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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.

##### MSC:
 35R35 Free boundary problems for PDEs 35J60 Nonlinear elliptic equations
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##### References:
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