کد مقاله | کد نشریه | سال انتشار | مقاله انگلیسی | نسخه تمام متن |
---|---|---|---|---|
839309 | 1470464 | 2016 | 21 صفحه PDF | دانلود رایگان |
![عکس صفحه اول مقاله: The first nontrivial eigenvalue for a system of pp-Laplacians with Neumann and Dirichlet boundary conditions The first nontrivial eigenvalue for a system of pp-Laplacians with Neumann and Dirichlet boundary conditions](/preview/png/839309.png)
We deal with the first eigenvalue for a system of two pp-Laplacians with Dirichlet and Neumann boundary conditions. If Δpw=div(|∇w|p−2∇w)Δpw=div(|∇w|p−2∇w) stands for the pp-Laplacian and αp+βq=1, we consider {−Δpu=λα|u|α−2u|v|β in Ω,−Δqv=λβ|u|α|v|β−2v in Ω, with mixed boundary conditions u=0,|∇v|q−2∂v∂ν=0,on ∂Ω. We show that there is a first non trivial eigenvalue that can be characterized by the variational minimization problem λp,qα,β=min{∫Ω|∇u|ppdx+∫Ω|∇v|qqdx∫Ω|u|α|v|βdx:(u,v)∈Ap,qα,β}, where Ap,qα,β={(u,v)∈W01,p(Ω)×W1,q(Ω):uv≢0 and ∫Ω|u|α|v|β−2vdx=0}. We also study the limit of λp,qα,β as p,q→∞p,q→∞ assuming that αp→Γ∈(0,1), and qp→Q∈(0,∞) as p,q→∞p,q→∞. We find that this limit problem interpolates between the pure Dirichlet and Neumann cases for a single equation when we take Q=1Q=1 and the limits Γ→1Γ→1 and Γ→0Γ→0.
Journal: Nonlinear Analysis: Theory, Methods & Applications - Volume 137, May 2016, Pages 381–401