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](http://paperity.org/p/6194912/on-some-nonlinear-elliptic-problems-for-p-laplacian-in-mathbb-r-n) is a link to the paper.



Nonlinear elliptic problem involving the p-laplacian, Hölder inequality
Some typing mistakes are detected, especially in these two lines. First, it should be mentioned that at this step, we have to consider the case where $p-1< \alpha.$ Take $t=\frac{p^\*qp}{\alpha(q-p^\*)+q+(p-1)p^\*}.$ One can easily check that $\frac{t}{p}>1$ and $t<p^\*.$ Then write in the integral $u^{\alpha+kp+1}$ as $u^{\alpha+1-p+(k+1)p}$ and use H\"older inequality with $w, \frac{q}{\alpha+1-p}$ and $\frac{t}{p}$ as exponents in the right side of the inequality. We get 
$\int f(x)u^{\alpha+kp+1}dx=\int f(x)u^{\alpha+1-p}u^{(k+1)p}dx\leq \left(\int (f(x))^wdx\right)^{\frac{1}{w}} \left(\int u^qdx\right)^{\frac{\alpha+1-p}{q}}\left(\int u^{(k+1)t}dx\right)^{\frac{p}{t}},$
with $\frac{1}{w}+\frac{\alpha+1-p}{q}+\frac{p}{t}=1.$


