article

Title: Solutions concentrating on higher dimensional subsets for singularly perturbed elliptic equations I

Authors: Thomas Bartsch and Shuangjie Peng

Issue: Volume 57 (2008), Issue 4, 1599-1632

Abstract:

$We consider the singularly perturbed equation \[ -\epsilon^{2} \Delta u + V(x)u = K(x)u^{p-1} \] on a domain $\Omega \subset \mathbb{R}^{N}$ which may be bounded or unbounded. Under suitable hypotheses on $V$, $K$ we construct layered solutions $u \in H^{1}_{0}(\Omega)$ which concentrate on certain high-dimensional subsets of $\Omega$. This gives a positive answer to a problem proposed by Ambrosetti, Malchiodi and Ni in [A. Ambrosetti, A. Malchiodi, and W.-M. Ni, Singularly perturbed elliptic equations with symmetry: Existence of solutions concentrating on spheres. I, Comm. Math. Phys. 235 (2003), 427-466].$