On a nonlinear PDE involving weighted $p$-Laplacian
2019 ◽
Vol 38
(5)
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pp. 131-145
Keyword(s):
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.
2018 ◽
2016 ◽
Vol 1
(1)
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pp. 229-238
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2010 ◽
Vol 12
(2)
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pp. 125-131
2020 ◽
Vol 5
(1)
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pp. 1-7
2013 ◽
Vol 5
(04)
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pp. 407-422
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2014 ◽
Vol 9
(8)
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pp. 238-248
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