## 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

### Keywords:

eigenvalue problems; fractional operators; $$p$$-Laplacian
Full Text:

### References:

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