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Multi-bump bound states of Schrödinger equations with a critical frequency. (English) Zbl 1123.35061

Summary: We consider existence and qualitative properties of standing wave solutions \(\Psi(x,t) = e^{-iEt/h}u(x)\) to the nonlinear Schrödinger equation \[ ih\frac{\partial\psi}{\partial t} = -\frac{h^2}{2m}\Delta\psi + W(x)\psi - |\psi|^{p-1}\psi = 0 \] with \(E\) being a critical frequency in the sense that \(\inf_{x\in\mathbb{R}^N} W(x) = E\). We verify that if the zero set of \(W-E\) has several isolated points \(x_{i}\) (\(i=1,\ldots,m\)) near which \(W-E\) is almost exponentially flat with approximately the same behavior, then for \(h>0\) small enough, there exists, for any integer \(k\), \(1\leq k\leq m\), a standing wave solution which concentrates simultaneously on \(\{x_j\mid j=1,\ldots,k\}\), where \(\{x_j\mid j=1,\ldots,k\}\) is any given subset of \(\{x_i\mid i=1,\ldots,m\}\). This generalizes the result of J. Byeon and Z. Wang in [Arch. Ration. Mech. Anal. 165, No. 4, 295–316 (2002; Zbl 1022.35064)].

MSC:

35Q55 NLS equations (nonlinear Schrödinger equations)
35J60 Nonlinear elliptic equations
35B33 Critical exponents in context of PDEs
35A15 Variational methods applied to PDEs
47J30 Variational methods involving nonlinear operators
58E05 Abstract critical point theory (Morse theory, Lyusternik-Shnirel’man theory, etc.) in infinite-dimensional spaces

Citations:

Zbl 1022.35064
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References:

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