Liouville type results and eventual flatness of positive solutions for p-Laplacian equations

Advances in Differential Equations

Volumn 7, Issue 12, 2002, Pages 1479-1512

  Du, Y ^a   Guo, Z ^b  

^a UNIVERSITY OF NEW ENGLAND   (Australia)

^b HENAN NORMAL UNIVERSITY   (China)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 0037777206     PISSN: 10799389     EISSN: None     Source Type: Journal    
DOI: None     Document Type: Article

References (34)

1. A. Aftalion and W. Reichel, Existence of two boundary blow-up solutions for semilinear elliptic equations, J. Diff. Eqns., 141 (1997), 400-421.
2. C. Bandle and M. Marcus, Asymptotic behaviour of solutions and derivatives for semilinear elliptic problems with blow-up on the boundary, Ann. Inst. Henri Poincaré 12 (1995), 155-171.
3. A. Cañada, P. Drábek, and J.L. Gamez, Existence of positive solutions for some problems with nonlinear diffusion, Trans. Amer. Math. Soc., 349 (1997), 4231-4249.
4. E.N. Dancer, On the uniqueness of the positive solution of a singularly perturbed problem, Rocky Mountain J. Math., 25 (1995), 957-975.
5. E.N. Dancer, On the number of positive solutions of weakly nonlinear elliptic equations when a parameter is large, Proc. London math. Soc., 53 (1986), 429-452.
6. E.N. Dancer and Y. Du, Some remarks on Liouville type results for quasilinear elliptic equations, Proc. Amer. Math. Soc., (to appear).
7. L. Damascelli and F. Pacella, Monotonicity and symmetry of solutions of p-Laplace equations, 1 < p < 2, via the moving plane method, Ann. Scuola Norm. Sup. Pisa Sci., XXVI (1998), 689-707.
8. L. Damascelli, F. Pacella and M. Ramaswamy, Symmetry of ground states of p-Laplace equations via the moving plane method, Arch. Rational Mech. Anal., 148 (1999), 291- 308.
9. G. Diaz and R. Letelier, Explosive solutions of quasilinear elliptic equations: Existence and uniqueness, Nonlinear Anal. 20 (1993), 97-125.
10. J. Diaz, "Nonlinear Partial Differential Equations and Free Boundary Problems, Vol. 1, Elliptic Equations," Pitman research notes in math., Vol. 106, Boston, 1985.
11. Y. Du and L. Ma, Logistic type equations on RN by a squeezing method involving boundary blow-up solutions, J. London Math. Soc., 64 (2001), 107-124.
12. Y. Du and Z. Guo, Boundary blow-up solutions and their applications in quasilinear elliptic equations, J. d'Analyse Math., (to appear).
13. B. Franchi, E. Lanconelli, and J. Serrin, Existence and uniqueness of non-negative solutions of quasilinear equations in RN, Adv. in Math. 118 (1996), 177-243.
14. J.Garcia-Melián and J. Sabina de Lis, Stationary profiles of degenerate problems when a parameter is large, Diff. Integral Eqns., 13 (2000), 1201-1232.
15. Z.M. Guo, Some existence and multiplicity results for a class of quasilinear elliptic eigenvalue problems, Nonlinear Anal. 18 (1992), 957-971.
16. Z.M. Guo, Uniqueness and flat core of positive solutions for quasilinear eigenvalue problems in general smooth domains, Math. Nachr., (to appear).
17. Z.M. Guo, Structure of large positive solutions of some semilinear elliptic problems where the nonlinearity changes sign, Top. Methods Nonlinear Anal., 18 (2001), 41-71.
18. Z.M. Guo and J.R.L. Webb, Uniqueness of positive solutions for quasilinear elliptic equations when a parameter is large, Proc. Royal Soc. Edinburgh, 124A (1994), 189- 198.
19. Z.M. Guo and J.R.L. Webb, Large and small solutions of a class of quasilinear elliptic eigenvalue problems, J. Diff. Eqns., 180 (2002), 1-50.
20. M.G. Huidobro, R. Manasevich, J. Serrin, M. Tang, and C.S. Yarur, Ground states and free boundary value problems for the n-Laplacian in n dimensional space, J. Funct. Anal., 172 (2000), 177-201.
21. S. Kamin and L. Veron, Flat core properties associated to the p-Laplace operator, Proc. Amer. Math. Soc., 118 (1993), 1079-1085.
22. J.B. Keller, On solutions of Δu = f(u), Comm. Pure Appl. Math. 10 (1957), 503-510.
23. A.C. Lazer and P.J. McKenna, On a problem of Bieberbach and Redemacher, Nonl. Anal. TMA., 21
24. M. Marcus and L. Véron, Uniqueness and asymptotic behaviour of solutions with boundary blow-up for a class of nonlinear elliptic equations, Ann. Inst. H. Poincaré Anal. Non Linéaire, 14 (1997), 237-274.
25. J. Matero, Quasilinear elliptic equations with boundary blow-up, J. Analyse Math. 69 (1996), 229-246.
26. P.J. McKenna, W. Reichel and W. Walter, Symmetry and multiplicity for nonlinear elliptic differential equations with boundary blow-up, Nonl. Anal., 28 (1997), 1213- 1225.
27. P. Pucci and J. Serrin, A note on the strong maximum principle for elliptic differential inequalities, J. Math. Pures Appl., 79 (2000), 57-71.
28. J. Serrin, Nonlinear elliptic equations of second order, Lectures at AMS Symposium on Partial Differential Equations, Berkeley, 1971.
29. J. Serrin and H. Zou, Symmetry of ground states of quasilinear elliptic equations, Arch. Rational Mech. Anal., 148 (1999), 265-290.
30. J. Serrin and H. Zou, Cauchy-Liouville and universal boundedness theorems for quasilinear elliptic equations and inequalities, Acta Math., (to appear).
31. S. Takeuchi, Positive solutions of a degenerate elliptic equation with logistic reaction, Proc. Amer. Math. Soc., 129 (2000), 433-441.
32. P. Tolksdorf, On the Dirichlet problem for quasilinear equations in domains with conical boundary points, Comm. P.D.E., 8 (1983), 773-817.
33. P. Tolksdorf, Regularity for more general class of quasilinear elliptic equations, J. Diff. Eqns., 51 (1984), 126-150.
34. J.L. Vazquez, A strong maximum principle for some quasilinear elliptic equations, Appl. Math. Optim., 12 (1984), 191-202.