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Behaviour of the first eigenvalue of the p-Laplacian in a domain with a hole

M. Sango (2001)

Colloquium Mathematicae

We investigate the behaviour of a sequence λ s , s = 1,2,..., of eigenvalues of the Dirichlet problem for the p-Laplacian in the domains Ω s , s = 1,2,..., obtained by removing from a given domain Ω a set E s whose diameter vanishes when s → ∞. We estimate the deviation of λ s from the eigenvalue of the limit problem. For the derivation of our results we construct an appropriate asymptotic expansion for the sequence of solutions of the original eigenvalue problem.

Bifurcation for nonlinear elliptic boundary value problems I.

Kazuaki Taira (1996)

Collectanea Mathematica

This paper is devoted to local static bifurcation theory for a class of degenerate boundary value problems for nonlinear second-order elliptic differential operators. The purpose of this paper is twofold. The first purpose is to prove that the first eigenvalue of the linearized boundary value problem is simple and its associated eigenfunction is positive. The second purpose is to discuss the changes that occur in the structure of the solutions as a parameter varies near the first eigenvalue of the...

Bifurcation theorems of Rabinowitz type for certain differential operators of the fourth order

Jolanta Przybycin (1992)

Annales Polonici Mathematici

This paper was inspired by the works of P. H. Rabinowitz. We study nonlinear eigenvalue problems for some fourth order elliptic partial differential equations with nonlinear perturbation of Rabinowitz type. We show the existence of an unbounded continuum of nontrivial positive solutions bifurcating from (μ₁,0), where μ₁ is the first eigenvalue of the linearization about 0 of the considered problem. We also prove the related theorem for bifurcation from infinity. The results obtained are similar...

Bifurcations for a problem with jumping nonlinearities

Lucie Kárná, Milan Kučera (2002)

Mathematica Bohemica

A bifurcation problem for the equation Δ u + λ u - α u + + β u - + g ( λ , u ) = 0 in a bounded domain in N with mixed boundary conditions, given nonnegative functions α , β ∈ L ∞ and a small perturbation g is considered. The existence of a global bifurcation between two given simple eigenvalues λ ( 1 ) , λ ( 2 ) of the Laplacian is proved under some assumptions about the supports of the functions α , β . These assumptions are given by the character of the eigenfunctions of the Laplacian corresponding to λ ( 1 ) , λ ( 2 ) .

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