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Positive versus free boundary solutions to a singular elliptic equation. (English) Zbl 1173.35743
Summary: The equation \(-\delta u = \chi_{u>o}(-1/u^{\beta } + \lambda f(x, u))\) in \(\Omega \) with Dirichlet boundary condition on \(\partial \Omega \) has a maximal solution \(u_{\lambda} \geq 0\) for every \(\lambda 0\). For \(\lambda \) less than a constant \(\lambda^*\), the solution vanishes inside the domain; and for \(\lambda \lambda *\), the solution is positive. We obtain optimal regularity of \(u_{\lambda}\) even in the presence of the free boundary.

35R35 Free boundary problems for PDEs
35J60 Nonlinear elliptic equations
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