An asymmetric Steklov problem with weights

An Asymmetric Steklov Problem With Weights: the singular case

“…To state our main result let us introduce the following family of paths in M m,n Γ = {γ ∈ C([0, 1], M m,n ) : γ(0) ≤ 0 and γ(1) ≥ 0}, which is nonempty (see [1]), and the finite minimax value…”

“…The situation here is again simpler in the non singular case (see Preliminaries). We start by giving an important proposition, which is proved in [1]. Proposition 3.2 If ∂Ω mdσ = 0, then ϕ m ∈ M m,n is a strict local minimum of f , with in addition for some ǫ 0 > 0 and all 0 < ǫ < ǫ 0 ,…”

“…Since ∂Ω mdσ = 0 or ∂Ω ndσ = 0, we know (see Proposition 3.1) thatf satisfies (P SC) c for all c > 0 and the mountain pass theorem (see Proposition 2.3) yields the conclusion. To show that there is no eigenvalue between max{λ * (m), λ * (n)} and c(m, n), we follow the same proof from the nonsingular case (see [1] …”

“…In a previous work (see [1]), we proved the existence of a first nonprincipal eigenvalue for (1) are ∂Ω mdσ = 0 and ∂Ω ndσ = 0 by applying a version of the mountain pass theorem to the functional f (u) = in this case (P S) condition is satisfied and the geometry of the mountain pass was derived from observation that ϕ m and ϕ n where strict local minima (ϕ m denotes the positive first eigenvalue of (1) with m = n). Our purpose in this work is to prove the existence of a first nonprincipal eigenvalue for (1) where ∂Ω mdσ = 0 or ∂Ω ndσ = 0.…”

“…The solutions of (1) or of related equations are always understood in the weak sense, i.e., u ∈ W 1,p (Ω) with…”