Eigenvalues for systems of fractional $p-$Laplacians

Abstract

We study the eigenvalue problem for a system of fractional $p-$Laplacians, that is, $$ \begin{cases} (-Δ_p)^r u = λ\dfracαp|u|^{α-2}u|v|^β &\text{in } Ω,\vspace{.1cm} (-Δ_p)^s u = λ\dfracβp|u|^α|v|^{β-2}v &\text{in } Ω, u=v=0 &\text{in }Ω^c=\R^N\setminusΩ. \end{cases} $$ We show that there is a first (smallest) eigenvalue that is simple and has associated eigen-pairs 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 $ [λ_{1,p}]^{\nicefrac{1}{p}}\to Λ_{1,\infty} $ as $p\to \infty ,$ where $${\Lambda }_{1,\infty }=\underset{(u,v)}{\inf }\left\{\frac{\max \{[u{]}_{r,\infty };[v{]}_{s,\infty }\}}{\Vert |u{|}^{\Gamma }|v{|}^{1-\Gamma }{\Vert }_{L^{\infty }(\Omega )}}\right\}={\left[\frac{1}{R(\Omega )}\right]}^{(1-\Gamma )s+\Gamma r}.$$ Here $R(Ω):=\max_{x\inΩ}\dist(x,\partialΩ)$ 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 eigen-pairs.

19 pages