Some regularity results for a singular elliptic problem. (English) Zbl 1336.35146
Summary: In the present paper we investigate the following singular elliptic problem with \(p\)-Laplacian operator:
\[
\begin{cases} -\Delta_p u = \frac{K(x)}{u^{\alpha}}\text{ in }\Omega, \\ u = 0\text{ on }\partial\Omega, \,u>0\text{ on }\Omega, \end{cases} \tag{P}
\]
where \(\Omega\) is a regular bounded domain of \(\mathbb R^{N}\), \(\alpha\in\mathbb R\), \(K\in L^\infty_{\mathrm{loc}}(\Omega)\) a non-negative function. We discuss the existence, the regularity and the uniqueness of a weak solution \(u\) to the problem (P).
MSC:
| 35J35 | Variational methods for higher-order elliptic equations |
| 35J50 | Variational methods for elliptic systems |
| 35R05 | PDEs with low regular coefficients and/or low regular data |
| 35D30 | Weak solutions to PDEs |
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