Uniform continuity and Brézis-Lieb-type splitting for superposition operators in Sobolev space. (English) Zbl 06992610

Summary: 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.


47H30 Particular nonlinear operators (superposition, Hammerstein, Nemytskiĭ, Uryson, etc.)
58E40 Variational aspects of group actions in infinite-dimensional spaces
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[1] N. Ackermann, On a periodic Schrödinger equation with nonlocal superlinear part, Math. Z. 248 (2004), no. 2, 423-443. · Zbl 1059.35037
[2] N. Ackermann, A nonlinear superposition principle and multibump solutions of periodic Schrödinger equations, J. Funct. Anal. 234 (2006), no. 2, 277-320. · Zbl 1126.35057
[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. · Zbl 1070.35083
[4] J. Appell and P. P. Zabrejko, Nonlinear Superposition Operators, Cambridge Tracts in Math. 95, Cambridge University Press, Cambridge, 1990.
[5] Y. V. Bogdanskii, Laplacian with respect to a measure on a Hilbert space and an {L_{2}}-version of the Dirichlet problem for the Poisson equation, Ukrainian Math. J. 63 (2012), no. 9, 1336-1348. · Zbl 1260.35021
[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. · Zbl 0526.46037
[7] J. Chabrowski, Weak Convergence Methods for Semilinear Elliptic Equations, World Scientific, Singapore, 1999. · Zbl 1059.35038
[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. · Zbl 1012.35001
[9] G. Da Prato and J. Zabczyk, Stochastic Equations in Infinite Dimensions, 2nd ed., Encyclopedia Math. Appl. 152, Cambridge University Press, Cambridge, 2014. · Zbl 1317.60077
[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. · Zbl 1119.35082
[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. · Zbl 1131.35075
[12] L. Gross, Potential theory on Hilbert space, J. Funct. Anal. 1 (1967), 123-181. · Zbl 0165.16403
[13] W. Kryszewski and A. Szulkin, Infinite-dimensional homology and multibump solutions, J. Fixed Point Theory Appl. 5 (2009), no. 1, 1-35. · Zbl 1189.58004
[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. · Zbl 0541.49009
[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. · Zbl 0704.49004
[16] P.-L. Lions, Solutions of Hartree-Fock equations for Coulomb systems, Comm. Math. Phys. 109 (1987), no. 1, 33-97. · Zbl 0618.35111
[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. · Zbl 0955.47024
[18] K. Tintarev and K.-H. Fieseler, Concentration Compactness. Functional-Analytic Grounds and Applications, Imperial College Press, London, 2007. · Zbl 1118.49001
[19] M. Willem, Minimax Theorems, Progr. Nonlinear Differential Equations Appl. 24, Birkhäuser, Boston, 1996.
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