Existence and semi-classical limit of the least energy solution to a nonlinear Schrödinger equation with electromagnetic fields
The author studies a standing wave \(\varphi(x,t)=e^{-iEth^{-1}}u(x)\) to a time-dependent nonlinear Schrödinger equation with electromagnetic potentials: \[ ih\partial_t \varphi=\left( \frac{h}{i} \nabla -A(x) \right)^2+V(x)\varphi-f(|\varphi|^2)\varphi, \tag{1} \] where \(h\) is a positive constant, \(A(x)\) is a real-valued vector potential, \(V(x)\) is a real-valued electric potential. Here for simplicity, the author treats the case \(n \geq 3\) and a special nonlinearity \(f(t)=t^{(p-1)/2}\) for some \(1<p<(n+2)/(n-2)\). He notes that he can also treat the two-dimensional case with some modifications. Then \(u\) satisfies \[ \left( \frac{h}{i}\nabla -A(x) \right)^2 u+(V(x)-E)u-f(|u|^2)u =0 \quad\text{in } \mathbb{R}^n. \tag{2} \] The purpose of this paper is to show the existence of the least energy solution \(u_h\) to equation (2) and to study a semi-classical limit of \(u_h\) as \(h \rightarrow 0\). Let \(W(x)=V(x)-E\) and assume that \(W\) is strictly positive; \(W(x) \geq \inf W(x) \equiv W^0 >0\), \(x \in \mathbb{R}^n\). Let \(F(t)=\int_0^t f(s) ds\) and the energy \[ J(u)=\frac{1}{2}\int_{\mathbb{R}^n} \{|D^h u|^2+W(x)|u|^2 \} dx -\int_{\mathbb{R}^n}F(|u|^2) dx, \] where \(D^h=(D^h_1 u,\dots,D^h_n u)\) and \(D_j^h=i^{-1} h \partial_j-A(x)\). Define the Hilbert space \(H\) to be the closure of \(C_0^{\infty}(\mathbb{R}^n)\) with respect to the norm \(\|u \|^2_H=\int_{\mathbb{R}^n}\{|D^hu|^2+W(x)|u|^2\} dx.\) The author first proves, under some assumptions on \(W\) and a magnetic field, the existence of a least energy solution of (2) \(u_h \in H\). Then he studies the profile of the solution \(u_h\) in the semi-classical limit of \(h \rightarrow 0\).
nonlinear Schrödinger equation magnetic field
scientific article; zbMATH DE number 1512506