Journal
ARCHIVE FOR RATIONAL MECHANICS AND ANALYSIS
Volume 231, Issue 1, Pages 45-61Publisher
SPRINGER
DOI: 10.1007/s00205-018-1270-0
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- [NSFC-11771324]
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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 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 W-1,W-2(RN) into L-p (RN) 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 etal. (Ann Inst Henri Poincare Anal Non Lineaire 25(1): 105-119, 2008).
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