| Abstract
| - For 1 < p< ∞, we consider the following problem . − Δpu = f( u), u> 0 in Ω, ∂νu = 0 on ∂Ω, whereΩ ⊂ 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) = − 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 Linéaire29 (2012) 573−588], that is to say, if p = 2 and f′ (1) > λ radk + 1 , with λ radk + 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.
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