On singular elliptic equations with measure sources

Oliva, Francescantonio¹ ; Petitta, Francesco¹

¹ Dipartimento di Scienze di Base e Applicate per l’ Ingegneria, “Sapienza”, Università di Roma, Via Scarpa 16, 00161 Roma, Italy

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

Résumé

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 $\Omega$.

Détail

Reçu le : 2014-04-15
DOI : 10.1051/cocv/2015004

Classification : 35J60, 35J61, 35J75, 35R06
Keywords: Nonlinear elliptic equations, singular elliptic equations, measure data