On Strong Comparison Principles for positive solutions of the Dirichlet p-Laplacian

Mabel Cuesta Université du Littoral-Côte d’Opale Bâtiment Poincaré, 50 rue F. Buisson, BP 699 F-62228 Calais France

              cuesta@lma.univ-littoral.fr

A strong comparison principle for weak solutions u ∈ W_{0}^{1,p}(Ω) of the following degenerate or singular Dirichlet problem is proved,

(P)  − div(|∇u|^{p−2}∇u) = λ|u|^{p−2}u + f(x) in Ω; u = 0 on ∂Ω.

Here, p ∈ (1, ∞) is a given number, Ω ⊂ IR^{N} is a bounded domain with a connected C^{2,α}-boundary, for some α ∈ (0, 1), λ is a constant, and 0 ≤ f ∈ L^{∞}(Ω). It is assumed that 0 ≤ λ < λ_{1}, where λ_{1} denotes the first eigenvalue associated with the corresponding eigenvalue problem. The following result is proved: Let u, v ∈ W_{0}^{1,p}(Ω) be any weak solutions of (P) corresponding to f and g (in place of f), respectively, where 0 ≤ f ≤ g and f ≢ g in Ω. Then 0 ≤ u < v in Ω, and ∂v/∂ν < ∂u/∂ν ≤ 0 on ∂Ω holds for the outer normal derivatives. We also investigate the case λ < 0.

AMS subject classifications: Primary 35B50, 47H07; Secondary 35J70, 47H05

Key words: Strong comparison principle; positive weak solution; p-Laplacian; first eigenvalue; divergence theorem