Some singularly perturbed problems on annuli and a counterexample to a problem of Gidas, Ni and Nirenberg

In this note we discuss the radical positive solutions of -epsilon(2) Delta U = g(U) in D, U = 0 on partial derivative D, where D is the annulus {s is an element of R-n:b < parallel to s parallel to < 1}. Here 0 < b < 1, n greater than or equal to 2, and g:R --> R is a suitable C-1 function with g(0) greater than or equal to 0 but with g changing sign. We answer a question on page 223 of the original Gidas-Ni-Nirenberg paper [8], by showing that the maximum of these solutions occurs at a point s(epsilon), where parallel to S(epsilon)parallel to --> b as epsilon --> 0. This also shows that the non-negativity condition in the result on page 223 of [8] is necessary.