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Existence and uniqueness of multi-bump bound states of nonlinear Schrödinger equations with electromagnetic fields. (English) Zbl 1357.35258

Summary: We are concerned with the existence and uniqueness of multi-bump bound states of the nonlinear Schrödinger equations with electromagnetic potential \[ i\hbar\frac{\partial\psi}{\partial t}=\left (\frac {\hbar}{i}\nabla-A(x)\right )^2\psi +V(x)\psi - |\psi |^{p-2}\psi ,\quad x\in\mathbb R^N, \] for sufficiently small \(\hbar>0\), where \(i\) is the imaginary unit, \(2<p<\frac{2N}{N-2}\) for \(N\geq 3\) and \(2<p<+\infty\) for \(N=1,2\). \(V(x)\) is a bounded real function on \(\mathbb{R}^N\), and \(A(x)=(A_1(x),A_2(x),\dots,A_N(x))\) is such that \(A_j(x)\) is a bounded real function on \(\mathbb{R}^N\) for \(j=1,2,\dots N\). For any finite collection of \(\{a_1,a_2,\dots,a_k\}\) of non-degenerate critical points of \(V(x)\), we show that there exists a solution \(\psi(x,t)=\exp(-iEt/\hbar)u(x)\) which is a small perturbation of a sum of one-bump solutions concentrated at \(a_1,a_2,\dots,a_k\) and \(u(x)\) is unique up to rotations for small \(\hbar>0\).

MSC:

35Q55 NLS equations (nonlinear Schrödinger equations)
81V10 Electromagnetic interaction; quantum electrodynamics
35J60 Nonlinear elliptic equations
35B33 Critical exponents in context of PDEs
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