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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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