## 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

### Keywords:

$$p$$-Laplacian radial systems; semipositone problems

### Citations:

Zbl 1067.35026; Zbl 1330.35132
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