Liouville results for m -Laplace equations of Lane-Emden-Fowler type

Lucio Damascelli; Alberto Farina; Berardino Sciunzi; Enrico Valdinoci

Annales de l'I.H.P. Analyse non linéaire (2009)

  • Volume: 26, Issue: 4, page 1099-1119
  • ISSN: 0294-1449

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Damascelli, Lucio, et al. "Liouville results for $m$-Laplace equations of Lane-Emden-Fowler type." Annales de l'I.H.P. Analyse non linéaire 26.4 (2009): 1099-1119. <http://eudml.org/doc/78881>.

@article{Damascelli2009,
author = {Damascelli, Lucio, Farina, Alberto, Sciunzi, Berardino, Valdinoci, Enrico},
journal = {Annales de l'I.H.P. Analyse non linéaire},
keywords = {degenerate PDEs; stable solutions; critical exponents; rigidity results},
language = {eng},
number = {4},
pages = {1099-1119},
publisher = {Elsevier},
title = {Liouville results for $m$-Laplace equations of Lane-Emden-Fowler type},
url = {http://eudml.org/doc/78881},
volume = {26},
year = {2009},
}

TY - JOUR
AU - Damascelli, Lucio
AU - Farina, Alberto
AU - Sciunzi, Berardino
AU - Valdinoci, Enrico
TI - Liouville results for $m$-Laplace equations of Lane-Emden-Fowler type
JO - Annales de l'I.H.P. Analyse non linéaire
PY - 2009
PB - Elsevier
VL - 26
IS - 4
SP - 1099
EP - 1119
LA - eng
KW - degenerate PDEs; stable solutions; critical exponents; rigidity results
UR - http://eudml.org/doc/78881
ER -

References

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  1. [1] Bidaut-Véron M.F., Local and global behavior of solutions of quasilinear equations of Emden–Fowler type, Arch. Rational Mech. Anal.107 (4) (1989) 293-324. Zbl0696.35022MR1004713
  2. [2] Bidaut-Véron M.F., Véron L., Nonlinear elliptic equations on compact Riemannian manifolds and asymptotics of Emden equations, Invent. Math.106 (3) (1991) 489-539. Zbl0755.35036MR1134481
  3. [3] D. Castorina, P. Esposito, B. Sciunzi, Degenerate elliptic equations with singular nonlinearities, preprint. Zbl1168.35020MR2471138
  4. [4] Chen W., Li C., Classification of solutions of some nonlinear elliptic equations, Duke Math. J.63 (3) (1991) 615-622. Zbl0768.35025MR1121147
  5. [5] Cuesta M., Takáč P., A strong comparison principle for positive solutions of degenerate elliptic equations, Differential Integral Equations13 (4–6) (2000) 721-746. Zbl0973.35077MR1750048
  6. [6] Damascelli L., Ramaswamy M., Symmetry of C 1 solutions ofp-Laplace equations in R N , Adv. Nonlinear Stud.1 (1) (2001) 40-64. Zbl0998.35016MR1850203
  7. [7] Damascelli L., Sciunzi B., Regularity, monotonicity and symmetry of positive solutions of m-Laplace equations, J. Differential Equations206 (2) (2004) 483-515. Zbl1108.35069MR2096703
  8. [8] Damascelli L., Sciunzi B., Harnack inequalities, maximum and comparison principles, and regularity of positive solutions of m-Laplace equations, Calc. Var. Partial Differential Equations25 (2) (2006) 139-159. Zbl1331.35130MR2188744
  9. [9] Dancer E.N., Some notes on the method of moving planes, Bull. Austral. Math. Soc.46 (3) (1992) 425-434. Zbl0777.35005MR1190345
  10. [10] Degiovanni M., Musesti A., Squassina M., On the regularity of solutions in the Pucci–Serrin identity, Calc. Var. Partial Differential Equations18 (3) (2003) 317-334. Zbl1046.35039MR2018671
  11. [11] DiBenedetto E., C 1 + α local regularity of weak solutions of degenerate elliptic equations, Nonlinear Anal.7 (8) (1983) 827-850. Zbl0539.35027MR709038
  12. [12] Esteban M.J., Lions P.-L., Existence and nonexistence results for semilinear elliptic problems in unbounded domains, Proc. Roy. Soc. Edinburgh Sect. A93 (1–2) (1982/83) 1-14. Zbl0506.35035MR688279
  13. [13] Fowler R.H., Further studies of Emden's and similar equations, Quart. J. Math. Oxford Ser. 22 (1931) 259-288. Zbl0003.23502
  14. [14] Farina A., Liouville-type results for solutions of - Δ u = u p - 1 u on unbounded domains of R N , C. R. Math. Acad. Sci. Paris, Ser. I341 (7) (2005) 415-418. Zbl1236.35057MR2168740
  15. [15] Farina A., On the classification of solutions of the Lane–Emden equation on unbounded domains of R N , J. Math. Pures Appl.87 (5) (2007) 537-561. Zbl1143.35041MR2322150
  16. [16] A. Farina, B. Sciunzi, E. Valdinoci, Bernstein and De Giorgi type problems: New results via a geometric approach, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (2008), in press. Zbl1180.35251MR2483642
  17. [17] Gidas B., Spruck J., Global and local behavior of positive solutions of nonlinear elliptic equations, Comm. Pure Appl. Math.34 (4) (1981) 525-598. Zbl0465.35003MR615628
  18. [18] Gidas B., Spruck J., Global a priori bounds for positive solutions of nonlinear elliptic equations, Comm. Partial Differential Equations6 (8) (1981) 883-901. Zbl0462.35041MR619749
  19. [19] Joseph D.D., Lundgren T.S., Quasilinear Dirichlet problems driven by positive sources, Arch. Rational Mech. Anal.49 (1–2) (1972/73) 241-269. Zbl0266.34021MR340701
  20. [20] Murthy M.K.V., Stampacchia G., Boundary value problems for some degenerate-elliptic operators, Ann. Mat. Pura Appl. (4)80 (1968) 1-122. Zbl0185.19201MR249828
  21. [21] Lieberman G.M., Boundary regularity for solutions of degenerate elliptic equations, Nonlinear Anal.12 (11) (1988) 1203-1219. Zbl0675.35042MR969499
  22. [22] Pohozaev S.I., On the eigenfunctions of the equation Δ u + λ f u = 0 , Dokl. Akad. Nauk SSSR165 (1965) 36-39. Zbl0141.30202MR192184
  23. [23] Pucci P., Serrin J., A general variational identity, Indiana Univ. Math. J.35 (3) (1986) 681-703. Zbl0625.35027MR855181
  24. [24] Pucci P., Serrin J., The strong maximum principle revisited, J. Differential Equations196 (1) (2004) 1-66. Zbl1109.35022MR2025185
  25. [25] Serrin J., Local behavior of solutions of quasi-linear elliptic equations, Acta Math.111 (1964) 247-302. Zbl0128.09101MR170096
  26. [26] Serrin J., Zou H., Cauchy–Liouville and universal boundedness theorems for quasilinear elliptic equations and inequalities, Acta Math.189 (2002). Zbl1059.35040MR1946918
  27. [27] Tolksdorf P., Regularity for a more general class of quasilinear elliptic equations, J. Differential Equations51 (1984) 126-150. Zbl0488.35017MR727034
  28. [28] Vazquez J.L., A strong maximum principle for some quasilinear elliptic equations, Appl. Math. Optim. (1984) 191-202. Zbl0561.35003MR768629

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