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On the number of sign-changing solutions of a semiclassical nonlinear Schrödinger equation. (English) Zbl 1207.35040

Summary: We study the following nonlinear Schrödinger equation:
\[ \varepsilon^2\Delta u-V(x)u+ |u|^{p-1}u=0 \quad\text{in }\mathbb R^N, \]
where \(\varepsilon>0\), \(u,V:\mathbb R^N\to\mathbb R\), \(p>1\). We prove that, under suitable conditions on the symmetry of \(V\), the set of sign-changing solutions has a rich structure in the semiclassical limit \(\varepsilon\to0\): we construct multipeak solutions with an arbitrarily large number of positive and negative peaks which collapse to either a local minimum or a local maximum of \(V\). The proof relies on a local approach and is based on the finite-dimensional reduction, in the spirit of the arguments employed in [X. Kang and J. Wei, Adv. Differ. Equ. 5, No. 7–9, 899–928 (2000; Zbl 1217.35065)].

MSC:

35B25 Singular perturbations in context of PDEs
35B40 Asymptotic behavior of solutions to PDEs
35J20 Variational methods for second-order elliptic equations
35J60 Nonlinear elliptic equations

Citations:

Zbl 1217.35065
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