BUBBLING ON BOUNDARY SUBMANIFOLDS FOR A SEMILINEAR NEUMANN PROBLEM NEAR HIGH CRITICAL EXPONENTS

In this paper we consider the following problem {-Delta u+u=u(n-k+2/n-k-2 +/-epsilon) in Omega u>0 in Omega partial derivative u/partial derivative v = 0 on partial derivative Omega where Omega is a smooth bounded domain in R-n, n >= 7, k is an integer with k >= 1, and epsilon > 0 is a small parameter. Assume there exists a k-dimensional closed, embedded, non degenerate minimal submanifold K in partial derivative Omega. Under a sign condition on a certain weighted avarage of sectional curvatures of partial derivative Omega along K, we prove the existence of a sequence epsilon = epsilon j -> 0 and of solutions u, to (0.1) such that vertical bar Delta u(epsilon)vertical bar(2) -> S delta(K), as epsilon -> 0 in the sense of measure, where SK denotes a Dirac delta along K and S is a universal positive constant.