A weighted eigencurve for Steklov problems with a potential

Abstract

In this work, we study the existence of the principal eigencurves for a Steklov problem with an indefinite weight for homogeneous perturbation of the p-Laplacian operator. We then establish many properties of these eigencurves: continuity, differentiability and asymptotic behaviors. We also use our approach to get similar result when mixed Dirichlet-Steklov boundary condition is considered.

On eigenvalues of p-biharmonic operator and associated concave-convex type equation

“…It is well established that (see for example [5], [6], [8], [17]), in order to prove the existence of principal eigensurfaces of (1.1), one fixes λ ∈ R and embeds the problem into the new eigenvalue problem of parameter µ:…”

Principal Eigenvalue for Cooperative (p,q)-biharmonic Systems

“…It is well established that (see, e.g., [7][8][9][10][11]), in order to prove the existence of strictly principal eigenvalue or semitrivial principal eigenvalue of (Q), one fixes λ and embeds the problem into the new eigenvalue problem of parameter µ ∈ R:…”

Weighted Steklov problem under nonresonance conditions

“…Furthermore, if β(V, m) < 0 then λ ±1 (V, m) = −∞ (see [14]). It can be therefore seen that problem (P V,m,n ) has a nontrivial and one-signed solutions under suitable assumptions (see details in [10]) if and only if λ = λ 1 (V, m) or λ = λ 1 (V, n).…”

“…Since the boundary weights lie in C α (∂Ω), every solution of (2.4) belongs to C 1,α (Ω), for 0 < α < 1 (see [12,14]). We note that if an eigenfunction u is positive in Ω, it is shown that u remains positive on ∂Ω (see the first part of the proof of Theorem 3.1 in [14]). Furthermore, one can state using Proposition 5.8 in [15] that if u changes sign in Ω then it is also a sign-changing function on ∂Ω.…”

“…and easily check that B 1,1 (γ) = 1 refering to the problem tackled in [14] with constant weight 1 on the boundary. The functionγ is therefore admissible for λ 1 (V, 1) and plugging a right ε > 0 in (3.32), we obtain again …”