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Existence of positive solution for a semi positone radial \(p\)-Laplacian system. (English) Zbl 1412.35101

Summary: In this paper, we prove, for \(\lambda\) and \(\mu\) large, the existence of a positive solution for the semi-positone elliptic system \[ \begin{cases} -\Delta_p u = \lambda\omega(x)f(v)\quad&\text{in}\,\,\Omega,\\-\Delta_q v=\mu\rho(x)g(u)\quad &\text{in}\,\,\Omega, \\ (u,v)=(0,0)\quad &\text{on}\,\,\partial \Omega,\end{cases} \] 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, Proc. R. Soc. Edinb., Sect. A, Math. 134, No. 1, 137–141 (2004; Zbl 1067.35026); Nonlinear Anal., Theory Methods Appl., Ser. A, Theory Methods 56, No. 7, 1007–1010 (2004; Zbl 1330.35132)]. 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.

MSC:

35J47 Second-order elliptic systems
35J57 Boundary value problems for second-order elliptic systems
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