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

Abstract

We deal with the first eigenvalue for a system of two $p-$Laplacians with Dirichlet and Neumann boundary conditions. If $Δ_{p}w=\mbox{div}(|\nabla w|^{p-2}w)$ stands for the $p-$Laplacian and $\frac{\alpha }{p}+\frac{\beta }{q}=1,$ we consider $$\{\begin{array}{ll}-{\Delta }_{p}u=\lambda \alpha |u{|}^{\alpha -2}u|v{|}^{\beta }& \text{ in }\Omega ,\\ -{\Delta }_{q}v=\lambda \beta |u{|}^{\alpha }|v{|}^{\beta -2}v& \text{ in }\Omega ,\end{array}$$ with mixed boundary conditions $$u=0,|\nabla v{|}^{q-2}\frac{\partial v}{\partial \nu }=0,\text{on }\partial \Omega .$$ We show that there is a first non trivial eigenvalue that can be characterized by the variational minimization problem $${\lambda }_{p,q}^{\alpha ,\beta }=\min \left\{\frac{{\int }_{\Omega }\frac{|\nabla u{|}^p}{p}dx+{\int }_{\Omega }\frac{|\nabla v{|}^q}{q}dx}{{\int }_{\Omega }|u{|}^{\alpha }|v{|}^{\beta }dx}:(u,v)\in {\mathcal{A}}_{p,q}^{\alpha ,\beta }\right\},$$ where $${\mathcal{A}}_{p,q}^{\alpha ,\beta }=\left\{(u,v)\in W_0^{1,p}(\Omega )\times W^{1,q}(\Omega ):uv\not\equiv 0\text{ and }{\int }_{\Omega }|u{|}^{\alpha }|v{|}^{\beta -2}vdx=0\right\}.$$ We also study the limit of ${\lambda }_{p,q}^{\alpha ,\beta }$ as $p,q\to \infty$ assuming that $\frac{\alpha }{p}\to \Gamma \in (0,1)$, and $\frac{q}{p}\to Q\in (0,\infty )$ as $p,q\to \infty .$ We find that this limit problem interpolates between the pure Dirichlet and Neumann cases for a single equation when we take $Q=1$ and the limits $\Gamma \to 1$ and $\Gamma \to 0$.

21 pages, 1 figure

References in corpus (2)

• The first non-zero Neumann $p-$fractional eigenvalue
• An optimal mass transport approach for limits of eigenvalue problems for the fractional $p$-Laplacian