Journal of the Korean Mathematical Society (대한수학회지)

Korean Mathematical Society (대한수학회)
권호 표지
DOI QR코드

DOI QR Code

GLOBAL BIFURCATION FOR GENERALIZED LAPLACIAN OPERATORS

  • Kim, In-Sook (DEPARTMENT OF MATHEMATICS SUNGKYUNKWAN UNIVERSITY)
  • Published : 2009.01.31
Download PDF
⟨ Previous Next ⟩

Abstract

A bifurcation problem for nonlinear partial differential equations of the form $$div({\varphi}(|{\nabla}u|){\nabla}u+{\mu}_0{\varphi}(|u|)u=q({\lambda},\;x,\;u,\;{\nabla}u)$$ subject to Dirichlet boundary conditions is discussed. Using a global bifurcation theorem of Rabinowitz type, we show that under certain conditions on $\varphi$ and q, the above equation has an unbounded connected set of solutions (u, $\lambda$).

Keywords

References

  1. R. A. Adams, Sobolev Spaces, Academic Press, New York, 1975.
  2. J. Appell, E. Giorgieri, and M. Vath, Nonlinear spectral theory for homogeneous oper-ators, Nonlinear Funct. Anal. Appl. 7 (2002), no. 4, 589–618.
  3. M. A. Del Pino and R. F. Man´asevich, Global bifurcation from the eigenvalues of the p-Laplacian, J. Differential Equations 92 (1991), no. 2, 226–251.
  4. N. Fukagai, M. Ito, and K. Narukawa, A bifurcation problem of some nonlinear degen- erate elliptic equations, Adv. Differential Equations 2 (1997), no. 6, 895-926.
  5. M. Furi and M. P. Pera, On the existence of an unbounded connected set of solutions for nonlinear equations in Banach spaces, Atti Accad. Naz. Lincei Rend. Cl. Sci. Fis. Mat. Natur. (8) 67 (1979), no. 1-2, 31–38.
  6. V. K. Le, Global bifurcation in some degenerate quasilinear elliptic equations by a vari-ational inequality approach, Nonlinear Anal. 46 (2001), no. 4, 567–589. https://doi.org/10.1016/S0362-546X(00)00134-6
  7. V. K. Le and K. Schmitt, Global Bifurcation in Variational Inequalities, Applications to obstacle and unilateral problems. Applied Mathematical Sciences, 123. Springer-Verlag, New York, 1997.
  8. K. Narukawa, Global bifurcation for quasilinear elliptic equations, Nonlinear Anal. 30 (1997), no. 8, 5241–5249. https://doi.org/10.1016/S0362-546X(97)00067-9
  9. P. H. Rabinowitz, A note on a nonlinear elliptic equation, Indiana Univ. Math. J. 22 (1972/73), 43.49.
  10. K. Schmitt and I. Sim, Bifurcation problems associated with generalized Laplacians, Adv. Differential Equations 9 (2004), no. 7-8, 797–828.
  11. M. Vath, Global bifurcation of the p-Laplacian and related operators, J. Differential Equations 213 (2005), no. 2, 389–409. https://doi.org/10.1016/j.jde.2004.10.005
  12. E. Zeidler, Nonlinear Functional Analysis and its Applications. III, Variational methods and optimization. Translated from the German by Leo F. Boron. Springer-Verlag, New York, 1985.
  13. E. Zeidler, Nonlinear Functional Analysis and its Applications. II/B, Nonlinear monotone operators. Translated from the German by the author and Leo F. Boron. Springer-Verlag, New York, 1990.