Multiple solutions for a class of elliptic equations with jumping nonlinearities

Molle, Riccardo; Passaseo, Donato

Annales de l'I.H.P. Analyse non linéaire, Tome 27 (2010) no. 2, pp. 529-553.

Nous considérons un problème de Dirichlet semi-linéaire avec le terme non linéaire qui interfère avec les valeurs propres de l'opérateur linéaire. Avec des méthodes variationnelles, nous montrons que le nombre de solutions est arbitrairement grand pourvu que le nombre de valeurs propres qui interfèrent avec le terme non linéaire soit suffisamment grand. Pour la démonstration nous prouvons que pour tout $k\in \mathbb{N}$ le problème a une solution qui présente k pics quand un paramètre est suffisamment grand. Nous décrivons aussi le comportement asymptotique et la forme de cette solution quand ce paramètre tend à l'infini.

We consider a semilinear elliptic Dirichlet problem with jumping nonlinearity and, using variational methods, we show that the number of solutions tends to infinity as the number of jumped eigenvalues tends to infinity. In order to prove this fact, for every positive integer k we prove that, when a parameter is large enough, there exists a solution which presents k interior peaks. We also describe the asymptotic behaviour and the profile of this solution as the parameter tends to infinity.

Classification : 35J20, 35J60, 35J65
Mots clés : Jumping nonlinearities, Multiplicity of solutions, Variational methods

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