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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
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