A quasilinear elliptic equation with absorption term and Hardy potential
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arXiv
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| Format: | Preprint |
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2024
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| author | Chen, Marie-Françoise Bidaut-Véron Huyuan |
| author_facet | Chen, Marie-Françoise Bidaut-Véron Huyuan |
| contents | Here we study the positive solutions of the equation \begin{equation*} -Δ_{p}u+μ\frac{u^{p-1}}{\left\vert x\right\vert ^{p}}+\left\vert x\right\vert ^{θ}u^{q}=0,\qquad x\in \mathbb{R}^{N}\backslash \left\{ 0\right\} \end{equation*}% where $Δ_{p}u={div}(\left\vert \nabla u\right\vert ^{p-2}\nabla u) $ and $1<p<N,q>p-1,μ,θ\in \mathbb{R}.$ We give a complete description of the existence and the asymptotic behaviour of the solutions near the singularity $0,$ or in an exterior domain. We show that the global solutions $\mathbb{R}^{N}\backslash \left\{ 0\right\} $ are radial and give their expression according to the position of the Hardy coefficient $μ$ with respect to the critical exponent $μ_{0}=-(\frac{N-p}{p})^{p}.$ Our method consists into proving that any nonradial solution can be compared to a radial one, then making exhaustive radial study by phase-plane techniques. Our results are optimal, extending the known results when $μ=0$ or $p=2$, with new simpler proofs.They make in evidence interesting phenomena of nonuniqueness when $θ+p=0$, and of existence of locally constant solutions when moreover $p>2$ . |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2410_11659 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | A quasilinear elliptic equation with absorption term and Hardy potential Chen, Marie-Françoise Bidaut-Véron Huyuan Analysis of PDEs 35J92, 35J75 Here we study the positive solutions of the equation \begin{equation*} -Δ_{p}u+μ\frac{u^{p-1}}{\left\vert x\right\vert ^{p}}+\left\vert x\right\vert ^{θ}u^{q}=0,\qquad x\in \mathbb{R}^{N}\backslash \left\{ 0\right\} \end{equation*}% where $Δ_{p}u={div}(\left\vert \nabla u\right\vert ^{p-2}\nabla u) $ and $1<p<N,q>p-1,μ,θ\in \mathbb{R}.$ We give a complete description of the existence and the asymptotic behaviour of the solutions near the singularity $0,$ or in an exterior domain. We show that the global solutions $\mathbb{R}^{N}\backslash \left\{ 0\right\} $ are radial and give their expression according to the position of the Hardy coefficient $μ$ with respect to the critical exponent $μ_{0}=-(\frac{N-p}{p})^{p}.$ Our method consists into proving that any nonradial solution can be compared to a radial one, then making exhaustive radial study by phase-plane techniques. Our results are optimal, extending the known results when $μ=0$ or $p=2$, with new simpler proofs.They make in evidence interesting phenomena of nonuniqueness when $θ+p=0$, and of existence of locally constant solutions when moreover $p>2$ . |
| title | A quasilinear elliptic equation with absorption term and Hardy potential |
| topic | Analysis of PDEs 35J92, 35J75 |
| url | https://arxiv.org/abs/2410.11659 |