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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
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