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Cauchy-Liouville and universal boundedness theorems for quasilinear elliptic equations and inequalities

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References

  1. Acerbi, E. &Fusco, N., Regularity for minimizers of non-quadratic functionals: the case 1<p<2.J. Math. Anal. Appl., 140 (1989), 115–135.

    Article  MathSciNet  MATH  Google Scholar 

  2. Astarita, G. & Marrucci, G.,Principles of Non-Newtonian Fluid Mechanics. McGraw-Hill, 1974.

  3. Bidaut-Veron, M.-F., Local and global behavior of solutions of quasilinear equations of Emden-Fowler type.Arch. Rational Mech. Anal., 107 (1989), 293–324.

    Article  MATH  MathSciNet  Google Scholar 

  4. Bidaut-Veron, M.-F. &Pohozaev, S., Nonexistence results and estimates for some nonlinear elliptic problems.J. Anal. Math., 84 (2001), 1–49.

    Article  MathSciNet  MATH  Google Scholar 

  5. Bidaut-Veron, M.-F. &Veron, L., Nonlinear elliptic equations on compact Riemannain manifolds and asymptotics of Emden equations.Invent. Math., 106 (1991), 489–539.

    Article  MathSciNet  MATH  Google Scholar 

  6. Caffarelli, L., Garofalo, N. &Segala, F., A gradient bound for entire solutions of quasi-linear equations and its consequences.Comm. Pure Appl. Math., 47 (1994), 1457–1473.

    MathSciNet  MATH  Google Scholar 

  7. Caffarelli, L., Gidas, B. &Spruck, J., Asymptotic symmetry and local behavior of semilinear elliptic equations with critical Sobolev growth.Comm. Pure Appl. Math., 42 (1989), 271–297.

    MathSciNet  MATH  Google Scholar 

  8. Cauchy, A., Mémoire sur les fonctions complémentaires.C. R. Acad. Sci. Paris, 19 (1844), 1377–1384; also (Euvres complètes, Ire série, tome VIII, 378–383.

    Google Scholar 

  9. Díaz, J. I.,Nonlinear Partial Differential Equations and Free Boundaries, Vol. 1. Elliptic Equations. Pitman Res. Notes Math. Ser., 106. Pitman, Boston, MA, 1985.

    Google Scholar 

  10. DiBenedetto, E.,C 1+α local regularity of weak solutions of degenerate elliptic equations.Nonlinear Anal., 7 (1983), 827–850.

    Article  MATH  MathSciNet  Google Scholar 

  11. Evans, L. C., A new proof of localC 1,α regularity for solutions of certain degenerate elliptic P.D.E..J. Differential Equations, 45 (1982), 356–373.

    Article  MATH  MathSciNet  Google Scholar 

  12. Gidas, B. &Spruck, J., Global and local behavior of positive solutions of nonlinear elliptic equations.Comm. Pure Appl. Math., 34 (1981), 525–598.

    MathSciNet  MATH  Google Scholar 

  13. Lewis, J. L., Regularity of the derivatives of solutions to certain degenerate elliptic equations.Indiana Univ. Math. J., 32 (1983), 849–858.

    Article  MATH  MathSciNet  Google Scholar 

  14. Martinson, L. K. &Pavlov, K. B., The effect of magnetic plasticity in non-Newtonian fluids.Magnit. Gidrodinamika, 2 (1969), 69–75.

    Google Scholar 

  15. —, Unsteady shear flows of a conducting fluid with a rheological power flow.Magnit. Gidrodinamika, 3 (1970), 5869–5875.

    Google Scholar 

  16. Mazzeo, R. &Pacard, F., A construction of singular solutions for a semilinear elliptic equation using asymptotic analysis.J. Differential Geom., 44 (1996), 331–370.

    MathSciNet  MATH  Google Scholar 

  17. Mikljukov, V. M., On the asymptotic properties of subsolutions of quasilinear equations of elliptic type and mappings with bounded distortion.Mat. Sb. (N.S.), 111 (1980), 42–66 (Russian).

    MATH  MathSciNet  Google Scholar 

  18. Mitidieri, È. &Pokhozhaev, S. I., the absence of global positive solutions to quasilinear elliptic inequalities.Dokl. Akad. Nauk, 359 (1998), 456–460; English translation inDokl. Math., 57 (1998), 250–253.

    MathSciNet  MATH  Google Scholar 

  19. Ni, W.-M. &Serrin, J., Existence and non-existence theorems for ground states of quasilinear partial differential equations: The anomalous case.Atti Convegni Lincei, 77 (1986), 231–257.

    Google Scholar 

  20. Peletier, L. A. &Serrin, J., Gradient bounds and Liouville theorems for quasilinear elliptic equations.Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4), 5 (1978), 65–104.

    MathSciNet  MATH  Google Scholar 

  21. Protter, M. H. &Weinberger, H. F.,Maximum Principles in Differential Equations. Springer-Verlag, New York, 1984 (reprinted).

    MATH  Google Scholar 

  22. Reshetnyak, Yu. G., Index boundedness condition for mappings with bounded distortion.Siberian Math. J., 9 (1968), 281–285.

    Article  MATH  Google Scholar 

  23. Serrin, J., Local behavior of solutions of quasi-linear equations.Acta Math., 111 (1964), 247–302.

    Article  MATH  MathSciNet  Google Scholar 

  24. —, Isolated singularities of solutions of quasi-linear equations.Acta Math., 113 (1965), 219–240.

    Article  MATH  MathSciNet  Google Scholar 

  25. —, Entire solutions of nonlinear Poisson equations.Proc. London Math. Soc. (3), 24 (1972), 348–366.

    MATH  MathSciNet  Google Scholar 

  26. —, Liouville theorems for quasilinear elliptic equations, inAtti del Convegno Internazionale sui Metodi Valutativi nella Fisica-Matematica (Rome, 1975), pp. 207–215. Accad. Naz. Lincei, Quaderno 217. Accad. Naz. Lincei, Rome, 1975.

    Google Scholar 

  27. Tolksdorf, P., Regularity for a more general class of quasilinear elliptic equations.J. Differential Equations, 51 (1984), 126–150.

    Article  MATH  MathSciNet  Google Scholar 

  28. Trudinger, N. S., On Harnack type inequalities and their applications to quasilinear elliptic equations.Comm. Pure Appl. Math., 20 (1967), 721–747.

    MATH  MathSciNet  Google Scholar 

  29. Uhlenbeck, K., Regularity for a class of non-linear elliptic systems.Acta Math., 138 (1977), 219–240.

    MATH  MathSciNet  Google Scholar 

  30. Ural'tseva, N. N., Degenerate quasilinear elliptic systems.Zap. Nauchn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI), 7 (1968), 184–222 (Russian).

    MATH  Google Scholar 

  31. Veron, L.,Singularities of Solutions of Second Order Quasilinear Equations. Pitman Res. Notes Math. Ser., 353. Longman, Harlow, 1996.

    MATH  Google Scholar 

  32. Zou, H., Slow decay and the Harnack inequality for positive solutions of Δu+up=0 inR n.Differential Integral Equations, 8 (1995), 1355–1368.

    MATH  MathSciNet  Google Scholar 

  33. —, Symmetry of positive solutions of Δu+up=0 inR n.J. Differential Equations, 120 (1995), 46–88.

    Article  MATH  MathSciNet  Google Scholar 

  34. Dancer, E. N., Superlinear problems on domains with holes of asymptotic shape and exterior problems.Math. Z., 229 (1998), 475–491.

    Article  MATH  MathSciNet  Google Scholar 

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Serrin, J., Zou, H. Cauchy-Liouville and universal boundedness theorems for quasilinear elliptic equations and inequalities. Acta Math. 189, 79–142 (2002). https://doi.org/10.1007/BF02392645

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