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Singularity estimates for elliptic systems of $m$-Laplacians

Yayun Li|Bei Liu
대한수학회 (The Korean Mathematical Society)|2018.11|대한수학회지 (Journal of the KMS)|제55권 제6호|pp. 1423-1433

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참고문헌

키워드

Liouville theoremdecay rateelliptic system of $m$-Laplaciandoubling lemmasingularity estimate
주제분류수학

초록

This paper is concerned about several quasilinear elliptic systems with $m$-Laplacians. According to the Liouville theorems of those systems on $\mathbb R^n$, we obtain the singularity estimates of the positive $C^1$-weak solutions on bounded or unbounded domain (but it is not $\mathbb R^n$) and their decay rates on the exterior domain when $|x| \to \infty$. The doubling lemma which is developed by Polacik-Quittner-Souplet plays a key role in this paper. In addition, the corresponding results of several special examples are presented.

Abstract

This paper is concerned about several quasilinear elliptic systems with $m$-Laplacians. According to the Liouville theorems of those systems on $\mathbb R^n$, we obtain the singularity estimates of the positive $C^1$-weak solutions on bounded or unbounded domain (but it is not $\mathbb R^n$) and their decay rates on the exterior domain when $|x| \to \infty$. The doubling lemma which is developed by Polacik-Quittner-Souplet plays a key role in this paper. In addition, the corresponding results of several special examples are presented.

참고문헌 (27)

  1. M. -F. Bidaut-Veron, "Local behaviour of the solutions of a class of nonlinear elliptic systems", Adv. Differential Equations, 5(1-3), pp.147~192 (2000)
  2. M.-F. Bidaut-Veron, "Asymptotics of solutions of some nonlinear elliptic systems", Comm. Partial Differential Equations, 21(7-8), pp.1035~1086 (1996)
  3. J. Byeon, "Symmetry and monotonicity of least energy solutions", Calc. Var. Partial Differential Equations, 36(4), pp.481~492 (2009)
  4. P. Clement, "Existence of positive solutions for a nonvariational quasilinear elliptic system", J. Differential Equations, 166(2), pp.455~477 (2000)
  5. E. N. Dancer, "Superlinear problems on domains with holes of asymptotic shape and exterior problems", Math. Z, 229(3), pp.475~491 (1998)
  6. W. Chen, "An integral system and the Lane-Emden conjecture", Discrete Contin. Dyn. Syst, 24(4), pp.1167~1184 (2009)
  7. J. Hulshof, "Asymptotic behaviour of ground states", Proc. Amer. Math. Soc, 124(8), pp.2423~2431 (1996)
  8. B. Hu, "Remarks on the blowup estimate for solution of the heat equation with a nonlinear boundary condition", Differential Integral Equations, 9(5), pp.891~901 (1996)
  9. Y. Lei, "Integrability and asymptotics of positive solutions of a γ-Laplace system", J. Differential Equations, 252(3), pp.2739~2758 (2012)
  10. Y. Lei, "Sharp criteria of Liouville type for some nonlinear systems", Discrete Contin. Dyn. Syst, 36(6), pp.3277~3315 (2016)
  11. C. Li, "Uniqueness of positive bound states to Schrodinger systems with critical exponents", SIAM J. Math. Anal, 40(3), pp.1049~1057 (2008)
  12. Y. Li, "On existence and nonexistence of positive solutions of an elliptic system with coupled terms", Commun. Pure Appl. Anal.
  13. T.-C. Lin, "Ground state of N coupled nonlinear Schrodinger equations in Rn, n ≤ 3", Comm. Math. Phys, 255(3), pp.629~653 (2005)
  14. T. -C. Lin, "Spikes in two coupled nonlinear Schrodinger equations", Ann. Inst. H. Poincare Anal. Non Lineaire, 22(4), pp.403~439 (2005)
  15. L. Ma, "Symmetry results for decay solutions of elliptic systems in the whole space", Adv. Math, 225(6), pp.3052~3063 (2010)
  16. E. Mitidieri, "Nonexistence of positive solutions of semilinear elliptic systems in RN", Differential Integral Equations, 9(3), pp.465~479 (1996)
  17. Y. Miyamoto, "Intersection properties of radial solutions and global bifurcation diagrams for supercritical quasilinear elliptic equations", NoDEA Nonlinear Differential Equations Appl, 23(2), pp.24~ (2016)
  18. P. Polacik, "Singularity and decay estimates in superlinear problems via Liouville-type theorems. I. Elliptic equations and systems", Duke Math. J, 139(3), pp.555~579 (2007)
  19. P. Quittner, "Symmetry of components for semilinear elliptic systems", SIAM J. Math. Anal, 44(4), pp.2545~2559 (2012)
  20. W. Reichel, "Non-existence results for semilinear cooperative elliptic systems via moving spheres", J. Differential Equations, 161(1), pp.219~243 (2000)
  21. J. Serrin, "Non-existence of positive solutions of Lane-Emden systems", Dif-ferential Integral Equations, 9(4), pp.635~653 (1996)
  22. J. Serrin, "Cauchy-Liouville and universal boundedness theorems for quasilinear elliptic equations and inequalities", Acta Math, 189(1), pp.79~142 (2002)
  23. P. Souplet, "The proof of the Lane-Emden conjecture in four space dimensions", Adv. Math, 221(5), pp.1409~1427 (2009)
  24. S. Sun, "Fast decay estimates for integrable solutions of the Lane-Emden type integral systems involving the Wolff potentials", J. Funct. Anal, 263(12), pp.3857~3882 (2012)
  25. P. Tolksdorf, "Regularity for a more general class of quasilinear elliptic equations", J. Differential Equations, 51(1), pp.126~150 (1984)
  26. Yun Wang, "Symmetry of components for radial solutions of $\gamma$-Laplacian systems", 대한수학회지, 53(2), pp.305~313 (2016), 대한수학회
  27. Y. Wang, "On critical exponents of a k-Hessian equation in the whole space", Proc. Roy. Soc. Edinburgh Sect. A

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