A guide to the Choquard equation

Abstract

We survey old and recent results dealing with the existence and properties of solutions to the Choquard type equations

$$\begin{aligned} -\Delta u + V(x)u = \left( |x|^{-(N-\alpha )} *|u |^p\right) |u |^{p - 2} u \quad \text {in} \ \mathbb {R}^N, \end{aligned}$$

and some of its variants and extensions.

Appendix: About the Riesz potentials

The Riesz potential has been introduced by Riesz in the 1930s [173] and was systematically studied in his fundamental paper [174]. An exposition on basic functional–analytic properties of the Riesz potentials could be found in Stein’s monograph [189, §5.1] and also in many places in [129]. A systematic potential theoretic study of the Riesz potentials is presented in the monographs by Landkof [120] (see also du Plessis [77, chapter 3] for a shorter exposition).

1.1 Definition and semigroup property

The Riesz potential $I_\alpha$ of order $\alpha \in (0, N)$ on the Euclidean space $\mathbb {R}^N$ of dimension $N \ge 1$ is defined for each $x \in \mathbb {R}^N {\setminus } \{0\}$ by

$$\begin{aligned} I_\alpha (x) = \frac{A_\alpha }{|x |^{N - \alpha }}, \quad \text {where}\quad A_\alpha = \frac{\Gamma (\tfrac{N-\alpha }{2})}{\Gamma (\tfrac{\alpha }{2})\pi ^{N/2}2^{\alpha }}. \end{aligned}$$

The choice of normalisation constant $A_\alpha$ ensures the semigroup property

$$\begin{aligned} I_\alpha *I_\beta =I_{\alpha +\beta },\quad \forall \alpha ,\beta >0 \quad \text {such that}\quad \alpha +\beta <N, \end{aligned}$$

and, for $N\ge 3$, the property

$$\begin{aligned}-\Delta I_\alpha =I_{\alpha -2},\quad \forall \alpha \in (2,N). \end{aligned}$$

In addition, $-\Delta I_2=\delta$, where $\delta$ is the Dirac delta function, that is $I_2$ is the Green function of the Laplacian $-\Delta$ on $\mathbb {R}^N$. More generally, $I_\alpha$ could be interpreted as the inverse of the fractional Laplacian operator $(-\Delta )^{\alpha /2}$. See, e.g. ([120, §I.1]; [174, 189, §V.1.1]) for the study of these fundamental properties of the Riesz potentials.

When $\alpha \rightarrow 0$, $I_\alpha \rightarrow \delta$, in the vague sense [120, p.46]. When $\alpha \rightarrow N$, $I_\alpha *f\rightarrow A_N\log \big (\frac{1}{|x|}\big )*f$ for every $f\in C^\infty _c(\mathbb {R}^N)$ such that $\int _{\mathbb {R}^N}f=0$, where $A_N=\lim _{\alpha \rightarrow N}(N-\alpha )A_\alpha =1/\big (\Gamma (d/2)\pi ^{d/2}2^{d-1}\big )$ [120, p.50].

The definition of the Riesz potentials as well as the semigroup property could be extended from $\alpha \in (0,N)$ to arbitrary complex $\alpha$ with $\mathrm {Re}(\alpha )>0$ and $\frac{\alpha -N}{2}\not \in \mathbb {N}\cup \{0\}$; however then the convolution $I_\alpha *f$ should be interpreted in the distributional sense; see, for example ([120] or [182, chapter 2]) for a more recent exposition. In this survey we always assume that $\alpha \in (0,N)$ and the convolution $I_\alpha *f$ is understood in the sense of the Lebesgue integral.

1.2 $L^p$–estimates

The Riesz potential of order $\alpha \in (0,N)$ of a function $f\in L^1_{\mathrm {loc}}(\mathbb {R}^N)$ is defined as

$$\begin{aligned} I_\alpha *f(x):=A_\alpha \int _{\mathbb {R}^N}\frac{f(y)}{|x-y |^{N - \alpha }}\mathrm{d}y. \end{aligned}$$

The latter integral converges in the classical Lebesgue sense for a.e. $x\in \mathbb {R}^N$ if and only if

$$\begin{aligned} f\in L^1\big (\mathbb {R}^N,(1+|x|)^{-(N-\alpha )}\mathrm{d}x\big ), \end{aligned}$$

(A.1)

Moreover, if (A.1) does not hold then $I_\alpha *|f|=+\infty$ everywhere in $\mathbb {R}^N$ [120, p.61–62].

The Riesz potential $I_\alpha$ is well–defined as an operator on the whole space $L^q(\mathbb {R}^N)$ if and only if $q\in [1,\frac{N}{\alpha })$. The Hardy–Littlewood–Sobolev inequality ([107, theorem 382]; [187]) (see also [129, theorem V.1]; [189, theorem 4.3]), which states that if $q \in (1, \infty )$ and if $\alpha < \frac{N}{q}$, then for every $f \in L^q (\mathbb {R}^N)$, we have $I_\alpha *f \in L^{\frac{Nq}{N - \alpha q}} (\mathbb {R}^N)$ and

$$\begin{aligned} \left( \int _{\mathbb {R}^N} \bigl |I_\alpha *f \bigr |^\frac{Nq}{N - \alpha q}\right) ^{\frac{1}{q} - \frac{\alpha }{N}} \le C_{N, \alpha , q} \left( \int _{\mathbb {R}^N} |f |^q\right) ^\frac{1}{q}. \end{aligned}$$

(A.2)

If $q=\frac{2N}{N + \alpha }$, then the optimal constant is given by ([84, 128]; [129, theorem 4.3])

$$\begin{aligned} C_{N, \alpha , q} = \frac{\Gamma (\frac{N - \alpha }{2})}{2^\alpha \pi ^{\alpha /2} \Gamma (\frac{N + \alpha }{2}) } \biggl (\frac{\Gamma (\frac{N}{2})}{\Gamma (N)}\biggr )^\frac{\alpha }{N}. \end{aligned}$$

(A.3)

The inequality (A.2) implies that if $q \in (1, \infty )$, $\alpha < \frac{N}{q}$ and $\frac{1}{r}=\frac{1}{q}-\frac{\alpha }{N}$, then

$$\begin{aligned}I_\alpha : L^q (\mathbb {R}^N)\rightarrow L^{r} (\mathbb {R}^N)\end{aligned}$$

is a bounded linear operator. If $f\in L^1(\mathbb {R}^N)$, then in general $I_\alpha *f\not \in L^\frac{N}{N-\alpha }(\mathbb {R}^N)$ (see for example [189, §V.1.2]); however,

$$\begin{aligned} I_\alpha : L^1 (\mathbb {R}^N)\rightarrow L^{r}\big (\mathbb {R}^N,(1+|x|)^{-\lambda }dx\big ) \end{aligned}$$

is a bounded operator for any $r\in [1,\frac{N}{N-\alpha })$ and $\lambda >N-r(N-\alpha )$ [182, p.38]. When $q\ge \frac{N}{\alpha }$ then $I_\alpha$ is not well defined on the whole space $L^q(\mathbb {R}^N)$. However, if $f\in L^{\frac{N}{\alpha }}(\mathbb {R}^N)$ and we additionally assume that $I_\alpha *f$ is almost everywhere finite on $\mathbb {R}^N$, then $I_\alpha *f$ is a function of bounded mean oscillation (BMO) ([155, theorem 7]; [191, theorem 2]). If $I_\alpha *f$ is almost everywhere finite on $\mathbb {R}^N$, $f\in L^{q}(\mathbb {R}^N)$ and $\frac{N}{q}<\alpha <\frac{N}{q}+1$, then $I_\alpha *f$ is Hölder continuous of order $\alpha -\frac{N}{q}$ [76, theorem 2].

These mapping properties of the Riesz potentials are important not only as a tool to control the domain of definition of the action functional of the Choquard equation but also in the study of the regularity properties of solutions by bootstrap type procedures (see Sect. 3.3.1).

The weighted version of the Hardy–Littlewood–Sobolev inequality due to Stein and Weiss [190] states that if $q \in (1, \infty )$, $s<N (1 - \frac{1}{q})$, $t<\frac{N}{r}$, $s+t\ge 0$, $q\le r<+\infty$ and $\frac{1}{r}=\frac{1}{q}+\frac{s+t-\alpha }{N}$, then for any $f\in L^q(\mathbb {R}^N,|x|^{sq}\,\mathrm {d}x)$,

$$\begin{aligned} \left( \int _{\mathbb {R}^N} \bigl |I_\alpha *f(x) \bigr |^r|x|^{-rt} \mathrm{d}x\right) ^{\frac{1}{r}} \le C \left( \int _{\mathbb {R}^N} |f(x) |^q|x|^{sq} \mathrm{d}x\right) ^\frac{1}{q}. \end{aligned}$$

(A.4)

When $r=q$ the estimate (A.4) is also known as the Hardy–Rellich inequality. Optimal constants for (A.4) are available in some special cases, see [19, 20, 108, 149, 183, 214].

Rubin [176] has proved that for radial functions $f\in L^q_{\mathrm {rad}}(\mathbb {R}^N,|x|^{sq}\,\mathrm {d}x)$ the same inequality (A.4) holds for a wider range $s+t\ge -(N-1)\big (\frac{1}{q}-\frac{1}{r}\big )$ (see also [69, 75]). Weighted inequalities of Stein–Weiss type become important in the analysis of nonautonomous Choquard equations with confining (Sect. 4.1.3) or decaying (Sect. 4.1.7) potentials.

1.3 Energy properties

The Riesz potential $I_\alpha$ naturally induces the quadratic form $D_\alpha$ defined by

$$\begin{aligned} D_{\alpha }(f,g):=A_\alpha \iint _{\mathbb {R}^N \times \mathbb {R}^N}\frac{f(x)g(y)}{|x-y |^{N - \alpha }}\,\mathrm {d}y\,\mathrm {d}x. \end{aligned}$$

A direct consequence of the semigroup property (A.1) is the inequality

$$\begin{aligned} D_{\alpha }(f,f) =\int _{\mathbb {R}^N} \bigl (I_\alpha *f) f =\int _{\mathbb {R}^N}\big |I_{\alpha /2}*f|^2\ge 0, \end{aligned}$$

(A.5)

valid for all functions f such that $D_\alpha (|f|,|f|)<+\infty$. Moreover, $D_\alpha (f,f)=0$ if and only if $f\equiv 0$ [120, theorem 1.15].

The Riesz–Sobolev rearrangement inequality states that for any two nonnegative functions f, g such that $D(f,g)<+\infty$,

$$\begin{aligned} D_\alpha (f^*,g^*)\le D_\alpha (f,f), \end{aligned}$$

(A.6)

where $f^*$ denotes the symmetric decreasing rearrangement of f. It was first established by Riesz [172] in one dimension; then Sobolev [187] extended the result to $\mathbb {R}^N$ (see also [31]). The equality in (A.6) occurs if and only if f is the translation of radially symmetric and nonincreasing function ([38]; [127, lemma 3]; [129, §3.7–3.9]). Inequality (A.6) is fundamental in the study of radial symmetry of groundstates of Choquard equation (Sect. 3.3.3).

1.4 Positivity and decay estimates

When the function $f\in L^1_{\mathrm {loc}}(\mathbb {R}^N)$ is nonnegative, an elementary estimate shows that for every $x\in \mathbb {R}^N$,

$$\begin{aligned}I_\alpha *f(x)\ge \frac{A_\alpha }{R^{N-\alpha }}\int _{B_R(x)}f(y)\,\mathrm {d}y.\end{aligned}$$

In particular, if the function f is positive on a set of positive measure of $\mathbb {R}^N$, then $I_\alpha *f$ is everywhere strictly positive on $\mathbb {R}^N$. Similarly, for each $x\in \mathbb {R}^N$ one can estimate

$$\begin{aligned} I_\alpha *f(x)\ge \frac{A_\alpha }{(2|x|)^{N-\alpha }}\int _{B_{2|x|}(x)}f(y)\,\mathrm {d}y\ge \frac{c}{|x|^{N-\alpha }}, \end{aligned}$$

(A.7)

that is, $I_\alpha *f$ cannot decay faster than $I_\alpha$ at infinity, even if the function f is compactly supported.

These decay properties of the Riesz potential are essential for the study of asymptotic decay of groundstates of Choquard equations (Sect. 3.3.4) and Liouville’s theorems (Sect. 4.1.6). To illustrate the decay of the Riesz potentials at infinity, assume that the pointwise bound

$$\begin{aligned} \limsup _{|x | \rightarrow \infty } f(x) |x |^\gamma <+\infty \end{aligned}$$

holds. Then, by a direct computation [150, lemma A.1],

$$\begin{aligned}&\limsup _{|x | \rightarrow \infty }\, (I_\alpha *f)(x)\,|x |^{\gamma -\alpha }<+\infty \quad \text {if }\alpha<\gamma<N,\\&\limsup _{|x | \rightarrow \infty }\, (I_\alpha *f)(x) \frac{|x |^{N - \alpha }}{\log |x |}<+\infty \quad \text {if } \gamma =N,\\&\limsup _{|x | \rightarrow \infty }\, (I_\alpha *f)(x)\,|x |^{N - \alpha } <+\infty \quad \text {if }\gamma >N, \end{aligned}$$

and the bounds are optimal, as can be seen by choosing $f(x)=(1+|x|)^{-\gamma }$ in the case $\gamma \ge N$, and by comparing with (A.7) in the case $\gamma >N$.

If $\gamma >N$, then the decay of $I_\alpha *f$ explicitly depends on $\Vert f\Vert _{L^1(\mathbb {R}^N)}$. More specifically [151, lemma 6.2], assume that

$$\begin{aligned} \sup _{\mathbb {R}^N}|f(x)|(1+|x|)^\gamma <+\infty , \end{aligned}$$

(A.8)

for some $\gamma >N$. Then

$$\begin{aligned} I_\alpha *f(x)=\Big (I_\alpha (x)\int _{\mathbb {R}^N} f (y) \,\mathrm {d}y\Big )\big (1+o(1)\big )\quad \text {as }\quad |x|\rightarrow \infty . \end{aligned}$$

(A.9)

Note that the assumption $f\in L^1(\mathbb {R}^N)$ alone does not imply that $I_\alpha *f=O(|x|^{-(N-\alpha )})$ even if f is radial: some additional control on the decay of f at infinity is always needed [186]. However, if f is radial and $\alpha >1$, then $I_\alpha *f=O(|x|^{-(N-\alpha )})$ if and only if $f\in L^1_{\mathrm {rad}}(\mathbb {R}^N)$ [186, theorem 5.A]. Radial estimates on the Riesz potentials could be found in [69, 75, 145, 176, 199].