I am studying the paper On some nonlinear elliptic problems for $p$-Laplacian in $\mathbb{R}^n$ by Abdelouahed El Khalil and Said El Manouni Mohammed Ouanan. I have a problem understanding one step of the proof needed to get a $ L^\infty$ estimate for the solution $u$. The step goes like this:

$$ \int f(x)u^{a+kp+1} dx \leq \left(\int (f(x))^w\right)^{\frac{1}{w}} \left(\int u^q\right)^{\frac{q}{a}} \left(\int (u)^{(k+1)t}\right)^{\frac{p}{t}}\,dx. $$

where $w:=p^*/(p^* -(a+1))$ , $t:=p^* q/(a(q-p^*) +q)$ and $q > p^*$ fixed, here $p^*$ is the Sobolev exponent of $p$. $f$ is in $ L^\infty \cap L^w $ and $u$ is in $L^s$ for all $p^* \leq s < \infty $. Also $1/w +1/t +a/q =1$ and $0 < a < p^*-1$.

I think it has to be with the Hölder inequality but I can not figure out how. Is there a mistake in this? Here is a link to the paper.