# Existence and Multiplicity of Solutions for the Equation with Nonlocal Integrodifferential Operator

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## Abstract

We are concerned with the following elliptic equation with a general nonlocal integrodifferential operator $${\mathcal {L}}_K$$

\begin{aligned} \begin{aligned} \left\{ \begin{array}{ll} -{\mathcal {L}}_Ku=\lambda u+f(x,u), &{}\quad \text {in}\quad \Omega ,\\ u=0, &{} \quad \text {in}\quad {\mathbb {R}}^n{\setminus }\Omega , \end{array}\right. \end{aligned} \end{aligned}

where $$\Omega$$ be an open-bounded set of $${\mathbb {R}}^n$$ with continuous boundary, $$\lambda \in {\mathbb {R}}$$ is a real parameter, and f is a nonlinear term with subcritical growth. We show the existence of a ground state and infinitely many pairs of solutions. The proof is based on the method of Nehari manifold for the equation with $$\lambda <\lambda _1$$, where $$\lambda _1$$ is the first eigenvalue of the nonlocal operator $$-{\mathcal {L}}_K$$ with homogeneous Dirichlet boundary condition, and the method of generalized Nehari manifold for the equation with $$\lambda \ge \lambda _1$$. As a concrete example, we derive the existence and multiplicity of solutions for the equation driven by fractional Laplacian

\begin{aligned} \begin{aligned} \left\{ \begin{array}{ll} (-\Delta )^\alpha u=\lambda u+f(x,u),&{}\quad \text {in}\quad \Omega ,\\ u=0, &{}\quad \text {in}\quad {\mathbb {R}}^n{\setminus }\Omega , \end{array}\right. \end{aligned} \end{aligned}

where $$0<\alpha <1$$. The results presented here may be viewed as the extension of some classical results for the Laplacian to nonlocal fractional setting.

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## References

1. Abatangelo, N., Valdinoci, E.: Getting acquainted with the fractional Laplacian. In: Dipierro, S. (ed.) Contemporary Research in Elliptic PDEs and Related Topics. Springer INdAM Series, pp. 1–105. Springer, Cham (2019)

2. Alberti, G., Bellettini, G.: A nonlocal anisotropic model for phase transitions. Math. Ann. 310, 527–560 (1998)

3. Bisci, G.M., Radulescu, V.D., Servadei, R.: Variational Methods for Nonlocal Fractional Problems, Volume 162 of Encyclopedia of Mathematics and Its Applications. Cambridge University Press, Cambridge (2016)

4. Bucur, C., Valdinoci, E.: Nonlocal diffusion and applications. Lecture Notes of the Unione Matematica Italiana, vol. 20. Springer, Cham, Unione Matematica Italiana, Bologna (2016). xii+155 pp

5. Cabré, X., Solà-Morales, J.: Layer solutions in a half-space for boundary reactions. Commun. Pure Appl. Math. 58, 1678–1732 (2005)

6. Caffarelli, L., Silvestre, L.: An extension problem related to the fractional Laplacian. Commun. Partial Differ. Equ. 32, 1245–1260 (2007)

7. Chen, W.J., Deng, S.B.: The Nehari manifold for nonlocal elliptic operators involving concave–convex nonlinearities. Z. Angew. Math. Phys. 66, 1387–1400 (2015)

8. Di Nezza, E., Palatucci, G., Valdinoci, E.: Hitchhiker’s guide to the fractional Sobolev spaces. Bull. Sci. Math. 136, 521–573 (2012)

9. Dipierro, S., Medina, M., Valdinoci, E.: Fractional elliptic problems with critical growth in the whole of $$\mathbb{R}^n$$, Appunti. Scuola Normale Superiore di Pisa (Nuova Serie) [Lecture Notes. Scuola Normale Superiore di Pisa (New Series)], vol. 15, Edizioni della Normale, Pisa (2017)

10. Dipierro, S., Palatucci, G., Valdinoci, E.: Dislocation dynamics in crystals: a macroscopic theory in a fractional Laplace setting. Commun. Math. Phys. 333, 1061–1105 (2015)

11. Felmer, P., Quaas, A., Tan, J.: Positive solutions of nonlinear Schrödinger equation with the fractional Laplacian. Proc. R. Soc. Edinb. Sect. A 142, 1237–1262 (2012)

12. Fiscella, A., Bisci, G.M., Servadei, R.: Bifurcation and multiplicity results for critical nonlocal fractional Laplacian problems. Bull. Sci. Math. 140, 14–35 (2016)

13. Gilboa, G., Osher, S.: Nonlocal operators with applications to image processing. Multiscale Model. Simul. 7, 1005–1028 (2008)

14. Gu, G.Z., Zhang, W., Zhao, F.K.: Infinitely many sign-changing solutions for a nonlocal problem. Ann. Mat. Pura Appl. 197, 1429–1444 (2018)

15. He, X.M., Zou, W.M.: Existence and concentration result for the fractional Schrödinger equations with critical nonlinearities. Calc. Var. Partial Differ. Equ. 55, 91 (2016)

16. Laskin, N.: Fractional quantum mechanics and Levy path integrals. Phys. Lett. A 268, 298–305 (2000)

17. Li, Y.Q., Wang, Z.Q., Zeng, J.: Ground states of nonlinear Schrödinger equations with potentials. Ann. Inst. H. Poincare Anal. Non Lineaire 23, 829–837 (2006)

18. Luo, H.X., Tang, X.H., Gao, Z.: Sign-changing solutions for non-local elliptic equations with asymptotically linear term. Commun. Pure Appl. Anal. 17, 1147–1159 (2018)

19. Nehari, Z.: On a class of nonlinear second-order differential equations. Trans. Am. Math. Soc. 95, 101–123 (1960)

20. Nehari, Z.: Characteristic values associated with a class of non-linear second-order differential equations. Acta Math. 105, 141–175 (1961)

21. Pankov, A.: Periodic nonlinear Schrödinger equation with application to photonic crystals. Milan J. Math. 73, 259–287 (2005)

22. Rabinowitz, P.H.: Minimax methods in critical point theory with applications to differential equations. In: CBMS Reg. Conf. Ser. Math., vol. 65. American Mathematical Society, Providence, RI (1986)

23. Servadei, R., Valdinoci, E.: The Brezis–Nirenberg result for the fractional Laplacian. Trans. Am. Math. Soc. 367, 67–102 (2015)

24. Servadei, R., Valdinoci, E.: A Brezis–Nirenberg result for non-local critical equations in low dimension. Commun. Pure Appl. Anal. 12, 2445–2464 (2013)

25. Servadei, R., Valdinoci, E.: Mountain pass solutions for non-local elliptic operators. J. Math. Anal. Appl. 389, 887–898 (2012)

26. Servadei, R., Valdinoci, E.: Variational methods for non-local operators of elliptic type. Discrete Contin. Dyn. Syst. 33, 2105–2137 (2013)

27. Shang, X.D., Zhang, J.H.: Ground states for fractional Schrödinger equations with critical growth. Nonlinearity 27, 187–207 (2014)

28. Szulkin, A., Weth, T.: The method of nehari manifold. In: Gao, D.Y., Motreanu, D. (eds.) Handbook of Nonconvex Analysis and Applications, pp. 597–632. International Press, Boston (2010)

29. Szulkin, A., Weth, T.: Ground state solutions for some indefinite variational problems. J. Funct. Anal. 257, 3802–3822 (2009)

30. Tan, J.: The Brezis–Nirenberg type problem involving the square root of the Laplacian. Calc. Var. Partial Differ. Equ. 42, 21–41 (2011)

31. Willem, M.: Minimax Theorems. Progress Nonlinear Differential Equations Appl, vol. 24. Birkhäuser, Basel (1996)

32. Zhang, B.L., Bisci, G.M., Servadei, R.: Superlinear nonlocal fractional problems with infinitely many solutions. Nonlinearity 28, 2247–2264 (2015)

33. Zhang, H., Xu, J.X., Zhang, F.B.: Existence and multiplicity of solutions for superlinear fractional Schrödinger equations in $$\mathbb{R}^N$$. J. Math. Phys. 56, 091502 (2015). 13 pp

## Acknowledgements

Part of this work was carried out in the period of a pleasant visit of the first author to the University of Western Australia, and the first author thanks Professors Enrico Valdinoci and Serena Dipierro for useful and helpful suggestions on the paper.

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Correspondence to Hui Zhang.

Communicated by See Keong Lee.

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The work was supported by the National Natural Science Foundation of China (Nos. 11601204, 11671077, 11571140), and Jiangsu Overseas Visiting Scholar Program for University Prominent Young and Middle-aged Teachers and Presidents.

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Zhang, H., Zhang, F. Existence and Multiplicity of Solutions for the Equation with Nonlocal Integrodifferential Operator. Bull. Malays. Math. Sci. Soc. 44, 1135–1154 (2021). https://doi.org/10.1007/s40840-020-00995-8

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• DOI: https://doi.org/10.1007/s40840-020-00995-8