We study the radially symmetric Schrödinger equation
$$ - \varepsilon ^{2} \Delta u + V{\left( {|x|} \right)}u = W{\left( {|x|} \right)}u^{p} ,\quad u > 0,\;\;u \in H^{1} ({\mathbb{R}}^{N} ), $$
with N ≥ 1, ɛ > 0 and p > 1. As ɛ→ 0, we prove the existence of positive radially symmetric solutions concentrating simultaneously on k spheres. The radii are localized near non-degenerate critical points of the function
$$\Gamma (r) = r^{{N - 1}} {\left[ {V(r)} \right]}^{{\frac{{p + 1}}{{p - 1}} - \frac{1}{2}}} {\left[ {W(r)} \...
