| Abstract: |
| We study the homogeneous Dirichlet problem for a class of nonlinear elliptic equations of the form:
\begin{equation*}
-\text{div} \big( [a(x) + |u|^\theta] |Du|^{p-2} Du \big) = f \quad \text{in } \Omega,
\end{equation*}
where $\Omega$ is a bounded open set in $\mathbb{R}^N$ ($N > 2$), $1 < p < N$, and $\theta > 0$. The coefficient matrix $a(x)$ is assumed to be elliptic with bounded entries, while the source term $f$ satisfies minimal summability assumptions.
We prove the existence of finite energy solutions, highlighting how the results depend on both the summability of $f$ and the value of the parameter $\theta$. Furthermore, we discuss the existence of $W_0^{1,1}(\Omega)$ solutions for small values of $\theta$ and for $p$ close to $1$. |
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