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

Marco Ghimenti, Anna Maria Micheletti

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.

Keywords


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

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