Multiple radial positive solutions of semilinear elliptic problems with Neumann boundary conditions

Denis Bonheure 1, 2 Christophe Grumiau 3 Christophe Troestler 3
1 MEPHYSTO - Quantitative methods for stochastic models in physics
LPP - Laboratoire Paul Painlevé - UMR 8524, ULB - Université Libre de Bruxelles [Bruxelles], Inria Lille - Nord Europe
Abstract : Assuming B R is a ball in R N , we analyze the positive solutions of the problem −∆u + u = |u| p−2 u, in B R , ∂ ν u = 0, on ∂ B R , that branch out from the constant solution u = 1 as p grows from 2 to +∞. The non-zero constant positive solution is the unique positive solution for p close to 2. We show that there exist arbitrarily many positive solutions as p → ∞ (in particular, for supercritical exponents) or as R → ∞ for any fixed value of p > 2, answering partially a conjecture in [12]. We give the explicit lower bounds for p and R so that a given number of solutions exist. The geometrical properties of those solutions are studied and illustrated numerically. Our simulations motivate additional conjectures. The structure of the least energy solutions (among all or only among radial solutions) and other related problems are also discussed.
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Denis Bonheure, Christophe Grumiau, Christophe Troestler. Multiple radial positive solutions of semilinear elliptic problems with Neumann boundary conditions. Nonlinear Analysis: Theory, Methods and Applications, Elsevier, 2016, 147, pp.236-273. ⟨10.1016/j.na.2016.09.010⟩. ⟨hal-01408548⟩

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