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arXiv 0905.1940math.AP
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一个负指数四阶椭圆型问题的临界维数

On the critical dimension of a fourth order elliptic problem with negative exponent

Amir Moradifam

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摘要：

研究了球$B \subset \R^N$上半线性双调和方程$βΔ^2 u-τΔu=\fracλ{(1-u)^2}$在Navier边界条件$u=Δu=0$ on $\partial B$下极值解的正则性，其中$λ>0$为参数，$τ>0$，$β>0$为固定常数.已知存在$λ^{*}$，使得对于$λ>λ^{*}$没有解，而对于$λ<λ^{*}$存在极小解的分支。我们的主要结果断言，$u^{*}$是正规的（$\sup_{B}u^{*}<1$）$N\leq 8$和$β, τ>0$和它是奇异的（$\sup_{B}u^{*}=1$）$N\geq 9$，$β>0$和$τ>0$与$\fracτβ$小。利用改进的Hardy-Rellich不等式证明了$N\geq 9$维方程极值解的奇异性.

英文摘要：

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.

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