Advertisement

The concentration-compactness principle for fractional order Sobolev spaces in unbounded domains and applications to the generalized fractional Brezis–Nirenberg problem

  • Julián Fernández Bonder
  • Nicolas Saintier
  • Analía Silva
Article
  • 62 Downloads

Abstract

In this paper we extend the well-known concentration-compactness principle for the Fractional Laplacian operator in unbounded domains. As an application we show sufficient conditions for the existence of solutions to some critical equations involving the fractional p-Laplacian in the whole \({\mathbb {R}}^n\).

Keywords

Concentration-compactness principle Unbounded domains Fractional elliptic-type problems 

Mathematics Subject Classification

35R11 46E25 45G05 

Notes

Acknowledgements

This paper was supported by Grants UBACyT 20020130100283BA, CONICET PIP 11220150100032CO and ANPCyT PICT 2012-0153. The authors are members of CONICET.

References

  1. 1.
    Akgiray, V., Geoffrey Booth, G.: The siable-law model of stock returns. J. Bus. Econ. Stat. 6(1), 51–57 (1988)Google Scholar
  2. 2.
    Aubin, T.: Problèmes isopérimétriques et espaces de Sobolev. C. R. Acad. Sci. Paris Sér. A-B, 280(5):Aii, A279–A281 (1975)Google Scholar
  3. 3.
    Barrios, B., Colorado, E., de Pablo, A., Sánchez, U.: On some critical problems for the fractional Laplacian operator. J. Differ. Equ. 252(11), 6133–6162 (2012)MathSciNetzbMATHGoogle Scholar
  4. 4.
    Barrios, B., Colorado, E., Servadei, R., Soria, F.: A critical fractional equation with concave–convex power nonlinearities. Ann. Inst. H. Poincaré Anal. Non Linéaire 32(4), 875–900 (2015)MathSciNetzbMATHGoogle Scholar
  5. 5.
    Brasco, L., Mosconi, S., Squassina, M.: Optimal decay of extremals for the fractional Sobolev inequality. Calc. Var. Partial Differ. Equ. 55(2), Art. 23–32 (2016)MathSciNetzbMATHGoogle Scholar
  6. 6.
    Brézis, H., Lieb, E.: A relation between pointwise convergence of functions and convergence of functionals. Proc. Am. Math. Soc. 88(3), 486–490 (1983)MathSciNetzbMATHGoogle Scholar
  7. 7.
    Brézis, H., Nirenberg, L.: Positive solutions of nonlinear elliptic equations involving critical Sobolev exponents. Commun. Pure Appl. Math. 36(4), 437–477 (1983)MathSciNetzbMATHGoogle Scholar
  8. 8.
    Bucur, C., Medina, M.: A fractional elliptic problem in \({\mathbb{R}}^n\) with critical growth and convex nonlinearities. ArXiv e-prints (2016)Google Scholar
  9. 9.
    Caffarelli, L., Silvestre, L.: An extension problem related to the fractional Laplacian. Commun. Partial Differ. Equ. 32(7–9), 1245–1260 (2007)MathSciNetzbMATHGoogle Scholar
  10. 10.
    Candito, P., Marano, S.A., Perera, K.: On a class of critical \((p, q)\)-Laplacian problems. Nonlinear Differ. Equ. Appl. NoDEA 22(6), 1959–1972 (2015)MathSciNetzbMATHGoogle Scholar
  11. 11.
    Chabrowski, J.: Concentration-compactness principle at infinity and semilinear elliptic equations involving critical and subcritical Sobolev exponents. Calc. Var. Partial Differ. Equ. 3(4), 493–512 (1995)MathSciNetzbMATHGoogle Scholar
  12. 12.
    Choi, W., Kim, S., Lee, K.-A.: Asymptotic behavior of solutions for nonlinear elliptic problems with the fractional Laplacian. J. Funct. Anal. 266(11), 6531–6598 (2014)MathSciNetzbMATHGoogle Scholar
  13. 13.
    Constantin, P.: Euler equations, Navier–Stokes equations and turbulence. In Mathematical foundation of turbulent viscous flows, volume 1871 of Lecture Notes in Math., pp. 1–43. Springer, Berlin (2006)Google Scholar
  14. 14.
    Di Nezza, E., Palatucci, G., Valdinoci, E.: Hitchhiker’s guide to the fractional Sobolev spaces. Bull. Sci. Math. 136(5), 521–573 (2012)MathSciNetzbMATHGoogle Scholar
  15. 15.
    Dipierro, S., Medina, M., Peral, I., Valdinoci, E.: Bifurcation results for a fractional elliptic equation with critical exponent in \(\mathbb{R}^n\). Manuscr. Math. 153(1–2), 183–230 (2017)zbMATHGoogle Scholar
  16. 16.
    Du, Q., Gunzburger, M., Lehoucq, R.B., Zhou, K.: Analysis and approximation of nonlocal diffusion problems with volume constraints. SIAM Rev. 54(4), 667–696 (2012)MathSciNetzbMATHGoogle Scholar
  17. 17.
    Eringen, A.C.: Nonlocal Continuum Field Theories. Springer, New York (2002)zbMATHGoogle Scholar
  18. 18.
    García Azorero, J., Peral Alonso, I.: Multiplicity of solutions for elliptic problems with critical exponent or with a nonsymmetric term. Trans. Am. Math. Soc 323(2), 877–895 (1991)MathSciNetzbMATHGoogle Scholar
  19. 19.
    Giacomin, G., Lebowitz, J.L.: Phase segregation dynamics in particle systems with long range interactions. I. Macroscopic limits. J. Stat. Phys. 87(1–2), 37–61 (1997)MathSciNetzbMATHGoogle Scholar
  20. 20.
    Gilboa, G., Osher, S.: Nonlocal operators with applications to image processing. Multiscale Model. Simul. 7(3), 1005–1028 (2008)MathSciNetzbMATHGoogle Scholar
  21. 21.
    Humphries, N., et al.: Environmental context explains Lévy and Brownian movement patterns of marine predators. Nature 465, 1066–1069 (2010)Google Scholar
  22. 22.
    Laskin, N.: Fractional quantum mechanics and Lévy path integrals. Phys. Lett. A 268(4–6), 298–305 (2000)MathSciNetzbMATHGoogle Scholar
  23. 23.
    Levendorski, S.: Pricing of the american put under Lévy processes. Int. J. Theor. Appl. Finance 7(03), 303–335 (2004)MathSciNetGoogle Scholar
  24. 24.
    Lieb, E.H.: Sharp constants in the Hardy–Littlewood–Sobolev and related inequalities. Ann. of Math. (2) 118(2), 349–374 (1983)MathSciNetzbMATHGoogle Scholar
  25. 25.
    Lions, P.-L.: The concentration-compactness principle in the calculus of variations. The limit case. I. Rev. Mat. Iberoam. 1(1), 145–201 (1985)MathSciNetzbMATHGoogle Scholar
  26. 26.
    Massaccesi, A., Valdinoci, E.: Is a nonlocal diffusion strategy convenient for biological populations in competition? J. Math. Biol. 74(1–2), 113–147 (2017)MathSciNetzbMATHGoogle Scholar
  27. 27.
    Metzler, R., Klafter, J.: The random walk’s guide to anomalous diffusion: a fractional dynamics approach. Phys. Rep. 339(1), 77 (2000)MathSciNetzbMATHGoogle Scholar
  28. 28.
    Mosconi, S., Perera, K., Squassina, M., Yang, Y.: The Brezis–Nirenberg problem for the fractional \(p\)-Laplacian. Calc. Var. Partial Differ. Equ. 55(4), Art. 105–25 (2016)MathSciNetzbMATHGoogle Scholar
  29. 29.
    Palatucci, G., Pisante, A.: Improved Sobolev embeddings, profile decomposition, and concentration-compactness for fractional Sobolev spaces. Calc. Var. Partial Differ. Equ. 50(3–4), 799–829 (2014)MathSciNetzbMATHGoogle Scholar
  30. 30.
    Perera, K., Squassina, M., Yang, Y.: Critical fractional \(p\)-Laplacian problems with possibly vanishing potentials. J. Math. Anal. Appl. 433(2), 818–831 (2016)MathSciNetzbMATHGoogle Scholar
  31. 31.
    Ponce, A.C.: Elliptic PDEs, measures and capacities, volume 23 of EMS Tracts in Mathematics. European Mathematical Society (EMS), Zürich, 2016. From the Poisson equations to nonlinear Thomas-Fermi problemsGoogle Scholar
  32. 32.
    Reynolds, A.M., Rhodes, C.J.: The Lévy flight paradigm: random search patterns and mechanisms. Ecology 90(4), 877–887 (2009)Google Scholar
  33. 33.
    Saintier, N., Silva, A.: Local existence conditions for an equations involving the \(p(x)\)-laplacian with critical exponent in \(\mathbb{R}^n\). Nonlinear Differ. Equ. Appl. 24, 19 (2017).  https://doi.org/10.1007/s00030-017-0441-2 zbMATHGoogle Scholar
  34. 34.
    Schoutens, W.: Lévy Processes Finance: Pricing Financial Derivatives. Willey Series in Probability and Statistics. Willey, New York (2003)Google Scholar
  35. 35.
    Servadei, R.: The Yamabe equation in a non-local setting. Adv. Nonlinear Anal. 2(3), 235–270 (2013)MathSciNetzbMATHGoogle Scholar
  36. 36.
    Servadei, R.: A critical fractional Laplace equation in the resonant case. Topol. Methods Nonlinear Anal. 43(1), 251–267 (2014)MathSciNetzbMATHGoogle Scholar
  37. 37.
    Servadei, R., Valdinoci, E.: A Brezis–Nirenberg result for non-local critical equations in low dimension. Commun. Pure Appl. Anal. 12(6), 2445–2464 (2013)MathSciNetzbMATHGoogle Scholar
  38. 38.
    Servadei, R., Valdinoci, E.: The Brezis–Nirenberg result for the fractional Laplacian. Trans. Am. Math. Soc. 367(1), 67–102 (2015)MathSciNetzbMATHGoogle Scholar
  39. 39.
    Servadei, R., Valdinoci, E.: Fractional Laplacian equations with critical Sobolev exponent. Rev. Mat. Complut. 28(3), 655–676 (2015)MathSciNetzbMATHGoogle Scholar
  40. 40.
    Shang, X., Zhang, J.: Ground states for fractional Schrödinger equations with critical growth. Nonlinearity 27(2), 187–207 (2014)MathSciNetzbMATHGoogle Scholar
  41. 41.
    Tan, J.: The Brezis–Nirenberg type problem involving the square root of the Laplacian. Calc. Var. Partial Differ. Equ. 42(1–2), 21–41 (2011)MathSciNetzbMATHGoogle Scholar
  42. 42.
    Triebel, H.: Theory of function spaces. Modern Birkhäuser Classics. Birkhäuser/Springer Basel AG, Basel, 2010. Reprint of 1983 edition [MR0730762], Also published in 1983 by Birkhäuser Verlag [MR0781540]Google Scholar
  43. 43.
    Yang, Y., Perera, K.: \(N\)-Laplacian problems with critical Trudinger–Moser nonlinearities. Ann. Sc. Norm. Super. Pisa Cl. Sci. (5) 16(4), 1123–1138 (2016)MathSciNetzbMATHGoogle Scholar
  44. 44.
    Zhou, K., Qiang, D.: Mathematical and numerical analysis of linear peridynamic models with nonlocal boundary conditions. SIAM J. Numer. Anal. 48(5), 1759–1780 (2010)MathSciNetzbMATHGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2018

Authors and Affiliations

  1. 1.Departamento de Matemática, Instituto de Matemática Luis Santaló, IMAS - CONICET Ciudad Universitaria, FCEyNUniversidad de Buenos AiresBuenos AiresArgentina
  2. 2.Departamento de Matemática, Ciudad Universitaria, FCEyNUniversidad de Buenos AiresBuenos AiresArgentina
  3. 3.Departamento de Matemática, Instituto de Matemática Aplicada de San Luis, IMASL, CONICET, FCFMyNUniversidad Nacional de San LuisSan LuisArgentina

Personalised recommendations