Stable and periodic solutions to nonlinear equations with fractional diffusion
The aim of this thesis is to study stable solutions to nonlinear elliptic equations involving the fractional Lapacian. More precisely, we study the extremal solution for the problem $(\Delta )^s u = \lambda f(u)$ in $\Omega$, $u \equiv 0 $ in $\R^n \setminus \Omega$, where $\lambda > 0$ is a para...
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| Tipo de recurso: | tesis de maestría |
| Fecha de publicación: | 2016 |
| País: | España |
| Institución: | Universitat Politècnica de Catalunya (UPC) |
| Repositorio: | UPCommons. Portal del coneixement obert de la UPC |
| Idioma: | inglés |
| OAI Identifier: | oai:upcommons.upc.edu:2117/89944 |
| Acceso en línea: | https://hdl.handle.net/2117/89944 |
| Access Level: | acceso abierto |
| Palabra clave: | Differential equations, Partial Nonlocal equations Fractional Laplacian Stable solutions Extremal solutions Equacions en derivades parcials Classificació AMS::35 Partial differential equations::35B Qualitative properties of solutions Àrees temàtiques de la UPC::Matemàtiques i estadística::Equacions diferencials i integrals::Equacions en derivades parcials |
| Sumario: | The aim of this thesis is to study stable solutions to nonlinear elliptic equations involving the fractional Lapacian. More precisely, we study the extremal solution for the problem $(\Delta )^s u = \lambda f(u)$ in $\Omega$, $u \equiv 0 $ in $\R^n \setminus \Omega$, where $\lambda > 0$ is a parameter and $s \in (0,1)$. The main result of this work, which is new, is the following: we prove that when $s=1/2$ and $\Omega = B_1$, then the extremal solution is bounded whenever $n \leq 8$. |
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