Normalized ground states solutions for nonautonomous Choquard equations^††thanks: ThisworkwaspartiallysupportedbyNSFC(11901532,11901531).

Consider the time dependent nonautonomous Choquard equation

where $N\in\mathbb{N}$ denotes space dimension, $0<\mu<N$, $A\in L^{\infty}(\mathbb{R}^{N},\mathbb{R})$, $p\in(2_{*,\mu},2^{*}_{\mu})$, where

Equation $(\ref{t1.1.0})$ has several physical origins. In particular, when $N=3$, $p=2$, $\mu=1$ and $A(x)\equiv 1$, (1.1) appeared at least as early as in 1954, in a work by S. I. Pekar [10, 18] describing the quantum mechanics of a polaron at rest. In 1976, P. Choquard [12] used $(\ref{t1.1.0})$ to describe an electron trapped in its own hole, in a certain approximation to Hartree-Fock theory of one component plasma. Twenty years later, R. Penrose proposed $(\ref{t1.1.0})$ as a model of self-gravitating matter, in a programme in which quantum state reduction is understood as a gravitational phenomenon, see [16].

For our setting, $A(x)$ is a real-valued bounded function and not necessarily a constant function. However, according to [4, 8], by testing equation (1.1) against $\bar{\psi}$ (the complex conjugate of $\psi$) and $\partial_{t}\bar{\psi}$, it is easy to obtain the conservation property of mass $\int_{\mathbb{R}^{N}}|\psi|^{2}dx$ and of energy

And similar to [8, Theorem 1.1], for $0<\mu<\min\{N,4\}$ and $2\leq p<2^{*}_{\mu}$, we have from Hardy-Littlewood-Sobolev inequality and Hölder inequality that

where $\|\cdot\|_{q}$ denotes the usual norm of the Lebesgue space $L^{q}(\mathbb{R}^{N},\mathbb{R})$. Then by Strichartz estimate and fixed point argument [4, Theorem 3.3.9], (1.1) is local well-posedness in $H^{1}(\mathbb{R}^{N},\mathbb{C})$. Moreover, if $2\leq p<\bar{p}$, it is standard to get global existence by the conservation of mass and energy and Gagliardo-Nirenberg inequality of convolutional type, see [8, Theorem 1.2]. Here, $\bar{p}:=\frac{2N-\mu+2}{N}$ denotes the $L^{2}$-critical (mass-critical) exponent.

In general, equation (1.1) admits special regular solutions, which are called solitary (standing) waves. More precisely, these solutions have the form $\psi(t,x)=e^{-i\lambda t}u(x)$, where $-\lambda\in\mathbb{R}$ is the frequency and $u(x)$ solves the following elliptic equation

Here the constraint $\int_{\mathbb{R}^{N}}|u|^{2}dx=c$ is natural due to the conservation of mass.

To study (P) variationally, we need to recall the following Hardy–Littlewood–Sobolev inequality [13, Theorem 4.3].

If $A(x)$ is bounded in $\mathbb{R}^{N}$, then by (1.2) and Sobolev inequality, the integral

is well defined in $H^{1}(\mathbb{R}^{N},\mathbb{R})$ for

As a result, the functional $I:H^{1}(\mathbb{R}^{N},\mathbb{R})\mapsto\mathbb{R}$,

is well defined. Furthermore, by a standard argument, we have $I\in C^{1}(H^{1}(\mathbb{R}^{N},\mathbb{R}),\mathbb{R})$.

Due to the constraint $\int_{\mathbb{R}^{N}}|u|^{2}dx=c$, the solution for (P) is called normalized solution, which can be found by looking for critical points of the functional $I$ on the constraint

In this situation, the frequency $-\lambda\in\mathbb{R}$ can no longer be fixed but instead appears as a Lagrange multiplier, and each critical point $u_{c}\in\mathcal{S}(c)$ of $I|_{\mathcal{S}(c)}$ corresponds a Lagrange multiplier $\lambda_{c}\in\mathbb{R}$ such that $(u_{c},\lambda_{c})$ solves (weakly) (P). Due to physical application, we are particularly interested in normalized ground state solutions, defined as follows:

For any $c>0$, we set

If the minimizers of $\sigma(c)$ exist, then all minimizers are critical points of $I|_{\mathcal{S}(c)}$ as well as normalized ground state solutions to (P).

There is a lot of literature studying ground states to the autonomous Choquard equations. For example, the existence of ground states to the autonomous Choquard equation

is established by Moroz and Van Schaftingen [17] under $\lambda=-1$ and $2_{*,\mu}<p<2^{*}_{\mu}$. In [19], Ye obtained sharp existence results of the normalized solution to (1.4). Precisely,

• (i)

  If $p\in\left(2_{*,\mu},\bar{p}\right)$, $\sigma(c)$ has at least one minimizer for each $c>0$ and $\sigma(c)>-\infty$;

• (ii)

  If $p\in\left(\bar{p},2^{*}_{\mu}\right)$, $\sigma(c)$ has no minimizer for each $c>0$ and $\sigma(c)=-\infty$;

• (iii)

  The $L^{2}-$critical case $p=\bar{p}$ is complicated, see [19] for details.

As far as we know, normalized solution of nonautonomous Choquard equation (P) has not been studied. In this paper, we are interested in normalized solutions for the nonautonomous Choquard equation (P) under two cases: (i) $L^{2}-$subcritical case, i.e., $p\in(2_{*,\mu},\bar{p})$; (ii) $L^{2}-$supercritical case, i.e., $p\in(\bar{p},2^{*}_{\mu})$.

For the $L^{2}-$subcritical case, we generalize the result in [19] to the nonautonomous setting.

Compared with [19], the proof of Theorem 1.3 is more complex due to more general nonlinearity in (P). The main difficulty is to prove the compactness of a minimizing sequence of $\sigma(c)=\inf\limits_{u\in\mathcal{S}(c)}I$. To do that, inspired by [1, 7, 14, 15], we shall establish the following subadditivity inequality:

with the help of the scaling

Let $Z_{c}$ denote the set of the normalized ground state solutions for (1.1). We also interest in the stability and instability of normalized ground state solutions, defined as follows:

Following the same argument as in [5], we can deduce that $Z_{c}$ is orbitally stable provided that any minimizing sequence to $\sigma(c)$ is compact in $H^{1}(\mathbb{R}^{N},\mathbb{R})$.

Note that due to the presence of the coefficients $A(x)$ in (P), our minimization problems are not invariant by the action of the translations. To overcome this difficult, we adopt the method of studying the nonautonomous Schrödinger equation in [2]. The main point is the analysis of the compactness of minimizing sequences to suitable constrained minimization problem related to (P).

More precisely,

In the second part of this article, we consider the $L^{2}-$supercritical case. Since $\sigma(c)=-\infty$ for $p\in(\bar{p},2^{*}_{\mu})$, it is impossible to search for a minimum of $I$ on $\mathcal{S}(c)$. So it is nature to look for a critical point of $I$ having a minimax characterization. For example, for the following Schrödinger equation

Jeanjean [9] constructed mountain-pass geometrical structure on $\mathcal{S}(c)\times\mathbb{R}$ to an auxiliary functional

where $F(u)=\int_{0}^{u}f(t)dt$. Then applying the Ekeland principle to the auxiliary functional, the author obtained a sequence $\{(v_{n},s_{n})\}\subset\mathcal{S}(c)\times\mathbb{R}$ which can be used to construct a bounded Palais-Smale sequence $\{u_{n}\}\subset\mathcal{S}(c)$ for $I$ at the M-P level.

By using Jeanjean’s method [9], Li and Ye [11] obtained the normalized solutions to the Choquard equation:

where $\lambda\in\mathbb{R}$, $N\geq 3$, $\mu\in(0,N)$, and $F(u)$ behaves like $|u|^{p}$ for $\frac{2N-\mu+2}{N}<p<\frac{2N-\mu}{N-2}$.

However, for nonautonomous equation (P), the method of constructing a Pohoz̆aev-Palais-Smale sequence in [9] fails. To overcome this difficulty, we adopt the method in [7]. More precisely, we assume that

• ($A_{3}$)

  $t\mapsto(Np-2N+\mu)A(tx)-2\nabla A(tx)\cdot(tx)$ is nonincreasing on $(0,\infty)$ for every $x\in\mathbb{R}^{N}$;

• ($A_{4}$)

  $t^{\frac{2p-(Np-2N+\mu)}{2}}A(tx)$ is strictly increasing on $t\in(0,\infty)$ for every $x\in\mathbb{R}^{N}$.

Besides $A\equiv$constant, there are indeed many functions which satisfy $(A_{1}),(A_{3})$ and $(A_{4})$. For example

• ($i$)

  $A_{1}(x)=1+be^{-\tau|x|}$ with $0<b\leq e\cdot\frac{2p-(Np-2N+\mu)}{2}$ and $\tau>0$;

• ($ii$)

  $A_{2}(x)=1+\frac{b}{1+|x|}$ with $0<b\leq 2[2p-(Np-2N+\mu)]$.

Under $(A_{1}),(A_{3})$ and $(A_{4})$, we shall establish the existence of normalized ground state solutions to the nonautonomous Choquard equation (P) by taking a minimum on the manifold

where $u^{t}(x):=t^{N/2}u(tx)$ for all $t>0$ and $x\in\mathbb{R}^{N}$, and $u^{t}\in\mathcal{S}(c)$ if $u\in\mathcal{S}(c)$.

To address the lack of compactness, we should consider the limit equation of (P):

The energy functional is defined as follows:

Similar to (1.9), we define

Finally, we give our future research directions about this article:
For $A\equiv 1$ and $\bar{p}<p<2^{*}_{\mu}$, by using the blow up for a class of initial data with nonnegative energy, Chen and Guo [6] proved that the standing wave of (1.1) must be strongly unstable. For nonautonomous situation ($A\not\equiv$constant), the method in [6] is invalid. We will study the problem in the future.

This paper is organized as follows. In section 2, we prove Theorem 1.3 and Theorem 1.5. In section 3, we show Theorem 1.6.

First, we prove a nonlocal version of Brezis-Lieb lemma, which will be used in the proof below both $L^{2}-$subcritical case and $L^{2}-$supercritical case. We need the following classical Brezis-Lieb lemma [3].