Existence in the nonlinear Schrödinger equation with bounded magnetic field

Abstract

The paper studies existence of ground states for the nonlinear Schrödinger equation

$$\begin{aligned} -(\nabla + {\mathbf {i}}A(x))^2u+V(x)u=|u|^{p-1}u ,\quad 2<p<2^*, \end{aligned}$$

(0.1)

with a general external magnetic field. In particular, no lattice periodicity or symmetry of the magnetic field, or presence of external electric field is required.

Appendix

In homogeneous Sobolev spaces equipped with the group of shifts and dilations, one has the following profile decomposition of Solimini which we quote in a slightly refined version of [20, Theorem 4.6.4].

Theorem 5.2

(Sergio Solimini, [18]) Let $(u_{k})$ be a bounded sequence in ${\dot{H}}^{m,p}(\mathbb {R}^{N})$, $m\in \mathbb {N}$, $1<p<N/m$. Then it has a renamed subsequence and there exist sequences of isometries on ${\dot{H}}^{m,p}(\mathbb {R}^{N})$, $(g_{k}^{(n)})_{k\in \mathbb {N}})_{n\in \mathbb {N}}$, and functions $w^{(n)}\in {\dot{H}}^{m,p}(\mathbb {R}^{N})$, such that $g_{k}^{(n)}u{\mathop {=}\limits ^{\mathrm {def}}}2^{ \frac{N-mp}{p}j_{k}^{(n)}}u(2^{j_{k}^{(n)}}(\cdot -y_{k}^{(n)}))$, $j_{k}^{(n)}\in \mathbb {Z}$, $y_{k}^{(n)}\in \mathbb {R}^{N}$, $n\in \mathbb {N}$, with

$$\begin{aligned} j_{k}^{(0)}=0, y_{k}^{(0)}=0, \bigl | j_{k}^{(n)}-j_{k}^{(m)} \bigr | +\bigl (2^{j_{k}^{(n)}}+2^{j _{k}^{(m)}}\bigr ) \bigl | y_{k}^{(n)}-y_{k}^{(m)} \bigr | \rightarrow \infty ,m\ne n,\nonumber \\ \end{aligned}$$

(5.5)

such that $[g_{k}^{(n)}]^{-1}u_{k}\rightharpoonup w^{(n)}$ in ${\dot{H}}^{m,p}(\mathbb {R}^{N})$,

$$\begin{aligned} u_{k}-\sum _{n\in \mathbb {N}} g_{k}^{(n)}w^{(n)}\rightarrow 0 \text { in } L^{p_{m} ^{*}} \bigl (\mathbb {R}^{N}\bigr ), \end{aligned}$$

(5.6)

the series $\sum _{n\in \mathbb {N}} g_{k}^{(n)}w^{(n)}$ converges in ${\dot{H}}^{m,p}(\mathbb {R}^{N})$ unconditionally with respect to n and uniformly in k, and

$$\begin{aligned} \sum _{n\in \mathbb {N}}\int _{\mathbb {R}^{N}} \bigl | \nabla ^{m}w^{(n)} \bigr | ^{p} \mathrm {d}{x} \le \liminf _{k \rightarrow \infty } \int _{\mathbb {R}^{N}} \bigl | \nabla ^{m}u_{k} \bigr | ^{p}\mathrm {d}{x} . \end{aligned}$$

(5.7)

This theorem cannot be directly applied to sequences bounded in $\dot{H}^{1,2}_A(\mathbb {R}^N)$ with the magnetic potential as in (5.1). However its proof can be trivially modified for this space, given that $\dot{H}^{1,2}_A(\mathbb {R}^N)$ is continuously embedded into $H^{1,2}_\mathrm{loc}(\mathbb {R}^N)$ and into $L^{p^*}(\mathbb {R}^N)$. The modifications are analogous to the argument used in the proof of Theorem 3.2 in the subcritical case and are omitted. We have:

Theorem 5.3

Let $(u_{k})$ be a bounded sequence in $\dot{H}^{1,2}_A(\mathbb {R}^N)$, $N\ge 3$. Then it has a renamed subsequence and there exist functions $w^{(n)}\in {H}_\mathrm{loc}^{1,2}(\mathbb {R}^{N})$ and sequences of bounded operators in ${H}_\mathrm{loc}^{1,2}(\mathbb {R}^{N})$, $(g_{k}^{(n)})_{k\in \mathbb {N}}$, such that $g_{k}^{(n)}u{\mathop {=}\limits ^{\mathrm {def}}}2^{ \frac{N-2}{2}j_{k}^{(n)}}u(2^{j_{k}^{(n)}}(\cdot -y_{k}^{(n)}))$, $j_{k}^{(n)}\in \mathbb {Z}$, $y_{k}^{(n)}\in \mathbb {R}^{N}$, $n\in \mathbb {N}$, with

$$\begin{aligned} j_{k}^{(0)}=0, y_{k}^{(0)}=0, \bigl | j_{k}^{(n)}-j_{k}^{(m)} \bigr | +\bigl (2^{j_{k}^{(n)}}+2^{j _{k}^{(m)}}\bigr ) \bigl | y_{k}^{(n)}-y_{k}^{(m)} \bigr | \rightarrow \infty ,m\ne n,\nonumber \\ \end{aligned}$$

(5.8)

such that $[g_{k}^{(n)}]^{-1}u_{k}\rightharpoonup w^{(n)}$ in ${H}^{1,2}_\mathrm{loc}(\mathbb {R}^{N})$,

$$\begin{aligned} u_{k}-\sum _{n\in \mathbb {N}} g_{k}^{(n)}w^{(n)}\rightarrow 0 \text { in } L^{p ^{*}} \bigl (\mathbb {R}^{N}\bigr ), \end{aligned}$$

(5.9)

the series $\sum _{n\in \mathbb {N}} g_{k}^{(n)}w^{(n)}$ converges in ${H}^{1,2}_\mathrm{loc}(\mathbb {R}^{N})$ unconditionally and uniformly in k, and

$$\begin{aligned} \sum _{n\in \mathbb {N}}\int _{\mathbb {R}^{N}} \bigl | \nabla _{A^{(n)}}w^{(n)} \bigr | ^{2} \mathrm {d}{x} \le \liminf _{k \rightarrow \infty } \int _{\mathbb {R}^{N}} \bigl | \nabla _A u_{k} \bigr | ^{2}\mathrm {d}{x}, \end{aligned}$$

(5.10)

where

$$\begin{aligned} A^{(n)}=\lim _{k\rightarrow \infty } 2^{-j_{k}^{(n)}}A(2^{j_{k}^{(n)}}\cdot +y_{k}^{(n)}). \end{aligned}$$

(5.11)

Moreover under conditions of Theorem 5.2 one has the following “iterated Brezis-Lieb lemma” [20, Theorem 4.7.1]:

$$\begin{aligned} \int _{\mathbb {R}^{N}} \bigl |u_{k} \bigr | ^{2^*}\mathrm {d}{x} \longrightarrow \sum _{n\in \mathbb {N}}\int _{\mathbb {R}^{N}} \bigl |w^{(n)} \bigr | ^{2^*} \mathrm {d}{x} . \end{aligned}$$

(5.12)

An elementary modification of the argument from [20, Theorem 4.7.1] also gives

$$\begin{aligned} \int _{\mathbb {R}^{N}} V(x)\bigl |u_{k} \bigr | ^{2}\mathrm {d}{x} \longrightarrow \sum _{n\in \mathbb {N}}\int _{\mathbb {R}^{N}}V^{(n)}(x) \bigl |w^{(n)} \bigr | ^{2} \mathrm {d}{x}, \end{aligned}$$

(5.13)

where

$$\begin{aligned} V^{(n)}=\lim _{k\rightarrow \infty }2^{-2j_{k}^{(n)}}V(2^{j_{k}^{(n)}}\cdot +y_{k}^{(n)}). \end{aligned}$$

(5.14)