[Paper Review] Normalized solutions for the fractional NLS with mass supercritical nonlinearity

Luigi Appolloni, Simone Secchi|arXiv (Cornell University)|May 30, 2020

Advanced Mathematical Physics Problems20 references32 citations

TL;DR

The paper proves existence of ground state solutions with prescribed L2-norm for a fractional NLS with mass supercritical nonlinearity, and shows multiplicity in the radial setting, along with detailed behavior of the ground-state energy as a function of mass.

ABSTRACT

We investigate the existence of solutions to the fractional nonlinear Schr\\"{o}dinger equation $(-\\Delta)^s u = f(u)$ with prescribed $L^2$-norm $\\int_{\\mathbb{R}^N} |u|^2 \\, dx =m$ in the Sobolev space $H^s(\\mathbb{R}^N)$. Under fairly general assumptions on the nonlinearity $f$, we prove the existence of a ground state solution and a multiplicity result in the radially symmetric case.

Motivation & Objective

• Motivate the study of bound-state solutions with fixed mass for the fractional NLS with nonlinearity f(u).
• Develop a variational framework on the L2-sphere to obtain ground states and multiplicity results.
• Determine the parameter dependence of the ground-state energy and Lagrange multiplier.
• Extend known local case strategies to the nonlocal fractional setting (0<s<1).
• Provide a radial-solution multiplicity result under symmetry assumptions.

Proposed method

• Formulate the constrained variational problem for I(u) on the L2-sphere S_m and its Pohozaev manifold P_m.
• Define the energy I(u)=1/2 [u]_{H^s}^2 - ∫ F(u) dx with F(t)=∫_0^t f(σ)dσ and the auxiliary tilde F(t)=f(t)t-2F(t).
• Impose standing assumptions (f0)-(f6) on the nonlinearity f to ensure subcritical Sobolev growth and mass supercriticality.
• Use a scaling (ρ*u)(x)=e^{Nρ/2}u(e^{ρ}x) and the associated fiber map to locate the Pohozaev constraint P(ρ*u)=0.
• Employ min-max/ linking-type variational methods and a Brezis-Lieb-type splitting to obtain PS sequences and convergence on P_m.
• Prove regularity and monotonicity properties of the ground-state energy E_m and analyze its asymptotics as m→0+ and m→∞.
• Extend the radial compactness argument to obtain infinitely many radial solutions when N>2.

Experimental results

Research questions

• RQ1 Can a ground-state solution with prescribed L2-norm m exist for the fractional NLS with mass supercritical nonlinearity?
• RQ2 Does a positive Lagrange multiplier μ arise for ground-state solutions under the given nonlinearity assumptions?
• RQ3 How does the ground-state energy E_m behave as a function of the prescribed mass m (continuity, monotonicity, and asymptotics)?
• RQ4 Is there a multiplicity of radially symmetric solutions in the radial setting for each m>0?
• RQ5 To what extent can Pohozaev-type constraints and fiber-map techniques be extended from local to nonlocal (fractional) NLS?

Key findings

• There exists a positive ground state for any m>0 under assumptions (f0)-(f5).
• For every ground state, the associated Lagrange multiplier μ is positive.
• The ground-state energy E_m is positive, continuous, and strictly decreasing in m, with E_m → ∞ as m→0+ and E_m → 0 as m→∞.
• There exist infinitely many radial solutions for any m>0 when N>2, with the energy I(u_k)→∞ as k→∞.
• The paper develops a variational framework and a fiber map that yield a constrained critical point system on P_m and extracts PS sequences.
• The results extend known strategies from the local case (s=1) to the nonlocal fractional setting (0<s<1).