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: email@example.com
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
Current usage metrics show cumulative count of Article Views (full-text article views including HTML views, PDF and ePub downloads, according to the available data) and Abstracts Views on Vision4Press platform.
Data correspond to usage on the plateform after 2015. The current usage metrics is available 48-96 hours after online publication and is updated daily on week days.
Initial download of the metrics may take a while.