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Power concavity and boundary value problems. (English) Zbl 0549.35025
Let \(n\geq 2\), and \(\Omega\) be a bounded convex domain in \({\mathbb{R}}^ n\). Suppose u satisfies \(\Delta u+f(x,u)=0\) in \(\Omega\) and \(u=0\) on \(\partial\Omega \). Suppose also that \(f\geq 0\), \(u^{\alpha -1}f(x,u)\) is decreasing in u, and \(u^{(3\alpha -1)/\alpha}f(x,u^{1/\alpha})\) is jointly concave in (x,u) for some \(0<\alpha\leq 1\). Then it is shown that \(u^{\alpha}\) is concave in \(\Omega\). In particular, if f is independent of u, and \(f^{\beta}\) is concave for some \(\beta\leq 1\), then \(u^{\alpha}\) is concave for \(0<\alpha\leq \beta /(2\beta +1)\). This is shown to be sharp. Similarly the solution to \(\Delta u=e^ u\) for \(n=2\) with infinite boundary data is shown to be convex. The result extends to suitable quasilinear equations.

35G30 Boundary value problems for nonlinear higher-order PDEs
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