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Infinitely many sign-changing solutions for modified Kirchhoff-type equations in \(\mathbb{R}^3\). (English) Zbl 1481.35210

Summary: Consider a class of modified Kirchhoff-type equations \[ -(1+b \displaystyle\int_{\mathbb{R}^3} |\nabla u|^2 \mathrm{d}x)\Delta u+V(x)u - \frac{1}{2} u\Delta (u^2)=f(u),\quad \text{in } \mathbb{R}^3, \] where the nonlinear term \(f\) is 4-superlinear at infinity. By using the method of invariant sets of descending flow, the existence of a sign-changing solution is obtained. When \(f\) is assumed to be odd, we prove that the above problem admits infinitely many sign-changing solutions. Moreover, when \(V(x) \equiv 1\) and the nonlinearity of power growth \(f(u) = |u|^{p-2} u\) with \(1<p\), we establish some existence and non-existence results. For \(p\in (1,2] \cup [12, \infty)\), the non-existence result relies on the deduction of some suitable Pohozaev identity. For \(p\in (3,4]\), using the Nehari-Pohozaev manifold, we prove that the above problem admits a ground state solution.

MSC:

35J62 Quasilinear elliptic equations
35A01 Existence problems for PDEs: global existence, local existence, non-existence
35A02 Uniqueness problems for PDEs: global uniqueness, local uniqueness, non-uniqueness
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[1] Kirchhoff, G., Mechanik (1883), Teubner: Leipzig, Teubner
[2] Chen, C.; Kuo, Y.; Wu, T., The Nehari manifold for a Kirchhoff type problem involving sign-changing weight functions, J Differ Equ, 250, 1876-1908 (2011) · Zbl 1214.35077
[3] Naimen, D., The critical problem of Kirchhoff type elliptic equations in dimension four, J Differ Equ, 257, 1168-1193 (2014) · Zbl 1301.35022
[4] Wu, X., Existence of nontrivial solutions and high energy solutions for Schrödinger-Kirchhoff-type equations in \(####\), Nonlinear Anal RWA, 12, 1278-1287 (2011) · Zbl 1208.35034
[5] Li, G.; Ye, H., Existence of positive groundstate solutions for the nonlinear Kirchhoff type equations in \(####\), J Differ Equ, 257, 566-600 (2014) · Zbl 1290.35051
[6] Guo, Z., Ground states for Kirchhoff equations without compact condition, J Differ Equ, 259, 2884-2902 (2015) · Zbl 1319.35018
[7] Borovskii, A.; Galkin, A., Dynamical modulation of an ultrashort high-intensity laser pulse in matter, J Exp Theor Phys, 77, 562-573 (1983)
[8] Brandi, H.; Manus, C.; Mainfray, G., Relativistic and ponderomotive self-focusing of a laser beam in a radially inhomogeneous plasma, Phys Fluids, 5, 3539-3550 (1993)
[9] Makhankov, V.; Fedyanin, V., Nonlinear effects in quasi-one-dimensional models of condensed matter theory, Phys Rep, 104, 1-86 (1984)
[10] Liu, H., Positive solution for a quasilinear elliptic equation involving critical or supercritical exponent, J Math Phys, 57, 159-180 (2016) · Zbl 1338.35180
[11] Liu, J.; Wang, Y.; Wang, Z-Q., Soliton solutions for quasilinear Schrödinger equations II, J Differ Equ, 187, 473-493 (2003) · Zbl 1229.35268
[12] Liu, J.; Wang, Y.; Wang, Z-Q., Solutions for quasilinear Schrödinger equations via the Nehari method, Commun Partial Differ Equ, 29, 879-901 (2004) · Zbl 1140.35399
[13] Liu, X.; Liu, J.; Wang, Z-Q., Quasilinear elliptic equations via pertubation method, Pro Am Math Soc, 141, 253-263 (2013) · Zbl 1267.35096
[14] Liu, X.; Liu, J.; Wang, Z-Q., Multiple sign-changing solutions for quasilinear elliptic equations via perturbation method, Commun Partia Differ Equ, 39, 2216-2239 (2014) · Zbl 1304.35290
[15] Liu, J.; Liu, X.; Wang, Z-Q., Multiple mixed states of nodal solutions for nonlinear Schrödinger systems, Calc Var Partial Differ Equ, 52, 565-586 (2015) · Zbl 1311.35291
[16] Wang, Y., Multiplicity of solutions for singular quasilinear Schrödinger equations with critical exponents, J Math Anal Appl, 58, 1027-1043 (2018) · Zbl 1379.35085
[17] Wang, Y.; Li, Z., Existence of solutions to quasilinear Schrödinger equations involving critical Sobolev exponent, Taiwan J Math, 22, 401-420 (2018) · Zbl 1401.35050
[18] Huang, C.; Jia, G., Existence of positive solutions for supercritical quasilinear Schrödinger elliptic equations, J Math Anal Appl, 472, 705-727 (2019) · Zbl 1418.35097
[19] Deng, Y.; Peng, S.; Shuai, W., Existence and asymptotic behavior of nodal solutions for the Kirchhoff-type problems in \(####\), J Funct Anal, 269, 3500-3527 (2015) · Zbl 1343.35081
[20] Shuai, W., Sign-changing solutions for a class of Kirchhoff-type problem in bounded domains, J Differ Equ, 259, 1256-1274 (2015) · Zbl 1319.35023
[21] Ye, H., The existence of least energy nodal solutions for some class of Kirchoff equations and Choquard equations in \(####\), J Math Anal Appl, 431, 935-954 (2015) · Zbl 1329.35203
[22] Sun, J.; Li, L.; Cencelj, M., Infinitely many sign-changing solutions for Kirchhoff type probelms in \(####\), Nonlinear Anal, 186, 33-54 (2019) · Zbl 1421.35119
[23] Huang, W.; Wang, L., Infinitely many sign-changing solutions for Kirchhoff type equations, Complex Var Elliptic Equ, 65, 920-935 (2020) · Zbl 1437.35215
[24] Feng, Z.; Wu, X.; Li, H., Multiple solutions for a modified Kirchoff-type equation in RN, Math Methods Appl Sci, 38, 708-725 (2015) · Zbl 1319.35040
[25] Wu, K.; Wu, X., Infinitely many small energy solutions for a modified Kirchhoff-type equation in \(####\), Comput Math Appl, 70, 592-602 (2015) · Zbl 1443.35042
[26] Chen, J.; Tang, X.; Gao, Z., Existence of multiple solutions for modified Schrödinger-Kirchhoff-Poisson type systems via perturbation method with sign-changing potential, Comput Math Appl, 73, 505-519 (2017) · Zbl 1375.35175
[27] Chen, L.; Feng, X.; Hao, X., The existence of sign-changing solution for a class of quasilinear Schrödinger-Poisson systems via perturbation method, Boundary Value Problems, 2019, 159 (2019) · Zbl 1513.35163
[28] Liu, Z.; Wang, Z-Q; Zhang, J., Infinitely many sign-changing solutions for the nonlinear Schrödinger-Poisson system, Ann Mat Pura Appl, 195, 775-794 (2016) · Zbl 1341.35041
[29] Zhang, W.; Liu, X., Infinitely many sign-changing solutions for a quasilinear elliptic equation in \(####\), J Math Anal Appl, 427, 722-740 (2015) · Zbl 1317.35086
[30] Bartsch, T.; Wang, Z-Q., Existence and multiple results for some superlinear elliptic problems on \(####\), Commun Partial Differ Equ, 20, 9-10, 1725-1741 (1995) · Zbl 0837.35043
[31] Ruiz, D.; Siciliano, G., Existence of ground states for a modified nonlinear Schrödinger equation, Nonlinearity, 23, 1221-1233 (2010) · Zbl 1189.35316
[32] Bartsch, T.; Liu, Z., On a superlinear elliptic p-Laplacian equation, J Differ Equ, 198, 149-175 (2004) · Zbl 1087.35034
[33] Bartsch, T.; Liu, Z.; Weth, T., Nodal solutions of a p-Laplacian equation, Proc Lond Math Soc, 91, 129-152 (2005) · Zbl 1162.35364
[34] Nie, J.; Wu, X., Existence and multiplicity of non-trivial solutions for a class of modified Schrödinger-Poisson systems, J Math Anal Appl, 408, 713-724 (2013) · Zbl 1308.81082
[35] Rabinowitz, PHMinimax Methods in Critical Points Theory with Application to Differential Equations. CBMS Regional Conf. Ser. Math. vol. 65. Am. Math.Soc. Providence:1986. · Zbl 0609.58002
[36] Ruiz, D., The Schrödinger-Poisson equation under the effect of a nonlinear local term, J Funct Anal, 237, 655-674 (2006) · Zbl 1136.35037
[37] Willem, MMinimax Theorems. Birkhäuser;1996.
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