Nonlinear elliptic problem involving the p-laplacian, Hölder inequality

Question

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.

Answer 1

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 $\frac{1}{w}+\frac{\alpha+1-p}{q}+\frac{p}{t}=1.$