BOUNDARY AND INTERIOR SPIKE LAYER FORMATION FOR AN ELLIPTIC EQUATION WITH SYMMETRY

Consider the problem del[k(\\x\\)delU(lambda)]+ lambdaf(U(lambda))=0, x is-an-element-of B: unit ball in R(n), delU(lambda).n/partial derivative B = 0. Conditions are given on k and f that guarantee the formation of boundary layer and also of a spike layer around the origin as lambda --> infinity for some particular radially symmetric solutions to the above problem. In the first case, the nodal curve of the solution approaches partial derivative B as lambda --> infinity, while in the second one it shrinks toward the origin as lambda --> infinity. Range location for the value of the solution on partial derivative B, in the first case, and at the origin in the second one, is given for lambda sufficiently large. Only radially symmetric solutions are considered since they are the only ones that can be stable equilibria of the corresponding parabolic equation.