Improved Sobolev inequalities and critical problems

Chen, Xiaoli; Jiang, Yang

doi:10.3934/cpaa.2020162

Cited by 4 publications

(8 citation statements)

References 14 publications

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“…Remark Theorems 1.2 and 1.3 extend Theorem 1.1 and Theorem 1.5 of Chen and Yang 10 to the product spaces

{\overset{H}{true}}^{s} false(ℝ^{n} false) \times {\overset{H}{true}}^{s} false(ℝ^{n} false)

and

D^{1, p} false(ℝ^{n} false) \times D^{1, p} false(ℝ^{n} false)

, respectively. Moreover, Theorems 1.2 and 1.3 are more general than Theorems 1 and 2 by Palatucci and Pisante 33 …”

Section: Introduction and Main Resultsmentioning

confidence: 58%

“…Recently, Chen and Yang 10 studied the doubly critical problems in

ℝ^{n}

{false(- normalΔ false)}^{s} u = \frac{false| u {false|}^{2_{s}^{*} false(α false) - 2} u}{false| x^{'} {false|}^{α}} + \frac{false| u {false|}^{2_{s}^{*} false(β false) - 2} u}{false| x^{'} {false|}^{β}},

where s ∈ (0, 1), 2 s < m < n ,

x = false(x^{'}, x^{''} false) \in ℝ^{m} \times ℝ^{n - m}

, and

0 < α, β < \min ({}, m, 2 n m s false/ n^{2} - 2 s false(n - m false))

. To search a nontrivial weak solutions of (), they revised () as the following: there exists C > 0 such that

{((), \int_{ℝ^{n}} \frac{false| u {false|}^{2_{s}^{*} false(α false)}}{false| x^{'} {false|}^{α}} d x)}^{\frac{1}{2_{s}^{*} false(α false)}} \leq C false‖ u {false‖}_{{\overset{H}{true}}^{s} false(ℝ^{n} false)}^{\frac{q}{2_{s}^{*} false(α false)}} (‖‖, \frac{u}{false| x^{'}})

…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

“…Remark 1.1. We mainly extend the results in Chen and Yang 10 which deal with the doubly critical equations to the doubly critical coupled systems. Theorem 1.1 indicates that we can relax the lower bound of 𝛾 1 , 𝛾 2 in system (1.1) provided 𝛼, 𝛽 > 0 since we need the extra condition 𝛾 1 , 𝛾 2 ≥ 0 if 𝛽 = 0.…”

mentioning

confidence: 83%

“…The method adopted in previous works 8,10,15,20,27 is not applicable to (1.1) since the appearance of the coupled terms (1.1). For this reason, we develop a new tool which is based on a weighted Morrey space; see Section 3.…”

mentioning

confidence: 99%

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On doubly critical coupled systems involving fractional Laplacian with partial singular weight

Yang

2021

Math Methods in App Sciences

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In this paper, we consider the doubly critical coupled systems involving fractional Laplacian in with partial singular weight: where s ∈ (0, 1), 0 ≤ α, β < 2s < n, 0 < m < n, , η1, η2 > 1, , γ1, γ2 < γH, and is some explicit constant. By establishing new embedding results involving partially weighted Morrey norms in the product space , we provide sufficient conditions under which a weak nontrivial solution of (0.1) exists via variational methods. We also extend these results to p‐Laplacian systems especially.

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“…Remark Theorems 1.2 and 1.3 extend Theorem 1.1 and Theorem 1.5 of Chen and Yang 10 to the product spaces

{\overset{H}{true}}^{s} false(ℝ^{n} false) \times {\overset{H}{true}}^{s} false(ℝ^{n} false)

and

D^{1, p} false(ℝ^{n} false) \times D^{1, p} false(ℝ^{n} false)

, respectively. Moreover, Theorems 1.2 and 1.3 are more general than Theorems 1 and 2 by Palatucci and Pisante 33 …”

Section: Introduction and Main Resultsmentioning

confidence: 58%

“…Recently, Chen and Yang 10 studied the doubly critical problems in

ℝ^{n}

{false(- normalΔ false)}^{s} u = \frac{false| u {false|}^{2_{s}^{*} false(α false) - 2} u}{false| x^{'} {false|}^{α}} + \frac{false| u {false|}^{2_{s}^{*} false(β false) - 2} u}{false| x^{'} {false|}^{β}},

where s ∈ (0, 1), 2 s < m < n ,

x = false(x^{'}, x^{''} false) \in ℝ^{m} \times ℝ^{n - m}

, and

0 < α, β < \min ({}, m, 2 n m s false/ n^{2} - 2 s false(n - m false))

. To search a nontrivial weak solutions of (), they revised () as the following: there exists C > 0 such that

{((), \int_{ℝ^{n}} \frac{false| u {false|}^{2_{s}^{*} false(α false)}}{false| x^{'} {false|}^{α}} d x)}^{\frac{1}{2_{s}^{*} false(α false)}} \leq C false‖ u {false‖}_{{\overset{H}{true}}^{s} false(ℝ^{n} false)}^{\frac{q}{2_{s}^{*} false(α false)}} (‖‖, \frac{u}{false| x^{'}})

…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

mentioning

confidence: 83%

mentioning

confidence: 99%

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On doubly critical coupled systems involving fractional Laplacian with partial singular weight

Yang

2021

Math Methods in App Sciences

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“…Regarding attainability, S q (R N ) is achieved when p < q < p * by the concentration-compactness principle while S p (R N ) and S p * (R N ) are never attained by the Pohožaev non-existence result, morever, S p * (R N ) = S. As a consequence of the above, from another point of view, the critical Sobolev exponent p * yields the sharp threshold for the existence and nonexistence of solutions to the related Euler-Lagrange equation. For further related results and refinements of optimal Sobolev embeddings see for instance [8,11] and references therein.…”

mentioning

confidence: 99%

Bounds for subcritical best Sobolev constants in W1, p

Du¹

2021

CPAA

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This paper aims at establishing fine bounds for subcritical best Sobolev constants of the embeddings<disp-formula> <label/> <tex-math id="FE1"> \begin{document}$ W_{0}^{1,p}(\Omega)\hookrightarrow L^{q}(\Omega),\quad 1\leq q< \begin{cases} \frac{Np}{N-p},& 1\leq p<N\\ \infty,& p = N \end{cases} $\end{document} </tex-math></disp-formula>where <inline-formula><tex-math id="M2">\begin{document}$ N\geq p\geq1 $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M3">\begin{document}$ \Omega $\end{document}</tex-math></inline-formula> is a bounded smooth domain in <inline-formula><tex-math id="M4">\begin{document}$ \mathbb{R}^{N} $\end{document}</tex-math></inline-formula> or the whole space. The Sobolev limiting case <inline-formula><tex-math id="M5">\begin{document}$ p = N $\end{document}</tex-math></inline-formula> is also covered by means of a limiting procedure.

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On a critical biharmonic system involving p-Laplacian and Hardy potential

Yang

2021

Applied Mathematics Letters

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Improved Sobolev inequalities and critical problems

Abstract: In this paper, we establish two refinement of Sobolev-Hardy inequalities in terms of Morrey spaces. Then, with help of these inequalities, we show the existence of nontrivial solutions for doubly critical problems in R N involving p-Laplacian

Cited by 4 publications

References 14 publications

On doubly critical coupled systems involving fractional Laplacian with partial singular weight

On doubly critical coupled systems involving fractional Laplacian with partial singular weight

Bounds for subcritical best Sobolev constants in <i>W</i><sup>1, <i>p</i></sup>

On a critical biharmonic system involving p-Laplacian and Hardy potential

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