We review some more and less recent results concerning bounds on nonlinear eigenvalues (NLEV) for gradient operators. In particular, we discuss the asymptotic behaviour of NLEV (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 Lusternik-Schnirelmann theory of critical points on one side and on the Lyapounov-Schmidt reduction to the relevant finite-dimensional kernel on the other side. The results are applied to some semilinear elliptic operators in bounded domains of RN. A section reviewing some general facts about eigenvalues of linear and nonlinear operators is included.

Chiappinelli, R. (2012). Variational Methods for NLEV Approximation Near a Bifurcation Point. INTERNATIONAL JOURNAL OF MATHEMATICS AND MATHEMATICAL SCIENCES, 2012 [10.1155/2012/102489].

Variational Methods for NLEV Approximation Near a Bifurcation Point

CHIAPPINELLI, RAFFAELE
2012-01-01

Abstract

We review some more and less recent results concerning bounds on nonlinear eigenvalues (NLEV) for gradient operators. In particular, we discuss the asymptotic behaviour of NLEV (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 Lusternik-Schnirelmann theory of critical points on one side and on the Lyapounov-Schmidt reduction to the relevant finite-dimensional kernel on the other side. The results are applied to some semilinear elliptic operators in bounded domains of RN. A section reviewing some general facts about eigenvalues of linear and nonlinear operators is included.
2012
Chiappinelli, R. (2012). Variational Methods for NLEV Approximation Near a Bifurcation Point. INTERNATIONAL JOURNAL OF MATHEMATICS AND MATHEMATICAL SCIENCES, 2012 [10.1155/2012/102489].
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/11365/42478
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