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

Abstract

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.