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Stability of half-degree point defect profiles for 2-D nematic liquid crystal
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 |
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.
References:
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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.
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L. Brasco, E. Parini and M. Squassina,
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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. |
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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. |
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E. Di Nezza, G. Palatucci and E. Valdinoci,
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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. |
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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. |
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G. Franzina and G. Palatucci,
Fractional $p$-eigenvalues, Riv. Mat. Univ. Parma, 5 (2014), 373-386.
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P. Grisvard,
Elliptic Problems in Nonsmooth Domains, Pitman, Boston, MA, 1985. |
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E. Lindgren and P. Lindqvist,
Fractional eigenvalues, Calc. Var., 49 (2014), 795-826.
doi: 10.1007/s00526-013-0600-1. |
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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. |
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. |
| [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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