### Positive solutions of singularly perturbed nonlinear elliptic problem on Riemannian manifolds with boundary

#### Abstract

Let $(M,g)$ be a smooth connected compact Riemannian manifold of finite

dimension $n\geq 2$\ with a smooth boundary $\partial M$. We consider the

problem

$$

\cases

-\varepsilon ^{2}\Delta _{g}u+u=|u|^{p-2}u,\quad u> 0 &\text{ on }M,\\

\displaystyle

\frac{\partial u}{\partial \nu }=0 & \text{on }\partial M,

\endcases

$$

where $\nu $ is an exterior normal to $\partial M$.

The number of solutions of this problem depends on the topological

properties of the manifold. In particular we consider the Lusternik

Schnirelmann category of the boundary.

dimension $n\geq 2$\ with a smooth boundary $\partial M$. We consider the

problem

$$

\cases

-\varepsilon ^{2}\Delta _{g}u+u=|u|^{p-2}u,\quad u> 0 &\text{ on }M,\\

\displaystyle

\frac{\partial u}{\partial \nu }=0 & \text{on }\partial M,

\endcases

$$

where $\nu $ is an exterior normal to $\partial M$.

The number of solutions of this problem depends on the topological

properties of the manifold. In particular we consider the Lusternik

Schnirelmann category of the boundary.

#### Keywords

Remannian manifold with boundary; semiclassical limit; Lusternik-Schnirelmann category

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