## The second eigenvalue of the fractional $$p$$-Laplacian.(English)Zbl 1349.35263

For a bounded, open set $$\Omega\subset\mathbb R^N$$ the authors consider the nonlinear eigenvalue problem $(- \Delta_p)^su=\lambda |u|^{p-2}u\text{ in }\Omega,\quad u=0\text{ in }\mathbb R^N\setminus\Omega,$ where $$0<s<1$$ and $$1<p<\infty$$ and the operator $$(-\Delta_p)^s$$ is defined as $(-\Delta_p)^su(x):=2\lim_{\delta\to 0+}\int_{\{y\in\mathbb R^N:|y-x|\geq\delta\}}\frac{|u(x)-u(y)|^{p-2}(u(x)-u(y))}{|x-y|^{N+sp}}dy,$ called by the authors fractional $$p$$-Laplacian. The main aim of the paper is to study the second eigenvalue $$\lambda_2(\Omega)$$ of the problem. It is shown that the second eigenvalue in fact exists (in particular, there is no accumulation of eigenvalues to the first eigenvalue $$\lambda_1(\Omega)$$). A mountain pass variational characterization of $$\lambda_2(\Omega)$$ is proved. Moreover, the estimate $\lambda_2 (\Omega)>\left(\frac{2|B|}{|\Omega|}\right)^{\frac{sp}{N}}\lambda_1(B)$ is proven, where $$B$$ is an arbitrary $$N$$-dimensional ball, and the authors show sharpness of the above estimate by verifying that for $$\Omega$$ being the union of two disjoint balls of equal volume whose mutual distance tends to infinity, the eigenvalue $$\lambda_2 (\Omega)$$ converges to the given bound. Consequently, for $$c>0$$ the shape optimization problem of finding a domain $$\Omega_0$$ such that $$\lambda_2 (\Omega_0) = \inf \{\lambda_2 (\Omega) : |\Omega| = c\}$$ does not admit a solution.

### MSC:

 35P30 Nonlinear eigenvalue problems and nonlinear spectral theory for PDEs 47J10 Nonlinear spectral theory, nonlinear eigenvalue problems 35R09 Integro-partial differential equations
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### References:

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