We consider the problem
$$\begin{aligned} \epsilon ^2 \Delta u-V(y)u+u^p\,=\,0,\quad u>0\quad \text{ in }\quad \Omega , \quad \frac{\partial u}{\partial \nu }\,=\,0\quad \text{ on }\quad \partial \Omega , \end{aligned}$$
where
$$\Omega $$
is a bounded domain in
$${\mathbb {R}}^2$$
with smooth boundary, the exponent p is greater than 1,
$$\epsilon >0$$
is a small parameter, V is a uniformly positive, smooth potential on
$$\bar{\Omega }$$
, and
$$\nu $$
deno...
