JournalsrlmVol. 17 , No. 3DOI 10.4171/rlm/469

Concentration at manifolds of arbitrary dimension for a singularly perturbed Neumann problem

  • Andrea Malchiodi

    Scuola Normale Superiore, Pisa, Italy
  • Fethi Mahmoudi

    Universidad de Chile, Santiago, Chile
Concentration at manifolds of arbitrary dimension for a singularly perturbed Neumann problem cover

Abstract

We consider the equation \e2\Du+u=up- \e^2 \D u + u = u^p in \ORN\O \subseteq \R^N, where \O\O is open, smooth and bounded, and we prove concentration of solutions along kk-dimensional minimal submanifolds of \pa\O\pa \O, for N3N \geq 3 and for k{1,,N2}k \in \{1, \dots, N-2\}. We impose Neumann boundary conditions, assuming 1<p<Nk+2Nk21<p <\frac{N-k+2}{N-k-2} and \e0+\e \to 0^+. This result settles in full generality a phenomenon previously considered only in the particular case N=3N = 3 and k=1k = 1.