Tokyo Journal of Mathematics

Positive Solutions for Non-cooperative Singular $p$-Laplacian Systems

D. D. HAI

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Abstract

We prove the existence of\ positive solutions for the $p$-Laplacian system \[ \left\{ \begin{array}{@{\,}c@{\enskip}c} -\Delta _{p}u_{1}=\lambda f_{1}(u_{2}) &\text{in}~\Omega \,, \\ -\Delta _{p}u_{2}=\lambda f_{2}(u_{1}) &\text{in}~\Omega \,, \\ \ \ \ \quad u_{1}=u_{2}=0 & \text{on}~\partial \Omega \,, \end{array} \right. \] where $\Delta _{p}u=\mbox{div}(|\nabla u|^{p-2}\nabla u),p>1, \Omega$ is a bounded domain in $\mathbf{R}^{n}$ with smooth boundary $\partial \Omega ,f_{i}:(0,\infty) \rightarrow \mathbf{R}$ are possibly singular at 0 and are not required to be positive or nondecreasing, and $\lambda $ is a large parameter.

Article information

Source
Tokyo J. Math., Volume 35, Number 2 (2012), 321-331.

Dates
First available in Project Euclid: 23 January 2013

Permanent link to this document
https://projecteuclid.org/euclid.tjm/1358951321

Digital Object Identifier
doi:10.3836/tjm/1358951321

Mathematical Reviews number (MathSciNet)
MR3058709

Zentralblatt MATH identifier
1279.35036

Subjects
Primary: 35J25: Boundary value problems for second-order elliptic equations
Secondary: 35J70: Degenerate elliptic equations

Citation

HAI, D. D. Positive Solutions for Non-cooperative Singular $p$-Laplacian Systems. Tokyo J. Math. 35 (2012), no. 2, 321--331. doi:10.3836/tjm/1358951321. https://projecteuclid.org/euclid.tjm/1358951321


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References

  • C. Azizieh, Ph. Clement and E. Mitidieri, Existence and a priori estimates for positive solutions for $p$-Laplace systems, J. Differential Equations 184 (2002), 422–442.
  • H. Brezis, Analyse fonctionnelle, théorie et applications, second edition, Masson, Paris (1983).
  • Ph. Clement, J. Fleckinger, E. Mitidieri and F. de Thelin, Existence of positive solutions for a nonvariational quasilinear elliptic system, J. Differential Equations 166 (2000), 455–477.
  • A. Canada, P. Drabek and J. L. Gamez, Existence of positive solutions for some problems with nonlinear diffusion, Trans. Amer. Math. Soc. 349 (1997), 4231–4249.
  • M. G. Crandall, P. H. Rabinowitz and L. Tartar, On a Dirichlet problem with a singular nonlinearity, Comm. Partial Differential Equations 2 (1977), 193–222.
  • C. Decoster and S. Nicaise, Lower and upper solutions for elliptic problems in nonsmooth domains, J. Differential Equations 244 (2008), 599–629.
  • J. Giacomoni, I. Schindler and P. Takáč, Sobolev versus Hölder local minimizers and existence of multiple solutions for a singular quasilinear equation, Ann. Sc. Norm. Super. Pisa, Cl. Sci. (5) 6 (2007), no. 1, 117–158.
  • D. D. Hai, Singular boundary value problems for the $p$-Laplacian, Nonlinear Anal. 73 (2010), 2876–2881.
  • D. D. Hai, On positive solutions for $p$-Laplacian systems with sign-changing nonlinearities, Hokaiddo Math. J. 39 (2010), 67–84.
  • D. D. Hai, On a class of singular $p$-Laplacian boundary value problems, J. Math. Anal. Appl. 383 (2011), 619–626.
  • D. D. Hai and R. Shivaji, An existence result on positive solutions for a class of $p$-Laplacian systems, Nonlinear Anal. 56 (2004), 1007–1010.
  • G. M. Lieberman, Boundary regularity for solutions of degenerate elliptic equations, Nonlinear Anal. 12 (1988), 1203–1219.
  • T. Oden, Qualitative Methods in Nonlinear Mechanics, Prentice-Hall, Inc., Englewood Cliffs, New Jersey (1986).
  • S. Sakaguchi, Concavity properties of solutions to some degenerate quasilinear elliptic Dirichlet problems, Ann. Sc. Norm. Super. Pisa, Cl. Sci. (4) 14 (1987), 403–421.
  • J. L. Vazquez, A strong maximum principle for some quasilinear elliptic equations, Appl. Math. Optim. 12 (1984), 191–202.