Boundedness of solutions to Ginzburg–Landau fractional Laplacian equation

Ma, Li

doi:10.1142/s0129167x16500488

Cited by 14 publications

(9 citation statements)

References 11 publications

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“…Corollary 2.3 was previously proved by Ma [11,Lemma 3] under an extra condition that u ∈ L q (R N ) for some q > 1. Here, we remove this unnecessary condition.…”

Section: Lemma 22 Let P > 1 and Cmentioning

confidence: 77%

“…Here, we remove this unnecessary condition. In [11], Corollary 2.3 plays a key role in studying the boundedness of solutions u : R N → R N of the Ginzburg-Landau fractional Laplacian equation…”

Section: Lemma 22 Let P > 1 and Cmentioning

confidence: 99%

“…We believe that this method can be used to study elliptic and parabolic problems with different nonlinearities. Furthermore, our result can be used to improve a result in [11] by removing the global L q assumption of solutions (see the comments below Corollary 2.3).…”

Section: Introductionmentioning

confidence: 98%

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Uniform Lower Bound and Liouville Type Theorem for Fractional Lichnerowicz Equations

Duong

Nguyen²,

Nguyen³

2021

Bull. Aust. Math. Soc.

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We study the fractional parabolic Lichnerowicz equation $$ \begin{align*} v_t+(-\Delta)^s v=v^{-p-2}-v^p \quad\mbox{in } \mathbb R^N\times\mathbb R \end{align*} $$ where $p>0$ and $ 0<s<1 $ . We establish a Liouville-type theorem for positive solutions in the case $p>1$ and give a uniform lower bound of positive solutions when $0<p\leq 1$ . In particular, when v is independent of the time variable, we obtain a similar result for the fractional elliptic Lichnerowicz equation $$ \begin{align*} (-\Delta)^s u=u^{-p-2}-u^p \quad\mbox{in }\mathbb R^N \end{align*} $$ with $p>0$ and $0<s<1$ . This extends the result of Brézis [‘Comments on two notes by L. Ma and X. Xu’, C. R. Math. Acad. Sci. Paris349(5–6) (2011), 269–271] to the fractional Laplacian.

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“…Corollary 2.3 was previously proved by Ma [11,Lemma 3] under an extra condition that u ∈ L q (R N ) for some q > 1. Here, we remove this unnecessary condition.…”

Section: Lemma 22 Let P > 1 and Cmentioning

confidence: 77%

Section: Lemma 22 Let P > 1 and Cmentioning

confidence: 99%

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Uniform Lower Bound and Liouville Type Theorem for Fractional Lichnerowicz Equations

Duong

Nguyen²,

Nguyen³

2021

Bull. Aust. Math. Soc.

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“…To prove Theorem 1.1, we need to prepare some lemmata about the fractional Poisson equation in R n . As we have used them in reference [9] and they may be useful to other situations, we present the proofs in great details.…”

Section: Preliminarymentioning

confidence: 99%

“…This idea may be useful for other problems such as the stationary nonlinear nonlocal Schrodinger systems. Note that solutions to the fractional Ginzburg-Landau model have different behavior [9].…”

Section: Introductionmentioning

confidence: 99%

On Nonlocal Nonlinear Elliptic Problems With the Fractional Laplacian

2019

Glasgow Math. J.

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In this paper, we study the existence of positive solutions to a semilinear nonlocal elliptic problem with the fractional α-Laplacian on Rn, 0 < α < n. We show that the problem has infinitely many positive solutions in $ {C^\tau}({R^n})\bigcap H_{loc}^{\alpha /2}({R^n}) $. Moreover, each of these solutions tends to some positive constant limit at infinity. We can extend our previous result about sub-elliptic problem to the nonlocal problem on Rn. We also show for α ∊ (0, 2) that in some cases, by the use of Hardy’s inequality, there is a nontrivial non-negative $ H_{loc}^{\alpha /2}({R^n}) $ weak solution to the problem $$ {( - \Delta )^{\alpha /2}}u(x) = K(x){u^p} \quad {\rm{ in}} \ {R^n}, $$ where K(x) = K(|x|) is a non-negative non-increasing continuous radial function in Rn and p > 1.

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On the Poisson equation of p-Laplacian and the nonlinear Hardy-type problems

2021

Z. Angew. Math. Phys.

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Boundedness of solutions to Ginzburg–Landau fractional Laplacian equation

Cited by 14 publications

References 11 publications

Uniform Lower Bound and Liouville Type Theorem for Fractional Lichnerowicz Equations

Uniform Lower Bound and Liouville Type Theorem for Fractional Lichnerowicz Equations

On Nonlocal Nonlinear Elliptic Problems With the Fractional Laplacian

On the Poisson equation of p-Laplacian and the nonlinear Hardy-type problems

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