In Chapter 3 of his monograph (available on Researchgate), Kavian applies the Mountain Pass Theorem to a semilinear elliptic equation. To this aim, he needs to check that a functional satisfies the Palais-Smale condition.
On page 159, he introduces a technical Lemma 6.6, in which a weighted inequality is proved in the Sobolev space $H_0^1(\Omega)$. The weight function $m$ must satisfy a condition of the form $$m(x)|s|^\theta \leq b_0(x)|s|+b_1 \left(1+|s|^{p+1} \right), \tag{1}$$ where $b_0 \in L^{p_0}$ with $p_0 \geq \frac{2N}{N+2}$ and $b_1$ is a real number.
Then he proves Proposition 6.7, namely the validity of the Palais-Smale condition under suitable assumptions on the nonlinearity $g$. For $G(x,s)=\int_0^s g(x,t)\, dt$, Kavian shows that (the number $R$ is fixed, no problem here) $$ G(x,s) \geq \min \{G(x,-R),G(x,R)\} R^{-\theta} |s|^\theta - c_1(x), $$ for some $c_1 \in L^{p_0}$. Then he defines $m(x) = \min \{G(x,-R),G(x,R),1\} R^{-\theta}$ and applies Lemma 6.6.
Here comes my question: is it legitimate to apply the technical Lemma? My feeling is that $m$ does not satisty (1), unless $b_1$ is allowed to be a function of $x$. I can see no way to get rid of $c_1$ with a uniform constant.
