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Existence and non-existence for strongly coupled quasi-linear cooperative elliptic systems. (English) Zbl 1197.35099

Summary: We study the prototype model of the boundary value problem \[ \mathrm{div}(|\nabla u|^{m-2}\nabla u) + u^av^b = 0 \text{ in } \Omega, \quad\mathrm{div}(|\nabla v|^{m-2}\nabla v) + u^cv^d = 0 \text{ in }\Omega, \quad u = v = 0\text{ on }\partial\Omega, \] where \(\Omega\in\mathbb R^n\), \(n\geq 2\) is a connected smooth domain, and the exponents \(m>1\) and \(a,b,c,d\geq 0\) are non-negative numbers. Under appropriate conditions on the exponents \(m\), \(a\), \(b\), \(c\) and \(d\), and on the domain \(\Omega\), a variety of results on a priori estimates, existence and non-existence of positive solutions are established.

MSC:

35J57 Boundary value problems for second-order elliptic systems
35J65 Nonlinear boundary value problems for linear elliptic equations
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