On a nonlinear PDE involving weighted $p$-Laplacian
On a nonlinear PDE involving weighted $p$-Laplacian
In the present paper, we study the nonlinear partial differential equation with the weighted $p$-Laplacian operator\begin{gather*}- \operatorname{div}(w(x)|\nabla u|^{p-2}\nabla u) = \frac{ f(x)}{(1-u)^{2}},\end{gather*}on a ball ${B}_{r}\subset \mathbb{R}^{N}(N\geq 2)$. Under some appropriate conditionson the functions $f, w$ and the nonlinearity $\frac{1}{(1-u)^{2}}$, we prove the existence and the uniqueness of solutions of the …