Existence of multiple non-trivial solutions for a nonlocal problem

Xianyong Yang; Zhipeng Yang; Xianyong Yang; Zhipeng Yang

doi:10.3934/math.2018.2.299

AIMS Mathematics

2019, Volume 4, Issue 2: 299-307. doi: 10.3934/math.2018.2.299

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Research article

Existence of multiple non-trivial solutions for a nonlocal problem

Xianyong Yang ^1,2,
Zhipeng Yang ^{3
,
,}

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

Received: 08 January 2019 Accepted: 12 March 2019 Published: 25 March 2019

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*} $
- integro-differential operators,
- multiple solutions,
- variational method
Citation: Xianyong Yang, Zhipeng Yang. Existence of multiple non-trivial solutions for a nonlocal problem[J]. AIMS Mathematics, 2019, 4(2): 299-307. doi: 10.3934/math.2018.2.299

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Abstract

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*} $

References

[1]	R. Servadei and E. Valdinoci, Mountain pass solutions for nonlocal elliptic operators, J. Math. Anal. Appl., 389 (2012), 887-898. doi: 10.1016/j.jmaa.2011.12.032
[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. doi: 10.1007/s00222-007-0086-6
[3]	L. Silvestre, Regularity of the obstacle problem for a fractional power of the Laplace operator, Commun. Pur. Appl. Math., 60 (2007), 67-112. doi: 10.1002/cpa.20153
[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. doi: 10.1007/s00526-010-0359-6
[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. doi: 10.1016/j.jfa.2009.01.020
[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. doi: 10.1007/s00526-013-0613-9
[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. doi: 10.1016/j.bulsci.2011.12.004
[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. doi: 10.1016/S0370-1573(00)00070-3
[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. doi: 10.1016/j.jde.2012.02.023
[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. doi: 10.1088/0951-7715/26/2/479
[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. doi: 10.1016/j.aml.2017.08.004
[22]	B. Zhang, G. Bisci and R. Servadei, Superlinear nonlocal fractional problems with infinitely many solutions, Nonlinearity, 28 (2015), 2247-2264. doi: 10.1088/0951-7715/28/7/2247
[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. doi: 10.1007/s10231-018-0731-2
[26]	G. Gu, W. Zhang and F. Zhao, Infinitely many positive solutions for a nonlocal problem, Appl. Math. Lett., 84 (2018), 49-55. doi: 10.1016/j.aml.2018.04.010
[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. doi: 10.1112/S0025579300015916

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