A system of local/nonlocal p -Laplacians: The eigenvalue problem and its asymptotic limit as p → ∞

Abstract

In this work, given $\mathit{p}\in \mathrm{(}1\mathrm{,}\infty \mathrm{)}$ , we prove the existence and simplicity of the first eigenvalue ${\mathit{\lambda }}_{\mathit{p}}$ and its corresponding eigenvector $\mathrm{(}{\mathit{u}}_{\mathit{p}}\mathrm{,}{\mathit{v}}_{\mathit{p}}\mathrm{)}$ , for the following local/nonlocal PDE system $$\begin{array}{}\text{(0.1)}& \left\{\begin{array}{cc}-{\mathrm{\Delta }}_{\mathit{p}}\mathit{u}+{\mathrm{(}-\mathrm{\Delta }\mathrm{)}}_{\mathit{p}}^{\mathit{r}}\mathit{u}=\frac{2\mathit{\alpha }}{\mathit{\alpha }+\mathit{\beta }}\mathit{\lambda }|\mathit{u}{|}^{\mathit{\alpha }-2}|\mathit{v}{|}^{\mathit{\beta }}\mathit{u}\hfill & \text{in }\mathrm{\Omega }\hfill \\ -{\mathrm{\Delta }}_{\mathit{p}}\mathit{v}+{\mathrm{(}-\mathrm{\Delta }\mathrm{)}}_{\mathit{p}}^{\mathit{s}}\mathit{v}=\frac{2\mathit{\beta }}{\mathit{\alpha }+\mathit{\beta }}\mathit{\lambda }|\mathit{u}{|}^{\mathit{\alpha }}|\mathit{v}{|}^{\mathit{\beta }-2}\mathit{v}\hfill & \text{in }\mathrm{\Omega }\hfill \\ \mathit{u}=0\hfill & \text{on }{\mathbb{R}}^{\mathit{N}}\setminus \mathrm{\Omega }\hfill \\ \mathit{v}=0\hfill & \text{on }{\mathbb{R}}^{\mathit{N}}\setminus \mathrm{\Omega }\mathrm{,}\hfill \end{array}\right.\end{array}$$ where $\mathrm{\Omega }\subset {\mathbb{R}}^{\mathit{N}}$ is a bounded open domain, $0\mathrm{<}\mathit{r}\mathrm{,}\mathit{s}\mathrm{<}1$ and $\mathit{\alpha }\mathrm{(}\mathit{p}\mathrm{)}+\mathit{\beta }\mathrm{(}\mathit{p}\mathrm{)}=\mathit{p}$ . Moreover, we address the asymptotic limit as $\mathit{p}\to \infty$ , proving the explicit geometric characterization of the corresponding first ∞-eigenvalue, namely ${\mathit{\lambda }}_{\infty }$ , and the uniformly convergence of the pair $\mathrm{(}{\mathit{u}}_{\mathit{p}}\mathrm{,}{\mathit{v}}_{\mathit{p}}\mathrm{)}$ to the ∞-eigenvector $\mathrm{(}{\mathit{u}}_{\infty }\mathrm{,}{\mathit{v}}_{\infty }\mathrm{)}$ . Finally, the triple $\mathrm{(}{\mathit{u}}_{\infty }\mathrm{,}{\mathit{v}}_{\infty }\mathrm{,}{\mathit{\lambda }}_{\infty }\mathrm{)}$ verifies, in the viscosity sense, a limiting PDE system.