Construction of two-bubble solutions for the energy-critical NLS in dimension 6

Abstract: We construct pure two-bubble solutions for the energy-critical focusing nonlinear Schrödinger equation in space dimension $N = 6$. They are global in (at least) one time direction and approach a superposition of two stationary states, both centered at the origin. One of the bubbles develops at scale $1$, whereas the length scale of the other converges to $0$ at rate $e<sup>{-|t|}$. The phases of the two bubbles form the right angle. Such solutions were previously constructed in dimension $N \geq 7$. The six-dimension case presents specific difficulties, as the ground state does not belong to $\dot H<sup>{-1}$. This prevents the use of the standard method of removing linear terms in modulation equations via suitable orthogonality conditions, due to loss of coercivity of the energy functional. The main novelty of this work is the introduction of modified modulation parameters to overcome this issue; these can be viewed as an analog of a normal form transformation in the context of modulation analysis. We also establish new coercivity estimates for the linearized energy, whose positive constants depend explicitly on the choice of the orthogonality conditions.

• The paper constructs a global radial pure two-bubble solution in six dimensions that approaches two Aubin–Talenti ground states at right-angle phases, with energy exactly 2E(W) and concentration rate e^{-|t|}.
• The authors overcome the failure of W to belong to H^{-1}(R^6) by combining localized orthogonality, R-dependent coercivity estimates, modified modulation parameters, spectral analysis, and a virial correction.
• A topological shooting argument controls the final unstable mode and completes the construction, while the paper leaves open nonexistence below six dimensions, uniqueness of the right-angle configuration, and multi-bubble dynamics.

The main result

The paper constructs a global radial solution $u: (-\infty, T_0] \to \mathcal{E}$ of the focusing energy-critical nonlinear Schrödinger equation in dimension $N = 6$,

$i\partial_t u + \Delta u + |u|u = 0,$

such that, as $t \to -\infty$,

$$\big\|u(t) - (-iW + W_{e^{-|t|}})\big\|_{\mathcal{E}} \leq C e^{-\frac{1}{2}|t|},$$

where $W(x) = (1 + |x|^2/24)^{-2}$ is the Aubin–Talenti ground state. The solution is a pure two-bubble: it approaches a superposition of two stationary states with no radiation, both centered at the origin, one at scale $\mu(t) \simeq 1$ and the other at scale $\lambda(t) \to 0$ at the exponential rate $e^{-|t|}$. The phases satisfy $\zeta(t) \simeq -\pi/2$ and $N = 6$0, i.e., they form a right angle. By energy conservation and decoupling, necessarily $N = 6$1.

This result completes the program initiated by Jendrej [Jacek:nls], who constructed such solutions for $N = 6$2. The authors state that a heuristic argument (given in an appendix via a reduced Lagrangian) indicates pure two-bubble solutions should not exist for $N = 6$3, so dimension 6 is the only remaining case. They also conjecture that the right-angle phase configuration is the only one compatible with a pure two-bubble.

Why dimension 6 is different

The central obstruction is that in $N = 6$4 the ground state $N = 6$5 does not belong to $N = 6$6. In dimensions $N = 6$7, standard modulation theory removes linear terms from the modulation equations by imposing global orthogonality conditions against the kernel elements of the adjoint linearized operator; this relies on coercivity of the linearized energy. In six dimensions this coercivity fails, so the authors must work with localized orthogonality conditions involving cut-offs $N = 6$8, $N = 6$9, where $i\partial_t u + \Delta u + |u|u = 0,$0 is a large fixed constant. A second structural difference is that the modulation parameters now decay exponentially rather than polynomially, which the authors justify by a Lagrangian reduction: plugging the two-bubble ansatz into the Lagrangian yields the reduced system $i\partial_t u + \Delta u + |u|u = 0,$1, $i\partial_t u + \Delta u + |u|u = 0,$2, predicting $i\partial_t u + \Delta u + |u|u = 0,$3.

The proof combines four ingredients carried over from prior work—modulation analysis, spectral theory of the linearized operator, a topological (Ważewski-type) argument, and concentration-compactness rigidity—with two genuinely new components: modified modulation parameters acting as an analog of a normal form transformation, and new coercivity estimates whose constants depend explicitly on the localization parameter $i\partial_t u + \Delta u + |u|u = 0,$4.

Variational estimates

The linearization around $i\partial_t u + \Delta u + |u|u = 0,$5 is governed by the operators $i\partial_t u + \Delta u + |u|u = 0,$6 and $i\partial_t u + \Delta u + |u|u = 0,$7, with $i\partial_t u + \Delta u + |u|u = 0,$8 and $i\partial_t u + \Delta u + |u|u = 0,$9 radially. There exist Schwartz functions $t \to -\infty$0 satisfying $t \to -\infty$1, $t \to -\infty$2, defining the unstable/stable modes. A key spectral input for closing the bootstrap on the unstable directions is the explicit bound

$t \to -\infty$3

proved by an elementary self-adjointness argument using $t \to -\infty$4. This bound ensures $t \to -\infty$5, which is exactly what allows the fundamental-solution estimate for the unstable mode $t \to -\infty$6 to close.

The coercivity analysis proceeds through localized bounds of the form

$t \to -\infty$7

with analogous estimates for $t \to -\infty$8, plus refined versions on annuli $t \to -\infty$9 or $$\big\|u(t) - (-iW + W_{e^{-|t|}})\big\|_{\mathcal{E}} \leq C e^{-\frac{1}{2}|t|},$$0 with $$\big\|u(t) - (-iW + W_{e^{-|t|}})\big\|_{\mathcal{E}} \leq C e^{-\frac{1}{2}|t|},$$1 and $$\big\|u(t) - (-iW + W_{e^{-|t|}})\big\|_{\mathcal{E}} \leq C e^{-\frac{1}{2}|t|},$$2. These are combined into a coercivity estimate near the two-bubble configuration:

$$\big\|u(t) - (-iW + W_{e^{-|t|}})\big\|_{\mathcal{E}} \leq C e^{-\frac{1}{2}|t|},$$3

A notable technical point is the logarithmic loss $$\big\|u(t) - (-iW + W_{e^{-|t|}})\big\|_{\mathcal{E}} \leq C e^{-\frac{1}{2}|t|},$$4, which arises from the borderline integrability of $$\big\|u(t) - (-iW + W_{e^{-|t|}})\big\|_{\mathcal{E}} \leq C e^{-\frac{1}{2}|t|},$$5 in $$\big\|u(t) - (-iW + W_{e^{-|t|}})\big\|_{\mathcal{E}} \leq C e^{-\frac{1}{2}|t|},$$6-type norms in six dimensions; it is ultimately absorbed by choosing $$\big\|u(t) - (-iW + W_{e^{-|t|}})\big\|_{\mathcal{E}} \leq C e^{-\frac{1}{2}|t|},$$7 large enough relative to the bootstrap rates.

Modulation analysis and the normal-form correction

Writing $$\big\|u(t) - (-iW + W_{e^{-|t|}})\big\|_{\mathcal{E}} \leq C e^{-\frac{1}{2}|t|},$$8 with the four localized orthogonality conditions, differentiating them yields a $$\big\|u(t) - (-iW + W_{e^{-|t|}})\big\|_{\mathcal{E}} \leq C e^{-\frac{1}{2}|t|},$$9 linear system for the modulation parameters. Under bootstrap assumptions ($W(x) = (1 + |x|^2/24)^{-2}$0, $W(x) = (1 + |x|^2/24)^{-2}$1, $W(x) = (1 + |x|^2/24)^{-2}$2, $W(x) = (1 + |x|^2/24)^{-2}$3), the rough estimates give $W(x) = (1 + |x|^2/24)^{-2}$4 but only $W(x) = (1 + |x|^2/24)^{-2}$5 and $W(x) = (1 + |x|^2/24)^{-2}$6—insufficient to close the bootstrap.

The remedy is the introduction of modified parameters $W(x) = (1 + |x|^2/24)^{-2}$7 defined by subtracting integral correction terms $W(x) = (1 + |x|^2/24)^{-2}$8 built from inner products of $W(x) = (1 + |x|^2/24)^{-2}$9 against the localized kernel directions, integrated from $\mu(t) \simeq 1$0 to $\mu(t) \simeq 1$1 with a time-dependent cut-off scale $\mu(t) \simeq 1$2. The authors show these corrections are small ($\mu(t) \simeq 1$3, etc.) and that the modified parameters satisfy improved equations:

$\mu(t) \simeq 1$4

$\mu(t) \simeq 1$5

where $\mu(t) \simeq 1$6 is the quadratic-in-$\mu(t) \simeq 1$7 interaction term. This mechanism is explicitly described as an analog of a normal form transformation within modulation analysis, and it is the paper's principal methodological contribution. The unstable-mode coordinates obey, up to small errors,

$\mu(t) \simeq 1$8

Closing the bootstrap

The bootstrap closes in stages. The parameters $\mu(t) \simeq 1$9 and $\lambda(t) \to 0$0 are controlled by direct integration. The pair $\lambda(t) \to 0$1 is handled simultaneously via the fundamental solutions of their linear ODEs, using $\lambda(t) \to 0$2 to guarantee decay of $\lambda(t) \to 0$3 at rate $\lambda(t) \to 0$4.

The most delicate step is the control of $\lambda(t) \to 0$5. The term $\lambda(t) \to 0$6 in the refined equation for $\lambda(t) \to 0$7 is quadratic in $\lambda(t) \to 0$8—the critical size—and cannot be absorbed directly. The authors introduce a virial functional: a function $\lambda(t) \to 0$9 approximating $e^{-|t|}$0 inside radius $e^{-|t|}$1 and constant outside, together with associated operators $e^{-|t|}$2, $e^{-|t|}$3 approximating scaled versions of $e^{-|t|}$4 and $e^{-|t|}$5. Defining

$e^{-|t|}$6

a Pohozaev-type computation shows $e^{-|t|}$7, which after integration from $e^{-|t|}$8 (using $e^{-|t|}$9 and $\zeta(t) \simeq -\pi/2$0) yields the two-sided bound $\zeta(t) \simeq -\pi/2$1. This estimate, fed back into the coercivity inequality, then gives $\zeta(t) \simeq -\pi/2$2, closing all estimates except $\zeta(t) \simeq -\pi/2$3.

The remaining unstable mode $\zeta(t) \simeq -\pi/2$4 is controlled by a shooting argument: initial data are chosen in a one-parameter family indexed by $\zeta(t) \simeq -\pi/2$5, and if no parameter produced a solution satisfying the full bootstrap up to $\zeta(t) \simeq -\pi/2$6, one would obtain a continuous retraction $\zeta(t) \simeq -\pi/2$7 fixing the boundary—a topological impossibility. Finally, a sequence of solutions on intervals $\zeta(t) \simeq -\pi/2$8 with $\zeta(t) \simeq -\pi/2$9 converges weakly along a subsequence, and the weak stability lemma upgrades this to a genuine solution on $N = 6$00, completing the proof.

Limitations and open questions

Several restrictions are inherent to the construction. It is confined to radial data, and the solution is constructed only globally in the past time direction; behavior as $N = 6$01 is not addressed. The right-angle phase configuration is asserted to be the only viable one, but this is stated as an expectation rather than proved. The nonexistence heuristic for $N = 6$02 presented in the appendix is formal, not a theorem. More broadly, the Soliton Resolution Conjecture for the energy-critical NLS remains completely open even in this setting; the present construction provides a single example of nondispersive dynamics rather than a classification. Whether multi-bubble solutions with more than two bubbles exist in dimension 6, and whether the exponential concentration rate $N = 6$03 is sharp among all such solutions, are questions the paper leaves unanswered.

Conclusion

The paper extends the construction of pure two-bubble solutions for the energy-critical focusing NLS from dimensions $N = 6$04 to the critical case $N = 6$05, where the failure of $N = 6$06 invalidates the standard modulation framework. The two innovations—localized orthogonality conditions with $N = 6$07-dependent coercivity constants, and modified modulation parameters functioning as a normal form transformation—resolve the loss of coercivity and yield the refined decay rates needed to close the bootstrap, supplemented by a virial correction to control the phase parameter and a Ważewski argument for the last unstable mode. The result settles the last case left open by the earlier high-dimensional construction and provides a model instance of nondispersive, radiation-free dynamics at twice the ground-state energy.