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

Abstract

We study the regularity of the extremal solution of the semilinear biharmonic equation $\beta {\Delta }^{2}u-\tau \Delta u=\frac{\lambda }{(1-u{)}^2}$ on a ball $B\subset {\mathbb{R}}^N$, under Navier boundary conditions $u=\Delta u=0$ on $\partial B$, where $\lambda >0$ is a parameter, while $\tau >0$, $\beta >0$ are fixed constants. It is known that there exists a ${\lambda }^{\ast }$ such that for $\lambda >{\lambda }^{\ast }$ there is no solution while for $\lambda <{\lambda }^{\ast }$ there is a branch of minimal solutions. Our main result asserts that the extremal solution $u^{\ast }$ is regular (${\sup }_B{u}^{\ast }<1$) for $N\le 8$ and $\beta ,\tau >0$ and it is singular (${\sup }_B{u}^{\ast }=1$) for $N\ge 9$, $\beta >0$, and $\tau >0$ with $\frac{\tau }{\beta }$ small. Our proof for the singularity of extremal solutions in dimensions $N\ge 9$ is based on certain improved Hardy-Rellich inequalities.

References in corpus (1)

• Regularity of the extremal solution in a MEMS model with advection