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 ofℝN. A section reviewing some general facts about eigenvalues of linear and nonlinear operators is included.
Highlights
Introduction and ExamplesThe term “nonlinear eigenvalue” NLEV is a frequent shorthand for “eigenvalue of a nonlinear problem,” see, for instance 1–3
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
Summary
The term “nonlinear eigenvalue” NLEV is a frequent shorthand for “eigenvalue of a nonlinear problem,” see, for instance 1–3. The remaining parts of this paper are organised as follows We complete this introductory section presenting as a matter of example some boundary-value problems for nonlinear differential equations, depending on a real parameter λ and admitting the zero solution for all values of λ, that can be cast in the form 1.1 with an appropriate choice of the function space E and of the operators A, B. A fundamental exception is that of the first eigenvalue of a homogeneous operator see Theorem 2.4 and Remark 2.6 in Section 2 which possesses— under additional assumptions on the operator itself—remarkable properties such as the positivity of the associated eigenfunctions, see 35 These properties have been extensively used in 36, 37 , e.g. in order to prove global bifurcation results for 1.24 from the first eigenvalue of the p-Laplacian. This does not change the essence of our remark, nor the results for 1.24 would be much different from those in 12
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