The first nontrivial eigenvalue for a system of -Laplacians with Neumann and Dirichlet boundary conditions

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We deal with the first eigenvalue for a system of two 35b9487e228e" title="Click to view the MathML source">p

p

-Laplacians with Dirichlet and Neumann boundary conditions. If Δ_pw=div(|∇w|^p−2∇w)

Δ_{p} w = div ({| \nabla w |}^{p - 2} \nabla w)

stands for the 35b9487e228e" title="Click to view the MathML source">p

p

-Laplacian and

\frac{α}{p} + \frac{β}{q} = 1

, we consider

{\begin{cases} - Δ_{p} u = λ α {| u |}^{α - 2} u {| v |}^{β} & in Ω, \\ - Δ_{q} v = λ β {| u |}^{α} {| v |}^{β - 2} v & in Ω, \end{cases}

with mixed boundary conditions

u = 0, {| \nabla v |}^{q - 2} \frac{\partial v}{\partial ν} = 0, on \partial Ω .

We show that there is a first non trivial eigenvalue that can be characterized by the variational minimization problem

λ_{p, q}^{α, β} = min {\frac{\int_{Ω} \frac{{| \nabla u |}^{p}}{p} d x + \int_{Ω} \frac{{| \nabla v |}^{q}}{q} d x}{\int_{Ω} {| u |}^{α} {| v |}^{β} d x} : (u, v) \in A_{p, q}^{α, β}},

where

A_{p, q}^{α, β} = {(u, v) \in W_{0}^{1, p} (Ω) \times W^{1, q} (Ω) : u v ≢ 0 and \int_{Ω} {| u |}^{α} {| v |}^{β - 2} v d x = 0} .

We also study the limit of 359c6ac">

λ_{p, q}^{α, β}

as p,q→∞

p, q \to \infty

assuming that

\frac{α}{p} \to Γ \in (0, 1)

, and

\frac{q}{p} \to Q \in (0, \infty)

as p,q→∞

p, q \to \infty

. We find that this limit problem interpolates between the pure Dirichlet and Neumann cases for a single equation when we take Q=1

Q = 1

and the limits Γ→1

Γ \to 1

and Γ→0

Γ \to 0

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