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Boundedness of solutions to Ginzburg–Landau fractional Laplacian equation

Li Ma

Li Ma

Zhongyuan Institute of Mathematics, Henan Normal University, Xinxiang 453007, P. R. China

Department of Mathematics, Henan Normal University, Xinxiang 453007, P. R. China

E-mail Address: lma@tsinghua.edu.cn

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https://doi.org/10.1142/S0129167X16500488Cited by:12 (Source: Crossref)

Abstract

In this paper, we give the boundedness of solutions to Ginzburg–Landau fractional Laplacian equation, which extends the Herve–Herve theorem into the nonlinear fractional Laplacian equation. We follow Brezis’ idea to use the Kato inequality. A related linear fractional Schrödinger equation is also studied.

Keywords:

AMSC: 35R11, 35B53, 35J61, 35B45

References

1. H. Brezis, Comments on two notes by L. Ma and X. Xu, C. R. Math. Acad. Sci. Paris 349(5–6) (2011) 269–271. Web of Science, Google Scholar
2. L. Caffarelli and L. Silvestre, Regularity theory for fully nonlinear integro-differential equations, Comm. Pure Appl. Math. 72 (2009) 597–638. Web of Science, Google Scholar
3. W. Chen, Y. Fang and R. Yang, Liouville theorems involving the fractional Laplacian on a half space, Adv. Math. 274 (2015) 167–198. Web of Science, Google Scholar
4. W. Chen and C. Li, An integral system and the Lane–Emden conjecture, Discrete Contin. Dyn. Syst. 4(24) (2009) 1167–1184. Web of Science, Google Scholar
5. R. L. Frank, E. Lenzmann and L. Silvestre, Uniqueness of radial solutions for the fractional laplacian, preprint (2013), arXiv:1302.2652v1. Google Scholar
6. M. Hervé and R. M. Hervé, Quelques proprietés des solutions de l’equation de Ginzburg–Landau sur un ouvert de $R^{2}$ $R^{2}$ , Potential Anal. 5 (1996) 591–609. Web of Science, Google Scholar
7. S. Jarohs and T. Weth, Symmetry via antisymmetric maximum principles in nonlocal problems of variable order, Ann. Mat., doi: 10.1007/s10231-014-0462-y. Google Scholar
8. N. S. Landkof, Foundations of Modern Potential Theory (Springer-Verlag, Berlin, 1972). Translated from the Russian by A. P. Doohovskoy; Die Grundlehren der mathematischen Wissenschaften, Band 180. Google Scholar
9. L. Ma, Liouville type theorem and uniform bound for the Lichnerowicz equation and the Ginzburg–Landau equation, C. R. Acad. Sci. Paris, Ser. I 348 (2010) 993–996. Web of Science, Google Scholar
10. L. Ma, Liouville type theorems for Lichnerowicz equations and Ginzburg–Landau equation: Survey, Adv. Pure Math. 1 (2011) 99–104. Google Scholar
11. L. Ma, On nonlocal nonlinear elliptic problem with the fractional Laplacian, preprint (2015), arXiv: 1501.00070. Google Scholar
12. L. Ma and X. Y. Wang, Kato inequality and Ginzburg–Landau equation in locally finite graphs, Sci. China Math. 56(4) (2013) 771–776. Web of Science, Google Scholar
13. L. Ma and X. Xu, Uniform bound and a non-existence result for Lichnerowicz equation in the whole n-space, C. R. Math. Ser. I 347 (2009) 805–808. Web of Science, Google Scholar