On the eigenvalues of an elliptic operator \( a(x,H(u)) \)

Campanato, Sergio

Atti della Accademia nazionale dei Lincei. Rendiconti Lincei. Matematica e applicazioni, Série 9, Tome 3 (1992) no. 2, pp. 107-110

Résumé

Let \( \Omega \) be a bounded open convex set of class \( C^{2} \). Let \( a(x,H(u)) \) be a non linear operator satisfying the condition (A) (elliptic) with constants \( \alpha \), \( \gamma \), \( \delta \). We prove that a number \( \lambda \ge 0 \) is an eigenvalue for the operator \( a(x,H(u)) \) if and only if the number \( \alpha \lambda \) is an eigen-value for the operator \( \Delta u \). If \( \lambda \ge 0 \) , the two systems \( a(x,H(u)) = \lambda u \) and \( \Delta u = \alpha \lambda u \) have the same solutions. In particular, also the eventual eigen-values of the operator \( a(x,H(u)) \) should all be negative. Finally, we obtain a sufficient condition for the existence of solutions \( u \in H^{2} \cap H_{0}^{1} (\Omega) \) of the system \( a(x,H(u)) = b(x,u,Du) \) where \( b(x,u,p) \) is a vector in \( \mathbb{R}^{N} \) with a controlled growth.