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

Abstract : 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.
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https://hal-u-picardie.archives-ouvertes.fr/hal-03623064
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Soumis le : mardi 29 mars 2022 - 14:27:24
Dernière modification le : vendredi 16 septembre 2022 - 18:12:28

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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, EDP Sciences, 2018, 24 (4), pp.1625-1644. ⟨10.1051/cocv/2017074⟩. ⟨hal-03623064⟩

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