[Paper Review] Normalized solutions of mass supercritical Schr\\"odinger equations with potential

Thomas Bartsch, Riccardo Molle+2 more|arXiv (Cornell University)|Aug 17, 2020

Advanced Mathematical Physics ProblemsMathematics30 references112 citations

TL;DR

This paper establishes the existence of normalized solutions to mass supercritical nonlinear Schrödinger equations with potentials decaying at infinity, using a novel min-max argument on the L2-sphere. It proves that under explicit, non-perturbative conditions on the potential V (including singularities), a solution (u, λ) ∈ H¹(R^N) × R⁺ exists for any ρ > 0 with ‖u‖₂ = ρ, even when the functional is unbounded below on the constraint manifold. The key contribution is a new linking-type variational method overcoming non-compactness and lack of Palais-Smale condition in the supercritical regime.

ABSTRACT

This paper is concerned with the existence of normalized solutions of the nonlinear Schrodinger equation-Delta u + V(x)u + lambda u = vertical bar u vertical bar(p-2)u in R-Nin the mass supercritical and Sobolev subcritical case 2 + 4/N < p < 2*. We prove the existence of a solution (u, lambda) is an element of H-1 (R-N) x R+ with prescribed L-2-norm parallel to u parallel to(2) = rho under various conditions on the potential V : R-N -> R, positive and vanishing at infinity, including potentials with singularities. The proof is based on a new min-max argument.

Motivation & Objective

• To establish the existence of normalized solutions (u, λ) ∈ H¹(R^N) × R⁺ to the nonlinear Schrödinger equation −Δu + V(x)u + λu = |u|^{p−2}u with fixed L²-norm ‖u‖₂ = ρ.
• To address the challenge of the functional being unbounded below on the L²-sphere Sρ in the mass supercritical regime (2 + 4/N < p < 2*), where standard minimization fails.
• To develop and apply a new min-max argument based on a linking structure to find critical points of the energy functional constrained to Sρ.
• To extend existence results to potentials V ≥ 0 with V(x) → 0 as |x| → ∞, including those with singularities (poles), under explicit, non-perturbative conditions on V and W(x) = V(x)|x|.

Proposed method

• Introduce a new min-max argument on the L²-sphere Sρ = {u ∈ H¹(R^N) : ‖u‖₂ = ρ} to find critical points of the energy functional F(u) = ½∫(|∇u|² + V(x)u²)dx − 1/p ∫|u|ᵖdx.
• Construct a family of scaled ground states Zρ from the autonomous problem (−Δ + 1)U = |U|^{p−2}U to serve as comparison profiles.
• Define a linking structure using the transformation h ⋆ Zρ(x) = e^{h} Zρ(e^{h}x), which shifts the profile in scale and translates in space.
• Prove that the functional F achieves a mountain pass level mρ on the autonomous case (V ≡ 0), and use this as a reference to build a linking geometry when V ≥ 0 is small or decaying.
• Establish boundedness of Palais-Smale sequences on Sρ by deriving a priori estimates using the Aubin-Talenti sharp constant A and the L^N/2 norm of V, ensuring convergence to a solution.
• Use the radial case as a model: apply a mountain pass argument on the radial subspace H¹_rad(R^N) ∩ Sρ to obtain a solution when V is radial or satisfies radial-type assumptions.

Experimental results

Research questions

• RQ1 Under what conditions on the potential V ≥ 0 with V(x) → 0 as |x| → ∞ does the normalized Schrödinger equation have a solution in H¹(R^N) × R⁺ for any ρ > 0 in the mass supercritical regime 2 + 4/N < p < 2*?
• RQ2 Can a variational method be constructed to find normalized solutions when the energy functional is unbounded below on the L²-sphere, precluding minimization?
• RQ3 How can the presence of singularities in V (e.g., poles) be accommodated in the existence theory for normalized solutions?
• RQ4 What explicit, non-perturbative conditions on V and W(x) = V(x)|x| ensure the existence of a solution, even when the L∞-norms of V and W are large for small ρ?
• RQ5 Can the new linking argument developed in this paper be adapted to handle non-radial potentials where the embedding H¹(R^N) ↪ L^p(R^N) is not compact?

Key findings

• The paper proves the existence of a normalized solution (u, λ) ∈ H¹(R^N) × R⁺ for the mass supercritical Schrödinger equation with V ≥ 0 and V(x) → 0 as |x| → ∞, under explicit conditions (V1) or (V2), for any ρ > 0.
• The solution is obtained via a novel min-max argument on the L²-sphere, which overcomes the lack of Palais-Smale condition and unboundedness of the functional in the supercritical regime.
• For potentials satisfying (V1), the existence is guaranteed if the L∞-norms of V and W(x) = V(x)|x| are bounded by explicit expressions involving mρ, ρ, and p, with the bounds allowing large V for small ρ.
• For potentials satisfying (V2) in dimension N ≥ 3, existence holds if ∥V∥_{N/2} and ∥W∥_N are bounded by expressions involving the Aubin-Talenti constant A and explicit functions of p and N, allowing singularities.
• In the radial case, a mountain pass argument on the radial subspace Sρ ∩ H¹_rad(R^N) yields a solution, and the method is robust under radial assumptions.
• The paper shows that the mountain pass level mρ for the autonomous case (V ≡ 0) is achieved by the scaled ground state Zρ, and this value is used as a reference in the linking construction.