Uniform bounds for higher-order semilinear problems in conformal dimension
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arXiv
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| Hauptverfasser: | , |
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| Format: | Preprint |
| Veröffentlicht: |
2017
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| _version_ | 1866916856457068544 |
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| author | Mancini, Gabriele Romani, Giulio |
| author_facet | Mancini, Gabriele Romani, Giulio |
| contents | We establish uniform a-priori estimates for solutions of the semilinear Dirichlet problem \begin{equation} \begin{cases} (-Δ)^m u=h(x,u)\quad&\mbox{in }Ω,\\ u=\partial_nu=\cdots=\partial_n^{m-1}u=0\quad&\mbox{on }\partialΩ, \end{cases} \end{equation} where $h$ is a positive superlinear and subcritical nonlinearity in the sense of the Trudinger-Moser-Adams inequality, either when $Ω$ is a ball or, provided an energy control on solutions is prescribed, when $Ω$ is a smooth bounded domain. The analogue problem with Navier boundary conditions is also studied. Finally, as a consequence of our results, existence of a positive solution is shown by degree theory. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_1710_05354 |
| institution | arXiv |
| publishDate | 2017 |
| record_format | arxiv |
| spellingShingle | Uniform bounds for higher-order semilinear problems in conformal dimension Mancini, Gabriele Romani, Giulio Analysis of PDEs 35J40, 35B45, 35J91, 35B44 We establish uniform a-priori estimates for solutions of the semilinear Dirichlet problem \begin{equation} \begin{cases} (-Δ)^m u=h(x,u)\quad&\mbox{in }Ω,\\ u=\partial_nu=\cdots=\partial_n^{m-1}u=0\quad&\mbox{on }\partialΩ, \end{cases} \end{equation} where $h$ is a positive superlinear and subcritical nonlinearity in the sense of the Trudinger-Moser-Adams inequality, either when $Ω$ is a ball or, provided an energy control on solutions is prescribed, when $Ω$ is a smooth bounded domain. The analogue problem with Navier boundary conditions is also studied. Finally, as a consequence of our results, existence of a positive solution is shown by degree theory. |
| title | Uniform bounds for higher-order semilinear problems in conformal dimension |
| topic | Analysis of PDEs 35J40, 35B45, 35J91, 35B44 |
| url | https://arxiv.org/abs/1710.05354 |