Remarks on bifurcation for elliptic operators with odd nonlinearity

We consider the semilinear elliptic eigenvalue problem (1) −Δu+f(x,u)=μu in Ω , u| ∂Ω =0 , where Ω⊂R N is a bounded smooth domain, f:Ω×R→R a Carathéodory function and μ∈R . By assuming that f(x,u) is odd in u and satisfies suitable growth conditions, we show that, given any r>0 , (1) has a sequence μ n (r) of eigenvalues with eigenfunctions u n (r) satisfying ∫ Ω u 2 n =r 2 . We also obtain results on bifurcation and comparison of the eigenvalues μ n (r) with eigenvalues of some linear problem.