Positive solutions for a Kirchhoff type problem with fast increasing weight and critical nonlinearity

Qian, Xiaotao; Chao, Wen

doi:10.14232/ejqtde.2019.1.27

Cited by 5 publications

(3 citation statements)

References 22 publications

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“…By using the variational method, there are many interesting results of positive solutions to (1.2), see e.g. [1,4,5,9,11] and the references therein.…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

New multiplicity of positive solutions for some class of nonlocal problems

Shi

Qian

2021

Bound Value Probl

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In this paper, we study the following nonlocal problem: $$ \textstyle\begin{cases} - (a-b \int _{\Omega } \vert \nabla u \vert ^{2}\,dx ) \Delta u= \lambda \vert u \vert ^{q-2}u, & x\in \Omega , \\ u=0, & x\in \partial \Omega , \end{cases} $$ { − ( a − b ∫ Ω | ∇ u | 2 d x ) Δ u = λ | u | q − 2 u , x ∈ Ω , u = 0 , x ∈ ∂ Ω , where Ω is a smooth bounded domain in $\mathbb{R}^{N}$ R N with $N\ge 3$ N ≥ 3 , $a,b>0$ a , b > 0 , $1< q<2$ 1 < q < 2 and $\lambda >0$ λ > 0 is a parameter. By virtue of the variational method and Nehari manifold, we prove the existence of multiple positive solutions for the nonlocal problem. As a co-product of our arguments, we also obtain the blow-up and the asymptotic behavior of these solutions as $b\searrow 0$ b ↘ 0 .

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“…By using the variational method, there are many interesting results of positive solutions to (1.2), see e.g. [1,4,5,9,11] and the references therein.…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

New multiplicity of positive solutions for some class of nonlocal problems

Shi

Qian

2021

Bound Value Probl

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“…Under different assumptions on f ðx, uÞ, many interesting results on the existence of solutions to (2) were obtained. We refer the interested readers to [2][3][4][5][6][7][8][9][10][11][12][13][14] and the references therein. However, we now face a new nonlocal term a − b Ð Ω j∇uj 2 dx, which is different from the well-known Kirchhoff type nonlocal term a + b Ð Ω j∇uj 2 dx.…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

Multiple Nontrivial Solutions for a Nonlocal Problem with Sublinear Nonlinearity

Shi

Qian

2021

Advances in Mathematical Physics

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In this paper, we study the following nonlocal problem − a − b ∫ Ω ∇ u 2 d x Δ u = λ u + f x u p − 2 u , x ∈ Ω , u = 0 , x ∈ ∂ Ω , where a , b > 0 are constants, 1 < p < 2 , λ > 0 , f ∈ L ∞ Ω is a positive function, and Ω is a smooth bounded domain in ℝ N with N ≥ 3 . By variational methods, we obtain a pair of nontrivial solutions for the considered problem provided f ∞ is small enough.

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“…There are also many studies for problem (1.2) on the whole space R N (N ≥ 3). For this case, we refer the interested readers to [10][11][12][13][14][15][16][17]. In particular, Chen [10] obtained the existence and multiplicity of positive solutions to the Kirchhoff problem with indefinite nonlinearity by using mountain pass theorem and minimization argument.…”

Section: Introduction and Main Reslutsmentioning

confidence: 99%