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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
DOI: https://doi.org/10.1090/S0002-9939-1995-1285981-7
MathSciNet review: 1285981
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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$.


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  • [1] L. Boccardo, F. Murat, and J.P. Puel, Existence de solutions fables pour des équations elliptiques quasi-linéaire à croissance quadratique, Nonlinear Partial Differential Equations and their Applications (London) (H. Brezis and J. L. Lions, eds.), Pitman, London, 1983, pp. 19-73. MR 716511 (84k:35064)
  • [2] -, $ {l^\infty }$ estimate for some nonlinear elliptic partial differential equations and application to an existence result, SIAM J. Math. Anal. 23 (1992), 326-333. MR 1147866 (93d:35049)
  • [3] Vincenzo Ferone and M. Rosaria Posteraro, On a class of quasilinear elliptic equations with quadratic growth in the gradient, Nonlinear Anal. TMA 20 (1993), 703-711. MR 1214736 (94c:35074)
  • [4] M. Giaquinta, Multiple integrals in the calculus of variations and nonlinear elliptic systems, Princeton Univ. Press, Princeton, NJ, 1983. MR 717034 (86b:49003)
  • [5] D. Gilbarg and N.S. Trudinger, Elliptic partial differential equations of second order, 2nd ed., Springer-Verlag, Berlin, 1983. MR 737190 (86c:35035)
  • [6] Bernhard Kawohl, Rearrangements and convexity of level sets in pde, Lecture Notes in Math., vol. 1150, Springer-Verlag, Berlin, 1985. MR 810619 (87a:35001)
  • [7] J.L. Lions, Quelques méthodes de résolution des problèmes aux limites nonlinéaire, Dunod, Paris, 1969.
  • [8] G. Talenti, Elliptic equations and rearrangements, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 3 (1976), 697-718. MR 0601601 (58:29170)

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Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1995-1285981-7
Keywords: Existence, degenerate, critical growth condition, rearrangements, spherical symmetric unbounded solution
Article copyright: © Copyright 1995 American Mathematical Society

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