A priori bounds and multiplicity of positive solutions for p-Laplacian Neumann problems with sub-critical growth

Alberto Boscaggin , Francesca Colasuonno , Benedetta Noris (1)

1 LAMFA - Laboratoire Amiénois de Mathématique Fondamentale et Appliquée - UMR CNRS 7352

Let 1 < p < +8 and let O. RN be either a ball or an annulus. We continue the analysis started in [Boscaggin, Colasuonno, Noris, ESAIM Control Optim. Calc. Var. (2017)], concerning quasilinear Neumann problems of the type -Delta pu = f(u), u> 0 in O,..u = 0 on. O. We suppose that f(0) = f(1) = 0 and that f is negative between the two zeros and positive after. In case O is a ball, we also require that f grows less than the Sobolev-critical power at infinity. We prove a priori bounds of radial solutions, focussing in particular on solutions which start above 1. As an application, we use the shooting technique to get existence, multiplicity and oscillatory behaviour (around 1) of non-constant radial solutions.

Domaines: Mathématiques [math]

Alberto Boscaggin, Francesca Colasuonno, Benedetta Noris. A priori bounds and multiplicity of positive solutions for p-Laplacian Neumann problems with sub-critical growth. Proceedings of the Royal Society of Edinburgh: Section A, Mathematics, 2020, 150 (1), pp.73-102. ⟨10.1017/prm.2018.143⟩. ⟨hal-03623061⟩