Anisotropic elliptic equations involving unbounded coefficients and singular nonlinearities
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866909934135803904 |
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| author | achhoud, Fessel Khelifi, Hichem |
| author_facet | achhoud, Fessel Khelifi, Hichem |
| contents | In this paper, we study the existence and regularity of solutions for a class of nonlinear singular elliptic equations involving unbounded coefficients and a singular right-hand side. Specifically, we are interested to problem whose simplest model is \begin{equation*} -\sum_{j=1}^N\partial_{j}\left([1+u^{q}]\vert \partial_{j} u \vert^{p_{j}-2} \partial_{j} u\right)= \frac{f}{u^γ}\text{ in $\mathcal{D},$}\quad u>0 \text{ in $\mathcal{D},$} \quad u=0 \hbox{ on}\;\; \partial\mathcal{D}, \end{equation*} where $\mathcal{D}$ is a bounded open subset of $\mathbb{R}^{N}$ with $N>2$, $ γ\geq0$, $q >0 $, $p_{j}>2$ for all $j=1,...,N$ and the source term $f$ belongs to $L^1(\mathcal{D})$, with $f \geq 0$ and $f \not\equiv 0$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2512_00485 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Anisotropic elliptic equations involving unbounded coefficients and singular nonlinearities achhoud, Fessel Khelifi, Hichem Analysis of PDEs In this paper, we study the existence and regularity of solutions for a class of nonlinear singular elliptic equations involving unbounded coefficients and a singular right-hand side. Specifically, we are interested to problem whose simplest model is \begin{equation*} -\sum_{j=1}^N\partial_{j}\left([1+u^{q}]\vert \partial_{j} u \vert^{p_{j}-2} \partial_{j} u\right)= \frac{f}{u^γ}\text{ in $\mathcal{D},$}\quad u>0 \text{ in $\mathcal{D},$} \quad u=0 \hbox{ on}\;\; \partial\mathcal{D}, \end{equation*} where $\mathcal{D}$ is a bounded open subset of $\mathbb{R}^{N}$ with $N>2$, $ γ\geq0$, $q >0 $, $p_{j}>2$ for all $j=1,...,N$ and the source term $f$ belongs to $L^1(\mathcal{D})$, with $f \geq 0$ and $f \not\equiv 0$. |
| title | Anisotropic elliptic equations involving unbounded coefficients and singular nonlinearities |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2512.00485 |