Degenerate (p,r)-Laplacian elliptic equations under weighted boundedness conditions

This paper establishes the existence and uniqueness of weak solutions for a class of double-degenerate singular elliptic equations involving $(p,r)$-Laplacian operator with weight functions $\omega \left(x\right)$ and $\vartheta \left(x\right)$ and a gradient-dependent nonlinearity. We introduce a novel weighted boundedness condition based on $\omega \left(x\right)$ to handle singular coefficients and relax regularity requirements. To the best of our knowledge, such conditions have not been previously addressed in the literature. Working in the weighted Sobolev space $W_0^{1,p}(\omega ,\Omega )$, we prove the associated operator is bounded, coercive, semicontinuous, and strictly monotone. Applying the Minty–Browder theorem, we obtain an explicit parameter range for $\lambda$ ensuring a unique weak solution.