On the uniqueness and simplicity of the principal eigenvalue

Lucia, Marcello

Atti della Accademia nazionale dei Lincei. Rendiconti Lincei. Matematica e applicazioni, Série 9, Tome 16 (2005) no. 2, pp. 133-142

Résumé

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.