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A priori estimates and blow-up behavior for solutions of −Q N u = Veu in bounded domain in ℝN

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Abstract

Let Q N be an N-anisotropic Laplacian operator, which contains the ordinary Laplacian operator, N-Laplacian operator and the anisotropic Laplacian operator. We firstly obtain the properties of Q N , which contain the weak maximal principle, the comparison principle and the mean value property. Then a priori estimates and blow-up analysis for solutions of Q N u in bounded domain in ℝN, N ≥ 2 are established. Finally, the blow-up behavior of the only singular point is also considered.

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References

  1. Alvino A, Ferone V, Trombetti G, et al. Convex symmetrization and applications. Ann Inst H Poincaré Anal Non Linéaire, 1997; 14: 275–293

    Article  MathSciNet  MATH  Google Scholar 

  2. Bartolucci D, Tarantello G. Liouville type equations with singular data and their applications to periodic multivortices for the electroweak theory. Comm Math Phys, 2002; 229: 3–47

    Article  MathSciNet  MATH  Google Scholar 

  3. Bellettini G, Paolini M. Anisotropic motion by mean curvature in the context of Finsler geometry. Hokkaido Math J, 1996; 25: 537–566

    Article  MathSciNet  MATH  Google Scholar 

  4. Belloni M, Ferone V, Kawohl B. Isoperimetric inequalities, Wulff shape and related questions for strongly nonlinear elliptic operators. Z Angew Math Phys, 2003; 54: 771–783

    Article  MathSciNet  MATH  Google Scholar 

  5. Brezis H, Merle F. Uniform estimates and blow-up behavior for solutions of -Δu = V (x)eu in two dimensions. Comm Partial Differential Equations, 1991; 16: 1223–1253

    Article  MathSciNet  MATH  Google Scholar 

  6. Chang S, Yang P. Prescribing Gaussian curvature on S 2. Acta Math, 1987; 159: 215–259

    Article  MathSciNet  MATH  Google Scholar 

  7. Chen C, Lin C. Sharp estimates for solutions of multi-bubbles in compact Riemann surfaces. Comm Pure Appl Math, 2002; 55: 728–771

    Article  MathSciNet  MATH  Google Scholar 

  8. Chen W, Li C. Prescribing Gaussian curvatures on surfaces with conical singularities. J Geom Anal, 1991; 4: 359–372

    Article  MathSciNet  MATH  Google Scholar 

  9. Ding W, Jost J, Li J, et al. The differential equation Δu = 8p - 8peu on a compact Riemann surface. Asian J Math, 1997; 1: 230–248

    Article  MathSciNet  Google Scholar 

  10. Ding W, Jost J, Li J, et al. Existence results for mean field equations. Ann Inst H Poincare Anal Non Lineaire, 1999; 16: 653–666

    Article  MathSciNet  MATH  Google Scholar 

  11. Djadli Z. Existence result for the mean field problem on Riemann surfaces of all genuses. Commun Contemp Math, 2008; 10: 205–220

    Article  MathSciNet  MATH  Google Scholar 

  12. Ferone V, Kawohl B. Remarks on a Finsler-Laplacian. Proc Amer Math Soc, 2009; 137: 247–253

    Article  MathSciNet  MATH  Google Scholar 

  13. Fonseca I, Muller S. A uniqueness proof for the Wulff theorem. Proc Roy Soc Edinburgh Sect A, 1991; 119: 125–136

    Article  MathSciNet  MATH  Google Scholar 

  14. Jost J, Wang G. Analytic aspects of the Toda system, I: A Moser-Trudinger inequality. Comm Pure Appl Math, 2001; 54: 1289–1319

    Article  MathSciNet  MATH  Google Scholar 

  15. Li J, Li Y. Solutions for Toda systems on Riemann surfaces. Annali della Scuola Normale Superiore di Pisa-Classe di Scienze-Serie V, 2005; 4: 703–728

    MathSciNet  MATH  Google Scholar 

  16. Li Y. Harnack type inequality: The method of moving planes. Comm Math Phys, 1999; 200: 421–444

    Article  MathSciNet  MATH  Google Scholar 

  17. Li Y, Shafrir I. Blow-up analysis for solutions of -Δu = V eu in dimension two. Indiana Univ Math J, 1994; 43: 1255–1270

    Article  MathSciNet  MATH  Google Scholar 

  18. Lin C. An expository survey of the recent development of mean field equations. Discrete Contin Dyn Syst, 2007; 19: 387–410

    Article  MathSciNet  MATH  Google Scholar 

  19. Liouville J. Sur léquation aux derivees partielles \(frac{{{\partial ^2}\log \lambda }}{{\partial u\partial u}} \pm \frac{\lambda }{{2{a^2}}} = 0\). J Math Pures Appl, 1853; 18: 71–72

  20. Ren X, Wei J. Counting peaks of solutions to some quasilinear elliptic equations with large exponents. J Differ Equations, 1995; 117: 28–55

    Article  MathSciNet  MATH  Google Scholar 

  21. Serrin J. Local behavior of solutions of quasi-linear equations. Acta Math, 1964; 111: 247–302

    Article  MathSciNet  MATH  Google Scholar 

  22. Spruck J, Yang Y. On multivortices in the electroweak theory I: Existence of periodic solutions. Comm Math Phys, 1992; 144: 1–16

    Article  MathSciNet  MATH  Google Scholar 

  23. Struwe M, Tarantello G. On the multivortex solutions in the Chern-Simons gauge theory. Boll Unione Mat Ital Sez B Artic Ric Mat, 1998; 1: 109–121

    MathSciNet  MATH  Google Scholar 

  24. Talenti G. Elliptic equations and rearrangements. Ann Sc Norm Super Pisa Cl Sci, 1976; 3: 697–718

    MathSciNet  MATH  Google Scholar 

  25. Tarantello G. Multiple condensate solutions for the Chern-Simons-Higgs theory. J Math Phys, 1996; 37: 3769–3796

    Article  MathSciNet  MATH  Google Scholar 

  26. Wang G, Xia C. A Brunn. Minkowski inequality for a Finsler-Laplacian. Analysis, 2011; 31: 103–115

    Article  MathSciNet  MATH  Google Scholar 

  27. Wang G, Xia C. A characterization of the Wulff shape by an overdetermined anisotropic PDE. Arch Ration Mech Anal, 2011; 99: 99–115

    Article  MathSciNet  MATH  Google Scholar 

  28. Wang G, Xia C. Blow-up analysis of a Finsler-Liouville equation in two dimensions. J Differ Equations, 2012; 252: 1668–1700

    Article  MathSciNet  MATH  Google Scholar 

  29. Wulff G. Zur Frage der Geschwindigkeit des Wachstums und der Auflosung der Kristallflachen. Z Krist, 1901; 34: 449–530

    Google Scholar 

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Authors and Affiliations

  1. School of Mathematical Sciences, University of Science and Technology of China, Hefei, 230026, China

    RuLong Xie & HuaJun Gong

  2. Department of Mathematics, Chaohu University, Hefei, 238000, China

    RuLong Xie

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  1. RuLong Xie
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  2. HuaJun Gong
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Correspondence to HuaJun Gong.

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Xie, R., Gong, H. A priori estimates and blow-up behavior for solutions of −Q N u = Veu in bounded domain in ℝN . Sci. China Math. 59, 479–492 (2016). https://doi.org/10.1007/s11425-015-5060-y

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  • DOI: https://doi.org/10.1007/s11425-015-5060-y

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