BOUNDARY CONCENTRATIONS ON SEGMENTS FOR A NEUMANN AMBROSETTI-PRODI PROBLEM

Given a smooth bounded domain Omega subset of R-2, we consider the following Ambrosetti-Prodi problem with Neumann boundary: {-triangle u = |u|(p) - sigma in Omega, partial derivative u/partial derivative v = 0 on partial derivative Omega. where p > 2, sigma > 0 is a large parameter and nu denotes the outward normal of partial derivative Omega. We constructed a new class of solutions comprised of a large number of spikes concentrated on a segment of the boundary containing a local minimum point of the mean curvature function and having the same mean curvature at the endpoints. A similar boundary-concentrating phenomenon was obtained for the Lin-Ni-Takagi problem by Ao et al. [3].