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The power concavity of solutions of some semilinear elliptic boundary- value problems. (English) Zbl 0554.35047
Let $$\Omega$$ be a bounded convex domain in $${\mathbb{R}}^ 2$$ with a smooth boundary. Let $$0<\gamma <1$$. Let $$u\in C^ 2(\Omega)\cap C({\bar \Omega})$$ be a solution, positive in $$\Omega$$, of $$-\Delta u=u^{\gamma}$$ in $$\Omega$$, $$u=0$$ on $$\partial \Omega$$. Then the function $$u^{\alpha}$$ is concave for $$\alpha =(1-\gamma)/2$$.

##### MSC:
 35J65 Nonlinear boundary value problems for linear elliptic equations 35J25 Boundary value problems for second-order elliptic equations 35B05 Oscillation, zeros of solutions, mean value theorems, etc. in context of PDEs 35A30 Geometric theory, characteristics, transformations in context of PDEs
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##### References:
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