Ground state solutions for Kirchhoff-type problems with convolution nonlinearity and Berestycki-Lions type conditions. (English) Zbl 1481.35211
Summary: In this paper, we discuss the following Kirchhoff-type problem with convolution nonlinearity
\[
-\left(1+ b\int_{\mathbb{R}^3}|\nabla u|^2 dx \right) \triangle u+ V(x)u=(I_{\alpha}*F(u))f(u),\;x\in \mathbb{R}^3,\;u\in H^1(\mathbb{R}^3),
\]
where \(b>0\), \(I_\alpha:\mathbb{R}^3\rightarrow \mathbb{R}\), with \(\alpha\in(0, 3)\), is the Riesz potential, \(V\) is differentiable, \(f\in\mathbb{C}(\mathbb{R}, \mathbb{R})\) and \(F(t)=\int^t_0f(s)ds\). Let \(f\) satisfies some relatively weak conditions in the absence of the usual Ambrosetti-Rabinowitz or monotonicity conditions. We get two classes of ground state solutions under the general “Berestycki-Lions conditions” on the nonlinearity \(f\) and we also give a minimax characterization of the ground state energy.
MSC:
| 35J62 | Quasilinear elliptic equations |
| 35A01 | Existence problems for PDEs: global existence, local existence, non-existence |
| 35J20 | Variational methods for second-order elliptic equations |
References:
| [1] | Arosio, A.; Panizzi, S., On the well-posedness of Kirchhoff string, Trans. Am. Math. Soc., 348, 305-330 (1996) · Zbl 0858.35083 · doi:10.1090/S0002-9947-96-01532-2 |
| [2] | Azzollini, A., The elliptic Kirchhoff equation in \({\mathbb{R}}^N\) perturbed by a local nonlinearity, Differ. Integral Equ., 25, 543-554 (2012) · Zbl 1265.35069 |
| [3] | Berestycki, H.; Lions, P., Nonlinear scalar field equations, I. Existence of a ground state, Arch. Ration. Mech. Anal., 82, 313-345 (1983) · Zbl 0533.35029 · doi:10.1007/BF00250555 |
| [4] | Brézis, H.; Lieb, H., A relation between pointwise convergence of functions and convergence of functionals, Proc. Am. Math. Soc., 88, 486-490 (1983) · Zbl 0526.46037 · doi:10.1090/S0002-9939-1983-0699419-3 |
| [5] | Chen, S.; Tang, X., Existence and non-existence results for Kirchhoff-type problems with convolution nonlinearity, Adv. Nonlinear Anal., 9, 148-167 (2020) · Zbl 1421.35100 · doi:10.1515/anona-2018-0147 |
| [6] | Chen, S.; Tang, X., Berestycki-Lions conditions on ground state solutions for Kirchhoff-type problems with variable potentials, J. Math. Phys., 60, 121509 (2019) · Zbl 1432.35192 · doi:10.1063/1.5128177 |
| [7] | Chen, S., Tang, X.: Ground state solutions for general Choquard equations with a variable potential and a local nonlinearity. Rev. R. Acad. Cienc. Exactas Fis. Nat. Ser. A Mat. (2020). doi:10.1007/s13398-019-00775-5 · Zbl 1437.35209 |
| [8] | Chipot, M.; Lovat, B., Some remarks on nonlocal elliptic and parabolic problems, Nonlinear Anal., 30, 4619-4627 (1997) · Zbl 0894.35119 · doi:10.1016/S0362-546X(97)00169-7 |
| [9] | Ruiz, D., The Schrödinger-Poisson equation under the effect of a nonlinear local term, J. Funct. Anal., 237, 655-674 (2006) · Zbl 1136.35037 · doi:10.1016/j.jfa.2006.04.005 |
| [10] | Gu, G.; Tang, X., The concentration behavior of ground states for a class of Kirchhoff-type problems with Hartree-type nonlinearity, Adv. Nonlinear Stud., 19, 779-795 (2019) · Zbl 1427.35050 · doi:10.1515/ans-2019-2045 |
| [11] | Guo, Z., Ground states for Kirchhoff equations without compact condition, J. Differ. Equ., 259, 2884-2902 (2015) · Zbl 1319.35018 · doi:10.1016/j.jde.2015.04.005 |
| [12] | Hu, D.; Tang, X.; Zhang, Q., Existence of ground state solutions for Kirchhoff-type problem with variable potential, Appl. Anal. (2021) · Zbl 1514.35202 · doi:10.1080/00036811.2021.1947499 |
| [13] | He, X.; Zou, W., Existence and concentration behavior of positive solutions for a Kirchhoff equation in \({\mathbb{R}}^3 \), J. Differ. Equ., 252, 1813-1834 (2012) · Zbl 1235.35093 · doi:10.1016/j.jde.2011.08.035 |
| [14] | Jeanjean, L., Toland, J.: Bounded Palais-Smale mountain-pass sequences. C. R. Acad. Sci. Paris Sér. I Math. 327, 23-28 (1998) · Zbl 0996.47052 |
| [15] | Jeanjean, L., On the existence of bounded Palais-Smale sequences and application to a Landesman-Lazer-type problem set on \({\mathbb{R}}^N \), Proc. R. Soc. Edinb. Sect. A, 129, 787-809 (1999) · Zbl 0935.35044 · doi:10.1017/S0308210500013147 |
| [16] | Kirchhoff, G., Mechanik (1883), Leipzig: Teubner, Leipzig · JFM 08.0542.01 |
| [17] | Li, G.; Ye, H., Existence of positive ground state solutions for the nonlinear Kirchhoff type equations in \({\mathbb{R}}^3 \), J. Differ. Equ., 257, 566-600 (2014) · Zbl 1290.35051 · doi:10.1016/j.jde.2014.04.011 |
| [18] | Lü, D., Existence and concentration of ground state solutions for singularly perturbed nonlocal elliptic problems, Monatsh Math., 182, 335-358 (2017) · Zbl 1369.35023 · doi:10.1007/s00605-016-0889-x |
| [19] | Lü, D., A note on Kirchhoff-type equations with Hartree-type nonlinearities, Nonlinear Anal., 99, 35-48 (2014) · Zbl 1286.35108 · doi:10.1016/j.na.2013.12.022 |
| [20] | Li, Y.; Li, X.; Ma, S., Groundstates for Kirchhoff-type equations with Hartree-type nonlinearities, Results Math. (2019) · Zbl 1412.35124 · doi:10.1007/s00025-018-0943-1 |
| [21] | Luo, H., Ground state solutions of Pohožaev type and Nehari type for a class of nonlinear Choquard equations, J. Math. Anal. Appl., 467, 842-862 (2018) · Zbl 1398.35071 · doi:10.1016/j.jmaa.2018.07.055 |
| [22] | Lions, P.: The concentration-compactness principle in the calculus of variation. The locally compact case. Part II. Ann Inst H Poincaré Anal Non Linéaire. 1, 223-283 (1984) · Zbl 0704.49004 |
| [23] | Lions, P.: The concentration-compactness principle in the calculus of variation. The locally compact case. Part I. Ann Inst H Poincaré Anal Non Linéaire. 1, 109-145 (1984) · Zbl 0541.49009 |
| [24] | Moroz, V.; Van, S., Existence of groundstate for a class of nonlinear Choquard equations, Trans. Am. Math. Soc., 367, 6557-6579 (2015) · Zbl 1325.35052 · doi:10.1090/S0002-9947-2014-06289-2 |
| [25] | Moroz, V.; Van, J., Groundstates of nonlinear Choquard equations: existence, qualitative properties and decay asymptotics, J. Funct. Anal., 265, 153-184 (2013) · Zbl 1285.35048 · doi:10.1016/j.jfa.2013.04.007 |
| [26] | Sun, J., Wu. T.: Ground state solutions for an indefinite Kirchhoff type problem with steep potential well. J. Differ. Equ. 256, 1771-1792 (2014) · Zbl 1288.35219 |
| [27] | Tang, X.; Chen, S., Ground STSTE solutions of Nehari-Pohozaev type for Kirchhoff-type problems with general potentials, Calc. Var. Partial Diffe. Equ., 56, 110-134 (2017) · Zbl 1376.35056 · doi:10.1007/s00526-017-1214-9 |
| [28] | Tang, X., Chen, S.: Singularly perturbed Choquard equations with nonlinearity satisfying Berestycki-Lions assumptions (2019). arXiv:1903.10347 · Zbl 1421.35068 |
| [29] | Van, J.; Xia, J., Groundstates for a local nonlinear perturbation of the Choquard equations with lower critical exponent, J. Math. Anal. Appl., 464, 1184-1202 (2018) · Zbl 1398.35094 · doi:10.1016/j.jmaa.2018.04.047 |
| [30] | Willem, M., Minimax Theorems, Progress in Nonlinear Differential Equations and Their Applications (1996), Boston: Birkhäuser Boston Inc., Boston · Zbl 0856.49001 |
| [31] | Xia, J., Zhang, X.: Saddle solutions for the critical Choquard equation. Calc. Var. Partial Differ. Equ. (2021). doi:10.1007/s00526-021-01919-5 · Zbl 1459.35216 |
| [32] | Zhang, Q., Hu, D.: Existence of solutions for a class of quasilinear Schrödinger equation with a Kirchhoff-type. Complex Var. Elliptic Equ. (2021). doi:10.1080/17476933.2021.1916918 · Zbl 1497.35244 |
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