Multipeak solutions for some singularly perturbed nonlinear elliptic problems on Riemannian manifolds

Abstract

Given (M, g) a smooth compact Riemannian N-manifold, we prove that for any fixed positive integer K the problem $$-\varepsilon^2\Delta_g u +u=u^{p-1}\,{{\rm in}\, M}, \quad u > 0\,{{\rm in}\,M}$$ has a K-peaks solution, whose peaks collapse, as e goes to zero, to an isolated local minimum point of the scalar curvature. Here p > 2 if N = 2 and \({2 < p < 2^*={2N \over N-2}\,if\,N\ge3}\).