Electron. J. Diff. Eqns., Vol. 2004(2004), No. 125, pp. 1-8.

Existence of solutions to n-dimensional pendulum-like equations

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

Abstract:
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 $c\in I_p$.

Submitted June 3, 2004. Published October 20, 2004.
Math Subject Classifications: 35J25, 35J65.
Key Words: Pendulum-like equations; boundary value problems; topological methods.

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Pablo Amster
Universidad de Buenos Aires
FCEyN - Departamento de Matemática
Ciudad Universitaria, Pabellóon I
(1428) Buenos Aires, Argentina
and Consejo Nacional de Investigaciones Científicas y Técnicas (CONICET)
email: pamster@dm.uba.ar
Pablo L. De Nápoli
Universidad de Buenos Aires
FCEyN - Departamento de Matemáatica
Ciudad Universitaria, Pabellón I
(1428) Buenos Aires, Argentina
email: pdenapo@dm.uba.ar
María Cristina Mariani
Department of Mathematical Sciences
New Mexico State University
Las Cruces, NM 88003-0001, USA
email: mmariani@nmsu.edu

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