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https://doi.org/10.1142/S0219199719500287Cited by:2 (Source: Crossref)

We consider the fractional mean-field equation on the interval I=(1,1)

(Δ)12u=ρeuIeudx,
subject to Dirichlet boundary conditions, and prove that existence holds if and only if ρ<2π. This requires the study of blowing-up sequences of solutions. We provide a series of tools in particular which can be used (and extended) to higher-order mean field equations of nonlocal type.

AMSC: 35R11, 35C15, 35B44, 35J61

References

  • 1. S. Baraket and F. Pacard, Construction of singular limits for a semilinear elliptic equation in dimension 2, Calc. Var. Partial Differential Equations 6 (1998) 1–38. Crossref, Web of ScienceGoogle Scholar
  • 2. R. M. Blumenthal, R. K. Getoor and D. B. Ray, On the distribution of first hits for the symmetric stable processes, Trans. Amer. Math. Soc. 99 (1961) 540–554. Google Scholar
  • 3. H. Brézis and F. Merle, Uniform estimates and blow-up behaviour for solutions of Δu = V (x)eu in two dimensions, Comm. Partial Differential Equations 16 (1991) 1223–1253. Crossref, Web of ScienceGoogle Scholar
  • 4. E. Caglioti, P. L. Lions, C. Marchioro and M. Pulvirenti, A special class of stationary flows for two-dimensional Euler equations: A statistical mechanics description, Comm. Math. Phys. 143 (1992) 501–525. Crossref, Web of ScienceGoogle Scholar
  • 5. W. Chen and C. Li, A Hopf type lemma for fractional equations, preprint (2017); arXiv:1705.04889. Google Scholar
  • 6. F. Da Lio and L. Martinazzi, The nonlocal Liouville-type equation in and conformal immersions of the disk with boundary singularities, Calc. Var. Partial Differential Equations 56 (2017), Article 152, 31 pp. Crossref, Web of ScienceGoogle Scholar
  • 7. F. Da Lio, L. Martinazzi and T. Rivière, Blow-up analysis of a nonlocal Liouville-type equation, Anal. PDE 8(7) (2015) 1757–1805. Crossref, Web of ScienceGoogle Scholar
  • 8. W. Ding, J. Jost, J. Li and G. Wang, Existence results for mean field equations, Ann. Inst. H. Poincaré Anal. Non Linéaire 16 (1999) 653–666. Crossref, Web of ScienceGoogle Scholar
  • 9. Z. Djadli, Existence result for the mean field problem on Riemann surfaces of all genuses, Commun. Contemp. Math. 10 (2008) 205–220. Link, Web of ScienceGoogle Scholar
  • 10. O. Druet and P.-D. Thizy, Multi-bumps analysis for Trudinger–Moser nonlinearities I — Quantification and location of concentration points, preprint (2017); arXiv:1710.08811. Google Scholar
  • 11. P. Esposito, M. Grossi and A. Pistoia, On the existence of blowing-up solutions for a mean field equation, Ann. Inst. H. Poincaré Anal. Non Linéaire 22 (2005) 227–257. Crossref, Web of ScienceGoogle Scholar
  • 12. A. Greco and R. Servadei, Hopf’s lemma and constrained radial symmetry for the fractional Laplacian, Math. Res. Lett. 23 (2016) 863–885. Crossref, Web of ScienceGoogle Scholar
  • 13. A. Iannizzotto, S. Mosconi and M. Squassina, Hs versus C0-weighted norms, Nonlinear Differ. Equ. Appl. 22 (2015) 477–497. Crossref, Web of ScienceGoogle Scholar
  • 14. S. Iula, A. Maalaoui and L. Martinazzi, Critical points of a fractional Moser–Trudinger embedding in dimension 1, Differ. Integr. Equ. 29 (2016) 455–492. Web of ScienceGoogle Scholar
  • 15. M. K. H. Kiessling, Statistical mechanics of classical particles with logarithmic interactions, Comm. Pure Appl. Math. 46 (1993) 27–56. Crossref, Web of ScienceGoogle Scholar
  • 16. Y. Li and I. Shafrir, Blow-up analysis for solutions of Δu = V eu in dimension 2, Indiana Univ. Math. J. 43 (1994) 1255–1270. Crossref, Web of ScienceGoogle Scholar
  • 17. A. Maalaoui, L. Martinazzi and A. Schikorra, Blow-up behavior of a fractional Adams–Moser–Trudinger-type inequality in odd dimension, Comm. Partial Differential Equations 41(10) (2016) 1593–1618. Crossref, Web of ScienceGoogle Scholar
  • 18. A. Malchiodi, Topological methods for an elliptic equation with exponential non-linearities, Discrete Contin. Dyn. Syst. 21 (2008) 277–294. Crossref, Web of ScienceGoogle Scholar
  • 19. A. Malchiodi and L. Martinazzi, Critical points of the Moser–Trudinger functional on a disk, J. Eur. Math. Soc. (JEMS) 16 (2014) 893–908. Crossref, Web of ScienceGoogle Scholar
  • 20. G. Mancini and L. Martinazzi, Extremal functions for some fractional Moser–Trudinger inequalities in dimension 1, in preparation. Google Scholar
  • 21. L. Martinazzi, Concentration-compactness phenomena in higher order Liouville’s equation, J. Funct. Anal. 256 (2009) 3743–3771. Crossref, Web of ScienceGoogle Scholar
  • 22. L. Martinazzi, Fractional Adams–Moser–Trudinger type inequalities, Nonlinear Anal. 127 (2016) 263–278. Crossref, Web of ScienceGoogle Scholar
  • 23. L. Martinazzi and M. Petrache, Asymptotics and quantization for a mean-field equation of higher order, Comm. Partial Differential Equations 35(3) (2010) 443–464. Crossref, Web of ScienceGoogle Scholar
  • 24. F. Robert and J.-C. Wei, Asymptotic behavior of a fourth order mean field equation with Dirichlet boundary condition, Indiana Univ. Math. J. 57 (2008) 2039–2060. Crossref, Web of ScienceGoogle Scholar
  • 25. X. Ros-Oton and J. Serra, The Dirichlet problem for the fractional Laplacian: Regularity up to the boundary, J. Math. Pures Appl. 101(3) (2014) 275–302. Crossref, Web of ScienceGoogle Scholar
  • 26. M. Struwe and G. Tarantello, On multivortex solutions in Chern–Simons Gauge theory, Boll. UMI (8) 1-B (1998) 109–121. Google Scholar
  • 27. T. Suzuki, Global analysis for a two-dimensional eigenvalue problem with exponential nonlinearity, Ann. Inst. H. Poincaré Anal. Non-Linéaire 9 (1992) 367–398. Crossref, Web of ScienceGoogle Scholar
  • 28. J.-C. Wei, Asymptotic behavior of a nonlinear fourth order eigenvalue problem, Comm. Partial Differential Equations 21 (1996) 1451–1467. Crossref, Web of ScienceGoogle Scholar
  • 29. V. H. Weston, On the asymptotic solution of a partial differential equation with exponential nonlinearity, SIAM J. Math. Anal. 9 (1978) 1030–1053. Crossref, Web of ScienceGoogle Scholar
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