Existence Of Solution Of Some P,Q-Laplacian System Under Local Superlinear Conditions

We study existence of positive radial solutions for the following class of quasi-linear elliptic systems (Formula presented.) where the nonlinearities (Formula presented.) satisfy some local superlinear property at (Formula presented.). Here B is the unity ball in (Formula presented.) and (Formula presented.). Some difficulties here are that this system, in general, is non-variational and that (Formula presented.) might occur. Thus, our strategy to obtain a positive solution is through a priori bound methods. One of the important results in this article is that we obtain these estimates, assuming superlinearity hypotheses on the variables u and v at infinity just in some subset (Formula presented.) of the ball, allowing us assuming, for example, supercritical growths on (Formula presented.) outside of that subset. We also note that we can consider nonlinearities which are not necessary asymptotic to a power at (Formula presented.) and so it is not possible to use the classic Liouville type nonexistence results.