On radial solutions of semi-relativistic Hartree equations

We consider the semi-relativistic Hartree type equation with nonlocal nonlinearity F (u) = λ(|x|−γ ∗ |u|2)u, 0 < γ < n, n ≥ 1. In [2], the global well-posedness (GWP) was shown for the value of γ ∈ (0, 2n n+1 ), n ≥ 2 with large data and γ ∈ (2, n), n ≥ 3 with small data. In this paper, we extend the previous GWP result to the case for γ ∈ (1, 2n−1 n ), n ≥ 2 with radially symmetric large data. Solutions in a weighted Sobolev space are also studied.

If the solution u of (1) or (2) has sufficient decay at infinity and smoothness, it satisfies two conservation laws: where K(u) = √ 1 − ∆ u, u , V (u) = 1 4 F (u), u and , is the complex inner product in L 2 .
For the well-posedness results, we refer the readers to the papers [10,2]. Lenzmann in [10] established the global well-posedness in H 1 2 (R 3 ) for γ = 1. In [2], we considered the general case 0 < γ < n, n ≥ 1 and showed the local and global existence by utiliizing the Strichartz estimates. In particular, we showed the global existence for 0 < γ < 2n n+1 in H 1 2 with large data and for 2 < γ < n in H s , s > γ 2 − n−2 2n with small data. The first result of this paper is on the global existence of radially symmetric solutions of (2) for 2n n+1 ≤ γ < 2n−1 n , n ≥ 2. Theorem 1.1. Let γ satisfy 1 < γ < 2n−1 n , n ≥ 2, s ≥ 1 2 . If λ > 0, then for any radially symmetric function ϕ ∈ H s , (2) has a unique radially symmetric solution for q = 2n n−1 + ε and σ = 1 2 + ε with sufficiently small ε, ε > 0. For all time the energy and L 2 norm of u(t) are conserved. If λ < 0, then there exists ρ > 0 such that the same conclusion holds for ϕ with ϕ L 2 ≤ ρ. Moreover, Here are the usual Sobolev space and a hibrid Sobolev space, where P ≤1 and P >1 are frequency projection over frequency less than 1 and greater than 1. We mean H s by H s 2 andḢ s byḢ s 2 . Hereafter, we denote the space L q T (B) by L q (−T, T ; B) and its norm by · L q T B for some Banach space B, and also L q (B) with norm In order to prove Theorem 1.1, we pursue the contraction mapping argument. For this purpose, we use the energy and L 2 conservation laws and the Strichartz estimate for radial functions. By the Strichartz estimate we mean (see [11,12]): where (q i , r i ), i = 0, 1, satisfy that for any θ ∈ [0, 1] If ϕ and F are radially symmetric, then by the well-known decay property of the Fourier transform of measure on unit sphere the estimate (5) can be extended as: where s ∈ R and 2n n−1 < p < ∞, σ = n 2 − n+1 p . The second estimate does not follow simply from the first one. We prove this via low-diagonal operator estimate. These will be shown in Section 2.
Interpolating (5) and (7) 1 , we get wider range of pairs (q, r). For Theorem 1.1, we need only the pairs (q, 2n n−1 ) with q slightly larger than 2n n−1 . To put is another way, given ε > 0 we can find q and σ such that 2n n−1 < q < 2n n−1 + ε, 1 2n < σ < 1 2n + ε and With these pairs we can make the value of σ close to 1 2n and the value γ to 2n−1 n . Next we consider a radial solution in weighted Sobolev space Theorem 1.2. Let n ≥ 2 and 1 < γ < 2n−1 n . Let ϕ and u be as in Theorem 1.1.
, where q and σ are the numbers as stated in Theorem 1.1. Moreover, if n ≥ 3, then The essential parts of the proof for the global existence in H 1,1 are the estimate (4) and the following estimates For the inequality (12), we do not need the radial symmetry. To obtain (10) and (11), we need to estimate |∇V γ (u)||x|u L 2 for which we establish the pointwise estimate of fractional integral of radial function for n ≥ 3. See Lemma 3.1 below.
If 0 < γ ≤ 1, then in view of GWP in H s , s ≥ 1 2 of [10,2], from the estimate |∇V γ (u)||x|u L 2 ∇u L 2 u H 1 2 u H 1,1 we deduce the GWP in H 1,1 without radial symmetry condition. One can also prove the global existence of radial solutions H k,l with integers k, l > 1 by the same method as in Section 3.
If not specified, the notation A B and A B denote A ≤ CB and A ≥ C −1 B, respectively. Different positive constants possibly depending on n, λ and γ might be denoted by the same letter C. A ∼ B means that both A B and A B.

Proof of Theorem 1.1
In this section, we prove Theorem 1.1. For simplicity, we only consider the positive time because the proof for negative time can be treated in the same way. 1 We can proceed the complex interpolation after changing U (t) an operator mapping from 1-dimensional function space to n-dimensional space in a similar way to the proof of (7) below.
Let us first define a complete metric space As stated in the introduction, given ε > 0 we can find q and σ satisfying (8). Now we define a mapping N : u → N (u) on X T,ρ by For any u ∈ X T,ρ , N (u) is radially symmetric. By Strichartz estimate (8), we have To estimate of the second term on the RHS of (14), let us introduce a generalized Leibniz rule (see Lemma A1 ∼ Lemma A4 in Appendix of [8]).
Since γ ≤ 2, we use Lemma 2.1 with (r 1 , q 1 ) = (∞, 2), (r 2 , q 2 ) = ( 2n γ , 2n n−γ ) and (r 1 , q 1 ) = (2, 2n n−γ ) = (q 2 , r 2 ), and Hardy-Littlewood-Sobolev inequality to obtain where 0 < ε 0 < n − γ and we have used the inequality that for any x ∈ R n Now if we choose ε and ε 0 so small that γ < 2 − 2σ and 2n , then we have from (14), (15) and embeddings H for some constant C. Thus if we choose ρ and T so that C ϕ H s ≤ ρ 2 and C(T + T 1− 2 q )ρ 3 ≤ ρ 2 , then we conclude that N maps from X T,ρ to itself. For any u, v ∈ X T,ρ , we have By (16) and Hölder's inequality, we have for sufficiently small ε 0 > 0 Now by another Hölder's inequality in time, we have Similarly, Hence by Sobolev embedding we get Substituting these two estimates into (17) and then using the fact C(T +T 1− 2 q )ρ 2 ≤ 1 2 for small T , we conclude that N is a contraction mapping on X T,ρ . The energy and L 2 conservations follow by the method in [13].
Now we show that the local solutions can be extended globally in time. For this purpose we prove an a priori estimate in X T for any T > 0. Fixing T , since γ < 2, from the energy conservation we see that at any t ≤ T , the solution u satisfies that for λ > 0, and hence by Young's inequality and the smallness of ϕ L 2 From the estimates (20) and (16), we have for δ > 0 Thus for sufficiently small δ but equivalent to the value (1 + ϕ 2 Finally, we have from (15) Hence by Gronwall's inequality and (21), This completes the proof of Theorem 1.1.
Proof of Strichartz estimate (7) of radial functions. For the first inequality, we follow the proof of Proposition 6.3 in [14]. By the spherical coordinate, where r = |x|, ρ = |ξ| and Let us define a one-dimensional function f by f (ρ) = w(ρ) ϕ(ρ)ρ n− 1 2 for some positive function to be chosen later and an operator W (t) by (W (t)f )(r) = r n−1 p (U (t)ϕ)(r). Then since the space H we have only to show that for w = ρ − 1 2 (1 + ρ 2 ) 2p , then we prove the claim. Now we prove the second inequality. For this purpose, it suffices to show that To show (26), let us introduce the low-diagonal operator estimate. For instance, see [3,15,1].

Lemma 2.2. Let A and B be Banach spaces. Let K be an operator such that
In fact, since for any G ∈ L 1 T (L 2 (0, ∞)) we can find a unique radial function 1 4 . Then by the Strichartz estimate (7), we have This proves (27) and thus the claim (26).

Proof of Theorem 1.2
We proceed a similar line to the proof of Theorem 1.1(contraction scheme for local existence, energy and L 2 conservation for global time extension) except for H 1,1 estimate. For this purpose, we will prove only a priori estimates (10), (11) and (12).
Let us begin with proof of (11). Using the commutator relation we have By using the estimate (15), we have It remains to estimate the last term. To do this, let us introduce a fractional integration of radial function in R n , n ≥ 3 which will be proven later.