• Jung, Tack-Sun (Department of Mathematics Kunsan National University) ;
  • Choi, Q-Heung (Department of Mathematics Education Inha University)
  • 투고 : 2008.11.07
  • 발행 : 2012.01.31


Let ${\Omega}$ be a bounded subset of $\mathbb{R}^n$ with smooth boundary. We investigate the solvability for a class of the system of the nonlinear elliptic equations with Dirichlet boundary condition. Using the mountain pass theorem we prove that the system has at least one nontrivial solution.


  1. A. Ambrosetti and P. Rabinowitz, Dual variational methods in critical point theory and applications, J. Funct. Anal. 14 (1973), 349-381.
  2. A. Castro and J. Cossio, Multiple solutions for a nonlinear Dirichlet problem, SIAM J. Math. Anal. 25 (1994), no. 6, 1554-1561.
  3. A. Castro and A. C. Lazer, Critical point theory and the number of solutions of a nonlinear Dirichlet problem, Ann. Mat. Pura Appl. (4) 120 (1979), 113-137.
  4. K. C. Chang, Solutions of asymptotically linear operator equations via Morse theory, Comm. Pure Appl. Math. 34 (1981), no. 5, 693-712.
  5. K. C. Chang, Infinite Dimensional Morse Theory and Multiple Solution Problems, Birkhauser, 1993.
  6. Q. H. Choi and T. Jung, Multiple periodic solutions of a semilinear wave equation at double external resonances, Commun. Appl. Anal. 3 (1999), no. 1, 73-84.
  7. E. N. Dancer, On the Dirichlet problem for weakly nonlinear elliptic partial differential equations, Proc. Roy. Soc. Edinburgh Sect. A 76 (1976/77), no. 4, 283-300.
  8. M. Degiovanni, Homotopical properties of a class of nonsmooth functions, Ann. Mat. Pura Appl. (4) 156 (1990), 37-71.
  9. M. Degiovanni, A. Marino, and M. Tosques, Evolution equations with lack of convexity, Nonlinear Anal. 9 (1985), no. 12, 1401-1443.
  10. D. R. Dunninger and H. Wang, Multiplicity of positive radial solutions for an elliptic system on an annulus, Nonlinear Anal. 42 (2000), no. 5, Ser. A: Theory Methods, 803-811.
  11. G. Fournier, D. Lupo, M. Ramos, and M. Willem, Limit relative category and critical point theory, Dynam. Report 3 (1993), 1-23.
  12. A. Groli, A. Marino, and C. Saccon, Variational theorems of mixed type and asymptotically linear variational inequalities, Topol. Methods Nonlinear Anal. 12 (1998), no. 1, 109-136.
  13. K. S. Ha and Y. H. Lee, Existence of multiple positive solutions of singular boundary value problems, Nonlinear Anal. 28 (1997), no. 8, 1429-1438.
  14. H. Hofer, Variational and topological methods in partially ordered Hilbert spaces, Math. Ann. 261 (1982), no. 4, 493-514.
  15. K. Lan and R. L. Webb, Positive solutions of semilinear differential equations with singularities, J. Differential Equations 148 (1998), no. 2, 407-421.
  16. Y. H. Lee, Existence of multiple positive radial solutions for a semilinear elliptic system on an unbounded domain, Nonlinear Anal. 47 (2001), no. 6, 3649-3660.
  17. Y. H. Lee, A multiplicity result of positive radial solutions for a multiparameter elliptic system on an exterior domain, Nonlinear Anal. 45 (2001), no. 5, Ser. A: Theory Meth-ods, 597-611.
  18. Y. H. Lee, Multiplicity of positive radial solutions for multiparameter semilinear elliptic systems on an annulus, J. Differential Equations 174 (2001), no. 2, 420-441.
  19. A. Marino and C. Saccon, Nabla theorems and multiple solutions for some noncooper-ative elliptic systems, Sezione Di Annalisi Mathematica E Probabilita, Dipartimento di Mathematica, Universita di Pisa, 2000.
  20. A. Marino and C. Saccon, Some variational theorems of mixed type and elliptic problems with jumping nonlinearities, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 25 (1997), no. 3-4 631-665.
  21. A. M. Micheletti and A. Pistoia, Multiplicity results for a fourth-order semilinear elliptic problem, Nonlinear Anal. 31 (1998), no. 7, 895-908.
  22. P. H. Rabinowitz, Minimax Methods in Critical Point Theory with Applications to Differential Equations, C.B.M.S. Reg. Conf. Ser. in Math. 6, American Mathematical Society, Providence, R1, 1986.