The second eigenvalue of the fractional \(p\)-Laplacian (Q323093)

For a bounded, open set \(\Omega\subset\mathbb R^N\) the authors consider the nonlinear eigenvalue problem NEWLINE\[NEWLINE(- \Delta_p)^su=\lambda |u|^{p-2}u\text{ in }\Omega,\quad u=0\text{ in }\mathbb R^N\setminus\Omega,NEWLINE\]NEWLINE where \(0<s<1\) and \(1<p<\infty\) and the operator \((-\Delta_p)^s\) is defined as NEWLINE\[NEWLINE(-\Delta_p)^su(x):=2\lim_{\delta\to 0+}\int_{\{y\in\mathbb R^N:|y-x|\geq\delta\}}\frac{|u(x)-u(y)|^{p-2}(u(x)-u(y))}{|x-y|^{N+sp}}dy,NEWLINE\]NEWLINE called by the authors \textit{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 NEWLINE\[NEWLINE\lambda_2 (\Omega)>\left(\frac{2|B|}{|\Omega|}\right)^{\frac{sp}{N}}\lambda_1(B)NEWLINE\]NEWLINE 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.