一类Kirchhoff型椭圆方程的山路解和基态解

Abstract

Abstract:

This paper mainly considers a kind of nonlinear elliptic equation with a Kirchhoff type nonlocal term

$\begin{equation} -\left(a+b\int_{\mathbb{R}^3}\left| \nabla u\right|^2 \right)\Delta u+V(x)u=Q(x)\left| u\right|^{p-1}u, \quad x\in\mathbb{R}^3, \end{equation}$(0.1)

where$ a,b>0 $are constants,$ p\in(1,5) $,$ V(x) $and$ Q(x) $are$ L^\infty(\mathbb{R}^3) $functions. It is known that if we apply the mountain pass lemma directly to obtain solution (i.e., mountain pass solution) to the equation (0.1), we must require$ 3\le p<5 $because of the appearance of nonlocal terms. When$ p\in(1,3) $, the fundamental difficulty in applying the mountain pass lemma is that we are unable to verify the boundedness of the (PS) sequence. To overcome this difficulty, when$ Q(x)\equiv 1 $, paper [Acta Math Sci, 2025, 45B(2): 385-400] introduced a new technique to demonstrate the equation (0.1) exists a mountain pass solution for all$ p\in(1,5) $, and discussed the relationship between the mountain pass solution and the ground state solution obtained. The purpose of this paper is to extend the results of [Acta Math Sci, 2025, 45B(2): 385-400] to the general case$ Q(x)\not\equiv 1 $.

Key words: Kirchhoff type equation, elliptic equation, Mountain-pass solution, ground state

CLC Number:

O175.23

Wang Hanyi, Huang Shiyu, Xiang Jianlin. Mountain-pass Solution and Ground State Solution for a Kirchhoff Type Elliptic Equation[J].Acta mathematica scientia,Series A, 2025, 45(4): 1041-1057.

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References 25

[1]	Antonio A, Paul H R. Dual variational methods in critical point theory and applications. J Funct Anal, 1973, 14(4): 349-381
[2]	Berestycki H, Lions P L. Nonlinear scalar field equations. I. Arch Rational Mech Anal, 1983, 82(4): 313-346
[3]	Deng S, Luo W. Multiplicity and concentration of solutions for a fractional magnetic Kirchhoff equation with competing potentials. Ann Henri Poincaré, 2024, 25(7): 3499-3528
[4]	Guo Z. Ground states for Kirchhoff equations without compact condition. J Differential Equations, 2015, 2.9(7): 2884-2902
[5]	He X, Zou W. Infinitely many positive solutions for Kirchhoff-type problems. Nonlinear Anal, 2009, 70(3): 1407-1414
[6]	He X, Zou W. Existence and concentration behavior of positive solutions for a Kirchhoff equation in $ \mathbb{R}^3$. J Differential Equations, 2012, 2.2(2): 1813-1834
[7]	Jeanjean L. On the existence of bounded Palais-Smale sequences and application to a Landesman-Lazer type problem set on $ \mathbb{R}^N $. Proc Roy Soc Edinburgh Sect A, 1999, 1.9(4): 787-809
[8]	Kirchhoff G. Mechanik. Leipzig: Teubner, 1883
[9]	Li A, Su J. Existence and multiplicity of solutions for Kirchhoff-type equation with radial potentials in $ \mathbb{R}^3$. Z Angew Math Phys, 2015, 66(6): 3147-3158
[10]	Li G, Niu Y. The existence and local uniqueness of multi-peak positive solutions to a class of Kirchhoff equation. Acta Math Sci, 2020, 40B(1): 90-112
[11]	Li G, Ye H. Existence of positive ground state solutions for the nonlinear Kirchhoff type equations in $\mathbb{R}^3$. J Differential Equations, 2014, 2.7(2): 566-600
[12]	Liu Y, Zhao L. Least energy solutions of the Schrödinger-Kirchhoff equation with linearly bounded nonlinearities. Qual Theory Dyn Syst, 2024, 23(1): Article 4
[13]	Liu Z, Guo S. Existence of positive ground state solutions for Kirchhoff type problems. Nonlinear Anal, 2015, 1.0: 1-13
[14]	Luo Li, Tang C. Existence and concentration of ground state solutions for critical Kirchhoff -type equation with steep potential well. Complex Var Elliptic Equ, 2022, 67(7): 1756-1771
[15]	She L, Sun X, Duan Y. Multiple positive solutions for a class of Kirchhoff type equations in $ \Bbb R^N $. Bound Value Probl, 2018, 20.8: 1-13
[16]	Shu Z, Zhang J. Normalized multi-bump solutions of nonlinear Kirchhoff equations. AIMS Math, 2024, 9(6): 16790-16809
[17]	Sun D. Ground state solutions of Schrödinger-Kirchhoff equations with potentials vanishing at infinity. J Funct Spaces, 2023, 20.3(1): 8829268
[18]	Wei C, Li A. Multiplicity of nontrivial solutions for Kirchhoff type equations with zero mass and a critical term. Nonlinear Anal Front Math China, 2022, 17(5): 813-828
[19]	Weng L, Zhang X, Zhou H. Mountain-pass solution for a Kirchhoff type elliptic equation. Acta Math Sci, 2025, 45B(2): 385-400
[20]	Wu D, Suo H, Lei J. Multiple positive solutions for Kirchhoff-type problems involving supercritical and critical terms. Qual Theory Dyn Syst, 2024, 23(3): Article 139
[21]	Wu M, Tang C. The existence and concentration of ground state sign-changing solutions for Kirchhoff-type equations with a steep potential well. Acta Math Sci, 2023, 43B(4): 1781-1799
[22]	Wu X. Existence of nontrivial solutions and high energy solutions for Schrödinger-Kirchhoff-type equations in $ {\bf R}^N $. Nonlinear Anal Real World Appl, 2011, 12(2): 1278-1287
[23]	Xu H. Existence of positive solutions for the nonlinear Kirchhoff type equations in $ \mathbb{R}^n $. J Math Anal Appl, 2020, 4.2(2): 123593
[24]	Ye H. Positive high energy solution for Kirchhoff equation in $ \mathbb{R}^3 $ with superlinear nonlinearities via Nehari-Pohožaev manifold. Discrete Contin Dyn Syst, 2015, 35(8): 3857-3877
[25]	Zhang J, Zhang X. Multi-bump solutions to Kirchhoff type equations in the plane with the steep potential well vanishing at infinity. J Math Anal Appl, 2024, 5.0(2): 128669

Mountain-pass Solution and Ground State Solution for a Kirchhoff Type Elliptic Equation

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