On the critical dimension of a fourth order elliptic problem with negative exponent
| dc.creator | Moradifam, Amir | |
| dc.date | 2009-05-12 | |
| dc.date.accessioned | 2026-07-07T13:14:10Z | |
| dc.date.available | 2026-07-07T13:14:10Z | |
| dc.description | We study the regularity of the extremal solution of the semilinear biharmonic equation $βΔ^2 u-τΔu=\fracλ{(1-u)^2}$ on a ball $B \subset \R^N$, under Navier boundary conditions $u=Δu=0$ on $\partial B$, where $λ>0$ is a parameter, while $τ>0$, $β>0$ are fixed constants. It is known that there exists a $λ^{*}$ such that for $λ>λ^{*}$ there is no solution while for $λ<λ^{*}$ there is a branch of minimal solutions. Our main result asserts that the extremal solution $u^{*}$ is regular ($\sup_{B}u^{*}<1$) for $N\leq 8$ and $β, τ>0$ and it is singular ($\sup_{B}u^{*}=1$) for $N\geq 9$, $β>0$, and $τ>0$ with $\fracτβ$ small. Our proof for the singularity of extremal solutions in dimensions $N\geq 9$ is based on certain improved Hardy-Rellich inequalities. | |
| dc.identifier | https://arxiv.org/abs/0905.1940 | |
| dc.identifier | http://arxiv.org/abs/0905.1940 | |
| dc.identifier.uri | http://salesiana.dossiersoluciones.com/handle/123456789/230125 | |
| dc.subject | Analysis of PDEs | |
| dc.title | On the critical dimension of a fourth order elliptic problem with negative exponent | |
| dc.type | text |