive and non-decreasing function \(f_0\) on \([0,\infty)\) such that \(f_0(u)\leq f(u)\) for every \(u>0\), \(f_0^{1/N}\) is locally Lipschitz continuous on \([0,\infty)\) and satisfies the Keller-Osserman condition, and, finally, \(f_0^{-1/N}\) is convex in \((0,\infty)\). In particular, this generalizes the result by A. C. Lazer and P. J. McKenna [J. Math. Anal. Appl. 197, No. 2, 341–362 (1996; Zbl 0856.35042)] where existence was proven if \(f(u)=u^q\) with \(q>N\). Moreover the authors provide asymptotic estimates of the behaviour of such solutions near \(\partial \Omega\) and prove uniqueness when the variation of \(f\) at \(\infty\) is regular of index \(q>N\).
Reviewer: Paolo Cascini (Santa Barbara)
MSC:
| 35J60 | Nonlinear elliptic equations |
| 35B40 | Asymptotic behavior of solutions to PDEs |
| 35J67 | Boundary values of solutions to elliptic equations and elliptic systems |
| 35B65 | Smoothness and regularity of solutions to PDEs |
Citations:
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