Scalar field equation with non-local diffusion

  • Patricio Felmer
  • Ignacio Vergara


In this paper we are interested on the existence of ground state solutions for fractional field equations of the form
$$\left\{\begin{array}{ll} (I - \Delta)^{\alpha}u = f(x, u) & \quad {\rm in} \, I\!R^N,\\ u > 0 & \quad {\rm in} \, I\!R^N, \quad \displaystyle\lim_{|x| \to \infty}u(x) = 0,\end{array}\right.$$
where \({\alpha \in (0,1)}\) and f is an appropriate super-linear sub-critical nonlinearity. We prove regularity, exponential decay and symmetry properties for these solutions. We also prove the existence of infinitely many bound states and, through a non-local Pohozaev identity, we prove nonexistence results in the supercritical case.

Mathematics Subject Classification

35J60 35Q55 35S05 


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  1. 1.
    Berestycki H., Lions P.: Nonlinear scalar field equations (I)(II). Arch. Ration. Mech. Anal. 82, 313–376 (1983)zbMATHMathSciNetGoogle Scholar
  2. 2.
    Caffarelli L., Silvestre L.: An extension problem related to the fractional Laplacian. Commun. Partial Differ. Equ. 32, 1245–1260 (2007)zbMATHMathSciNetCrossRefGoogle Scholar
  3. 3.
    Cheng M.: Bound state for the fractional Schrödinger equation with unbounded potential. J. Math. Phys. 53, 043507 (2012)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Coti Zelati, V., Nolasco, M.: Existence of ground states for nonlinear, pseudorelativistic Schrodinger equations. Rend. Lincei Mat. Appl. 22, 51–72 (2011)Google Scholar
  5. 5.
    Di Nezza, E., Patalluci, G., Valdinoci, E.: Hitchhiker’s guide to the fractional Sobolev spaces. Bull. Sci. Math. 136(5), 521–573 (2012)Google Scholar
  6. 6.
    Elgart A., Schlein B.: Mean field dynamics of boson stars. Commun. Pure Appl. Math. 60, 500–545 (2007)zbMATHMathSciNetCrossRefGoogle Scholar
  7. 7.
    Felmer, P., Quaas, A., Tan, J.: Positive solutions of nonlinear Schrödinger equation with the fractional laplacian. Proc. R. Soc. Edinb. Sect. A Math. 142(6), 1237–1262 (2012)Google Scholar
  8. 8.
    Felmer, P., Torres, C.: Non-linear Schrödinger equation with non-local regional diffusion. Calc. Var. Partial Differ. Equ. (To appear)Google Scholar
  9. 9.
    Frohlich J., Jonsson B., Lenzmann E.: Boson stars as solitary waves. Commun. Math. Phys. 274, 1–30 (2007)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Guo X., Xu M.: Some physical applications of fractional Schrödinger equation. J. Math. Phys. 47, 082104 (2006)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Hislop, P.D.: Exponential decay of two-body eigenfunctions: a review. In: Mathematical Physics and Quantum Field Theory, Electronic Journal of Differential Equations, pp. 265–288 (2000)Google Scholar
  12. 12.
    Laskin N.: Fractional quantum mechanics and Lévy path integrals. Phys. Lett. A 268, 298–305 (2000)zbMATHMathSciNetCrossRefGoogle Scholar
  13. 13.
    Laskin N.: Fractional Schrödinger equation. Phys. Rev. E 66, 056–108 (2002)MathSciNetCrossRefGoogle Scholar
  14. 14.
    Lieb E.H., Thirring W.: Gravitational collapse in quantum mechanics with relativistic kinetic energy. Ann. Phys. 155(2), 494–512 (1984)MathSciNetCrossRefGoogle Scholar
  15. 15.
    Lieb E.H., Yau H.-T.: The Chandrasekhar theory of stellar collapse as the limit of quantum mechanics. Commun. Math. Phys. 112, 147–174 (1987)zbMATHMathSciNetCrossRefGoogle Scholar
  16. 16.
    Lions P.L.: The concentration-compactness principle in the calculus of variations. The locally compact case. I. Ann. Inst. H. Poincaré Anal. Non Linéaire 2, 109–145 (1984)Google Scholar
  17. 17.
    Ma L., Chen D.: Radial symmetry and monotonicity for an integral equation. J. Math. Anal. Appl. 342, 943–949 (2008)zbMATHMathSciNetCrossRefGoogle Scholar
  18. 18.
    Mawhin, J, Willem, M.: Critical point theory and Hamiltonian systems. In: Applied Mathematical Sciences, vol. 74. Springer, Berlin (1989)Google Scholar
  19. 19.
    Rabinowitz P.: On a class of nonlinear Schrödinger equations. ZAMP 43, 270–291 (1992)zbMATHMathSciNetCrossRefGoogle Scholar
  20. 20.
    Servadei R., Valdinoci E.: Mountain pass solutions for non-local elliptic operators. J. Math. Anal. Appl. 389, 887–898 (2012)zbMATHMathSciNetCrossRefGoogle Scholar
  21. 21.
    Servadei, R., Valdinoci, E.: Lewy–Stampacchia type estimates for variational for variational inequalities driven by nonlocal operators. Rev. Mat. Iberoam. 29, 1091–1126 (2013)Google Scholar
  22. 22.
    Sickel, W., Skrzypczak, L.: Radial subspaces of Besov and Lizorkin–Triebel classes: extended strauss lemma and compactness of embeddings. J. Fourier Anal. Appl. 6(6), 639–662 (2000)Google Scholar
  23. 23.
    Stein, E.: Singular integrals and differentiability properties of functions. In: Princeton Mathematical Series. Princeton University Press, Princeton (1971)Google Scholar
  24. 24.
    Tan J., Wang Y., Yang J.: Nonlinear fractional field equations. Nonlinear Anal. 75, 2098–2110 (2012)zbMATHMathSciNetCrossRefGoogle Scholar
  25. 25.
    Ziemer, W.P.: Weakly Differentiable Functions. Springer, New York (1989)Google Scholar

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© Springer Basel 2015

Authors and Affiliations

  1. 1.Departamento de Ingeniería Matemática, Centro de Modelamiento, Matemático UMR2071 CNRS-UChileUniversidad de ChileSantiagoChile

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