In this paper we study the existence and regularity of solutions to the following Dirichlet problem

$$
\begin{cases} 
-\operatorname{div}(a(x)|\nabla u|^{p-2} \nabla u) + u|u|^{r-1} = \frac{f(x)}{u^{\theta}} & \text{in }\Omega, \\ 
u > 0 & \text{in }\Omega, \\ 
u = 0 & \text{on }\partial\Omega 
\end{cases}
$$

proving that the lower order term $u|u|^{r-1}$ has some regularizing effects on the solutions.

The impact of a lower order term in a Dirichlet problem with a singular nonlinearity

Lucio Boccardo

Gisella Croce

Abstract

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