Bound state solutions of Kirchhoff type problems with critical exponent

Xie, Qian; Ma, Shiwang; Zhang, Xu

doi:10.1016/j.jde.2016.03.028

Cited by 61 publications

(28 citation statements)

References 19 publications

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“…The Kirchhoff type problem has been extensively studied. For examples, to our best knowledge, Ma and Muñoz Rivera were the first result by the variational method; a quasilinear elliptic equation of Kirchhoff type was considered in Alves et al; the Kirchhoff type problem with critical exponent was first investigated by Alves et al; for the uniqueness result, see Anello and Liao et al; for the multiplicity of solutions for a superlinear Kirchhoff type equations with critical Sobolev exponent in

R^{N}

, see Li and Liao; He and Zou considered the infinitely many solutions of the Kirchhoff type problem; the sign‐changing solution was studied by Mao and Zhang, Tang and Cheng and Zhang and Perera; for bound state solutions, see Xie et al; Naimen was the first that considered the Kirchhoff type problem in dimension four; the Kirchhoff type problems was considered by the Yang index in Perera and Zhang. …”

Section: Introduction and Main Resultsmentioning

confidence: 99%

Existence and multiplicity of positive solutions for a class of singular Kirchhoff type problems with sign‐changing potential

Tang

Liao

2018

Math Methods in App Sciences

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R^{N}

Section: Introduction and Main Resultsmentioning

confidence: 99%

Existence and multiplicity of positive solutions for a class of singular Kirchhoff type problems with sign‐changing potential

Tang

Liao

2018

Math Methods in App Sciences

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“…This, combined with (2.9), gives that the unique positive solution ϕ − δ,y is the least energy solution. Actually, there is no essential difference between N = 4 in the above situation and N = 3 in Proposition 2.1 of Xie et al [25]. The constant c a,λ,4 − has also been given in Naimen [18].…”

Section: Qilin Xie and Jianshe Yumentioning

confidence: 86%

“…Remark 1.2. It should be mentioned that the basic idea in the proof follows from those in Benci and Cerami [2] and Xie et al [25]. For problem (SK * ) with a = 1, λ = 0 in 4 dimension, Benci and Cerami [2] obtained a positive solution under the following assumptions: V (x) ≥ 0 for any x ∈ R 4 , there exist two positive constants p 1 < 2 < p 2 such that…”

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confidence: 92%

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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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“…where M(x, t) ∈ C(Ω × R + , R + ) is a positive continuous function and Ω ⊂ R 3 is a smooth domain (see previous studies [1][2][3][4][5][6][7][8][9] and the references therein). Such problem is often referred to be nonlocal because of the presence of the Kirchhoff term M ( x, ∫ Ω |∇u| 2 dx ) , which implies that (2) is no longer a pointwise identity.…”

Section: Introductionmentioning

confidence: 99%

On the existence of least energy solution for Kirchhoff equation in R3

2020

Math Meth Appl Sci

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Bound state solutions of Kirchhoff type problems with critical exponent

Cited by 61 publications

References 19 publications

Existence and multiplicity of positive solutions for a class of singular Kirchhoff type problems with sign‐changing potential

Existence and multiplicity of positive solutions for a class of singular Kirchhoff type problems with sign‐changing potential

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

On the existence of least energy solution for Kirchhoff equation in R3

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