On a new class of elliptic systems with nonlinearities of arbitrary growth

Abstract

We guarantee the existence of infinitely many different pairs of solutions to the system { − Δ u = v p in Ω ; − Δ v = f ( u ) in Ω ; u = v = 0 on ∂ Ω , where 0 < p < 2 N − 2 , Ω is a bounded domain in R N and the continuous nonlinear term f has an unusual oscillatory behavior. The sequence of solutions tends to zero (resp., infinity) with respect to certain norms and the nonlinear term f may enjoy an arbitrary growth at infinity (resp., at zero) whenever f oscillates near zero (resp., at infinity). Our results provide the first applications of Ricceri's variational principle in the theory of coupled elliptic systems.