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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.

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