A concentration phenomenon for semilinear elliptic equations

For a domain \Omega\subset\dR^N we consider the equation -\Delta u + V(x)u = Q_n(x)\abs{u}^{p-2}u with zero Dirichlet boundary conditions and $p\in(2,2^*)$. Here $V\ge 0$ and $Q_n$ are bounded functions that are positive in a region contained in $\Omega$ and negative outside, and such that the sets $\{Q_n>0\}$ shrink to a point $x_0\in\Omega$ as $n\to\infty$. We show that if $u_n$ is a nontrivial solution corresponding to $Q_n$, then the sequence $(u_n)$ concentrates at $x_0$ with respect to the $H^1$ and certain $L^q$-norms. We also show that if the sets $\{Q_n>0\}$ shrink to two points and $u_n$ are ground state solutions, then they concentrate at one of these points

A re-examination of the life and work of A.F.G. Kerr and of his colleagues and friends

Arthur Francis George Kerr’s life is reviewed and related to a previously published account. Kerr’s collecting activity is analysed using an expanded version of the Thai Biogeography Group’s database of collections. 8,666 of the total 48,970 collections are Kerr’s and 3,178 are those of his colleagues and friends. Therefore, the total number of collections made by Kerr and his acquaintances is likely to be larger and more diverse than previously believed. Mapping of these data using GIS show that Kerr’s collecting activities focussed on particular regions of Thailand at particular times. Also large areas of the country remained unexplored by Kerr and his acquaintances: a pattern that, to some extent, persists to this day. The large, but dispersed, archive of Kerr’s photographs, maps, living collections and correspondence indicate that he was a skilled photographer (taking at least 3,000 images), cartographer (producing many hand-drawn maps) and exceptionally acute, accurate and detailed observer (filling numerous notebooks and leaving other records). It is clear that digitising these collections to form an on-line dedicated website is highly desirable to further progress on the flora of Thailand and surrounding countries and would form an unique record of the social history of early 20thC Thailand