Landis-type conjecture for the half-Laplacian

In this paper, we consider the following equation with the half Laplacian

(1.1)

$$(-\Delta)^{\frac{1}{2}}u+{\bf b}({\bf x})\cdot\nabla u+q({\bf x})u=0\quad\text{in}\;\;\mathbb{R}^{n},$$

where $n\geq 1$. Our aim is to investigate the minimal decay rate of nontrivial solutions of (1.1). In other words, we consider the unique continuation property from infinity of (1.1). This problem is closely related to the conjecture proposed by Landis in the 60’s [KL88]. Landis conjectured that, if $u$ is a solution to the classical Schrödinger equation

(1.2)

$$-\Delta u+q({\bf x})u=0\quad\text{in}\;\;\mathbb{R}^{n},$$

with a bounded potential $q$, satisfying the decay estimate

$$|u({\bf x})|\leq\exp(-C|{\bf x}|^{1+}),$$

then $u\equiv 0$. Landis’ conjecture was disproved by Meshkov [Mes92], who constructed a complex-valued potential $q\in L^{\infty}(\mathbb{R}^{n})$ and a nontrivial solution $u$ of (1.2) such that

$$|u({\bf x})|\leq\exp(-C|{\bf x}|^{\frac{4}{3}}).$$

In the same work, Meshkov showed that if

$$|u({\bf x})|\leq\exp(-C|{\bf x}|^{\frac{4}{3}+}),$$

then $u\equiv 0$. Based on a suitable Carleman estimate, a quantitative version of Meshkov’s result was established in [BK05], see also [CS99, Dav14, DZ18, DZ19, KL19, LUW11, LW14] for related results. We also refer to [Zhu18, Theorem 2] for some decay estimates at infinity for higher order elliptic equations.

In view of Meshkov’s example, Kenig modified Landis’ original conjecture and asked that whether the Landis’ conjecture holds true for real-valued potentials $q$ in [Ken06]. The real version of Landis’ conjecture in the plane was resolved recently in [LMNN20]. We also refer to [Dav20, DKW17, DKW20, KSW15] for the early development of the real version of Landis’ conjecture.

For the fractional Schrödinger equation, the Landis-type conjecture was studied in [RW19]. The main theme of this paper is to extend the results in [RW19] to the fractional Schrödinger equation with the half Laplacian (1.1). Previously, the authors in [KW19] proved some partial results for the fractional Schrödinger equation

(1.3)

$$((-\Delta)^{s}+b({\bf x}){\bf x}\cdot\nabla+q({\bf x}))u=0\quad\text{in}\;\;\mathbb{R}^{n},$$

where $s\in(0,1)$ and $b$, $q$ are scalar-valued functions. The main tools used in [KW19] are the Caffarelli-Silvestre extension and the Carleman estimate. The particular form of the drift coefficient in (1.3) is due to the applicability of the Carleman estimate. It turns out when $s=\frac{1}{2}$, i.e., the case of half Laplacian, we can treat a general vector-valued drift coefficient ${\bf b}({\bf x})$ in (1.1). The underlying reason is that the Caffarelli-Silvestre extension solution of $(-\Delta)^{\frac{1}{2}}u=0$ in ${\mathbb{R}}^{n}$ is a harmonic function in ${\mathbb{R}}_{+}^{n+1}$. Inspired by this observation, we show that if both ${\bf b}$ and $q$ are differentiable, then any nontrivial solution of (1.1) can not decay exponentially at infinity. The detailed statement is described in the following theorem.

It is interesting to compare Theorem 1.1 with [RW19, Theorem 1]. Assume that $u\in H^{s}(\mathbb{R}^{n})$ is a solution to

(1.5)

$$(-\Delta)^{s}u+q({\bf x})u=0\quad\text{in}\,\,\mathbb{R}^{n}$$

such that $|q({\bf x})|\leq 1$ and $|{\bf x}\cdot\nabla q({\bf x})|\leq 1$. If

(1.6)

$$\int_{\mathbb{R}^{n}}e^{|{\bf x}|^{\alpha}}|u|^{2}\,\mathsf{d}{\bf x}<\infty\quad\text{for some}\,\,\alpha>1,$$

then $u\equiv 0$. Therefore, for $s=\frac{1}{2}$, Theorem 1.1 extends their results by slightly relaxing the condition on $q$ and also adding a drift term. Another key improvement is that the exponential decay rate $e^{-\lambda|{\bf x}|}$ is sharper than (1.6).

The proof of Theorem 1.1 consists of two steps. Inspired by [RW19], we first pass the boundary decay (1.4c) to the bulk decay of the Caffarelli-Silvestre extension solution (harmonic function) in the extended space $\mathbb{R}^{n}\times(0,\infty)$. In the second step, we apply the Liouville-type theorem (Theorem 6.1) to the harmonic function. It is noted that we do not use any Carleman estimate here. On the other hand, using the harmonic function in the unit ball ${v}_{0}({\bf z}):=\Re(e^{-1/{\bf z}^{\alpha}})$, ${\bf z}\in{\mathbb{C}}$, $0<\alpha<1$ (see [Jin93]), it is not difficult to construct an example to show the optimality of the Liouville-type theorem. In view of this example, we believe that the decay assumption (1.4c) is optimal.

When ${\bf b}\equiv 0$, the following theorem can be found in [Kow21, Theorem 1.1.9], which was obtained using similar ideas as in the proof of Theorem 1.1.

It is interesting to compare this result with [RW19, Theorem 2]. There, it was proved that if $u\in H^{s}(\mathbb{R}^{n})$ solves (1.5) with $|q(x)|\leq 1$ and

(1.8)

$$\int_{\mathbb{R}^{n}}e^{|{\bf x}|^{\alpha}}|u|^{2}\,\mathsf{d}{\bf x}<\infty\quad\text{for some}\,\,\alpha>\frac{4s}{4s-1},$$

then $u\equiv 0$. When $s=\frac{1}{2}$, (1.8) becomes

$$\int_{\mathbb{R}^{n}}e^{|{\bf x}|^{\alpha}}|u|^{2}\,\mathsf{d}{\bf x}<\infty$$

for $\alpha>2$, which is clearly stronger than (1.7). On the other hand, Theorem 1.3 holds regardless whether $q$ is real-valued or complex-valued.

This paper is organized as follows. In Section 2, we will study the decaying behavior of $\nabla u$. In Section 3, we localize the nonlocal operator $(-\Delta)^{\frac{1}{2}}$ by the Caffarelli-Silvestre extension. In Section 4, we derive some useful estimates about the Caffarelli-Silvestre extension $\tilde{u}$ of the solution $u$, which is harmonic. In Section 5, we obtain the decay rate of $\tilde{u}$ from that of $u$. Finally, we prove Theorem 1.1 in Section 6 by Armitage’s Liouville-type theorem. Furthermore, we provide another proof of this Liouville-type theorem in Appendix A.

Let $1<p<\infty$. For each $u\in L^{p}(\mathbb{R}^{n})$, let $\psi$ satsify $(-\Delta)^{\frac{1}{2}}\psi=u$ and let ${\bf u}:=\nabla\psi$. Using the $L^{p}$-boundedness of the Riesz transform [Ste16] (see also [BG13]), we can show that

(2.1)

$$\|{\bf u}\|_{L^{p}(\mathbb{R}^{n})}\leq C(n,p)\|u\|_{L^{p}(\mathbb{R}^{n})}.$$

We remark that this estimate is also used in the proof of [CCW01, Theorem 2.1]. Note that we can formally write ${\bf u}=\nabla(-\Delta)^{-\frac{1}{2}}u$. Plugging $(-\Delta)^{\frac{1}{2}}\psi=u$ and ${\bf u}=\nabla\psi$ into (2.1) implies

(2.2)

$$\|\nabla\psi\|_{L^{p}(\mathbb{R}^{n})}\leq C(n,p)\|(-\Delta)^{\frac{1}{2}}\psi\|_{L^{p}(\mathbb{R}^{n})}.$$

Thanks to (2.2), we can obtain the following lemma.

In this section, we briefly discuss the Caffarelli-Silvestre extension [CS07]. We also refer to [GFR19, Appendix A] for higher order fractional Laplacian.

Let $\mathbb{R}_{+}^{n+1}:=\mathbb{R}^{n}\times\mathbb{R}_{+}=\begin{Bmatrix}\begin{array}[]{l|l}{\bf x}=({\bf x}^{\prime},x_{n+1})&{\bf x}^{\prime}\in\mathbb{R}^{n},x_{n+1}>0\end{array}\end{Bmatrix}$ and ${\bf x}_{0}=({\bf x}^{\prime},0)\in\mathbb{R}^{n}\times\{0\}$. For $R>0$, we denote

	$\displaystyle B_{R}^{+}({\bf x}_{0})$	$\displaystyle:=\begin{Bmatrix}\begin{array}[]{l|l}{\bf x}\in\mathbb{R}_{+}^{n+1}&|{\bf x}-{\bf x}_{0}|\leq R\end{array}\end{Bmatrix},$
	$\displaystyle B_{R}^{\prime}({\bf x}_{0})$	$\displaystyle:=\begin{Bmatrix}\begin{array}[]{l|l}{\bf x}\in\mathbb{R}^{n}\times\{0\}&|{\bf x}-{\bf x}_{0}|\leq R\end{array}\end{Bmatrix}.$

To simplify the notations, we also denote $B_{R}^{+}:=B_{R}^{+}(0)$ and $B_{R}^{\prime}:=B_{R}^{\prime}(0)$. We define two Sobolev spaces

	$\displaystyle\dot{H}^{1}(\mathbb{R}_{+}^{n+1})$	$\displaystyle:=\begin{Bmatrix}\begin{array}[]{l|l}v:\mathbb{R}_{+}^{n+1}\rightarrow\mathbb{R}&\int_{\mathbb{R}_{+}^{n+1}}|\nabla v|^{2}\,\mathsf{d}{\bf x}<\infty\end{array}\end{Bmatrix},$
	$\displaystyle H_{{\rm loc}}^{1}(\mathbb{R}_{+}^{n+1})$	$\displaystyle:=\begin{Bmatrix}\begin{array}[]{l|l}v\in\dot{H}^{1}(\mathbb{R}_{+}^{n+1})&\int_{\mathbb{R}^{n}\times(0,r)}|v|^{2}\,\mathsf{d}{\bf x}<\infty\text{ for some constant }r>0\end{array}\end{Bmatrix}.$

Given any $\mu\in\mathbb{R}$ and $u\in H^{\mu}(\mathbb{R}^{n})$. Following from [GFR19, Lemma A.1], there exists $\tilde{u}\in\mathcal{C}^{\infty}(\mathbb{R}_{+}^{n+1})$ such that

(3.1)

$$\begin{cases}\Delta\tilde{u}=0\quad\text{in}\;\;\,\mathbb{R}_{+}^{n+1},\\ \displaystyle{\lim_{x_{n+1}\rightarrow 0}}\|\tilde{u}(\cdot,x_{n+1})-u\|_{H^{\mu}(\mathbb{R}^{n})}=0,\end{cases}$$

where $\nabla=(\nabla^{\prime},\partial_{n+1})=(\partial_{1},\cdots,\partial_{n},\partial_{n+1})$, and the half Laplacian is equivalent to the Dirichlet-to-Neumann map of the extension problem (3.1):

(3.2)

$$\lim_{x_{n+1}\rightarrow 0}\|\partial_{n+1}\tilde{u}(\cdot,x_{n+1})+(-\Delta)^{\frac{1}{2}}u\|_{H^{\mu-1}(\mathbb{R}^{n})}=0$$

(see [GFR19, (A.3)]). In particular, when $\mu=\frac{1}{2}$, it follows from [GFR19, Corollary A.2] that

$$\|\tilde{u}\|_{\dot{H}^{1}(\mathbb{R}_{+}^{n+1})}\leq C\|u\|_{H^{\frac{1}{2}}(\mathbb{R}^{n})}\quad\text{for some positive constant }C.$$

In view of this observation, if $u\in H^{1}(\mathbb{R}^{n})$ and both ${\bf b},q$ are bounded, we can reformulate (1.1) as the following local elliptic equation:

(3.3)

$$\begin{cases}\Delta\tilde{u}=0&\text{in}\;\;\mathbb{R}_{+}^{n+1},\\ \tilde{u}({\bf x}^{\prime},0)=u({\bf x}^{\prime})&\text{on}\;\;\mathbb{R}^{n}\quad(\text{in}\;\;H^{1}(\mathbb{R}^{n})\text{-sense}),\\ {\displaystyle\lim_{x_{n+1}\rightarrow 0}}\partial_{n+1}\tilde{u}({\bf x})={\bf b}({\bf x}^{\prime})\cdot\nabla^{\prime}u+q({\bf x}^{\prime})u&\text{on}\;\;\mathbb{R}^{n}\quad(\text{in}\;\;L^{2}(\mathbb{R}^{n})\text{-sense}).\end{cases}$$

Since $u\in H^{1}(\mathbb{R}^{n})\equiv{\rm dom}\,((-\Delta)^{\frac{1}{2}})$, from [Sti10, page 48–49], we have that $\tilde{u}\in H_{{\rm loc}}^{1}(\mathbb{R}_{+}^{n+1})$ and

(3.4)

$$\|\tilde{u}(\bullet,x_{n+1})\|_{L^{2}(\mathbb{R}^{n})}\leq\|u\|_{L^{2}(\mathbb{R}^{n})}.$$

The following lemma is a special case of [RW19, Equation (19)] (see also [KW19, Lemma 3.2]).

By choosing $\sigma=\frac{1}{2}$, $\nu=2$ and $a({\bf x}^{\prime})\equiv 0$ in [JLX14, Proposition 2.6(i)], we obtain the following version of De Giorgi-Nash-Moser type theorem.

Combining (4.1) and (4.2), together with some suitable scaling, we can obtain the following lemma.