# Infinitely Many Solutions for Fractional p-Kirchhoff Equations

Article

## Abstract

In this paper we consider the existence of infinitely many weak solutions for fractional Schrödinger–Kirchhoff problems. Precisely speaking, we investigate
\begin{aligned} \left\{ \begin{array}{cl} M\left( \int _{\mathbb {R}^{2n}}\frac{|u(x)-u(y)|^p}{|x-y|^{n+sp}}\mathrm{d}x\mathrm{d}y\right) (-\triangle )_p^su+V(x)|u|^{p-2}u=f(x,u), &{}\quad \mathrm{in}~\Omega ,\\ u=0, &{}\quad \mathrm{in}~\mathbb {R}^n\setminus \Omega , \end{array}\right. \end{aligned}
where $$\Omega$$ is a bounded subset with Lipshcitz boundary $$\partial \Omega$$, $$0<s<1$$ is fixed, and $$1<p<n$$, $$(-\triangle )_p^s$$ is the fractional p-Laplacian operator. Kirchhoff function M, potential function V and nonlinearity f satisfy some suitable assumptions. Under those conditions, some new multiplicity results are obtained by applying the fountain theorem and the dual fountain theorem.

## Keywords

Schrödinger–Kirchhoff-type equation fractional p-Laplacian fountain theorem dual fountain theorem

35R11 35A15

## References

1. 1.
Figueiredo, G.M., Santos Junior, J.R.: Multiplicity of solutions for a Kirchhoff equation with subcritical or critical growth. Differ. Integr. Equ. 25(9–10), 853–868 (2012)
2. 2.
Figueiredo, G.M.: Existence of a positive solution for a Kirchhoff problem type with critical growth via truncation argument. J. Math. Anal. Appl. 401, 706–713 (2013)
3. 3.
Colasuonno, F., Pucci, P.: Multiplicity of solutions for $$p(x)$$-polyharmonic Kirchhoff equations. Nonlinear Anal. 74, 5962–5974 (2011)
4. 4.
Autuori, G., Fiscella, A., Pucci, P.: Stationary Kirchhoff problems involving a fractional elliptic operator and a critical nonlinearity. Nonlinear Anal. 125, 699–714 (2015)
5. 5.
Servadei, R., Valdinoci, E.: The Brezis-Nirenberg result for the fractional Laplacian. Trans. Am. Math. Soc. 367, 67–102 (2015)
6. 6.
Chen, C.S., Song, H.X., Xiu, Z.H.: Multiple solution for $$p$$-Kirchhoff equations in $${\mathbb{R}}^N$$. Nonlinear Anal. 86, 146–156 (2013)
7. 7.
Nyamoradi, N.: Existence of three solutions for Kirchhoff nonlocal operators of elliptic type. Math. Commun. 18, 489–502 (2013)
8. 8.
Nyamoradi, N., Chung, N.T.: Existence of solutions to nonlocal Kirchhoff equations of elliptic type via genus theory. Electron. J. Differ. Equ. 2014, 1–12 (2014)
9. 9.
Nyamoradi, N., Teng, K.: Existence of solutions for a Kirchhoff-type-nonlocal operators of elliptic type. Commun. Pure Appl. Anal. 14, 361–371 (2015)
10. 10.
Xiang, M.Q., Bisci, G.M., Tian, G.H., Zhang, B.L.: Infinitely many solutions for the stationary Kirchhoff problems involving the fractional $$p$$-Laplacian. Nonlinearity 29, 357–374 (2016)
11. 11.
Fiscella, A., Valdinoci, E.: A critical Kirchhoff type problem involving a nonlocal operator. Nonlinear Anal. 94, 156–170 (2014)
12. 12.
Pucci, P., Saldi, S.: Critical stationary Kirchhoff equations in $${\mathbb{R}}^N$$ involving nonlocal operators. Rev. Mat. Iberoam. 32(1), 1–22 (2016)
13. 13.
Pucci, P., Xiang, M.Q., Zhang, B.L.: Multiple solutions for nonhomogeneous Schrödinger-Kirchhoff type equations involving the fractional $$p$$-Laplacian in $${\mathbb{R}}^N$$. Calc. Var. Partial Differ. Equ. 54, 2785–2806 (2015)
14. 14.
Xiang, M.Q., Zhang, B.L., Guo, X.Y.: Infinitely many solutions for a fractional Kirchhoff type problem via Fountain Theorem. Nonlinear Anal. 120, 299–313 (2015)
15. 15.
Di Nezza, E., Palatucci, G., Valdinoci, E.: Hitchhiker’s guide to the fractional Sobolev spaces. Bull. Sci. Math. 136, 521–573 (2012)
16. 16.
Rabinowitz, P.H.: On a class of nonlinear Schrödinger equations. Z. Angew. Math. Phys. 43, 270–291 (1992)
17. 17.
Bartsch, T., Willem, M.: On an elliptic equation with concave and convex nonlinearities. Proc. Am. Math. Soc. 123, 3555–3561 (1995)
18. 18.
Willem, M.: Minimax Theorems. Progress in Nonlinear Differential Equations and Their Applications, vol. 24. Birkhäuser Boston, lnc., Boston (1996)Google Scholar
19. 19.
Teng, K.M.: Multiple solutions for a class of fractional Schrödinger equations in $${\mathbb{R}}^N$$. Nonlinear Anal. Real World Appl. 21, 76–86 (2015)
20. 20.
Chang, X.J.: Ground state solutions of asymptotically linear fractional Schrödinger equations. J. Math. Phys. 54(6), 061504 (2013)
21. 21.
Autuori, G., Pucci, P.: Elliptic problems involving the fractional Laplacian in $${\mathbb{R}}^N$$. J. Differ. Equ. 255, 2340–2362 (2013)
22. 22.
Ge, B.: Multiple solutions of nonlinear Schrödinger equation with the fractional Laplacian. Nonlinear Anal. Real World Appl. 30, 236–247 (2016)
23. 23.
Bartsch, T.: Infinitely many solutions of a symmetric Dirichlet problem. Nonlinear Anal. 20, 1205–1216 (1993)
24. 24.
Barrios, B., Colorado, E., De Pablo, A., Sanchez, U.: On some critical problems for the fractional Laplacian operator. J. Differ. Equ. 252, 6133–6162 (2012)
25. 25.
Binlin, Z., Molica Bisci, G., Servadei, R.: Superlinear nonlocal fractional problems with infinitely many solutions. Nonlinearity 28, 2247–2264 (2015)

© Springer International Publishing AG, part of Springer Nature 2018

## Authors and Affiliations

• Libo Yang
• 1
• 2
• Tianqing An
• 1
1. 1.College of ScienceHohai UniversityNanjingPeople’s Republic of China
2. 2.Faculty of Mathematics and PhysicsHuaiyin Institute of TechnologyHuai’anPeople’s Republic of China