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A note on the moving hyperplane method. (English) Zbl 1109.35332
Summary: In [J. Differ. Equ. 179, No. 1, 213–245 (2002; Zbl 1109.35350)], Ph. Clément and the first author used continuation methods to establish existence results for problems of the form \(-\Delta_pu=f(u)\) in \(\Omega\), \(u=0\) on \(\partial\Omega\), \(u>0\) on \(\Omega\), where \(1<p\leq 2\), \(\Omega\subset\mathbb R^N\) is a bounded convex domain and \(f\colon \mathbb R\to[0,+\infty)\) is continuous. We make precise the domain regularity needed for having the monotonicity and symmetry results recently proved by L. Damascelli and F. Pacella [Ann. Sc. Norm. Super. Pisa, Cl. Sci. (4) 26, No. 4, 689–707 (1998; Zbl 0930.35070)] on \(p\)-Laplace equations. For this purpose, we study the continuity and semicontinuity of some parameters linked with the moving hyperplane method.

MSC:
35J60 Nonlinear elliptic equations
35B99 Qualitative properties of solutions to partial differential equations
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