Existence of positive solution of the equation 
                
                  
                
                $$(-\Delta )^{s}u+a(x)u=|u|^{2^{*}_{s}-2}u$$
                
                  
                    
                      
                        
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Correia, Jeziel N.; Figueiredo, Giovany M.

doi:10.1007/s00526-019-1502-7

Cited by 12 publications

(9 citation statements)

References 12 publications

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“…Instead of using the technique described in Alves [2], they constructed a barycenter function and combined it with the Quantitative deformation lemma to demonstrate the existence of positive high-energy solutions for the quasilinear problem (1.2) involving the critical exponent and Neumann boundary conditions. Furthermore, we would like to mention that a similar result for p = 2 can be found in earlier studies [4,5]. Besides, elliptic problems with the p-Laplacian operator and multiple critical nonlinearities in the sense of the Hardy-Sobolev embedding have received significant attention from scholars in the past decade.…”

Section: Introduction and Main Resultssupporting

confidence: 75%

Multiple high‐energy solutions for an elliptic system with critical Hardy–Sobolev nonlinearity

Ri,

2024

Math Methods in App Sciences

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This paper discusses the existence of multiple high‐energy solutions for a ‐Laplacian system involving critical Hardy‐Sobolev nonlinearity in . Considering that the “double” lack of compactness in the system is caused by the unboundedness of and the presence of the critical Hardy–Sobolev exponent, we demonstrate the version to of Struwe's classical global compactness result for double ‐Laplace operator. In virtue of the quantitative deformation lemma, a barycenter function, and the Brouwer degree theory, the existence of multiple high‐energy solutions is established. The results of this paper extend and complement the recent work.

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Section: Introduction and Main Resultssupporting

confidence: 75%

Multiple high‐energy solutions for an elliptic system with critical Hardy–Sobolev nonlinearity

Ri,

2024

Math Methods in App Sciences

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“…As far as we know, when f (x, u) = |u| 2 * −2 u, Benci and Cerami [12] first considered the existence of bound state solutions under the assumptions that V ≥ 0 and ||V || L N 2 is small enough. Later, Correia and Figueiredo [14] considered the nonlocal characteristic of fractional Laplacian operator and extend the results in [12] to fractional Schrödinger equation. Recently, Guo and Li [23] extend the results in [14] to two distinct bound state solutions by using quantitative deformation lemma and Brouwer degree theory, where they assumed that…”

Section: Introduction and Main Resultsmentioning

confidence: 87%

“…Later, Correia and Figueiredo [14] considered the nonlocal characteristic of fractional Laplacian operator and extend the results in [12] to fractional Schrödinger equation. Recently, Guo and Li [23] extend the results in [14] to two distinct bound state solutions by using quantitative deformation lemma and Brouwer degree theory, where they assumed that…”

Section: Introduction and Main Resultsmentioning

confidence: 87%

Existence of bound state solutions for p-Laplacian equation with critical growth

Dai

Liu

2023

Preprint

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Note: Please see pdf for full abstract with equations. In this paper, we study the following p-Laplacian problem −Δpu + V(x)|u|p−2u = |u|p*−2u + α(x)|u|q−2u, x ∈ RN, u ∈ W1,p(RN), where Δp := div(|∇u|p−2∇u) is the p-Laplacian operator, V is a nonnegative function, N > p2, 1 < p < q < p* and p* = Np/N−p is the Sobolev critical exponent. Under appropriate assumptions on V and a, we prove that this problem has at least two distinct bound state solutions.

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“…In that article, they proved that if N ≥ 3 and a N/2 is small enough, then the problem (1.2) has at least one positive solution. After this pioneering work, several other authors studied problems related to (1.2); see for example [2,7,8,11,13,19,21,23] and references therein. Correia and Figueiredo [13] studied the following version of problem (1.2) for the fractional Laplacian,…”

Section: Introductionmentioning

confidence: 99%

Existence of positive solutions for fractional Laplacian systems with critical growth

Correia¹,

Oliveira²

2022

ejde

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In this article, we show the existence of positive solution to the nonlocal system $$\displaylines{ (-\Delta)^s u +a(x)u=\frac{1}{2_s^*} H_u(u,v) \quad \hbox{in }\mathbb{R}^N,\cr (-\Delta)^s v +b(x)v=\frac{1}{2_s^*}H_v(u,v) \quad \hbox{in } \mathbb{R}^N,\cr u,v>0 \quad \text{in } \mathbb{R}^N,\cr u,v\in \mathcal{D}^{s,2 }(\mathbb{R}^N). }$$ We also prove a global compactness result for the associated energy functionalsimilar to that due to Struwe in [26]. The basic tools are some information from a limit systemwith $a(x) = b(x) = 0$, a variant of the Lion's principle of concentration and compactness for fractional systems, and Brouwer degree theory.

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Existence of positive solution of the equation $$(-\Delta )^{s}u+a(x)u=|u|^{2^{*}_{s}-2}u$$ ( - Δ ) s u +

Cited by 12 publications

References 12 publications

Multiple high‐energy solutions for an elliptic system with critical Hardy–Sobolev nonlinearity

Multiple high‐energy solutions for an elliptic system with critical Hardy–Sobolev nonlinearity

Existence of bound state solutions for p-Laplacian equation with critical growth

Existence of positive solutions for fractional Laplacian systems with critical growth

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