In this paper, we study the following Kirchhoff-Carrier type equation
$ -\left(a+bM\left(|\nabla u|_{2}, |u|_{\tau}\right)\right)\Delta u-\lambda u = |u|^{p-2}u, \quad \ {\rm in}\ \mathbb{R}^{3}, $
where $ a, \ b > 0 $ are constants, $ \lambda\in \mathbb{R}, \ p\in (2, 6) $. By using a minimax procedure, we obtain infinitely solutions $ (v^{b}_{n}, \lambda_{n}) $ with $ v^{b}_{n} $ having a prescribed $ L^{2} $-norm. Moreover, we give a convergence property of $ v_{n}^{b} $ as $ b\rightarrow 0^{+} $.
Citation: Jie Yang, Haibo Chen. Normalized solutions for Kirchhoff-Carrier type equation[J]. AIMS Mathematics, 2023, 8(9): 21622-21635. doi: 10.3934/math.20231102
In this paper, we study the following Kirchhoff-Carrier type equation
$ -\left(a+bM\left(|\nabla u|_{2}, |u|_{\tau}\right)\right)\Delta u-\lambda u = |u|^{p-2}u, \quad \ {\rm in}\ \mathbb{R}^{3}, $
where $ a, \ b > 0 $ are constants, $ \lambda\in \mathbb{R}, \ p\in (2, 6) $. By using a minimax procedure, we obtain infinitely solutions $ (v^{b}_{n}, \lambda_{n}) $ with $ v^{b}_{n} $ having a prescribed $ L^{2} $-norm. Moreover, we give a convergence property of $ v_{n}^{b} $ as $ b\rightarrow 0^{+} $.
| [1] |
C. O. Alves, D. P. Covei, Existence of solution for a class of nonlocall elliptic problem via sub-supersolution method, Nonlinear Anal.-Real, 23 (2015), 1–8. https://doi.org/10.1016/j.nonrwa.2014.11.003 doi: 10.1016/j.nonrwa.2014.11.003
|
| [2] | H. Berestycki, P. L. Lions, Nonlinear scalar field equations, II existence of infinitely many solutions, Arch. Ration. Mech. An., 82 (1983), 347–375. Available from: https://link.springer.com/article/10.1007/BF00250556. |
| [3] | T. Bartsch, S. De Valeriola, Normalized solutions of nonlinear Schrödinger equations, Arch. Math., 100 (2013), 75–83. Available from: https://arXiv.org/abs/1209.0950v1. |
| [4] |
G. Che, H. Chen, Existence and concentration result for Kirchhoff equations with critical exponent and hartree nonlinearity, J. Appl. Anal. Comput., 10 (2020), 2121–2144. https://doi.org/10.11948/20190338 doi: 10.11948/20190338
|
| [5] |
G. Che, H. Chen, Existence and multiplicity of systems of Kirchhoff-type equations with general potentials, Math. Method. Appl. Sci., 40 (2019), 775–785. https://doi.org/10.1002/mma.4007 doi: 10.1002/mma.4007
|
| [6] |
M. Chipot, B. Lovat, Some remarks on nonlocal elliptic and parabolic problems, Nonlinear Anal.-Theor., 30 (1997), 4619–4627. https://doi.org/10.1016/S0362-546X(97)00169-7 doi: 10.1016/S0362-546X(97)00169-7
|
| [7] | G. F. Carrier, On the nonlinear vibration problem of the elastic string, Q. J. Appl. Math., 3 (1945), 157–165. |
| [8] | L. Jeanjean, Existence of solutions with prescribed norm for semilinear elliptic equations, Nonlinear Anal., 28 (1997), 1633–1659. |
| [9] |
F. Jin, B. Yan, The sign-changing solutions for nonlinear elliptic problem with Carrier type, J. Math. Anal. Appl., 487 (2020), 124002. https://doi.org/10.1016/j.jmaa.2020.124002 doi: 10.1016/j.jmaa.2020.124002
|
| [10] | G. Kirchhoff, Mechanik, Teubner, Leipzig, 1883. |
| [11] |
L. Kong, H. Chen, Normalized solutions for nonlinear Kirchhoff type equations in high dimensions, Electron. Res. Arch., 30 (2022), 1282–1295. https://doi.org/10.3934/era.2022067 doi: 10.3934/era.2022067
|
| [12] |
J. L. Lions, On some questions in boundary value problems of mathematical physics, North-Holland Math. Stud., 30 (1978), 284–346. https://doi.org/10.1016/S0304-0208(08)70870-3 doi: 10.1016/S0304-0208(08)70870-3
|
| [13] |
T. Luo, Multiplicity of normalized solutions for a class of nonlinear Schrödinger-Poisson-Slater equations, J. Math. Anal. Appl., 416 (2014), 195–204. https://doi.org/10.1016/j.jmaa.2014.02.038 doi: 10.1016/j.jmaa.2014.02.038
|
| [14] |
Z. Liu, Y. Lou, J. Zhang, A perturbation approach to studying sign-changing solutions of Kirchhoff equations with a general nonlinearity, Ann. Mat. Pur. Appl., 201 (2022), 1229–1255. https://doi.org/10.1007/s10231-021-011 doi: 10.1007/s10231-021-011
|
| [15] |
Z. Liu, H. Luo, J. Zhang, Existence and multiplicity of bound state solutions to a Kirchhoff type equation with a general nonlinearity, J. Geom. Anal., 32 (2022), 125. https://doi.org/10.1007/s12220-021-00849-0 doi: 10.1007/s12220-021-00849-0
|
| [16] |
X. Luo, Q. Wang, Existence and asymptotic behavior of high energy normalized solutions for the Kirchhoff type equations in $ \mathbb{R}^{3}$, Nonlinear Anal.-Real, 33 (2017), 19–32. https://doi.org/10.1016/j.nonrwa.2016.06.001 doi: 10.1016/j.nonrwa.2016.06.001
|
| [17] |
S. Yao, H. Chen, V. Radulescu, J. Sun, Normalized solutions for lower critical Choquard equations with critical sobolev perturbation, Siam. J. Math. Anal., 54 (2022), 3696–3723. https://doi.org/10.1137/21M1463136 doi: 10.1137/21M1463136
|
| [18] |
Y. Su, Z. Feng, Fractional sobolev embedding with radial potential, J. Differ. Equations, 340 (2022), 1–44. https://doi.org/10.1016/j.jde.2022.08.030 doi: 10.1016/j.jde.2022.08.030
|
| [19] |
Y. Su, S. Liu, Nehari-Pohozaev-type ground state solutions of Kirchhoff-type equation with singular potential and critical exponent, Can. Math. Bull., 65 (2022), 473–495. https://doi.org/10.4153/S0008439521000436 doi: 10.4153/S0008439521000436
|
| [20] |
J. Sun, K. Wang, T. Wu, On indefinite Kirchhoff-type equations under the combined effect of linear and superlinear terms, J. Math. Phys., 62 (2021), 031505. https://doi.org/10.1063/5.0030427 doi: 10.1063/5.0030427
|
| [21] |
X. Xu, B. Qin, A variational approach for Kirchhoff-Carrier type non-local equation boundary value problems, J. Math. Anal. Appl., 508 (2022), 125885. https://doi.org/10.1016/j.jmaa.2021.125885 doi: 10.1016/j.jmaa.2021.125885
|
| [22] |
W. Xie, H. Chen, Existence and multiplicity of normalized solutions for a class of Schrodinger-Poisson equations with general nonlinearities, Math. Method. Appl. Sci., 43 (2020), 3602–3616. https://doi.org/10.1002/mma.6140 doi: 10.1002/mma.6140
|
Jie Yang, Haibo Chen. Normalized solutions for Kirchhoff-Carrier type equation[J]. AIMS Mathematics, 2023, 8(9): 21622-21635. doi: 10.3934/math.20231102
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