In this paper, we studied the existence of normalized solutions to the following Kirchhoff equation with a perturbation:
$ \left\{ \begin{aligned} &-\left(a+b\int _{\mathbb{R}^{N}}\left | \nabla u \right|^{2} dx\right)\Delta u+\lambda u = |u|^{p-2} u+h(x)\left |u\right |^{q-2}u, \quad \text{ in } \mathbb{R}^{N}, \\ &\int_{\mathbb{R}^{N}}\left|u\right|^{2}dx = c, \quad u \in H^{1}(\mathbb{R}^{N}), \end{aligned} \right. $
where $ 1\le N\le 3, a, b, c > 0, 1\leq q < 2 $, $ \lambda \in \mathbb{R} $. We treated three cases:
(i) When $ 2 < p < 2+\frac{4}{N}, h(x)\ge0 $, we obtained the existence of a global constraint minimizer.
(ii) When $ 2+\frac{8}{N} < p < 2^{*}, h(x)\ge0 $, we proved the existence of a mountain pass solution.
(iii) When $ 2+\frac{8}{N} < p < 2^{*}, h(x)\leq0 $, we established the existence of a bound state solution.
[1] | G. Kirchhoff, Mechanik, Teubner, Leipzig, 1883. |
[2] | A. Arosio, S. Panizzi, On the well-posedness of the Kirchhoff string, Trans. Amer. Math. Soc., 348 (1996), 305–330. https://doi.org/10.1090/S0002-9947-96-01532-2 doi: 10.1090/S0002-9947-96-01532-2 |
[3] | M. Cavalcanti, V. Cavalcanti, J. Soriano, Global existence and uniform decay rates for the Kirchhoff-Carrier equation with nonlinear dissipation, Adv. Differential Equations, 6 (2001), 701–730. https://doi.org/10.57262/ade/1357140586 doi: 10.57262/ade/1357140586 |
[4] | P. D'Ancona, S. Spagnolo, Global solvability for the degenerate Kirchhoff equation with real analytic data, Invent. Math., 108 (1992), 247–262. https://doi.org/10.1007/BF02100605 doi: 10.1007/BF02100605 |
[5] | J. L. Lions, On some questions in boundary value problems of mathmatical 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 |
[6] | G. Figueiredo, J. R. Santos, Multiplicity and concentration behavior of positive solutions for a Schrödinger-Kirchhoff type problem via penalization method, ESAIM Control Optim. Calc. Var., 20 (2014), 389–415. https://doi.org/10.1051/cocv/2013068 doi: 10.1051/cocv/2013068 |
[7] | Z. J. Guo, Ground states for Kirchhoff equations without compact condition, J. Differential Equations, 259 (2015), 2884–2902. https://doi.org/10.1016/j.jde.2015.04.005 doi: 10.1016/j.jde.2015.04.005 |
[8] | X. M. He, W. M. Zou, Ground states for nonlinear kirchhoff equations with critical growth, Ann. Mat. Pura Appl., 193 (2014), 473–500. https://doi.org/10.1007/s10231-012-0286-6 doi: 10.1007/s10231-012-0286-6 |
[9] | A. M. Mao, Z. T. Zhang, Sign-changing and multiple solutions of Kirchhoff type problems without the P.S. condition, Nonlinear Anal., 70 (2009), 1275–1287. https://doi.org/10.1016/j.na.2008.02.011 doi: 10.1016/j.na.2008.02.011 |
[10] | C. A. Stuart, Bifurcation for Dirichlet problems without eigenvalues, Proc. London Math. Soc., 45 (1982), 169–192. https://doi.org/10.1112/plms/s3-45.1.169 doi: 10.1112/plms/s3-45.1.169 |
[11] | L. Jeanjean, Existence of solutions with prescribed norm for semilinear elliptic equations, Nonlinear Anal., 28 (1997), 1633–1659. https://doi.org/10.1016/S0362-546X(96)00021-1 doi: 10.1016/S0362-546X(96)00021-1 |
[12] | T. Bartsch, N. Soave, A natural constraint approach to normalized solutions of nonlinear Schrödinger equations and systems, J. Funct. Anal., 272 (2017), 4998–5037. https://doi.org/10.1016/j.jfa.2017.01.025 doi: 10.1016/j.jfa.2017.01.025 |
[13] | T. Bartsch, N. Soave, Multiple normalized solutions for a competing system of Schrödinger equations, Calc. Var. Partial Differential Equations, 58 (2019). https://doi.org/10.1007/s00526-018-1476-x |
[14] | N. Soave, Normalized ground states for the NLS equation with combined nonlinearities, J. Differential Equations, 269 (2020), 6941–6987. https://doi.org/10.1016/j.jde.2020.05.016 doi: 10.1016/j.jde.2020.05.016 |
[15] | S. T. Chen, X. H. Tang, Normalized solutions for nonautonomous Schrödinger equations on a suitable manifold, J. Geom. Anal., 30 (2020), 1637–1660. https://doi.org/10.1007/s12220-019-00274-4 doi: 10.1007/s12220-019-00274-4 |
[16] | Z. Chen, W. M. Zou, Existence of Normalized Positive Solutions for a Class of Nonhomogeneous Elliptic Equations, J. Geom. Anal., 33 (2023). https://doi.org/10.1007/s12220-023-01199-9 |
[17] | C. O. Alves, On existence of multiple normalized solutions to a class of elliptic problems in whole $\mathbb{R}^{N}$, Z. Angew. Math. Phys., 73 (2022). https://doi.org/10.1007/s00033-022-01741-9 |
[18] | D. M. Cao, E. S. Noussair, Multiplicity of positive and nodal solutions for nonlinear elliptic problems in $\mathbb{R}^{N}$, Ann. Inst. H. Poincar$\acute{e}$ C Anal. Non Lin$\acute{e}$aire., 13 (1996), 567–588. https://doi.org/10.1016/S0294-1449(16)30115-9 |
[19] | P. H. Zhang, Z. Q. Han, Normalized ground states for Kirchhoff equations in $\mathbb{R}^{3}$ with a critical nonlinearity, J. Math. Phys., 63 (2022). https://doi.org/10.1063/5.0067520 |
[20] | M. I. Weinstein, Nonlinear Schrödinger equations and sharp interpolation estimates, Commun. Math. Phys., 87 (1983), 567–576. https://doi.org/10.1007/BF01208265 doi: 10.1007/BF01208265 |
[21] | H. Y. Ye, The sharp existence of constrained minimizers for a class of nonlinear Kirchhoff equations, Math. Methods Appl. Sci., 38 (2015), 2663–2679. https://doi.org/10.1002/mma.3247 doi: 10.1002/mma.3247 |
[22] | H. Y. Ye, The mass concentration phenomenon for $L^{2}$-critical constrained problems related to Kirchhoff equations, Z. Angew. Math. Phys., 67 (2016). https://doi.org/10.1007/s00033-016-0624-4 |
[23] | X. Y. Zeng, Y. M. Zhang, Existence and uniqueness of normalized solutions for the Kirchhoff equation, Appl. Math. Lett., 74 (2017), 52–59. https://doi.org/10.1016/j.aml.2017.05.012 doi: 10.1016/j.aml.2017.05.012 |
[24] | G. B. Li, X. Luo, T. Yang, Normalized solutions to a class of Kirchhoff equations with Sobolev critical exponent, Ann. Fenn. Math., 47 (2022), 895–925. https://doi.org/10.54330/afm.120247 doi: 10.54330/afm.120247 |
[25] | P. C. Carri$\tilde{a}$o, O. H. Miyagaki, A. Vicente, Normalized solutions of Kirchhoff equations with critical and subcritical nonlinearities: the defocusing case, Partial Differ. Equ. Appl., 3 (2022). https://doi.org/10.1007/s42985-022-00201-3 |
[26] | H. Y. Ye, The existence of normalized solutions for $L^{2}$-critical constrained problems related to Kirchhoff equations, Z. Angew. Math. Phys., 66 (2015), 1483–1497. https://doi.org/10.1007/s00033-014-0474-x doi: 10.1007/s00033-014-0474-x |
[27] | S. T. Chen, V. R$\check{a}$dulescu, X. H. Tang, Normalized Solutions of nonautonomous Kirchhoff equations: sub- and super-critical cases, Appl. Math. Optim., 84 (2021), 773–806. https://doi.org/10.1007/s00245-020-09661-8 doi: 10.1007/s00245-020-09661-8 |
[28] | L. Cai, F. B. Zhang, Normalized Solutions of Mass Supercritical Kirchhoff Equation with Potential, J. Geom. Anal., 33 (2023). https://doi.org/10.1007/s12220-022-01148-y |
[29] | A. Fiscella, A. Pinamonti, Existence and multiplicity results for Kirchhoff-type problems on a double-phase setting, Mediterr. J. Math., 20 (2023). https://doi.org/10.1007/s00009-022-02245-6 |
[30] | A. Fiscella, G. Marino, A. Pinamonti, S. Verzellesi, Multiple solutions for nonlinear boundary value problems of Kirchhoff type on a double phase setting, Rev. Mat. Complut., 37 (2024), 205–236. https://doi.org/10.1007/s13163-022-00453-y doi: 10.1007/s13163-022-00453-y |
[31] | W. H. Xie, H. B. Chen, Existence and multiplicity of normalized solutions for the nonlinear Kirchhoff type problems, Comput. Math. Appl., 76 (2018), 579–591. https://doi.org/10.1016/j.camwa.2018.04.038 doi: 10.1016/j.camwa.2018.04.038 |
[32] | T. Bartsch, R. Molle, M. Rizzi, M. Verzini, Normalized solutions of mass supercritical Schrödinger equations with potential, Comm. Partial Differential Equations, 46 (2021), 1729–1756. https://doi.org/10.1080/03605302.2021.1893747 doi: 10.1080/03605302.2021.1893747 |
[33] | J. Bellazzini, G. Siciliano, Scaling properties of functionals and existence of constrained minimizers, J. Funct. Anal., 261 (2011), 2486–2507. https://doi.org/10.1016/j.jfa.2011.06.014 doi: 10.1016/j.jfa.2011.06.014 |
[34] | Q. L. Xie, S. W. Ma, X. Zhang, Bound state solutions of Kirchhoff type problems with critical exponent, J. Differential Equations, 261 (2016), 890–924. https://doi.org/10.1016/j.jde.2016.03.028 doi: 10.1016/j.jde.2016.03.028 |
[35] | Q. Wang, A. Qian, Normalized Solutions to the Kirchhoff Equation with Potential Term: Mass Super-Critical Case, Bull. Malays. Math. Sci. Soc., 46 (2023). https://doi.org/10.1007/s40840-022-01444-4 |
[36] | T. Bartsch, T. Weth, Three nodal solutions of singularly perturbed elliptic equations on domains without topology, Ann. Inst. H. Poincar$\acute{e}$ C Anal. Non Lin$\acute{e}$aire., 22 (2005), 259–281. https://doi.org/10.1016/j.anihpc.2004.07.005 |
[37] | G. Cerami, D. Passaseo, The effect of concentrating potentials in some singularly perturbed problems, Calc. Var. Partial Differential Equations, 17 (2003), 257–281. https://doi.org/10.1007/s00526-002-0169-6 doi: 10.1007/s00526-002-0169-6 |