Ja n 20 01 A note on the moving hyperplane method

Let us consider the problem:    −∆ p u = f (u) in Ω, u = 0 on ∂Ω, u ∈ C 1 (Ω), u > 0 in Ω (1) where 1 < p ≤ 2, Ω ⊂ R N is a bounded convex domain, ∆ p is the p-Laplacian operator defined by ∆ p u = div(|∇u| p−2 ∇u) and f : R → [0, +∞) is continuous on R, locally Lipschitz continuous on (0, +∞) and satisfies ∃C 0 , C 1 > 0 such that C 0 |u| q ≤ f (u) ≤ C 1 |u| q ∀u ∈ R + where q > p − 1. In [1], Ph. Clément and the first author proved the existence of a nontrivial positive solution to (1) by using continuation methods and establishing a priori estimates for the solutions of some nonlinear eigenvalue problem associated with (1). The desired a priori estimates use a blow up argument as well as some monotonicity and symmetry results proved by Damascelli and Pacella in [3] and generalizing to the p-Laplacian operator with 1 < p < 2 the well known results of Gidas–Ni–Nirenberg from [4] and Berestycki–Nirenberg in [2]. In their proof, Damascelli and Pacella use a new technique consisting in moving hyperplanes orthogonal to directions close to a fixed one. To be efficient, this procedure needs some continuity of some parameters linked with the moving plane method (see the functions λ 1 (ν) and a(ν) defined below). Therefore they assume in their result that ∂Ω is smooth to insure this continuity (and only for that reason). However, such a smoothness hypothesis does not appear in the case p = 2 in the classical moving plane procedure (see [2]). Our purpose here is to give more precision on the regularity of the domain Ω that is needed to have the continuity of the function a(ν) and the lower semicontinuity of λ 1 (ν), and so to have the monotonicity and symmetry results of [3]. This question is also important concerning the existence result from [1]. Specifically, we ask that the domain be of class C 1 , and we also discuss convexity conditions relating to the continuity of λ 1 (ν). In this paper, Ω will denote an open bounded domain in R N with C 1 boundary. We will say that Ω is strictly convex if for all x, …