## Rocky Mountain Journal of Mathematics

### Infinitely many solutions of systems of Kirchhoff-type equations with general potentials

#### Abstract

This paper is concerned with the following systems of Kirchhoff-type equations: $$\begin{cases} -\left (a+b\int _{\mathbb {R}^{N}}|\nabla u|^{2}\mathrm {d}x\right )\Delta u\\ \quad +V(x)u=F_{u}(x, u, v)\quad & x\in \mathbb {R}^{N},\\ -\left (c+d\int _{\mathbb {R}^{N}}|\nabla v|^{2}\mathrm {d}x\right )\Delta v\\ \quad +V(x)v=F_{v}(x, u, v) & x\in \mathbb {R}^{N},\\ u(x)\rightarrow 0,\quad v(x)\rightarrow 0 &\mbox {as } |x|\rightarrow \infty . \end{cases}$$ Under some more relaxed assumptions on $V(x)$ and $F(x, u, v)$, we prove the existence of infinitely many negative-energy solutions for the above system via the genus properties in critical point theory. Some recent results from the literature are greatly improved and extended.

#### Article information

Source
Rocky Mountain J. Math., Volume 48, Number 7 (2018), 2187-2209.

Dates
First available in Project Euclid: 14 December 2018

https://projecteuclid.org/euclid.rmjm/1544756807

Digital Object Identifier
doi:10.1216/RMJ-2018-48-7-2187

Mathematical Reviews number (MathSciNet)
MR3892130

Zentralblatt MATH identifier
06999260

#### Citation

Che, Guofeng; Chen, Haibo. Infinitely many solutions of systems of Kirchhoff-type equations with general potentials. Rocky Mountain J. Math. 48 (2018), no. 7, 2187--2209. doi:10.1216/RMJ-2018-48-7-2187. https://projecteuclid.org/euclid.rmjm/1544756807

#### References

• A. Azzollini, The Kirchhoff equation in $\mathbb{R}^{N}$ perturbed by a local nonlinearity, Diff. Int. Eqs. 25 (2015), 543–554.
• T. Bartsch and Z. Wang, Existence and multiplicity results for some superlinear elliptic problems on $\mathbb{R}^{N}$, Comm. Part. Diff. Eqs. 20 (1995), 1725–1741.
• G.F. Che and H.B. Chen, Existence and multiplicity of systems of Kirchhoff-type equations with general potentials, Math. Meth. Appl. Sci. 40 (2017), 775–785.
• ––––, Infinitely many solutions for a class of modified nonlinear fourth-order equation on $\mathbb{R}^{N}$, Bull. Korean Math. Soc. 54 (2017), 895–909.
• C. Chen, Y. Kuo and T.F. Wu, The Nehari manifold for a Kirchhoff type problem involving sign-changing weight functions, J. Diff. Eqs. 250 (2011), 1876–1908.
• J. Chen and X.H. Tang, Infinitely many solutions for a class of sublinear Schrödinger equations, Taiwanese J. Math. 19 (2015), 381–396.
• B.T. Cheng and X.H. Tang, Ground state sign-changing solutions for asymptotically $3$-linear Kirchhoff-type problems, Compl. Var. Ellip. Eqs. 62 (2017), 1093–1116.
• Y. Huang and Z. Liu, On a class of Kirchhoff type problems, Arch. Math. 102 (2014), 127–139.
• J. Jin and X. Wu, Infinitely many radial solutions for Kirchhoff-type problems in $\mathbb{R}^{N}$, J. Math. Anal. Appl. 369 (2010), 564–574.
• G. Kirchhoff, Mechanik, Teubner, Leipzig, 1883.
• S. Liang and J. Zhang, Existence of solutions for Kirchhoff type problems with critical nonlinearity in $\mathbb{R}^{3}$, Nonlin. Anal. 17 (2014), 126–136.
• J.L. Lions, On some questions in boundary value problems in mathematical physics, North-Holland Math. Stud. 30 (1978), 284–346.
• T.F. Ma and J.E. Muñoz Rivera, Positive solutions for a nonlinear nonlocal elliptic transmission problem, Appl. Math. Lett. 16 (2003), 243–248.
• J. Mawhin and M. Willem, Critical point theory and hamiltonian systems, Appl. Math. Sci. 74 (1989).
• J.J. Nie, Existence and multiplicity of nontrivial solutions for a class of Schrödinger-Kirchhoff-type equations, J. Math. Anal. Appl. 417 (2014), 65–79.
• P.H. Rabinowitz, Minimax methods in critical point theory with applications to differential equations, CBMS Reg. Conf. Ser. Math. 65 (1986).
• H.X. Shi and H.B. Chen, Ground state solutions for asymptotically periodic coupled Kirchhoff-type systems with critical growth, Math. Meth. Appl. Sci. 39 (2016), 2193–2201.
• J.T. Sun and T.F. Wu, Ground state solutions for an indefinite Kirchhoff type problem with steep potential well, J. Diff. Eqs. 256 (2014), 1771–1792.
• ––––, Existence and multiplicity of solutions for an indefinite Kirchhoff-type equation in bounded domains, Proc. Roy. Soc. Edinburgh 146 (2016), 435–448.
• X. Tang and S.T. Chen, Ground state solutions of Nehari-Pohozaev type for Kirchhoff-type problems with general potentials, Calc. Var. Part. Diff. Eqs. 56 (2017), 1–25.
• X.H. Tang and B.T. Cheng, Ground state sign-changing solutions for Kirchhoff type problems in bounded domains, J. Diff. Eqs. 261 (2016), 2384–2402.
• X.H. Tang and X.Y. Lin, Infinitely many homoclinic orbits for Hamiltonian systems with indefinite sign subquadratic potentials, Nonlin. Anal. 74 (2011) 6314–6325.
• M. Willem, Minimax theorems, Birkhäuser, Berlin, 1996.
• X. Wu, Existence of nontrivial solutions and high energy solutions Schrödinger-Kirchhoff-type equations in $\mathbb{R}^{N}$, Nonlin. Anal. 12 (2012), 1278–1287.
• ––––, High energy solutions of systems of Kirchhoff-type equations in $\mathbb{R}^{N}$, J. Math. Phys. 53 (2012), 1–18.
• L.P. Xu and H.B. Chen, Existence and multiplicity of solutions for fourth-order elliptic equations of Kirchhoff type via genus theory, Bound. Value Prob. 212 (2014), 1–12.
• ––––, Multiplicity results for fourth order elliptic equations of Kirchhoff-type, Acta Math. Sci. 35 (2015), 1067–1076.
• ––––, Multiple solutions for the nonhomogeneous fourth order elliptic equations of Kirchhoff-type, Taiwanese J. Math. 19 (2015), 1215–1226.
• F. Zhou, K. Wu and X. Wu, High energy solutions of systems of Kirchhoff-type equations on $\mathbb{R}^{N}$, Comp. Math. Appl. 66 (2013), 1299–1305.
• W.M. Zou and M. Schechter, Critical point theory and its applications, Springer, New York, 2006.