An existence result for a linear-superlinear elliptic system with Neumann boundary conditions (Q2497409)

The author discusses an elliptic system of two equations in one dimension on the bounded interval \((0,1)\), with Neumann boundary conditions. He assumes that the nonlinearities are asymptotically linear at \(-\infty\) and superlinear at \(+\infty\). He states the existence of a solution for any couple of forcing terms in \(L^2\), under rather general assumptions on the nonlinearities. For the proof, he uses variational methods and the Galerkin procedure: an approximation method which is useful to overcome the difficulties arising from the strong indefiniteness of the associated functional. The author obtains results also for the resonant problem, and the case in which one of the two equations is asymptotically linear both at \(-\infty\) and at \(+\infty\).