Volume 24, Number 4, October–December 2018
|Page(s)||1625 - 1644|
|Published online||25 January 2019|
Multiple positive solutions for a class of p-Laplacian Neumann problems without growth conditions
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: firstname.lastname@example.org
Accepted: 10 November 2017
For 1 < p < ∞, we consider the following problem
−Δpu = f(u), u > 0 in Ω, ∂νu = 0 on ∂Ω,
where Ω ⊂ ℝN 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) = −sp−1 + sq−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) > λradk + 1, with λradk + 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.
Mathematics Subject Classification: 35J92 / 35A24 / 35B05 / 35B09
Key words: Quasilinear elliptic equations / Shooting method / Sobolev-supercritical nonlinearities / Neumann boundary / conditions
© EDP Sciences, SMAI 2018
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