Ground states for nonlinear Kirchhoff equations with critical growth

He, Xiaoming; Zou, Wenming

doi:10.1007/s10231-012-0286-6

Cited by 80 publications

(30 citation statements)

References 44 publications

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“…引进了一种抽象的泛函分析框架后, 方程 (1.3) 得到人们的广泛关注和研究, 详细结果参见文献 [3][4][5][6][7]. 另一方面, 自 Brézis 和 Nirenberg 在文献 [8] 中提出了关于临界增长半线性椭圆方程的问题后, 许多学者都热衷于研究临界增长椭圆方程 (参见文献 [9][10][11][12][13][14][15][16]). 文献 [9][10][11][12][13] 通过变分的方法得到了众多临界增长椭圆方程的解的存在性的结果.…”

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The existence of nontrivial solutions for $\boldsymbol{p}$-Kirchhoff type equations with critical exponent in $\boldsymbol{\mathbb{R}^{N}}$

Li¹,

Yahui²

2017

Sci. Sin.-Math.

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Section: )unclassified

“…Figueiredo 和 Santos Junior [14] 利用亏格理论得到了临界增长 Kirchhoff 型方程的多重非平凡解的存在性结果. He 和 Zou [15] 用变分的方法得到了一类临界增长…”

Section: )unclassified

The existence of nontrivial solutions for $\boldsymbol{p}$-Kirchhoff type equations with critical exponent in $\boldsymbol{\mathbb{R}^{N}}$

Li¹,

Yahui²

2017

Sci. Sin.-Math.

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“…In recent decades, the problem of Kirchhoff type has been extensively studied by many researchers, and many interesting results are obtained in . They mainly focused on the Kirchhoff‐type problem with critical nonlinear term in bounded domain or whole space.…”

Section: Introductionmentioning

confidence: 99%

“…However, Kirchhoff's model received great attention only after Lions [4] proposed an abstract framework for the model (1.2). In recent decades, the problem of Kirchhoff type has been extensively studied by many researchers, and many interesting results are obtained in [5][6][7][8][9][10][11][12][13][14][15][16][17]. They mainly focused on the Kirchhoff-type problem with critical nonlinear term in bounded domain or whole space.…”

Section: Introductionmentioning

confidence: 99%

Multiplicity of solutions for Kirchhoff type equations involving critical Sobolev exponents in high dimension

Yao

2016

Math Methods in App Sciences

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In this paper, we study the following Kirchhoff-type elliptic problemwhere R N .N 4/ is a bounded domain with smooth boundary @ , a, b, , > 0 and 1 < q < 2 D 2N=.N 2/. When N D 4, we obtain that there is a ground state solution to the problem for q 2 .2, 4/ by using a variational methods constrained on the Nehari manifold and also show the problem possesses infinitely many negative energy solutions for q 2 .1, 2/ by applying usual Krasnoselskii genus theory. In addition, we admit that there is a positive solution to the equations for N 5 under some suitable conditions.

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“…has been widely studied by many authors under different conditions on V and f . We refer the readers to [1,4,7,8,11,12,13,17,21,22,23,24] for related results.…”

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confidence: 99%

Bounded state solutions of Kirchhoff type problems with a critical exponent in high dimension

Xie¹,

Yu²

2019

Communications on Pure &Amp; Applied Analysis

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In the present paper, we consider the following Kirchhoff type problemwhere a is a positive constant, λ is a positive parameter, V ∈ L N 2 (R N ) is a given nonnegative function and 2 * is the critical exponent. The existence of bounded state solutions for Kirchhoff type problem with critical exponents in the whole R N (N ≥ 5) has never been considered so far. We obtain sufficient conditions on the existence of bounded state solutions in high dimension N ≥ 4, and especially it is the fist time to consider the case when N ≥ 5 in the literature. 129 130 QILIN XIE AND JIANSHE YU (i) N = 4 and λ > S −2 ; (ii) N ≥ 5 and λ > Λ 0 ; then problem (SK * ) has no nontrivial solution.Remark 1.6. Now, we have partially solved the existence of nontrivial solutions for the Kirchhoff type problem (SK * ). When N = 4, it follows from Theorem 1.1 and Theorem 1.5 that problem has at least one nontrivial solution for 0 < λ < S −2 ; and problem has no nontrivial solution for λ > S −2 . We do not know what will happen for λ = S −2 . So does the case when N ≥ 5 and λ = Λ 0 . Because of the limitation of our methods and technique, Theorem 1.3 only obtains nontrivial solutions for problem (SK * ) with Λ 0 − µ < λ < Λ 0 . A natural and interesting question is whether we can establish multiplicity theorems for problem (SK * ) with small positive parameter λ. We guess that the answer to the above problem is positive. The main difficulties when we investigate problem (SK * ) are that for Kirchhoff type problems with critical exponents, it is not easy to verify a range where Palais-Smale condition holds, especially when the energy level may be negative. This paper is organized as follows. In Section 2, we give the variational setting for our problem and investigate the solutions for the limit equation of (SK * ). After that, we prove that problem (SK * ) can not be solved by minimization on Nehari manifold. At the end of Section 2, we prove our Theorem 1.5 indirectly. In Section 3, by a description of Palais-Smale sequences, we obtain a local compactness result Proposition 3.2 with N = 4 and a global compactness result Proposition 3.3 with N ≥ 5. By a standard argument, we obtain the existence of the bounded state solutions in Section 4. In Section 5, we prove some necessary lemmas, which have been used in our main context. 2.Preliminaries. In this section, we give the variational setting for problem (SK * ) and some basic information on the limit equation of (SK * ). The main works in this paper are considered in the Hilbert space D 1,2 (R N ) = {u ∈ L 2 * (R N ) : QILIN XIE AND JIANSHE YU

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Ground states for nonlinear Kirchhoff equations with critical growth

Cited by 80 publications

References 44 publications

The existence of nontrivial solutions for $\boldsymbol{p}$-Kirchhoff type equations with critical exponent in $\boldsymbol{\mathbb{R}^{N}}$

The existence of nontrivial solutions for $\boldsymbol{p}$-Kirchhoff type equations with critical exponent in $\boldsymbol{\mathbb{R}^{N}}$

Multiplicity of solutions for Kirchhoff type equations involving critical Sobolev exponents in high dimension

Bounded state solutions of Kirchhoff type problems with a critical exponent in high dimension

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