Abstract

We study existence of positive solutions of the nonlinear system - (p1 (t, u, v) u′)′ = h1 (t)f1 (t, u, v) in (0, 1); - (p2 (t, u, v) v′)′ = h2 (t)f2 (t, u, v) in (0, 1); u(0) = u(1) = v(0) = v(1) = 0, where p1 (t, u, v) = 1/(a1 (t) + c1 g1 (u, v)) and p2 (t, u, v) = 1/( a2 (t) + c2g2 (u, v)). Here, it is assumed that g1, g2 are nonnegative continuous functions, a1 (t), a2 (t) are positive continuous functions, c1, c2 ≥ 0, h1, h2 ∈ L1 (0, 1), and that the nonlinearities f1, f2 satisfy superlinear hypotheses at zero and + ∞. The existence of solutions will be obtained using a combination among the method of truncation, apriori bounded and Krasnosel'skii well-known result on fixed point indices in cones. The main contribution here is that we provide a treatment to the above system considering differential operators with nonlinear coefficients. Observe that these coefficients maynot necessarily be bounded from below by a positive bound which is independent of u and v.