1,720,980 research outputs found
Locating the point spectrum of some semilinear elliptic operators
No full textSemilinear elliptic eigenvalue problems formulated in terms of a uniformly elliptic formally selfadjoint operator were studied. Constrained critical point theory was used to construct a family of eigenvalues with each of the eigenvalues carrying an eigenfunction. The eigenvalue problem was used to describe the vibrations of a nonlinearly distorted membrane having an additional noise term
A-priori bounds and asymptotics on the eigenvalues in bifurcation problems for perturbed self-adjoint operators
No full textWe 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 RN and in particular to Sturm–Liouville operators
On spectral asymptotics and bifurcation for elliptic operators with odd superlinear term
No full textSurjectivity of coercive gradient operators in Hilbert space and nonlinear spectral theory
No full textWe consider continuous gradient operators F acting in a real Hilbert space H, and we study their surjectivity under the basic assumption that the corresponding functional 〈F(x), x〉-where 〈· 〉 is the scalar product in H-is coercive. While this condition is sufficient in the case of a linear operator (where one in fact deals with a bounded self-adjoint operator), in the general case we supplement it with a compactness condition involving the number ω(F) introduced by Furi, Martelli, and Vignoli, whose positivity indeed guarantees that F is proper on closed bounded sets of H.We then use Ekeland's variational principle to reach the desired conclusion. In the second part of this article, we apply the surjectivity result to give a perspective on the spectrum of these kinds of operators-ones not considered by Feng or the above authors-when they are further assumed to be sublinear and positively homogeneous
An estimate on the eigenvalues in bifurcation for gradient mappings
No full textThis paper provides supplementary information on the Krasnoselʹskiĭ bifurcation theorem for gradient mappings ∇φ(u)=λu in a real Hilbert space. Strengthening the differentiability assumption, we prove a better estimate of the form λ r =λ 0 +O(r p−1 ) for the bifurcating solutions (u r ,λ r ) on the sphere S r . Applications are given to nonlinear elliptic eigenvalue problems of the form −Δu=μ(u+f(x,u)), x∈Ω, u=0, x∈∂Ω
Remarks on bifurcation for elliptic operators with odd nonlinearity
No full textWe 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
Upper and lower bounds for higher order eigenvalues of some semilinear elliptic equations
No full textWe prove upper and lower bounds on the eigenvalues (as the H(0)(1)(Omega) norm of the eigenfunction tends to zero) in bifurcation problems for a class of semilinear elliptic equations in bounded domains of R(N). It is shown that these bounds are computable in terms of the eigenvalues of the associated linear equation. (C) 2010 Elsevier Inc. All rights reserved
Variational Methods for NLEV Approximation Near a Bifurcation Point
No full textWe 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 . A section reviewing some general facts about eigenvalues of linear and nonlinear operators is included
Nonlinear rayleigh quotients and nonlinear spectral theory
Full text linkWe give a new and simplified definition of spectrum for a nonlinear operator F acting in a real Banach space X, and study some of its features in terms of (qualitative and) quantitative properties of F such as the measure of noncompactness, α(F), of F. Then, using as a main tool the Ekeland Variational Principle, we focus our attention on the spectral properties of F when F is a gradient operator in a real Hilbert space, and in particular on the role played by its Rayleigh quotient R(F) and by the best lower and upper bounds, m(F) and M(F), of R(F)
Nonlinear homogeneous perturbation of the discrete spectrum of a self-adjoint operator and a new Constrained Saddle Point Theorem
No full textWe prove a Critical Point Theorem for C1 functionals on the unit sphere of a separable Hilbert space H which improves a previous result of ours. This is applied in nonlinear eigenvalue theory
to study the effect of suitably restricted homogeneous perturbations upon the discrete spectrum of a
bounded self-adjoint operator in H
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