Singularly perturbed nonlinear elliptic problems on manifolds

Abstract

Let \({\cal M}\) be a connected compact smooth Riemannian manifold of dimension \(n \ge 3\) with or without smooth boundary \(\partial {\cal M}.\) We consider the following singularly perturbed nonlinear elliptic problem on \({\cal M}\) $$ \varepsilon^2 \Delta_{{\cal M}} u - u + f(u)=0, \ \ u > 0 \quad {\rm on} \quad {\cal M}, \quad \frac{\partial u}{\partial \nu}=0 {\rm on } \partial {\cal M} $$ where \(\Delta_{{\cal M}}\) is the Laplace-Beltrami operator on \({\cal M} \), \(\nu\) is an exterior normal to \(\partial {\cal M}\) and a nonlinearity \(f\) of subcritical growth. For certain \(f,\) there exists a mountain pass solution \(u_\varepsilon\) 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 \(\partial {\cal M} =\emptyset(\partial {\cal M} \ne \emptyset),\) the peak point \(x_\varepsilon\) of the solution \(u_\varepsilon\) converges to a maximum point of the scalar curvature \(S\) on \({\cal M}\)(the mean curvature \(H\) on \(\partial {\cal M})\) as \(\varepsilon \to 0,\)respectively.