[Paper Review] Normalized solutions to the mixed dispersion nonlinear Schr\\"odinger equation in the mass critical and supercritical regime

Denis Bonheure, Jean‐Baptiste Casteras+2 more|arXiv (Cornell University)|Feb 26, 2018

Advanced Mathematical Physics Problems54 references79 citations

TL;DR

The paper studies existence, multiplicity, and stability of normalized (mass-constrained) solutions to the mixed dispersion NLS with biharmonic and Laplacian terms under mass constraint, focusing on mass-critical and mass-supercritical regimes, and analyzes ground states, radial solutions, concentration, and dynamics.

ABSTRACT

In this paper, we study the existence of solutions to the mixed dispersion nonlinear Schr\\"odinger equation $$ \\gamma \\Delta ^2 u -\\Delta u + \\alpha u=|u|^{2 \\sigma} u, \\quad u \\in H^2(\\R^N), $$ under the constraint $$ \\int_{\\R^N}|u|^2 \\, dx =c>0. $$ We assume $\\gamma >0, N \\geq 1, 4 \\leq \\sigma N < \\frac{4N}{(N-4)^+}$, whereas the parameter $\\alpha \\in \\R$ will appear as a Lagrange multiplier. Given $c \\in \\R^+$, we consider several questions including the existence of ground states, of positive solutions and the multiplicity of radial solutions. We also discuss the stability of the standing waves of the associated dispersive equation.

Motivation & Objective

• Investigate existence of normalized (mass-constrained) solutions to the mixed dispersion NLS with gamma > 0 under the constraint ∫|u|^2 = c, identifying ground states and multiplicity.
• Analyze the role of the Lagrange multiplier α and the constraint set M(c) in obtaining constrained critical points.
• Study the behavior of ground states, radial solutions, and concentration phenomena as mass c varies, including stability/instability of standing waves.

Proposed method

• Analyze the elliptic equation gamma Δ^2 u - Δu + αu = |u|^{2σ}u with ∫|u|^2 = c, via energy functional E(u) on S(c).
• Define the constrained manifold M(c) = {u ∈ S(c) : Q(u) = 0} with Derrick-Pohozaev type identity Q(u).
• Prove coercivity and boundedness from below of E on M(c) (and radii R) to obtain minimizers Γ(c) and ground states for c in (c0, cN,σ).
• Show existence of a Palais–Smale sequence on S(c) at level Γ(c) and derive strong convergence to a ground state u_c with α_c > 0.
• Employ concentration-compactness and genus theory to obtain infinitely many radial solutions when 4 < σN < 4*, and results for σN = 4, including radial ground states and sign-changing solutions.
• Prove stability results for the dispersive equation and instability by blow-up for radial ground states.

Experimental results

Research questions

• RQ1 Does there exist a ground state solution of gamma Δ^2 u - Δu + αu = |u|^{2σ}u under the mass constraint ∫|u|^2 = c for c in (c0, cN,σ)?
• RQ2 Can one obtain infinitely many radial solutions via variational methods for the mass-constrained problem, and how does the exponent σN affect this multiplicity?
• RQ3 What is the behavior of ground states as c approaches critical mass cN* (concentration phenomenon) in the mass-critical case?
• RQ4 Are the standing waves associated with ground states stable or unstable under the time-dependent mixed dispersion NLS, and in which regimes?
• RQ5 How does the presence of the Lagrange multiplier α influence positivity, sign-changing nature, and symmetry of constrained minimizers?

Key findings

• For mass-critical σN = 4, there exists cN* > 0 with m(c) = 0 for 0 < c ≤ cN* and m(c) = -∞ for c > cN*, and no nontrivial solution on S(c) for c ≤ cN*.
• There exists cN,σ > c0 such that for c ∈ (c0, cN,σ) the constrained problem (Pc) has a ground state u_c with E(u_c) = Γ(c) and α_c > 0.
• If 4 < σN < 4* and N ≥ 2, (Pc) possesses infinitely many radial solutions for c in (0, cN,σ); in the mass-critical case (σN = 4) there are infinitely many radial solutions for suitable large c in certain dimensions.
• In dimensions N = 3 or 4, there exist radial ground states for any c > c0 when σN ≥ 4, with a positive Lagrange multiplier and, for given c, sign-changing radial ground states can occur.
• For the dispersive equation, radial ground states are unstable by blow-up (finite or infinite time) in the mass-constrained setting with 4 ≤ σN < 4*; when α is fixed, similar instability results hold in the mass-supercritical/subcritical regimes.