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.