On the inter-critical inhomogeneous generalized Hartree equation

It is the purpose of this work to study the Choquard equation iu˙-(-Δ)su=±|x|γ(Iα∗|·|γ|u|p)|u|p-2u\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} i\dot{u}-(-\Delta )^s u=\pm |x|^{\gamma }(I_\alpha *|\cdot |^\gamma |u|^p)|u|^{p-2}u \end{aligned}$$\end{document}in the space H˙s∩H˙sc\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\dot{H}^s\cap \dot{H}^{s_c}$$\end{document}, where 0<sc<s\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$0<s_c<s$$\end{document} corresponds to the scale invariant homogeneous Sobolev norm. Here, one considers to two separate cases. The first one is the classical case s=1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$s=1$$\end{document} and the second one is the fractional regime 0<s<1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$0<s<1$$\end{document} with radial data. One tries to develop a local theory using a new adapted sharp Gagliardo–Nirenberg estimate. Moreover, one investigates the concentration of non-global solutions in LT∗∞(H˙sc)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L^\infty _{T^*}(\dot{H}^{s_c})$$\end{document}. One needs to deal with the lack of a mass conservation, since the data are not supposed to be in L2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L^2$$\end{document}. This note gives a complementary to the previous works about the same problem in the energy space H1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$H^1$$\end{document}.

The Eq. (1.1) enjoys the scaling invariance Thanks to the identity the unique critical exponent giving an invariant Sobolev norm under the previous scaling is Two particular cases are widely investigated in the literature. The first one s c = 0 is related to the mass conservation and gives the L 2 -critical exponent p * := 1 + 2+2γ +α N . The second one s c = 1 gives theḢ 1critical exponent This case is also related to the energy conservation law Denote also the continuous Sobolev injectionḢ 1 The critical Sobolev exponent in the fractional case reads Thus, the mass-critical and energy-critical exponents are q * := 1 + α + 2s + 2γ N , q * := 1 + 2s + 2γ + α N − 2s .
Denote also the continuous Sobolev injectionḢ s (R N ) → L q c (R N ), where The conserved energy reads To the authors knowledge, there is few papers dealing with the inhomogeneous generalized Hartree problem. Indeed, a sharp threshold of global well-posedness and scattering vs finite time blow-up of energy solutions was obtained by the first author in [1,16]. The spherically symmetric assumption for the energy scattering was removed in [18]. The instability of standing waves was treated in [7]. The local theory was developed in [15] for the fractional inhomogeneous Choquard problem with radial data. The Hartree problems (1.1) and (1.3) are locally well posed in the respective energy spaces H 1 and H s rd , see [1,Theorem 5.2] and [15,Theorem 2.4]. Here, one studies these problems respectively inḢ 1 ∩Ḣ s c anḋ H s rd ∩Ḣ s c . The spherically symmetric assumption in the fractional case is due to the loss of regularity in the non-radial fractional Strichartz estimate [8]. It is the aim of this paper to develop a local theory by removing the mass conservation since the data is no longer in L 2 . There exist mainly three technical difficulties to handle. The first one is related to the presence of a singular inhomogeneous term | · | γ , the second one is the non-local source term and the last one is the presence of a non-local fractional Laplace operator. In the case of a Schrödinger problem with an inhomogeneous local source term, the local theory inḢ 1 ∩ H s c was developed recently in [2]. It seems that [2, Proposition 4.2] needs the restriction σ < 2−b N −(2−b) . This gives some supplementary conditions in the given results.
The rest of this paper is organized as follows. The next section contains the main results and some useful estimates needed in the sequel. Section 3 contains the proofs of the results concerning the inhomogeneous Choquard problem (1.1). The last section contains the proofs of the results concerning the fractional inhomogeneous Choquard problem (1.3).
Denote for simplicity the Lebesgue space L r := L r (R N ) with the usual norm · r := · L r and · := · 2 . Take H s := H s (R N ) be the usual inhomogeneous Sobolev space endowed with the complete norm If X is an abstract space C T (X ) := C([0, T ], X ) stands for the set of continuous functions valued in X and X rd is the set of radial elements in X , moreover for an eventual solution to (1.1), T * > 0 denotes its lifespan.
Finally, x ± are two real numbers near to x satisfying x + > x and x − < x.

Background material
Let us start with the contribution of this note.

Main results
This subsection contains two parts.

Results about the Choquard problem (1.1)
First, one gives a sharp Gagliardo-Nirenberg-type inequality.
The best constant in the above estimate is attained in some ψ ∈Ḣ 1 rd satisfying C opt = P [ψ] and (2.8)

Remark 2.2
The restriction p < 1 + 2+2γ +α N −2−2γ is needed in the proof of a compactness embedding, which gives the convergence of a minimizing sequence. See Lemma 3.1. This restriction does not appear [1] in the estimate on H 1 , available for 1 + α+2γ Second, one considers the evolution regime. The Schrödinger problem (1.1) is locally well posed inḢ 1 ∩ H s c .

Results about the fractional Choquard problem (1.3)
First, one gives a sharp Gagliardo-Nirenberg-type inequality. (2.12) Remark 2.8 The restriction q < 1 + 2+2γ +α N −2s−2γ is needed in the proof of a compactness embedding, which gives the convergence of a minimizing sequence. See Lemma 4.1. This restriction does not appear [15] in the estimate on H s , available for 1 + α+2γ Second, one considers the evolution regime. The Schrödinger problem (1.3) is locally well posed inḢ s rd ∩ H s c .
Assume that if N = 2, 2s + 2γ + α > 1 or 2s + 2γ + α < 1 and q < 1 + 1 1−(2s+2γ +α) . Then, there exists T * > 0 and a unique maximal solution to (1.3), where (q, r ) ∈ and (q 1 , r 1 ) ∈ s c . Moreover, where φ is a ground state of (2.11); 3. the energy is conserved. Remarks 2.10 Note that 1. The assumption q ≥ 2 avoids a singularity in the source term; 2. There is an extra restriction in the bi-dimensional case; 3. In the second point the blow-up is due to the fractional s-order gradient concentration in L 2 ; 4. The radial assumption avoids a loss of regularity in Strichartz estimates [8].
Finally, one is concerned with the finite time blow-up of the focusing solutions. Thanks to the Sobolev injection H s rd →Ḣ s rd ∩Ḣ s c and the localized variance identity [15], there is non-global solutions to (1.3) with data inḢ s rd ∩Ḣ s c . In the following, one gives a concentration result about finite time blow-up solutions in L ∞ T * (Ḣ s c ). Theorem 2.11 Take the same conditions in Theorem 2.9. Let a maximal solution to (1.3), whereφ is a ground state of (2.11).
Remark 2.12 Using the scaling

Useful estimates
This sub-section contains some estimates needed in the sequel. First, recall a Hardy-Littlewood-Sobolev inequality [12].
Then, for any u ∈ W s, p , one has Let us write a fractional chain rule [3].
gives an essential estimate in the Schrödinger context.
For simplicity, one denotes μ the set of μ admissible pairs. Let also Take the particular case := 0 and S(I ) := S 0 (I ).
A standard tool to control solutions of (1.1) is the Strichartz estimate [4,6].

Proposition 2.17
Let N ≥ 1 and γ ∈ R. Then, there exists C > 0 such that Recall some Strichartz estimates [8] for the fractional Schrödinger problem.

The Choquard equation
In this section, one proves the results about the generalized Hartree problem (1.1). In this section, one defines the real numbers and the source term

Gagliardo-Nirenberg inequality
The goal of this sub-section is to prove Theorem 2.1. Let us start with a compact injection.
Proof Take a sequence of function such that sup n u n p c + ∇u n < ∞, converging weakly to zero. One On the other hand, with Poincaré inequality and compact Sobolev injections, one has .
The proof is ended.

Proof of the interpolation inequality (2.5)
First, using Lemmas 2.13 and 2.14, one has Denoting for a, b ∈ R, the scaling u a,b := au(b·), one computes It follows that
Then, ψ n ψ inḢ 1 ∩ L p c and using Sobolev injection (3.13) via Lemma 2.13, we get for a sub-sequence denoted also (ψ n ), This implies that, when n goes to infinity .
Using lower semi continuity of theḢ 1 ∩ L p c norm, we get ψ p c ≤ 1 and ∇ψ ≤ 1.
It follows that .
The minimizer satisfies the Euler equation Hence, ψ satisfies This completes the proof.

Proof of the Eq. (2.8)
Thanks to the previous subsection, we know that where ψ is given in (2.6). Take, for a, b ∈ R, the scaling ψ = φ a,b := aφ(b·). Then, the previous equation gives The proof is closed.

Well-posedness
In this sub-section, one proves Theorem 2.3. Let us give some estimates of the source term.
Then, there exist c, θ > 0 and 0 < θ 2 < 2( p − 1) such that for any real interval |I | ≤ 1, one has The first and last terms are controlled similarly. Also the second and third one. Let us decompose the first term as follows By Lemma 2.13 via Sobolev injections, one has The integrability condition reads Let the admissible pair Then, where one takes θ := 1 − 2 p q > 0. For the second term, one computes with Strichartz estimates and Lemma 2.13 via Sobolev injections Take N c = −γ − 2 and N d = −γ + , for some 0 < << 1. Letting 2 a := 1 c + 1 d , one gets This is the same condition (3.14). Thus, one keeps the same admissible pair (q, r ) = (q 1 , r 1 ). Thus, . The notations of the real numbers a, c, q, r, θ may change from term to another. Let us decompose the second term as follows .
The integrability condition reads This is the same (3.14). Thus taking previous admissible pairs, one gets For the second term, one computes with Strichartz and Hardy-Littlewood-Sobolev estimates via Sobolev injections Here, Take N c = −γ − 2 and N d = −γ + , for some 0 < << 1. Letting 2 This is the same condition (3.14). Thus, The estimates on the unit ball follow similarly. This finishes the proof of the first point. 2. Using Strichartz estimates and Lemma 2.14, one has with previous notations Compute, using Lemma 2.13 and Sobolev injections Here, Take the admissible pair (q, r ) ∈ such that With a direct computation, one has 2 < r < 2N N −2 . So, for N ≥ 4, one has s c r < r < 2N N −2 ≤ N . In the case N = 2, this condition reads This is satisfied if 2 + 2γ + α > 1 or 2 + 2γ + α < 1 and p < 1 + 1 1−(2+2γ +α) . For N = 3, the condition N > r is equivalent to This is satisfied if p ≥ 2. In conclusion, the condition p < p * gives 1 2 − 2 p−1 q > 0 and Let us control (I 2 ) . Compute, using Lemma 2.13, Here, taking for some 0 < << 1, N c = −γ + 2 and N d = −γ − , one gets Taking account of (3.15), one keeps the above admissible couple and Let us estimate the following term as above Here, This is the same condition as (3.15). Thus, similarly Let us estimate the following quantity.
Here, taking N a = −γ + 1 − and N b = −γ + 2 , for some 0 < << 1, one gets the condition This is the same condition as (3.15). Thus, similarly The estimation of the other terms follow similarly.

Taking account of Hardy-Littlewood-Sobolev and Soblev inequalities, one has for (q, r ) ∈ −s c and (q, r ) ∈ s c ,
Here, one takes θ = θ , 2 The first inequality is equivalent to q > 2 p−θ 1−s c . Let us take 0 < θ << 1 and A direct computation gives (3.16). Thus, Let us do similar estimations on the complementary of the unit ball. Take the same notations and write Here, one takes θ = θ , 2 The following choice satisfies the above conditions This closes the proof. Let, for T, R > 0 the centered ball B T (R) with radius R of the space Using the triangular inequality, let us write for w := u − v, Thanks to Lemma 3.2 via Strichartz estimate, one has Moreover, taking account of Stritarz estimates via Lemma 3.2, one writes Choose R := 2c u 0 Ḣ 1 ∩Ḣ sc and T > 0 such that c(T θ + T θ 2 ) < 1 2R 2( p−1) , one gets a contraction of B T (R). The proof follows with a Picard argument.

Non-global solutions
In this sub-section, one proves Theorem 2.5. Take a sequence of positive real numbers t n → T * and the sequences A computation gives Take a weak limit of v n inḢ s c ∩Ḣ 1 denoted by v. With the weak limit lower semi-continuity and λ(t n ) >> β n , one has for any R > 0, Now, with the lower semi-continuity and the compact embedding (3.13), one has The proof is complete.

Gagliardo-Nirenberg inequality
The goal of this sub-section is to prove Theorem 2.7. Let us start with a compact injection.

Lemma 4.1 Let N ≥ 2, γ, α satisfying (1.4) and 1 + α+2γ
Then, the following injection is compactḢ Proof Take a sequence of function such that sup n u n q c + (− ) s 2 u n < ∞, converging weakly to zero.
One proves that u n → 0. Since q < q * , one has q c < 2N N −2s and an interpolation argument gives ).
On the other hand, with Poincare inequality and compcat Sobolev injections, one has ).
The proof is ended.

Proof of the interpolation inequality (2.9)
First, using Lemmas 2.13 and 2.14, one has The estimate (2.9) is proved.

Proof of the Eq. (2.10)
Denote Using (2.5), there exists a sequence (v n ) inḢ s ∩ L p c such that Denoting for a, b ∈ R, the scaling u a,b := au(b·), one computes It follows that
Then, ψ n ψ inḢ s ∩ L q c and using Sobolev injection (4.17), we get for a sub-sequence denoted also (ψ n ), This implies that, when n goes to infinity .
The minimizer satisfies the Euler equation This completes the proof.

Proof of the Eq. (2.12)
Thanks to the previous sub-section, we know that C opt,s = 1 β s = Q[ψ], where ψ is given in (2.10). Take, for a, b ∈ R, the scaling ψ = φ a,b := aφ(b·). Then, the previous elliptic equation gives The proof is closed.

Proof of Theorem 2.9
In this sub-section, one proves Theorem 2.9. The next result regroups some local estimates about the source term.

Now, one computes with Strichartz and Hölder estimates via Sobolev injections
Thus, The integrability condition reads This is the same condition (4.18). So, following similarly, one gets . Now, let us estimate the term Here taking 0 < << 1, N a 2 = −γ + and N a 1 = −γ − 2 , such that 2N a := N a 1 + N a 2 < −2γ and 1 + α Thus, with previous computation, one has .
With the same reasoning, one gets .
Let us do similar estimations on the complementary of the unit ball. Take the same notations and write .

Global existence
Assume that u L ∞ T * (Ḣ s c ) < ∞ and u L ∞ T * (L qc ) < φ q c and T * < ∞. Then, Theorem 2.7 gives The proof is complete.
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