Multiple positive solutions for a class of p-Laplacian Neumann problems without growth conditions

¹ Dipartimento di Matematica, Università di Torino, via Carlo Alberto 10, 10123 Torinio, Italia
² Département de Mathématique, Université Libre de Bruxelles, Campus de la Plaine - CP214, boulevard du Triomphe – 1050 Bruxelles, Belgique
³ Laboratoire Amiénois de Mathématique Fondamentale et Appliquée, Université de Picardie Jules Verne, 33 rue Saint- Leu, 80039 Amiens, France

^* Corresponding author: benedetta.noris@u-picardie.fr

Received: 10 April 2017
Accepted: 10 November 2017

Abstract

For 1 < p < ∞, we consider the following problem

     −Δ_pu = f(u),   u > 0 in Ω,   ∂_νu = 0 on ∂Ω,

where Ω ⊂ ℝ^N is either a ball or an annulus. The nonlinearity f is possibly supercritical in the sense of Sobolev embeddings; in particular our assumptions allow to include the prototype nonlinearity f(s) = −s^p−1 + s^q−1 for every q > p. We use the shooting method to get existence and multiplicity of non-constant radial solutions. With the same technique, we also detect the oscillatory behavior of the solutions around the constant solution u ≡ 1. In particular, we prove a conjecture proposed in [D. Bonheure, B. Noris and T. Weth, Ann. Inst. Henri Poincaré Anal. Non Linéaire 29 (2012) 573−588], that is to say, if p = 2 and f′ (1) > λ^rad_{k + 1}, with λ^rad_{k + 1} the (k + 1)-th radial eigenvalue of the Neumann Laplacian, there exists a radial solution of the problem having exactly k intersections with u ≡ 1, for a large class of nonlinearities.