Ducomet, Bernard; Nečasová, Šárka; Vasseur, Alexis. On global motions of a compressible barotropic and selfgravitating gas with density-dependent viscosities. Z. Angew. Math. Phys. 61 (2010), no. 3, 479--491.
By Eq. (12), we see readily that the authors concerns about the finite mass case, and thus $$ex int ho d xleq C. eex$$ Moreover, $$ex int ho ln ho =int ho frac{1}{ve}ln ho^ve leq int frac{1}{ve} ho ( ho^ve-1) leq frac{1}{ve} ho^{1+ve},quadforall ve>0, eex$$ where we have used the following fundamental inequality $$ex ln xleq x-1,quad forall x>0. eex$$ Taking $ve=frac{gm-1}{2}$, we have $$eex ea int ho ln ho &leq frac{2}{gm-1}int ho^frac{1+gm}{2}\ &=frac{2}{gm-1}int frac{1}{delta} ho^frac{1}{2}cdot delta ho^frac{gm}{2}\ &leq frac{1}{gm-1}int sex{frac{1}{delta^2} ho+delta^2 ho^gm}\ &leq C+frac{delta^2}{gm-1}int ho^gm,quad foralldelta>0. eea eeex$$ Choosing $delta$ sufficiently small, we can then absorb the term $int ho ln ho$.
Remark. In the above calculations, only the restriction $gm>1$ was used!