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Multiplicity of nontrivial solutions for Kirchhoff type equations with zero mass and a critical term

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Frontiers of Mathematics in China 2022年第5期17卷 813-828页

作者： Chongqing WEI Anran LI School of Mathematical Sciences Shanxi UniversityTaiyuan 030006China

In this paper,a class of Kirchhoff type equations in R^(N)(N≥3)with zero mass and a critical term is *** some appropriate conditions,the existence of multiple solutions is obtained by using variational methods and a variant of symmetric mountain pass *** Second Concentration Compactness lemma is used to overcome the lack of compactness in critical *** to the usual Kirchhoff-type problems,we only require the nonlinearity to satisfy the classical superquadratic condition(Ambrosetti-Rabinowitz condition).

关键词： Kirchhoff type equations with a critical term variational methods symmetric mountain pass theorem Second Concentration Compactness lemma

MULTIPLICITY RESULTS FOR FOURTH ORDER ELLIPTIC EQUATIONS OF KIRCHHOFF-TYPE

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Acta Mathematica Scientia 2015年第5期35卷 1067-1076页

作者：许丽萍陈海波 Department of Mathematics and Statistics Henan University of Science and Technology School of Mathematics and Statistics Central South University

In this paper, we concern with the following fourth order elliptic equations of Kirchhoff type {Δ^2u-（a＋bfR^3｜↓△u｜^2dx）△u＋V（x）u=f（x,u）,x∈R^3, u∈H^2（R3）,where a, b 〉 0 are constants and the primitive of the nonlinearity f is of superlinear growth near infinity in u and is also allowed to be sign-changing. By using variational methods, we establish the existence and multiplicity of solutions. Our conditions weaken the Ambrosetti- Rabinowitz type condition.

关键词： fourth order elliptic equations of Kirchhoff type symmetric mountain pass theorem variational methods

Existence theorem for a Class of Nonlinear Fourth-order Schrodinger-Kirchhoff-Type Equations

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Journal of Partial Differential Equations 2017年第2期30卷 146-164页

作者： TANG Shiqiang CHEN Peng LIU Xiaochun School of Mathematics and Statistics Wuhan University Wuhan 430072 China

This paper is concerned with the existence of nontrivial solutions for the following fourth-order equations of Kirchhoff type {Δ^2u-（a＋b∫R^N｜▽u｜^2dx）Δu＋λv（x）u=f（x,u）,x∈R^N,u∈H^2（R^N）,where a,b are positive constants, λ≥ 1 is a parameter, and the nonlinearity f is either superlinear or sublinear at infinity in u. With the help of the variational methods, we obtain the existence and multiplicity results in the working spaces.

关键词： Fourth-order elliptic equations symmetric mountain pass theorem Morse theory.

Existence of Infinitely Many High Energy Solutions for a Fourth-Order Kirchhoff Type Elliptic Equation in R<sup>3</sup>

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Journal of Applied Mathematics and Physics 2020年第8期8卷 1550-1559页

作者： Ting Xiao Canlin Gan Qiongfen Zhang College of Science Guilin University of Technology Guilin China

In this paper, we consider the following fourth-order equation of Kirchhoff type where a, b > 0 are constants, 3 p V ∈ C (R3, R);Δ2: = Δ (Δ) is the biharmonic operator. By using symmetric mountain pass theorem and variational methods, we prove that the above equation admits infinitely many high energy solutions under some sufficient assumptions on V (x). We make some assumptions on the potential V (x) to solve the difficulty of lack of compactness of the Sobolev embedding. Our results improve some related results in the literature.

关键词： Fourth-Order Kirchhoff Type Elliptic Equation Infinitely Many Solutions symmetric mountain pass theorem Variational Methods