The effect of Q-condition in elliptic equations involving Hardy potential and singular convection term
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866912239080964096 |
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| author | Achhoud, Fessel Bouajaja, Abdelkader Redwane, Hicham |
| author_facet | Achhoud, Fessel Bouajaja, Abdelkader Redwane, Hicham |
| contents | Using an approach by contradiction we prove the existence and uniqueness of a weak solution to a quasi-linear elliptic boundary value problem with singular convection term and Hardy Potential. Whose simplest model is
\begin{equation*} \Scale[0.8]{\ds u \in W_0^{1,2}(\mathcal{O})\cap L^\infty(\mathcal{O}) : -Δu=-\mathcal{A}\text{div}\left(\frac{x}{\vert x\vert^2}u\right)+λ\frac{u}{\vert x\vert^2}+f(x),} \end{equation*} where \(\mathcal{O}\) is a bounded open set in \(\mathbb{R}^N\), $\left(\mathcal{A},λ\right) \in \left(0, \infty\right)^2$ and \(f\in W^{-1,2}(\mathcal{O})\).
Additionally, by taking advantage of the regularizing effect of the interaction between the coefficient of the zero order term and the datum, we establish the existence, uniqueness and regularity of a weak solution to a quasi-linear boundary value problem whose simplest example is \begin{equation*} \Scale[0.8]{\ds u \in W_0^{1,2}(\mathcal{O})\cap L^\infty(\mathcal{O}) : -Δu +a(x)\vert u\vert^{p-2}u=-\mathcal{A}\text{div}\left(\frac{x}{\vert x\vert^2}u\right)+λ\frac{u}{\vert x\vert^2}+f(x),} \end{equation*} under suitable assumptions on $a$ and $f$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2502_14492 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | The effect of Q-condition in elliptic equations involving Hardy potential and singular convection term Achhoud, Fessel Bouajaja, Abdelkader Redwane, Hicham Analysis of PDEs 35J60, 35K05, 35K67, 35R09 Using an approach by contradiction we prove the existence and uniqueness of a weak solution to a quasi-linear elliptic boundary value problem with singular convection term and Hardy Potential. Whose simplest model is \begin{equation*} \Scale[0.8]{\ds u \in W_0^{1,2}(\mathcal{O})\cap L^\infty(\mathcal{O}) : -Δu=-\mathcal{A}\text{div}\left(\frac{x}{\vert x\vert^2}u\right)+λ\frac{u}{\vert x\vert^2}+f(x),} \end{equation*} where \(\mathcal{O}\) is a bounded open set in \(\mathbb{R}^N\), $\left(\mathcal{A},λ\right) \in \left(0, \infty\right)^2$ and \(f\in W^{-1,2}(\mathcal{O})\). Additionally, by taking advantage of the regularizing effect of the interaction between the coefficient of the zero order term and the datum, we establish the existence, uniqueness and regularity of a weak solution to a quasi-linear boundary value problem whose simplest example is \begin{equation*} \Scale[0.8]{\ds u \in W_0^{1,2}(\mathcal{O})\cap L^\infty(\mathcal{O}) : -Δu +a(x)\vert u\vert^{p-2}u=-\mathcal{A}\text{div}\left(\frac{x}{\vert x\vert^2}u\right)+λ\frac{u}{\vert x\vert^2}+f(x),} \end{equation*} under suitable assumptions on $a$ and $f$. |
| title | The effect of Q-condition in elliptic equations involving Hardy potential and singular convection term |
| topic | Analysis of PDEs 35J60, 35K05, 35K67, 35R09 |
| url | https://arxiv.org/abs/2502.14492 |