zbMATH — the first resource for mathematics

Positive solutions for nonlinear Schrödinger-Kirchhoff equations in \(\mathbb{R}^3\). (English) Zbl 1437.35287
Summary: In this paper, we study the nonlinear Schrödinger-Kirchhoff-type equation with pure power nonlinearities in \(\mathbb{R}^3\) by variational methods. By carrying out the constrained minimization on a special manifold which is a combination of the Nehari manifold and Pohozaev manifold, we proved the existence of positive radial solutions of this equation for the power \(p \in (1, 5)\). The results of this paper extend some existing conclusions, especially for \(p \in (1, 2]\).
35J60 Nonlinear elliptic equations
35B09 Positive solutions to PDEs
35A01 Existence problems for PDEs: global existence, local existence, non-existence
35A15 Variational methods applied to PDEs
Full Text: DOI
[1] He, X.; Zou, W., Existence and concentration behavior of positive solutions for a Kirchhoff equation in \(\mathbb{R}^3\), J. Differential Equations, 252, 1813-1834 (2012)
[2] Alves, C. O.; Figueiredo, G. M., Nonlinear perturbations of a periodic Kirchhoff equation in \(\mathbb{R}^N\), Nonlinear Anal., 75, 2750-2759 (2012)
[3] Azzollini, A., The elliptic Kirchhoff equation in \(\mathbb{R}^N\) perturbed by a local nonlinearity, Differ. Integr. Equ., 25, 543-554 (2012)
[4] Tang, Z.; Wang, L., Optimal number of solutions for nonlinear coupled Schrödinger systems, part I: Synchronized case, J. Differential Equations., 266, 3601-3653 (2019)
[5] Figueiredo, G. M.; Ikoma, N.; Santos Júnior, J. R., Existence and concentration result for the Kirchhoff type equations with general nonlinearities, Arch. Ration. Mech. Anal., 213, 931-979 (2014)
[6] Chen, S.; Liu, S., Standing waves for 4-superlinear Schrödinger-Kirchhoff equations, Math. Methods Appl. Sci., 38, 2185-2193 (2015)
[7] Xu, H., Existence of positive solutions for the nonlinear Kirchhoff type equations in \(\mathbb{R}^n\), J. Math. Anal. Appl., 482, 2, Article 123593 pp. (2020)
[8] Sun, J.; Liu, S., Nontrivial solutions of Kirchhoff type problems, Appl. Math. Lett., 25, 500-504 (2012)
[9] Sun, J.; Wu, T.-F., Ground state solutions for an indefinite Kirchhoff type problem with steep potential well, J. Differential Equations, 256, 1771-1792 (2014)
[10] Ye, Y.; Tang, C.-L., Multiple solutions for Kirchhoff-type equations in \(\mathbb{R}^N\), J. Math. Phys., 54, Article 081508 pp. (2013), 16
[11] Kirchhoff, G., Mechanik (1883), Teubner: Teubner Leipzig
[12] Ruiz, D., The Schrödinger-Poisson equation under the effect of a nonlinear local term, J. Funct. Anal., 237, 655-674 (2006)
[13] Berestycki, H.; Lions, P.-L., Nonlinear scalar field equations. I. Existence of a ground state, Arch. Ration. Mech. Anal., 82, 313-345 (1983)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.