Infinitely many sign-changing solutions for the Brézis-Nirenberg problem involving the fractional Laplacian

Liu, Lin; Sun, Jijiang; Tersian, Stepan

doi:10.1515/fca-2017-0061

Cited by 15 publications

(18 citation statements)

References 33 publications

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“…Let us compare the "solutions" in the two cases. For the "spectral formulation", the behavior of the solution (11) around the boundary |x| = 1 is quite different than (6) for the "Riesz formulation" (see Section 1.1). Also, we give this simple example to illustrate the following Caffarelli -Stinga result (see [8])…”

Section: Homogeneous Dirichlet Conditionsmentioning

confidence: 96%

“…then, comparing (9) with (10), it follows that (u, e k ) = 0 for k = 2m and (u, e 2m+1 ) = 2 2 π 2s+1 1 (2m + 1) 2s+1 , m = 0, 1, 2... Thus, (11) u s (x) = 2 2…”

Section: Homogeneous Dirichlet Conditionsmentioning

confidence: 97%

“…w) at x 0 ∈ ∂ Ω. (11). Because of the symmetry, it is enough to study the behavior of the solution (11) only around x = −1.…”

Section: Homogeneous Dirichlet Conditionsmentioning

confidence: 99%

See 2 more Smart Citations

Spectral Fractional Laplacian with Inhomogeneous Dirichlet Data: Questions, Problems, Solutions

Harizanov¹,

Margenov²,

Popivanov³

2020

Preprint

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In this paper we discuss the topic of correct setting for the equation (−∆) s u = f , with 0 < s < 1. The definition of the fractional Laplacian on the whole space R n , n = 1, 2, 3 is understood through the Fourier transform, see, e.g., Karniadakis et.al. (arXiv, 2018). The real challenge however represents the case when this equation is posed in a bounded domain Ω and proper boundary conditions are needed for the correctness of the corresponding problem. Let us mention here that the case of inhomogeneous boundary data has been neglected up to the last years. The reason is that imposing nonzero boundary conditions in the nonlocal setting is highly nontrivial. There exist at least two different definitions of fractional Laplacian, and there is still ongoing research about the relations of them. They are not equivalent. The focus of our study is a new characterization of the spectral fractional Laplacian. One of the major contributions concerns the case when the right hand side f is a Dirac δ function. For comparing the differences between the solutions in the spectral and Riesz formulations, we consider an inhomogeneous fractional Dirichlet problem. The provided theoretical analysis is supported by model numerical tests.

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Section: Homogeneous Dirichlet Conditionsmentioning

confidence: 96%

“…then, comparing (9) with (10), it follows that (u, e k ) = 0 for k = 2m and (u, e 2m+1 ) = 2 2 π 2s+1 1 (2m + 1) 2s+1 , m = 0, 1, 2... Thus, (11) u s (x) = 2 2…”

Section: Homogeneous Dirichlet Conditionsmentioning

confidence: 97%

See 1 more Smart Citation

Spectral Fractional Laplacian with Inhomogeneous Dirichlet Data: Questions, Problems, Solutions

Harizanov¹,

Margenov²,

Popivanov³

2020

Preprint

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“…One is the spectral fractional operator (cf. [28,58] and the references therein), the other is the integral fractional operator (cf. [17]), which is one of main topics in this paper.…”

Section: 2mentioning

confidence: 99%

“…where α 0 = λ − µ as in (28). Since φ ∈ L p (R N ) ∩ H s (R N ), then for every η > 0 there exists a R 1 = R 1 (η) ≥ 1 such that for all k ≥ R 1 ,…”

Section: 1mentioning

confidence: 99%

Pullback attractors for bi-spatial continuous random dynamical systems and application to stochastic fractional power dissipative equation on an unbounded domain

Zhao¹

2019

Discrete &Amp; Continuous Dynamical Systems - B

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In this article, a notion of bi-spatial continuous random dynamical system is introduced between two completely separable metric spaces. It is show that roughly speaking, if such a random dynamical system is asymptotically compact and random absorbing in the initial space, then it admits a bispatial pullback attractor which is measurable in two spaces. The measurability of pullback attractor in the regular spaces is completely solved theoretically. As applications, we study the dynamical behaviour of solutions of the nonautonomous stochastic fractional power dissipative equation on R N with additive white noise and a polynomial-like growth nonlinearity of order p, p ≥ 2. We prove that this equation generates a bi-spatial (L 2 (R N ), H s (R N ) ∩ L p (R N ))continuous random dynamical system, and the random dynamics for this system is captured by a bi-spatial pullback attractor which is compact and attracting in H s (R N ) ∩ L p (R N ), where H s (R N ) is a fractional Sobolev space with s ∈ (0, 1). Especially, the measurability of pullback attractor is individually derived by proving the the continuity of solutions in H s (R N ) and L p (R N ) with respect to the sample. A difference estimates approach, rather than the usual truncation estimate and spectral decomposition technique, is employed to overcome the loss of Sobolev compact embedding in H s (R N ) ∩ L p (R N ), s ∈ (0, 1), N ≥ 1.2010 Mathematics Subject Classification. Primary: 35R60, 35B40, 35B41; Secondary: 35B65. Key words and phrases. Bi-spatial continuous random dynamical system, fractional power dissipative equation, fractional Sobolev space, pullback attractor, measurability.The operator (−∆) s with s ∈ (0, 1) is called a fractional power Laplacian, whose limit as s ↑ 1 is the classic Laplacian ∆, see [17]. It is a nonlocal generalization of the classic Laplacian that is often used to model the diffusive processes, see [17,27,40] and the references therein. As for the long-time dynamics, Hu et al [32] discussed the existence of a random attractor in L 2 (R N ) for the fractional power dissipative stochastic equation with additive noises. Wang [46] discussed the existence and uniqueness of random attractor for the same equation with multiplicative noise on bounded domain. The dynamics of 3D Ginzburg-Landau equation involving fractional power Laplacian ware studied in [33,34]. Most recently, Gu et al [21] obtained the regularity of the random attractor for problem (1) in H s (R N ) in the case of multiplicative noise, where the authors employed the tail estimate and spectral decomposition technique to overcome the loss of compactness on unbounded domains. But little is known about the continuity and measurability of its solutions in H s (R N ) and L p (R N ) for s ∈ (0, 1) and p ≥ 2. In this paper, we prove the existence, regularity and measurability of pullback attractors for the stochastic fractional power dissipative equation (1) defined on the whole space R N .Most of the stochastic partial differential equations (SPDEs) present a regular solut...

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Sign‐changing solutions of fractional Laplacian system with critical exponent

Li,

Wen

2024

Math Methods in App Sciences

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This paper deals with the following fractional Laplacian system with critical exponent: where is a bounded smooth open connected set in is the fractional critical Sobolev exponent, and is the first eigenvalue of fractional Laplacian under the condition in . We prove that, for each fixed and slightly smaller than , the above system with admits a sign‐changing solution in the following sense: one component changes sign, while the other one is positive. Our result includes the lower dimensional case . Compared with the classical Laplacian case, our problem is nonlocal and the first component of the solution is sign‐changing, some new difficulties arise and new arguments and estimates should be introduced.

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Infinitely many sign-changing solutions for the Brézis-Nirenberg problem involving the fractional Laplacian

Cited by 15 publications

References 33 publications

Spectral Fractional Laplacian with Inhomogeneous Dirichlet Data: Questions, Problems, Solutions

Spectral Fractional Laplacian with Inhomogeneous Dirichlet Data: Questions, Problems, Solutions

Pullback attractors for bi-spatial continuous random dynamical systems and application to stochastic fractional power dissipative equation on an unbounded domain

Sign‐changing solutions of fractional Laplacian system with critical exponent

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