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