# Ground states in the diffusion-dominated regime

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## Abstract

We consider macroscopic descriptions of particles where repulsion is modelled by non-linear power-law diffusion and attraction by a homogeneous singular kernel leading to variants of the Keller–Segel model of chemotaxis. We analyse the regime in which diffusive forces are stronger than attraction between particles, known as the diffusion-dominated regime, and show that all stationary states of the system are radially symmetric non-increasing and compactly supported. The model can be formulated as a gradient flow of a free energy functional for which the overall convexity properties are not known. We show that global minimisers of the free energy always exist. Further, they are radially symmetric, compactly supported, uniformly bounded and \(C^\infty \) inside their support. Global minimisers enjoy certain regularity properties if the diffusion is not too slow, and in this case, provide stationary states of the system. In one dimension, stationary states are characterised as optimisers of a functional inequality which establishes equivalence between global minimisers and stationary states, and allows to deduce uniqueness.

## Mathematics Subject Classification

35K55 35K65 49K20## 1 Introduction

*fair-competition regime*. In the

*diffusion-dominated regime*we choose \(m>m_c\), which means that the diffusion part of the functional (1.3) dominates as \(\lambda \rightarrow \infty \). In other words, concentrations are not energetically favourable for any value of \(\chi >0\) and \(m>m_c\). The range \(0<m<m_c\) is referred to as the

*attraction-dominated regime*. In this work, we focus on the diffusion-dominated regime \(m>m_c\).

### Theorem 1

Let \(N\ge 1\), \(\chi >0\) and \(k \in (-N,0)\). All stationary states of Eq. (1.1) are radially symmetric non-increasing. If \(m>m_c\), then there exists a global minimiser \(\rho \) of \({{\mathcal {F}}}\) on \({\mathcal {Y}}\). Further, all global minimisers \(\rho \in {{\mathcal {Y}}}\) are radially symmetric non-increasing, compactly supported, uniformly bounded and \(C^{\infty }\) inside their support. Moreover, all global minimisers of \({{\mathcal {F}}}\) are stationary states of (1.1), according to Definition 1, whenever \(m_c<m < m^*\). Finally, if \(m_c<m\le 2\), we have \(\rho \in {{\mathcal {W}}}^{1,\infty }\left( {\mathbb {R}}^N\right) \).

### Theorem 2

Let \(N=1\), \(\chi >0\), \(k \in (-1,0)\) and \(m>m_c\). All stationary states of (1.1) are global minimisers of the energy functional \({{\mathcal {F}}}\) on \({{\mathcal {Y}}}\). Further, stationary states of (1.1) in \({{\mathcal {Y}}}\) are unique.

Diffusion-aggregation at the top equations of the form (1.1) are ubiquitous as macroscopic models of cell motility due to cell adhesion and/or chemotaxis phenomena while taking into account volume filling constraints [10, 29, 45]. The non-linear diffusion models the very strong localised repulsion between cells while the attractive non-local term models either cell movement toward chemosubstance sources or attractive interaction between cells due to cell adhension by long filipodia. They encounter applications in cancer invasion models, organogenesis and pattern formation [18, 24, 28, 42, 46].

The archetypical example of the Keller–Segel model in two dimensions corresponding to the logarithmic case \((m = 1,k=0)\) has been deeply studied by many authors [2, 3, 5, 6, 15, 19, 23, 30, 31, 32, 43, 44, 47], although there are still plenty of open problems. In this case, there is an interesting dichotomy based on a critical parameter \(\chi _c>0\): the density exists globally in time if \(0<\chi <\chi _c\) (diffusion overcomes self-attraction) and expands self-similarly [14, 27], whereas blow-up occurs in finite time when \(\chi >\chi _c\) (self-attraction overwhelms diffusion), while for \(\chi =\chi _c\) infinitely many stationary solutions exist with intricated basins of attraction [3]. The three-dimensional configuration with Newtonian interaction \((m = 1,k = 2-N)\) appears in gravitational physics [20, 21], although it does not have this dichotomy, belonging to the attraction-dominated regime. However, the dichotomy does happen for the particular exponent \(m=4/3\) of the non-linear diffusion for the 3D Newtonian potential as discovered in [4]. This was subsequently generalised for the fair-competition regime where \(m=m_c\) for a given \(k\in (-N,0)\) in [12, 13].

In the diffusion-dominated case, it was already proven in [16] that global minimisers exist in the particular case of \(m>1=m_c\) for the logarithmic interaction kernel \(k=0\). Their uniqueness up to translation and mass normalisation is a consequence of the important symmetrisation result in [17] asserting that all stationary states to (1.1) for \(2-N\le k<0\) are radially symmetric. We will generalise this result to our present framework for the range \(-N<k<2-N\) not included in [17] due to the special treatment needed for the arising singular integral terms. This is the main goal of Sect. 2 where we remind the reader the precise definition and basic properties of stationary states for (1.1). In short, we show that stationary solutions are continuous compactly supported radially non-increasing functions with respect to their centre of mass. Some of these results are in fact generalisations of previous results in [12, 17] and we skip some of the details.

Let us finally comment that the symmetrisation result reduces the uniqueness of stationary states to uniqueness of radial stationary states that eventually leads to a full equivalence between stationary states and global minimisers of the free energy (1.3). This was used in [17] to solve completely the 2D case with \(m>1=m_c\) for the logarithmic interaction kernel \(k=0\), and it was the new ingredient to fully characterise the long-time asymptotics of (1.1) in that particular case.

In view of the main results already announced above, we show in Sect. 3 the existence of global minimisers for the full range \(m>m_c\) and \(k\in (-N,0)\) which are steady states of the Eq. (1.1) as soon as \(m<m^*\). This additional constraint on the range of non-linearities appears only in the most singular range \(-N<k<1-N\) and allows us to get the right Hölder regularity on the minimisers in order to make sense of the singular integral in the gradient of the attractive non-local potential force (1.2).

Besides existence of minimisers, Sect. 3 contains some of the main novelties of this paper. First, in order to prove boundedness of minimisers, we develop a fine estimate on the interaction term based on the asymptotics of the Riesz potential of radial functions, and show that this estimate is well suited exactly for the diffusion dominated regime (see Lemma 2 and Theorem 7). Moreover, thanks to the Schauder estimates for the fractional Laplacian, we improve the regularity results for minimisers in [12] and show that they are smooth inside their support, see Theorem 10. This result applies both to the diffusion-dominated and fair-competition regime.

These global minimisers are candidates to play an important role in the long-time asymptotics of (1.1). We show their uniqueness in one dimension by optimal transportation techniques in Sect. 4. The challenging open problems remaining are uniqueness of radially non-increasing stationary solutions to (1.1) in its full generality and the long-time asymptotics of (1.1) in the whole diffusion-dominated regime, even for non-singular kernels within the fast diffusion case.

Plan of the paper: In Sect. 2 we define and analyse stationary states, showing that they are radially symmetric and compactly supported. Section 3 is devoted to global minimisers. We show that global minimisers exist, are bounded and we provide their regularity properties. Eventually, Sect. 4 proves uniqueness of stationary states in the one-dimensional case.

## 2 Stationary states

Let us define precisely the notion of stationary states to the diffusion–aggregation equation (1.1).

### Definition 1

**stationary state**for the evolution equation (1.1) if \({\bar{\rho }}^{m} \in {{\mathcal {W}}}_{loc}^{1,2}\left( {\mathbb {R}}^N\right) \), \(\nabla {\bar{S}}_k[{\bar{\rho }}]\in L^1_{loc}\left( {\mathbb {R}}^N\right) \), and it satisfies

In fact, as shown in [12] via a near-far field decomposition argument of the drift term, the function \(S_k[\rho ]\) and its gradient defined in (1.2) satisfy even more than the regularity \(\nabla S_k[\rho ] \in L_{loc}^1\left( {\mathbb {R}}^N\right) \) required in Definition 1:

### Lemma 1

- (i)
\( S_k[\rho ] \in L^{\infty }\left( {\mathbb {R}}^N\right) \).

- (ii)
\(\nabla S_k[\rho ] \in L^{\infty }\left( {\mathbb {R}}^N\right) \), assuming additionally \(\rho \in C^{0,\alpha }\left( {\mathbb {R}}^N\right) \) with \(\alpha \in (1-k-N,1)\) in the range \(k \in (-N,1-N]\).

Lemma 1 implies further regularity properties for stationary states of (1.1). For precise proofs, see [12].

### Proposition 1

It follows from Proposition 1 that \({\bar{\rho }} \in {{\mathcal {W}}}^{1,\infty }\left( {\mathbb {R}}^N\right) \) in the case \(m_c<m\le 2\).

### 2.1 Radial symmetry of stationary states

The aim of this section is to prove that stationary states of (1.1) are radially symmetric. This is one of the main results of [17], and is achieved there under the assumption that the interaction kernel is not more singular than the Newtonian potential close to the origin. As we will briefly describe in the proof of the next result, the main arguments continue to hold even for the more singular Riesz kernels \(W_{k}\).

### Theorem 3

(Radiality of stationary states) Let \(\chi >0\) and \(m>m_c\). If \({\bar{\rho }} \in L^1_+({\mathbb {R}}^N) \cap L^\infty ({\mathbb {R}}^N)\) with \(\Vert {\bar{\rho }}\Vert _1=1\) is a stationary state of (1.1) in the sense of Definition 1, then \({\bar{\rho }}\) is radially symmetric non-increasing up to a translation.

### Proof

*not*radially non-increasing up to

*any*translation. By Proposition 1, we have

*c*in \(\text {supp}({\bar{\rho }})\). Let us now introduce the

*continuous Steiner symmetrisation*\(S^\tau {\bar{\rho }}\) in direction \(e_1 = (1,0,\cdots ,0)\) of \({\bar{\rho }}\) as follows. For any \(x_1 \in {\mathbb {R}}, x'\in {\mathbb {R}}^{N-1}, h>0\), let

*v*(

*h*) defined as

*v*(

*h*), we do not have \(M^{v(h_1)\tau }(U_{x'}^{h_1}) \subset M^{v(h_2)\tau }(U_{x'}^{h_2}) \) for all \(h_1>h_2\), but it is still possible to prove that (see [17, Proposition 2.7])

### 2.2 Stationary states are compactly supported

In this section, we will prove that all stationary states of Eq. (1.1) have compact support, which agrees with the properties shown in [16, 17, 33]. We begin by stating a useful asymptotic estimate on the Riesz potential inspired by [50, § 4]. For the proof of Proposition 2, see Appendix 1.

### Proposition 2

- (i)
If \(1-N<k<0\), then \(|x|^k*\rho (x)\le C_1 |x|^{k}\) on \({\mathbb {R}}^N\).

- (ii)If \(-N<k\le 1-N\) and if \(\rho \) is supported on a ball \(B_R\) for some \(R<\infty \), thenwhere$$\begin{aligned} |x|^k*\rho (x)\le C_2 T_k(|x|,R)\, |x|^{k}, \quad \forall \, \, |x|>R, \end{aligned}$$$$\begin{aligned} T_k(|x|,R):=\left\{ \begin{array}{l@{\quad }l} \left( \frac{|x|+R}{|x|-R}\right) ^{1-k-N}&{}\text{ if } \, k\in (-N,1-N),\\ \\ \left( 1+\log \left( \frac{|x|+R}{|x|-R}\right) \right) &{}\text{ if }\, k=1-N \end{array}\right. \end{aligned}$$(2.9)

*k*and

*N*.

From the above estimate, we can derive the expected asymptotic behaviour at infinity.

### Corollary 1

Let \(\rho \in {{\mathcal {Y}}}\) be radially non-increasing. Then \(W_k*\rho \) vanishes at infinity, with decay not faster than that of \(|x|^{k}\).

### Proof

*r*yields \(k+r>1-N\).

As a rather simple consequence of Corollary 1, we obtain:

### Corollary 2

Let \({\bar{\rho }}\) be a stationary state of (1.1). Then \({\bar{\rho }}\) is compactly supported.

### Proof

*C*. Then we necessarily have \(C=0\). Indeed, \({\bar{\rho }}^{m-1}\) vanishes at infinity since it is radially non-increasing and integrable, and by Corollary 1 we have that \(S_{k}[{\bar{\rho }}]=W_k*{\bar{\rho }}\) vanishes at infinity as well. Therefore

## 3 Global minimisers

### Theorem 4

### Proof

The inequality is a direct consequence of the standard sharp HLS inequality and of Hölder’s inequality. It follows that \(C_*\) is finite and bounded from above by the optimal constant in the HLS inequality. \(\square \)

### 3.1 Existence of global minimisers

### Theorem 5

(Existence of global minimisers) For all \(\chi >0\) and \(k \in (-N,0)\), there exists a global minimiser \(\rho \) of \({{\mathcal {F}}}\) in \({{\mathcal {Y}}}\). Moreover, all global minimisers of \({{\mathcal {F}}}\) in \({{\mathcal {Y}}}\) are radially non-increasing.

We follow the concentration compactness argument as applied in Appendix A.1 of [33]. Our proof is based on [38, Theorem II.1, Corollary II.1]. Let us denote by \({\mathcal {M}}^p({\mathbb {R}}^N)\) the Marcinkiewicz space or weak \(L^p\) space.

### Theorem 6

### Proposition 3

### Proof of Theorem 5

*N*-dimensional unit ball. Then

### Remark 1

Global minimisers of \({{\mathcal {F}}}\) satisfy a corresponding Euler–Lagrange condition. The proof can be directly adapted from [16, Theorem 3.1] or [12, Proposition 3.6], and we omit it here.

### Proposition 4

### 3.2 Boundedness of global minimisers

This section is devoted to showing that all global minimisers of \({{\mathcal {F}}}\) in \({{\mathcal {Y}}}\) are uniformly bounded. In the following, for a radial function \(\rho \in L^1({\mathbb {R}}^N)\) we denote by \(M_\rho (R):=\int _{B_R}\rho \, dx\) the corresponding mass function, where \(B_R\) is a ball of radius *R*, centered at the origin. We start with the following technical lemma:

### Lemma 2

*k*and

*N*.

### Proof

Notice that the result is trivial if \(\rho \) is bounded. The interesting case here is \(\rho \) unbounded, implying that \(A_H>0\) for any \(H>0\).

Let us make use of Proposition 2, which we apply to the compactly supported function \(\rho _H:=\rho {\mathbb {1}}_{\{\rho \ge H\}}/M_\rho \left( A_H\right) \).

We are now in a position to prove that any minimiser of \({{\mathcal {F}}}\) is bounded.

### Theorem 7

Let \(\chi >0\), \(k \in (-N,0)\) and \(m >m_c\). Then any global minimiser of \({{\mathcal {F}}}\) over \({{\mathcal {Y}}}\) is uniformly bounded and compactly supported.

### Proof

*q*are defined as in Lemma 2, \(B_{A_H}^C\) denotes the complement of \(B_{A_H}\) and \({\mathbb {1}}_{D_r}\) is the characteristic function of a ball \(D_r:=B_r(x_0)\) of radius \(r>0\), centered at some \(x_0\ne 0\) and such that \(D_r\cap B_{A_H}=\emptyset \). Note that \(A_H\le H^{-q}/2< H_0^{-q}/2< 1/2\). Hence, we can take \(r>1\) and \(D_r\) centered at the point \(x_0=(2r,0, \dots , 0) \in {\mathbb {R}}^N\). Notice in particular that since \(\rho \) is unbounded, for any \(H>0\) we have that \(B_{A_H}\) has non-empty interior. On the other hand, \(B_{A_H}\) shrinks to the origin as \(H\rightarrow \infty \) since \(\rho \) is integrable.

*a*,

*b*as above, the HLS inequality implies

*H*as specified in Lemma 2.

*H*,

*q*. On the one hand, notice that for a choice \(\eta >0\) small enough such that \(m>m_c+\eta \), we have

*q*such that

*q*satisfies (3.18), it follows that \(-kq<m-1-\eta \) and at the same time \(1-q(k+N)<m-1-\eta \), showing that \({\mathcal {K}}_{k,q,N}(H) \) from Lemma 2 grows slower than \(H^{m-1-\eta }\) as \(H\rightarrow \infty \) for \(k \ne 1-N\). If \(k=1-N\), we have that for any \(C>0\) there exists \(H>H_0\) large enough such that \(CH^{1-q}\log (1+H^q)<H^{m-1-\eta }\) since \(q>2-m+\eta \), and so the same result follows. Hence, for any large enough

*H*we have

*H*. First of all, notice that \(\int _{B_{A_H}}\rho ^{m}\, dx\) is strictly positive since we are assuming that \(\rho \) is unbounded. We can therefore fix

*H*large enough such that the constant in front of \(\int _{B_{A_H}}\rho ^{m}\) is strictly positive. Secondly, we have already proven that \(\varepsilon _r\) and \(I_3(r)\) vanish as \(r\rightarrow \infty \), so we can choose

*r*large enough such that

Finally, we can just use the Euler–Lagrange Eq. (3.5) and the same argument as for Corollary 2 to prove that \(\rho \) is compactly supported. \(\square \)

### 3.3 Regularity properties of global minimisers

This section is devoted to the regularity properties of global minimisers. With enough regularity, global minimisers satisfy the conditions of Definition 1, and are therefore stationary states of Eq. (1.1). This will allow us to complete the proof of Theorem 1.

*s*.

*Bessel potential space*\({\mathcal {L}}^{2s,p}({\mathbb {R}}^{N})\) as made by all functions \(f\in L^{p}({\mathbb {R}}^{N})\) such that \((I-{\varDelta })^{s}f\in L^{p}({\mathbb {R}}^{N})\), meaning that

*f*is the Bessel potential of an \(L^{p}({\mathbb {R}}^N)\) function (see [52, pag. 135]). Since we are working with the operator \((-{\varDelta })^s\) instead of \((I-{\varDelta })^s\), we make use of a characterisation of the space \({\mathcal {L}}^{2s,p}({\mathbb {R}}^N)\) in terms of Riesz potentials. For \(1<p<\infty \) and \(0<s<1\) we have

*fractional Sobolev*space \({{\mathcal {W}}}^{2s,p}({\mathbb {R}}^{N})\) by

*U*in \({\mathbb {R}}^{N}\), then \(C^{0,\alpha }({\overline{U}}):=C^{\alpha ',\alpha ''}({\overline{U}})\), where \(\alpha '+\alpha ''=\alpha \), \(\alpha ''\in [0,1)\) and \(\alpha '\) is the greatest integer less than or equal to \(\alpha \). With this notation, we have \(C^{0,1}({\mathbb {R}}^N)=C^{1,0}({\mathbb {R}}^N)={{\mathcal {W}}}^{1,\infty }({\mathbb {R}}^N)\). In particular, using (3.23) it follows that for \(\alpha > 0\), \(s \in (0,1)\) and \(\alpha +2s\) not an integer,

*s*: \(m_c:=2-\frac{2s}{N}\) and

### Theorem 8

Let \(\chi >0\) and \(s \in (0,N/2)\). If \(m_c<m< m^*\), then any global minimiser \(\rho \in {{\mathcal {Y}}}\) of \({{\mathcal {F}}}\) satisfies \(S_k[\rho ]=W_k*\rho \in {\mathcal {W}}^{1,\infty }({\mathbb {R}}^N)\), \(\rho ^{m-1}\in {\mathcal {W}}^{1,\infty }({\mathbb {R}}^N)\) and \(\rho \in C^{0,\alpha }({\mathbb {R}}^N)\) with \(\alpha =\min \{1,\tfrac{1}{m-1}\}\).

### Proof

Recall that the global minimiser \(\rho \in {{\mathcal {Y}}}\) of \({{\mathcal {F}}}\) is radially symmetric non-increasing and compactly supported by Theorem 5 and Theorem 7. Since \(\rho \in L^1\left( {\mathbb {R}}^N\right) \cap L^\infty \left( {\mathbb {R}}^N\right) \) by Theorem 7, we have \(\rho \in L^p\left( {\mathbb {R}}^N\right) \) for any \(1< p<\infty \). Since \(\rho =c_{N,s}(-{\varDelta })^s S_k[\rho ]\), it follows from (3.19) that \(S_k[\rho ]\in L^{q}({\mathbb {R}}^N)\), \(q=\tfrac{Np}{N-2sp}\) for all \(1<p<\tfrac{N}{2s}\), that is \(S_k[\rho ]\in L^{p}({\mathbb {R}}^N)\) for all \(p \in (\tfrac{N}{N-2s},\infty )\). Then, if \(s\in (0,1)\), since \(S_{k}\) is the Riesz potential of the density \(\rho \) in \(L^{p}\), by the characterisation (3.20) of the Bessel potential space, we conclude that \(S_k[\rho ] \in {{\mathcal {L}}}^{2s,p}({\mathbb {R}}^N)\) for all \(p>\tfrac{N}{N-2s}\). Let us first consider \(s<1/2\), as the cases \(1/2<s<N/2\) and \(s=1/2\) follow as a corollary.

*p*there hold \(1-2s<\beta _n<1\). Note that \(S_k[\rho ] \in L^\infty \left( {\mathbb {R}}^N\right) \) by Lemma 1, and if \(\rho \in C^{0,\gamma }\left( {\mathbb {R}}^N\right) \) for some \(\gamma \in (0,1)\) such that \(\gamma +2s<1\), then \(S_k[\rho ] \in C^{0,\gamma +2s}\left( {\mathbb {R}}^N\right) \) by (3.24), implying \(\rho ^{m-1} \in C^{0,\gamma +2s}\left( {\mathbb {R}}^N\right) \) using the Euler–Lagrange conditions (3.5). Therefore \(\rho \in C^{0,\gamma +2s}\left( {\mathbb {R}}^N\right) \) since \(m \in (m_c,2]\). Iterating this argument \((n-1)\) times starting with \(\gamma =\beta \) gives \(\rho \in C^{0,\beta _n}\left( {\mathbb {R}}^N\right) \) . Since \(\beta _n<1\) and \(\beta _n+2s>1\), a last application of (3.24) yields \(S_k[\rho ]\in {\mathcal {W}}^{1,\infty }({\mathbb {R}}^N)\), so that \(\rho ^{m-1}\in {\mathcal {W}}^{1,\infty }({\mathbb {R}}^N)\), thus \(\rho \in {\mathcal {W}}^{1,\infty }({\mathbb {R}}^N)\). This concludes the proof in the case \(m\le 2\).

\(N\ge 2\), \(1/2\le s<N/2\): We start with the case \(s=1/2\). We have \(S_k[\rho ]\in L^p({\mathbb {R}}^N)\) for any \(p>\tfrac{N}{N-1}\) as shown at the beginning of this proof. By (3.20) we get \(S_k[\rho ] \in {{\mathcal {L}}}^{1,p}\left( {\mathbb {R}}^N\right) \) for all \(p>\tfrac{N}{N-1}\). Then we also have \(S_k[\rho ] \in {{\mathcal {L}}}^{2r,p}({\mathbb {R}}^N)\) for all \(p>\tfrac{N}{N-1}\) and for all \(r \in (0,1/2)\) by the embeddings between Bessel potential spaces, see [52, pag. 135]. Noting that \(2\ge \tfrac{N}{N-1}\) for \(N\ge 2\), by (3.21) and (3.22) we get \(S_k[\rho ]\in C^{0,2r-N/p}({\mathbb {R}}^N)\) for any \(r\in (0,1/2)\) and any \(p>\tfrac{N}{2r}\). That is, \(S_k[\rho ]\in C^{0,\gamma }({\mathbb {R}}^N)\) for any \(\gamma \in (0,1)\). By the Euler–Lagrange Eq. (3.5), \(\rho \in C^{0,\gamma \alpha }({\mathbb {R}}^N)\) with \(\alpha =\min \{1,\tfrac{1}{m-1}\}\), and so (3.24) for \(s=1/2\) implies \(S_k[\rho ]\in {\mathcal {W}}^{1,\infty }({\mathbb {R}}^N)\). Again by the Euler–Lagrange Eq. (3.5), we obtain \(\rho ^{m-1}\in {\mathcal {W}}^{1,\infty }({\mathbb {R}}^N)\).

If \(1/2<s<N/2\) on the other hand, we obtain directly that \(S_k[\rho ] \in {{\mathcal {W}}}^{1,\infty }({\mathbb {R}}^N)\) by Lemma 1, and so \(\rho ^{m-1}\in {\mathcal {W}}^{1,\infty }({\mathbb {R}}^N)\).

We conclude that \(\rho \in C^{0,\alpha }({\mathbb {R}}^N)\) with \(\alpha =\min \{1,\tfrac{1}{m-1}\}\) for any \(1/2\le s<N/2\). \(\square \)

### Remark 2

If \(m\ge m^*\) and \(s<1/2\), we recover some Hölder regularity, but it is not enough to show that global minimisers of \({{\mathcal {F}}}\) are stationary states of (1.1). More precisely, \(m \ge m^*\) means \(\tfrac{2s(m-1)}{m-2} \le 1 \), and so it follows from (3.28) that \(\rho \in C^{0,\gamma }\left( {\mathbb {R}}^N\right) \) for any \(\gamma <\tfrac{2s}{m-2}\). Note that \(m \ge m^*\) also implies \(\tfrac{2s}{m-2}\le 1-2s\), and we are therefore not able to go above the desired Hölder exponent \(1-2s\).

### Remark 3

*p*, hence (3.22) implies that \(\rho \) has the Hölder regularity stated in Remark 2.

We are now ready to show that global minimisers possess the good regularity properties to be stationary states of equation (1.1) according to Definition 1.

### Theorem 9

Let \(\chi >0\), \(s \in (0,N/2)\) and \(m_c<m < m^*\). Then all global minimisers of \({{\mathcal {F}}}\) in \({{\mathcal {Y}}}\) are stationary states of equation (1.1) according to Definition 1.

### Proof

Note that \(m<m^*\) means \(1-2s<1/(m-1)\), and so thanks to Theorem 8, \(S_k[\rho ]\) and \(\rho \) satisfy the regularity conditions of Definition 1. Further, since \(\rho ^{m-1} \in {\mathcal {W}}^{1,\infty }\left( {\mathbb {R}}^N\right) \), we can take gradients on both sides of the Euler–Lagrange condition (3.5). Multiplying by \(\rho \) and writing \(\rho \nabla \rho ^{m-1}=\tfrac{m-1}{m}\nabla \rho ^{m}\), we conclude that global minimisers of \({{\mathcal {F}}}\) in \({{\mathcal {Y}}}\) satisfy relation (2.1) for stationary states of Eq. (1.1). \(\square \)

In fact, we can show that global minimisers have even more regularity inside their support.

### Theorem 10

Let \(\chi >0\), \(m_c<m\) and \(s \in (0,N/2)\). If \(\rho \in {{\mathcal {Y}}}\) is a global minimiser of \({{\mathcal {F}}}\), then \(\rho \) is \(C^\infty \) in the interior of its support.

### Proof

By Theorem 8 and Remark 2, we have \(\rho \in C^{0,\alpha }({\mathbb {R}}^N)\) for some \(\alpha \in (0,1)\). Since \(\rho \) is radially symmetric non-increasing, the interior of \(\mathrm{supp\ }(\rho )\) is a ball centered at the origin, which we denote by *B*. Note also that \(\rho \in L^1({\mathbb {R}}^N) \cap L^\infty ({\mathbb {R}}^N)\) by Theorem 7, and so \(S_k[\rho ] \in L^\infty ({\mathbb {R}}^N)\) by Lemma 1.

Assume first that \(s\in (0,1)\cap (0,N/2)\). Applying (3.25) with \(B_R\) centered at a point within *B* and such that \(B_R\subset \subset B\), we obtain \(S_k[\rho ]\in C^{0,\gamma }(B_{R/2})\) for any \(\gamma <\alpha +2s\). It follows from the Euler–Lagrange condition (3.5) that \(\rho ^{m-1}\) has the same regularity as \(S_k[\rho ]\) on \(B_{R/2}\), and since \(\rho \) is bounded away from zero on \(B_{R/2}\), we conclude \(\rho \in C^{0,\gamma }(B_{R/2})\) for any \(\gamma <\alpha +2s\). Repeating the previous step now on \(B_{R/2}\), we get the improved regularity \(S_k[\rho ]\in C^{0,\gamma }(B_{R/4})\) for any \(\gamma <\alpha +4s\) by (3.25), which we can again transfer onto \(\rho \) using (3.5), obtaining \(\rho \in C^{0,\gamma }(B_{R/4})\) for any \(\gamma <\alpha +4s\). Iterating, any order \(\ell \) of differentiability for \(S_{k}\) (and then for \(\rho \)) can be reached in a neighborhood of the center of \(B_R\). We notice that the argument can be applied starting from any point \(x_{0}\in B\), and hence \(\rho \in C^{\infty }(B)\).

*B*such that \(B_R\subset \subset B\), and using the iteration rule

*s*in Hölder regularity for \(\rho \) each time we divide the radius

*R*by \(2^l\). In this way, we can reach any differentiability exponent for \(\rho \) around any point of

*B*, and thus \(\rho \in C^{\infty }(B)\). \(\square \)

### Remark 4

We observe that the smoothness of minimisers in the interior of their support also holds in the fair-competition regime \(m=m_c\). In such case global Hölder regularity was obtained in [12].

The main result Theorem 1 follows from Theorem 3, Corollary 2, Theorem 5, Proposition 4, Theorem 7, Theorem 9 and Theorem 10.

## 4 Uniqueness

### 4.1 Optimal transport tools

Optimal transport is a powerful tool for reducing functional inequalities onto pointwise inequalities. In other words, to pass from microscopic inequalities between particle locations to macroscopic inequalities involving densities. This sub-section summarises the main results of optimal transportation we will need in the one-dimensional setting. They were already used in [11] and in [13], where we refer for detailed proofs.

*U*, bounded below such that \(U(0) = 0\) we have [40]

### Lemma 3

*a*and

*b*is given by \([a,b]_s=(1-s)a+sb\). Equality is achieved in (4.2) if and only if the distributional derivative of the transport map \(\psi ''\) is a constant function.

### 4.2 Functional inequality in one dimension

In what follows, we will make use of a characterisation of stationary states based on some integral reformulation of the necessary condition stated in Proposition 4. This characterisation was also the key idea in [11, 13] to analyse the asymptotic stability of steady states and the functional inequalities behind.

### Lemma 4

The proof follows the same methodology as for the fair-competition regime [13, Lemma 2.8] and we omit it here.

### Theorem 11

### Proof

In fact, the result in Theorem 11 implies that all critical points of \({{\mathcal {F}}}\) in \({{\mathcal {Y}}}\) are global minimisers. Further, we obtain the following uniqueness result:

### Corollary 3

(Uniqueness) Let \(\chi >0\) and \(k \in (-1,0)\). If \(m_c<m\), then there exists at most one stationary state in \({{\mathcal {Y}}}\) to equation (1.1). If \(m_c<m<m^*\), then there exists a unique global minimiser for \({{\mathcal {F}}}\) in \({{\mathcal {Y}}}\).

### Proof

Assume there are two stationary states to Eq. (1.1): \({\bar{\rho }}_1, {\bar{\rho }}_2 \in {{\mathcal {Y}}}\). Then Theorem 11 implies that \({{\mathcal {F}}}[{\bar{\rho }}_1]={{\mathcal {F}}}[{\bar{\rho }}_2]\), and so \({\bar{\rho }}_1\) is a dilation of \({\bar{\rho }}_2\). By Theorem 5, there exists a minimiser of \({{\mathcal {F}}}\) in \({{\mathcal {Y}}}\), which is a stationary state of Eq. (1.1) if \(m_c<m<m^*\) by Theorem 9, and so uniqueness follows. \(\square \)

Theorem 11 and Corollary 3 complete the proof of the main result Theorem 2.

## Notes

### Acknowledgements

We thank Y. Yao and F. Brock for useful discussion about the continuous Steiner symmetrisation. We thank X. Ros-Otón, P. R. Stinga and P. Mironescu for some fruitful explanations concerning the regularity properties of fractional elliptic equations used in this work. We are grateful to R. Frank for suggesting the alternative proof for the existence of minimisers in Remark 1. JAC was partially supported by the Royal Society via a Wolfson Research Merit Award and by the EPSRC grant number EP/P031587/1. FH acknowledges support from the EPSRC grant number EP/H023348/1 for the Cambridge Centre for Analysis. EM was partially supported by the FWF project M1733-N20. BV was partially supported by GNAMPA of INdAM, “Programma triennale della Ricerca dell’Università degli Studi di Napoli “Parthenope”- Sostegno alla ricerca individuale 2015-2017”. EM and BV are members of the GNAMPA group of the Istituto Nazionale di Alta Matematica (INdAM). This work was partially supported by the Simons-Foundation grant 346300 and the Polish Government MNiSW 2015-2019 matching fund. The authors are very grateful to the Mittag-Leffler Institute for providing a fruitful working environment during the special semester *Interactions between Partial Differential Equations & Functional Inequalities*.

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