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Fractional eigenvalues. (English) Zbl 1292.35193

The authors investigate the nonlocal eigenvalue problem \[ 2\int_{R^{n}}\frac{|u(y)-u(x)|^{p-2}(u(y)-u(x))}{|y-x|^{\alpha p}}dy+\lambda|u(x)|^{p-2}u(x)=0 \] in a bounded domain \(\Omega\subset \mathbb R^{n},\) \(p>2,\) \(n<\alpha p<n+p,\) as Euler-Lagrange equation, arising from the minimization of the fractional Rayleigh quotient and especially the cases of large \(p\) and the limit case \(p\rightarrow \infty.\) The solutions are treated in the viscosity sense. It is proved that positive viscosity solutions are unique up to a normalization and the first eigenvalue is isolated. For sign changing solutions strange phenomena are detected, caused by the influence of points far away appearing in the domain of integration for the non-local operator.
For sign changing solutions of the \(\infty\)-eigenvalue equation \(\max\{\mathcal{L}_{\infty}\) \({ u(x)},\) \( \mathcal{L}_{\infty}^{-}\) \(u(x)+\) \(\lambda u(x)\}=0\) it is needed to introduce the open sets \(\{u<0\}\) and the nodal line \(\{u=0\}.\) Strange phenomena occur in the case of the \(\infty\)-eigenvalue equation, such as nodal domains – which are the connected components of the open sets \(\{u>0\}\) and \(\{u<0\}\) – not having the same first \(\infty\)-eigenvalue, yet they all come from the same higher \(\infty\)-eigenfunction; the restriction of a higher eigenfunction to one of its nodal domains is not an \(\infty\)-eigenfunction for the nodal domain. The last phenomenon arises even in one-dimensional examples.

MSC:

35P30 Nonlinear eigenvalue problems and nonlinear spectral theory for PDEs
35R11 Fractional partial differential equations
35J60 Nonlinear elliptic equations
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[1] Anane, A.: Simplicité et isolation de la première valeur propre du p-laplacien avec poids. C. R. Acad. Sci. Paris Sér. I Math. 305(16), 725-728 (1987) · Zbl 0633.35061
[2] Bardi, M., Capuzzo-Dolcetta, I.: Optimal Control and Viscosity Solutions of Hamilton-Jacobi-Bellman Equations, Systems & Control: Foundations & Applications. Birkhäuser Boston Inc., Boston, MA (1997). With appendices by Maurizio Falcone and Pierpaolo Soravia · Zbl 0890.49011
[3] Belloni, M; Kawohl, B, A direct uniqueness proof for equations involving the p-Laplace operator, Manuscripta Math., 109, 229-231, (2002) · Zbl 1100.35032
[4] Bourgain, J., Brezis, H., Mironescu, P.: Limiting embedding theorems for \({\rm W^{s, p}}\) when \(s↑ 1\) and applications. J. Anal. Math. 87, 77-101. Dedicated to the memory of Thomas H. Wolff (2002) · Zbl 1029.46030
[5] Chambolle, A., Lindgren, E., Monneau, R.: A Hölder infinity Laplacian, accepted for publication in ESAIM: Control, Optimisation and Calculus of Variations (2011)
[6] Champion, T., De Pascale, L., Jimenez, C.: The \(∞ \)-eigenvalue problem and a problem of optimal transportation. Commun. Appl. Anal. 13(4), 547-565 (2009) · Zbl 1189.35214
[7] Nezza, E; Palatucci, G; Valdinoci, E, Hitchhikers guide to the fractional Sobolev spaces, Bull. Sci. Math., 136, 521-573, (2012) · Zbl 1252.46023
[8] Frank, R.L., Leander, G.: Refined semiclassical asymptotics for fractional powers of the Laplace operator. preprint (2011) · Zbl 1337.35163
[9] Fukagai, N; Ito, M; Narukawa, K, Limit as \(p→ ∞ \) of p-Laplace eigenvalue problems and L\(^∞ \)-inequality of the Poincaré type, Differ. Integr. Equ., 12, 183-206, (1999) · Zbl 1064.35512
[10] Hynd, R., Smart, C.K., Yu, Y.: Nonuniqueness of infinity ground states, preprint (2012) · Zbl 1277.35272
[11] Ishii, H; Nakamura, G, A class of integral equations and approximation of p-Laplace equations, Calc. Var. Partial Differ. Equ., 37, 485-522, (2010) · Zbl 1198.45005
[12] Juutinen, P; Lindqvist, P; Manfredi, JJ, The \(∞ \)-eigenvalue problem, Arch. Ration. Mech. Anal., 148, 89-105, (1999) · Zbl 0947.35104
[13] Kassmann, M.: The classical Harnack inequality fails for nonlocal operators. preprint No. 360, Sonderforschungsbereich 611 · Zbl 0981.31005
[14] Kawohl, B; Lindqvist, P, Positive eigenfunctions for the p-Laplace operator revisited, Analysis (Munich), 26, 545-550, (2006) · Zbl 1136.35063
[15] Kellogg, O.D.: Foundations of Potential Theory, Reprint from the first edition of 1929: Die Grundlehren der Mathematischen Wissenschaften, Band 31. Springer, Berlin (1967)
[16] Koike, S.: A Beginner’s Guide to the Theory of Viscosity Solutions, vol. 13. MSJ Memoirs, Mathematical Society of Japan, Tokyo (2004) · Zbl 1056.49027
[17] Kwaśnicki, M, Eigenvalues of the fractional Laplace operator in the interval, J. Funct. Anal., 262, 2379-2402, (2012) · Zbl 1234.35164
[18] Ôtani, M., Teshima, T.: On the first eigenvalue of some quasilinear elliptic equations. Proc. Jpn. Acad. Ser. A Math. Sci. 64(1), 8-10 (1988) · Zbl 0662.35080
[19] Yifeng, Y, Some properties of the ground states of the infinity Laplacian, Indiana Univ. Math. J., 56, 947-964, (2007) · Zbl 1114.49032
[20] Zoia, A., Rosso, A., Kardar, M.: Fractional Laplacian in bounded domains. Phys. Rev. E (3) 76(2), 021116, 11 (2007) · Zbl 1064.35512
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