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

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Abstract

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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Authors and Affiliations

  1. Center for Applied Mathematics, Tianjin University, Tianjin, 300072, People’s Republic of China

    Zhi-Qiang Wang

  2. Department of Mathematics and Statistics, Utah State University, Logan, UT, 84322, USA

    Zhi-Qiang Wang

  3. Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing, 100190, People’s Republic of China

    Chengxiang Zhang

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  1. Zhi-Qiang Wang
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  2. Chengxiang Zhang
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Correspondence to Zhi-Qiang Wang.

Additional information

Communicated by P. Rabinowitz

Dedicated to Professor Yiming Long on the occasion of his 70th birthday

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Wang, ZQ., Zhang, C. Convergence From Power-Law to Logarithm-Law in Nonlinear Scalar Field Equations. Arch Rational Mech Anal 231, 45–61 (2019). https://doi.org/10.1007/s00205-018-1270-0

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