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

Perturbed fractional eigenvalue problems

Maria Fărcăşeanu 1,2, , Mihai Mihăilescu 1,2,, and  Denisa Stancu-Dumitru 3,4,

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.

Keywords: Perturbed eigenvalue problem, non-local operator, variational methods, fractional Sobolev space.
Mathematics Subject Classification: Primary: 35P30; Secondary: 49J35, 47J30, 46E35.
Citation: Maria Fărcăşeanu, Mihai Mihăilescu, Denisa Stancu-Dumitru. Perturbed fractional eigenvalue problems. Discrete & Continuous Dynamical Systems - A, 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. Google Scholar

[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. Google Scholar

[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.Google Scholar

[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. Google Scholar

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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. Google Scholar

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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. Google Scholar

[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. Google Scholar

[8]

G. Franzina and G. Palatucci, Fractional $p$-eigenvalues, Riv. Mat. Univ. Parma, 5 (2014), 373-386. Google Scholar

[9]

P. Grisvard, Elliptic Problems in Nonsmooth Domains, Pitman, Boston, MA, 1985. Google Scholar

[10]

E. Lindgren and P. Lindqvist, Fractional eigenvalues, Calc. Var., 49 (2014), 795-826. doi: 10.1007/s00526-013-0600-1. Google Scholar

[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. Google Scholar

[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. Google Scholar

show all references

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. Google Scholar

[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. Google Scholar

[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.Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[8]

G. Franzina and G. Palatucci, Fractional $p$-eigenvalues, Riv. Mat. Univ. Parma, 5 (2014), 373-386. Google Scholar

[9]

P. Grisvard, Elliptic Problems in Nonsmooth Domains, Pitman, Boston, MA, 1985. Google Scholar

[10]

E. Lindgren and P. Lindqvist, Fractional eigenvalues, Calc. Var., 49 (2014), 795-826. doi: 10.1007/s00526-013-0600-1. Google Scholar

[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. Google Scholar

[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. Google Scholar

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