Existence of solutions to \(n\)-dimensional pendulum-like equations

Amster, Pablo ; De Napoli, Pablo L. ; Mariani, Maria Cristina

Electronic journal of differential equations, Tome 2004 (2004)

Résumé

We study the elliptic boundary-value problem

$\displaylines{ \Delta u + g(x,u) = p(x) \quad \hbox{in } \Omega \cr u\big|_{\partial \Omega} = \hbox{\rm constant}, \quad \int_{\partial\Omega} \frac {\partial u}{\partial \nu} = 0, }$

where $g$ is $T$-periodic in $u$, and $\Omega \subset \mathbb{R}^n$ is a bounded domain. We prove the existence of a solution under a condition on the average of the forcing term $p$. Also, we prove the existence of a compact interval $I_p \subset \mathbb{R}$ such that the problem is solvable for $\tilde p(x) = p(x) + c$ if and only if $c\in I_p$.

Détail

Classification : 35J25, 35J65
Keywords: pendulum-like equations, boundary value problems, topological methods