I am reviewing the proof of Theorem 8.15 in Gilbarg-Trudinger (see below).
In equation (8.36), how can we take $N\to\infty$? Writing $2^* := \frac{2\hat{n}}{\hat{n}-2}$ and $q^\ast := \frac{2q}{q-2}$, equation (8.36) is presicely $$ \left(\int_{w \le N}\left(w^\beta - k^\beta\right)^{2^\ast} + \int_{w > N}\left(\beta N^{\beta-1}w - \beta N^\beta +N^\beta-k^\beta\right)^{2^\ast}\right)^{1/2^\ast} \le C\left(\int_{w \le N}\left(\beta w^{\beta}\right)^{q^\ast} + \int_{w > N}\left(\beta N^{\beta-1}w\right)^{q^\ast}\right)^{1/q^\ast}. $$ By the monotone convergence theorem, I can easily handle the integrals over $w \ge N$. Clearly, since $w\in L^{2^\ast}$ we also have $w\in L^{q^\ast}$; so $\int_{w> N} w^{q^\ast} \to 0$ as $N\to\infty$. However, I do not know whether $$ \int_{w > N}\left(\beta N^{\beta-1}w\right)^{q^\ast} \to 0 $$ as $N\to\infty$ due to the multiple $N^{(\beta-1)q^\ast}$.
Any advice on how to handle this integral (as well as the integral $\int_{w > N}\left(\beta N^{\beta-1}w - \beta N^\beta +N^\beta-k^\beta\right)^{2^\ast}$) would be appeciated. I thought of trying to combine them but I am not sure how to do that either.
Equation 8.3
\begin{equation} Lu=D_i(a^{ij}(x)D_ju+b^i(x)u)+c^i(x)D_iu+d(x)u \end{equation}.
Equation 8.29 \begin{equation} \begin{aligned} A^i(x,z,p) &= a^{ij}(x)p_j + b^i(x)z - f^i(x),\\ B(x,z,p) &= c^i(x)p_i + d(x)z - g(x) \end{aligned} \end{equation} Equation 8.30 For $v\ge 0, v\in C_0^1(\Omega)$
\begin{equation} \int_{\Omega}\left(D_ivA^i(x,u,Du)-vB(x,u,Du)\right)dx \le(\ge,=)0 \end{equation}
Equation 8.32
\begin{equation} \bar z=|z|+k,\qquad \bar b=\lambda^{-2}(|b|^2+|c|^2+k^{-2}|f|^2)+\lambda^{-1}(|d|+k^{-1}|g|) \end{equation}
Equation 8.33
\begin{align} p_iA^i(x,z,p) & \ge \frac{\lambda}{2}(|p|^2-2\bar b\bar z^2) \\ | \bar zB(x,z,p) | &\le \frac{\lambda}{2}\left( \epsilon|p|^2+\frac{\bar b}{\epsilon}\bar z^2\right) \end{align}
Any help will be greatly appreciated
I feel a little silly but I'll leave my answer here anyway.
If we don't split the integral between where $w \ge N$ and $w<N$, we can just apply the monotone convergence theorem to $$ f_N(x) = \left[\mathbb{1}_{w\le N}\left(w^\beta - k^\beta\right) + \mathbb{1}_{w>N}\left(\beta N^{\beta-1}w - \beta N^\beta + N^\beta - k^\beta\right)\right]^{2^{\ast}}. $$ Indeed, $(f_N)$ is an increasing sequence of non-negative functions converging pointwise to $\left(w^\beta - k^\beta\right)^{2^\ast}$.
Same idea on the right-hand side.