Let $\Omega \subset \mathbb{R}^{N}$ be a smooth bounded domain , $g:\Omega\times\mathbb{R}\rightarrow\mathbb{R}$ is a Caratheodory function such that $g(x,t)=0$ for $t\leq0$ . Suppose that there exist function $a\in L^{r}$ and $d\in L^{p'}$ such that
$\left|g(x,t)\right|\leq a(x)t^{p-1}+d(x)$
with $r>N/p$ if $1<p\leq N$ and $r=1$ if $p>N$ ; $p'$ is Holder conjugate of $p$
Let $\left\{ u_{n}\right\} \subset W_{0}^{1,p}$ be a sequence such that $\left\Vert u_{n}\right\Vert \rightarrow\infty$ as $n\rightarrow\infty$ . Let us define $v_{n}=u_{n}/\left\Vert u_{n}\right\Vert $ . Hence $\left\Vert v_{n}\right\Vert =1$ and we may assume that $v_{n}\rightarrow v$ weakly in $W_{0}^{1,p}$ . Prove that $\dfrac{g(x,u_{n})}{\left\Vert u_{n}\right\Vert ^{p-1}}\rightarrow g_{0}$ weakly in $L^{\overline{p}}$ for some $\overline{p}>p*'$ if $p<N$ and $\overline{p}=1$ if $p\geq N$ .
Here are my efforts:
Firstly consider $p<N$ , my intension is: prove that $\dfrac{g(x,u_{n})}{\left\Vert u_{n}\right\Vert ^{p-1}}$ is bounded in $L^{\overline{p}}$ for some $\overline{p}>p*'$
$\dfrac{g(x,u_{n})}{\left\Vert u_{n}\right\Vert ^{p-1}}\leq a(x)\left|v_{n}\right|^{p-1}+\dfrac{d(x)}{\left\Vert u_{n}\right\Vert ^{p-1}}$
${\displaystyle \int_{\Omega}}\left|a(x)\left|v_{n}\right|^{p-1}\right|^{\delta}dx\leq{\displaystyle \int_{\Omega}}\left|a(x)\right|^{\delta}\left|v_{n}\right|^{(p-1)\delta}dx$
$\leq\left\Vert a(x)^{\delta}\right\Vert _{L^{\frac{N}{p\delta}}}\left\Vert \left|v_{n}\right|^{(p-1)\delta}\right\Vert _{L^{\frac{N}{N-p\delta}}}$
$\leq\left\Vert a(x)^{\delta}\right\Vert _{L^{\frac{N}{p\delta}}}\left\Vert v_{n}\right\Vert _{L^{\frac{N(p-1)\delta}{N-p\delta}}}^{(p-1)\delta}$
I expect that by Sobolev embedding, $\left\Vert a(x)^{\delta}\right\Vert _{L^{\frac{N}{p\delta}}}\left\Vert v_{n}\right\Vert _{L^{\frac{N(p-1)\delta}{N-p\delta}}}^{(p-1)\delta}\leq C\left\Vert a(x)^{\delta}\right\Vert _{L^{\frac{N}{p\delta}}}\left\Vert v_{n}\right\Vert ^{(p-1)\delta}=C\left\Vert a(x)^{\delta}\right\Vert _{L^{\frac{N}{p\delta}}}$ , so we are done.
Thus we have to pick $\delta$ such that ${\displaystyle \frac{N(p-1)\delta}{N-p\delta}}<\dfrac{Np}{N-p} \Longleftrightarrow\delta<\dfrac{pN}{Np-N+p}=p*'$
My trouble is when choose $\overline{p}=\delta$ , we are done that $\dfrac{g(x,u_{n})}{\left\Vert u_{n}\right\Vert ^{p-1}}$ is bounded in $L^{\overline{p}}$ with $\overline{p}<p*'$ (not $\overline{p}>p*'$ ). On the other hand if choose $\overline{p}'=\delta$ , we are done that $\overline{p}>p*'$ but $\dfrac{g(x,u_{n})}{\left\Vert u_{n}\right\Vert ^{p-1}}$ is bounded in $L^{\overline{p}'}$ , not $L^{\overline{p}}$ .
Please help me to take out that trouble. I appreciate your help.