Proceedings of the Royal Society of Edinburgh, 107A, 27-42, 1987 A variational approach for a class of singular perturbation problems and applications Nicholas D. Alikakos Mathematics Department, University of Tennessee, Knoxville, TN 37996, U.S.A.; Lefschetz Center for Dynamical Systems, Division of Applied Mathematics, Brown University, Providence, RI 02912, U.S.A. and Henry C. Simpson Mathematics Department, University of Tennessee, Knoxville, TN 37996, U.S.A. (MS received 4 September 1986) Synopsis We study the limit as e —> 0 of global minimisers of functionals of the type U») = W [ Wu\ 2 dx+ \ F(\x\,u)dx, where Q is an annul us or a ball in R". Introduction We begin with some general remarks and later in Section I we introduce more restrictive hypotheses. Consider the family of equations s 2 Au E -f(r, u e ) = 0, r=\x\, (P e ) on Q = {x e W/O^Ri =i |JC| = i? 2 ) with homogeneous Dirichlet or Neumann conditions on 3Q and take / to have three roots, a(r), j3(r), y(r), with two of them strongly stable, fjr, a(r)) i= c 2 > 0, f u (r, j8(r)) ^ c 2 > 0, a(r) < y(r) < j8(r). The problem we investigate concerns the relationship as e —> 0 between solutions to (P E ) and those to the limiting equation -f(r,u) = 0. (P O ) Equation (P o ) can be thought of as an approximation to (P E ) for e small and in a number of physical situations the solutions to (P e ) can be thought of as microscopic versions of (P o ). One well established approach for dealing with this problem is based on the construction of upper and lower solutions to (P E ) with ii e > u E , that converge to the desired limiting state as e ^ O . We refer the reader to [9] where the method was introduced and to [6] for a recent application. Another approach employs unconventional implicit function theorems by means of which one obtains a family of solutions to (P E ) converging to the required limiting state ([10], [11]). In the present paper we adopt a different point of view.