Branches of solutions to semilinear elliptic equations on \mathbb{R}^N

For a large class of functions f, we consider the nonlinear elliptic eigenvalue problem We describe the behaviour of the branch of solutions emanating from an eigenvalue of odd multiplicity below the essential spectrum of the linearized problem. A sharper result is obtained in the case of the lowest eigenvalue. The discussion is based on the degree theory for proper Fredholm maps developed by P.M Fitzpatrick, J. Pejsachowicz and P.J. Rabier. Received November 13, 1996; in final form March 24, 1997