Abstract

Let Omega be a bounded domain in R-n with smooth boundary partial derivative Omega. We consider the equation d(2)Delta u - u + u(n-k+2/n-k-2) = 0 in Omega, under zero Neumann boundary conditions, where d is a small positive parameter. We assume that there is a k-dimensional closed, embedded minimal submanifold K of partial derivative Omega which is nondegenerate, and a certain weighted average of sectional curvatures of partial derivative Omega is positive along K. Then we prove the existence of a sequence d = d(j) -> 0 and a positive solution u(d) such that d(2)vertical bar del ud vertical bar(2) -> S delta(K) as d -> 0 in the sense of measures, where SK stands for the Dirac measure supported on K and S is a positive constant.