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Radial solutions of an elliptic equation with singular nonlinearity. (English) Zbl 1163.35016
Summary: For the equation
$-\Delta u+u^{-\beta}= u^p, \qquad u>0\quad\text{in }B_R, \qquad u=0\quad\text{on }\partial B_R,$
where $$B_R\subseteq\mathbb R^N$$, $$0<\beta<1$$ and $$1<p< \frac{N+2}{N-2}$$ if $$N\geq 3$$, $$1<p<+\infty$$ if $$N=2$$, we show that there is $$\overline{R}>0$$ such that a radial solution $$u_R$$ exists if and only if $$0<R\leq\overline{R}$$. It is unique in the class of radial solutions and $$u_R'(R)<0$$ if $$R<\overline{R}$$, while $$u_{\overline{R}}'(\overline{R})=0$$. We also give a variational characterization of $$u_{\overline{R}}$$.

##### MSC:
 35J60 Nonlinear elliptic equations 35J20 Variational methods for second-order elliptic equations 35B05 Oscillation, zeros of solutions, mean value theorems, etc. in context of PDEs
##### Keywords:
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