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Eigenvalues for systems of fractional $p-$Laplacians (1605.02926v1)

Published 10 May 2016 in math.AP

Abstract: We study the eigenvalue problem for a system of fractional $p-$Laplacians, that is, $$ \begin{cases} (-\Delta_p)r u = \lambda\dfrac{\alpha}p|u|{\alpha-2}u|v|{\beta} &\text{in } \Omega,\vspace{.1cm} (-\Delta_p)s u = \lambda\dfrac{\beta}p|u|{\alpha}|v|{\beta-2}v &\text{in } \Omega, u=v=0 &\text{in }\Omegac=\RN\setminus\Omega. \end{cases} $$ 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 $ [\lambda_{1,p}]{\nicefrac{1}{p}}\to \Lambda_{1,\infty} $ as $p\to\infty,$ where $$ \Lambda_{1,\infty} = \inf_{(u,v)} \left{ \frac{\max { [u]{r,\infty} ; [v]{s,\infty} } }{ | |u|{\Gamma} |v|{1-\Gamma} |{L\infty (\Omega)} } \right} = \left[ \frac{1}{R(\Omega)} \right]{ (1-\Gamma) s + \Gamma r }. $$ Here $R(\Omega):=\max{x\in\Omega}\dist(x,\partial\Omega)$ and $[w]{t,\infty} \coloneqq \sup{(x,y)\in\overline{\Omega}} \frac{| w(y) - w(x)|}{|x-y|{t}}.$ Finally, we identify a PDE problem satisfied, in the viscosity sense, by any possible uniform limit along subsequences of the eigen-pairs.

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