[Paper Review] Prescribed mass ground states for a doubly nonlinear Schr\"odinger equation in dimension one

Filippo Boni, Simone Dovetta|arXiv (Cornell University)|Jul 18, 2019

Advanced Mathematical Physics ProblemsMathematics54 references22 citations

TL;DR

This paper establishes existence and uniqueness of ground states for a one-dimensional doubly nonlinear Schrödinger equation with both a standard power nonlinearity and a pointwise nonlinearity localized at the origin, under various criticality regimes. It proves that ground states exist and are unique at every mass when both nonlinearities are subcritical, at a threshold mass when one is critical and the other subcritical, and only at a specific critical mass in the doubly critical case, where the critical mass is lower than in the individual critical cases due to nonlinear interaction.

ABSTRACT

We investigate the problem of existence and uniqueness of ground states at fixed mass for two families of focusing nonlinear Schr\"odinger equations on the line. The first family consists of NLS with power nonlinearities concentrated at a point. For such model, we prove existence and uniqueness of ground states at every mass when the nonlinearity power is $L^2-$subcritical and at a threshold value of the mass in the $L^2-$critical regime. The second family is obtained by adding a standard power nonlinearity to the previous setting. In this case, we prove existence and uniqueness at every mass in the doubly subcritical case, namely when both the powers related to the pointwise and the standard nonlinearity are subcritical. If only one power is critical, then existence and uniqueness hold only at masses lower than the critical mass associated to the critical nonlinearity. Finally, in the doubly critical case ground states exist only at critical mass, whose value results from a non--trivial interplay between the two nonlinearities.

Motivation & Objective

• To establish the existence and uniqueness of ground states at fixed mass for a doubly nonlinear Schrödinger equation on the real line with a standard power nonlinearity and a pointwise nonlinearity at the origin.
• To analyze the behavior of the energy functional under different criticality regimes: doubly subcritical, single critical (one critical, one subcritical), and doubly critical (both critical).
• To determine the precise threshold mass values at which ground states exist, particularly in the critical regimes, and to investigate the interplay between the two nonlinearities in shaping the critical mass.
• To prove that in the doubly critical case (p=6, q=4), ground states exist only at a unique critical mass µ∗, which is strictly less than the critical masses for the individual standard and pointwise nonlinearities.

Proposed method

• The study analyzes the energy functional Fp,q(u) = 1/2∥u′∥²₂ − 1/p∥u∥ᵖₚ − 1/q|u(0)|^q on H¹(R) under the mass constraint ∥u∥²₂ = µ.
• It employs variational methods, including coercivity arguments and Gagliardo-Nirenberg inequalities, to establish boundedness and existence of minimizers.
• For the doubly critical case (p=6, q=4), the paper uses scaling invariance (F6,4(uλ) = λ²F6,4(u)) and upper semicontinuity of the energy functional to characterize the critical mass µ∗.
• It proves existence of ground states at µ∗ by contradiction and weak convergence arguments, showing that any weak limit of minimizing sequences at µ∗ must be nontrivial and achieve the infimum.
• The paper constructs solutions by pasting two soliton pieces (solutions to u'' + |u|^{p-2}u = ωu) satisfying the nonlinear jump condition at x=0: u′(0⁻) − u′(0⁺) = |u(0)|^{q−2}u(0).
• It uses the explicit form of the soliton φ(x) = [p/(2ω)(1 − tanh²(...))]^{1/(p−2)} to derive the critical mass µ∗ via integration of the matching condition.

Experimental results

Research questions

• RQ1 Under what conditions on the nonlinearities (p, q) do ground states exist at every fixed mass µ > 0?
• RQ2 What happens to the existence and uniqueness of ground states when one or both nonlinearities are at their L²-critical power?
• RQ3 How does the interaction between a standard power nonlinearity and a pointwise nonlinearity at the origin affect the critical mass threshold?
• RQ4 In the doubly critical case (p=6, q=4), is the critical mass µ∗ equal to, greater than, or less than the critical masses for the individual nonlinearities?

Key findings

• For 2 < p < 6 and 2 < q < 4 (doubly subcritical regime), ground states exist and are unique at every µ > 0.
• When p = 6 (L²-critical) and 2 < q < 4 (subcritical), ground states exist and are unique only for µ < √3 π / 2, with energy F6,q(µ) = −∞ for µ ≥ √3 π / 2.
• When 2 < p < 6 (subcritical) and q = 4 (L²-critical), ground states exist and are unique only for µ < 2, with energy Fp,4(µ) = −∞ for µ ≥ 2.
• In the doubly critical case (p=6, q=4), ground states exist only at a unique critical mass µ∗ = √3 (π/2 − arcsin(√(3/7))), and F6,4(µ) = 0 for µ ≤ µ∗, F6,4(µ) = −∞ for µ > µ∗.
• The critical mass µ∗ is strictly less than both the individual critical masses √3 π / 2 and 2, demonstrating a nontrivial interaction between the two nonlinearities that reduces the threshold.
• All solutions of the stationary equation in the doubly critical case share the same mass µ∗, and F6,4(ϕω) = 0 for all ω > 0, confirming that the entire family {ϕω}ω>0 are ground states at µ∗.