39 research outputs found
On the best possible remaining term in the Hardy Inequality
We give a necessary and sufficient condition on a radially symmetric
potential on that makes it an admissible candidate for an improved
Hardy inequality of the following form:
\begin{equation}\label{gen-hardy.0} \hbox{ \quad for all .} \end{equation}Comment: 13 pages. Updated versions --if any-- of this author's papers can be
downloaded at http://pims.math.ca/~nassif
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 on a ball , under Navier boundary conditions on ,
where is a parameter, while , are fixed
constants. It is known that there exists a such that for
there is no solution while for
there is a branch of minimal solutions. Our main result asserts that the
extremal solution is regular () for and
and it is singular () for ,
, and with small. Our proof for the
singularity of extremal solutions in dimensions is based on certain
improved Hardy-Rellich inequalities