Multi-bump standing waves for nonlinear Schrödinger equations with a general nonlinearity: the topological effect of potential wells. (English) Zbl 1472.35110
Summary: In this article, we are interested in multi-bump solutions of the singularly perturbed problem
\[-\varepsilon^2\Delta v+V(x)v=f(v)\quad\text{in }\mathbb{R}^N.\]
Extending previous results, we prove the existence of multi-bump solutions for an optimal class of nonlinearities \(f\) satisfying the Berestycki-Lions conditions and, notably, also for more general classes of potential wells than those previously studied. We devise two novel topological arguments to deal with general classes of potential wells. Our results prove the existence of multi-bump solutions in which the centers of bumps converge toward potential wells as \(\varepsilon\rightarrow 0\). Examples of potential wells include the following: the union of two compact smooth submanifolds of \(\mathbb{R}^N\) where these two submanifolds meet at the origin and an embedded topological submanifold of \(\mathbb{R}^N \).
MSC:
| 35J10 | Schrödinger operator, Schrödinger equation |
| 35Q55 | NLS equations (nonlinear Schrödinger equations) |
| 35B25 | Singular perturbations in context of PDEs |
| 58E05 | Abstract critical point theory (Morse theory, Lyusternik-Shnirel’man theory, etc.) in infinite-dimensional spaces |
| 35J20 | Variational methods for second-order elliptic equations |
Keywords:
nonlinear Schrödinger equations; singular perturbation; semi-classical standing waves; variational methodsReferences:
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