A Class of Non-linear Anisotropic Elliptic problems with Unbounded Coefficients and Singular Quadratic Lower Order Terms
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arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| author | Achhoud, Fessel Khelifi, Hichem |
| author_facet | Achhoud, Fessel Khelifi, Hichem |
| contents | In this work, we study the existence and regularity results of anisotropic elliptic equations with a singular lower order term that grows naturally with respect to the gradient and unbounded coefficients. We take up the following model problem \begin{equation*} \left\{\begin{array}{ll}-\displaystyle\sum\limits_{j\in J} D_{j}\left(\left[ 1+ u^{q}\right]\vert D_{j}u\vert^{p_{j}-2} D_{j}u\right)+\sum\limits_{j\in J}\frac{\vert D_{j}u\vert^{p_{j}}}{ u^θ}=f& \hbox{in}\;Ω, \\ u>0& \hbox{in}\;Ω,
u =0 & \hbox{on}\; \partialΩ, \end{array}
\right. \end{equation*} $Ω$ is a bounded domain in $\mathbb{R}^{N}$, $j\in J=\{1,2,\ldots,N\},$ $q>0$, $0< θ<1$, $2\leq p_{1}\leq p_{2}\leq... \leq p_{N}$ and $f\in L^{1}(Ω)$. Our study's conclusions will depend on the values of $q$ and $θ$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2512_08391 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | A Class of Non-linear Anisotropic Elliptic problems with Unbounded Coefficients and Singular Quadratic Lower Order Terms Achhoud, Fessel Khelifi, Hichem Analysis of PDEs In this work, we study the existence and regularity results of anisotropic elliptic equations with a singular lower order term that grows naturally with respect to the gradient and unbounded coefficients. We take up the following model problem \begin{equation*} \left\{\begin{array}{ll}-\displaystyle\sum\limits_{j\in J} D_{j}\left(\left[ 1+ u^{q}\right]\vert D_{j}u\vert^{p_{j}-2} D_{j}u\right)+\sum\limits_{j\in J}\frac{\vert D_{j}u\vert^{p_{j}}}{ u^θ}=f& \hbox{in}\;Ω, \\ u>0& \hbox{in}\;Ω, u =0 & \hbox{on}\; \partialΩ, \end{array} \right. \end{equation*} $Ω$ is a bounded domain in $\mathbb{R}^{N}$, $j\in J=\{1,2,\ldots,N\},$ $q>0$, $0< θ<1$, $2\leq p_{1}\leq p_{2}\leq... \leq p_{N}$ and $f\in L^{1}(Ω)$. Our study's conclusions will depend on the values of $q$ and $θ$. |
| title | A Class of Non-linear Anisotropic Elliptic problems with Unbounded Coefficients and Singular Quadratic Lower Order Terms |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2512.08391 |