Existence and non-uniqueness of stationary states for the Vlasov–Poisson equation on \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{\mathbb {R}}}^3$$\end{document}R3 subject to attractive background charges

We prove the existence of stationary solutions for the density of an infinitely extended plasma interacting with an arbitrary configuration of background charges. Furthermore, we show that the solution cannot be unique if the total charge of the background is attractive. In this case, infinitely many different stationary solutions exist. The non-uniqueness can be explained by the presence of trapped particles orbiting the attractive background charge.


Introduction
We consider the response of a spatially homogeneous plasma to a given background distribution of charges. This phenomenon can be modeled by the following nonlinear stationary Vlasov-Poisson equation for the plasma electron density f (x, v), on the three dimensional Here μ describes the distribution of background charges and f 0 the electron distribution far away from the perturbation. As usual, ρ[ f ] denotes the spatial density of electrons given by The system (1.1)-(1.3) is often considered in plasma physics, since the onset of Debye screening can quickly be derived for the linearized system. A detailed discussion can be found in the plasma physics textbooks [5,6]. The model (1.1)-(1.3) is also used in other contexts in plasma physics, for instance for characterizing plasma waves (cf. [10]). For a plasma interacting with a repulsive point charge, screening has been proved rigorously in [1]. A number of results have been shown for the nonlinear Vlasov-Poisson equation in the case of finite mass and finite energy. In this case, the conserved quantities can be used to study existence and stability of stationary solutions (cf. [9]). We also refer to recent results on the stability of a point charge interacting with a plasma of finite mass [7,8].
A closely related problem is the existence and stability of stationary solutions to the Vlasov-Poisson Boltzmann equation. In the presence of collisions, the natural boundary condition f 0 are Maxwellian distributions. We refer to [2][3][4] for details.
The key point of this paper is to show that stationary solutions to (1.1)-(1.3) exist for general background measures μ, and infinitely many stationary states f exist as soon as the total charge of the background measure is attractive.
The main result is contained in the following theorem.
for some function F 0 ∈ C 1 (R) satisfying Assumption 1.2 below, and μ ∈ M(R 3 ) be a measure with finite total variation, i.e.
If the total charge θ ∈ R given by is negative, then there exist infinitely many different solutions to the stationary problem (1.1)-(1.3).
We split the proof of Theorem 1.1 in parts. The existence of solutions is shown in Proposition 3.2, using the extensions F of F 0 constructed in Sect. 2. The non-uniqueness of solutions is given by Proposition 4.1.
In [1], the existence of stationary solutions and their screening properties have been investigated for The existence proof in [1] relies on radial symmetry of the constructed solution f and the compact embedding H 1 r (R 3 ) ⊂ L p r (R 3 ), 2 < p < 6 under radial symmetry. Most importantly, the proof only applies to repulsive interaction, i.e. θ > 0.
The remainder of the paper is devoted to the proof of Theorem 1.1. Some parts of the proof follow similar to [1]. New ideas are needed to deal with general, non-radial solutions, the attractive case θ < 0 and non-uniqueness of solutions.
As in [1], we look for solutions f of the form In the case of a repulsive point charge, we can show Q ≥ 0. Therefore, the function F in (1.6) is completely determined by the boundary condition f 0 in (1.3). In the presence of an attractive test charge, Q also attains negative values and the function F is not uniquely determined by f 0 . Physically, this can be explained by the presence of electrons trapped on orbits around the attractive background charge. This phonomenon is also discussed in [10].
In order to prove that infinitely many solutions exist for θ < 0 in (1.5), we construct a family of admissible extensions F of F 0 The set of admissible extensions F is characterized by the function We make the following assumption on the distribution f 0 of the plasma at |x| → ∞, which is also assumed in [1].

Assumption 1.2 (Velocity distribution at infinity)
The boundary condition f 0 can be repre- (1.9) (iii) stability condition: F 0 satisfies: For future reference, we define

Extension to the negative half-line
We show the existence of solutions if the function g defined in (1.7) satisfies the following four properties: (i) normalization: (ii) differentiability and monotonicity: g ∈ C 2 (R) and (iii) sub-differential at zero: Before we establish the existence of solutions under the conditions (2.1)-(2.4), we demonstrate that it is possible to extend the function F 0 to the negative half-line such that these conditions are met. This is the content of the following lemma. 1 2 ) and c β > 0 consider the functioñ and let F β,c β be the extension of F 0 byF β,c β . Here r = 1 + |r | 2 is the Japanese bracket. Then F β,c β ∈ C 1 b (R; R + ) and for c β > 0 large enough, the function g β,c β defined by (1.7) satisfies the conditions (2.1), (2.2), (2.3) and (2.4) with α = 3 2 − β.
Step 2. The normalization condition (2.1) follows since F is an extension of F 0 and F 0 satisfies (1.8). Furthermore, the derivative of g β,c β can be represented as Since F β,c β is positive, (2.2) follows.
Step 3. For the proof of (2.3), we first remark that the condition holds for r ≥ 0. As observed in [1] this follows since F 0 satisfies (1.10), and therefore g β,c β is convex on the positive half-line.
Step 4. Since g β,c β ∈ C 2 (R), the condition (2.4) holds locally. It remains to check the asymptotics for r → −∞. We again use (2.5). From the decay condition on F 0 (1.9) we easily obtain Similarly, the estimate follows for the integral term in (2.5) by a straightforward computation. (3.1)

Existence of solutions
Then we have

Moreover, B is a continuous operator B
Proof The non-negativity of B follows from the subdifferential condition (2.3) on g. For the upper bound, we first recall that g ∈ C 2 , and σ is defined by (1.11). Since α < 2 this yields For |P| → ∞ the inequality follows from the growth condition (2.4).

Continuity of the operator
and finishes the proof.

Proof
Step 1. Recall σ > 0 defined in (1.11), and let σ be the fundamental solution to .

2), B[S] satisfies
By the Green's function property of σ , S is a weak solution to Let us further introduce the functions H 1 , H by where C 0 > 0 is the constant appearing in (3.2) and . (3.6) Using that φ defined in (3.6) is a Riesz potential, we find that Step 2. We pick the coefficient q 0 as q 0 = 2α, and define the operator We claim that K is a continuous compact operator. Continuity follows from the continuity of B shown in Lemma 3.1. It remains to prove that the image of K is precompact. To this end, consider a sequence R k ∈ L q 0 (R 3 ), and G k := K [R k ]. Then for some M > 0 we have

8)
with σ as introduced in (3.3). We observe that due to (3.7) the upper bound H satisfies Compactness now follows from the Riesz criterion for precompactness in L p (R 3 ) since For the first and second condition, we use (3.8) and H ∈ L q 0 (R 3 ), the last line follows from (3.9).
Step 3. We infer from Step 2 the existence of a non-negative solution R ∈ W 1,q 0 (R 3 )∩L 2 (R 3 ) to the equation This follows from Schaefer's fixed point theorem, since K maps into a bounded set in L q 0 (R 3 ), so in particular there exists M > 0 such that for P ∈ L q 0 (R 3 ) and λ ∈ [0, 1] with P = λK (P) we have Together with Step 1 and Schaefer's fixed point theorem, this allows us to conclude the existence of a fixed point to the mapping K , i.e. R satisfying (3.11). Non-negativity of R follows by construction of K .
Step 4. There exists R ∈ L q 0 (R 3 ) such that R = σ * B(R + S). (3.12) Notice that (3.12) follows from (3.11) if we can show B[R + S] ≤ H . To this end, let R ∈ L q 0 (R 3 ) be any solution to (3.11). Then R is a weak solution to Unpacking the definition of B (cf. (3.1)) yields Since R ≥ 0 and g is monotone decreasing due to (2.2), we infer and therefore R ≤ φ * B[S] = H 1 . Courtesy of (3.2) we conclude B[R + S] ≤ H . From R satisfying (3.12) we also infer R ∈ W 1,1 (R 3 ).
Step 5. We define Q by However, we have g 1 (r ) > g 2 (r ) on the negative half-axis r < 0 since β 1 < β 2 . Since Q 1 takes negative values on a set of positive measure we reach a contradiction.
Step 3. Finally | f 1 − f 2 | = 0 ∈ L 1 loc (R 3 ). Assume the contrary, and let A ⊂ R 3 the set on which Q 1 < 0. By Step 1, A has positive measure. Then we estimate On the other hand, for any y < 0 the integrand is positive. This leads to a contradiction to the assumption.
Acknowledgements R.W. acknowledges financial support from the Austrian Science Fund (FWF) project F65. Furthermore, R.W. would like to thank the Isaac Newton Institute for Mathematical Sciences for support and hospitality during the programme "Frontiers in kinetic theory: connecting microscopic to macroscopic scales-KineCon 2022" when work on this paper was undertaken. This work was supported by EPSRC Grant number EP/R014604/1.

Funding Open access funding provided by Austrian Science Fund (FWF).
Data Availability Statement Data sharing is not applicable to this article as no new data were created or analysed in this study.

Conflict of interest
The author has no conflicts to disclose.
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