On the uniqueness and simplicity of the principal eigenvalue

Lucia, Marcello

Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni, Tome 16 (2005), p. 133-142 / Harvested from Biblioteca Digitale Italiana di Matematica

Abstract

Given an open set $\Omega$ of $\mathbb{R}^{N}$$(N > 2)$, bounded or unbounded, and a function $w \in L^{\frac{N}{2}} (\Omega)$ with $w^{+}\neq 0$ but allowed to change sign, we give a short proof that the positive principal eigenvalue of the problem $$ - \triangle u = \lambda w (x) u, \qquad u \in \mathcal{D}^{1,2}_{0} (\Omega)$$ is unique and simple. We apply this result to study unbounded Palais-Smale sequences as well as to give a priori estimates on the set of critical points of functionals of the type $$I(u) = \frac{1}{2}\int_{\Omega} |\nabla u|^{2} \, dx - \int_{\Omega} G(x,u) \, dx, \quad u \in \mathcal{D}^{1,2}_{0} (\Omega),$$ when $G$ has a subquadratic growth at infinity.

Résumé

Dato un aperto connesso $\Omega$ di $\mathbb{R}^{N}$$(N > 2)$, limitato o illimitato, e una funzione $w \in L^{\frac{N}{2}} (\Omega)$ con $w^{+}\neq 0$ cui è consentito cambiare segno, si dimostra che l'autovalore principale positivo del problema $$ - \triangle u = \lambda w (x) u, \qquad u \in \mathcal{D}^{1,2}_{0} (\Omega)$$ è unico e semplice. Tale risultato viene applicato allo studio delle successioni di Palais-Smale illimitate ed utilizzato per costruire stime a priori sull'insieme dei punti critici di funzionali del tipo $$I(u) = \frac{1}{2}\int_{\Omega} |\nabla u|^{2} \, dx - \int_{\Omega} G(x,u) \, dx, \quad u \in \mathcal{D}^{1,2}_{0} (\Omega),$$ dove $G$ ha un andamento subquadratico all'infinito.

Détail

Publié le : 2005-06-01

MR 2225507 | Zbl 1225.35159

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