## Semiclassical symmetric Schrödinger equations: existence of solutions concentrating simultaneously on several spheres.(English)Zbl 1133.35087

Authors’ abstract: We study the radially symmetric Schrödinger equation
$-\varepsilon ^{2} \Delta u + V(|x|)u= W(|x|)u^p ,\quad u > 0,\;u \in H^1 (\mathbb R^N),$ with $$N \geq 1$$, $$\varepsilon > 0$$ and $$p > 1$$. As $$\varepsilon \rightarrow 0$$, 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 $$\Gamma(r)= r^{N-1} [V(r)]^{\frac{p+1}{p-1}- \frac 12} [W(r)]^{-\frac{2}{p-1}}$$.

### MSC:

 35Q55 NLS equations (nonlinear Schrödinger equations) 35J70 Degenerate elliptic equations 35D05 Existence of generalized solutions of PDE (MSC2000)
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