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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:
[1] Sperb, Maximum principles and their applications (1981) · Zbl 0454.35001
[2] Protter, Maximum principles in differential equations (1967) · Zbl 0153.13602
[3] Makar-Limanov, Math. Notes 9 pp 52– (1971) · Zbl 0222.31004
[4] DOI: 10.1007/BF00946979 · Zbl 0508.73051
[5] Keady, The concavity of solutions of -{\(\Delta\)} 30 (1984) · Zbl 0568.35040
[6] DOI: 10.1080/03605308208820254 · Zbl 0508.49013
[7] DOI: 10.1512/iumj.1983.32.32042 · Zbl 0481.35024
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