[Paper Review] Single--peaks for a magnetic Schrödinger equation with critic al growth

Sara Barile, Silvia Cingolani+1 more|ArXiv.org|Jun 27, 2006

Nonlinear Partial Differential Equations29 references20 citations

TL;DR

This paper establishes the existence of single-peak complex-valued solutions to a magnetic Schrödinger equation with critical growth in the semiclassical limit, using a perturbation method in critical point theory. It proves that for small ε > 0, at least one solution exists when the electric potential V is non-vanishing, and two solutions exist if V changes sign.

ABSTRACT

We prove existence results of complex-valued solutions for a semilinear Schrödinger equation with critical growth under the perturbation of an external electromagnetic field. Solutions are found via an abstract perturbation result in critical point theory.

Motivation & Objective

• To establish the existence of complex-valued solutions for a semilinear Schrödinger equation with critical nonlinearity under the influence of a small external electromagnetic field.
• To analyze the behavior of solutions in the semiclassical limit (ε → 0+), where ε parametrizes the strength of the magnetic and electric potentials.
• To extend variational methods to the critical growth case (p = (N+2)/(N-2)) with non-periodic magnetic and electric potentials.
• To prove the existence of at least one solution when the electric potential V is non-zero, and two solutions when V changes sign.

Proposed method

• Employ a perturbation framework in critical point theory, adapted from [1, 2, 5], to handle the lack of compactness due to critical growth.
• Use a finite-dimensional reduction to reduce the infinite-dimensional problem to finding critical points of a functional Γ on the set of extremal functions z_{μ,ξ} = κ_N μ^{(N/2−1)} / (μ² + |x−ξ|²)^{(N−2)/2}.
• Construct a Melnikov-type functional Γ(μ,ξ) that captures the interaction between the magnetic potential A and the electric potential V near the concentration point (μ,ξ).
• Analyze the asymptotic behavior of Γ as μ → 0+ to relate the critical points of Γ to bifurcating solutions of the original equation.
• Apply an abstract critical point theorem (Theorem 5.1) to deduce the existence of solutions u_ε for small ε > 0 based on the existence of a strict local minimum or maximum of Γ.
• Use reflection and symmetry arguments to extend Γ across μ = 0 and ensure non-constancy of the functional under non-trivial V.

Experimental results

Research questions

• RQ1 Under what conditions does a magnetic Schrödinger equation with critical growth admit single-peak solutions in the semiclassical limit?
• RQ2 How does the interplay between the magnetic potential A and the electric potential V affect the existence and multiplicity of solutions?
• RQ3 Can the perturbation method in critical point theory be extended to the critical case when the nonlinearity is not subcritical?
• RQ4 What role does the sign of the electric potential V play in the multiplicity of solutions?
• RQ5 Is the Melnikov functional Γ sufficient to detect bifurcating solutions when A and V are non-periodic?

Key findings

• For sufficiently small ε > 0, the equation admits at least one complex-valued solution u_ε that concentrates at a point in space as ε → 0.
• If the electric potential V changes sign, then at least two distinct solutions exist for small ε > 0, due to the existence of both a local minimum and maximum of the Melnikov functional Γ.
• The asymptotic behavior of the functional Γ as μ → 0+ is determined by the integral −½∫|A(ξ)|² z₀² dx, which captures the magnetic contribution to the energy correction.
• The leading-order term in the energy expansion of u_ε is ε² ∫V(x) z₀² dx, showing that the electric potential V governs the energy shift in the perturbation.
• The Melnikov functional Γ is not constant when V is non-vanishing, ensuring the existence of critical points that generate solutions via bifurcation.
• The result holds for all α ∈ [1,2), with the magnetic potential A affecting the reduction only in the case α = 1, while higher-order terms in ε do not alter the leading-order analysis.