Nonlinear elliptic equations defined by a class of monotone operators with a singular nonlinearity having variable exponent

Abstract

We prove existence and regularity of solutions to the following problem, defined by a class of monotone operators with singular nonlinearity
\begin{equation*}
\left\{
\begin{array}{c}
-div(a(x,Du))+\nu\left \vert u \right \vert^{p-2}u=\dfrac{f}{u^{\gamma(x)}}%
\text{ \ \ \ \ in \ }\Omega \\

u>0\text{ \ \ \ \ in \ }\Omega \\
u=0\text{ \ \ \ \ on \ }\partial{\Omega}
\end{array}%
\right.
\end{equation*}
where $\Omega$ is a bounded open set in $%
%TCIMACRO{\U{211d} }%
%BeginExpansion
\mathbb{R}
%EndExpansion
^{N},1<p<N$. $\gamma(x)>0$ is a smooth function,
having a convenient behavior near $\partial{\Omega}$ and $\nu>0$ is a real number and $f$ is a non-negative function belonging to some Lebesgue space $L^m(\Omega)$.