Elliptic systems involving sublinear and superlinear nonlinearities

Abstract

Combining topological methods and a priori estimates of solutions of some auxiliary problem, we establish the existence and multiplicity of solutions of some class of elliptic systems. We give here some relevant applications to elliptic systems in dimension N≥2, whose are not possible to study from the variational point of view or by using the blow-up technique coupled with Liouville-type results. For instance, we establish new results for some Hamiltonian systems without the Ambrosetti-Rabinowitz condition, that is usually required in this type of problems, even more, the nonlinearities could have arbitrary growth in some parts of the domain. We also give a relevant application for some biharmonic equation with very weak assumptions.