On singular elliptic equations with measure sources

Abstract

We prove existence of solutions for a class of singular elliptic problems with a general measure as source term whose model is $$\{\begin{array}{llcc}-\Delta u=\frac{f(x)}{u^{\gamma }}+\mu & \text{in}\Omega ,u=0& \text{on}\partial \Omega ,u>0& \text{on}\Omega ,\end{array}$$ where $\Omega$ is an open bounded subset of ${\mathbb{R}}^N$. Here $\gamma >0$, $f$ is a nonnegative function on $\Omega$, and $\mu$ is a nonnegative bounded Radon measure on $\Omega$.

References in corpus (4)

• Inverse Problem for a Curved Quantum Guide
• Existence and nonexistence of solutions for singular quadratic quasilinear equations
• Selected problems on elliptic equations involving measures
• Homogenization of a Dirichlet semilinear elliptic problem with a strong singularity at $u=0$ in a domain with many small holes

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