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Sibirsk. Mat. Zh., 2005, Volume 46, Number 5, Pages 963–984 (Mi smj1032)  

This article is cited in 23 scientific papers (total in 23 papers)

On localization of solutions of elliptic equations with nonhomogeneous anisotropic degeneracy

S. N. Antontseva, S. I. Shmarevb

a University of Beira Interior
b Universidad de Oviedo

Abstract: The work deals with the Dirichlet problem for elliptic equations with nonhomogeneous anisotropic degeneracy in a possibly unbounded domain of multidimensional Euclidean space. The existence of weak solutions is proved. Some conditions are established connecting the character of nonlinearity of the equation and the geometric characteristics of the domain which guarantee the one-dimensional localization (vanishing) of weak solutions. The equation with anisotropic degeneracy is shown to admit localized solutions even in the absence of absorption.

Keywords: nonlinear elliptic equation, nonhomogeneous degeneracy, localization of solutions

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English version:
Siberian Mathematical Journal, 2005, 46:5, 765–782

Bibliographic databases:

UDC: 517.957
Received: 05.04.2005

Citation: S. N. Antontsev, S. I. Shmarev, “On localization of solutions of elliptic equations with nonhomogeneous anisotropic degeneracy”, Sibirsk. Mat. Zh., 46:5 (2005), 963–984; Siberian Math. J., 46:5 (2005), 765–782

Citation in format AMSBIB
\by S.~N.~Antontsev, S.~I.~Shmarev
\paper On localization of solutions of elliptic equations with nonhomogeneous anisotropic degeneracy
\jour Sibirsk. Mat. Zh.
\yr 2005
\vol 46
\issue 5
\pages 963--984
\jour Siberian Math. J.
\yr 2005
\vol 46
\issue 5
\pages 765--782

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    This publication is cited in the following articles:
    1. Antontsev S., Shmarev S., “Elliptic equations and systems with nonstandard growth conditions: Existence, uniqueness and localization properties of solutions”, Nonlinear Anal., 65:4 (2006), 728–761  crossref  mathscinet  zmath  isi  elib  scopus
    2. Antontsev S., Shmarev S., “Parabolic equations with anisotropic nonstandard growth conditions”, Free Boundary Problems: Theory and Applications, International Series of Numerical Mathematics, 154, 2007, 33–44  crossref  mathscinet  zmath  isi
    3. Proc. Steklov Inst. Math., 261 (2008), 11–21  mathnet  crossref  mathscinet  zmath  isi  elib  elib
    4. Alves C.O., “Existence of solution for a degenerate $p(x)$-Laplacian equation in $\mathbb R^N$”, J. Math. Anal. Appl., 345:2 (2008), 731–742  crossref  mathscinet  zmath  adsnasa  isi  scopus
    5. Amaziane B., Antontsev S., Pankratov L., Piatnitski A., “Homogenization of p–Laplacian in perforated domain”, Ann. Inst. H. Poincaré Anal. Non Linéaire, 26:6 (2009), 2457–2479  crossref  mathscinet  zmath  adsnasa  isi  scopus
    6. Vétois J., “A priori estimates for solutions of anisotropic elliptic equations”, Nonlinear Anal., 71:9 (2009), 3881–3905  crossref  mathscinet  zmath  isi  elib  scopus
    7. Amaziane B., Pankratov L., Prytula V., “Nonlocal effects in homogenization of p(epsilon)(x)–Laplacian in perforated domains”, Nonlinear Anal., 71:9 (2009), 4078–4097  crossref  mathscinet  zmath  isi  elib  scopus
    8. Hamidi A.E., Vétois J., “Sharp Sobolev Asymptotics for Critical Anisotropic Equations”, Arch. Ration. Mech. Anal., 192:1 (2009), 1–36  crossref  zmath  adsnasa  isi  scopus
    9. Antontsev S., Shmarev S., “Localization of solutions of anisotropic parabolic equations”, Nonlinear Analysis-Theory Methods & Applications, 71:12 (2009), E725–E737  crossref  mathscinet  zmath  isi  scopus
    10. Jérôme, Vétois, “Asymptotic stablility, convexity, and Lipschitz regularity of domains in the anisotropic regime”, Commun. Contemp. Math., 12:1 (2010), 35–53  crossref  mathscinet  zmath  isi  scopus
    11. Antontsev S., Shmarev S., “Vanishing solutions of anisotropic parabolic equations with variable nonlinearity”, J. Math. Anal. Appl., 361:2 (2010), 371–391  crossref  mathscinet  zmath  isi  elib  scopus
    12. P. V. Sadchikov, A. D. Baev, “O nekotorykh kraevykh zadachakh v poluprostranstve dlya odnogo klassa psevdodifferentsialnykh uravnenii s vyrozhdeniem”, Izv. Sarat. un-ta. Nov. ser. Ser. Matematika. Mekhanika. Informatika, 10:2 (2010), 34–41  mathnet  elib
    13. Alves C.O., “Existence of radial solutions for a class of $p(x)$-Laplacian equations with critical growth”, Differential and Integral Equations, 23:1-2 (2010), 113–123  mathscinet  zmath  isi
    14. Amaziane B., Pankratov L., Piatnitski A., “Homogenization of variational functionals with nonstandard growth in perforated domains”, Netw Heterog Media, 5:2 (2010), 189–215  crossref  mathscinet  zmath  isi  elib  scopus
    15. Kandilakis D.A., Sidiropoulos N., “Elliptic problems involving the p(x)-Laplacian with competing nonlinearities”, J Math Anal Appl, 379:1 (2011), 378–387  crossref  mathscinet  zmath  isi  scopus
    16. Antontsev S., Shmarev S., “Elliptic equations with triple variable nonlinearity”, Complex Variables and Elliptic Equations, 56:7–9 (2011), 573–597  crossref  mathscinet  zmath  isi  scopus
    17. Di Castro A., “Anisotropic elliptic problems with natural growth terms”, Manuscripta Math, 135:3–4 (2011), 521–543  crossref  mathscinet  zmath  isi  elib  scopus
    18. Ar. S. Tersenov, “New a priori estimates of solutions to anisotropic elliptic equations”, Siberian Math. J., 53:3 (2012), 539–550  mathnet  crossref  mathscinet  isi
    19. Antontsev S., Shmarev S., “Doubly Degenerate Parabolic Equations with Variable Nonlinearity I: Existence of Bounded Strong Solutions”, Adv. Differ. Equat., 17:11-12 (2012), 1181–1212  mathscinet  zmath  isi  elib
    20. S. V. Pikulin, “Ob otsenke razmera zony lokalizatsii nositelya resheniya polulineinogo ellipticheskogo uravneniya”, Vestn. SamGU. Estestvennonauchn. ser., 2013, no. 9/1(110), 28–34  mathnet
    21. Alves C.O., Ferreira M.C., “Nonlinear Perturbations of a P(X)-Laplacian Equation With Critical Growth in R-N”, Math. Nachr., 287:8-9 (2014), 849–868  crossref  mathscinet  zmath  isi  scopus
    22. Sert U., Soltanov K., “On Solvability of a Class of Nonlinear Elliptic Type Equation With Variable Exponent”, J. Appl. Anal. Comput., 7:3 (2017), 1139–1160  crossref  mathscinet  isi  scopus
    23. Sert U., Soltanov K., “Solvability of Nonlinear Elliptic Type Equation With Two Unrelated Non Standard Growths”, J. Korean. Math. Soc., 55:6 (2018), 1337–1358  crossref  mathscinet  zmath  isi  scopus
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