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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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