Lokalisierung der niederenergetischen Lösungen eines singulär gestörten elliptischen Neumann-Problems mittels der Geometrie des Gebietsrandes

Abstract

For a smooth domain $\Omega\subseteq\mathbb{R}^N$, $N \ge 2$, with compact boundary $\partial\Omega$, we consider the singularly perturbed elliptic problem

$-d^2\Delta u + u = f(u)$ in $\Omega$,

with homogeneous Neumann boundary conditions. Here $f$ is taken to be sufficiently smooth, superlinear and to have subcritical growth. The parameter $d$ is small and positive. We are only interested in positive solutions $u$.

There is a formulation as a variational problem with energy functional $J_d$. It is known that low energy solutions have a unique maximum which is achieved on $\partial\Omega$. Our interest lies in the connection between the location of these maxima and the mean curvature $H$ of $\partial\Omega$. We show that low energy solutions concentrate near critical points of $H$, and that nondegenerate critical points of $H$ lead to the existence of low energy solutions concentrated nearby.

The method is as follows: There is a finite dimensional Manifold $Z_d$, diffeomorphic to $\partial\Omega$, such that the critical points of the restriction of $J_d$ to $Z_d$ are exactly the low energy solutions. The construction of $Z_d$ uses the Nehari manifold of $J_d$ and the ground state solution of the equation above on $\mathbb{R}^N$. The theorem follows from asymptotically, in $C^1$, expressing the values of $J_d$ on $Z_d$ in terms of $H$, as $d$ tends to 0.