On an elliptic problem with boundary blow-up and a singular weight: the radial case. (English) Zbl 1039.35036

The authors study a semilinear elliptic problem with boundary blow-up of the form \[ \Delta u=a(x)u^m\quad \text{ in}\;\Omega,\quad u=+\infty\quad \text{ on}\;\partial\Omega. \] Assuming that \(a\) is a continuous radial function with \(a(x)\sim C_0\text{ dist\,}(x,\partial B)^{-\gamma}\) as
\(\text{ dist\,}(x,\partial B)\to 0,\) for some \(C_0>0,\) \(\gamma>0,\) the authors determine the issues of existence, multiplicity and behaviour near the boundary for radial positive solutions, in terms of the values of \(m\) and \(\gamma.\) The case \(0<m\leq 1,\) as well as estimates for solutions to the linear problem \(m=1\) are also considered.


35J60 Nonlinear elliptic equations
35B45 A priori estimates in context of PDEs
35J25 Boundary value problems for second-order elliptic equations
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