Isolation and simplicity for the first eigenvalue of the p-Laplacian with a nonlinear boundary condition (original) (raw)

Two-Parameter Eigenvalues Steklov Problem involving the p-Laplacian

2013

We study the existence of eigenvalues for a two parameter Steklov eigenvalues problem with weights. Moreover, we prove the simplicity and the isolation results of the principal eigenvalue. Finally, we obtain the continuity and the differentiability of this principal eigenvalue.

Existence Results for the p-Laplacian with Nonlinear Boundary Conditions

Journal of Mathematical Analysis and Applications, 2001

In this paper we study the existence of nontrivial solutions for the problem, ∆pu = |u| p−2 u in a bounded smooth domain Ω ⊂ R N , with a nonlinear boundary condition given by |∇u| p−2 ∂u/∂ν = f (u) on the boundary of the domain. The proofs are based on variational and topological arguments.

Multiple solutions for p-Laplacian eigenproblem with nonlinear boundary conditions

Boletim da Sociedade Paranaense de Matemática, 2016

In this paper we study the existence of at least two nontrivial solutions for the nonlinear problem p-Laplacian, with nonlinear boundary conditions. We establish that there exist at least two solutions, which are opposite signs. For this reason, we characterize the first eigenvalue of an intermediary eigenvalue problem by the minimization method. In fact, in some sense, we establish the non-resonance below the first eigenvalues of nonlinear Steklov-Robin.

On the first eigenvalue of the Steklov eigenvalue problem for the infinity Laplacian

Electronic Journal of Differential Equations, 2006

Let Λ p p be the best Sobolev embedding constant of W 1,p (Ω) → L p (∂Ω), where Ω is a smooth bounded domain in R N. We prove that as p → ∞ the sequence Λp converges to a constant independent of the shape and the volume of Ω, namely 1. Moreover, for any sequence of eigenfunctions up (associated with Λp), normalized by up L ∞ (∂Ω) = 1, there is a subsequence converging to a limit function u∞ which satisfies, in the viscosity sense, an ∞-Laplacian equation with a boundary condition.

A weighted eigenvalue problem for the p-Laplacian plus a potential

Nonlinear Differential Equations and Applications NoDEA, 2009

Let ∆p denote the p-Laplacian operator and Ω be a bounded domain in R N. We consider the eigenvalue problem −∆pu + V (x)|u| p−2 u = λm(x)|u| p−2 u, u ∈ W 1,p 0 (Ω) for a potential V and a weight function m that may change sign and be unbounded. Therefore the functional to be minimized is indefinite and may be unbounded from below. The main feature here is the introduction of a value α(V, m) that guarantees the boundedness of the energy over the weighted sphere M = {u ∈ W 1,p 0 (Ω); Ω m|u| p dx = 1}. We show that the above equation has a principal eigenvalue if and only if either m ≥ 0 and α(V, m) > 0 or m changes sign and α(V, m) ≥ 0. The existence of further eigenvalues is also treated here, mainly a second eigenvalue (to the right) and their dependence with respect to V and m.

Eigenvalue problems for the p-Laplacian with indefinite weights

Electronic Journal of Differential Equations, 2001

We consider the eigenvalue problem −∆pu = λV (x)|u| p−2 u, u ∈ W 1,p 0 (Ω) where p > 1, ∆p is the p-Laplacian operator, λ > 0, Ω is a bounded domain in R N and V is a given function in L s (Ω) (s depending on p and N). The weight function V may change sign and has nontrivial positive part. We prove that the least positive eigenvalue is simple, isolated in the spectrum and it is the unique eigenvalue associated to a nonnegative eigenfunction. Furthermore, we prove the strict monotonicity of the least positive eigenvalue with respect to the domain and the weight.

Steklov problems for the p−Laplace operator involving Lq -norm

Moroccan Journal of pure and applied analysis, 2022

In this paper, we are concerned with the study of the spectrum for the nonlinear Steklov problem of the form ∆ p u = |u| p−2 u in Ω, |∇u| p−2 ∂u ∂ν = λ u p−q q,∂Ω |u| q−2 u on ∂Ω, where Ω is a smooth bounded domain in R N (N ≥ 1), λ is a real number which plays the role of eigenvalue and the unknowns u ∈ W 1,p (Ω). Using the Ljusterneck-Shnirelmann theory on C 1 manifold and Sobolev trace embedding we prove the existence of an increasing sequence positive of eigenvalues (λ k) k≥1 , for the above problem. We then establish that the first eigenvalue is simple and isolated.

On a Dirichlet problem involving p-Laplacian

International Mathematical Forum, 2007

In this paper, the existence of at least three weak solutions for Dirichlet problem Δ p u + λf (x, u) = 0 in Ω, u = 0 on ∂Ω, where Δ p u =div(|∇u| p−2 ∇u) is the p-Laplacian operator, Ω ⊂ R N (N ≥ 1) is non-empty bounded open set with smooth boundary ∂Ω , p > N, λ > 0 and f : Ω × R → R is a L 1-Caratheodory function, is established. The approach is based on variational methods and critical points.

EIGENVALUE PROBLEMS WITH p-LAPLACIAN OPERATORS

2014

In this article, we study eigenvalue problems with the p-Laplacian operator: −(|y′|p−2y′)′ = (p− 1)(λρ(x)− q(x))|y|p−2y on (0, πp), where p > 1 and πp ≡ 2π/(p sin(π/p)). We show that if ρ ≡ 1 and q is singlewell with transition point a = πp/2, then the second Neumann eigenvalue is greater than or equal to the first Dirichlet eigenvalue; the equality holds if and only if q is constant. The same result also holds for p-Laplacian problem with single-barrier ρ and q ≡ 0. Applying these results, we extend and improve a result by [24] by using finitely many eigenvalues and by generalizing the string equation to p-Laplacian problem. Moreover, our results also extend a result of Huang [14] on the estimate of the first instability interval for Hill equation to single-well function q.