A priori estimates and application to the symmetry of solutions for critical $p$-Laplace equations
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arXiv
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| Format: | Preprint |
| Veröffentlicht: |
2014
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| _version_ | 1866913238769205248 |
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| author | Vétois, Jérôme |
| author_facet | Vétois, Jérôme |
| contents | We establish pointwise a priori estimates for solutions in $D^{1,p}(\mathbb{R}^n)$ of equations of type $-Δ_pu=f(x,u)$, where $p\in(1,n)$, $Δ_p:=\mbox{div}\big(\left|\nabla u\right|^{p-2}\nabla u\big)$ is the $p$-Laplace operator, and $f$ is a Caratheodory function with critical Sobolev growth. In the case of positive solutions, our estimates allow us to extend previous radial symmetry results. In particular, by combining our results and a result of Damascelli-Ramaswamy, we are able to extend a recent result of Damascelli-Merchán-Montoro-Sciunzi on the symmetry of positive solutions in $D^{1,p}(\mathbb{R}^n)$ of the equation $-Δ_pu=u^{p^*-1}$, where $p^*:=np/(n-p)$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_1407_6336 |
| institution | arXiv |
| publishDate | 2014 |
| record_format | arxiv |
| spellingShingle | A priori estimates and application to the symmetry of solutions for critical $p$-Laplace equations Vétois, Jérôme Analysis of PDEs We establish pointwise a priori estimates for solutions in $D^{1,p}(\mathbb{R}^n)$ of equations of type $-Δ_pu=f(x,u)$, where $p\in(1,n)$, $Δ_p:=\mbox{div}\big(\left|\nabla u\right|^{p-2}\nabla u\big)$ is the $p$-Laplace operator, and $f$ is a Caratheodory function with critical Sobolev growth. In the case of positive solutions, our estimates allow us to extend previous radial symmetry results. In particular, by combining our results and a result of Damascelli-Ramaswamy, we are able to extend a recent result of Damascelli-Merchán-Montoro-Sciunzi on the symmetry of positive solutions in $D^{1,p}(\mathbb{R}^n)$ of the equation $-Δ_pu=u^{p^*-1}$, where $p^*:=np/(n-p)$. |
| title | A priori estimates and application to the symmetry of solutions for critical $p$-Laplace equations |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/1407.6336 |