[1]
|
C. J. Amick and J. F. Toland, Uniqueness and related analytic properties for the Benjamin-Ono equation--a nonlinear Neumann problem in the plane, Acta Math., 1991, 167, 107-126. doi: 10.1007/BF02392447
CrossRef Google Scholar
|
[2]
|
A. Arosio and S. Panizzi, On the well-posedness of the Kirchhoff string, Trans. Amer. Math. Soc., 1996, 348(1), 305-330. doi: 10.1090/S0002-9947-96-01532-2
CrossRef Google Scholar
|
[3]
|
G. Autuori, A. Fiscella and P. Pucci, Stationary Kirchhoff problems involving a fractional operator and a critical nonlinearity, Nonlinear Anal., 2015, 125, 669-714.
Google Scholar
|
[4]
|
M. Caponi and P. Pucci, Existence theorems for entire solutions of stationary Kirchhoff fractional $p$-Laplacian equations, Annali di Matematica, 2016, 195, 2099-2129. doi: 10.1007/s10231-016-0555-x
CrossRef Google Scholar
|
[5]
|
C. V. Coffman, Uniqueness of the ground state solution for $\Delta u-u+u^{3} = 0$ and a variational characterization of other solutions, Arch. Rational Mech. Anal., 1972, 46, 81-95. doi: 10.1007/BF00250684
CrossRef Google Scholar
|
[6]
|
P. D'Ancona and S. Spagnolo, Global solvability for the degenerate Kirchhoff equation with real analytic data, Invent. Math., 1992, 108(2), 247-262.
Google Scholar
|
[7]
|
Y. Deng, S. Peng and W. Shuai, Existence and asymptotic behavior of nodal solutions for the kirchhoff-type problems in $\mathbb{R} ^{3}$, J. Funct. Anal., 2015, 269, 3500-3527. doi: 10.1016/j.jfa.2015.09.012
CrossRef Google Scholar
|
[8]
|
G. M. Figueiredo, N. Ikoma and J. R. S. Júnior, Existence and concentration result for the Kirchhoff type equations with general nonlinearities, Arch. Ration. Mech. Anal., 2014, 213, 931-979. doi: 10.1007/s00205-014-0747-8
CrossRef Google Scholar
|
[9]
|
A. Fiscella, R. Servadei and E. Valdinoci, Asymptotically linear problems driven by fractional Laplacian operators, Math. Methods in Appl. Sci., 2015, 38(16), 3551-3563. doi: 10.1002/mma.3438
CrossRef Google Scholar
|
[10]
|
A. Fiscella and E. Valdinoci, A critical Kirchhoff type problem involving a nonlocal operator, Nonlinear Anal., 2014, 94, 156-170. doi: 10.1016/j.na.2013.08.011
CrossRef Google Scholar
|
[11]
|
R. L. Frank and E. Lenzmann, Uniqueness of non-linear ground states for fractional Laplacians in $\mathbb{R}$, Acta Math., 2013, 210, 261-318. doi: 10.1007/s11511-013-0095-9
CrossRef Google Scholar
|
[12]
|
R. L. Frank, E. Lenzmann and L. Silvestre, Uniqueness of radial solutions for the fractional laplacian, Comm. on Pure and Appl. Math., 2016, 69, 1671-1726. doi: 10.1002/cpa.21591
CrossRef Google Scholar
|
[13]
|
Z. Guo, Ground states for Kirchhoff equations without compact condition, J. Diff. Eqs., 2015, 259, 2884-2902. doi: 10.1016/j.jde.2015.04.005
CrossRef Google Scholar
|
[14]
|
J. Guo, S. Ma and G. Zhang, Solutions of the autonomous Kirchhoff type equations in $\mathbb{R}^{N}$, Appl. Math. Lett., 2018, 82, 14-17. doi: 10.1016/j.aml.2018.02.011
CrossRef Google Scholar
|
[15]
|
X. He and W. Zou, Existence and concentration behavior of positive solutions for a Kirchhoff equation in $\mathbb{R}^{3}$, J. Diff. Eqs., 2012, 252, 1813-1834. doi: 10.1016/j.jde.2011.08.035
CrossRef Google Scholar
|
[16]
|
C. E. Kenig, Y. Martel, and L. Robbiano, Local well-posedness and blow-up in the energy space for a class of $L^{2}$ critical dispersion generalized Benjamin-Ono equations, Ann. Inst. H. Poincar é Anal. Non Linéaire, 2011, 28, 853-887. doi: 10.1016/j.anihpc.2011.06.005
CrossRef Google Scholar
|
[17]
|
G. Kirchhoff, Mechanik, Teubner, Leipzig, 1883.
Google Scholar
|
[18]
|
M. K. Kwong, Uniqueness of positive solutions of $\Delta u-u+u^{p} = 0$ in $\mathbb{R}^{n}$, Arch. Rational Mech. Anal., 1989, 105, 243-266. doi: 10.1007/BF00251502
CrossRef Google Scholar
|
[19]
|
G. Li, S. Peng and C. Xiang, Uniqueness and nondegeneracy of positive solutions to a class of Kirchhoff equations in $ \mathbb{R}^{3}$, arXiv preprint arXiv: 1610.07807, 2016-arxiv.org.
Google Scholar
|
[20]
|
G. Li and C. Xiang, Nondegeneracy of positive solutions to a Kirchhoff problem with critical Sobolev growth, Appl. Math. Lett., 2018, 86, 270-275. doi: 10.1016/j.aml.2018.07.010
CrossRef Google Scholar
|
[21]
|
T. Lin and J. Wei, Ground state of $N$ coupled nonlinear Schrödinger Equations in $\mathbb{R}^{n}$, $n\leq 3$, Comm. Math. Phys., 2005, 255, 629-653. doi: 10.1007/s00220-005-1313-x
CrossRef Google Scholar
|
[22]
|
J. L. Lions, On some questions in boundary value problems of mathematical physics, in: Contemporary Developments in Continuum Mechanics and Partial Differential Equations, Proceedings of International Symposium, Inst. Mat., Univ. Fed. Rio de Janeiro, Rio de Janeiro, 1977, in: North-Holland Math. Stud., vol. 30, North-Holland, Amsterdam, 1978, 284-346.
Google Scholar
|
[23]
|
S. Lu, An autonomous Kirchhoff-type equation with general nonlinearity in $\mathbb{R}^{N}$, Nonlinear Anal. RWA, 2017, 34, 233-249. doi: 10.1016/j.nonrwa.2016.09.003
CrossRef Google Scholar
|
[24]
|
K. McLeod, Uniqueness of positive radial solutions of $ \Delta u+f\left(u\right) = 0$ in $\mathbb{R}^{n}$. Ⅱ, Trans. Amer. Math. Soc., 1993, 339, 495-505.
Google Scholar
|
[25]
|
V. Moroz and J. Van Schaftingen, A guide to the Choquard equation, J. Fixed Point Theory Appl., 2017, 19(1), 773-813. doi: 10.1007/s11784-016-0373-1
CrossRef Google Scholar
|
[26]
|
E. Di Nezza, G. Palatucci and E. Valdinoci, Hitchhiker's guide to the fractional Sobolev spaces, Bull. Sci. Math., 2012, 136, 521-573. doi: 10.1016/j.bulsci.2011.12.004
CrossRef Google Scholar
|
[27]
|
P. Pucci, M. Xiang and B. Zhang, Existence and multiplicity of entire solutions for fractional $p$-Kirchhoff equations, Adv. Nonlinear Anal., 2016, 5(1), 27-55.
Google Scholar
|
[28]
|
J. V. Schaftingen and J. Xia, Standing waves with a critical frequency for nonlinear Choquard equations, Nonlinear Anal., 2017, 161, 87-107. doi: 10.1016/j.na.2017.05.014
CrossRef Google Scholar
|
[29]
|
L. Shao and H. Chen, Existence and concentration result for a class of fractional Kirchhoff equations with Hartree-type nonlinearities and steep potential well, C. R. Acad. Sci. Paris, Ser. I, 2018, 356, 489-497. doi: 10.1016/j.crma.2018.03.008
CrossRef Google Scholar
|
[30]
|
M. I. Weinstein, Existence and dynamic stability of solitary wave solutions of equations arising in long wave propagation, Comm. Partial Diff. Eqs., 1987, 12, 1133-1173. doi: 10.1080/03605308708820522
CrossRef Google Scholar
|
[31]
|
K. Wu and F. Zhou, Nodal solutions for a Kirchhoff type problem in $\mathbb{R}^{N}$, Appl. Math. Lett., 2019, 88, 58-63. doi: 10.1016/j.aml.2018.08.008
CrossRef Google Scholar
|
[32]
|
M. Xiang and F. Wang, Fractional Schrö dinger-Poisson-Kirchhoff type systems involving critical nonlinearities, Nonlinear Anal., 2017, 164, 1-26. doi: 10.1016/j.na.2017.07.012
CrossRef Google Scholar
|
[33]
|
Q. Xie, Singular perturbed Kirchhoff type problem with critical exponent, J. Math. Anal. Appl., 2017, 454, 144-180. doi: 10.1016/j.jmaa.2017.04.048
CrossRef Google Scholar
|
[34]
|
Q. Xie, S. Ma and X. Zhang, Infinitely many bound state solutions of Kirchhoff problem in $\mathbb{R}^{3}$, Nonlinear Anal. RWA, 2016, 29, 80-97. doi: 10.1016/j.nonrwa.2015.10.010
CrossRef Google Scholar
|
[35]
|
Q. Xie, S. Ma and X. Zhang, Bound state solutions of Kirchhoff type problems with critical exponent, J. Diff. Eqs., 2016, 261(2), 890-924. doi: 10.1016/j.jde.2016.03.028
CrossRef Google Scholar
|
[36]
|
J. Zhang, Z. Lou, Y. Jia and W. Shao, Ground state of Kirchhoff type fractional Schrödinger equations with critical growth, J. Math. Anal. Appl., 2018, 462, 57-83. doi: 10.1016/j.jmaa.2018.01.060
CrossRef Google Scholar
|