# American Institute of Mathematical Sciences

December  2017, 37(12): 6243-6255. doi: 10.3934/dcds.2017270

## Perturbed fractional eigenvalue problems

 1 Department of Mathematics, University of Craiova, 200585 Craiova, Romania 2 "Simion Stoilow" Institute of Mathematics of the Romanian Academy, 010702 Bucharest, Romania 3 Department of Mathematics and Computer Science, University Politehnica of Bucharest, 060042 Bucharest, Romania 4 "Simion Stoilow" Institute of Mathematics of the Romanian Academy, 010702 Bucharest, Romania

* Corresponding author: Mihai Mihăilescu

Received  January 2017 Revised  June 2017 Published  August 2017

Fund Project: The research of M. Fărcăşeanu and M. Mihăilescu was partially supported by CNCS-UEFISCDI Grant No. PN-II-RU-TE- 2014-4-0007. D. Stancu-Dumitru has been partially supported by CNCS-UEFISCDI Grant No. PN-III-P1-1.1-PD-2016-0202.

Let $Ω\subset\mathbb{R}^N$ ($N≥2$) be a bounded domain with Lipschitz boundary. For each $p∈(1,∞)$ and $s∈ (0,1)$ we denote by $(-Δ_p)^s$ the fractional $(s,p)$-Laplacian operator. In this paper we study the existence of nontrivial solutions for a perturbation of the eigenvalue problem $(-Δ_p)^s u=λ |u|^{p-2}u$, in $Ω$, $u=0$, in $\mathbb{R}^N\backslash Ω$, with a fractional $(t,q)$-Laplacian operator in the left-hand side of the equation, when $t∈(0,1)$ and $q∈(1,∞)$ are such that $s-N/p=t-N/q$. We show that nontrivial solutions for the perturbed eigenvalue problem exists if and only if parameter $λ$ is strictly larger than the first eigenvalue of the $(s,p)$-Laplacian.

Citation: Maria Fărcăşeanu, Mihai Mihăilescu, Denisa Stancu-Dumitru. Perturbed fractional eigenvalue problems. Discrete and Continuous Dynamical Systems, 2017, 37 (12) : 6243-6255. doi: 10.3934/dcds.2017270
##### References:
 [1] M. Bocea and M. Mihăilescu, Existence of nonnegative viscosity solutions for a class of problems involving the $∞$-Laplacian, Nonlinear Differential Equations and Applications (NoDEA), 23 (2016), Art. 11, 21 pp. doi: 10.1007/s00030-016-0373-2. [2] L. Brasco, E. Parini and M. Squassina, Stability of variational eigenvalues for the fractional $p$-Laplacian, Discrete Continuous Dynam. Systems -A, 36 (2016), 1813-1845.  doi: 10.3934/dcds.2016.36.1813. [3] L. Del Pezzo, J. Fernandez Bonder and L. Lopez Rios, An optimization problem for the first eigenvalue of the $p$-fractional Laplacian, preprint, arXiv: 1601.03019v1. [4] L. Del Pezzo and A. Quaas, Global bifurcation for fractional $p$-Laplacian and an application, Z. Anal. Anwend., 35 (2016), 411-447.  doi: 10.4171/ZAA/1572. [5] E. Di Nezza, G. Palatucci and E. Valdinoci, Hitchhiker's guide to the fractional Sobolev spaces, Bull. Sci. Math., 136 (2012), 521-573.  doi: 10.1016/j.bulsci.2011.12.004. [6] M. Fărcăşeanu, M. Mihăilescu and D. Stancu-Dumitru, On the set of eigenvalues of some PDEs with homogeneous Neumann boundary condition, Nonlinear Analysis, 116 (2015), 19-25.  doi: 10.1016/j.na.2014.12.019. [7] R. Ferreira and M. Perez-Llanos, Limit problems for a Fractional $p$-Laplacian as $p \to \infty$, Nonlinear Differential Equations and Applications (NoDEA), 23 (2016), Art. 14, 28 pp. doi: 10.1007/s00030-016-0368-z. [8] G. Franzina and G. Palatucci, Fractional $p$-eigenvalues, Riv. Mat. Univ. Parma, 5 (2014), 373-386. [9] P. Grisvard, Elliptic Problems in Nonsmooth Domains, Pitman, Boston, MA, 1985. [10] E. Lindgren and P. Lindqvist, Fractional eigenvalues, Calc. Var., 49 (2014), 795-826.  doi: 10.1007/s00526-013-0600-1. [11] M. Mihăilescu, An eigenvalue problem possessing a continuous family of eigenvalues plus an isolated eigenvalue, Communications on Pure and Applied Analysis, 10 (2011), 701-708.  doi: 10.3934/cpaa.2011.10.701. [12] M. Mihăilescu and G. Moroşanu, Eigenvalues of $-Δ_p -Δ_q$ under Neumann boundary condition, Canadian Mathematical Bulletin, 59 (2016), 606-616.  doi: 10.4153/CMB-2016-025-2.

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##### References:
 [1] M. Bocea and M. Mihăilescu, Existence of nonnegative viscosity solutions for a class of problems involving the $∞$-Laplacian, Nonlinear Differential Equations and Applications (NoDEA), 23 (2016), Art. 11, 21 pp. doi: 10.1007/s00030-016-0373-2. [2] L. Brasco, E. Parini and M. Squassina, Stability of variational eigenvalues for the fractional $p$-Laplacian, Discrete Continuous Dynam. Systems -A, 36 (2016), 1813-1845.  doi: 10.3934/dcds.2016.36.1813. [3] L. Del Pezzo, J. Fernandez Bonder and L. Lopez Rios, An optimization problem for the first eigenvalue of the $p$-fractional Laplacian, preprint, arXiv: 1601.03019v1. [4] L. Del Pezzo and A. Quaas, Global bifurcation for fractional $p$-Laplacian and an application, Z. Anal. Anwend., 35 (2016), 411-447.  doi: 10.4171/ZAA/1572. [5] E. Di Nezza, G. Palatucci and E. Valdinoci, Hitchhiker's guide to the fractional Sobolev spaces, Bull. Sci. Math., 136 (2012), 521-573.  doi: 10.1016/j.bulsci.2011.12.004. [6] M. Fărcăşeanu, M. Mihăilescu and D. Stancu-Dumitru, On the set of eigenvalues of some PDEs with homogeneous Neumann boundary condition, Nonlinear Analysis, 116 (2015), 19-25.  doi: 10.1016/j.na.2014.12.019. [7] R. Ferreira and M. Perez-Llanos, Limit problems for a Fractional $p$-Laplacian as $p \to \infty$, Nonlinear Differential Equations and Applications (NoDEA), 23 (2016), Art. 14, 28 pp. doi: 10.1007/s00030-016-0368-z. [8] G. Franzina and G. Palatucci, Fractional $p$-eigenvalues, Riv. Mat. Univ. Parma, 5 (2014), 373-386. [9] P. Grisvard, Elliptic Problems in Nonsmooth Domains, Pitman, Boston, MA, 1985. [10] E. Lindgren and P. Lindqvist, Fractional eigenvalues, Calc. Var., 49 (2014), 795-826.  doi: 10.1007/s00526-013-0600-1. [11] M. Mihăilescu, An eigenvalue problem possessing a continuous family of eigenvalues plus an isolated eigenvalue, Communications on Pure and Applied Analysis, 10 (2011), 701-708.  doi: 10.3934/cpaa.2011.10.701. [12] M. Mihăilescu and G. Moroşanu, Eigenvalues of $-Δ_p -Δ_q$ under Neumann boundary condition, Canadian Mathematical Bulletin, 59 (2016), 606-616.  doi: 10.4153/CMB-2016-025-2.
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