Abstract and Applied Analysis

Nonexistence theorems for weak solutions of quasilinear elliptic equations

A. G. Kartsatos and V. V. Kurta

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Abstract

New nonexistence results are obtained for entire bounded (either from above or from below) weak solutions of wide classes of quasilinear elliptic equations and inequalities. It should be stressed that these solutions belong only locally to the corresponding Sobolev spaces. Important examples of the situations considered herein are the following: Σi=1n(a(x)|u|p2uxi)=|u|q1u,Σi=1n(a(x)|uxi|p2uxi)xi=|u|q1u,Σi=1n(a(x)|u|p2uxi/1+|u|2)xi=|u|q1u, where n1,p>1,q>0 are fixed real numbers, and a(x) is a nonnegative measurable locally bounded function. The methods involve the use of capacity theory in connection with special types of test functions and new integral inequalities. Various results, involving mainly classical solutions, are improved and/or extended to the present cases.

Article information

Source
Abstr. Appl. Anal., Volume 6, Number 3 (2001), 163-189.

Dates
First available in Project Euclid: 13 April 2003

Permanent link to this document
https://projecteuclid.org/euclid.aaa/1050266745

Digital Object Identifier
doi:10.1155/S1085337501000549

Mathematical Reviews number (MathSciNet)
MR1861245

Zentralblatt MATH identifier
1119.35322

Subjects
Primary: 35J60: Nonlinear elliptic equations 35R45: Partial differential inequalities

Citation

Kartsatos, A. G.; Kurta, V. V. Nonexistence theorems for weak solutions of quasilinear elliptic equations. Abstr. Appl. Anal. 6 (2001), no. 3, 163--189. doi:10.1155/S1085337501000549. https://projecteuclid.org/euclid.aaa/1050266745


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References

  • H. Berestycki, I. Capuzzo-Dolcetta, and L. Nirenberg, Superlinear indefinite elliptic problems and nonlinear Liouville theorems, Topol. Methods Nonlinear Anal. 4 (1994), no. 1, 59–78.
  • M.-F. Bidaut-Véron, Local and global behavior of solutions of quasilinear equations of Emden-Fowler type, Arch. Rational Mech. Anal. 107 (1989), no. 4, 293–324.
  • P. Clément, R. Manásevich, and E. Mitidieri, Positive solutions for a quasilinear system via blow up, Comm. Partial Differential Equations 18 (1993), no. 12, 2071–2106.
  • J. Frehse, Capacity methods in the theory of partial differential equations, Jahresber. Deutsch. Math.-Verein. 84 (1982), no. 1, 1–44.
  • B. Gidas and J. Spruck, Global and local behavior of positive solutions of nonlinear elliptic equations, Comm. Pure Appl. Math. 34 (1981), no. 4, 525–598.
  • A. G. Kartsatos and V. V. Kurta, On the problem of the nonexistence of entire solutions of quasilinear elliptic equations, Dokl. Akad. Nauk 371 (2000), no. 5, 591–593 (Russian).
  • V. V. Kurta, On the behavior of solutions of quasilinear elliptic equations of second order in unbounded domains, Ukrainian Math. J. 44 (1992), no. 2, 245–248, [translated from Ukrain. Mat. Zh. 44 (1992), 279–283. MR93f:3506693f:35066].
  • ––––, Phragmén-Lindelöf theorems for semilinear equations, Soviet Math. Dokl. 45 (1992), no. 1, 31–33, [translated from Dokl. Akad. Nauk SSSR 322 (1992), 38–40. MR93c:3502293c:35022].
  • ––––, Some problems of the qualitative theory of second order nonlinear equations, Ph.D. thesis, Steklov Math. Inst., Moscow, 1994.
  • ––––, On the questions of the absence of entire positive solutions for semilinear elliptic equations, Russian Math. Surveys 50 (1995), 783.
  • ––––, The nonexistence of positive solutions of some elliptic equations, Math. Notes 65 (1999), no. 4, 462–469.
  • ––––, On the nonexistence of positive solutions to semilinear elliptic equations, Proc. Steklov Inst. Math. 227 (1999), 155–162.
  • J. L. Lions, Quelques Méthodes de Résolution des Problèmes aux Limites Non Linéaires, Dunod, Paris, 1969 (French).
  • V. M. Mikljukov, A new approach to the Bernstein theorem and to related questions of equations of minimal surface type, Mat. Sb. (N.S.) 108(150) (1979), no. 2, 268–289. \MR80e:53005
  • E. Mitidieri, Nonexistence of positive solutions of semilinear elliptic systems in ${\bf {R}}\sp {N}$, Differential Integral Equations 9 (1996), no. 3, 465–479.
  • E. Mitidieri and S. I. Pokhozhaev, Absence of global positive solutions of quasilinear elliptic inequalities, Dokl. Akad. Nauk 359 (1998), no. 4, 456–460 (Russian).
  • ––––, Nonexistence of positive solutions for quasilinear elliptic problems in ${\bf {R}}\sp {N}$, Proc. Steklov Inst. Math. 227 (1999), 186–216, [translated from Tr. Mat. Inst. Steklova 227 (1999), 192–222. MR2001g:350822001g:35082].
  • W.-M. Ni and J. Serrin, Nonexistence theorems for quasilinear partial differential equations, Rend. Circ. Mat. Palermo (2) Suppl. (1985), no. 8, 171–185 (Italian).
  • J. Serrin and H. Zou, Non-existence of positive solutions of Lane-Emden systems, Differential Integral Equations 9 (1996), no. 4, 635–653.