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]

Mostafa Mbekhta. Representation and approximation of the polar factor of an operator on a Hilbert space. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020463
[2]

Fioralba Cakoni, Pu-Zhao Kow, Jenn-Nan Wang. The interior transmission eigenvalue problem for elastic waves in media with obstacles. Inverse Problems & Imaging, , () : -. doi: 10.3934/ipi.2020075
[3]

Gloria Paoli, Gianpaolo Piscitelli, Rossanno Sannipoli. A stability result for the Steklov Laplacian Eigenvalue Problem with a spherical obstacle. Communications on Pure & Applied Analysis, 2021, 20 (1) : 145-158. doi: 10.3934/cpaa.2020261
[4]

Alessandro Carbotti, Giovanni E. Comi. A note on Riemann-Liouville fractional Sobolev spaces. Communications on Pure & Applied Analysis, 2021, 20 (1) : 17-54. doi: 10.3934/cpaa.2020255
[5]

Marco Ghimenti, Anna Maria Micheletti. Compactness results for linearly perturbed Yamabe problem on manifolds with boundary. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020453
[6]

Shun Zhang, Jianlin Jiang, Su Zhang, Yibing Lv, Yuzhen Guo. ADMM-type methods for generalized multi-facility Weber problem. Journal of Industrial & Management Optimization, 2020  doi: 10.3934/jimo.2020171
[7]

Lingfeng Li, Shousheng Luo, Xue-Cheng Tai, Jiang Yang. A new variational approach based on level-set function for convex hull problem with outliers. Inverse Problems & Imaging, , () : -. doi: 10.3934/ipi.2020070
[8]

Leilei Wei, Yinnian He. A fully discrete local discontinuous Galerkin method with the generalized numerical flux to solve the tempered fractional reaction-diffusion equation. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020319
[9]

Abdollah Borhanifar, Maria Alessandra Ragusa, Sohrab Valizadeh. High-order numerical method for two-dimensional Riesz space fractional advection-dispersion equation. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020355
[10]

Nguyen Huy Tuan. On an initial and final value problem for fractional nonclassical diffusion equations of Kirchhoff type. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020354
[11]

Gongbao Li, Tao Yang. Improved Sobolev inequalities involving weighted Morrey norms and the existence of nontrivial solutions to doubly critical elliptic systems involving fractional Laplacian and Hardy terms. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020469
[12]

Andrew D. Lewis. Erratum for "nonholonomic and constrained variational mechanics". Journal of Geometric Mechanics, 2020, 12 (4) : 671-675. doi: 10.3934/jgm.2020033
[13]

Andrea Braides, Antonio Tribuzio. Perturbed minimizing movements of families of functionals. Discrete & Continuous Dynamical Systems - S, 2021, 14 (1) : 373-393. doi: 10.3934/dcdss.2020324
[14]

Qiao Liu. Local rigidity of certain solvable group actions on tori. Discrete & Continuous Dynamical Systems - A, 2021, 41 (2) : 553-567. doi: 10.3934/dcds.2020269
[15]

Shipra Singh, Aviv Gibali, Xiaolong Qin. Cooperation in traffic network problems via evolutionary split variational inequalities. Journal of Industrial & Management Optimization, 2020  doi: 10.3934/jimo.2020170
[16]

Predrag S. Stanimirović, Branislav Ivanov, Haifeng Ma, Dijana Mosić. A survey of gradient methods for solving nonlinear optimization. Electronic Research Archive, 2020, 28 (4) : 1573-1624. doi: 10.3934/era.2020115
[17]

Li-Bin Liu, Ying Liang, Jian Zhang, Xiaobing Bao. A robust adaptive grid method for singularly perturbed Burger-Huxley equations. Electronic Research Archive, 2020, 28 (4) : 1439-1457. doi: 10.3934/era.2020076
[18]

Hirokazu Ninomiya. Entire solutions of the Allen–Cahn–Nagumo equation in a multi-dimensional space. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 395-412. doi: 10.3934/dcds.2020364
[19]

Thierry Cazenave, Ivan Naumkin. Local smooth solutions of the nonlinear Klein-gordon equation. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020448
[20]

Zedong Yang, Guotao Wang, Ravi P. Agarwal, Haiyong Xu. Existence and nonexistence of entire positive radial solutions for a class of Schrödinger elliptic systems involving a nonlinear operator. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020436

2019 Impact Factor: 1.338

Tools

Metrics

  • PDF downloads (65)
  • HTML views (70)
  • Cited by (5)

Other articles
by authors

[Back to Top]

Copyright © 2020 American Institute of Mathematical Sciences