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On a critical curl-curl system of Lane-Emden type

  • Ioannis Arkoudis
  • Jarosław Mederski
  • Andrzej Szulkin
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Abstract

We study the system ∇×∇× u = |v|q−2v, ∇×∇× v = |u|p−2u of Lane-Emden type in ℝ3, with exponents p,q>3 on the critical hyperbola 1/p+1/q=1/3. We prove the existence of a ground state solution by reducing the system to a quasilinear fourth-order curl-curl equation and minimizing the associated Sobolev-curl quotient. A concentration-compactness argument shows there are finitely many concentration points for a Palais-Smale sequence (un). This yields a.e. convergence of ∇×∇× un to ∇×∇× u where u is the weak limit of (un). We also prove that the optimal Sobolev-curl constant is attained and is strictly larger than the usual second-order Sobolev constant. If p=q, we show u and v are necessarily equal for every solution. In an appendix we show that our concentration argument also applies to a scalar quasilinear fourth order equation.

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Cite

BibTeXRIS
BibTeX
@misc{arkoudis2026critical,
  title = {{On a critical curl-curl system of Lane-Emden type}},
  author = {Ioannis Arkoudis and Jarosław Mederski and Andrzej Szulkin},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.21747},
  eprint = {2610.21747},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.21747},
}
RIS
TY  - UNPB
TI  - On a critical curl-curl system of Lane-Emden type
AU  - Ioannis Arkoudis
AU  - Jarosław Mederski
AU  - Andrzej Szulkin
PY  - 2026
DA  - 2026/10/15
PB  - arXiv
DO  - 10.48550/arxiv.2610.21747
UR  - https://arxiv.org/abs/2610.21747
AB  - We study the system ∇×∇× u = |v|^q−2v, ∇×∇× v = |u|^p−2u of Lane-Emden type in ℝ^3, with exponents p,q>3 on the critical hyperbola 1/p+1/q=1/3. We prove the existence of a ground state solution by reducing the system to a quasilinear fourth-order curl-curl equation and minimizing the associated Sobolev-curl quotient. A concentration-compactness argument shows there are finitely many concentration points for a Palais-Smale sequence (u_n). This yields a.e. convergence of ∇×∇× u_n to ∇×∇× u where u is the weak limit of (u_n). We also prove that the optimal Sobolev-curl constant is attained and is strictly larger than the usual second-order Sobolev constant. If p=q, we show u and v are necessarily equal for every solution. In an appendix we show that our concentration argument also applies to a scalar quasilinear fourth order equation.
ER  -