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Eigenvalues for systems of fractional \(p\)-Laplacians. (English) Zbl 1400.35197

The paper deals with the nonlocal nonlinear eigenvalue problem for a system of fractional \(p\)-Laplacians \[ \begin{cases} (-\Delta_p)^ru=\lambda\frac{\alpha}{p}|u|^{\alpha-2}u|v|^{\beta} & \text{ in }\Omega,\\ (-\Delta_p)^sv=\lambda\frac{\beta}{p}|u|^{\alpha}|v|^{\beta-2}v & \text{ in }\Omega,\\ u=v=0 & \text{ in }\Omega^c=\mathbb{R}^N\setminus \Omega, \end{cases} \] where \(\Omega\) is a bounded smooth domain in \(\mathbb{R}^N\), \(p>2\), \(r,\,s\in(0,1)\), \(\alpha,\,\beta\in (0,p)\), \(\alpha+\beta=p\), \(\min\{\alpha,\beta\}\geq 1\) and \(\lambda\) is the eigenvalue. The authors investigate the existence of the smallest eigenvalue \(\lambda_{1,p}\) which is simple, and show that the associated eigenpairs are positive and bounded. Then they prove the existence of a sequence of eigenvalues \(\lambda_n\) such that \(\lambda_n\to\infty\) as \(n\to\infty\). The limit of \(\lambda_{1,p}^{1/p}\) as \(p\to\infty\) is also studied, and the authors show that this limit verified a variational characterization and a simple geometrical characterization. In addition, they obtain that a uniform limit (along with subsequences) for the eigenfunctions is a viscous solution to a limit PDE eigenvalue problem.

MSC:

35P30 Nonlinear eigenvalue problems and nonlinear spectral theory for PDEs
35R11 Fractional partial differential equations
45G05 Singular nonlinear integral equations
47G20 Integro-differential operators
35J57 Boundary value problems for second-order elliptic systems
35J92 Quasilinear elliptic equations with \(p\)-Laplacian
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