Existence of positive solution for a semi positone radial $p$-Laplacian system

Abstract

In this paper, we prove, for $\lambda $ and $\mu $ large, the existence of a positive solution for the semi-positone elliptic system \[ {\mathrm (P)} \qquad \left \{\begin{aligned} & - \Delta _p u = \lambda \omega (x) f(v) &&\mbox {in } \Omega , \\ &- \Delta _q v = \mu \rho (x) g(u) &&\mbox {in } \Omega , \\ &(u,v) = (0,0) &&\mbox {on } \partial \Omega , \end{aligned} \right . \] where $\Omega = B_1 (0) = \{ x\in \mathbb {R}^N: |x|\leq 1 \} $, and, for $m>1$, $\Delta _m$ denotes the $m$-Laplacian operator $p,q>1$. The weight functions $\omega $, $\rho \colon \overline {\Omega } \rightarrow \mathbb {R}$ are radial, continuous, nonnegative and not identically null, and the non-linearities $f,g\colon [0,\infty ) \rightarrow \mathbb {R}$ are continuous functions such that $f(t)$, $g(t)\geq -\sigma $. The result presented extends, for the radial case, some results in the literature [D. D. Hai and R. Shivaji]. In particular, we do not impose any monotonic condition on $f$ or $g$. The result is obtained as an application of the Schauder fixed point theorem and the maximum principle.