1,720,979 research outputs found
Constrained critical points and eigenvalue approximation for semilinear elliptic operators
No full textLet Ω⊂R N (N>2) be a bounded open set with smooth boundary ∂Ω , and letv L_0 be a uniformly elliptic operator acting in the Sobolev space H 1 0 (Ω) . Denote by (μ 0 n) the sequence of eigenvalues of the problem L_0 u=μu in Ω, u=0 on ∂Ω . The purpose of this paper is to study the stability of an eigenvalue μ 0 n (n fixed), under addition to L_0 of a nonlinear term of the form m(x,s) , under various conditions on the function m:Ω×R→R . We establish bounds on the perturbed eigenvalue μ r (associated with eigenfunctions u with ∥u∥ L 2 (Ω) =r) , and discuss the connection with bifurcation from the trivial solutions
Remarks on Krasnoselʹskiĭ bifurcation theorem
No full textWe consider bifurcation from the trivial branch for nondifferentiable perturbations of linear operators with compact resolvent. Comparing the growth of the nonlinearity at 0 with the norm of the resolvent for values of μ close to an eigenvalue μ 0 of the operator with odd multiplicity, conditions are obtained ensuring that some neighborhood of μ 0 contains a bifurcation point
Isolated connected eigenvalues in nonlinear spectral theory
No full textThe existence of eigenvalues for nonlinear homogeneous operators is discussed, considering perturbations of linear operators having a simple isolated eigenvalue.
It is shown in particular that the nonlinear eigenvalues themselves are isolated. The proof is based on the Lyapounov-Schmidt reduction. The result is applied to
a class of semilinear elliptic operators in bounded domains of R^n
On the eigenvalue problem for some nonlinear perturbations of compact selfadjoint operators. PDF Clipboard Journal Article
No full textWe discuss the equation (1) F(u)≡Tu+N(u)=λu, where T is a compact selfadjoint linear operator, the nonlinear perturbation N:H→H is assumed to be a positively homogeneous and completely continuous gradient operator. We discuss the existence and location of eigenvalues of F , bifurcating from unperturbed eigenvalues of T. The critical points theory of smooth functionals is used
Eigenvalues of homogeneous gradient mappings in Hilbert space and the Birkhoff-Kellogg theorem
No full textAbstract. It is well known that any (nontrivial) linear compact self-adjoint operator acting in a Hilbert space possesses at least one non-zero eigenvalue. We present a generalization of this to nonlinear mappings as in the title, and discuss the relations of our results with the Birkhoff-Kellogg Theorem on one side, and with the spectral properties of self-adjoint operators on the other
On the number of critical points of a C1 function on the sphere
Get PDFFor a C1 function f:Rn →R (n≥2) , we consider the least number k of distinct critical points that f must possess when restricted to the sphere S={x∈Rn :∥x∥=1} . Clearly k≥2 (for f attains its absolute minimum and maximum on S), and a result of Lyusternik and Shnirelʹman establishes that k=n if f is even. Here we prove that k=n if, for a given orthonormal system (ei ), max S∩Vi
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