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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.

##### MSC:
 35G30 Boundary value problems for nonlinear higher-order PDEs
##### Keywords:
infinite boundary data; quasilinear equations
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