Archive for Rational Mechanics and Analysis

, Volume 231, Issue 1, pp 45–61 | Cite as

Convergence From Power-Law to Logarithm-Law in Nonlinear Scalar Field Equations

  • Zhi-Qiang WangEmail author
  • Chengxiang Zhang


In this note, we uncover a relation between power-law nonlinear scalar field equations and logarithmic-law scalar field equations.We show that the ground state solutions, as p\({\downarrow}\) 2 for the power-law scalar field equations, converge to the ground state solutions of the logarithmic-law equations. As an application of this relation, we show that the associated Sobolev inequalities for imbedding from W1,2(\({\mathbb{R}^{N}}\)) into Lp (\({\mathbb{R}^{N}}\)) converge to an associated logarithmic Sobolev inequality, giving a new proof of the latter inequality due to Lieb–Loss (Analysis, 2nd edn, Graduate studies in mathematics, 14, American Mathematical Society, Providence, 2001). Using this relation, we also derive a Liouville type theorem for positive solutions of the nonlinear scalar field equation with power-law nonlinearity, giving a sharp version of an earlier result in Felmer et al. (Ann Inst Henri Poincaré Anal Non Linéaire 25(1): 105–119, 2008).


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© Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Center for Applied MathematicsTianjin UniversityTianjinPeople’s Republic of China
  2. 2.Department of Mathematics and StatisticsUtah State UniversityLoganUSA
  3. 3.Academy of Mathematics and Systems ScienceChinese Academy of SciencesBeijingPeople’s Republic of China

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