Topological Methods in Nonlinear Analysis

A global bifurcation result for quasilinear elliptic equations in Orlicz-Sobolev spaces

Vy Khoi Le

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The paper is concerned with a global bifurcation result for the equation $$ -\text{div} (A(|\nabla u|) \nabla u) = g(x,u,\lambda) $$ in a general domain $\Omega$ with non necessarily radial solutions. Using a variational inequality formulation together with calculations of the Leray-Schauder degrees for mappings in Orlicz-Sobolev spaces, we show a global behavior (the Rabinowitz alternative) of the bifurcating branches.

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Topol. Methods Nonlinear Anal., Volume 15, Number 2 (2000), 301-327.

First available in Project Euclid: 22 August 2016

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Le, Vy Khoi. A global bifurcation result for quasilinear elliptic equations in Orlicz-Sobolev spaces. Topol. Methods Nonlinear Anal. 15 (2000), no. 2, 301--327.

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  • R. Adams, Sobolev Spaces, New York, Academic Press (1975) \ref
  • H. Attouch, Variational Convergence for Functions and Operators, London, Pitman (1984) \ref
  • J. Ball, J. Currie and P. Olver, Null Lagrangians, weak continuity, and variational problems of arbitrary order , J. Funct. Anal. (1981, 41 ), 135–174 \ref
  • A. Benkirane and A. Elmahi, Strongly nonlinear elliptic unilateral problems having natural growth terms and $L^1$ data , Rend. Mat. Appl. (1998, 18 ), 289–303 \ref ––––, An existence theorem for a strongly nonlinear elliptic problem in Orlicz spaces , Nonlinear Anal. (1999, 36 ), 11–24 \ref
  • F. E. Browder, Existence theorems for nonlinear partial differential equations (S. S. Chern and S. Smale, eds.), 16, Proc. Sympos. Pure Math. , 1–60, Amer. Math. Soc., Providence (1970) \ref
  • P. Clément, M. García-Huidobro, R. Manásevich and K. Schmitt, Solutions of Mountain Pass type of Quasilinear Elliptic Equations, Preprint \ref
  • M. Degiovanni, Bifurcation problems for nonlinear elliptic variational inequalities , Ann. Fac. Sci. Toulouse Math. (2) (1989, 10 ), 215–258 \ref
  • M. A. del Pino and R. F. Manásevich, Global bifurcation from the eigenvalues of the p-Laplacian , J. Differential Equations (1991, 92 ), 226–251 \ref
  • T. Donaldson, Nonlinear elliptic boundary value problems in Orlicz–Sobolev spaces , J. Differential Equations (1971, 10 ), 507–528 \ref
  • P. Drábek, Solvability and Bifurcations of Nonlinear Equations, Essex, Longman Scientific and Technical Series (1992) \ref
  • N. Fukagai, M. Ito and K. Narukawa, A bifurcation problem of some nonlinear degenerate elliptic equations , Adv. Differential Equations (1997, 2 ), 895–926 \ref
  • M. Garc\' ia-Huidobro, V. K. Le, R. Manásevich and K. Schmitt, On pricipal eigenvalues for quasilinear elliptic differential operators: An Orlicz–Sobolev space setting , Nonlinear Differential Equations Appl. (1999, 6 ), 207–225 \ref
  • M. Garc\' ia-Huidobro, R. Manásevich and K. Schmitt, Some bifurcation results for a class of $p$-Laplacian like operators , Differential Integral Equations (1997, 10 ), 51–66 \ref
  • J. Gossez, Nonlinear elliptic boundary value problems for equations with rapidly (or slowly) increasing coefficients , Trans. Amer. Math. Soc., (1974, 190 ), 163–205 \ref
  • J. P. Gossez and V. Mustonen, Variational inequalities in Orlicz–Sobolev spaces , Nonlinear Anal. (1987, 11 ), 379–392 \ref
  • E. Hewitt and K. Stromberg, Real and Abstract Analysis. A Modern Treatment of the Theory of Functions of a Real Variable, Springer (1965) \ref
  • D. Kinderlehrer and G. Stampacchia, An Introduction to Variational Inequalities and Their Applications, New York, Academic Press (1980) \ref
  • M. A. Krasnosels'kiĭ and J. Rutic'kiĭ, Convex Functions and Orlicz Spaces, Gröningen, Noorhoff (1961) \ref
  • A. Kufner, O. John and S. Fučic, Function Spaces, Noorhoff (1997, Leyden) \ref
  • M. Kučera, Bifurcation points of varational inequalities , Czech. Math. J. (1982, 32 ), 208–226 \ref
  • V. K. Le and K. Schmitt, Global Bifurcation in Variational Inequalities: Applications to Obstacle and Unilateral Problems, Appl. Math. Sci., 123 , New York, Springer (1997) \ref ––––, Quasilinear elliptic equations and inequalities with rapidly growing coefficients , J. London Math. Soc., to appear \ref
  • Vy Khoi Le, Global bifurcation in some degenerate quasilinear elliptic equations by a variational inequality approach , Nonlinear Anal., to appear \ref
  • J. L. Lions, Quelques Méthodes de Résolution des Problèmes aux Limites non Linéaires, Paris, Dunod (1969) \ref
  • K. Narukawa and T. Suzuki, Nonlinear Eigenvalue Problem for a Modified Capillary Surface Equation , Funkcial. Ekvac. (1994, 37 ), 81–100 \ref
  • F. Schuricht, Bifurcation from minimax solutions by variational inequalities , Math. Nachr. (1991, 154 ), 67–88 \ref
  • I. V. Skrypnik, Methods for Analysis of Nonlinear Elliptic Boundary Value Problems, American Mathematical Society (1994, 139 , Translations of Mathematical Monographs, Providence)