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Existence and Multiplicity of Solutions for Nonlinear Elliptic Problems with the Fractional Laplacian

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Published/Copyright: February 10, 2015
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Fractional Calculus and Applied Analysis
From the journal Fractional Calculus and Applied Analysis Volume 18 Issue 1

Abstract

In this paper we consider a nonlinear eigenvalue problem driven by the fractional Laplacian. By applying a version of the three-critical-points theorem we obtain the existence of three solutions of the problem in Hα(Ω). In addition the existence of at least two nontrivial solutions are also been obtained when 2<μ<2α*.

MSC 2010: 35J60; 47J30
Key Words and Phrases: fractional Laplacian; three critical points theorem; fractional Sobolev spaces; Dirichlet problem

References

[1] M. D.M. Gonz'alez, R. Monneau, Slow motion of particle systems as a limit of a reaction-diffusion equation with half-Laplacian in dimension one. Discrete Contin. Dyn. Syst. 32, No 4 (2012), 1255-1286.Search in Google Scholar

[2] L. A. Caffarelli, S. Salsa, L. Silvestre, Regularity estimates for the solution and the free boundary of the obstacle problem for the fractional Laplacian. Invent. Math. 171, No 2 (2008), 425-461.Search in Google Scholar

[3] Q. Y. Guan, Z. M. Ma, Boundary problems for fractional Laplacians. Stoch. Dyn. 5, No 3 (2005), 385-424.Search in Google Scholar

[4] Y. Sire, E. Valdinoci, Fractional Laplacian phase transitions and boundary reactions: a geometric inequality and a symmetry result. J. Funct. Anal. 256, No 6 (2009), 1842-1864.Search in Google Scholar

[5] N. Laskin, Fractional quantum mechanics and L'evy path integrals. Phys. Lett. A. 268, No 4-6 (2000), 298-305.10.1016/S0375-9601(00)00201-2Search in Google Scholar

[6] E. Di Nezza, G. Palatucci, E. Valdinoci, Hitchhiker's guide to the fractional Sobolev spaces. Bull. Sci. Math. 136, No 5 (2012), 521-573.Search in Google Scholar

[7] B. Barrios, E. Colorado, A. de Pablo, U. S'anchez, On some critical problems for the fractional Laplacian operator. J. Differ. Equ. 252, No 11 (2012), 6133-6162.Search in Google Scholar

[8] C. Mouhot, E. Russ, Y. Sire, Fractional Poincar'e inequalities for general measures. J. Math. Pures Appl. 95, No 1 (2011), 72-84.Search in Google Scholar

[9] G. Bonanno, Some remarks on a three critical points theorem. Nonlinear Anal. 54, No 4 (2003), 651-665.Search in Google Scholar

[10] X. Cabr'e, Y. Sire, Nonlinear equations for fractional Laplacians I: Regularity, maximum principles, and Hamiltonian estimates. Ann. Inst. H. Poincar'e Anal. Non Lin'eaire31, No 1 (2014), 23-53.Search in Google Scholar

[11] J. G. Tan, The Brezis-Nirenberg type problem involving the square root of the Laplacian. Calc. Var. Partial Differ. Equ. 42, No 1-2 (2011), 21-41.10.1007/s00526-010-0378-3Search in Google Scholar

[12] R. Servadei, E. Valdinoci, Mountain pass solutions for non-local elliptic operators. J. Math. Anal. Appl. 389, No 2 (2012), 887-898.Search in Google Scholar

[13] K. Perera, R. P. Agarwal, D. O'Regan, Morse Theoretic Aspects of p-Laplacian Type Operators. American Mathematical Society, Providence (2010).10.1090/surv/161Search in Google Scholar

[14] P. Rabinowitz, 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).Search in Google Scholar

[15] X. J. Chang, Z. Q. Wang, Nodal and multiple solutions of nonlinear problems involving the fractional Laplacian. J. Differ. Equ. 256, No 8 (2014), 2965-2992.Search in Google Scholar

[16] R. Servadei, E. Valdinoci, Variational methods for non-local operators of elliptic type. Discrete Contin. Dyn. Syst. 33, No 5 (2013), 2105-2137.Search in Google Scholar

[17] M. M. Fall, T. Weth, Nonexistence results for a class of fractional elliptic boundary value problems. J. Funct. Anal. 263, No 8 (2012), 2205-2227.Search in Google Scholar

Received: 2014-4-30
Published Online: 2015-2-10

© 2015 Diogenes Co., Sofia

Articles in the same Issue

  1. FCAA Related News, Events and Books (FCAA-Volume 18-1-2015)
  2. Electrolytic Fractional Derivative and Integral of Two-Terminal Element
  3. Initial Value Problem of Fractional Integro-Differential Equations in Banach Space
  4. Examples of Analytical Solutions by Means of Mittag-Leffler Function of Fractional Black-Scholes Option Pricing Equation
  5. Multiple Solutions for a Class of Fractional Hamiltonian Systems
  6. Some Analytical and Numerical Properties of the Mittag-Leffler Functions
  7. Sobolev Type Fractional Dynamic Equations and Optimal Multi-Integral Controls with Fractional Nonlocal Conditions
  8. Systems of Nonlinear Fractional Differential Equations
  9. Existence and Multiplicity of Solutions for Nonlinear Elliptic Problems with the Fractional Laplacian
  10. Invariant Subspace Method and Exact Solutions of Certain Nonlinear Time Fractional Partial Differential Equations
  11. Filippov Lemma for a Class of Hadamard-Type Fractional Differential Inclusions
  12. New Stability Results for Partial Fractional Differential Inclusions with Not Instantaneous Impulses
  13. Several Results of Fractional Derivatives in Dʹ(R+)
  14. Modeling Extreme-Event Precursors with the Fractional Diffusion Equation
  15. Multiplicity Results for Integral Boundary Value Problems of Fractional Order with Parametric Dependence
  16. Improvements on Flat Output Characterization for Fractional Systems
  17. Nonlocal Fractional Boundary Value Problems with Slit-Strips Boundary Conditions
  18. Remarks to the Paper “On the Existence of Blow UP Solutions for a Class of Fractional Differential Equations” by Z. Bai et al., In “FCAA”, VOL. 17, NO 4 (2014), 1175-1187
https://doi.org/10.1515/fca-2015-0009
Keywords for this article
fractional Laplacian; three critical points theorem; fractional Sobolev spaces; Dirichlet problem

Articles in the same Issue

  1. FCAA Related News, Events and Books (FCAA-Volume 18-1-2015)
  2. Electrolytic Fractional Derivative and Integral of Two-Terminal Element
  3. Initial Value Problem of Fractional Integro-Differential Equations in Banach Space
  4. Examples of Analytical Solutions by Means of Mittag-Leffler Function of Fractional Black-Scholes Option Pricing Equation
  5. Multiple Solutions for a Class of Fractional Hamiltonian Systems
  6. Some Analytical and Numerical Properties of the Mittag-Leffler Functions
  7. Sobolev Type Fractional Dynamic Equations and Optimal Multi-Integral Controls with Fractional Nonlocal Conditions
  8. Systems of Nonlinear Fractional Differential Equations
  9. Existence and Multiplicity of Solutions for Nonlinear Elliptic Problems with the Fractional Laplacian
  10. Invariant Subspace Method and Exact Solutions of Certain Nonlinear Time Fractional Partial Differential Equations
  11. Filippov Lemma for a Class of Hadamard-Type Fractional Differential Inclusions
  12. New Stability Results for Partial Fractional Differential Inclusions with Not Instantaneous Impulses
  13. Several Results of Fractional Derivatives in Dʹ(R+)
  14. Modeling Extreme-Event Precursors with the Fractional Diffusion Equation
  15. Multiplicity Results for Integral Boundary Value Problems of Fractional Order with Parametric Dependence
  16. Improvements on Flat Output Characterization for Fractional Systems
  17. Nonlocal Fractional Boundary Value Problems with Slit-Strips Boundary Conditions
  18. Remarks to the Paper “On the Existence of Blow UP Solutions for a Class of Fractional Differential Equations” by Z. Bai et al., In “FCAA”, VOL. 17, NO 4 (2014), 1175-1187
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