## 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:

  Brasco L. and Franzina G., An anisotropic eigenvalue problem of Stekloff type and weighted Wulff inequalities, NoDEA Nonlinear Differential Equations Appl. 20 (2013), 1795-1830. · Zbl 1292.35192  Brasco L. and Franzina G., On the Hong-Krahn-Szego inequality for the p-Laplace operator, Manuscripta Math. 141 (2013), 537-557. · Zbl 1317.35149  Brasco L. and Franzina G., Convexity properties of Dirichlet integrals and Picone-type inequalities, Kodai Math. J. 37 (2014), 769-799. · Zbl 1315.47054  Brasco L., Lindgren E. and Parini E., The fractional Cheeger problem, Interfaces Free Bound. 16 (2014), 419-458. · Zbl 1301.49115  Caffarelli L. and Silvestre L., An extension problem related to the fractional Laplacian, Comm. Partial Differential Equations 32 (2007), 1245-1260. · Zbl 1143.26002  Cuesta M., Minimax theorems on $${C^{1}}$$ manifolds via Ekeland variational principle, Abstr. Appl. Anal. 13 (2003), 757-768. · Zbl 1072.58004  Cuesta M., De Figueiredo D. G. and Gossez J.-P., The beginning of the Fučik spectrum for the p-Laplacian, J. Differential Equations 159 (1999), 212-238. · Zbl 0947.35068  Di Castro A., Kuusi T. and Palatucci G., Local behavior of fractional p-minimizers, preprint (2014), ; to appear in Ann. Inst. H. Poincaré Anal. Non Linéaire. · Zbl 1355.35192  Di Castro A., Kuusi T. and Palatucci G., Nonlocal Harnack inequalities, J. Funct. Anal. 267 (2014), 1807-1836. · Zbl 1302.35082  Di Nezza E., Palatucci G. and Valdinoci E., Hitchhiker’s guide to the fractional Sobolev spaces, Bull. Sci. Math. 136 (2012), 521-573. · Zbl 1252.46023  Drábek P. and Robinson S. B., Resonance problems for the p-Laplacian, J. Funct. Anal. 169 (1999), 189-200. · Zbl 0940.35087  Dyda B., A fractional order Hardy inequality, Illinois J. Math. 48 (2004), 575-588. · Zbl 1068.26014  Franzina G. and Palatucci G., Fractional p-eigenvalues, Riv. Mat. Univ. Parma (N.S.) 5 (2014), 315-328.  Franzina G. and Valdinoci E., Geometric analysis of fractional phase transition interfaces, Geometric Properties for Parabolic and Elliptic PDEs, Springer INdAM Ser. 2, Springer, New York (2013), 117-130. · Zbl 1302.35402  Goyal S. and Sreenadh K., On the Fučik spectrum of non-local elliptic operators, NoDEA Nonlinear Differential Equations Appl. 21 (2014), 567-588. · Zbl 1296.35106  Hong I., On an inequality concerning the eigenvalue problem of membrane, Kōdai Math. Semin. Rep. 6 (1954), 113-114. · Zbl 0057.08805  Iannizzotto A. and Squassina M., Weyl-type laws for fractional p-eigenvalue problems, Asymptot. Anal. 88 (2014), 233-245. · Zbl 1296.35103  Kassmann M., A priori estimates for integro-differential operators with measurable kernels, Calc. Var. Partial Differential Equations 34 (2009), 1-21. · Zbl 1158.35019  Krahn E., Über Minimaleigenschaften der Kugel in drei und mehr Dimensionen, Acta Comm. Univ. Dorpat. A9 (1926), 1-44. · JFM 52.0510.03  Kuusi T., Mingione G. and Sire Y., Nonlocal equations with measure data, Comm. Math. Phys. 337 (2015), 1317-1368. · Zbl 1323.45007  Lindgren E. and Lindqvist P., Fractional eigenvalues, Calc. Var. Partial Differential Equations 49 (2014), 795-826. · Zbl 1292.35193  Lindqvist P., Notes on the p-Laplace equation, Report 102, University of Jyvaskyla, Department of Mathematics and Statistics, Jyvaskyla, 2006. · Zbl 1256.35017  Maz’ya V. and Shaposhnikova T., On the Bourgain, Brezis, and Mironescu theorem concerning limiting embeddings of fractional Sobolev spaces, J. Funct. Anal. 195 (2002), 230-238. · Zbl 1028.46050  Pólya G., On the characteristic frequencies of a symmetric membrane, Math. Z. 63 (1955), 331-337. · Zbl 0065.08703
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