Mathematische Zeitschrift

, Volume 237, Issue 1, pp 85–124 | Cite as

Global bifurcation for quasilinear elliptic equations on $\mathbb{R}^{N}$

  • P.J. Rabier
  • C.A. Stuart
Original articles


In this paper we discuss the global behaviour of some connected sets of solutions \((\lambda,u)\) of a broad class of second order quasilinear elliptic equations


for \(x\in\mathbb{R}^{N}\) where \(\lambda\) is a real parameter and the function u is required to satisfy the condition


The basic tool is the degree for proper Fredholm maps of index zero in the form due to Fitzpatrick, Pejsachowicz and Rabier. To use this degree the problem must be expressed in the form \(F:J\times X\rightarrow Y\) where J is an interval, X and Y are Banach spaces and F is a \(C^{1}\) map which is Fredholm and proper on closed bounded subsets. We use the usual spaces \(X=W^{2,p}(\mathbb{R}^{N})\) and \(Y=L^{p}(\mathbb{R}^{N})\). Then the main difficulty involves finding general conditions on \(a_{\alpha\beta}\) and b which ensure the properness of F. Our approach to this is based on some recent work where, under the assumption that \(a_{\alpha\beta} \) and b are asymptotically periodic in x as $\left| x\right| \rightarrow\infty$, we have obtained simple conditions which are necessary and sufficient for \(F(\lambda,\cdot):X\rightarrow Y\) to be Fredholm and proper on closed bounded subsets of X. In particular, the nonexistence of nonzero solutions in X of the asymptotic problem plays a crucial role in this issue. Our results establish the bifurcation of global branches of solutions for the general problem. Various special cases are also discussed. Even for semilinear equations of the form


our results cover situations outside the scope of other methods in the literature.


Banach Space Broad Class Real Parameter Main Difficulty Simple Condition 
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Copyright information

© Springer-Verlag Berlin Heidelberg 2001

Authors and Affiliations

  • P.J. Rabier
    • 1
  • C.A. Stuart
    • 2
  1. 1.Department of Mathematics, University of Pittsburgh, Pittsburgh, PA 15260, USA (e-mail: US
  2. 2.Département de Mathématiques, Ecole Polytechnique Fédérale de Lausanne, CH-1015 Lausanne, Switzerland (e-mail: CH

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