## Positive solution of Laplacian noncooperative system with potential control.(English)Zbl 1210.35077

Summary: We are concerned with the uniform positivity preserving property on a domain $$D$$ of $$\mathbb R^d$$ ($$d\geq 3$$) for the noncooperative system $\begin{cases} -\Delta u=f(\cdot ,u)-\mu av \quad \text{in } D,\\ -\Delta v=bu \quad \text{in } D,\\ \lim \limits _{x\to \partial _\infty D}u(x)=\lim \limits _{x\to \partial _\infty D}v(x)=0, \end{cases}$ where $$\partial _\infty D=\partial D$$ if $$D$$ is bounded, $$\partial _\infty D=\partial D\cup \{+\infty \}$$ if $$D$$ is unbounded. We give appropriate conditions on $$a$$, $$b$$ and $$f$$ to get existence and positivity of the solutions with potential control.

### MSC:

 35J56 Boundary value problems for first-order elliptic systems 35B05 Oscillation, zeros of solutions, mean value theorems, etc. in context of PDEs

### Keywords:

noncooperative system; uniform positivity