Patricio Cerda and Pedro Ubilla

Copyright © 2008 Patricio Cerda and Pedro Ubilla. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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)+c1g1(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, a priori 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 may not necessarily be bounded from below by a positive bound which is independent of u and v.