Esistenza di infinite soluzioni per problemi non lineari in assenza di parametro

Ambrosetti, Antonio

Atti della Accademia nazionale dei Lincei. Rendiconti della Classe di scienze fisiche, matematiche e naturali, Série 8, Tome 52 (1972) no. 5, pp. 660-667

Résumé

Consider the variational, non-linear boundary value problem \begin{equation}\tag{1} \sum_{i,k} \frac{\partial}{\partial x_{i}} \left( a_{i,k}(x) \frac{\partial u}{\partial x_{k}} \right) + \psi(x,u(x)) = 0\qquad u |_{\partial \Omega} = 0\end{equation} in a bounded open set $\Omega \subset \mathbf{R}^{n}$. If $a_{i,k}(x)$ and $\psi(x,t)$ satisfy suitable conditions (see § 1), we prove that (1) has an infinite number of solutions, which are the critical points of a functional on a suitable manifold. This critical points are studied by means of Lusternik-Schnirelman theory.