Advertisement

Ukrainian Mathematical Journal

, Volume 71, Issue 4, pp 651–662 | Cite as

Existence Results for a Class of Kirchhoff-Type Systems with Combined Nonlinear Effects

  • G. A. AfrouziEmail author
  • S. Shakeri
  • H. Zahmatkesh
BRIEF COMMUNICATIONS
  • 10 Downloads
We study the existence of positive solutions for a nonlinear system
$$ {\displaystyle \begin{array}{l}-{M}_1\left(\underset{\varOmega }{\int }{\left|{\nabla}_u\right|}^p dx\right)\operatorname{div}\left({\left|x\right|}^{- ap}{\left|{\nabla}_u\right|}^{p-2}{\nabla}_u\right)=\lambda {\left|x\right|}^{-\left(a+1\right)p+{c}_1}f\left(u,\upsilon \right),\kern1em x\in \varOmega, \\ {}-{M}_2\left(\underset{\varOmega }{\int }{\left|{\nabla}_{\upsilon}\right|}^q dx\right)\operatorname{div}\left({\left|x\right|}^{- bq}{\left|{\nabla}_{\upsilon}\right|}^{q-2}{\nabla}_{\upsilon}\right)=\lambda {\left|x\right|}^{-\left(b+1\right)q+{c}_2}g\left(u,\upsilon \right),\kern1em x\in \varOmega, \\ {}u=\upsilon =0,\kern1em x\in \vartheta \varOmega, \end{array}} $$

where Ω is a bounded smooth domain in ℝN with \( 0\in \varOmega, 1\kern0.33em <\kern0.33em p,q\kern0.33em <N,0\le a<\frac{N-p}{p},0\le b<\frac{N-q}{q}, \) and c1, c2, and λ are positive parameters. Here, M1, M2, f, and g satisfy certain conditions. We use the method of sub- and supersolutions to establish our results.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    G. A. Afrouzi, N. T. Chung, and S. Shakeri, “Existence of positive solutions for Kirchhoff type equations,” Electron. J. Different. Equat., 180, 1–8 (2013).MathSciNetzbMATHGoogle Scholar
  2. 2.
    G. A. Afrouzi, N. T. Chung, and S. Shakeri, “Positive solutions for a infinite semipositone problem involving nonlocal operator,” Rend. Sem. Mat. Univ. Padova, 132, 25–32 (2014).CrossRefGoogle Scholar
  3. 3.
    C. O. Alves and F. J. S. A. Corrêa, “On existence of solutions for a class of problem involving a nonlinear operator,” Comm. Appl. Nonlin. Anal., 8, 43–56 (2001).MathSciNetzbMATHGoogle Scholar
  4. 4.
    C. Atkinson and K. El Kalli, “Some boundary-value problems for the Bingham model,” J. Non-Newton. Fluid Mech., 41, 339–363 (1992).CrossRefGoogle Scholar
  5. 5.
    H. Bueno, G. Ercole, W. Ferreira, and A. Zumpano, “Existence and multiplicity of positive solutions for the p-Laplacian with nonlocal coefficient,” J. Math. Anal. Appl., 343, 151–158 (2008).MathSciNetCrossRefGoogle Scholar
  6. 6.
    L. Caffarelli, R. Kohn, and L. Nirenberg, “First order interpolation inequalities with weights,” Compos. Math., 53, 259–275 (1984).MathSciNetzbMATHGoogle Scholar
  7. 7.
    A. Canada, P. Drabek, and J. L. Gamez, “Existence of positive solutions for some problems with nonlinear diffusion,” Trans. Amer. Math. Soc., 349, 4231–4249 (1997).MathSciNetCrossRefGoogle Scholar
  8. 8.
    F. Cistea, D. Motreanu, and V. Radulescu, “Weak solutions of quasilinear problems with nonlinear boundary condition,” Nonlin. Anal., 43, 623–636 (2001).MathSciNetCrossRefGoogle Scholar
  9. 9.
    N. T. Chung, “An existence result for a class of Kirchhoff type systems via sub- and supersolutions method,” Appl. Math. Lett., 35, 95–101 (2014).MathSciNetCrossRefGoogle Scholar
  10. 10.
    E. N. Dancer, “Competing species systems with diffusion and large interaction,” Rend. Sem. Mat. Fis. Milano, 65, 23–33 (1995).MathSciNetCrossRefGoogle Scholar
  11. 11.
    P. Drabek and J. Hernandez, “Existence and uniqueness of positive solutions for some quasilinear elliptic problem,” Nonlin. Anal., 44, No. 2, 189–204 (2001).MathSciNetCrossRefGoogle Scholar
  12. 12.
    P. Drabek and S. H. Rasouli, “A quasilinear eigenvalue problem with Robin conditions on the nonsmooth domain of finite measure,” Z. Anal. Anwend., 29, No. 4, 469–485 (2010).MathSciNetCrossRefGoogle Scholar
  13. 13.
    J. F. Escobar, “Uniqueness theorems on conformal deformations of metrics, Sobolev inequalities, and an eigenvalue estimate,” Comm. Pure Appl. Math., 43, 857–883 (1990).MathSciNetCrossRefGoogle Scholar
  14. 14.
    F. Fang and Sh. Liu, “Nontrivial solutions of superlinear p-Laplacian equations,” J. Math. Anal. Appl., 351, 138–146 (2009).MathSciNetCrossRefGoogle Scholar
  15. 15.
    X. Han and G. Dai, “On the sub-supersolution method for p(x)-Kirchhoff type equations,” J. Inequal. Appl., 2012 (2012).Google Scholar
  16. 16.
    G. Kirchhoff, Mechanik, Teubner, Leipzig (1883).zbMATHGoogle Scholar
  17. 17.
    G. S. Ladde, V. Lakshmikantham, and A. S. Vatsal, “Existence of coupled quasisolutions of systems of nonlinear elliptic boundary value problems,” Nonlin. Anal., 8, No. 5, 501–515 (1984).MathSciNetCrossRefGoogle Scholar
  18. 18.
    O. H. Miyagaki and R. S. Rodrigues, “On positive solutions for a class of singular quasilinear elliptic systems,” J. Math. Anal. Appl., 334, 818–833 (2007).MathSciNetCrossRefGoogle Scholar
  19. 19.
    M. Nagumo, “Über die Differentialgleichung y″ = f(x, y, y′),” Proc. Phys.-Math. Soc. Japan, 19, 861–866 (1937).zbMATHGoogle Scholar
  20. 20.
    H. Poincaré, “Les fonctions fuchsiennes et l’équation Δu = e u,” J. Math. Pures Appl. (9), 4, 137–230 (1898).Google Scholar
  21. 21.
    S. H. Rasouli, “On a class of singular elliptic system with combined nonlinear effects,” Acta Univ. Apulensis Math. Inform., 38, 187–195 (2014).MathSciNetzbMATHGoogle Scholar
  22. 22.
    S. H. Rasouli and G. A. Afrouzi, “The Nehari manifold for a class of concave-convex elliptic systems involving the p-Laplacian and nonlinear boundary condition,” Nonlin. Anal., 73, 3390–3401 (2010).MathSciNetCrossRefGoogle Scholar
  23. 23.
    P. Tolksdorf, “Regularity for a more general class of quasilinear elliptic equations,” J. Different. Equat., 51, 126–150 (1984).MathSciNetCrossRefGoogle Scholar
  24. 24.
    B. Xuan, “The solvability of quasilinear Brezis–Nirenberg-type problems with singular weights,” Nonlin. Anal., 62, 703–725 (2005).MathSciNetCrossRefGoogle Scholar
  25. 25.
    B. Xuan, “The eigenvalue problem for a singular quasilinear elliptic equation,” Electron. J. Different. Equat., 16 (2004).Google Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of MazandaranBabolsarIran
  2. 2.Department of Mathematics, Ayatollah Amoli BranchIslamic Azad UniversityAmolIran

Personalised recommendations