In this paper, we consider the following nonlinear Kirchhoff-type problem with sublinear perturbation and steep potential well
$ \begin{eqnarray*} \left \{\begin{array}{ll} -\Big(a+b\int_{\mathbb{R}^3}|\nabla u|^2dx\Big)\Delta u+\lambda V(x)u = f(x,u)+g(x)|u|^{q-2}u\ \ \mbox{in}\ \mathbb{R}^3,\\ \\ u\in H^1(\mathbb{R}^3), \\ \end{array} \right. \label{1} \end{eqnarray*} $
where $ a $ and $ b $ are positive constants, $ \lambda > 0 $ is a parameter, $ 1 < q < 2 $, the potential $ V\in C(\mathbb{R}^3, \mathbb{R}) $ and $ V^{-1}(0) $ has a nonempty interior. The functions $ f $ and $ g $ are assumed to obey a certain set of conditions. The existence of two nontrivial solutions are obtained by using variational methods. Furthermore, the concentration behavior of solutions as $ \lambda\rightarrow \infty $ is also explored.
Citation: Shuwen He, Xiaobo Wen. Existence and concentration of solutions for a Kirchhoff-type problem with sublinear perturbation and steep potential well[J]. AIMS Mathematics, 2023, 8(3): 6432-6446. doi: 10.3934/math.2023325
In this paper, we consider the following nonlinear Kirchhoff-type problem with sublinear perturbation and steep potential well
$ \begin{eqnarray*} \left \{\begin{array}{ll} -\Big(a+b\int_{\mathbb{R}^3}|\nabla u|^2dx\Big)\Delta u+\lambda V(x)u = f(x,u)+g(x)|u|^{q-2}u\ \ \mbox{in}\ \mathbb{R}^3,\\ \\ u\in H^1(\mathbb{R}^3), \\ \end{array} \right. \label{1} \end{eqnarray*} $
where $ a $ and $ b $ are positive constants, $ \lambda > 0 $ is a parameter, $ 1 < q < 2 $, the potential $ V\in C(\mathbb{R}^3, \mathbb{R}) $ and $ V^{-1}(0) $ has a nonempty interior. The functions $ f $ and $ g $ are assumed to obey a certain set of conditions. The existence of two nontrivial solutions are obtained by using variational methods. Furthermore, the concentration behavior of solutions as $ \lambda\rightarrow \infty $ is also explored.
| [1] |
T. Bartsch, Z. Q. Wang, Existence and multiplicity results for superlinear elliptic problems on $\mathbb{R}^N$, Commun. Part. Differ. Eq., 20 (1995), 1725–1741. https://doi.org/10.1080/03605309508821149 doi: 10.1080/03605309508821149
|
| [2] |
T. Bartsch, A. Pankov, Z. Q. Wang, Nonlinear Schrödinger equations with steep potential well, Commun. Contemp. Math., 3 (2001), 549–569. https://doi.org/10.1142/S0219199701000494 doi: 10.1142/S0219199701000494
|
| [3] |
Y. H. Ding, A. Szulkin, Bound states for semilinear Schrödinger equations with sign-changing potential, Calc. Var. Partial Dif., 29 (2007), 397–419. https://doi.org/10.1007/s00526-006-0071-8 doi: 10.1007/s00526-006-0071-8
|
| [4] | G. Kirchhoff, Mechanik, Leipzig: Teubner, 1883. |
| [5] | A. Arosio, S. Panizzi, On the well-posedness of the Kirchhoff string, Trans. Am. Math. Soc., 348 (1996), 305–330. |
| [6] |
M. Chipot, B. Lovat, Some remarks on non local elliptic and parabolic problems, Nonlinear Anal., 30 (1997), 4619–4627. https://doi.org/10.1016/S0362-546X(97)00169-7 doi: 10.1016/S0362-546X(97)00169-7
|
| [7] |
Y. H. Li, F. Y. Li, Existence of a positive solution to Kirchhoff type problems without compactness conditions, J. Differ. Equ., 253 (2012), 2285–2294. https://doi.org/10.1016/j.jde.2012.05.017 doi: 10.1016/j.jde.2012.05.017
|
| [8] |
G. B. Li, H. Y. Ye, Existence of positive ground state solutions for the nonlinear Kirchhoff type equations in $\mathbb{R}^3$, J. Differ. Equ., 257 (2014), 566–600. https://doi.org/10.1016/j.jde.2014.04.011 doi: 10.1016/j.jde.2014.04.011
|
| [9] |
Z. P. Wang, H. S. Zhou, Positive solutions for nonlinear Schrödinger equations with deepening potential well, J. Eur. Math. Soc., 11 (2009), 545–573. https://doi.org/10.4171/JEMS/160 doi: 10.4171/JEMS/160
|
| [10] |
H. Y. Ye, Positive high energy solution for Kirchhoff equation in $\mathbb{R}^3$ with superlinear nonlinearities via Nehari-Pohozaev manifold, Discret. Contin. Dyn. Syst., 35 (2015), 3857–3877. https://doi.org/10.3934/dcds.2015.35.3857 doi: 10.3934/dcds.2015.35.3857
|
| [11] |
Z. S. Liu, S. J. Guo, Positive solutions for asymptotically linear Schrödinger-Kirchhoff-type equations, Math. Meth. Appl. Sci., 37 (2014), 571–580. https://doi.org/10.1002/mma.2815 doi: 10.1002/mma.2815
|
| [12] |
Y. S. Jiang, H. S. Zhou, Schrödinger-Poisson system with steep potential well, J. Differ. Equ., 251 (2011), 582–608. http://dx.doi.org/10.1016/j.jde.2011.05.006 doi: 10.1016/j.jde.2011.05.006
|
| [13] |
L. G. Zhao, H. D. Liu, F. K. Zhao, Existence and concentration of solutions for the Schrödinger-Poisson equations with steep well potential, J. Differ. Equ., 255 (2013), 1–23. https://doi.org/10.1016/j.jde.2013.03.005 doi: 10.1016/j.jde.2013.03.005
|
| [14] |
H. Zhang, F. B. Zhang, Ground states for the nonlinear Kirchhoff type problems, J. Math. Anal. Appl., 423 (2015), 1671–1692. http://dx.doi.org/10.1016/j.jmaa.2014.10.062 doi: 10.1016/j.jmaa.2014.10.062
|
| [15] |
Z. J. Guo, Ground states for Kirchhoff equations without compact condition, J. Differ. Equ., 259 (2015), 2884–2902. http://dx.doi.org/10.1016/j.jde.2015.04.005 doi: 10.1016/j.jde.2015.04.005
|
| [16] |
C. T. Ledesma, Existence and concentration of solutions for a non-linear fractional Schrödinger equation with steep potential well, Commun. Pur. Appl. Anal., 15 (2016), 535–547. https://doi.org/10.3934/cpaa.2016.15.535 doi: 10.3934/cpaa.2016.15.535
|
| [17] |
Y. H. Li, Q. Geng, The existence of nontrivial solution to a class of nonlinear Kirchhoff equations without any growth and Ambrosetti-Rabinowitz, Appl. Math. Lett., 96 (2019), 153–158. http://dx.doi.org/10.1016/j.aml.2019.04.027 doi: 10.1016/j.aml.2019.04.027
|
| [18] |
M. Du, L. X. Tian, J. Wang, F. B. Zhang, Existence of ground state solutions for a super-biquadratic Kirchhoff-type equation with steep potential well, Appl. Anal., 95 (2016), 627–645. https://doi.org/10.1080/00036811.2015.1022312 doi: 10.1080/00036811.2015.1022312
|
| [19] |
D. Q. Zhang, G. Q. Chai, W. M. Liu, Existence and concentration of solutions for the nonlinear Kirchhoff type equations with steep well potential, Bound. Value Probl., 2017 (2017), 142. https://doi.org/10.1186/s13661-017-0875-9 doi: 10.1186/s13661-017-0875-9
|
| [20] |
F. B. Zhang, M. Du, Existence and asymptotic behavior of positive solutions for Kirchhoff type problems with steep potential well, J. Differ. Equ., 269 (2020), 10085–10106. https://doi.org/10.1016/j.jde.2020.07.013 doi: 10.1016/j.jde.2020.07.013
|
| [21] |
J. T. Sun, T. F. Wu, Steep potential well may help Kirchhoff type equations to generate multiple solutions, Nonlinear Anal., 190 (2020), 111609. https://doi.org/10.1016/j.na.2019.111609 doi: 10.1016/j.na.2019.111609
|
| [22] |
L. Zhou, C. X. Zhu, Existence and asymptotic behavior of ground state solutions to Kirchhoff-type equations of general convolution nonlinearity with a steep potential well, Mathematics, 10 (2022), 812. https://doi.org/10.3390/math10050812 doi: 10.3390/math10050812
|
| [23] |
D. Choudhuri, Existence and Hölder regularity of infinitely many solutions to a $p$-Kirchhoff type problem involving a singular nonlinearity without the Ambrosetti-Rabinowitz (AR) condition, Z. Angew. Math. Phys., 72 (2021), 36. https://doi.org/10.1007/s00033-020-01464-9 doi: 10.1007/s00033-020-01464-9
|
| [24] | M. Schechter, A variation of the mountain pass lemma and applications, J. London Math. Soc., 44 (1991), 491–502. https://doi.org/101112/jlms/s2-44.3.491 |
| [25] |
J. T. Sun, T. F. Wu, Multiplicity and concentration of homoclinic solutions for some second order Hamiltonian systems, Nonlinear Anal., 114 (2015), 105–115. https://doi.org/10.1016/j.na.2014.11.009 doi: 10.1016/j.na.2014.11.009
|
| [26] | M. Willem, Minimax theorems, Boston: Birkhäuser, 1996. |
| [27] | R. K. Giri, D. Choudhuri, S. Pradhan, Existence and concentration of solutions for a class of elliptic PDEs involving $p$-biharmonic operator, Mat. Vestn., 70 (2018), 147–154. |
Shuwen He, Xiaobo Wen. Existence and concentration of solutions for a Kirchhoff-type problem with sublinear perturbation and steep potential well[J]. AIMS Mathematics, 2023, 8(3): 6432-6446. doi: 10.3934/math.2023325
DownLoad: