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

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 Poincare Anal Non Lineaire 25(1): 105–119, 2008).