Eigenvalue problems for the p-Laplacian

An alternative approach to the Faber–Krahn inequality for Robin problems

“…The corresponding eigenfunction ψ is simple and can be chosen such that ψ(x) > 0 for all x ∈ Ω and normalised such that ψ ∞ = 1. The proofs in [17] using the direct method in the calculus of variations are easily adapted to our situation (see [6,Section 2] or [16,Theorem 3.4]). Note that the very elegant and simple proof from [13,2] could be adapted.…”

Positive solutions for perturbations of the eigenvalue problem for the Robin p-Laplacian

“…We can view problem (P λ ) as a perturbation of the classical eigenvalue problem for the Robin p-Laplacian, investigated by Lê [12] and Papageorgiou and Rădulescu [15]. Similar studies concerning positive solutions, were conducted by Brezis and Oswald [5] (for problems driven by the Dirichlet Laplacian) and by Diaz and Saa [6] (for problems driven by the Dirichlet p-Laplacian).…”

“…This eigenvalue problem was studied by Lê [12] and Papageorgiou and Rădulescu [15]. We say that λ ∈ R is an eigenvalue of the negative Robin p-Laplacian (denoted by −∆ R p ), if problem (E λ ) admits a nontrivial solution u, known as an eigenfunction corresponding to the eigenvalue λ.…”

“…All the other eigenvalues have nodal (sign-changing) eigenfunctions. For more about the higher parts of the spectrum of −∆ R p , we refer to Lê [12] and Papageorgiou and Rădulescu [15].…”

On the spectrum of one dimensional p-Laplacian for an eigenvalue problem with Neumann boundary conditions

“…[1,6,15,13,7,8,3,10,5,11,2,12,16,4,9] and the references therein). In particular, we present here some results that motivated this work: First, we mention the result in [8], that we consider as a principal key for the development of our results.…”