New linking theorem and elliptic systems with nonlinear boundary condition (Q1863639)

The authors establish a new linking theorem without (PS)-type assumption. The new theorem produces bounded (PS) sequences. The result is applied to the study of the existence of nontrivial (positive) solutions to the elliptic system \[ \Delta u=u,\qquad \delta v=v,\qquad \text{ in} \Omega \] with nonlinear boundary condition \[ \partial u/\partial\eta=G_v,\qquad \partial v/\partial\eta =G_u,\qquad \text{ on} \partial\Omega, \] where \(\Omega\) is a bounded domain of \(\mathbb{R}^N\) with smooth boundary, \(\partial/\partial\eta\) is the outer normal derivative. The ``weak'' superlinear and asymptotically linear cases are considered.