SHARP VARIATIONAL CHARACTERIZATION AND A SCHR¨ODINGER EQUATION WITH HARTREE TYPE NONLINEARITY

. In this paper, we ﬁrst give a sharp variational characterization to the smallest positive constant C V GN in the following Variant Gagliardo- Nirenberg interpolation inequality: where u ∈ W 1 , 2 ( R N ) and N ≥ 1. Then we use this characterization to determine the sharp threshold of (cid:107) ϕ 0 (cid:107) L 2 such that the solution of iϕ t = −(cid:52) ϕ + | x | 2 ϕ − ϕ | ϕ | p − 2 ( | x | − α ∗| ϕ | p ) with initial condition ϕ (0 ,x ) = ϕ 0 exists globally or blows up in a ﬁnite time. We also outline some results on the appli- cations of C V GN to the Cauchy problem of iϕ t = −(cid:52) ϕ − ϕ | ϕ | p − 2 ( | x | − α ∗| ϕ | p ).


Dedicated to Professor Boling Guo on the occasion of his 80th birthday
Abstract. In this paper, we first give a sharp variational characterization to the smallest positive constant C V GN in the following Variant Gagliardo-Nirenberg interpolation inequality: where u ∈ W 1,2 (R N ) and N ≥ 1. Then we use this characterization to determine the sharp threshold of ϕ 0 L 2 such that the solution of iϕt = − ϕ + |x| 2 ϕ − ϕ|ϕ| p−2 (|x| −α * |ϕ| p ) with initial condition ϕ(0, x) = ϕ 0 exists globally or blows up in a finite time. We also outline some results on the applications of C V GN to the Cauchy problem of iϕt = − ϕ−ϕ|ϕ| p−2 (|x| −α * |ϕ| p ).

Introduction.
In this note, we study the following semilinear Schrödinger equation with a Hartree type nonlinearity as well as Schrödinger equation with harmonic potential and Hartree type nonlinearity where N ≥ 1, 0 < α < min{N, 4}, ϕ := ϕ(t, x) : R + × R N → C is a complex-valued function and * is the standard convolution in R N . Problems of this kind arise naturally from various physical situations. For example, Eq.(1) can be considered as the classical limit of the field equations describing a quantum mechanical nonrelativistic many boson system [10]. Eq. (2) can be used to describe average pulse propagation in dispersion-managed fibers, see e. g. [13]. When α = 1, p = 2 and N = 3, Eq.(2) is also equivalent to the Schrödinger-Poisson system with harmonic potential, This equation arises typically in the mean field approximation of the many body effects, modeled by the Poisson equation with a confinement modeled by the quadratic potential of the harmonic oscillator. For other variant of Schrödinger equation, we refer the interested readers to [2] and the references therein.
Partial results for radially symmetric functions in W 1,2 (R N ) has been obtained in [7] (where we assume N ≥ 2). But general results for N ≥ 1 are still unknown. In the present paper, we have two goals. One is to give a sharp variational characterization of C V GN related to the following Variant Gagliardo-Nirenberg interpolation inequality: there is a positive constant C such that for all u ∈ W 1,2 (R N ), where A = N (p−2)+α 2 and B = p − A. Note that the inequality (4) has been shown in [7] and we denote The other is to use the characterization of C V GN to solve the unknown problems mentioned above. We will study Eq.(2) in details but only outline some counterpart for Eq.(1).
To explain our ideas and strategies of solving these problems, we recall first that for the semilinear Schrödinger equation iϕ t + ϕ + |ϕ| q−1 ϕ = 0, (t, x) ∈ R + × R N . Weinstein [14] has originally solved similar problems mentioned above by studying the following Gagliardo-Nirenberg inequality Weinstein [14] got a sharp estimate on the smallest positive constant C GN by solving the minimization problem
Noticing that in the case of N = 1, we do not know if the embedding (5) is compact or not. It seems that the methods of Weinstein [14] can not be used directly to study this for N = 1. In Section 2, we develop a scaling method and characterize C V GN for all spatial dimension N ≥ 1. The main result is contained in Theorem 2.4. We emphasize here that the characterization of this C V GN not only has independent interests, but also can be applied to study some other properties of solutions of Eqs.(1)+(3) and Eqs.(2)+(3). The variational characterization of C V GN is the main contribution of the present paper. We also point out that this scaling method can be used to study a class of interpolation inequality involving p-Laplace operator, which will be published elsewhere. In Section 3, we summarize some local existence results on the Eqs. (2)+(3) for initial data whose L 2 (R N ) norm can be as large as you want. In Section 6, we outline some counterpart results for the Eqs. Notations. Throughout this paper, all integrals are taken over R N unless stated otherwise. The W 1,2 (R N ) is the standard Sobolev space with the norm u 2 = (|∇u| 2 + |u| 2 )dx. The norm in L q (R N ) (1 ≤ q ≤ ∞) is denoted by · L q . M j (j ∈ N ) denote various positive constants whose exact value are not important.

2.
Variational characterization of C V GN . This section is devoted to the characterization of C V GN . We start with the following Variant Gagliardo-Nirenberg interpolation inequality. Proposition 1. [7] Let 0 < α < N and (2N − α)/N < p < 2 * (α). Then there is a positive constant C(p, α, N ) depending on p, α and N such that for any u ∈ W 1,2 (R N ), where A = N (p−2)+α 2 and B = p−A. The C V GN denotes the infimum of all possible C(p, α, N ).
The characterization of C V GN will be divided into several steps. In the first place, we consider the following auxiliary maximization problem with constraint. For any λ > 0, we denote S λ = {u ∈ W 1,2 (R N ) : u 2 = λ} and set We will adapt the method of Lions [11] to prove that for any given λ > 0, the m λ is achieved. Note that m λ > 0 for any λ > 0. Moreover we have the following lemma.
Lemma 2.1. Let (u n ) n∈N be a maximizing sequence of the m λ . Then for any ε > 0, , and a subsequence (u n k ) k∈N such that Proof. According to Lions' concentration compactness lemma [11], we divide the proof into two steps. Denote ρ n (x) = |∇u n (x)| 2 + |u n (x)| 2 . In the first step, we claim that for any given R > 0, We prove this claim indirectly. Suppose that there is a subsequence of (ρ n ) n∈N , still denoted by (ρ n ) n∈N , such that Then from an inequality of [12], see also [7, Lemma 2.2], we have that By the assumption of p, one deduces from Sobolev inequality that where M 2 is independent of z, R and u n . Now covering R N by balls of radius R, in such a way that each point of R N is contained in at most N + 1 balls, we hence obtain that Therefore g(u n ) → 0 as n → ∞, which is a contradiction since u n satisfies lim n→∞ g(u n ) = m λ > 0.
In the second step, we prove that the case of does not occur either. Arguing by a contradiction, we may assume that (7) holds for some β ∈ (0, λ). Then for any given ε > 0 and ε small enough, there is R 0 > 0 such that for all R ≥ R 0 , there are n 0 ∈ N and a sequence z n ∈ R N (n ∈ N) such that Moreover, we also have that for any R 1 > R 0 , In particular, Therefore we have that (8) and Now we define a cut-off function η . From a direct calculation and using the fact of Similarly, we can deduce that Note that for any a, b ∈ R and q > 1, there holds (8), Hölder inequality and an elementary calculation that Therefore, by letting n → ∞ and then ε → 0 (since R 0 → ∞ as ε → 0). Which is a contradiction.
It is now deduced from Lions' Concentration Compactness lemma [11] that Lemma 2.1 holds. The proof is complete.
. Then for any λ > 0, the m λ is achieved by some u ∈ W 1,2 (R N ).
Proof. Let (u n ) n∈N be a maximizing sequence of m λ , i.e., u n 2 = λ and g(u n ) → m λ . Then by Lemma 2.1, we have that for any ε > 0, there is R 0 < +∞ and a sequence z k ∈ R N and a subsequence (u n k ) k∈N ⊂ (u n ) n∈N such that This and Proposition 1 imply that g(u n k ) → g(u) and u is a maximizer of the m λ .
In the second place, we consider another auxiliary problem. On W 1,2 (R N ), we define the functionals To proceed, we point out first that for any λ > 0, m λ = λ p m 1 . From the Proposition 2 and Lagrange multiplier rule, we know that for λ = 1, there is A simple scaling argument implies that θ Moreover, we have θ V 1 ) and the following proposition.
Then a maximizer U of the m Λ is a minimal action solution of Eq. (13). That is to say, U is a solution of Eq.(13) and for any v ∈ N , one has that L(U ) ≤ L(v).
Proof. Firstly we prove that U 2 = g(U ). Indeed, since U is a maximizer of the m Λ , there is θ Λ such that We obtain that U 2 = g(U ). Therefore θ Λ = 1. Secondly, we prove that U is a minimal action solution of Eq. (13). For any v ∈ N , we have that √ Λv v ∈ S Λ . Since U is a maximizer of the m Λ , It is now deduced from U 2 = g(U ) and I(v) = 0 that U 2 ≤ v 2 . Therefore, This implies that U is a minimizer of the minimum d. We obtain from [15,Theorem 4.3] that U is a minimal action solution of Eq.(13).
In the third place, we give another characterization to a minimal action solution of Eq.(13), which will paly an important role in the study of C V GN . Proof. Clearly, from the definition we know that m r0 ≥ g(φ). It remains to prove that m r0 ≤ g(φ). For any u ∈ W 1,2 (R N ) satisfying 2p−2 such that I(s 0 u) = 0, which implies that s 0 u 2 = s 2p 0 g(u). Since φ is a minimal action solution of Eq. (13), we obtain that According to the fact of g(φ) = φ 2 = u 2 , we deduce that Since u is chosen arbitrarily, we get that m r0 ≤ φ 2 = g(φ). This completes the proof.
We are now in a position to give a variational characterization of C V GN . To simplify the notations, we denote K := N (p − 2) + α in the rest of this section. Define the following functional Proof. Since φ is a minimal action solution of Eq.(13), one has that I(φ) = 0. Combining this with Lemma 2.2, one can get the conclusion immediately by direct calculation.
Proof. We prove this theorem by studying the following minimization problem where In the first step, we claim that Indeed from φ ∈ W 1,2 (R N )\{0} and Lemma 2.3, one deduces that Therefore (15) holds since φ is chosen arbitrarily.
In the second step, we prove that To prove this, for any u ∈ W 1,2 (R N )\{0}, we define w(x) := ξu(µx), where ξ and µ are positive parameters which will be determined later. By direct computation, one obtains that and Now we determine ξ and µ by the following equations and Noting that from Lemma 2.3 we know that It is deduced from Eq. (19) and Eq.(20) that and From the choice of w, we have that w 2 = φ 2 . Since φ is a minimal action solution of Eq. (13), we obtain from Proposition 4 that Combining this with Eq.(21) and Eq.(22), we deduce that The expression of J(u) implies that Since u is chosen arbitrarily, we obtain that From the above two steps, we get that The proof is complete.

4.
Critical mass for critical nonlinearity. In this section, we will use "our estimate" of C V GN to give a sharp condition on the solution of Eqs.(2)+(3) which exists globally in time or blows up in a finite time. The new aspect is in the case of N = 1, since for N ≥ 2, we have used the radial symmetric properties of W 1,2 r (R N ) to study similar phenomena in [7]. Here with the help of Theorem 2.4, we can deal with Eq.(2) generally for N ≥ 1. We give an answer on the question: how small an initial data can ensure the existence of global solution of Eqs.(2)+(3) in the case of p = 2 + (2 − α)/N . The answer is simple as we see below.
Proof. For N ≥ 2, this proposition has been proven in [7]. While for N ≥ 1, the proof is the same and we omit the details here. Proof. For N ≥ 2, this lemma has been proven in [7]. While for N ≥ 1, the proof is the same and we omit the details here.
Proof of Theorem 4.2. For any positive constant λ and complex number c with |c| ≥ 1, a direct computation yields that On the other hand, since the function w(x) makes the inequality (4) with constant C V GN into equality, one obtains that So it is very reasonable to conjecture that for some class of initial data ϕ 0 with ϕ 0 L 2 ≥ w L 2 , the solutions of Eqs.

It follows from
(2)+(3) exist globally in time. In fact, this conjecture is true in the case of 2 + (2 − α)/N < p < 2 * (α). Furthermore, we can prove that when 2 + (2 − α)/N < p < 2 * (α), the solutions of Eqs.(2)+(3) exist globally in time for a large class of initial data whose norm can be taken as large as one wants. (2)+(3) in the case of supercritical nonlinearity 2 + (2 − α)/N < p < 2 * (α). An interesting aspect is that we can get the global solutions for arbitrarily large data. We emphasize that the use of Theorem 2.4 is essential. We also point out that for N ≥ 2, we have studied these in [7]. Here we give very general results for N ≥ 1. The following lemma 5.1 will be useful in what follows.
Proof. The proof is the same as [7, lemma 5.2].
Remark 3. By Lemma 5.2, we get that Eqs.(2)+(3) possesses global solutions for a large class of initial data whose norm can be as large as we want. On the other hand, from the definition of V (λ) and Theorem 5.3, we know that V (λ) → w L 2 as p → 2 + (2 − α)/N . So we obtain the sharp condition for global existence in the case of initial data ϕ 0 L 2 < w L 2 , which coincides with Theorem 4.1. In the case of critical nonlinearity p = 2 + (2 − α)/N , the condition (25) is sharp. However, we do not know whether or not the condition (30) is sharp in the case of super critical nonlinearity 2 + (2 − α)/N < p < 2 * (α). 6. Remarks on the Cauchy problem of Eq.(1). In this section, we outline some applications of C V GN to the Cauchy problem of Eq.(1). Theorem 6.1. Let N ≥ 1, 0 < α < min{N, 4} and p = 2 + (2 − α)/N . If ϕ 0 ∈ W 1,2 (R N ) and ϕ 0 L 2 < w L 2 , where w is a minimal action solution of Eq.(13), then Eqs.(1)+(3) has a global solution ϕ(x, t) ∈ C(R + , W 1,2 (R)).  is a solution of iϕ t + ϕ + ϕ|ϕ| p−2 (|x| −α * |ϕ| p ) = 0 and ϕ blows up at finite time. But for the general p, the blow-up derived by self-similarity is still open and we can not solve it at this moment.