Multi-bump bound states of nonlinear Schrödinger equations with electromagnetic fields and critical frequency
The author deals with nonlinear Schrödinger equations with an electromagnetic potential \[ i\frac{\partial\psi}{\partial t}=-(\nabla+i A(x))^2 \psi+\lambda W(x)\psi-f(|\psi|^2)\psi,\;(t,x)\in\mathbb{R} \times\mathbb{R}^N,\tag{1} \] where \(i\) is the imaginary unit, \(\lambda\) is the parameter corresponding to Plank constant, \(A(x)=(A_1(x),\dots,A_N (x))\) is a real vector (magnetic) potential and \(W(x)\) is a scalar electric potential. The author is interested in standing wave solutions, that is \[ \psi(t,x)=\exp(-i\lambda Et)u(t)\tag{2} \] to (1) when \(\lambda\) is sufficiently large, where \(E\) is a real number and \(u(x)\) is a complex-valued function which satisfies \[ -(\nabla+iA(x))^2u(x)+\lambda (W(x)-E)u(x)=f\left(| u(x)|^2\right)u(x),\;x\in\mathbb{R}^N.\tag{3} \] In the present study the author considers the existence of multi-bump solutions to (3).
scientific article; zbMATH DE number 5365149