Fractional equations with indefinite nonlinearities

Chen, Wenxiong; Li, Congming; Zhu, Jiuyi

doi:10.3934/dcds.2019054

Cited by 36 publications

(15 citation statements)

References 17 publications

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“…For the classical BECs problem when s = 1, p = 3 and N = 2, 3 in [30], the standard comparison method from the ODE theory is sufficed to show such a non-existence result, which, however, is invalid in the fractional case. We take the sliding method to overcome this obstacle, which, developed by Chen, Li and Zhu [11], would be typically available in the study of some other similar problems when dealing with the fractional operator.…”

Section: Proof Of Theorem 11 By a Non-existence Resultsmentioning

confidence: 99%

Concentrated solutions to fractional Schrödinger equations with prescribed $L^2$-norm

Guo,

Luo,

Wang

et al. 2021

Preprint

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In this work, we investigate the existence and local uniqueness of normalized k-peak solutions for the fractional Schrödinger equations with attractive interactions with a class of degenerated trapping potential with non-isolated critical points.Precisely, applying the finite dimensional reduction method, we first obtain the existence of k-peak concentrated solutions and especially describe the relationship between the chemical potential µ and the attractive interaction a. Second, after precise analysis of the concentrated points and the Lagrange multiplier, we prove the local uniqueness of the kpeak solutions with prescribed L 2 -norm, by use of the local Pohozaev identities, the blow-up analysis and the maximum principle associated to the nonlocal operator (−∆) s .To our best knowledge, there is few results on the excited normalized solutions of the fractional Schrödinger equations before this present work. The main difficulty lies in the non-local property of the operator (−∆) s . First, it makes the standard comparison argument in the ODE theory invalid to use in our analysis. Second, because of the algebraic decay involving the approximate solutions, the estimates, on the Lagrange multiplier for example, would become more subtle. Moreover, when studying the corresponding harmonic extension problem, several local Pohozaev identities are constructed and we have to estimate several kinds of integrals that never appear in the classic local Schrödinger problems. In addition, throughout our discussion, we need to distinct the different cases of p − 1 < 4s N , p − 1 = 4s N , and p − 1 > 4s N , which are called respectively that the mass-subcritical, the mass-critical, and the mass-supercritical case, due to the mass-constraint condition. Another difficulty comes from the influence of the different degenerate rates along different directions at the critical points of the potential V (x).

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Section: Proof Of Theorem 11 By a Non-existence Resultsmentioning

confidence: 99%

Concentrated solutions to fractional Schrödinger equations with prescribed $L^2$-norm

Guo,

Luo,

Wang

et al. 2021

Preprint

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“…where e 1 = (1, 0, ..., 0), and σ is a small positive number to be chosen as in the proof of Theorem 1 in [12]. Obviously, wλ (x, t) and w λ (x, t) have the same sign and lim |x|→+∞ wλ (x, t) = 0.…”

Section: More Relevant Liouville Type Theoremsmentioning

confidence: 99%

Liouville theorems for fractional parabolic equations

Chen

2021

Preprint

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In this paper, we establish several Liouville type theorems for entire solutions to fractional parabolic equations. We first obtain the key ingredients needed in the proof of Liouville theorems, such as narrow region principles and maximum principles for antisymmetric functions in unbounded domains, in which we remarkably weaken the usual decay condition u → 0 at infinity with respect to the spacial variables to a polynomial growth on u by constructing auxiliary functions. Then we derive monotonicity for the solutions in a half space R n + × R and obtain some new connections between the nonexistence of solutions in a half space R n + × R and in the whole space R n−1 × R and therefore prove the corresponding Liouville type theorems.To overcome the difficulty caused by the non-locality of the fractional Laplacian, we introduce several new ideas which will become useful tools in investigating qualitative properties of solutions for a variety of non-local parabolic problems.

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“…Hence it is reasonable to assume that u is bounded when we consider equation (40). Without the condition lim |x|→∞ u(x) = 0, in order to use the method of moving planes, in [22], an auxiliary function was introduced, so that the planes can still be moved along x 1 direction all the way up to ∞ to derive…”

Section: 1mentioning

confidence: 99%

“…At a minimum ofw λ , the middle term on the right hand side (the integral) neither vanishes nor has a definite sign. This is the main difficulty encountered by the fractional nonlocal operator, and to circumvent which, the authors in [22] introduced a different auxiliary function and estimate (− ) α/2 w λ in an entirely different approach.…”

Section: 1mentioning

confidence: 99%

Direct methods on fractional equations

Chen¹,

Qi²

2019

Discrete &Amp; Continuous Dynamical Systems - A

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In this paper, we summarize some of the recent developments in the area of fractional equations with focus on the ideas and direct methods on fractional non-local operators. These results have more or less appeared in a series of previous literature, in which the ideas were usually submerged in detailed calculations. What we are trying to do here is to single out these ideas and illustrate the inner connections among them, so that the readers can see the whole picture and quickly grasp the essence of these useful methods and apply them to a variety of problems in this area.

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Fractional equations with indefinite nonlinearities

Cited by 36 publications

References 17 publications

Concentrated solutions to fractional Schrödinger equations with prescribed $L^2$-norm

Concentrated solutions to fractional Schrödinger equations with prescribed $L^2$-norm

Liouville theorems for fractional parabolic equations

Direct methods on fractional equations

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