Title :

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

Author(s) :

Bonheure, Denis [Auteur]
Département de mathématiques Université Libre de Bruxelles
Quantitative methods for stochastic models in physics [MEPHYSTO]
Grumiau, Christophe [Auteur]
Institut de Mathématiques [Mons]
Troestler, Christophe [Auteur]
Institut de Mathématiques [Mons]

Journal title :

Nonlinear Analysis: Theory, Methods and Applications

Pages :

236-273

Publisher :

Elsevier

Publication date :

2016

ISSN :

0362-546X

HAL domain(s) :

Mathématiques [math]/Equations aux dérivées partielles [math.AP]

English abstract : [en]

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 ...
Show more >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.Show less >

Language :

Anglais

Popular science :

Non

Collections :

• Laboratoire Paul Painlevé - UMR 8524

Source :

Harvested from HAL