Limit profiles for singularly perturbed Choquard equations with local repulsion

We study Choquard type equation of the form where N≥3\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$N\ge 3$$\end{document}, Iα\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$I_\alpha $$\end{document} is the Riesz potential with α∈(0,N)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\alpha \in (0,N)$$\end{document}, p>1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$p>1$$\end{document}, q>2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$q>2$$\end{document} and ε≥0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varepsilon \ge 0$$\end{document}. Equations of this type describe collective behaviour of self-interacting many-body systems. The nonlocal nonlinear term represents long-range attraction while the local nonlinear term represents short-range repulsion. In the first part of the paper for a nearly optimal range of parameters we prove the existence and study regularity and qualitative properties of positive groundstates of (P0)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(P_0)$$\end{document} and of (Pε)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(P_\varepsilon )$$\end{document} with ε>0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varepsilon >0$$\end{document}. We also study the existence of a compactly supported groundstate for an integral Thomas–Fermi type equation associated to (Pε)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(P_{\varepsilon })$$\end{document}. In the second part of the paper, for ε→0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varepsilon \rightarrow 0$$\end{document} we identify six different asymptotic regimes and provide a characterisation of the limit profiles of the groundstates of (Pε)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(P_\varepsilon )$$\end{document} in each of the regimes. We also outline three different asymptotic regimes in the case ε→∞\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varepsilon \rightarrow \infty $$\end{document}. In one of the asymptotic regimes positive groundstates of (Pε)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(P_\varepsilon )$$\end{document} converge to a compactly supported Thomas–Fermi limit profile. This is a new and purely nonlocal phenomenon that can not be observed in the local prototype case of (Pε)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(P_\varepsilon )$$\end{document} with α=0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\alpha =0$$\end{document}. In particular, this provides a justification for the Thomas–Fermi approximation in astrophysical models of self-gravitating Bose–Einstein condensate.


Background
We are concerned with the asymptotic properties of positive groundstate solutions of the Choquard type equation If u ε is a solution of (P ε ) with N = 3, α = 2, p = 2 and q = 4 then ψ(t, x) := e iεt u ε (x) is a standing wave solution of the time-dependent equation i∂ t ψ = − ψ − (I 2 * |ψ| 2 )ψ + |ψ| 2 ψ, (t, x) ∈ R × R 3 . (1.1) Equation of this form models, in particular, self-gravitating Bose-Einstein condensates with repulsive short-range interactions, which describe astrophysical objects such as boson stars and, presumably, dark matter galactic halos. In this context, (1.1) was introduced and studied under the name of Gross-Pitaevskii-Poisson equation in [8,16,53], see a survey paper [17]. More generally, Eq. (P ε ) can be seen as a stationary NLS with an attractive long range interaction, represented by the nonlocal term, coupled with a repulsive short range interaction, represented by the local nonlinearity. While for the most of the relevant physical applications p = 2, the values p = 2 may appear in several relativistic models of the density functional theory [2][3][4].
In this work we are specifically interested in the case where ε > 0 is a small (or large) parameter and all other parameters are fixed. Our main goal is to understand the behaviour of groundstate solutions of (P ε ) when ε → 0. We also discuss the case ε → ∞, which is to some extent dual to ε → 0. The local prototype of (P ε ) and a formal limit of (P ε ) as α → 0 is the equation It is well-known that this equation admits a unique positive solution in H 1 (R N ) for any 1 < 2 p < q < ∞ provided that ε > 0 is sufficiently small, and has no finite energy solutions for large ε. This result goes back to Strauss [51,Example 2] and Berestycki and Lions [6,Example 2] (see [41, Theorem A] for a precise existence statement and further references). A complete characterization of all possible asymptotic regimes in (1.2) as ε → 0 was obtained in [41], see also earlier work [46]. Essentially, three different limit regimes were identified in [41], depending on whether p is less, equal or bigger than the critical Sobolev exponent p * = N N −2 . Recently, (1.2) had been revisited in [28] where nondegeneracy of ground-sates and sharp asymptotics of the L 2 -norm of the ground states as ε → 0 had been described in connection with the uniqueness conjecture in the L 2 -constraint minimization problem associated to (1.2). See also [39], where the same problem is studied with the opposite sign of the |u| q−2 u-term.

Existence and properties of groundstates for (P " )
We are not aware of a systematic study of ground-sates of Choquard Eq. (P ε ). First existence results seem to appear in [45] in the case N = 3, α = 2, p = 2. See also [7,23,27,29] and references therein for further results which however do not cover the optimal ranges of parameters. The planar case with the logarithmic convolution kernel was studied in [19] but since the kernel is sign-changing this requires different techniques. Near optimal existence results for the Choquard equation of type (P ε ) with an attractive local perturbation (the opposite sign of the local nonlinear term) were recently obtained in [30,31].
Our first goal in this work is to establish the existence of ground-sate solutions of Choquard Eq. (P ε ) for an optimal range of parameters. By a groundstate solution of (P ε ) we understand a weak solution u ∈ H 1 (R N ) ∩ L q (R N ) which has a minimal energy amongst all nontrivial finite energy solutions of (P ε ). Remarkably, and in contrast with its local prototype (1.2), we prove that ground states for (P ε ) exist for every ε > 0. We also establish some qualitative properties of the solutions (P ε ) such as regularity and decay at infinity. These properties are similar to the standard Choquard Eq. (C ). Note that we do not study the uniqueness or non-degeneracy of the groundstates of (P ε ) and we are not aware of any even partial results in this direction. We believe this is a very difficult open problem. Our results do not rely and do not require the uniqueness or non-degeneracy. Essential tools to control the nonlocal term in I ε are the Hardy-Littlewood-Sobolev (HLS) inequality which is valid for any p ≥ 1 (and C α is independent of p); and the Sobolev inequality where 2 * = 2N N −2 is the critical Sobolev exponent and D 1 (R N ) denotes the homogeneous Sobolev space with the norm u D 1 (R N ) = ∇u L 2 . The values of the sharp constants S * > 0 and C α > 0 are known explicitly [34,35]. HLS and Sobolev inequalities can be used to control the nonlocal term in the two cases: The two cases have non-empty intersection but this is not significant for us at this moment.
In each of these two cases, I ε : H 1 (R N ) ∩ L q (R N ) → R is well defined and critical points of I ε are solutions of (P ε ). Our main existence result for (P ε ) is the following. Theorem 1.1 Let N +α N < p < N +α N −2 and q > 2, or p ≥ N +α N −2 and q > 2N p N +α . Then for each ε > 0, Eq. (P ε ) admits a positive spherically symmetric ground state solution u ε ∈ H 1 ∩ L 1 ∩ C 2 (R N ) that is a monotone decreasing function of |x|. Moreover, there exists C ε > 0 such that The existence range of Theorem 1.1 is optimal. This follows from the Pohožaev identity argument, see Corollary 4.1. We emphasise that no upper restrictions on p and q are needed and in particular, q could take Sobolev supercritical values, i.e. q > 2 * (see Fig. 1). The decay rates of ground states at infinity are exactly the same as in the standard Choquard case, compare Theorem 2.4 below or [43,Theorem 4]. For a discussion of the implicit exponential decay in the case p = 2 we refer to [43, pp. 157-158] or [44, Section 6.1]. Our main goal in this paper is to understand and classify the asymptotic profiles as ε → 0 and ε → ∞ of the groundstates u ε , constructed in Theorem 1.1. Remarkably, our study uncovers a novel and rather complicated limit structure of the problem, with six different limit equations (see Fig. 1) as ε → 0: • Formal limit when the family of ground states u ε converges to a groundstate of the formal limit equation − u − (I α * |u| p )|u| p−2 u + |u| q−2 u = 0 in R N .
The existence and qualitative properties of groundstate for (P 0 ) for the optimal range of parameters is new and is studied in Sect. 5, see Theorem 2.1. The convergence of the groundstates to the limit profile is proved in Theorem 2.2. converges to a groundstate of the standard Choquard equation which was studied in [43]. The convergence is proved in Theorem 2.5. (TF) The existence and qualitative properties of groundstate for (TF) for p = 2 will be studied in the forthcoming work [26]. In this paper we consider only the case p = 2 which is well known in the literature when α = 2 [1,5,36] and was studied recently in [11,12] for the general α ∈ (0, N ), yet for the range of powers q which is incompatible with our assumptions. In Theorem 2.6 we prove the existence and some qualitative properties of a groundstate for (TF) with p = 2 for the optimal range q > 4N N +α . This extends some of the existence results in [11,12]. The convergence of v ε to a groundstate of (TF) is proved in Theorem 2.7 for p = 2 and α = 2 (the general case p = 2 and α = 2 will be studied in [26]). Remarkably, for p = 2 the limit groundstates of (TF) are compactly supported, so the rescaled groundstates v ε develops a steep "corner layer" as ε → 0! • Critical Choquard regime, when the family of ground states u ε converges after an implicit rescaling to a groundstate of the critical Choquard equation A detailed characterisation of the ground states of (C H L ) was recently obtained in [18,21]. In Theorem 2.8 we derive a sharp two-sided asymptotic characterisation of the rescaling λ ε , following the ideas developed in the local case in [41]. • Self-similar regime q = 2 2 p+α 2+α , when ground states u ε are obtained as rescalings of the groundstate u 1 , i.e.
• the Critical Thomas-Fermi regime, when the family of ground states u ε converges after an implicit rescaling to a groundstate of the critical Thomas-Fermi equation Groundstates of this equation correspond to the minimizers of the Hardy-Littlewood-Sobolev inequality and completely characterised by Lieb in [34]. In Theorem 2.9 we derive a two-sided asymptotic characterisation of the rescaling λ ε .
Self-similar, Thomas-Fermi and critical Thomas-Fermi regimes are specific to the nonlocal case only. When α = 0 they all "collapse" into the case p = q, which is degenerate for the local prototype Eq. (1.2). Three other regimes could be traced back to the local Eq. (1.2) studied in [41].
When ε → ∞ the limit structure is simpler. Only Choquard, Thomas-Fermi and selfsimilar regimes are relevant (see Fig. 2) and there are no critical regimes. In particular, the Thomas-Fermi limit with ε → ∞ appears in the study of the stationary Gross-Pitaevskii-Poisson Eq. (1.1), see Remark 3.1.
The precise statements of our results for ε → 0 are given in Sect. 2. In Sect. 3 we outline the results for ε → ∞ and discuss the connection with astrophysical models of self-gravitating Bose-Einstein condensate. In Sect. 4 we prove Theorem 1.1. In Sects. 5 and 6 we establish the existence and basic properties of groundstates for the "zero-mass" limit equation (P 0 ) and for the Thomas-Fermi Eq. (TF). In Sects. 7 and 8 we study the asymptotic profiles of the groundstates of (P ε ) in the non-critical and critical regimes respectively. Finally, in the Appendix we discuss a contraction inequality which was communicated to us by Augusto Ponce and which we used as a key tool in several regularity proofs.

Asymptotic notations
For real valued functions f (t), g(t) ≥ 0 defined on a subset of R + , we write: g(t) = 0. As usual, B R = {x ∈ R N : |x| < R} and C, c, c 1 etc., denote generic positive constants.

Asymptotic profiles as " → 0
Our main goal in this work is to understand the asymptotic behaviour of the constructed in Theorem 1.1 groundstate solutions u ε of (P ε ) in the limits ε → 0 and ε → ∞.

Formal limit (P 0 )
Loosely speaking, the elliptic regularity implies that u ε converges as ε → 0 to a nonnegative radial solution of the formal limit equation However, (P 0 ) becomes a meaningful limit equation for (P ε ) only in the situation when (P 0 ) admits a nontrivial nonnegative solution. Otherwise the information that u ε converges to zero (trivial solution of (P 0 )) does not reveal any information about the limit profile of u ε . We prove in this work the following existence result for (P 0 ). Theorem 2.1 Let N +α N < p < N +α N −2 and 2 < q < 2N p N +α , or p > N +α N −2 and q > 2N p N +α . Then Eq. (P 0 ) admits a positive spherically symmetric groundstate solution u 0 ∈ D 1 ∩L q ∩C 2 (R N ) which is a monotone decreasing function of |x|. Moreover, The restrictions on p and q in the existence part of the theorem ensures that the energy I 0 which corresponds to (P 0 ) is well-defined on D 1 (R N ) ∩ L q (R N ), see (5.1) below. A Pohožaev identity argument (see Remark 5.2) confirms that the existence range in Theorem 2.1 is optimal, with the exception of the "double critical" point p = N +α N −2 and q = 2N p N +α = 2 * on the ( p, q)-plane (see Remark 5.3).
Note that The upper bound on u 0 in the case N +α N −2 < 2 and N +α N −2 < p ≤ 2 3 1 + N +α N −2 remains open. We conjecture that our restriction on p for the upper decay estimate is technical and that u 0 ∼ |x| −(N −2) as |x| → ∞ for all p > N +α N −2 . Observe that the energy I 0 is well posed in the space , small perturbation arguments in the spirit of the Lyapunov-Schmidt reduction are not directly applicable to the family I ε in the limit ε → 0. Using direct variational analysis based on the comparison of the groundstate energy levels for two problems, we establish the following result.
and q > 2N p N +α . Then as ε → 0, the family of ground states u ε of (P ε ) converges in D 1 (R N ) and L q (R N ) to a positive spherically symmetric ground state solution u 0 ∈ D 1 ∩ L q (R N ) of the formal limit Eq.
is related only to the upper decay bound on u 0 in Theorem 2.1, i.e. we could establish the convergence of u ε to u 0 for p > N +α N −2 as soon as we know that u 0 ∼ |x| −(N −2) as |x| → ∞.

Rescaled limits
When N +α N < p ≤ N +α N −2 and q ≥ 2N p N +α the formal limit problem (P 0 ) has no nontrivial sufficiently regular finite energy solutions (see Remark 5.3). As a consequence, u ε converges uniformly on compact sets to zero. We are going to show that in these regimes u ε converges to a positive solution of a limit equation after a rescaling v(x) := ε s u(ε t x), for specific choices of s, t ∈ R. The rescaling transforms (P ε ) into the equation (2.1) If q = 2 2 p+α 2+α then there are three natural possibilities to choose s and t, each achieving the balance of three different terms in (2.1). Note that the choice ε −s−2t = ε −(2 p−1)s+αt = ε −(q−1)s leads to s = t = 0, when (2.1) reduces to the original equation (P ε ).
I. First rescaling. The choice ε −s−2t = ε 1−s = ε −(2 p−1)s+αt leads to s = − 2+α 4( p−1) , t = −1/2 and rescaled equation = 0, and we obtain as a formal limit the Choquard equation = 0 and we obtain as a formal limit the = 0, and we obtain as a formal limit the nonlinear local equation Such equation has no nonzero finite energy solutions and we rule out the third rescaling as trivial.
2.3 Self-similar regime q = 2 2p+2 +Į n this special case the choice s = − 2+α 4( p−1) and t = −1/2 leads to the balance of all four and any solution of (P ε ) is a rescaling of a solution of (P 1 ). For completeness, we include this obvious observation in the following statement. Theorem 2.3 (Self-similar regime) Let N +α N < p < N +α N −2 and q = 2 2 p+α 2+α . Then for any ε > 0, where u 1 is a ground state solution of (P 1 ).

First rescaling: Choquard limit
The following result describes the existence region and some qualitative properties of the groundstates of (C ).
is a monotone decreasing function of |x|. Moreover, there exists C > 0 such that The existence interval in this theorem is sharp, in the sense that (C ) does not have finite energy solutions for p / ∈ N +α N , N +α N −2 . The uniqueness of the ground state solution is known only for N = 3, p = 2 and α = 2 [33] and several other special cases [38,49,54].
In this paper we prove that after the 1st rescaling, groundstates of (P ε ) converge to the groundstates of the Choquard Eq. (C ), as soon as (C ) admits a nontrivial groundstate. Theorem 2.5 (Choquard limit) Let N +α N < p < N +α N −2 and q > 2 2 p+α 2+α . As ε → 0, the rescaled family of ground states converges in D 1 (R N ) and L q (R N ) to a positive spherically symmetric ground state solution v 0 ∈ D 1 ∩ L q (R N ) of the Choquard Eq. (C ).

2nd rescaling: Thomas-Fermi limit for p = 2
In this paper we consider the 2nd rescaling regime only in the case p = 2. The general case p = 2 is studied in the forthcoming work [26]. When p = 2 the formal limit equation for (P ε ) in the 2nd rescaling is the Thomas-Fermi type integral equation (TF) One of the possible ways to write the variational problem that leads to (TF) after a rescaling is By a groundstate of (TF) we understand a rescaling of a nonnegative minimizer for s TF that satisfies the limit Eq. (TF).
To study nonnegative minimizers of the s TF it is convenient to substitute for an equivalent representation For m > m c := 2 − α/N it is not difficult to see that, after a rescaling, minimizers for s TF are in the one-to-one correspondence with the minimizers of The existence and qualitative properties of minimizers for σ TF in the case N = 3, α = 2 and for m > 4/3 is classical and goes back to [1,36]. The case N ≥ 2, α ∈ (0, N ) and m > m c it is a recent study by Carrillo et al. [12]. If m < m c then σ TF = −∞ by scaling, while m = m c is the L 1 -critical exponent for σ TF (this case is studied in [11]). Note that m c > 4+α 2+α so in the 2nd rescaling regime we always have σ TF = −∞ when ε → 0! In the next theorem we show that, unlike for σ TF , minimization for s TF is possible for all m > 2N N +α . The existence and qualitative properties of the minimizers are summarised below.
Only the existence part of the theorem requires a proof. The Euler-Lagrange equation, regularity and qualitative properties of the minimizers could be obtained by adaptations of the arguments developed for m > m c in [11,12]. We outline the arguments in Sect. 7.3.
In the case m ≥ m c the uniqueness of the minimizer for σ TF was recently proved in [10] for α < 2, see also [13] for α = 2 and a survey of earlier results in this direction. For the full range m > 2N N +α and for α < 2 the uniqueness of a bounded radially nonincreasing solution for the Euler-Lagrange Eq. (2.5) (and hence the uniqueness of the minimizer ρ * for s TF ) is the recent result in [14, Theorem 1.1 and Proposition 5.4]. For α = 2 the same follows from [24,Lemma 5]. For α ∈ (2, N ) the uniqueness of the minimizer for s TF or for σ TF seems to be open at present.
Next we prove that in the special case α = 2, groundstates of (P ε ) converge to a groundstate of the Thomas-Fermi Eq. (TF). Theorem 2.7 (Thomas-Fermi limit for α = 2) Let N ≤ 5, p = 2, α = 2 and 4N N +2 < q < 3. As ε → 0, the rescaled family of ground states converges in L 2 (R N ) and L q (R N ) to a nonnegative spherically symmetric compactly supported ground state solution v 0 ∈ L 2 ∩ L q (R N ) of the Thomas-Fermi Eq. (TF).

Remark 2.1
While the uniqueness of the Thomas-Fermi groundstate v 0 for α > 2 is generally open, it is clear from the proof of Theorem 2.6 that every ground state of (TF) must have the same regularity and compact support properties as stated in Theorem 2.6. In particular, v ε always exhibits as ε → 0 a "corner layer" near the boundary of the support of the limit groundstate of (TF).

−2
When p = N +α N −2 and q > 2N p N +α = 2 * , neither (C ) nor (P 0 ) have nontrivial solutions. We prove that in this case the limit equation for (P ε ) is given by the critical Choquard equation A variational problem that leads to (C H L ) can be written as It is known [21, Lemma 1.1] that where S * is the Sobolev constant in (1.4) and C α is the Hardy-Littlewood-Sobolv constant in (1.3) (with p = N +α N −2 ). By a groundstate of (C H L ) we understand a rescaling of a positive minimizer for S H L that satisfies Eq. (C H L ). Denote a groundstate of the Emden-Fowler equation − U * = U 2 * −1 * in R N . Then (see e.g. [21, Lemma 1.1]) all radial groundstates of (C H L ) are given by the function (2.10) and the family of its rescalings We prove that in the critical Choquard regime the family of ground states u ε converge in a suitable sense to V after an implicit rescaling λ ε . Note that V ∈ L 2 (R N ) only if N ≥ 5 and hence the lower dimensions should be handled differently, as the L 2 -norm of u ε must blow up when N = 3, 4. Our principal result is a sharp two-sided asymptotic estimate on the rescaling λ ε as ε → 0. Similar result in the local case α = 0 was first observed in [46] and then rigorously established in [41] Theorem 2.8 (Critical Choquard limit) Let p = N +α N −2 and q > 2N p N +2 = 2 * . There exists a rescaling λ ε : (0, ∞) → (0, ∞) such that as ε → 0, the rescaled family of ground states (2.12)

Critical Thomas-Fermi regime
We show that in this case the limit equation for (P ε ) is given by the critical Thomas-Fermi type equation By a groundstate of (TF * ) we understand a positive solution of (TF * ) which is a rescaling of a nonnegative minimizer for the Hardy-Littlewood-Sobolev minimization problem where C α is the optimal constant in (1.3). It is known [34,Theorem 4.3] that all radial groundstates of (TF * ) are given by (2.14) for a constant σ α,N > 0, and the family of rescalings We prove that, similarly to the critical Choquard regime, in the critical Thomas-Fermi regime the family of ground states u ε converge in a suitable sense to U after an implicit rescaling λ ε .
Our main result in the critical Thomas-Fermi regime is the following. (2.17)

Remark 2.3
We expect that the upper asymptotic bounds (2.17) with p ≥ 2(3+α) 3 could be refined to match the lower bounds, but this remains open at the moment.

Asymptotic profiles as " → ∞ and Gross-Pitaevskii-Poisson model
The behaviour of ground states u ε as ε → ∞ is less complex than in the case ε → 0. Only the 1st and the 2nd rescalings are meaningful, separated by the q = 2 2 p+α 2+α line, however the limit equations "switch" compared to the case ε → 0. There are no critical regimes.

Fig. 2 Three limit regimes for
The proofs of Theorems 3.2 and 3.1 are very similar to the proofs of Theorems 2.7 and 2.5. We only note that the proof on Theorem 3.2 will involve the estimate (7.25) with q ≥ 4 when the the right hand side of (7.25) blows-up. However the rate of the blow-up in (7.25) isn't strong enough and all quantities involved in the proof remain under control. We leave the details to the interested readers.

Remark 3.1
Note that the nature of rescaling (3.2) changes when q = 4: for q > 4 the mass of u ε concentrates near the origin, while for q < 4 it "escapes" to infinity. In particular, the stationary version of the Gross-Pitaevskii-Poisson Eq. (1.1) (q = 4, N = 3, α = 2) fits into the Thomas-Fermi regime as ε → ∞. The rescaling (3.2) in this case takes the , or we can say that u ε concentrates towards the compactly supported v 0 . This is precisely the phenomenon which was already observed in [8,53], where the radius of the support of v 0 has the meaning of the radius of self-gravitating Bose-Einstein condensate, see [16]. The limit minimization problem s TF in the Gross-Pitaevskii-Poisson Eq. (1.1) case becomes and the Euler-Lagrange Eq. (2.5) in this case is linear inside the support of ρ: To find explicitly the solution of (3.3) constructed in Theorem 2.6, we use the sin(|x|) |x| ansatz as in [8,53], [15, p. 92].
For λ > 0 and |x| ≤ π/λ consider the family Then − ρ λ = λ 2 ρ λ in B π/λ , and Optimising in λ > 0, we find that and the minimum occurs at the optimal λ * = π 2 2 1/5 . Taking into account the uniqueness of the spherically symmetric nonincreasing minimizer for s TF in the case α = 2, which follows from [13, Theorem 1.2] (see also [14,Lemma 5.2]), the function is the unique spherically symmetric nonincreasing minimizer for s TF and a solution of (3.3). The solution of the limit Thomas-Fermi Eq. (TF), which is written in this case as is given by the rescaled function in (2.6), This is (up to the physical constants) the Thomas-Fermi approximation solution for selfgravitating BEC observed in [8,16,53] and the support radius R 0 = π is the approximate radius of the BEC star. Our Theorem 3.2 provides a rigorous justification for the convergence of the Thomas-Fermi approximation.

Variational setup
It is a standard consequence of Sobolev and Hardy-Littlewood-Sobolev (HLS) inequalities [35, Theorems 4.3 and 8.3] that for N +α In this case an additional assumption q > 2N p N +α ensures the control of the nonlocal term by the L q and L 2 -norm via the HLS inequality and interpolation, i.e.
for a θ ∈ (0, 1). As a consequence, for p > N +α N −2 and q > 2N p N +α the energy I ε is well-defined on the space Clearly, H q endowed with the norm It is easy to check that I ε ∈ C 1 (H q , R) and the problem (P ε ) is variationaly well-posed, in the sense that weak solutions u ∈ H q of (P ε ) are critical points of I ε , i.e.
for all ϕ ∈ H q . In particular, weak solutions u ∈ H q of (P ε ) satisfy the Nehari identity It is standard to see that under minor regularity assumptions weak solutions of (P ε ) also satisfy the Pohožaev identity.
Proof The proof is an adaptation of [43, Proposition 3.1], we omit the details.
As a consequence, we conclude that the existence range stated in Theorem 1.1 is optimal.
Proof Follows from Pohožaev and Nehari identities.

Apriori regularity and decay at infinity
We show that all weak nonnegative solutions of (P ε ) are in fact bounded classical solutions with an L 1 -decay at infinity. We first prove a partial results which relies on the maximum principle for the Laplacian.
The proof of the next statement in the case p < N +α N −2 is an adaptation of the iteration arguments in [43,Proposition 4.1]. We only outline the main steps of the proof. The case p ≥ N +α N −2 is new and relies heavily on the contraction inequality (A.3), which is discussed in the appendix.
Proof Note that u ≥ 0 weakly satisfies the inequality Then, by the HLS and Hölder inequalities, Since p > N +α N , we start the s n -iteration with α , as in [43]. Then we achieve s n+1 ≥ 1 after a finite number of steps. This implies u ∈ L 1 (R N ).
Proof Assume that q ≤ N p α , otherwise we conclude by Lemma 4.1. We consider separately the cases p < N +α N −2 and p ≥ N +α N −2 , which use different structures within the Eq. (P ε ).
Then, by the HLS and Hölder inequalities, α+2 , as in [43]. Then we achieve s n+1 > N p α after a finite number of steps.
Then, by the HLS and Hölder inequalities, and by the contraction inequality A.
We start the s n -iteration with s 0 = q. If q ≥ 2 p we achieve s n+1 > N p α after a finite number of steps. If q < 2 p we note that since p ≥ N +α N −2 , we have s 0 = q > 2N p N +α . Then we again achieve s n+1 > N p α after a finite number of steps. (Or if s n+1 = N p α we readjust s 0 .) Step 3. u ∈ W 2,r (R N ) for every r > 1 and u ∈ C 2 (R N ).
. Then the conclusion follows by the standard Schauder estimates, see [43, p. 168] for details.

Proposition 4.3 (Decay asymptotics) Let N +α
be a radially symmetric solution of (P ε ). Then Moreover, there exists C ε > 0 such that Proof To simplify the notation, we drop the subscript ε for u ε in this proof. Let u ∈ L 1 ∩ C 2 (R N ) be a positive radially symmetric solution of (P ε ). By the Strauss' ). In particular, this implies (4.11).
Case p ≥ 2. The derivation of the decay asymptotic of u in the case(s) p ≥ 2 requires minimal modifications of the proofs of [43, Propositions 6.3, 6.5]. Indeed, (4.11) implies As in [43, Propositions 6.3] we conclude that The initial rough upper bound (4.13) implies that the term u q−2 (x) in the linearisation potential W ε (x) has an exponential decay and is negligible in the subsequent asymptotic analysis of Propositions 6.3 and 6.5 in [43]. We omit the details. Case p < 2. This proof is an adaptation of [43, Propositions 6.6].
To derive the upper bound, we note that by Young's inequality, By (4.12), we have Therefore, u satisfies the inequality We now apply [43, Lemma 6.7] twice and use the linearity of the operator − + ε(2 − p) to obtain To deduce the lower bound, note that by the chain rule, Since p ∈ (1, 2) and q > 2, by the equation satisfied by u and by (4.12) and (4.14), for some c ε > 0 we have We apply now [43, Lemma 6.7] to deduce By the comparison principle, (4.15) and the assertion follows from the combination of (4.15) and (4.14).

Proof of the existence
Throughout this section, we assume that either N +α N < p < N +α N −2 and q > 2 or p ≥ N +α N −2 and q > 2N p N +α . We construct a groundstate of (P ε ) by minimising over the Pohožaev manifold of (P ε ). Similar approach for Choquard's equations with different classes of nonlinearities was recently used in [25,32]. Set Clearly, there exists a unique t u > 0 such that f u (t u ) = max{ f u (t) : t > 0} and f u (t u )t u = 0, which means that u(x/t u ) ∈ P ε . Therefore P ε = ∅.
where C > 0 is independent of u ∈ H q .

Lemma 4.2
Assume that either N +α N < p < N +α N −2 and q > 2, or p ≥ N +α N −2 and q > 2N p N +α . Then there exists C > 0 such that for all u ∈ H q , To clarify the last inequality, note that M(u t ) ≤ t N M(u) = 1. Then using (4.17), we obtain Similarly as before, we conclude that which completes the proof.
To find a groundstate solution of (P ε ), we prove the existence of a spherically symmetric nontrivial nonegative minimizer of the minimization problem (4.20) and then show that P ε is a natural constraint for I ε , i.e. the minimizer u 0 ∈ P ε satisfies I ε (u 0 ) = 0. Such approach for the local equations goes back at least to [50] in the local case and to [48] in the case of nonlocal problems. We divide the proof of the existence of the groundstate into several steps.
Proof Indeed, for u ∈ P ε , we have, by using the HLS and Sobolev inequalities, which means that there exists C > 0 such that M(u) ≥ C for all u ∈ P ε .
Proof Since c ε is well defined, there exists a sequence {u n } ⊂ P ε such that I ε (u n ) → c ε . It follows from (4.21) that both { ∇u n 2 2 } and { R N (I α * |u n | p )|u n | p dx} are bounded. Note that P ε (u n ) = 0. Then we see that { u n 2 2 } and { u n q q } are bounded, and therefore {u n } is bounded in H q .
Let u * n be the Schwartz spherical rearrangement of |u n |. Then u * n ∈ H q,rad , the subspace of H q which consists of all spherically symmetric functions in H q , and cf. [35,Section 3]. For each u * n , there exists a unique t n ∈ (0, 1) such that v n := u * n ( x t n ) ∈ P ε . Therefore we obtain that Since U ∈ L s (R N ) for s ∈ (2, q * ), by the Lebesgue dominated convergence we conclude that for s ∈ (2, q * ), Note that q * > 2N p N +α and hence we can always choose s > 2N p N +α > p such that {v n } is bounded in L s (R N ). Then by the nonlocal Brezis-Lieb Lemma with high local integrability [40,Proposition 4.7] we conclude that This means that v = 0, since by Lemma 4.2 the sequence {M(v n )} has a positive lower bound. Then there exists a unique t 0 > 0 such that v( x t 0 ) ∈ P ε . By the weakly lower semi-continuity of the norm, we see that We conclude this step by taking u 0 (x) := v( x t 0 ).
Proof Arguing by contradiction, we assume that P ε (u 0 ) = 0. Then u 0 is a weak solution of the following equation, This, together with P ε (u 0 ) = 0, implies that which contradict u 0 = 0.

Existence and properties of groundstates for (P 0 )
In this section we study the existence and some qualitative properties of groundstate solutions for the equation where N ≥ 3, α ∈ (0, N ), p > 1 and q > 2. Equation (P 0 ) appears as a formal limit of (P ε ) with ε = 0. The natural domain for the formal limit energy I 0 which corresponds to (P 0 ) is the space Clearly, D q endowed with the norm can not be considered as a small perturbation of (P 0 ), since the domain of I 0 is strictly bigger than the domain of I ε . If p ∈ ( N +α N , N +α N −2 ) and q ∈ (2, 2N p N +α ) or p ≥ N +α N −2 and q > 2N p N +α then the HLS, Sobolev and interpolation inequalities ensure the control of the nonlocal term by the L q and D 1 -norms, with a θ ∈ (0, 1). Then it is standard to check that I 0 ∈ C 1 (D q , R) and the problem (P 0 ) is variationaly well-posed, in the sense that weak solutions u ∈ D q of (P ε ) are critical points of I ε , i.e.
for all ϕ ∈ D q . In particular, weak solutions u ∈ D q of (P 0 ) satisfy the Nehari identity We are going to prove the existence of a ground state of (P 0 ) by minimizing over the Pohožaev manifold P 0 . This requires apriori additional regularity and some decay properties of the weak solutions. Lemma 5.1 (L 1 -decay) Let N +α N < p < N +α N −2 and 2 < q < 2N p N +α . If 0 ≤ u ∈ D q is a weak solution of (P 0 ) then u ∈ L 1 (R N ).
Proof Note that u ≥ 0 weakly satisfies the inequality (4.9). Then, by the HLS and Hölder inequalities, and by the contraction inequality A.3, u ∈ L s n (R N ) with 0 < 2 p−1 We start the s n -iteration with s 0 = q < 2N p N +α . Then we achieve s n+1 ≤ 1 after a finite number of steps. Proposition 5.1 (Regularity) Let N +α N < p < N +α N −2 and 2 < q < 2N p N +α , or p ≥ N +α N −2 and q > 2N p N +α . If 0 ≤ u ∈ D q is a nontrivial weak solution of (P 0 ) then u ∈ C 2 (R N ) and u(x) > 0 for all x ∈ R N .
Proof Since 0 ≤ u ∈ D q we know that u ∈ L q ∩ L 2 * (R N ). Assume that q ≤ N p α , otherwise we conclude that u ∈ L ∞ (R N ) by a modification of the comparison argument of Lemma 4.1.
If p > N +α N −2 and q > 2N p N +α we can show that u ∈ L ∞ (R N ) by repeating the same iteration argument as in the proof of Proposition 4.2, Step 2(B).
If N +α N < p < N +α N −2 and 2 < q < 2N p N +α we know additionally that u ∈ L 1 ∩ L 2 * (R N ) by Lemma 5.1. Then we can conclude that u ∈ L ∞ (R N ) by repeating the iteration argument in the proof of Proposition 4.2, Step 2(A).
Finally, u ∈ L q * ∩ L ∞ (R N ) implies u ∈ C 2 (R N ) by the standard Hölder and Schauder estimates, while positivity of u(x) follows via the weak Harnack inequality, as in the proof of Proposition 4.2, Steps 3 and 4.
Unlike in the case ε > 0, for p > N +α N −2 we can not conclude that u ∈ L 1 (R N ) via a regularity type iteration arguments. In fact, the decay of groundstates of (P 0 ) is more complex.
Proof of Theorem 2. 1 We assume that either N +α As in the case ε > 0, it is standard to check that P 0 = ∅ (see (4.17)). To construct a groundstate solution of (P 0 ), we prove the existence of a spherically symmetric nontrivial nonegative minimizer of the minimization problem and then show that P 0 is a natural constraint for I 0 , i.e. the minimizer u 0 ∈ P 0 satisfies It is easy check that where C > 0 is independent of u ∈ D q . Similarly to Lemma 4.2, we also can prove that there exists C > 0 such that for all u ∈ D q , which allows to control the nonlocal term. The remaining arguments follow closely Steps 1-5 in the proof of Theorem 1.1. We omit further details.

Remark 5.1
An equivalent route to construct a groundstate solution of (P 0 ) is to prove the existence of a minimizer of the problem This does not require apriori regularity or decay properties of the weak solutions. It is standard but technical to establish the relation and to prove that the minimization problems for a 0 and c 0 are equivalent up to a rescaling. Moreover, if w 0 ∈ D q is a minimizer for a 0 then x is a solution of (P 0 ).

Remark 5.2 Combining (5.2) and (5.3), we conclude that
N −2 then (P 0 ) has no nontrivial solution for q = 2N p N +α . This confirms that the existence assumptions of Theorem 2.1 on p and q are optimal, with one exception of the double-critical case p = N +α N −2 and q = 2N p N +α .

Remark 5.3
In the double critical case p = N +α N −2 and q = 2N p N +α the Pohožaev argument does not lead to the nonexistence. In fact, it is not difficult to check (cf. [21, Lemma 1.1]) that the Emden-Fowler solution U * defined in (2.9) satisfies and the "Lagrange multiplier" can not be scaled out due to the scale invariance of the equation. It is an interesting open problem to show that a rescaling of U * is a minimizer of the variational problem (5.11) in the double-critical case.

The Thomas-Fermi groundstate
To simplify notation we set in this section m := q/2. Denote We first establish the following.

Proof of Theorem 2.6
Let ρ * ∈ L 1 ∩ L m (R N ) be a minimizer for s TF , as constructed in Proposition 6.1. It is standard to show that ρ * satisfies for a Lagrange multiplier λ ∈ R. The proof can be adapted from [1] or [11,Proposition 3.6] and we omit it here. Since ρ * ≥ 0, (6.2) is equivalent to the Thomas-Fermi equation Testing (6.3) against ρ * we conclude that m ρ * m m + ρ * 1 = λ, (6.4) or taking into account the definition of s TF we conclude that To prove the virial identity (2.4) consider the rescaling ρ t (x) = t − N +α Therefore we have It follows from (6.3) that ρ * satisfies and the virial identity To prove the L ∞ -bound on ρ * , in view of the Strauss' radial bound (6.1) we only need to show that ρ * is bounded near the origin. Observe that 0 ≤ ρ m−1 * ≤ I α * ρ * a.e. in R N . (6.8) If m > N α then I α * ρ * ∈ C 0,α− N m (R N ) and hence ρ * ∈ L ∞ (R N ) in view of (6.3). If m < N α we employ an L s -iteration of the same structure as in our proof of Proposition 4.2, Step 2(B). Indeed, by the HLS and Hölder inequalities applied to (6.8) We start the s n -iteration with s 0 = m. If m ≥ 2 we achieve s n+1 > N α after a finite number of steps. If m < 2 we note that s 0 = m > 2N N +α by the assumption. Then we again achieve s n+1 > N α after a finite number of steps. (Or if s n+1 = N α we readjust s 0 .) The compact support property is standard (cf. [11,Corollary 3.8]). We sketch the argument for completeness. Indeed, since ρ * ∈ L 1 (R N ) is a nonnegative radially nonincreasing function, it is known that for any α ∈ (0, N ), (and C = ρ * 1 if α > 1), cf. [20,Corollary 2.3]. This is incompatible with (6.2), unless ρ * has a compact support. Since ρ * is nonincreasing we also conclude that supp(ρ * ) is a connected set and hence must be a ball of radius R * > 0 (and if α > 1 then R * ρ * If α > m−2 m−1 + then the Hölder regularity ρ * ∈ C 0,γ (R N ) with γ = min{1, 1 m−1 } follows exactly as in [12,Theorem 8]. We only note that the iteration steps (3.26), (3.27) in [12, p. 127] remain valid for any m ≤ 2, as soon as ρ * ∈ L ∞ (R N ), which is ensured by our assumption m > 2N N +α . If m > 2 and α ≤ m−2 by the same argument as in [12,Remark 2]. Further, ρ * ∈ C ∞ (B R * ) can be deduced as in [12,Theorem 10]. Finally, keeping in mind that q = 2m, the function However once the existence of a minimizer for s TF is established (see Proposition 6.1), solutions constructed for α ∈ (0, 2] in [14,36] coincide with the minimizer for s TF in view of the uniqueness.

Asymptotic profiles: non-critical regimes
In this section we prove the convergence of rescaled groundstates u ε to the limit profiles in the three noncritical regimes.

Formal limit (P 0 )
Throughout this section we assume that p > N +α N −2 and q > 2N p N +α , or N +α N < p < N +α N −2 and 2 < q < 2N p N +α . Let u ε be the positive spherically symmetric groundstate solution of (P ε ) constructed in Theorem 1.1, and c ε = I ε (u ε ) > 0 denotes the corresponding energy level, defined in (4.20). We are going to show that u ε converges to the constructed in Theorem 2.1 positive spherically symmetric groundstate u 0 of the formal limit Eq. (P 0 ), which has the energy c 0 = I 0 (u 0 ) > 0, as defined in (C H L ).
Below we present the proof only in the supercritical case p > N +α N −2 and q > 2N p N +α . The subcritical case N +α N < p < N +α N −2 and 2 < q < 2N p N +α follows the same lines but easier, because in this case u 0 ∈ L 1 (R N ). The proof in the supercritical case relies on the decay estimate (5.4), which needs an additional restriction p > max N +α Proof First, we use u ε with ε > 0 as a test function for P 0 . We obtain Hence there exists a unique t ε ∈ (0, 1) such that u ε (x/t ε ) ∈ P 0 , and we have To show that c ε → c 0 as ε → 0 we shall use u 0 as a test function for P ε . According to (5.4), u 0 ∈ L 2 (R N ) iff N ≥ 5. The dimensions N = 3, 4 require a separate consideration.
and q > 2N p N +α . Then the quantities are uniformly bounded as ε → 0.

Proof of Theorem 2.2 (case
and q > 2N p N +α ). From Corollary 7.1 and (7.6), we see that {u ε (x/t ε )} is a minimizing sequence for c 0 which is bounded in and u ε (x/t ε ) → w 0 a.e. in R N , by the local compactness of the emebedding D 1 (R N ) → L 2 loc (R N ) on bounded domains. Using Strauss' radial L s -bounds with s = 2 * and s = q, we conclude that

Similarly to
Step 3 in Sect. 4, using Lebesgue dominated convergence and nonlocal Proposition 4.7], we can show that u ε (x/t ε ) → w 0 in D 1 ∩ L q (R N ) and w 0 is a groundstate solution of (P 0 ).

1st rescaling: Choquard limit
Throughout this section we assume that N +α N < p < N +α N −2 and q > 2 2 p+α 2+α . The energy corresponding to the 1st rescaling is given by and the corresponding Pohožaev functional is ε (v) and consider the rescaled minimization problem where P is the Pohžaev manifold of (C ε ). When ε = 0, we formally obtain c (1) It is known that c Let u ε be the positive spherically symmetric groundstate solution of (P ε ) constructed in Theorem 1.1. Then the rescaled groundstate is a groundstate solution of (C ε ), i.e. I ε . We are going to show that v ε converges to the groundstate v 0 of the Choquard Eq. (C ).
has a unique maximum and P (1) 0 (w ε,1 ) < 0, thus there exists t v ε ∈ (0, 1) such that w ε,t vε ∈ P (1) 0 . Therefore we have On the other hand, let v 0 ∈ P (1) 0 be a radially symmetric ground state of (C ), that is ε . This implies that t v 0 (ε) is bounded, up to subsequence, we assume that . We obtain which means that t α v 0 (ε) ≤ 1 + Cε κ , or equivalently Therefore It follows that c
Proof Follows from Pohožaev and Nehari identities for I (1) ε and Lemma 7.3, as in Corollary 7.1.
Let u ε be the positive spherically symmetric groundstate solution of (P ε ) constructed in Theorem 1.1. It is clear that the rescaled groundstate is a groundstate solution of (TF ε ), i.e. I (2) ε (v ε ) = c (2) ε . We are going to show that v ε converges to a groundstate of the Thomas-Fermi Eq. (TF), characterised in Theorem 2.6.
Before we do this, we deduce a two-sided estimate on the "boundary behaviour" of the the nonnegative radially symmetric Thomas-Fermi minimizer ρ * , constructed in Theorem 2.6. Recall that supp(ρ * ) =B R * for some R * > 0 and ρ * is C ∞ inside the support. Moreover, since we assume that p = 2, α = 2 (and denote m = q/2), we see that ρ m−1 * ∈ C 0,1 (R N ) and ρ * ∈ C 0,γ (R N ), where γ = min{1, 1 m−1 }. In particular, ρ * ∈ H 1 0 (B R * ) and we can apply − to the Euler-Lagrange Eq. (2.5) considered in B R * , to obtain We conclude that ρ m−1 * is superharmonic in B R * and by the boundary Hopf lemma Hence we deduce a two-sided bound Similar estimates should be available for α = 0, at least under the assumption α < 2. We will study this in the forthcoming paper [26].
Proof of Theorem 2.7 By Lemmas 7.4 and 7.5, we see that {v ε (x/t ε )} is a bounded minimizing sequence of c (2) 0 . Then similarly to the arguments in the proof of Proposition 6.1 and Theorem 2.6, there and v 0 is a weak solution of (TF).

Critical Choquard case
Throughout this sub-section we assume that p = N +α N −2 and q > 2 * = 2N N −2 . Consider the minimization problem and S H L is achieved by the function (8.2) and the family of rescalings here U * is the Emden-Fowler solution in (2.9). Up to a rescaling, V λ is a solution of the critical Choquard Eq. (C H L ) and satisfies H L . The energy functional associated to (C H L ) is We define where By a simple calculation, we see that c H L = α+2 Proof Note that u ε ∈ P ε is a solution of (P ε ) with I ε (u ε ) = c ε , then P H L (u ε ) < 0, thus there exists t ε ∈ (0, 1) such that u ε (x/t ε ) ∈ P H L and we have Therefore, σ ε = c ε − c H L > 0.

Conclusion of the proof for N = 3, 4. Combining previous estimates together, we obtain
which completes the proof of this lemma.
Proof Follows from Nehari and Pohožaev identities for (P ε ).
Proof By Lemma 8.1, we see that (8.9) This, together with Lemma 8.2, implies that there exists C > 0 such that for all small ε > 0. Note that P H L (u ε ) < 0, then there exists t ε ∈ (0, 1) such that P H L (u ε (x/t ε )) = 0, which means that If t ε → 0 as ε → 0, then we must have ∇u ε 2 2 → 0 as ε → 0, this contradicts Therefore, there exists C > 0 such that t ε > C for ε > 0 small. Thus we have This means that Moreover, we have ε u ε 2 2 → 0 as ε → 0. It follows from Lemma 8.2 that H L x), then we have, as ε → 0, On the other hand, by the HLS inequality, we have This means that lim ε→0 w ε which completes the proof. Set Then w ε 2 * = V 2 * = 1, ∇V 2 2 = S * and {w ε } is a minimizing sequence for the critical Sobolev constant S * . Similarly to the arguments in [41, p. 1094], we conclude that for ε > 0 small there exists λ ε > 0 such that We define the rescaled family Then v ε 2 * = 1, ∇v ε 2 2 = S * + o(1), i.e., {v ε } is a minimizing sequence for S * . Furthermore, which, together with the definitions of w ε and w ε , implies that By a simple calculation, we see that v ε solves the equation By the definition of v ε and w ε , we obtain It follows from Lemma 8.2 and Lemma 8.3 that Therefore we can deduce the following estimates on λ ε . H L |v ε | q−2 .
By Lemma 8.5 and since u ε ≤ C|x| −(N −2)/2 u ε 2 * , we conclude that where C > 0 does not depend on ε > 0 or x ∈ R N . Therefore, for small ε > 0, it follows from (8.10) that solutions v ε > 0 satisfy the linear inequality By [41,Lemma 4.8] we have the following lower estimates of v ε .
Proof For all N ≥ 3, (8.13) implies that v ε q q ≥ C for some C > 0.
Case N ≥ 5. By Lemmas 8.1 and 8.5, we see that for N ≥ 5 Moreover, v ε This complete the proof.

Critical Thomas-Fermi case
Throughout this section we assume that N +α N < p < N +α N −2 and q = 2N p N +α . Let where C α is the optimal constant in (1.3), as described in (2.13).
It is well-known [34,Theorem 4.3] that S TF is achieved by the function (8.17) and the family of rescalings (8.18) here U is the groundstate solution of (TF * ) defined in (2.14). It is clear that The energy functional which corresponds to (TF * ) is We define, By a simple calculation, we see that c TF = α 2N p S N +α α TF . Note that V λ ∈ L 2 (R N ) for N ≥ 4, and V λ ∈ D 1 (R N ) if N ≥ 3.