A-priori bounds and asymptotics on the eigenvalues in bifurcation problems for perturbed self-adjoint operators

We prove upper and lower bounds on the eigenvalues and discuss their asymptotic behaviour (as the norm of the eigenvector tends to zero) in bifurcation problems from the line of trivial solutions, considering perturbations of linear self-adjoint operators in a Hilbert space. The proofs are based on the Lyapounov–Schmidt reduction. The results are applied to a class of semilinear elliptic operators in bounded domains of R N and in particular to Sturm–Liouville operators.