# American Institute of Mathematical Sciences

June  2016, 9(3): 619-629. doi: 10.3934/dcdss.2016016

## A fractional eigenvalue problem in $\mathbb{R}^N$

 1 Department of Mathematics and Computer Sciences, University of Perugia, Via Vanvitelli 1, 06123 Perugia, Italy, Italy

Received  April 2015 Revised  June 2015 Published  April 2016

We prove that a linear fractional operator with an asymptotically constant lower order term in the whole space admits eigenvalues.
Citation: Giacomo Bocerani, Dimitri Mugnai. A fractional eigenvalue problem in $\mathbb{R}^N$. Discrete & Continuous Dynamical Systems - S, 2016, 9 (3) : 619-629. doi: 10.3934/dcdss.2016016
##### References:
 [1] W. Allegretto, Principal eigenvalues for indefinite-weight elliptic problems in $\mathbbR^n$,, Proc. Amer. Math. Soc., 116 (1992), 701. doi: 10.2307/2159436. Google Scholar [2] A. Ambrosetti and A. Malchiodi, Nonlinear Analysis and Semilinear Elliptic Problems,, Cambridge, (2007). doi: 10.1017/CBO9780511618260. Google Scholar [3] G. Bocerani and D. Mugnai, An asymptotically linear fractional problem in $\mathbbR^N$,, submitted., (). Google Scholar [4] K. J. Brown, C. Cosner and J. Fleckinger, Principal eigenvalues for problems with indefinite weight function on $\mathbbR^N$,, Proc. Amer. Math. Soc., 109 (1990), 147. doi: 10.2307/2048374. Google Scholar [5] X. Chang and Z.-Q. Wang, Ground state of scalar field equations involving a fractional Laplacian with general nonlinearity,, Nonlinearity, 26 (2013), 479. doi: 10.1088/0951-7715/26/2/479. Google Scholar [6] P. Drábek and Y. X. Huang, Bifurcation problems for the $p-$Laplacian in $\mathbbR^n$,, Trans. Amer. Math. Soc., 349 (1997), 171. doi: 10.1090/S0002-9947-97-01788-1. Google Scholar [7] P. Felmer, A. Quaas and J. Tan, Positive solutions of the nonlinear Schrödinger equation with the fractional Laplacian,, Proc. Roy. Soc. Edinburgh Sect. A, 142 (2012), 1237. doi: 10.1017/S0308210511000746. Google Scholar [8] R. L. Frank, E. H. Lieb and R. Seiringer, Hardy-Lieb-Thirring inequalities for fractional Schrödinger operators,, J. Amer. Math. Soc., 21 (2008), 925. doi: 10.1090/S0894-0347-07-00582-6. Google Scholar [9] L. Leadi and A. Yechoui, Principal eigenvalue in an unbounded domain with indefinite potential,, NoDEA Nonlinear Differential Equations Appl., 17 (2010), 391. doi: 10.1007/s00030-010-0059-0. Google Scholar [10] J. Liu, X. Liu and Y. Guo, On an asymptotically $p$-linear $p$-Laplacian equation in $\mathbbR^N$,, Nonlinear Anal., 74 (2011), 676. doi: 10.1016/j.na.2010.09.024. Google Scholar [11] P. L. Lions, The concentration compactness principle in the calculus of variations, the locally compact case, II,, Ann. Inst. H. Poincaré Anal. Non Linéaire, 1 (1984), 223. Google Scholar [12] D. Mugnai and N. S. Papageorgiou, Bifurcation for positive solutions of nonlinear diffusive logistic equations in $\mathbbR^N$ with Indefinite Weight,, Indiana Univ. Math. J., 63 (2014), 1397. doi: 10.1512/iumj.2014.63.5369. Google Scholar [13] D. Mugnai and N. S. Papageorgiou, Resonant nonlinear Neumann problems with indefinite weight,, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (5), 11 (2012), 729. Google Scholar [14] S. Secchi, On fractional Schrödinger equations in $\mathbbR^N$ without the Ambrosetti-Rabinowitz Condition,, Topol. Methods. Nonlinear Anal., (). Google Scholar [15] E. M. Stein, Singular Integrals and Differentiability Properties of Functions,, Princeton University Press, (1970). Google Scholar [16] M. Struwe, Variational Methods: Applications to Nonlinear Partial Differential Equations and Hamiltonian Systems,, Springer-Verlag, (2008). Google Scholar [17] A. Szulkin and M. Willem, Eigenvalue problems with indefinite weight,, Studia Mathematica, 135 (1999), 191. Google Scholar [18] J. Tan and J. Xiong, A Harnack inequality for fractional Laplace equations with lower order terms,, Discrete Contin. Dyn. Syst., 31 (2011), 975. doi: 10.3934/dcds.2011.31.975. Google Scholar [19] E. Valdinoci, From the long jump random walk to the fractional Laplacian,, Bol. Soc. Esp. Mat. Apl., 49 (2009), 33. Google Scholar [20] M. Willem, Minimax Theorems,, Progr. Nonlinear Differential Equations Appl. 24, 24 (1996). doi: 10.1007/978-1-4612-4146-1. Google Scholar

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##### References:
 [1] W. Allegretto, Principal eigenvalues for indefinite-weight elliptic problems in $\mathbbR^n$,, Proc. Amer. Math. Soc., 116 (1992), 701. doi: 10.2307/2159436. Google Scholar [2] A. Ambrosetti and A. Malchiodi, Nonlinear Analysis and Semilinear Elliptic Problems,, Cambridge, (2007). doi: 10.1017/CBO9780511618260. Google Scholar [3] G. Bocerani and D. Mugnai, An asymptotically linear fractional problem in $\mathbbR^N$,, submitted., (). Google Scholar [4] K. J. Brown, C. Cosner and J. Fleckinger, Principal eigenvalues for problems with indefinite weight function on $\mathbbR^N$,, Proc. Amer. Math. Soc., 109 (1990), 147. doi: 10.2307/2048374. Google Scholar [5] X. Chang and Z.-Q. Wang, Ground state of scalar field equations involving a fractional Laplacian with general nonlinearity,, Nonlinearity, 26 (2013), 479. doi: 10.1088/0951-7715/26/2/479. Google Scholar [6] P. Drábek and Y. X. Huang, Bifurcation problems for the $p-$Laplacian in $\mathbbR^n$,, Trans. Amer. Math. Soc., 349 (1997), 171. doi: 10.1090/S0002-9947-97-01788-1. Google Scholar [7] P. Felmer, A. Quaas and J. Tan, Positive solutions of the nonlinear Schrödinger equation with the fractional Laplacian,, Proc. Roy. Soc. Edinburgh Sect. A, 142 (2012), 1237. doi: 10.1017/S0308210511000746. Google Scholar [8] R. L. Frank, E. H. Lieb and R. Seiringer, Hardy-Lieb-Thirring inequalities for fractional Schrödinger operators,, J. Amer. Math. Soc., 21 (2008), 925. doi: 10.1090/S0894-0347-07-00582-6. Google Scholar [9] L. Leadi and A. Yechoui, Principal eigenvalue in an unbounded domain with indefinite potential,, NoDEA Nonlinear Differential Equations Appl., 17 (2010), 391. doi: 10.1007/s00030-010-0059-0. Google Scholar [10] J. Liu, X. Liu and Y. Guo, On an asymptotically $p$-linear $p$-Laplacian equation in $\mathbbR^N$,, Nonlinear Anal., 74 (2011), 676. doi: 10.1016/j.na.2010.09.024. Google Scholar [11] P. L. Lions, The concentration compactness principle in the calculus of variations, the locally compact case, II,, Ann. Inst. H. Poincaré Anal. Non Linéaire, 1 (1984), 223. Google Scholar [12] D. Mugnai and N. S. Papageorgiou, Bifurcation for positive solutions of nonlinear diffusive logistic equations in $\mathbbR^N$ with Indefinite Weight,, Indiana Univ. Math. J., 63 (2014), 1397. doi: 10.1512/iumj.2014.63.5369. Google Scholar [13] D. Mugnai and N. S. Papageorgiou, Resonant nonlinear Neumann problems with indefinite weight,, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (5), 11 (2012), 729. Google Scholar [14] S. Secchi, On fractional Schrödinger equations in $\mathbbR^N$ without the Ambrosetti-Rabinowitz Condition,, Topol. Methods. Nonlinear Anal., (). Google Scholar [15] E. M. Stein, Singular Integrals and Differentiability Properties of Functions,, Princeton University Press, (1970). Google Scholar [16] M. Struwe, Variational Methods: Applications to Nonlinear Partial Differential Equations and Hamiltonian Systems,, Springer-Verlag, (2008). Google Scholar [17] A. Szulkin and M. Willem, Eigenvalue problems with indefinite weight,, Studia Mathematica, 135 (1999), 191. Google Scholar [18] J. Tan and J. Xiong, A Harnack inequality for fractional Laplace equations with lower order terms,, Discrete Contin. Dyn. Syst., 31 (2011), 975. doi: 10.3934/dcds.2011.31.975. Google Scholar [19] E. Valdinoci, From the long jump random walk to the fractional Laplacian,, Bol. Soc. Esp. Mat. Apl., 49 (2009), 33. Google Scholar [20] M. Willem, Minimax Theorems,, Progr. Nonlinear Differential Equations Appl. 24, 24 (1996). doi: 10.1007/978-1-4612-4146-1. Google Scholar
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