MULTIPLE POSITIVE SOLUTIONS FOR A CLASS OF p-LAPLACIAN NEUMANN PROBLEMS WITHOUT GROWTH CONDITIONS

Alberto Boscaggin , Francesca Colasuonno , Benedetta Noris (1)

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

For 1 < p < infinity, we consider the following problem -Delta(p)u =f(u), u > 0 in Omega, partial derivative(nu)u = 0 on partial derivative Omega, where Omega subset of 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 and T. Weth, Ann. Inst. Henri Poincare Anal. Non Lineaire 29 (2012) 573 588], that is to say, if p = 2 and f' (1) > lambda(rad)(k+1) , with lambda(rad)(k+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.

Domaines: Mathématiques [math]

Alberto Boscaggin, Francesca Colasuonno, Benedetta Noris. MULTIPLE POSITIVE SOLUTIONS FOR A CLASS OF p-LAPLACIAN NEUMANN PROBLEMS WITHOUT GROWTH CONDITIONS. ESAIM: Control, Optimisation and Calculus of Variations, 2018, 24 (4), pp.1625-1644. ⟨10.1051/cocv/2017074⟩. ⟨hal-03623064⟩