Electronic Journal of Differential Equations,
Vol. 1999(1999), No. 40, pp. 1-15.
Title: Existence results for Hamiltonian elliptic systems with nonlinear
boundary conditions
Authors: Julian Fernandez Bonder (Univ. de Buenos Aires, Argentina)
Juan Pablo Pinasco (Univ. de San Andres, Argentina)
Julio D. Rossi (Univ. de Buenos Aires, Argentina)
Abstract:
We prove the existence of nontrivial solutions to the system
$$ \Delta u = u, \quad \Delta v = v, $$
on a bounded set of $R^N$, with nonlinear coupling at the
boundary given by
$$\partial u/\partial\eta = H_v,\quad \partial v/\partial\eta = H_u\,.$$
The proof is done under suitable assumptions on the
Hamiltonian $H$, and based on a
variational argument that is a generalization of the mountain pass theorem.
Under further assumptions on the Hamiltonian, we prove the existence of
positive solutions.
Submitted May 30, 1999. Published October 7, 1999.
Math Subject Classifications: 35J65, 35J20, 35J55
Key Words: elliptic systems; nonlinear boundary conditions; variational problems