Ground state of the mass-critical inhomogeneous nonlinear Schrödinger functional

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Abstract

We study the ground state problem of the nonlinear Schrödinger functional with a mass-critical inhomogeneous nonlinear term. We provide the optimal condition for the existence of ground states and show that in the critical focusing regime there is a universal blow-up profile given by the unique optimizer of a Gagliardo-Nirenberg interpolation inequality.

Introduction

We consider the mass-critical nonlinear Schrödinger equationitu(t,x)=Δxu(x)+2qk(x)|u(t,x)|q2u(t,x),q=2+4d with u(t,)H1(Rd) and a given continuous function k:RdR.

When k=k0 is a constant (the homogeneous case), (1) boils down to the usual nonlinear Schrödinger equation studied extensively in the literature of dispersive partial differential equations (see e.g. [17]). In particular, in d=2 dimensions it comes from the famous Gross-Pitaevskii theory describing the Bose-Einstein condensation in quantum Bose gases [3], [15]. In the homogeneous case, it is well-known that if the power of nonlinearity q is replaced by any power smaller than 2+4/d, then the (homogeneous) equation (1) is globally well-posed for any initial datum in H1(Rd). On the other hand, if q=2+4/d and k0<0, then the global well-posedness is only guaranteed if the mass of the initial datum is smaller than a critical value, namelyu(0,.)L2(Rd)2<m0:=(q2|k0|)2q2QL2(Rd)2, where Q is the unique radial positive solution to the equation (6). For this reason, the case q=2+4/d is called mass-critical. Moreover, if q=2+4/d andu(0,.)L2(Rd)2=m0, then even if u0 is very smooth, it is possible that the solution to (1) blows up in finite time, namely the solution exists in an interval [0,T) and the kinetic energy is unbounded at the critical time:limtTuL2=+. In fact, all possibilities of such blow-up solutions (called minimal-mass blow-up solutions) have been characterized in the seminal paper of Merle [11].

The non-constant potential k (the inhomogeneous case) corresponds to a inhomogeneous interacting effect and it arises naturally in nonlinear optics for the propagation of laser beams. Mathematically, this case is interesting as it breaks the large group of symmetries of the homogeneous case. The study of the nonlinear Schrödinger equation with inhomogeneous nonlinearity was initiated by Merle [12] where he obtained a sufficient condition for the nonexistence of minimal mass blow-up solutions. On the other hand, minimal mass blow-up solutions exist if k is sufficiently smooth and flat around its minima; see Banica-Carles-Duyckaerts [1] and Krieger-Schlag [5]. In d=2 dimensions, the full classification of minimal mass blow-up solutions in the inhomogeneous case was solved by Raphael-Szeftel [16].

In the present paper, we are interested in the ground state solution of (1). To be precise, we will study the variational problemEk=inf{Ek(u):Rd|u|2=1} associated to the nonlinear Schrödinger functionalEk(u)=Rd|u(x)|2dx+Rdk(x)|u(x)|2+4/ddx,uH1(Rd). By the standard techniques from calculus of variations, any minimizer u0 of Ek in (3) is a solution to the stationary nonlinear Schrödinger equationΔu(x)+2qk(x)|u(x)|q2u(x)=μu(x) with a constant μR (which is the Lagrange multiplier associated to the mass constraint uL2=1). Consequently,u(t,x)=eitμu0(x) is a solitary plane-wave solution to the time-dependent problem (1).

Similarly to time-dependent problem studied in [12], [1], [5], [16], a critical feature of the ground state problem (3) appears when infxk(x) crosses the threshold a which is the optimal constant in the Gagliardo-Nirenberg interpolation inequality:(Rd|u(x)|2dx)(Rd|u(x)|2dx)2/daRd|u(x)|2+4/ddx,uH1(Rd). This inequality has been well studied in [2], [18], [10], [6]. It is known that (5) has a unique optimizer Q up to translations and dilations. In fact, Q is the unique radial positive solution to the equationΔQ+QQ1+4/d=0.

Our first result concerns the existence and nonexistence of minimizers of the variational problem Ek in (3).

Theorem 1 Existence and nonexistence of minimizers

Assume that kC(Rd).

  • (i)

    (Subcritical case: existence.) If infk>a andRd1(k(x)+a)d/2dx< then Ek>0 and it has a minimizer.

  • (ii)

    (Subcritical case: nonexistence.) If infk>a andlim sup|x|k(x)|x|2<, then Ek=0 and it has no minimizer.

  • (iii)

    (Critical case.) If infk=a=k(x0) for some x0Rd andlimxx0k(x)k(x0)|xx0|2=0, then Ek=0 and it has no minimizer except the case ka.

  • (iv)

    (Supercritical case.) If infk<a, then Ek=.

Remark 2

In the subcritical case infk>a, it is remarkable that the growth of k(x) as |x| really matters the existence of minimizers. Note that in the integrability condition (7) holds iflim inf|x|k(x)|x|2+ε>0,for some ε>0. Thus the existence condition (7) in (i) and the nonexistence condition (8) in (ii) are mostly the complement to each other.

Remark 3

In the critical case infk=a, the condition (9) means that k is flat enough around its minimum point x0. If kC2(Rd), then (9) is equivalent to the degeneracy condition2k(x0)=0, where 2k is the Hessian matrix of k.

On the other hand, the opposite condition to (9) thatlimxx0k(x)k(x0)|xx0|2ε>0,for some ε>0, was assumed by Merle [12] when he proved the nonexistence of minimal mass blow-up solutions for the time-dependent problem. In fact, (10) implies the local integrability of (k(x)+a)d/2 (note that if (k(x)+a)d/2 is integrable, then by following the proof of Theorem 1 (i) we can prove that Ek>0; see Remark 5). However, (10) never happens if kC2. From our analysis, the casekC2,2k(x0)0 is still missing, and it is indeed related to an open question in [12, Remark after Prop. 5.4, page 76]. The difficult case (11) has been studied by Raphael-Szeftel [16] in the context of minimal mass blow-up solutions in R2, but it is not clear to us how to transfer their techniques to the ground state problem in the present paper.

Next, we concentrate on the existence case (i) in Theorem 1, and analyze the blow-up behavior when infk tends to a. To make the analysis rigorous, we need to impose some explicit behavior of k around its minima.

Assumption for the blow-up result. For the following blow-up theorem, we will assume thatk(x)=K(x)a with K:RdR a fixed function satisfying:

  • (i)

    infK=0 and K has finite minima {xj}j=1J.

  • (ii)

    For any j, there exists pj>0 such thatlimxxjK(x)|xxj|pj=λj>0.

  • (iii)

    Kd/2 is integrable away from {xj}j=1J, namely for any R>0,minj|xxj|>RK(x)d/2dx<.

Let us denote p=max{p1,...,pJ}, λ=min{λj:pj=p} and Z={zj:pj=p,λj=λ} (the set of flattest minima of K).

Theorem 4 Blow-up profile

We consider the variational problem (3) with k(x)=K(x)a, where K satisfies the above conditions with p>2. Let Q0=Q/QL2 with Q be the unique positive radial solution to (6). Then we havelimaaEk(aa)12/p=infξ>0[ξ2Rd|Q0|2+4/d+ξ2pλRd|x|p|Q0|2+4/d]=(λp2Rd|x|p|Q0|2+4/d)2/p(pp2Rd|Q0|2+4/d)12/p. Moreover, if ua is a minimizer for Ek, then for any sequence ana, there exist a subsequence ana and an element zZ (the set of flattest minima of k) such that up to a phaselimana(aan)d/(2p)uan(z+(aan)1/px)bd/2Q0(bx) strongly in H1(Rd), where b is the optimizer for the right side of (12):b=((p2)λRd|x|p|Q0|2+4/d2Rd|Q0|2+4/d)1/p. Moreover, if Z has a unique element, then (13) holds true for the whole family aa.

As we will see from the proof of Theorem 4, when aa, although the ground state energy tends to 0 as O((aa)12/p) by (12), the kinetic energy uaL22 tends to infinity as O((aa)2/p) (see (22) below). This blow-up phenomenon is analogous to (2) in the time-dependent problem. Actually, we are able to provide exact details of the blow-up phenomenon by (12).

On one hand, our study can be seen as a complement of the previous works in the time-dependent problem [12], [5], [1], [16]. On the other hand, the ground state problem is different and has its own difficulty. In the time-dependent problem, the energy functional is a constant in time, and the main interest is the blow-up behavior at the critical time. In this case, the a-priori information on the initial datum and the blow-up time is crucial for the analysis. In our problem, the energy functional has to be minimized under appropriate constraints, and the main interest is the blow-up behavior at the local minimum of the inhomogeneous function k(x). We have to work on the full functional space H1(Rd) where the minimization problem is formulated, and hence we are not granted any a-priori information on the ground states and the ground state energy. In particular, our results can not be obtained by following the techniques in the time-dependent problem.

In the following we will prove Theorem 1 in Section 2 and prove Theorem 4 in Section 3. The proof of Theorem 1 is based on the concentration-compactness method of Lions [8], [9]. The proof of Theorem 4 is obtained by a concentration argument, inspired from the paper of Guo-Seiringer [4] who studied the blow-up profile of the Bose-Einstein condensation in 2D with the homogeneous nonlinearity (k=const) and a trapping potential V(x)=|x|s with s>0 (see also [13] for a related result with attractive external potentials). Here our main task is to deal with the inhomogeneous nonlinearity, which makes the analysis both complicated and interesting in several places.

Acknowledgments

I would like to thank a referee for helpful suggestions.

Section snippets

Existence and nonexistence of minimizers

Proof of Theorem 1

(i) By the Gagliardo-Nirenberg inequality (5), for all uH1(Rd) with uL2=1 we haveEk(u)=Rd|u(x)|2+Rdk(x)|u(x)|2+4/ddxRd|u(x)|2+min{infk,0}Rd|u(x)|2+4/ddx(1+min{infk,0}a)Rd|u|20. Since infk>a, we deduce that Ek0. Moreover, if {un} is a minimizing sequence for Ek, then {un} is bounded in H1(Rd). By the Banach-Alaoglu theorem, up to a subsequence, we can assume that unu weakly in H1(Rd).

Let us prove that unu strongly in L2(Rd). First, since unu weakly in H1(Rd), Sobolev's

Blow-up analysis

Proof of Theorem 4

Step 1: Energy upper bound. This is done similarly as in the proof of Theorem 1. Without loss of generality let us assume that x1Z. Then for any ε>0, there exists r=rε>0 such thatk(x)(λ+ε)|xx1|pa,xB(x1,r). Then by the variational principle, for any uH1(Rd), supported on B(0,r) with uL2=1 we haveEkEk(u(.x1))Rd|u(x)|2+Rd((λ+ε)|x|pa)|u(x)|2+4/ddx.

Now we choose a trial function u. Let Q0=Q/QL2 be the (normalized) optimizer of the Gagliardo-Nirenberg inequality (5). Take a smooth

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