Solutions concentrating on higher dimensional subsets for singularly perturbed elliptic equations I Thomas BartschShuangjie Peng 35J6035J25singularly perturbed elliptic equationvariational methodcritical pointconcentrating solutionslayer solutions 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]. Indiana University Mathematics Journal 2008 text pdf 10.1512/iumj.2008.57.3243 10.1512/iumj.2008.57.3243 en Indiana Univ. Math. J. 57 (2008) 1599 - 1632 state-of-the-art mathematics http://iumj.org/access/