Singularly perturbed nonlinear elliptic problems on manifolds

Abstract

Let M be a connected compact smooth Riemannian manifold of dimension n >= 3 with or without smooth boundary aM. We consider the following singularly perturbed nonlinear elliptic problem on M epsilon(2)Delta(M)u - u + f(u) = 0, u > 0 on M, au/av = 0 on a M where Am is the Laplace-Beltrami operator on M, v is an exterior normal to aM and a nonlinearity f of subcritical growth. For certain f, there exists a mountain pass solution it, of above problem which exhibits a spike layer. We are interested in the asymptotic behaviour of the spike layer. Without any non-degeneracy condition and monotonicity of f (t)/t, we show that if aM = theta(aM not equal theta), the peak point x(epsilon) of the solution it, converges to a maximum point of the scalar curvature S on M(the mean curvature H on aM) as epsilon -> 0, respectively.