THE EXISTENCE AND LOCAL UNIQUENESS OF MULTI-PEAK POSITIVE SOLUTIONS TO A CLASS OF KIRCHHOFF EQUATION

doi:10.1007/s10473-020-0107-y

Acta mathematica scientia,Series B ›› 2020, Vol. 40 ›› Issue (1): 90-112.doi: 10.1007/s10473-020-0107-y

• Articles • Previous Articles Next Articles

THE EXISTENCE AND LOCAL UNIQUENESS OF MULTI-PEAK POSITIVE SOLUTIONS TO A CLASS OF KIRCHHOFF EQUATION

Gongbao LI, Yahui NIU

Hubei Key Laboratory of Mathematical Sciences and School of Mathematics and Statistics, Central China Normal University, Wuhan 430079, China

Received:2018-11-20 Revised:2019-03-27 Online:2020-02-25 Published:2020-04-14
Contact: Gongbao LI E-mail:ligb@mail.ccnu.edu.cn
Supported by:
This work was supported by Natural Science Foundation of China (11771166), Hubei Key Laboratory of Mathematical Sciences and Program for Changjiang Scholars and Innovative Research Team in University #IRT_17R46. Niu was financially supported by funding for basic research business in Central Universities (innovative funding projects) (2018CXZZ090).

Abstract

Abstract: In the present paper, we consider the nonlocal Kirchhoff problem
- (ε²a + εb∫_R³ |▽u|²)△u + u = Q(x)u^p, u > 0 in R³,
where a, b > 0, 1 < p < 5 and ε > 0 is a parameter. Under some assumptions on Q(x), we show the existence and local uniqueness of positive multi-peak solutions by LyapunovSchmidt reduction method and the local Pohozaev identity method, respectly.

Key words: Kirchhoff equations, multi-peak positive solutions, local uniqueness, local Pohozaev identity, Lyapunov-Schmidt reduction

CLC Number:

35A01

Gongbao LI, Yahui NIU. THE EXISTENCE AND LOCAL UNIQUENESS OF MULTI-PEAK POSITIVE SOLUTIONS TO A CLASS OF KIRCHHOFF EQUATION[J].Acta mathematica scientia,Series B, 2020, 40(1): 90-112.

Trendmd

References

[1] Arosio A, Panizzi S. On the well-posedness of the Kirchhoff string. Trans Amer Math Soc, 1996, 348:305-330
[2] Bartsch T, Peng S. Semiclassical symmetric Schrödinger equations:existence of solutions concentrating simultaneously on several spheres. Z Angew Math Phys, 2007, 58:778-804
[3] Berestycki H, Lions P L. Nonlinear scalar field equations. I. Existence of a ground state. Arch Rational Mech Anal, 1983, 82:313-345
[4] Bernstein S. Sur une classe d'équations fonctionelles aux dérivées partielles. Bull Acad Sci URSS Sér, 1940, 4:17-26
[5] Cao D, Li S, Luo P. Uniqueness of positive bound states with multi-bump for nonlinear Schrödinger equations. Calc Var Partial Differential Equations, 2015, 54:4037-4063
[6] Cao D, Noussair E S. Multiplicity of positive and nodal solutions for nonlinear elliptic solutions in R^N. Ann Inst H Poincaré Anal Non Linéaire, 1996, 13:567-588
[7] Cao D, Noussair E S, Yan S. Existence and uniqueness results on Single-Peaked solutions of a semilinear problem. Ann Inst H Poincaré Anal Non Linéaire, 1998, 15:73-111
[8] Cao D, Noussair E S, Yan S. Solutions with multiple peaks for nonlinear elliptic equations. Proc Royal Soc Edinburgh, 1999, 129:235-264
[9] Cao D, Peng S. Semi-classical bound states for Schrödinger equations with potentials vanishing or unbounded at infinity. Comm Partial Differential Equations, 2009, 34:1566-1591
[10] Chang S, Gustafson S, Nakanishi K, Tsai T P. Spectra of linearized operators for NLS solitary waves. SIAM J Math Anal, 2007, 39:1070-1111
[11] D'Ancona P, Spagnolo S. Global solvability for the degenerate Kirchhoff equation with real analytic data. Invent Math, 1992, 108:247-262
[12] Deng X, Lin C, Yan S. On the prescribed scalar curvature problem in R^N, local uniqueness and periodicity. J Math Pures Appl, 2015, 104:1013-1044
[13] Figueiredo G M, Ikoma N, Santos Júnior João R. Existence and concentration result for the Kirchhoff type equations with general nonlinearities. Arch Rational Mech Anal, 2014, 213:931-979
[14] Grossi M. On the number of single-peak solutions of the nonlinear Schrödinger equations. Ann Inst H Poincaré Anal Non Linéaire, 2002, 19:261-280
[15] He Y. Concentrating bounded states for a class of singularly perturbed Kirchhoff type equations with a general nonlinearity. J Differential Equations, 2016, 261:6178-6220
[16] He Y, Li G. Standing waves for a class of Kirchhoff type problems in R³ involving critical Sobolev exponents. Calc Var Partial Differential Equations, 2015, 54:3067-3106
[17] He Y, Li G, Peng S. Concentrating bound states for Kirchhoff type problems in R³ involving critical Sobolev exponents. Adv Nonlinear Stud, 2014, 14:483-510
[18] He X, Zou W. Existence and concentration behavior of positive solutions for a Kirchhoff equation in R³. J Differential Equations, 2012, 252:1813-1834
[19] Hu T, Shuai W. Multi-peak solutions to Kirchhoff equations in R³ with general nonlinearity. J Differential Equations, 2018, 265(8):3587-3617
[20] Kirchhoff G. Mechanik. Teubner, Leipzig, 1883
[21] Li G, Luo P, Peng S, Wang C, Xiang C. Uniqueness and Nondegeneracy of positive solutions to Kirchhoff equations and its applications in singular perturbation problems. Preprint at arXiv:1703.05459[math.AP]
[22] Lions J L. On some questions in boundary value problems of mathematical physics//Contemporary Development in Continuum Mechanics and Partial Differential Equations. North-Holland Math Stud. Amsterdam, New York:North-Holland, 1978:284-346
[23] Luo P, Peng S, Wang C, Xiang C. Multi-peak positive solutions to a class of Kirchhoff equations. Proc Royal Soc Edinburgh, A, 2019, 149(4):1097-1122
[24] Noussair E S, Yan S. On positive multi-peak solutions of a nonlinear elliptic problem. J London Math Soc, 200, 62:213-227
[25] Pohozaev S I. A certain class of quasilinear hyperbolic equations. Mat Sb (NS), 1975, 96:152-166(in Russian)
[26] Rabinowitz P H. On a class of nonlinear Schrödinger equations. Z Angew Math Phys, 1992, 43:270-291
[27] Wang J, Tian L, Xu J, Zhang F. Multiplicity and concentration of positive solutions for a Kirchhoff type problem with critical growth. J Differential Equations, 2012, 253:2314-2351

THE EXISTENCE AND LOCAL UNIQUENESS OF MULTI-PEAK POSITIVE SOLUTIONS TO A CLASS OF KIRCHHOFF EQUATION

PDF (PC)

Abstract

Cite this article

share this article

References

Related Articles 1

Metrics

Comments

Recommended 0