This paper concerns a wide class of singular perturbation problems arising from such diverse fields as phase transitions, chemotaxis, pattern formation, population dynamics and chemical reaction theory. The corresponding elliptic equations in a bounded domain without any symmetry assumptions are studied. It is assumed that the mean curvature of the boundary has M isolated, non-degenerate critical points. Then it is shown that for any positive integer M[les ]M there exists a stationary solution with M local peaks which are attained on the boundary and which lie close to these critical points. The method is based on Lyapunov–Schmidt reduction.