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

2017/03/16 by Alberto Boscaggin, Francesca Colasuonno, Boscaggin, Alberto +3
Computer Science · Mathematics · #Advanced Mathematical Modeling in Engineering #Analysis of PDEs (math.AP) #FOS: Mathematics #Nonlinear Differential Equations Analysis #Nonlinear Partial Differential Equations

paper · doi:10.48550/arxiv.1703.05727

openalex publication_date 2017/03/16 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

For 10 in Ω, ∂_νu=0 on ∂Ω, where Ω⊂\mathbb 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, T. Weth, \it Ann. Inst. H. Poincaré Anal. Non Lináire vol. 29, pp. 573-588 (2012)], that is to say, if p=2 and f'(1)>λ_k+1^rad, there exists a radial solution of the problem having exactly k intersections with u≡1 for a large class of nonlinearities.

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