THE EXISTENCE AND CONCENTRATION OF GROUND STATE SIGN-CHANGING SOLUTIONS FOR KIRCHHOFF-TYPE EQUATIONS WITH A STEEP POTENTIAL WELL∗

doi:10.1007/s10473-023-0419-6

Abstract

Abstract: In this paper, we consider the nonlinear Kirchhoff type equation with a steep potential well

$\begin{equation*} -\Big(a+b\int_{\mathbb{R}^{3}}|\nabla u|^{2}\,{\rm d}x\Big)\Delta u+\lambda V(x)u=f(u) \qquad {\rm in}\ \mathbb{R}^{3}, \end{equation*}$ where

$a,b>0$ are constants,

$\lambda$ is a positive parameter,

$V\in C(\mathbb{R}^{3}, \mathbb{R})$ is a steep potential well and the nonlinearity

$f\in C(\mathbb{R}, \mathbb{R})$ satisfies certain assumptions. By applying a sign-changing Nehari manifold combined with the method of constructing a sign-changing

$(PS)_{C}$ sequence, we obtain the existence of ground state sign-changing solutions with precisely two nodal domains when

$\lambda$ is large enough, and find that its energy is strictly larger than twice that of the ground state solutions. In addition, we also prove the concentration of ground state sign-changing solutions.

Key words: Kirchhoff-type equation, ground state sign-changing solutions, steep potential well

Menghui WU, Chunlei TANG. THE EXISTENCE AND CONCENTRATION OF GROUND STATE SIGN-CHANGING SOLUTIONS FOR KIRCHHOFF-TYPE EQUATIONS WITH A STEEP POTENTIAL WELL^∗[J].Acta mathematica scientia,Series B, 2023, 43(4): 1781-1799.

Trendmd

References

[1] Arosio A, Panizzi S. On the well-posedness of the Kirchhoff string. Trans Amer Math Soc, 1996, 348(1): 305-330
[2] Bartsch T, Pankov A, Wang Z Q. Nonlinear Schrödinger equations with steep potential well. Commun Contemp Math,2001, 3(4): 549-569
[3] Bartsch T, Wang Z Q. Existence and multiplicity results for superlinear elliptic problems on

$\mathbb{R}^{N}$ . Commun Partial Differential Equations, 1995, 20(10): 1725-1741
[4] Bellazzini J, Jeanjean L. On dipolar quantum gases in the unstable regime. SIAM J Math Anal, 2016, 48(3): 2028-2058
[5] Brezis H, Lieb E. A relation between pointwise convergence of functions and convergence functionals. Proc Amer Math Soc, 1983, 88(3): 486-490
[6] Cavalcanti M M, Domingos Cavalacanti V N, Soriano J A. Global existence and uniform decay rates for the Kirchhoff-Carrier equation with nonlinear dissipation. Adv Differential Equations, 2001, 6(6): 701-730
[7] Cerami G, Solimini S, Struwe M. Some existence results for superlinear elliptic boundary value problem involving critical exponents. J Funct Anal, 1986, 69(6): 289-306
[8] Chen J H, Tang X H, Cheng B T. Existence and concentration of ground state sign-changing solutions for Kirchhoff type equations with steep potential well and nonlinearity. Topol Methods Nonlinear Anal, 2018, 51(1): 111-133
[9] D'Ancona P, Spagnolo S. Global solvability for the degenerate Kirchhoff equation with real analytic data. Invent Math, 1992, 108(2): 247-262
[10] Ding Y H, Szulkin A. Bound states for semilinear Schrödinger equations with sign-changing potential. Calc Var Partial Differential Equations,2007, 29(3): 397-419
[11] Du M, Tian L, Wang J, Zhang F. Existence of ground state solutions for a super-biquadratic Kirchhoff-type equation with steep potential well. Appl Anal, 2016, 95(3): 627-645
[12] Feng R T, Tang C L. Ground state sign-changing solutions for a Kirchhoff equation with asymptotically 3-linear nonlinearity. Qual Theory Dyn Syst, 2021, 20(3): Art 91
[13] Hofer H. Variational and topological methods in partially odered Hilbert space. Math Ann, 1982, 261(4): 493-514
[14] Jia H F, Li G B. Multiplicity and concentration behaviour of positive solutions for Schrödinger-Kirchhoff type equations involving the

$p$ -Laplacian in

$\mathbb{R}^{N}$ . Acta Mathematica Scientia,2018, 38B(2): 391-418
[15] Jia H F, Luo X. Existence and concentrating behavior of solutions for Kirchhoff type equations with steep potential well. J Math Anal Appl, 2018, 467(2): 893-915
[16] Jiang Y, Zhou H S. Schrödinger-Poisson system with a steep well potential. J Differential Equations,2011, 251(3): 582-608
[17] Jin Q F. Multiple sign-changing solutions for nonlinear Schrödinger equations with potential well. Appl Anal,2020, 99(15): 2555-2570
[18] Kirchhoff G.Mechanik. Leipzig: Teubner, 1883
[19] Luo L P, Tang C L. 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
[20] Lushnikov P. Collaspse of Bose-Einstein codensates with dipole-dipole interactions. Phys Rev A, 2002, 66: 051601(R)
[21] Lushnikov P. Collaspse and stable self-trapping for Bose-Einstein condensates with

$1/r^{b}$ type attractive interatomic interaction potential. Phys Rev A, 2010, 82: 023615
[22] Mao A M, Zhang Z T. Sign-changing and multiple solutions of Kirchhoff type problems without the P.S. condition. Nonlinear Anal, 2009, 70(3): 1275-1287
[23] Miranda C. Unsservazione su un teorema di Brouwer. Boll Unione Mat Ital, 1940, 3(107): 5-7
[24] Pi H, Zeng Y. Existence results for the Kirchhoff type equation with a general nonlinear term. Acta Mathematica Scientia, 2022, 42B(5): 2063-2077
[25] Rabinowitz P H.Variational methods for nonlinear eigenvalue problems//Prodi G. Eigenvalues of Nonlinear Problems. Berlin: Springer, 2009: 139-195
[26] Shuai W. Sign-changing solutions for a class of Kirchhoff-type problem in bounded domains. J Differential Equations, 2015, 259(4): 1256-1274
[27] Sun J T, Wu T. Ground state solutions for an indefinite Kirchhoff type problem with steep potential well. J Differential Equations, 2014, 256(4): 1771-1792
[28] Tan J L, Li Y Y, Tang C L. The existence and concentration of ground state solutions for Chern-Simons-Schrödinger systems with a steep well potential. Acta Mathematica Scientia,2022, 42B(3): 1125-1140
[29] Wang Z P, Zhou H S. Positive solutions for nonlinear Schrödinger equations with deepening potential well. J Eur Math Soc,2009, 11(3): 545-573
[30] Willem M. Minimax Theorems.Boston: Birkháuser, 1996
[31] Xie Q L. Least energy nodal solution for Kirchhoff type problem with an asymptotically 4-linear nonlinearity. Appl Math Lett, 2020, 102: 106157
[32] Xie Q L, Ma S W. Existence and concentration of positive solutions for Kirchhoff-type problems with a steep well potential. J Math Anal Appl, 2015, 431(2): 1210-1223
[33] Zhang F, Du M. Existence and asymptotic behavior of positive solutions for Kirchhoff type problems with steep potential well. J Differential Equations, 2020, 269(11): 10085-10106
[34] Zhang J, Lou Z L. Existence and concentration behavior of solutions to Kirchhoff type equation with steep potential well and critical growth. J Math Phys, 2021, 62(1): 011506
[35] Zhang Z, Perera K. Sign changing solutions of Kirchhoff type problems via invariant sets of descent flow. J Math Anal Appl, 2006, 317(2): 456-463
[36] Zhao L, Liu H, Zhao F. Existence and concentration of solutions for the Schrödinger-Poisson equations with steep well potential. J Differential Equations,2013, 255(1): 1-23
[37] Zhong X J, Tang C L. The existence and nonexistence results of ground state nodal solutions for a Kirchhoff type problem. Commun Pure Appl Anal, 2017, 16(2): 611-627
[38] Zhong X J, Tang C L. Ground state sign-changing solutions for a Schrödinger-Poisson system with a critical nonlinearity in

$\mathbb{R}^3$ . Nonlinear Anal,2018, 39: 166-184

[1]	Fen HE, Zhen WANG, Tingting CHEN. THE SHOCK WAVES FOR A MIXED-TYPE SYSTEM FROM CHEMOTAXIS^∗[J]. Acta mathematica scientia,Series B, 2023, 43(4): 1717 -1734 .
[2]	Xia YE, Yanxia XU, Zejia WANG. THE ZERO LIMIT OF THERMAL DIFFUSIVITY FOR THE 2D DENSITY-DEPENDENT BOUSSINESQ EQUATIONS^∗[J]. Acta mathematica scientia,Series B, 2023, 43(4): 1800 -1818 .
[3]	Yansheng ZHONG, Yongqing LI. MULTIPLE POSITIVE SOLUTIONS TO A CLASS OF MODIFIED NONLINEAR SCHRÖDINGER EQUATION IN A HIGH DIMENSION^∗[J]. Acta mathematica scientia,Series B, 2023, 43(4): 1819 -1840 .
[4]	Yanyan CUI, Zunfeng LI, Yonghong XIE, Yuying QIAO. THE NONLINEAR BOUNDARY VALUE PROBLEM FOR k HOLOMORPHIC FUNCTIONS IN $\mathbb{C}^2$ ^∗[J]. Acta mathematica scientia,Series B, 2023, 43(4): 1571 -1586 .
[5]	Lili FAN, Meichen HOU. ASYMPTOTIC STABILITY OF SHOCK WAVES FOR THE OUTFLOW PROBLEM OF A HEAT-CONDUCTIVE IDEAL GAS WITHOUT VISCOSITY^∗[J]. Acta mathematica scientia,Series B, 2023, 43(4): 1735 -1766 .
[6]	Wei LUO, Zhaoyang YIN. LOCAL BIFURCATION OF STEADY ALMOST PERIODIC WATER WAVES WITH CONSTANT VORTICITY^∗[J]. Acta mathematica scientia,Series B, 2023, 43(4): 1633 -1644 .
[7]	Fei WU, Zejun WANG, Fangqi CHEN. THE GLOBAL EXISTENCE OF BV SOLUTIONS OF THE ISENTROPIC p-SYSTEM WITH LARGE INITIAL DATA^∗[J]. Acta mathematica scientia,Series B, 2023, 43(4): 1668 -1674 .
[8]	Liangzhen LEI, Yutao MA. LARGE DEVIATIONS FOR TOP EIGENVALUES OF ß -JACOBI ENSEMBLES AT SCALING TEMPERATURES^∗[J]. Acta mathematica scientia,Series B, 2023, 43(4): 1767 -1780 .
[9]	Jinxia CEN, Stanis law MIGÓRSKI, Emilio VILCHES, Shengda ZENG. TIME PERIODIC SOLUTIONS TO THE EVOLUTIONARY OSEEN MODEL FOR A GENERALIZED NEWTONIAN INCOMPRESSIBLE FLUID^∗[J]. Acta mathematica scientia,Series B, 2023, 43(4): 1645 -1667 .
[10]	Zhenhai Liu, Nikolaos S. Papageorgiou. SINGULAR DOUBLE PHASE EQUATIONS^*[J]. Acta mathematica scientia,Series B, 2023, 43(3): 1031 -1044 .

THE EXISTENCE AND CONCENTRATION OF GROUND STATE SIGN-CHANGING SOLUTIONS FOR KIRCHHOFF-TYPE EQUATIONS WITH A STEEP POTENTIAL WELL^∗

Abstract

Cite this article

share this article

References

Related Articles 1

Metrics

Comments

Recommended 10