AIMS Mathematics, 2019, 4(2): 299-307. doi: 10.3934/math.2018.2.299.

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Existence of multiple non-trivial solutions for a nonlocal problem

1 School of Preparatory Education, Yunnan Minzu University, Kunming 650500, P.R. China
2 School of Mathematics and Statistics, Central south University, Changsha 410205 P.R. China
3 Department of Mathematics, Yunnan Normal University, Kunming 650500, P.R. China

In this paper, we are concerned with the following general nonlocal problem\begin{equation*}\begin{cases}-\mathcal{L}_K u=\lambda_1u+f(x,u)& \text{in}\ \Omega,\\u=0& \text{in}\ \mathbb{R}^N\backslash\Omega,\end{cases}\end{equation*}where $\lambda_1$ denotes the first eigenvalue of the nonlocal integro-differential operator $-\mathcal{L}_K$, $\Omega\subset\mathbb{R}^N$ is open, bounded domain with smooth boundary. Under several structural assumptions on $f$, we verify that the problem possesses at least two non-trivial solutions and locate the region in different parts of the Hilbert space by variational method. As a particular case, we derive an existence theorem for the following equation driven by the fractional Laplacian\begin{equation*}\begin{cases}(-\Delta)^su=\lambda_1u+f(x,u)& \text{in}\ \Omega,\\u=0& \text{in}\ \mathbb{R}^N\backslash\Omega.\end{cases}\end{equation*}
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Keywords integro-differential operators; multiple solutions; variational method

Citation: Xianyong Yang, Zhipeng Yang. Existence of multiple non-trivial solutions for a nonlocal problem. AIMS Mathematics, 2019, 4(2): 299-307. doi: 10.3934/math.2018.2.299


  • 1.R. Servadei and E. Valdinoci, Mountain pass solutions for nonlocal elliptic operators, J. Math. Anal. Appl., 389 (2012), 887-898.    
  • 2.L. Caffarelli, S. Salsa and L. Silvestre, Regularity estimates for the solution and the free boundary of the obstacle problem for the fractional Laplacian, Invent. Math., 171 (2008), 425-461.    
  • 3.L. Silvestre, Regularity of the obstacle problem for a fractional power of the Laplace operator, Commun. Pur. Appl. Math., 60 (2007), 67-112.    
  • 4.L. Caffarelli, J. Roquejoffre and O. Savin, Nonlocal minimal surfaces, Commun. Pur. Appl. Math., 63 (2010), 1111-1144.
  • 5.L. Caffarelli and E. Valdinoci, Uniform estimates and limiting arguments for nonlocal minimal surfaces, Calc. Var. Partial Dif., 41 (2011), 203-240.    
  • 6.Y. Sire and E. Valdinoci, Fractional Laplacian phase transitions and boundary reactions: A geometric inequality and a symmetry result, J. Funct. Anal., 256 (2009), 1842-1864.    
  • 7.C. Bucur and E. Valdinoci, Nonlocal Diffusion and Applications, In book series: Lecture Notes of the Unione Matematica Italiana, volume 20, Springer, Heidelberg, 2016.
  • 8. S. Serfaty and J. Vázquez, A mean field equation as limit of nonlinear diffusions with fractional Laplacian operators, Calc. Var. Partial Dif., 49 (2014), 1091-1120.    
  • 9. J. Vázquez, Nonlinear diffusion with fractional Laplacian operators, In: Nonlinear Partial Differential Equations. The Abel symposium 2010, Springer, Heidelberg, 2012.
  • 10. R. Cont and P. Tankov, Financial Modelling with Jump Processes, Chapman & Hall/CRC Financial Mathematics Series, Boca Raton, FL: Chapman & Hall/CRC, 2004.
  • 11. E. Di Nezza, G. Palatucci and E. Valdinoci, Hitchhiker's guide to the fractional Sobolev spaces, Bull. Sci. Math., 136 (2012), 521-573.    
  • 12.G. Bisci, V. Radulescu and R. Servadei, Variational methods for nonlocal fractional problems, Cambridge University Press, 2016.
  • 13.R. Metzler and J. Klafter, The random walk's guide to anomalous diffusion: A fractional dynamics approach, Phys. Rep., 339 (2000), 1-77.    
  • 14.L. Silvestre, Regularity of the obstacle problem for a fractional power of the Laplace operator, Thesis (Ph.D.), The University of Texas at Austin, 2005.
  • 15.S. Dipierro, M. Medina and E. Valdinoci, 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)], 15, Pisa: Edizioni della Normale, 2017.
  • 16.B. Barrios, E. Colorado, A. de Pablo, et al. On some critical problems for the fractional Laplacian operator, J. Differ. Equations, 252 (2012), 6133-6162.    
  • 17.G. Bisci and B. Pansera, Three weak solutions for nonlocal fractional equations, Adv. Nonlinear Stud., 14 (2014), 591-601.
  • 18.L. Caffarelli, J. Roquejoffre and Y. Sire, Variational problems for free boundaries for the fractional Laplacian, J. Eur. Math. Soc., 12 (2010), 6133-6162.
  • 19.X. Chang and Z. Wang, Ground state of scalar field equations involving a fractional Laplacian with general nonlinearity, Nonlinearity, 26 (2013), 479-494.    
  • 20. K. Teng, Multiple solutions for a class of fractional Schrödinger equations in $R^N$, Nonlinear Anal.: Real World Appl., 21 (2015), 76-86.
  • 21.Z. Yang and F. Zhao, Three solutions for a fractional Schrödinger equation with vanishing potentials, Appl. Math. Lett., 76 (2018), 90-95.    
  • 22. B. Zhang, G. Bisci and R. Servadei, Superlinear nonlocal fractional problems with infinitely many solutions, Nonlinearity, 28 (2015), 2247-2264.    
  • 23.R. Servadei and E. Valdinoci, Variational methods for nonlocal operators of elliptic type, Discrete Contin. Dyn. Syst., 33 (2013), 2105-2137.
  • 24. R. Servadei and E. Valdinoci, The Brezis-Nirenberg result for the fractional Laplacian, Trans. Amer. Math. Soc., 367 (2015), 67-102.
  • 25. G. Gu, W. Zhang and F. Zhao, Infinitely many sign-changing solutions for a nonlocal problem, Ann. Mat. Pur. Appl., 197 (2018), 1429-1444.    
  • 26. G. Gu, W. Zhang and F. Zhao, Infinitely many positive solutions for a nonlocal problem, Appl. Math. Lett., 84 (2018), 49-55.    
  • 27.G. Gu, Y. Yu and F. Zhao, The least energy sign-changing solution for a nonlocal problem, J. Math. Phys., 58 (2017), Article ID 051505: 1-11.
  • 28. M. Schechter, Multiple solutions for semilinear elliptic problems, Mathematika, 47 (2000), 307-317.    


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