Abstract and Applied Analysis

Multiplicity of Positive Solutions for Semilinear Elliptic Systems

Tsing-San Hsu

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Abstract

We study the effect of the coefficient h ( x ) of the critical nonlinearity on the number of positive solutions for semilinear elliptic systems. Under suitable assumptions for f ( x ) , g ( x ) , and h ( x ) , we should prove that for sufficiently small λ , μ > 0 , there are at least k + 1 positive solutions of the semilinear elliptic systems - Δ u = λ f ( x ) | u | q - 2 u + / ( α + β) ) h ( x ) | u | α - 2 u | v | β , - Δ v = μ g ( x ) | v | q - 2 v + / ( α + β) ) h ( x ) | u | α | v | β - 2 v , where 0 Ω N is a bounded domain, α > 1 , β > 1 , and N / ( N - 2 ) < q < 2 < α + β = 2 * for N > 4 .

Article information

Source
Abstr. Appl. Anal., Volume 2013 (2013), Article ID 746380, 13 pages.

Dates
First available in Project Euclid: 27 February 2014

Permanent link to this document
https://projecteuclid.org/euclid.aaa/1393511842

Digital Object Identifier
doi:10.1155/2013/746380

Mathematical Reviews number (MathSciNet)
MR3044999

Zentralblatt MATH identifier
1277.35161

Citation

Hsu, Tsing-San. Multiplicity of Positive Solutions for Semilinear Elliptic Systems. Abstr. Appl. Anal. 2013 (2013), Article ID 746380, 13 pages. doi:10.1155/2013/746380. https://projecteuclid.org/euclid.aaa/1393511842


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References

  • C. O. Alves, D. C. de Morais Filho, and M. A. S. Souto, “On systems of elliptic equations involving subcritical or critical Sobolev exponents,” Nonlinear Analysis: Theory, Methods & Applications, vol. 42, no. 5, pp. 771–787, 2000.
  • K. J. Brown and T.-F. Wu, “A semilinear elliptic system involving nonlinear boundary condition and sign-changing weight function,” Journal of Mathematical Analysis and Applications, vol. 337, no. 2, pp. 1326–1336, 2008.
  • D. Cao and P. Han, “High energy positive solutions of Neumann problem for an elliptic system of equations with critical nonlinearities,” Calculus of Variations and Partial Differential Equations, vol. 25, no. 2, pp. 161–185, 2006.
  • J. Chabrowski and J. Yang, “On the Neumann problem for an elliptic system of equations involving the critical Sobolev exponent,” Colloquium Mathematicum, vol. 90, no. 1, pp. 19–35, 2001.
  • C.-M. Chu and C.-L. Tang, “Existence and multiplicity of positive solutions for semilinear elliptic systems with Sobolev critical exponents,” Nonlinear Analysis. Theory, Methods & Applications, vol. 71, no. 11, pp. 5118–5130, 2009.
  • P. Han, “Multiple positive solutions of nonhomogeneous elliptic systems involving critical Sobolev exponents,” Nonlinear Analysis: Theory, Methods & Applications, vol. 64, no. 4, pp. 869–886, 2006.
  • T.-S. Hsu, “Multiple positive solutions for a critical quasilinear elliptic system with concave-convex nonlinearities,” Nonlinear Analysis: Theory, Methods & Applications, vol. 71, no. 7-8, pp. 2688–2698, 2009.
  • T.-S. Hsu and H.-L. Lin, “Multiple positive solutions for a critical elliptic system with concave-convex nonlinearities,” Proceedings of the Royal Society of Edinburgh. Section A, vol. 139, no. 6, pp. 1163–1177, 2009.
  • Y. Shen and J. Zhang, “Multiplicity of positive solutions for a semilinear $p$-Laplacian system with Sobolev critical exponent,” Nonlinear Analysis: Theory, Methods & Applications, vol. 74, no. 4, pp. 1019–1030, 2011.
  • T.-F. Wu, “The Nehari manifold for a semilinear elliptic system involving sign-changing weight functions,” Nonlinear Analysis: Theory, Methods & Applications, vol. 68, no. 6, pp. 1733–1745, 2008.
  • H.-l. Lin, “Multiple positive solutions for semilinear elliptic systems,” Journal of Mathematical Analysis and Applications, vol. 391, no. 1, pp. 107–118, 2012.
  • G. Tarantello, “On nonhomogeneous elliptic equations involving critical Sobolev exponent,” Annales de l'Institut Henri Poincaré. Analyse Non Linéaire, vol. 9, no. 3, pp. 281–304, 1992.
  • G. Tarantello, “Multiplicity results for an inhomogeneous Neumann problem with critical exponent,” Manuscripta Mathematica, vol. 81, no. 1-2, pp. 57–78, 1993.
  • T.-S. Hsu, “Multiplicity results for $p$-Laplacian with critical nonlinearity of concave-convex type and sign-changing weight functions,” Abstract and Applied Analysis, vol. 2009, Article ID 652109, 24 pages, 2009.
  • K. J. Brown and Y. Zhang, “The Nehari manifold for a semilinear elliptic equation with a sign-changing weight function,” Journal of Differential Equations, vol. 193, no. 2, pp. 481–499, 2003.
  • T.-F. Wu, “On semilinear elliptic equations involving concave-convex nonlinearities and sign-changing weight function,” Journal of Mathematical Analysis and Applications, vol. 318, no. 1, pp. 253–270, 2006.
  • H. Brézis and E. Lieb, “A relation between pointwise convergence of functions and convergence of functionals,” Proceedings of the American Mathematical Society, vol. 88, no. 3, pp. 486–490, 1983.
  • T.-S. Hsu, “Existence and multiplicity of positive solutions to a perturbed singular elliptic system deriving from a strongly coupled critical potential,” Boundary Value Problems, vol. 116, 14 pages, 2012.
  • J. L. Vázquez, “A strong maximum principle for some quasilinear elliptic equations,” Applied Mathematics and Optimization, vol. 12, no. 3, pp. 191–202, 1984.
  • P. Han, “The effect of the domain topology on the number of positive solutions of an elliptic system involving critical Sobolev exponents,” Houston Journal of Mathematics, vol. 32, no. 4, pp. 1241–1257, 2006.
  • H. Brézis and L. Nirenberg, “Positive solutions of nonlinear elliptic equations involving critical Sobolev exponents,” Communications on Pure and Applied Mathematics, vol. 36, no. 4, pp. 437–477, 1983.
  • X. Cheng and S. Ma, “Existence of three nontrivial solutions for elliptic systems with critical exponents and weights,” Nonlinear Analysis: Theory, Methods & Applications, vol. 69, no. 10, pp. 3537–3548, 2008.
  • M. Struwe, Variational Methods, vol. 34 of Results in Mathematics and Related Areas (3), Springer, Berlin, Germany, 2nd edition, 1996.
  • M. Willem, Minimax Theorems, Progress in Nonlinear Differential Equations and Their Applications, 24, Birkhäuser, Boston, Mass, USA, 1996.
  • G. Bianchi, J. Chabrowski, and A. Szulkin, “On symmetric solutions of an elliptic equation with a nonlinearity involving critical Sobolev exponent,” Nonlinear Analysis: Theory, Methods & Applications, vol. 25, no. 1, pp. 41–59, 1995.
  • I. Ekeland, “On the variational principle,” Journal of Mathematical Analysis and Applications, vol. 47, pp. 324–353, 1974.