American Institute of Mathematical Sciences

Advanced
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

[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

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

[1]

Sabri Bahrouni, Hichem Ounaies. Embedding theorems in the fractional Orlicz-Sobolev space and applications to non-local problems. Discrete & Continuous Dynamical Systems - A, 2020, 40 (5) : 2917-2944. doi: 10.3934/dcds.2020155
[2]

Raffaella Servadei, Enrico Valdinoci. Variational methods for non-local operators of elliptic type. Discrete & Continuous Dynamical Systems - A, 2013, 33 (5) : 2105-2137. doi: 10.3934/dcds.2013.33.2105
[3]

Shixiu Zheng, Zhilei Xu, Huan Yang, Jintao Song, Zhenkuan Pan. Comparisons of different methods for balanced data classification under the discrete non-local total variational framework. Mathematical Foundations of Computing, 2019, 2 (1) : 11-28. doi: 10.3934/mfc.2019002
[4]

Rafael Abreu, Cristian Morales-Rodrigo, Antonio Suárez. Some eigenvalue problems with non-local boundary conditions and applications. Communications on Pure & Applied Analysis, 2014, 13 (6) : 2465-2474. doi: 10.3934/cpaa.2014.13.2465
[5]

Walter Allegretto, Yanping Lin, Shuqing Ma. On the box method for a non-local parabolic variational inequality. Discrete & Continuous Dynamical Systems - B, 2001, 1 (1) : 71-88. doi: 10.3934/dcdsb.2001.1.71
[6]

Anouar Bahrouni, VicenŢiu D. RĂdulescu. On a new fractional Sobolev space and applications to nonlocal variational problems with variable exponent. Discrete & Continuous Dynamical Systems - S, 2018, 11 (3) : 379-389. doi: 10.3934/dcdss.2018021
[7]

Bin Guo, Wenjie Gao. Finite-time blow-up and extinction rates of solutions to an initial Neumann problem involving the $p(x,t)-Laplace$ operator and a non-local term. Discrete & Continuous Dynamical Systems - A, 2016, 36 (2) : 715-730. doi: 10.3934/dcds.2016.36.715
[8]

Anouar Bahrouni. Trudinger-Moser type inequality and existence of solution for perturbed non-local elliptic operators with exponential nonlinearity. Communications on Pure & Applied Analysis, 2017, 16 (1) : 243-252. doi: 10.3934/cpaa.2017011
[9]

Alexander V. Rezounenko, Petr Zagalak. Non-local PDEs with discrete state-dependent delays: Well-posedness in a metric space. Discrete & Continuous Dynamical Systems - A, 2013, 33 (2) : 819-835. doi: 10.3934/dcds.2013.33.819
[10]

Massimiliano Ferrara, Giovanni Molica Bisci, Binlin Zhang. Existence of weak solutions for non-local fractional problems via Morse theory. Discrete & Continuous Dynamical Systems - B, 2014, 19 (8) : 2483-2499. doi: 10.3934/dcdsb.2014.19.2483
[11]

Xing-Bin Pan. Variational and operator methods for Maxwell-Stokes system. Discrete & Continuous Dynamical Systems - A, 2020, 40 (6) : 3909-3955. doi: 10.3934/dcds.2020036
[12]

Joseph G. Conlon, André Schlichting. A non-local problem for the Fokker-Planck equation related to the Becker-Döring model. Discrete & Continuous Dynamical Systems - A, 2019, 39 (4) : 1821-1889. doi: 10.3934/dcds.2019079
[13]

Christos V. Nikolopoulos, Georgios E. Zouraris. Numerical solution of a non-local elliptic problem modeling a thermistor with a finite element and a finite volume method. Conference Publications, 2007, 2007 (Special) : 768-778. doi: 10.3934/proc.2007.2007.768
[14]

Monica Marras, Nicola Pintus, Giuseppe Viglialoro. On the lifespan of classical solutions to a non-local porous medium problem with nonlinear boundary conditions. Discrete & Continuous Dynamical Systems - S, 2020, 13 (7) : 2033-2045. doi: 10.3934/dcdss.2020156
[15]

Qiyu Jin, Ion Grama, Quansheng Liu. Convergence theorems for the Non-Local Means filter. Inverse Problems & Imaging, 2018, 12 (4) : 853-881. doi: 10.3934/ipi.2018036
[16]

Gabriel Peyré, Sébastien Bougleux, Laurent Cohen. Non-local regularization of inverse problems. Inverse Problems & Imaging, 2011, 5 (2) : 511-530. doi: 10.3934/ipi.2011.5.511
[17]

Olivier Bonnefon, Jérôme Coville, Guillaume Legendre. Concentration phenomenon in some non-local equation. Discrete & Continuous Dynamical Systems - B, 2017, 22 (3) : 763-781. doi: 10.3934/dcdsb.2017037
[18]

Yuxia Guo, Jianjun Nie. Infinitely many non-radial solutions for the prescribed curvature problem of fractional operator. Discrete & Continuous Dynamical Systems - A, 2016, 36 (12) : 6873-6898. doi: 10.3934/dcds.2016099
[19]

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
[20]

Xing-Bin Pan. An eigenvalue variation problem of magnetic Schrödinger operator in three dimensions. Discrete & Continuous Dynamical Systems - A, 2009, 24 (3) : 933-978. doi: 10.3934/dcds.2009.24.933

2019 Impact Factor: 1.338

Tools

Metrics

  • PDF downloads (55)
  • HTML views (69)
  • Cited by (4)

Other articles
by authors

[Back to Top]

Copyright © 2020 American Institute of Mathematical Sciences