On Ambrosetti-Malchiodi-Ni conjecture on two-dimensional smooth bounded domains

Suting Wei, Bin Xu, Jun Yang

We consider the problem $$ ε^2 Δu-V(y)u+u^p\,=\,0,~~u>0~~\quad\mbox{in}\quadΩ,~~\quad\frac {\partial u}{\partial ν}\,=\,0\quad\mbox{on}~~~\partial Ω, $$ where $\Omega$ is a bounded doma…