Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)



Nonlinear degenerate elliptic partial differential equations with critical growth conditions on the gradient

Authors: Kwon Cho and Hi Jun Choe
Journal: Proc. Amer. Math. Soc. 123 (1995), 3789-3796
MSC: Primary 35J70; Secondary 35J60
MathSciNet review: 1285981
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: We consider a nonlinear degenerate elliptic partial differential equation $ - {\operatorname{div}}(\vert\nabla u{\vert^{p - 2}}\nabla u) = H(x,u,\nabla u)$ with the critical growth condition on $ H(x,u,\nabla u) \leq g(x) + \vert\nabla u{\vert^p}$, where g is sufficiently integrable and p is between 1 and $ \infty $. Our first goal of this paper is to prove the existence of the solution in $ W_0^{1,p} \cap {L^\infty }$. The main idea is to obtain the uniform $ {L^\infty }$-estimate of suitable approximate solutions, employing a truncation technique and radially decreasing symmetrization techniques based on rearrangements. We also find an example of unbounded weak solution of $ - {\operatorname{div}}(\vert\nabla u{\vert^{p - 2}}\nabla u) = \vert\nabla u{\vert^p}$ for $ 1 < p \leq n$.

References [Enhancements On Off] (What's this?)

  • [1] L. Boccardo, F. Murat, and J.-P. Puel, Existence de solutions faibles pour des équations elliptiques quasi-linéaires à croissance quadratique, Nonlinear partial differential equations and their applications. Collège de France Seminar, Vol. IV (Paris, 1981/1982) Res. Notes in Math., vol. 84, Pitman, Boston, Mass.-London, 1983, pp. 19–73 (French, with English summary). MR 716511
  • [2] L. Boccardo, F. Murat, and J.-P. Puel, 𝐿^{∞} estimate for some nonlinear elliptic partial differential equations and application to an existence result, SIAM J. Math. Anal. 23 (1992), no. 2, 326–333 (English, with French summary). MR 1147866,
  • [3] Vincenzo Ferone and M. Rosaria Posteraro, On a class of quasilinear elliptic equations with quadratic growth in the gradient, Nonlinear Anal. 20 (1993), no. 6, 703–711. MR 1214736,
  • [4] Mariano Giaquinta, Multiple integrals in the calculus of variations and nonlinear elliptic systems, Annals of Mathematics Studies, vol. 105, Princeton University Press, Princeton, NJ, 1983. MR 717034
  • [5] David Gilbarg and Neil S. Trudinger, Elliptic partial differential equations of second order, 2nd ed., Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 224, Springer-Verlag, Berlin, 1983. MR 737190
  • [6] Bernhard Kawohl, Rearrangements and convexity of level sets in PDE, Lecture Notes in Mathematics, vol. 1150, Springer-Verlag, Berlin, 1985. MR 810619
  • [7] J.L. Lions, Quelques méthodes de résolution des problèmes aux limites nonlinéaire, Dunod, Paris, 1969.
  • [8] Giorgio Talenti, Elliptic equations and rearrangements, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 3 (1976), no. 4, 697–718. MR 0601601

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC: 35J70, 35J60

Retrieve articles in all journals with MSC: 35J70, 35J60

Additional Information

Keywords: Existence, degenerate, critical growth condition, rearrangements, spherical symmetric unbounded solution
Article copyright: © Copyright 1995 American Mathematical Society