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

Abstract

Keywords:Bifurcating family Isolated eigenvalue of finite multiplicity Gradient operator Homogeneous operatorWe 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.

Local structure of bifurcation curves for nonlinear Sturm–Liouville problems

“…Then λ is parameterized by R such as λ = λ n (R), and it is easy to see that λ n (R) → n 2 as R → 0. Then Chiappinelli [5] obtained the following result as an example of the main theorem in [5].…”

“…Our method turns out to be useful also for the study of higher order bifurcation curves (that is, curves of nontrivial solutions bifurcating from higher order eigenvalues of the linear problem [9,10]). Rather than developing a systematic analysis, we show this in a specific case on considering a problem suggested by Chiappinelli [5] recently, in which solutions are parameterized via their L 2 gradient norm. Precisely, let f (λ, u) = λ(u + |u| p−1 u) (p > 1) and consider the following problem −u (t) = λ u(t) + |u| p−1 u(t) , t ∈ J := (0, π), (1.10) u(0) = u(π ) = 0, (1.11) where λ > 0 is a parameter.…”

“…Precisely, let f (λ, u) = λ(u + |u| p−1 u) (p > 1) and consider the following problem −u (t) = λ u(t) + |u| p−1 u(t) , t ∈ J := (0, π), (1.10) u(0) = u(π ) = 0, (1.11) where λ > 0 is a parameter. We consider the interval [0, π] rather than [0, 1] just to make easier the comparison with the results of [5]. Recall that the eigenvalues of the linear problem are n 2 (n ∈ N).…”

“…In [5], it is mentioned that to discover further term in (1.14) seems to be interesting. We here answer his problem.…”

“…We do not use the classical bifurcation theory by [9]. So our approach is different from that used in [5].…”