On singular elliptic equations with measure sources

Oliva, Francescantonio; Petitta, Francesco

ESAIM: Control, Optimisation and Calculus of Variations, Tome 22 (2016) no. 1, pp. 289-308.

We prove existence of solutions for a class of singular elliptic problems with a general measure as source term whose model is

$$\begin{array}{c}\hfill \left\{\begin{array}{cc}-\Delta u=\frac{f\left(x\right)}{u^{\gamma }}+\mu \hfill & \text{in}\Omega ,\hfill \\ u=0\hfill & \text{on}\partial \Omega ,\hfill \\ u>0\hfill & \text{on}\Omega ,\hfill \end{array}\right.\end{array}$$

where $\Omega$ is an open bounded subset of ${ℝ}^N$. Here $\gamma >0$, $f$ is a nonnegative function on $\Omega$, and $\mu$ is a nonnegative bounded Radon measure on $\Omega$.

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