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Advances in Nonlinear Analysis

Editor-in-Chief: Radulescu, Vicentiu / Squassina, Marco

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Uniform continuity and Brézis–Lieb-type splitting for superposition operators in Sobolev space

Nils Ackermann
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  • Instituto de Matemáticas, Universidad Nacional Autónoma de México, Circuito Exterior, C.U., 04510 México D.F., Mexico
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Published Online: 2016-10-11 | DOI: https://doi.org/10.1515/anona-2016-0123

Abstract

Using concentration-compactness arguments, we prove a variant of the Brézis–Lieb-Lemma under weaker assumptions on the nonlinearity than known before. An intermediate result on the uniform continuity of superposition operators in Sobolev space is of independent interest.

Keywords: Superposition operator; Sobolev space; uniform continuity; Brézis–Lieb splitting

MSC 2010: 47H30; 58E40

References

  • [1]

    N. Ackermann, On a periodic Schrödinger equation with nonlocal superlinear part, Math. Z. 248 (2004), no. 2, 423–443. Google Scholar

  • [2]

    N. Ackermann, A nonlinear superposition principle and multibump solutions of periodic Schrödinger equations, J. Funct. Anal. 234 (2006), no. 2, 277–320. Google Scholar

  • [3]

    N. Ackermann and T. Weth, Multibump solutions of nonlinear periodic Schrödinger equations in a degenerate setting, Commun. Contemp. Math. 7 (2005), no. 3, 269–298. Google Scholar

  • [4]

    J. Appell and P. P. Zabrejko, Nonlinear Superposition Operators, Cambridge Tracts in Math. 95, Cambridge University Press, Cambridge, 1990. Google Scholar

  • [5]

    Y. V. Bogdanskii, Laplacian with respect to a measure on a Hilbert space and an L2-version of the Dirichlet problem for the Poisson equation, Ukrainian Math. J. 63 (2012), no. 9, 1336–1348. Google Scholar

  • [6]

    H. Brézis and E. Lieb, A relation between pointwise convergence of functions and convergence of functionals, Proc. Amer. Math. Soc. 88 (1983), no. 3, 486–490. Google Scholar

  • [7]

    J. Chabrowski, Weak Convergence Methods for Semilinear Elliptic Equations, World Scientific, Singapore, 1999. Google Scholar

  • [8]

    G. Da Prato and J. Zabczyk, Second Order Partial Differential Equations in Hilbert Spaces, London Math. Soc. Lecture Note Ser. 293, Cambridge University Press, Cambridge, 2002. Google Scholar

  • [9]

    G. Da Prato and J. Zabczyk, Stochastic Equations in Infinite Dimensions, 2nd ed., Encyclopedia Math. Appl. 152, Cambridge University Press, Cambridge, 2014. Google Scholar

  • [10]

    Y. Ding and A. Szulkin, Bound states for semilinear Schrödinger equations with sign-changing potential, Calc. Var. Partial Differential Equations 29 (2007), no. 3, 397–419. Google Scholar

  • [11]

    Y. Ding and J. Wei, Semiclassical states for nonlinear Schrödinger equations with sign-changing potentials, J. Funct. Anal. 251 (2007), no. 2, 546–572. Google Scholar

  • [12]

    L. Gross, Potential theory on Hilbert space, J. Funct. Anal. 1 (1967), 123–181. Google Scholar

  • [13]

    W. Kryszewski and A. Szulkin, Infinite-dimensional homology and multibump solutions, J. Fixed Point Theory Appl. 5 (2009), no. 1, 1–35. Google Scholar

  • [14]

    P.-L. Lions, The concentration-compactness principle in the calculus of variations. The locally compact case. I, Ann. Inst. H. Poincaré Anal. Non Linéaire 1 (1984), no. 2, 109–145. Google Scholar

  • [15]

    P.-L. Lions, The concentration-compactness principle in the calculus of variations. The locally compact case. II, Ann. Inst. H. Poincaré Anal. Non Linéaire 1 (1984), no. 4, 223–283. Google Scholar

  • [16]

    P.-L. Lions, Solutions of Hartree-Fock equations for Coulomb systems, Comm. Math. Phys. 109 (1987), no. 1, 33–97. Google Scholar

  • [17]

    E. Priola, On a class of Markov type semigroups in spaces of uniformly continuous and bounded functions, Studia Math. 136 (1999), no. 3, 271–295. Google Scholar

  • [18]

    K. Tintarev and K.-H. Fieseler, Concentration Compactness. Functional-Analytic Grounds and Applications, Imperial College Press, London, 2007. Google Scholar

  • [19]

    M. Willem, Minimax Theorems, Progr. Nonlinear Differential Equations Appl. 24, Birkhäuser, Boston, 1996. Google Scholar

About the article

Received: 2016-06-02

Revised: 2016-08-15

Accepted: 2016-08-16

Published Online: 2016-10-11

Published in Print: 2018-11-01


Funding Source: Consejo Nacional de Ciencia y Tecnología

Award identifier / Grant number: 237661

Funding Source: Dirección General de Asuntos del Personal Académico, Universidad Nacional Autónoma de México

Award identifier / Grant number: IN104315

This research was partially supported by CONACYT grant 237661 and UNAM-DGAPA-PAPIIT grant IN104315 (Mexico).


Citation Information: Advances in Nonlinear Analysis, Volume 7, Issue 4, Pages 587–599, ISSN (Online) 2191-950X, ISSN (Print) 2191-9496, DOI: https://doi.org/10.1515/anona-2016-0123.

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