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 above problem. Our analysis mainly combines the variational method and critical point theory. Such solution is obtained as a minimizer for the energy functional associated with our problem in the setting of the weighted Sobolev spaces.
@article{33978,
title = {On a nonlinear PDE involving weighted $p$-Laplacian},
journal = {Boletim da Sociedade Paranaense de Matem\'atica},
volume = {38},
year = {2019},
doi = {10.5269/bspm.v38i5.33978},
language = {EN},
url = {http://dml.mathdoc.fr/item/33978}
}
El Khalil, A.; Alaoui, M. D. Morchid; Laghzal, Mohamed; Touzani, A. On a nonlinear PDE involving weighted $p$-Laplacian. Boletim da Sociedade Paranaense de Matemática, Tome 38 (2019) . doi : 10.5269/bspm.v38i5.33978. http://gdmltest.u-ga.fr/item/33978/