Ann. Henri Poincar´ e Online First c  2018 Springer Nature Switzerland AG https://doi.org/10.1007/s00023-018-0714-2 Annales Henri Poincar´ e Elliptic Systems with Some Superlinear Assumption Only Around the Origin Patricio Cerda, Marco Aurelio Souto and Pedro Ubilla Abstract. In this paper, using a priori bound techniques we study exis- tence of positive solutions of the elliptic system: ⎧ ⎨ ⎩ −div(|x| α 1 ∇u)= |x| β 1 f (|x|, u, v) x ∈ B, −div(|x| α 2 ∇v)= |x| β 2 g(|x|, u, v) x ∈ B, u(x)=0= v(x), x ∈ ∂B. where B is the unitary ball centered at the origin. Assuming that f,g are nonnegative nonlinearities and that f (|x|, u, v)+ g(|x|, u, v) is superlinear at 0 and at ∞, we establish some results of existence of one positive solution. As an application, we establish two positive solutions for some non-homogeneous elliptic system. The main novelties here are that the nonlinearities could have growth above the critical hyperbola on some part of the domain as well as only local superlinear hypotheses at ∞ 1. Introduction The study of existence of solutions of the nonlinear elliptic systems has been of great interest in recent years. For this type of result, see, among others, [1– 8] and the survey papers [9, 10]. There are two major classes of systems that can be treated variationally: Hamiltonian and gradient systems. In contrast to gradient-type systems, which resemble in many aspects to elliptic equations, Hamiltonian systems exhibit striking and challenging differences with these ones. A particular feature of these systems is the fact that they may be treated within several variational settings: fractional Sobolev spaces [11, 12], reduction by inversion [13], dual method [14], Lyapunov–Schmidt-type reductions [15], etc. The first author was supported by PAI-CONICYT Grant 79140015. The second author was partially supported by CNPq—Proc. 306.082/2017-9. The third author was partially supported by FONDECYT Grants 1120524 and 1161635.