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Original Articles

Complete study of the existence and uniqueness of solutions for semilinear elliptic equations involving measures concentrated on boundary

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Pages 1687-1729 | Received 14 Dec 2016, Accepted 24 Dec 2016, Published online: 06 Feb 2017

References

  • Gmira A, Véron L. Boundary singularities of solutions of some nonlinear elliptic equations. Duke Math J. 1991;64:271–324.
  • Marcus M, Véron L. The boundary trace of positive solutions of semilinear elliptic equations: the subcritical case. Arch Rat Mech Anal. 1998;144:201–231.
  • Marcus M, Véron L. The boundary trace of positive solutions of semilinear elliptic equations: the supercritical case. J Math Pures Appl. 1998;77:481–524.
  • Marcus M, Véron L. Removable singularities and boundary traces. J Math Pures Appl. 2001;80:879–900.
  • Marcus M, Véron L. The boundary trace and generalized B.V.P. for semilinear elliptic equations with coercive absorption. Comm Pure Appl Math. 2003;56:689–731.
  • Bidaut-Véron MF, Vivier L. An elliptic semilinear equation with source term involving boundary measures: the subcritical case. Rev Mat Iberoamericana. 2000;16:477–513.
  • Bénilan Ph, Brezis H. Nonlinear problems related to the Thomas-Fermi equation. J Evolution Equ. 2003;3:673–770.
  • Brezis H. Some variational problems of the Thomas-Fermi type variational inequalities and complementarity problems. In: Proceedings of International School, Erice. Chichester: Wiley; 1980. p. 53–73.
  • Marcus M, Ponce AC. Reduced limits for nonlinear equations with measures. J Funct Anal. 2010;258:2316–2372.
  • Ponce AC. Selected problems on elliptic equations involving measures. 2012, arXiv:1204.0668.
  • Vazquez J. On a semilinear equation in ℝ2 involving bounded measures. Proc Roy Soc Edinburgh. 1983;95A:181–202.
  • Chen H, Felmer P, Quaas A. Large solution to elliptic equations involving fractional Laplacian. Ann Inst H Poincaré Anal Non Linéaire. 2015;32:1199–1228.
  • Chen H, Véron L. Semilinear fractional elliptic equations involving measures. J Differ Equ. 2014;257(5):1457–1486.
  • Chen H, Véron L. Semilinear fractional elliptic equations with gradient nonlinearity involving measures. J Funct Anal. 2014;266(8):5467–5492.
  • Chen H, Véron L. Weakly and strongly singular solutions of semilinear fractional elliptic equations. Asymptotic Anal. 2014;88:165–184.
  • Chen H, Yang J. Semilinear fractional elliptic equations with measures in unbounded domain. Nonlinear Analysis. 2016;145:118–142.
  • Ros-Oton X, Serra J. The Dirichlet problem for the fractional laplacian: regularity up to the boundary. J Math Pures Appl. 2014;101(3):275–302.
  • Véron L. Elliptic equations involving measures, stationary partial differential equations. Vol. I, Handbook of differential equations. Amsterdam: North-Holland; 2004. p. 593-712.
  • Brezis H, Cabré X. Some simple PDE’s without solutions. Boll Unione Mat Italiana. 1998;8:223–262.
  • Bidaut-Véron MF, Yarur C. Semilinear elliptic equations and systems with measure data: existence and a priori estimates. Adv Differ Equ. 2002;7(3):257–296.
  • Kalton NJ, Verbitsky IE. Nonlinear equations and weighted norm inequalities. Trans AMS. 1999;351:3341–3397.
  • Baras P, Pierre M. Critéres d’existence de solutions positives pour des équations semi-linéaires non monotones [Criterion of the existence of positive solutions of equations with non monotone semi-linearities]. Ann Inst H Poincaré Anal Non Linéaire. 1985;2:185–212.
  • Chen H, Felmer P, Véron L. Elliptic equations involving general subcritical source nonlinearity and measures, arXiv:1409.3067.
  • Bénilan Ph, Brezis H, Crandall M. A semilinear elliptic equation in L1(ℝN). Ann Sc Norm Sup Pisa Cl Sci. 1975;2:523–555.
  • Cignoli R, Cottlar M. An introduction to functional analysis. Amsterdam: North-Holland; 1974.
  • Chen Z, Song R. Estimates on Green functions and Poisson kernels for symmetric stable process. Math Ann. 1998;312:465–501.
  • Felmer P, Quaas A. Fundamental solutions and Liouville type theorems for nonlinear integral operators. Adv Math. 2011;226:2712–2738.
  • Di Nezza E, Palatucci G, Valdinoci E. Hitchhiker’s guide to the fractional Sobolev spaces. Bull Sci Math. 2012;136:521–573.

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