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| Study of Certain Navier Problems in Sobolev Space with Weights |
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| KeyWord:Navier problem, degenerate quasilinear elliptic equations, weighted Sobolev spaces, weak solution |
| Author Name | Affiliation | | Y. Fadil | Laboratory LMACS, Faculty of Science and Technics, Sultan Moulay Slimane University, BP 523, 23000, Beni Mellal, Morocco | | M. El Ouaarabi | Fundamental and Applied Mathematics Laboratory, Faculty of Sciences Ain Chock, Hassan II University, BP 5366, 20100. Casablanca, Morocco | | M. Oukessou | Laboratory LMACS, Faculty of Science and Technics, Sultan Moulay Slimane University, BP 523, 23000, Beni Mellal, Morocco |
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| Abstract: |
| In this paper, we study the following Navier problem
\begin{gather*}-{\rm{div}}\Big[ v_{1}\mathcal{K}(z,\nabla w) +v_{2}\mathcal{L}(z,w,\nabla w)\Big]+\Delta\left[\phi_{1}|\Delta w|^{t-2} \Delta w+\phi_{2}|\Delta w|^{q-2} \Delta w\right]
+ \\ v_{3}b(z,w)+ v_{4}\vert w\vert^{p-2}w=h(z),
\end{gather*}
Here, $h \;\in\; L^{p'}(\mathcal{Q},v_{1}^{1-p'})$, $\mathcal{K}$, $\mathcal{L}$ and $b$ are Carath\'{e}odory functions and $\phi_{1}$,$\phi_{2}$,$v_1$, $v_{2}$, $v_{3}$ and $v_{4}$ are $A_p$-weights functions. By using the theory of monotone operators (Browder-Minty Theorem), we demonstrate the existence and uniqueness of weak solution to the above problem. |
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