Francesca Colasuonno

In this presentation, we will analyze a p-Laplacian problem set in a ball of R^N, with homogeneous Neumann boundary conditions. The equation involves a nonlinearity g which is (p-1)-superlinear at infinity, possibly supercritical in the sense of Sobolev embeddings. The nonlinearity allows the problem to have a constant non-zero solution.
In this setting, we prove via shooting method the existence, multiplicity, and oscillatory behavior (around the constant solution) of non-constant, positive, radial solutions. We show that the situation changes drastically depending on p>1. For example, in the prototype case g(s)=s^{q-1}, if p>2, the problem has infinitely many solutions for q>p. While, if p=2, the problem admits at least k non-constant solutions provided that q-2 is bigger than the (k+1)-th radial eigenvalue of the Laplacian with Neumann boundary conditions. Finally, for 1<p<2 a surprising result is found, as non-constant solutions with the same oscillatory behavior appear in couples when the radius of the domain is big enough. We will try to give a unified description and motivation for these three different situations.
This is a joint work with Alberto Boscaggin (Università di Torino) and Benedetta Noris (Universitè de Picardie Jules Verne).
[A. Boscaggin, F. Colasuonno, B. Noris, Multiple positive solutions for a class of p-Laplacian Neumann problems without growth conditions, preprint]
[F. Colasuonno, B. Noris, A p-Laplacian supercritical Neumann problem, Discrete
Contin. Dyn. Syst., Vol. 37 n. 6 (2017) 3025-3057]