Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: On the critical dimension of a fourth order elliptic problem with
  negative exponent   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 $B
\subset \R^N$, 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 $\lambda^{*}$ such that for
$\lambda>\lambda^{*}$ there is no solution while for $\lambda<\lambda^{*}$
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
$\beta, \tau>0$ and it is singular ($\sup_{B}u^{*}=1$) for $N\geq 9$,
$\beta>0$, and $\tau>0$ with $\frac{\tau}{\beta}$ 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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The categories of this paper are: chem-ph
9
The categories of this paper are: nlin
12
The categories of this paper are: nucl-th
14
The categories of this paper are: math-ph
19
The categories of this paper are: eess
18
The categories of this paper are: econ
29
The categories of this paper are: funct-an
31
The categories of this paper are: ao-sci
32
The categories of this paper are: atom-ph
4
The categories of this paper are: cond-mat
1
The categories of this paper are: cs
23
The categories of this paper are: solv-int
3
The categories of this paper are: physics
25
The categories of this paper are: patt-sol
24
The categories of this paper are: dg-ga
21
The categories of this paper are: adap-org