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A remark on uniqueness of large solutions for elliptic systems of competitive type. (English) Zbl 1131.35015

Summary: We prove that the semilinear system \(\Delta u=a(x)u^pv^q\), \(\Delta v=b(x)u^rv^s\) in a smooth bounded domain \(\Omega\subset\mathbb R^N\) has a unique positive solution with the boundary condition \(u=v=+\infty\) on \(\partial\Omega\), provided that \(p,s>1\), \(q,r>0\) and \((p-1)(s-1)-qr>0\). The main novelty is imposing a growth on the possibly singular weights \(a(x)\), \(b(x)\) near \(\partial\Omega\), rather than requiring them to have a precise asymptotic behaviour.

MSC:

35J60 Nonlinear elliptic equations
35J65 Nonlinear boundary value problems for linear elliptic equations
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