On the first eigensurface for the third order spectrum of p-biharmonic operator with weight

Abstract

In this work, we will study the simplicity and the isolation of the first eigensurface for the spectrum of the operator Δ 2 p u+2β.∇(|Δu| p−2 Δu)+ |β| 2 |Δu| p−2 Δu, where β ∈ IR N under Navier boundary conditions.

Strictly or Semitrivial Principal Eigensurface for p , q -Biharmonic Systems

“…and Γ p n ðβ, aÞ ⟶ ∞as n ⟶ ∞: The authors in [12] gave the first eigensurface Γ p 1 ð:,aÞ and showed that if a ≥ 0 a.e. in Ω, then Γ p 1 ð:,aÞ is simple (i. e., the associated eigenfunctions are a constant multiple of one another) and principal, i.e., the associated eigenfunction, denoted by φ p,a is positive or negative on Ω with…”

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

“…We may now assume the following conditions: Proof. An easy adaptation of the proof of Lemma 3.4 in [12]. □ Proposition 3.10.…”

“…Recently in the particular case of (1.1) where V ≡ 0 and m ≥ 0, the authors in [11], have showed that the spectrum of problem (1.1) contains at least one sequence of positive eigensurfaces. The authors in [12] have showed that the first eigensurface, in the same case, is simple and associated to positive eigenfunction. Also they showed that if m ∈ C( Ω) the first eigensurface is isolated and associated to every positive or negative eigenfunction.…”