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MULTIPLE SOLUTIONS FOR A p-LAPLACIAN SYSTEM WITH NONLINEAR BOUNDARY CONDITIONS

  • Zhou, Jun (School of Mathematics and Statistics Southwest University) ;
  • Kim, Chan-Gyun (Department of Mathematics College of William and Mary)
  • Received : 2012.11.14
  • Published : 2014.01.31

Abstract

A nonlinear elliptic problem involving p-Laplacian and nonlinear boundary condition is considered in this paper. By using the method of Nehari manifold, it is proved that the system possesses two nontrivial nonnegative solutions if the parameter is small enough.

Keywords

References

  1. R. P. Agarwal, M. B. Ghaemi, and S. Saiedinezhad, The Nehari manifold for the degenerate p-Laplacian quasilinear elliptic equations, Adv. Math. Sci. Appl. 20 (2010), no. 1, 37-50.
  2. C. O. Alves and A. El Hamidi, Nehari manifold and existence of positive solutions to a class of quasilinear problems, Nonlinear Anal. 60 (2005), no. 4, 611-624. https://doi.org/10.1016/j.na.2004.09.039
  3. A. Anane, Simplicite et isolation de la premiere valeur propre du p-Laplacien avec poids, C. R. Acad. Sci. Paris Ser. I Math. 305 (1987), no. 16, 725-728.
  4. A. Anane and J.-P. Gossez, Strongly nonlinear elliptic problems near resonance: a variational approach, Comm. Partial Differential Equations 15 (1990), no. 8, 1141-1159. https://doi.org/10.1080/03605309908820717
  5. G. Barles, Remarks on uniqueness results of the first eigenvalue of the p-Laplacian, Ann. Fac. Sci. Toulouse Math. (5) 9 (1988), no. 1, 65-75. https://doi.org/10.5802/afst.649
  6. A. Bechah, K. Chaib, and F. de Thelin, Existence and uniqueness of positive solution for subhomogeneous elliptic problems in $R^N$, Rev. Mat. Apl. 21 (2000), no. 1-2, 1-17.
  7. M.-F. Bidaut-Veron and T. Raoux, Proprietes locales des solutions d'un systeme elliptique non lineaire, C. R. Acad. Sci. Paris Ser. I Math. 320 (1995), no. 1, 35-40.
  8. L. Boccardo and D. Guedes de Figueiredo, Some remarks on a system of quasilinear elliptic equations, Nonlinear Differential Equations Appl. 9 (2002), no. 3, 309-323. https://doi.org/10.1007/s00030-002-8130-0
  9. J. F. Bonder and J. D. Rossi, Existence results for the p-Laplacian with nonlinear boundary conditions, J. Math. Anal. Appl. 263 (2001), no. 1, 195-223. https://doi.org/10.1006/jmaa.2001.7609
  10. K. J. Brown, The Nehari manifold for a semilinear elliptic equation involving a sublinear term, Calc. Var. Partial Differential Equations 22 (2005), no. 4, 483-494.
  11. K. J. Brown and T.-F. Wu, A fibering map approach to a semilinear elliptic boundary value problem, Electron. J. Differential Equations 2007 (2007), no. 69, 1-9.
  12. K. J. Brown and T.-F. Wu, A semilinear elliptic system involving nonlinear boundary condition and sign-changing weight function, J. Math. Anal. Appl. 337 (2008), no. 2, 1326-1336. https://doi.org/10.1016/j.jmaa.2007.04.064
  13. J. Chabrowski, On multiple solutions for nonhomogeneous system of elliptic equations, Rev. Mat. Univ. Complut. Madrid 9 (1996), no. 1, 207-234.
  14. C.-Y. Chen, Y.-C. Kuo, and T.-F.Wu, The Nehari manifold for a Kirchhoff type problem involving sign-changing weight functions, J. Differential Equations 250 (2011), no. 4, 1876-1908. https://doi.org/10.1016/j.jde.2010.11.017
  15. C.-Y. Chen and T.-F.Wu, The Nehari manifold for indefinite semilinear elliptic systems involving critical exponent, Appl. Math. Comput. 218 (2012), no. 22, 10817-10828. https://doi.org/10.1016/j.amc.2012.04.026
  16. M. Chipot, M. Chlebik, M. Fila, and I. Shafrir, Existence of positive solutions of a semilinear elliptic equation in ${\mathbb{R}}^n_+$ with a nonlinear boundary condition, J. Math. Anal. Appl. 223 (1998), no. 2, 429-471. https://doi.org/10.1006/jmaa.1998.5958
  17. M. Chipot, I. Shafrir, and M. Fila, On the solutions to some elliptic equations with nonlinear Neumann boundary conditions, Adv. Differential Equations 1 (1996), no. 1, 91-110.
  18. P. Clement, J. Fleckinger, E. Mitidieri, and F. de Thelin, Existence of positive solutions for a nonvariational quasilinear elliptic system, J. Differential Equations 166 (2000), no. 2, 455-477. https://doi.org/10.1006/jdeq.2000.3805
  19. F. de Thelin, Quelques resultats d'existence et de non-existence pour une EDP elliptique non lineaire, C. R. Acad. Sci. Paris Ser. I Math. 299 (1984), no. 18, 911-914.
  20. M. A. del Pino and R. F. Manasevich, Global bifurcation from the eigenvalues of the p-Laplacian, J. Differential Equations 92 (1991), no. 2, 226-251. https://doi.org/10.1016/0022-0396(91)90048-E
  21. J. I. Diaz, Nonlinear Partial Differential Equations and Free Boundaries. Vol. I, Volume 106 of Research Notes in Mathematics, Pitman (Advanced Publishing Program), Boston, MA, 1985.
  22. E. DiBenedetto, Degenerate Parabolic Equations, Springer-Verlag, New York, 1993.
  23. P. Drabek, Nonlinear eigenvalue problem for p-Laplacian in $R^N$, Math. Nachr. 173 (1995), 131-139. https://doi.org/10.1002/mana.19951730109
  24. P. Drabek and Y. X. Huang, Bifurcation problems for the p-Laplacian in $R^N$, Trans. Amer. Math. Soc. 349 (1997), no. 1, 171-188. https://doi.org/10.1090/S0002-9947-97-01788-1
  25. P. Drabek and S. I. Pohozaev, Positive solutions for the p-Laplacian: application of the fibering method, Proc. Roy. Soc. Edinburgh Sect. A 127 (1997), no. 4, 703-726. https://doi.org/10.1017/S0308210500023787
  26. J. F. Escobar, Uniqueness theorems on conformal deformation of metrics, Sobolev inequalities, and an eigenvalue estimate, Comm. Pure Appl. Math. 43 (1990), no. 7, 857-883. https://doi.org/10.1002/cpa.3160430703
  27. P. Felmer, R. F. Manasevich, and F. de Thelin, Existence and uniqueness of positive solutions for certain quasilinear elliptic systems, Comm. Partial Differential Equations 17 (1992), no. 11-12, 2013-2029.
  28. J. Garcia-Azorero, I. Peral, and J. D. Rossi, A convex-concave problem with a nonlinear boundary condition, J. Differential Equations 198 (2004), no. 1, 91-128. https://doi.org/10.1016/S0022-0396(03)00068-8
  29. Y. Li, Asymptotic behavior of positive solutions of equation ${\Delta}u$ + $K(x)u^p$ = 0 in $R^n$, J. Differential Equations 95 (1992), no. 2, 304-330. https://doi.org/10.1016/0022-0396(92)90034-K
  30. P. Lindqvist, On the equation div $(\left|{\nabla}u\right|^{p-2}{\nabla}u)$ + ${\lambda}\left|u\right|^{p-2}u$ = 0, Proc. Amer. Math. Soc. 109 (1990), no. 1, 157-164.
  31. Z. Nehari, On a class of nonlinear second-order differential equations, Trans. Amer. Math. Soc. 95 (1960), 101-123. https://doi.org/10.1090/S0002-9947-1960-0111898-8
  32. Z. Nehari, Characteristic values associated with a class of non-linear second-order differential equations, Acta Math. 105 (1961), 141-175. https://doi.org/10.1007/BF02559588
  33. W.-M. Ni and J. Serrin, Existence and non-existence theorems for ground states of quasilinear partial differential equations. The anomalous case, Rome, Acc. Naz. dei Lincei, Atti dei Convegni 77 (1986), 231-257.
  34. M. Otani, On certain second order ordinary differential equations associated with Sobolev-Poincare-type inequalities, Nonlinear Anal. 8 (1984), no. 11, 1255-1270. https://doi.org/10.1016/0362-546X(84)90014-2
  35. M. Otani, Existence and nonexistence of nontrivial solutions of some nonlinear degenerate elliptic equations, J. Funct. Anal. 76 (1988), no. 1 140-159. https://doi.org/10.1016/0022-1236(88)90053-5
  36. D. Pierotti and S. Terracini, On a Neumann problem with critical exponent and critical nonlinearity on the boundary, Comm. Partial Differential Equations 20 (1995), no. 7-8, 1155-11875. https://doi.org/10.1080/03605309508821128
  37. 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, Nonlinear Anal. 73 (2010), no. 10, 3390-3401. https://doi.org/10.1016/j.na.2010.07.021
  38. G. Tarantello, On nonhomogeneous elliptic equations involving critical Sobolev exponent, Ann. Inst. H. Poincare Anal. Non Lineaire 9 (1992), no. 3, 281-304. https://doi.org/10.1016/S0294-1449(16)30238-4
  39. P. Tolksdorf, Regularity for a more general class of quasilinear elliptic equations, J. Differential Equations 51 (1984), no. 1, 126-150. https://doi.org/10.1016/0022-0396(84)90105-0
  40. T.-F. Wu, On semilinear elliptic equations involving concave-convex nonlinearities and sign-changing weight function, J. Math. Anal. Appl. 318 (2006), no. 1, 253-270.
  41. T.-F. Wu, Multiple positive solutions for semilinear elliptic systems with nonlinear boundary condition, Appl. Math. Comput. 189 (2007), no. 2, 1712-1722. https://doi.org/10.1016/j.amc.2006.12.052
  42. Z. Wu, J. Zhao, J. Yin, and H. Li, Nonlinear Diffusion Equations, World Scientific Publishing Co. Inc., River Edge, NJ, 2001.