We investigate the existence of weak solutions for a class of nonlinear elliptic inclusion problems that involve singular terms, degenerate coercivity, and explicit dependence of the principal part on the unknown function. The problem under study is \( {\left\{ \begin{array}{ll} \beta (u) - \operatorname {div}\left( \dfrac{a(x, u, \nabla u)}{(1 + u)^{\theta (p-1)}} \right) \ni \dfrac{f}{u^{\gamma }} & \text {in } \Omega , \\ u \ge 0 & \text {in } \Omega , \\ u = 0 & \text {on } \partial \Omega , \end{array}\right. } \) where \(\Omega \subset \mathbb {R}^N\) ( \(N \ge 2\) ) is a bounded domain, \(\beta \) is a maximal monotone graph with \(0 \in \beta (0)\) , and \(a(x,s,\xi )\) is a Carathéodory function. The forcing term \(f\) is strictly positive and belongs to \(L^\infty (\Omega )\) , while the parameters satisfy \(0 \le \gamma \le 1\) and \(0 \le \theta (p-1) < \frac{N(p-1)}{N-1} - 1\) . The factor \((1+u)^{-\theta (p-1)}\) induces degenerate coercivity, and the term \(u^{-\gamma }\) introduces a strong singularity. Using an approximation scheme, uniform a priori estimates, and techniques from the theory of pseudomonotone operators, we establish the existence of a nonnegative weak solution.
