Branches of solutions to semilinear elliptic equations on IRN

For a large class of functions f, we consider the nonlinear elliptic eigenv alue problem [GRAPHICS] We describe the behaviour of the branch of solutions emanating from an eige nvalue of odd multiplicity below the essential spectrum of the linearized p roblem, A sharper result is obtained in the case of the lowest eigenvalue, The discussion is based on the degree theory for C-2 proper Fredholm maps d eveloped by P.M Fitzpatrick, J. Pejsachowicz and P.J. Rabier.