Displaying similar documents to “On the uniqueness and simplicity of the principal eigenvalue”

Eigenvalue problems with indefinite weight

Andrzej Szulkin, Michel Willem (1999)

Studia Mathematica

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We consider the linear eigenvalue problem -Δu = λV(x)u, u ∈ D 0 1 , 2 ( Ω ) , and its nonlinear generalization - Δ p u = λ V ( x ) | u | p - 2 u , u ∈ D 0 1 , p ( Ω ) . The set Ω need not be bounded, in particular, Ω = ℝ N is admitted. The weight function V may change sign and may have singular points. We show that there exists a sequence of eigenvalues λ n → ∞ .

On the existence of five nontrivial solutions for resonant problems with p-Laplacian

Leszek Gasiński, Nikolaos S. Papageorgiou (2010)

Discussiones Mathematicae, Differential Inclusions, Control and Optimization

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In this paper we study a nonlinear Dirichlet elliptic differential equation driven by the p-Laplacian and with a nonsmooth potential. The hypotheses on the nonsmooth potential allow resonance with respect to the principal eigenvalue λ₁ > 0 of ( - Δ ₚ , W ₀ 1 , p ( Z ) ) . We prove the existence of five nontrivial smooth solutions, two positive, two negative and the fifth nodal.

On the nodal set of the second eigenfunction of the laplacian in symmetric domains in R N

Lucio Damascelli (2000)

Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni

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We present a simple proof of the fact that if Ω is a bounded domain in R N , N ≥ 2 , which is convex and symmetric with respect to k orthogonal directions, 1 ≤ k ≤ N , then the nodal sets of the eigenfunctions of the laplacian corresponding to the eigenvalues λ 2 , ⋯ , λ k + 1 must intersect the boundary. This result was proved by Payne in the case N = 2 for the second eigenfunction, and by other authors in the case of convex domains in the plane, again for the second eigenfunction.

Eigenvalues of the p -Laplacian in 𝐑 N with indefinite weight

Yin Xi Huang (1995)

Commentationes Mathematicae Universitatis Carolinae

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We consider the nonlinear eigenvalue problem - div ( | ∇ u | p - 2 ∇ u ) = λ g ( x ) | u | p - 2 u in 𝐑 N with p > 1 . A condition on indefinite weight function g is given so that the problem has a sequence of eigenvalues tending to infinity with decaying eigenfunctions in W 1 , p ( 𝐑 N ) . A nonexistence result is also given for the case p ≥ N .

Symmetry breaking in the minimization of the first eigenvalue for the composite clamped punctured disk

Claudia Anedda, Fabrizio Cuccu (2015)

Applicationes Mathematicae

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Let D₀=x∈ ℝ²: 0<|x|<1 be the unit punctured disk. We consider the first eigenvalue λ₁(ρ ) of the problem Δ² u =λ ρ u in D₀ with Dirichlet boundary condition, where ρ is an arbitrary function that takes only two given values 0 < α < β and is subject to the constraint ∫ D ₀ ρ d x = α γ + β ( | D ₀ | - γ ) for a fixed 0 < γ < |D₀|. We will be concerned with the minimization problem ρ ↦ λ₁(ρ). We show that, under suitable conditions on α, β and γ, the minimizer does not inherit the radial symmetry of the...