Electron. J. Diff. Eqns., Vol. 1999(1999), No. 40, pp. 1-15.

Existence results for Hamiltonian elliptic systems with nonlinear boundary conditions

Julian Fernandez Bonder, Juan Pablo Pinasco, & Julio D. Rossi

Abstract:
We prove the existence of nontrivial solutions to the system
$$ \Delta u = u, \quad \Delta v = v, $$
on a bounded set of RN, 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

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Julian Fernandez Bonder
Departamento de Matematica, FCEyN
UBA (1428) Buenos Aires, Argentina.
e-mail: jfbonder@dm.uba.ar

Juan Pablo Pinasco
Universidad de San Andres
Vito Dumas 284 (1684), Prov. Buenos Aires, Argentina.
e-mail: jpinasco@udesa.edu.ar

Julio D. Rossi
Departamento de Matematica, FCEyN
UBA (1428) Buenos Aires, Argentina.
e-mail: jrossi@dm.uba.ar

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