Eigenvalues for systems of fractional $p$-Laplacians

Abstract

We study the eigenvalue problem for a system of fractional $p$-Laplacians, that is, \begin{equation} \begin{cases} (-\Delta _p)^r u = \lambda \tfrac {\alpha }p|u|^{\alpha -2}u|v|^{\beta } &\mbox{in } \Omega ,\\ (-\Delta _p)^s v = \lambda \tfrac {\beta }p|u|^{\alpha }|v|^{\beta -2}v &\mbox{in } \Omega ,\\ u=v=0 &\mbox{in } \Omega ^c=\mathbb{R} ^N\setminus \Omega. \end{cases} \end{equation} We show that there is a first (smallest) eigenvalue that is simple and has associated eigenpairs composed of positive and bounded functions. Moreover, there is a sequence of eigenvalues $\lambda _n$ such that $\lambda _n\to \infty $ as $n\to \infty $.

In addition, we study the limit as $p\to \infty $ of the first eigenvalue, $\lambda _{1,p}$, and we obtain $[\lambda _{1,p}]^{{1}/{p}}\to \Lambda _{1,\infty }$ as $p\to \infty ,$ where $$\Lambda _{1,\infty }= \inf _{(u,v)} \bigg \{\frac {\max \{ [u]_{r,\infty } ; [v]_{s,\infty } \} }{ \| |u|^{\Gamma } |v|^{1-\Gamma } \|_{L^\infty (\Omega )}}\bigg \} = \bigg [ \frac {1}{R(\Omega )} \bigg ]^{ (1-\Gamma ) s + \Gamma r }. $$ Here, $$R(\Omega ):=\max _{x\in \Omega }dist (x,\partial \Omega )\mbox { and } [w]_{t,\infty } := \!\sup _{(x,y)\in \overline {\Omega }} \frac {| w(y) - w(x)|}{|x-y|^{t}}.$$

Finally, we identify a PDE problem satisfied, in the viscosity sense, by any possible uniform limit along subsequences of the eigenpairs.