Increasing radial solutions for Neumann problems without growth restrictions

Bonheure, Denis; Noris, Benedetta; Weth, Tobias

Annales de l'I.H.P. Analyse non linéaire, Tome 29 (2012) no. 4, pp. 573-588.

Nous étudions lʼexistence de solutions radiales positives croissantes de problèmes de Neumann super linéaires dans la boule. Nous nʼimposons aucune restriction de croissance sur la non linéarité à lʼinfini et nos hypothèses permettent également une interaction avec le spectre. Notre approche combinne des arguments topologiques et variationnels. Nous contournons le manque de compacité en travaillant dans le cône des fonctions radiales, positives et croissantes de $H^1\left(B\right)$.

We study the existence of positive increasing radial solutions for superlinear Neumann problems in the ball. We do not impose any growth condition on the nonlinearity at infinity and our assumptions allow for interactions with the spectrum. In our approach we use both topological and variational arguments, and we overcome the lack of compactness by considering the cone of nonnegative, nondecreasing radial functions of $H^1\left(B\right)$.

Mots clés : Supercritical problems, Krasnoselʼskiĭ fixed point, Invariant cone, Gradient flow

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