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Boundary blow-up solutions to elliptic systems of competitive type. (English) Zbl 1162.35359

The elliptic system \(\Delta u=u^pv^q\), \(\Delta v=u^rv^s\), where \(p,s>1\), \(q,r>0\) is studied on smooth bounded domain \(\Omega\subset \mathbb R^n\). Dirichlet type boundary conditions: \(u=\lambda, v=\mu\) or \(u=v=+\infty\) or \(u=+\infty, v=\mu\) are considered with \(\lambda,\mu>0\). Under certain hypotheses the existence and nonexistence of positive solutions is proved together with discussion of uniqueness and nonuniqueness. Moreover, the exact asymptotic behaviour of the solutions and their normal derivatives near \(\partial \Omega\) is provided.

MSC:

35J55 Systems of elliptic equations, boundary value problems (MSC2000)
35B45 A priori estimates in context of PDEs
35J60 Nonlinear elliptic equations
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