[Paper Review] Improved estimates and a limit case for the electrostatic Klein-Gordon-Maxwell system

Antonio Azzollini, Lorenzo Pisani+1 more|ArXiv.org|Nov 30, 2009

Advanced Mathematical Physics Problems6 references3 citations

TL;DR

This paper improves existence results for the electrostatic Klein-Gordon-Maxwell system with homogeneous nonlinearity $ f(u) = \frac{1}{p}|u|^p $, establishing the existence of nontrivial solutions for $ p \in (2,4) $ under a sharper threshold $ \omega < g(p)m $, where $ g(p) = \sqrt{(p-2)(4-p)} $ for $ p \in (2,3) $ and $ g(p) = 1 $ for $ p \in [3,4) $. It further establishes the existence of solutions in the critical limit case $ \omega = m $, analogous to the zero-mass case, under stronger, inhomogeneous assumptions on $ f $. The analysis relies on variational reduction and concentration-compactness techniques to overcome strong indefiniteness.

ABSTRACT

We study the class of nonlinear Klein-Gordon-Maxwell systems describing a standing wave (charged matter field) in equilibrium with a purely electrostatic field. We improve some previous existence results in the case of an homogeneous nonlinearity. Moreover, we deal with a limit case, namely when the frequency of the standing wave is equal to the mass of the charged field; this case shows analogous features of the well known "zero mass case" for scalar field equations.

Motivation & Objective

• To improve previous existence results for the electrostatic Klein-Gordon-Maxwell system with homogeneous nonlinearity $ f(u) = \frac{1}{p}|u|^p $ in the range $ p \in (2,4) $.
• To establish the existence of nontrivial solutions in the limit case $ \omega = m $, where the frequency equals the mass, analogous to the zero-mass case in scalar field equations.
• To overcome the strong indefiniteness of the associated energy functional through a reduction method, reducing the problem to a single equation in $ u $.
• To prove existence under stronger, inhomogeneous assumptions on $ f $ in the $ \omega = m $ case, ensuring the functional is bounded below and satisfies the Palais-Smale condition.

Proposed method

• Use of a variational reduction: the system is reduced to a single equation in $ u $ by solving the Poisson equation for $ \phi_u $, leading to the reduced functional $ I(u) = \mathcal{S}(u, \phi_u) $.
• Application of the indirect method of Struwe and Jeanjean to handle the lack of compactness in the Palais-Smale condition for the reduced functional.
• Employment of the concentration-compactness principle to analyze weak convergence and recover compactness in the limit case $ \omega = m $.
• Use of Sobolev embeddings and interpolation estimates to control nonlinear terms, particularly $ \|u_n - u_0\|_{L^{3/2}(K)} \to 0 $ on compact sets $ K $.
• Introduction of a parameter $ \alpha \in \left(\frac{2-p}{2(6-p)}, \frac{1}{6}\right) $ to construct a suitable lower bound for the quadratic form involving $ \phi_u $, ensuring coercivity.
• Proof of the existence of a minimizer via the Ekeland variational principle and the boundedness of the Palais-Smale sequence in $ H^1(\mathbb{R}^3) $.

Experimental results

Research questions

• RQ1 Can the existence threshold for nontrivial solutions in the electrostatic Klein-Gordon-Maxwell system be improved for $ p \in (2,4) $ beyond previous results?
• RQ2 What happens to the system when the frequency $ \omega $ equals the mass $ m $, and does it admit solutions in this critical limit case?
• RQ3 How can the strong indefiniteness of the energy functional be overcome to prove existence of solutions in the $ \omega = m $ case?
• RQ4 What conditions on the nonlinearity $ f $ are necessary to ensure existence in the $ \omega = m $ limit case, and how do they differ from the $ \omega < m $ case?

Key findings

• For $ p \in (2,4) $, the system admits a nontrivial weak solution if $ \omega < g(p)m $, where $ g(p) = \sqrt{(p-2)(4-p)} $ for $ p \in (2,3) $ and $ g(p) = 1 $ for $ p \in [3,4) $, improving prior thresholds.
• In the limit case $ \omega = m $, the system admits a nontrivial solution under stronger, inhomogeneous assumptions on $ f $, including $ f(t) \geq C_1 \min(|t|^p, |t|^q) $ with $ 4 < \alpha \leq p < 6 < q $.
• The reduced functional $ I(u) $ is bounded from below and satisfies the Palais-Smale condition under the given assumptions, ensuring the existence of a minimizer.
• The proof relies on a concentration-compactness argument, showing that weak limits of Palais-Smale sequences are nontrivial solutions, with $ \phi_0 \neq 0 $.
• The existence of a suitable $ \alpha \in I_p $ ensures the quadratic form $ A_{p,\alpha}e^2\phi_u^2 + B_{p,\alpha}e\omega\phi_u + C_{p,\alpha}\Omega \geq 0 $, which is crucial for coercivity and compactness.
• The analysis confirms that the $ \omega = m $ case shares structural similarities with the zero-mass case in scalar field equations, despite the absence of a linear term in $ u $.