Taiwanese Journal of Mathematics

Multiplicity of Solutions for a Class of Quasilinear Elliptic Systems in Orlicz-Sobolev Spaces

Liben Wang, Xingyong Zhang, and Hui Fang

Full-text: Open access

Abstract

In this paper, we investigate the following nonlinear and non-homogeneous elliptic system \[ \begin{cases} -\operatorname{div}(a_1(|\nabla u|) \nabla u) = \lambda_1 F_u(x,u,v) - \lambda_2 G_u(x,u,v) - \lambda_3 H_u(x,u,v) &\textrm{in $\Omega$}, \\ -\operatorname{div}(a_2(|\nabla v|) \nabla v) = \lambda_1 F_v(x,u,v) - \lambda_2 G_v(x,u,v) - \lambda_3 H_v(x,u,v) &\textrm{in $\Omega$}, \\ u = v = 0 &\textrm{on $\partial \Omega$}, \end{cases} \] where $\Omega$ is a bounded domain in $\mathbb{R}^N$ ($N \geq 1$) with smooth boundary $\partial \Omega$, $\lambda_1$, $\lambda_2$, $\lambda_3$ are three parameters, $\phi_i(t) = a_i(|t|)t$ ($i = 1,2$) are two increasing homeomorphisms from $\mathbb{R}$ onto $\mathbb{R}$, and functions $F$, $G$, $H$ are of class $C^1(\Omega \times \mathbb{R}^2, \mathbb{R})$ and satisfy some reasonable growth conditions. By using a three critical points theorem due to B. Ricceri, we obtain that system has at least three solutions. With some additional conditions, by using a four critical points theorem due to G. Anello, we obtain that system has at least four solutions.

Article information

Source
Taiwanese J. Math., Volume 21, Number 4 (2017), 881-912.

Dates
Received: 4 September 2016
Revised: 24 November 2016
Accepted: 27 November 2016
First available in Project Euclid: 27 July 2017

Permanent link to this document
https://projecteuclid.org/euclid.twjm/1501120840

Digital Object Identifier
doi:10.11650/tjm/7887

Mathematical Reviews number (MathSciNet)
MR3684392

Zentralblatt MATH identifier
06871351

Subjects
Primary: 35J20: Variational methods for second-order elliptic equations 35J50: Variational methods for elliptic systems

Keywords
Orlicz-Sobolev spaces quasilinear weak solution critical point

Citation

Wang, Liben; Zhang, Xingyong; Fang, Hui. Multiplicity of Solutions for a Class of Quasilinear Elliptic Systems in Orlicz-Sobolev Spaces. Taiwanese J. Math. 21 (2017), no. 4, 881--912. doi:10.11650/tjm/7887. https://projecteuclid.org/euclid.twjm/1501120840


Export citation

References

  • R. A. Adams and J. J. F. Fournier, Sobolev Spaces, Second edition, Pure and Applied Mathematics (Amsterdam) 140, Academic Press, Amsterdam, 2003.
  • C. O. Alves, G. M. Figueiredo and J. A. Santos, Strauss and Lions type results for a class of Orlicz-Sobolev spaces and applications, Topol. Methods Nonlinear Anal. 44 (2014), no. 2, 435–456.
  • A. Ambrosetti and P. H. Rabinowitz, Dual variational methods in critical point theory and applications, J. Functional Analysis 14 (1973), no. 4, 349–381.
  • G. Anello, Multiple nonnegative solutions for an elliptic boundary value problem involving combined nonlinearities, Math. Comput. Modelling 52 (2010), no. 1-2, 400–408.
  • ––––, On the multiplicity of critical points for parameterized functionals on reflexive Banach spaces, Studia Math. 213 (2012), no. 1, 49–60.
  • G. Bonanno, Some remarks on a three critical points theorem, Nonlinear Anal. 54 (2003), no. 4, 651–665.
  • G. Bonanno and S. A. Marano, On the structure of the critical set of non-differentiable functions with a weak compactness condition, Appl. Anal. 89 (2010), no. 1, 1–10.
  • G. Bonanno, G. Molica Bisci and V. Rădulescu, Existence of three solutions for a non-homogeneous Neumann problem through Orlicz-Sobolev spaces, Nonlinear Anal. 74 (2011), no. 14, 4785–4795.
  • ––––, Quasilinear elliptic non-homogeneous Dirichlet problems through Orlicz-Sobolev spaces, Nonlinear Anal. 75 (2012), no. 12, 4441–4456.
  • H. Brezis, Functional Analysis, Sobolev Spaces and Partial Differential Equations, Universitext, Springer, New York, 2011.
  • F. Cammaroto and L. Vilasi, Multiple solutions for a nonhomogeneous Dirichlet problem in Orlicz-Sobolev spaces, Appl. Math. Comput. 218 (2012), no. 23, 11518–11527.
  • N. T. Chung, Three solutions for a class of nonlocal problems in Orlicz-Sobolev spaces, J. Korean Math. Soc. 50 (2013), no. 6, 1257–1269.
  • N. T. Chung and H. Q. Toan, On a nonlinear and non-homogeneous problem without (A-R) type condition in Orlicz-Sobolev spaces, Appl. Math. Comput. 219 (2013), no. 14, 7820–7829.
  • Ph. Clément, M. García-Huidobro, R. Manásevich and K. Schmitt, Mountain pass type solutions for quasilinear elliptic equations, Calc. Var. Partial Differential Equations 11 (2000), no. 1, 33–62.
  • F. Colasuonno, P. Pucci and C. Varga, Multiple solutions for an eigenvalue problem involving $p$-Laplacian type operators, Nonlinear Anal. 75 (2012), no. 12, 4496–4512.
  • P. De Nápoli and M. C. Mariani, Mountain pass solutions to equations of $p$-Laplacian type, Nonlinear Anal. 54 (2003), no. 7, 1205–1219.
  • G. Dinca, P. Jebelean and J. Mawhin, Variational and topological methods for Dirichlet problems with $p$-Laplacian, Port. Math. (N.S.) 58 (2001), no. 3, 339–378.
  • N. Fukagai, M. Ito and K. Narukawa, Positive solutions of quasilinear elliptic equations with critical Orlicz-Sobolev nonlinearity on $\mb{R}^{N}$, Funkcial. Ekcac. 49 (2006), no. 2, 235–267.
  • N. Fukagai and K. Narukawa, On the existence of multiple positive solutions of quasilinear elliptic eigenvalue problems, Ann. Mat. Pura Appl. (4) 186 (2007), no. 3, 539–564.
  • M. García-Huidobro, V. K. Le, R. Manásevich and K. Schmitt, On principal eigenvalues for quasilinear elliptic differential operators: an Orlicz-Sobolev space setting, NoDEA Nonlinear Differential Equations Appl. 6 (1999), no. 2, 207–225.
  • J.-P. Gossez, Orlicz-Sobolev spaces and nonlinear elliptic boundary value problems, in Nonlinear Analysis, Function Spaces and Applications (Proc. Spring School, Horni Bradlo, 1978), 59–94, Teubner, Leipzig, 1979.
  • J. Huentutripay and R. Manásevich, Nonlinear eigenvalues for a quasilinear elliptic system in Orlicz-Sobolev spaces, J. Dynam. Differential Equations 18 (2006), no. 4, 901–929.
  • Q. Jiu and J. Su, Existence and multiplicity results for Dirichlet problems with $p$-Laplacian, J. Math. Anal. Appl. 281 (2003), no. 2, 587–601.
  • S. Liu, Existence of solutions to a superlinear $p$-Laplacian equation, Electron. J. Differential Equations 2001 (2001), no. 66, 1–6.
  • M. Mihăilescu and V. Rădulescu, Existence and multiplicity of solutions for quasilinear nonhomogeneous problems: an Orlicz-Sobolev space setting, J. Math. Anal. Appl. 330 (2007), no. 1, 416–432.
  • M. Mihăilescu and D. Repovš, Multiple solutions for a nonlinear and non-homogeneous problem in Orlicz-Sobolev spaces, Appl. Math. Comput. 217 (2011), no. 14, 6624–6632.
  • M. M. Rao and Z. D. Ren, Applications of Orlicz Spaces, Monographs and Textbooks in Pure and Applied Mathematics 250, Marcel Dekker, New York, 2002.
  • B. Ricceri, On a three critical points theorem, Arch. Math. (Basel) 75 (2000), no. 3, 220–226.
  • ––––, A further three critical points theorem, Nonlinear Anal. 71 (2009), no. 9, 4151–4157.
  • ––––, A further refinement of a three critical points theorem, Nonlinear Anal. 74 (2011), no. 18, 7446–7454.
  • L. Wang, X. Zhang and H. Fang, Existence and multiplicity of solutions for a class of $(\phi_1,\phi_2)$-Laplacian elliptic system in $\mb{R}^N$ via genus theory, Comput. Math. Appl. 72 (2016), no. 1, 110–130.
  • F. Xia and G. Wang, Existence of solution for a class of elliptic systems, Journal of Hunan Agricultural University (Natural Sciences) 33 (2007), no. 3, 362–366.
  • E. Zeidler, Nonlinear Functional Analysis and its Applications II/B: Nonlinear Monotone Operators, Springer-Verlag, New York, 1990.
  • Z. Zhang, Existence of positive radial solutions for quasilinear elliptic equations and systems, Electron. J. Differential Equations 2016 (2016), no. 50, 9 pp.