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.


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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