For
\(t>0\),
\(h>0\), let
\(S_{t, h}\) be a real function defined by
$$ S_{t, h}(\eta)= \textstyle\begin{cases} 1,& \eta>t+h , \\ \frac{\eta-t}{h}, & t\leq\eta\leq t+h , \\ 0, & |\eta|\leq t, \\ \frac{\eta+t}{h}, & -t-h\leq\eta\leq-t , \\ -1,& \eta\leq-t-h. \end{cases} $$
(3.2)
It is easy to see that
\(S_{t, h}(\phi(u))\in W_{0}^{1,p}(\Omega)\cap L^{\infty}(\Omega)\) and so
\(S_{t, h}(\phi (u))e^{\gamma_{\theta}(|u|)}\in W_{0}^{1,p}(\Omega)\cap L^{\infty}(\Omega)\), where
ϕ and
\(\gamma _{\theta}\) are defined as in (
2.1) and (
2.3). Taking
\(v=e^{\gamma_{\theta}(|u|)}S_{t, h}(\phi(u))\) as a test function in (
2.5), we have
$$\begin{aligned} \begin{aligned} &\frac{1}{h}\int_{\{t< |\phi(u)|\leq t+h\}}\phi'(u)e^{\gamma_{\theta}(|u|)}a(x,u, \nabla u)\nabla u\,\mathrm{d}x \\ &\qquad {}+\int_{\{|\phi(u)|> t\}}\bigl\vert S_{t, h}\bigl( \phi(u)\bigr)\bigr\vert \frac{\beta(|u|)}{\alpha(|u|)+\theta}e^{\gamma_{\theta}(|u|)}a(x,u,\nabla u)\nabla u\,\mathrm{d}x \\ &\qquad {} +\int_{\{|\phi(u)|> t\}}F(x,u,\nabla u)e^{\gamma_{\theta}(|u|)}S_{t, h} \bigl(\phi(u)\bigr)\,\mathrm{d}x \\ &\quad =\int_{\{|\phi(u)|> t\}}fe^{\gamma_{\theta}(|u|)}S_{t, h}\bigl( \phi(u)\bigr)\,\mathrm{d}x. \end{aligned} \end{aligned}$$
Then letting
\(\theta\rightarrow0\), we obtain
$$\begin{aligned}& \frac{1}{h}\int_{\{t< |\phi(u)|\leq t+h\}} \phi'(u)e^{\gamma (|u|)}a(x,u,\nabla u)\nabla u\,\mathrm{d}x \\& \qquad {}+\int_{\{|\phi(u)|> t\}}\bigl\vert S_{t, h}\bigl( \phi(u)\bigr)\bigr\vert \frac{\beta(|u|)}{\alpha(|u|)}e^{\gamma(|u|)}a(x,u,\nabla u)\nabla u\,\mathrm{d}x \\& \qquad {} +\int_{\{|\phi(u)|> t\}}F(x,u,\nabla u)e^{\gamma(|u|)}S_{t, h} \bigl(\phi(u)\bigr)\,\mathrm{d}x \\& \quad =\int_{\{|\phi(u)|> t\}}fe^{\gamma(|u|)}S_{t, h}\bigl( \phi(u)\bigr)\,\mathrm{d}x, \end{aligned}$$
(3.3)
where
γ is defined as in (
2.1). Notice that
\(|S_{t, h}(\phi(u))|\leq1\), by (H
1), (H
3), and applying Hölder’s inequality, we deduce from (
3.3) that
$$ \frac{1}{h}\int_{\{t< \omega\leq t+h\}}|\nabla\omega|^{p}\, \mathrm{d}x \leq\int_{\{\omega> t\}}|f|e^{\gamma(|u|)}\,\mathrm{d}x\leq \|f\| _{L^{q}(\Omega)}\biggl(\int_{\{\omega> t\}}\bigl\vert e^{\gamma(|u|)}\bigr\vert ^{q'}\,\mathrm {d}x\biggr)^{\frac{1}{q'}}, $$
where
\(\omega=|\phi(u)|=\phi(|u|)\). Let
h tend to zero, we find that
$$ -\frac{d}{dt}\int_{\{\omega>t\}}|\nabla \omega|^{p}\,\mathrm{d}x \leq\int_{\{\omega> t\}}|f|e^{\gamma(|u|)} \,\mathrm{d}x\leq\|f\| _{L^{q}(\Omega)}\biggl(\int_{\{\omega> t\}}\bigl\vert e^{\gamma(|u|)}\bigr\vert ^{q'}\,\mathrm {d}x \biggr)^{\frac{1}{q'}}. $$
(3.4)
Setting
$$z(t)=\sup_{\{|s|>\phi^{-1}(t)\}}\frac{e^{\gamma(|s|)}}{(1+\phi (|s|))^{p-1}}, $$
since
ϕ is strictly increasing and
\(\lim_{s\rightarrow \pm\infty}\phi(s)=0\), we have
$$ \lim_{t\rightarrow +\infty}z(t)=0. $$
(3.5)
Concerning the term
\((\int_{\{\omega> t\}}|e^{\gamma(|u|)}|^{q'}\,\mathrm {d}x)^{\frac{1}{q}}\), we have
$$\begin{aligned} \biggl(\int_{\{\omega> t\}}\bigl\vert e^{\gamma(|u|)} \bigr\vert ^{q'}\,\mathrm{d}x\biggr)^{\frac{1}{q}}&= \biggl(\int _{\{\omega> t\}}\biggl(\frac{e^{\gamma(|u|)}}{(1+\omega )^{p-1}}\biggr)^{q'}(1+ \omega)^{q'(p-1)}\,\mathrm{d}x\biggr)^{\frac{1}{q'}} \\ &\leq C(p,q)z(t)\biggl[\biggl(\int_{\{\omega> t\}} \omega^{q'(p-1)}\,\mathrm {d}x\biggr)^{\frac{1}{q'}}+\bigl( \mu_{\omega}(t)\bigr)^{\frac{1}{q'}}\biggr] \\ &\leq C(p,q)z(t)\biggl[\biggl(\int_{0}^{\mu_{\omega}(t)} \omega_{*}^{q'(p-1)}\,\mathrm {d}s\biggr)^{\frac{1}{q'}}+\bigl( \mu_{\omega}(t)\bigr) ^{\frac{1}{q'}}\biggr]. \end{aligned}$$
(3.6)
By (
3.4), (
3.6), and Lemma
2.1, it follows that
$$\begin{aligned} \begin{aligned}[b] &NC_{N}^{1/N}\mu_{\omega}(t)^{1-1/N} \\ &\quad \leq \bigl(-\mu_{\omega}'(t)\bigr)^{1/p'}\biggl(- \frac{d}{dt}\int_{\{u>t\}}|\nabla \omega|^{p}\, \mathrm{d}x\biggr)^{\frac{1}{p}} \\ &\quad \leq\bigl(-\mu_{\omega}'(t)\bigr)^{1/p'}C(p,q)z^{\frac{1}{p}}(t) \biggl[\biggl(\int_{0}^{\mu _{\omega}(t)}\omega_{*}^{q'(p-1)} \,\mathrm{d}s\biggr)^{\frac{1}{pq'}}+\bigl(\mu_{\omega}(t) \bigr)^{\frac{1}{pq'}}\biggr], \end{aligned} \end{aligned}$$
(3.7)
which indicates that, for
\(0<\theta<\theta+h<|\Omega|\),
$$\begin{aligned} \frac{\omega_{*}(\theta)-\omega_{*}(\theta+h)}{h} \leq&\frac{C(p,q)}{hNC_{N}^{1/N}}\int_{\omega_{*}(\theta+h)}^{\omega _{*}(\theta)}z^{\frac{1}{p}}(t) \frac{(-\mu_{\omega}'(t))^{1/p'}}{\mu_{\omega}(t)^{1-1/N}} \\ &{}\times\biggl[\biggl(\int_{0}^{\mu _{\omega}(t)} \omega_{*}^{q'(p-1)} \,\mathrm{d}s\biggr)^{\frac{1}{pq'}}+\bigl( \mu_{\omega}(t)\bigr)^{\frac{1}{pq'}}\biggr]\,\mathrm {d}t \\ < &\frac{C(p,q,N)}{h}\sup_{s\in[\omega_{*}(\theta+h),+\infty]}z^{\frac{1}{p}}(s)\int _{\omega_{*}(\theta+h)}^{\omega_{*}(\theta)} \frac{(-\mu_{\omega}'(t))^{1/p'}}{\mu_{\omega}(t)^{1-1/N}} \\ &{}\times\biggl[\biggl(\int _{0}^{\mu _{\omega}(t)}\omega_{*}^{q'(p-1)} \,\mathrm{d}s \biggr)^{\frac{1}{pq'}}+\bigl(\mu_{\omega}(t)\bigr)^{\frac{1}{pq'}}\biggr]\, \mathrm {d}t. \end{aligned}$$
Then we employ (1.15) of [
9] to get
$$\begin{aligned} \frac{\omega_{*}(\theta)-\omega_{*}(\theta+h)}{h} < &\frac{C(p,q,N)}{h} \sup_{s\in[\omega_{*}(\theta+h),+\infty]}z^{\frac{1}{p}}(s) \int_{\theta}^{\theta+h} \frac{(-\omega'_{*}(\sigma))^{1/p}}{\sigma^{1-\frac{1}{N}}} \\ &{}\times\biggl[\biggl(\int _{0}^{\sigma}\omega_{*}^{q'(p-1)} \,\mathrm{d}s \biggr)^{\frac{1}{pq'}}+\sigma^{\frac{1}{pq'}}\biggr]\,\mathrm{d}\sigma. \end{aligned}$$
Then letting
h tend to zero, we deduce that, for almost
\(\theta\in [0,|\Omega|]\),
$$ -\omega'_{*}(\theta) < C(p,q,N)\sup_{s\in[\omega_{*}(\theta),+\infty]}z^{\frac {1}{p}}(s) \frac{(-\omega'_{*}(\theta))^{1/p}}{\theta^{1-\frac{1}{N}}}\biggl[\biggl(\int_{0}^{\theta} \omega_{*}^{q'(p-1)} \,\mathrm{d}s\biggr)^{\frac{1}{pq'}}+\theta^{\frac{1}{pq'}} \biggr], $$
which leads, after applying Young’s inequality, to
$$\begin{aligned} -\omega'_{*}(\theta) &< C(p,q,N)\Bigl[\sup _{s\in[\omega_{*}(\theta),+\infty]}z^{\frac {1}{p}}(s)\Bigr]^{p'} \frac{1}{\theta^{(1-\frac{1}{N})p'}}\biggl[\biggl(\int_{0}^{\theta} \omega_{*}^{q'(p-1)} \,\mathrm{d}s\biggr)^{\frac{p'}{pq'}}+\theta^{\frac{p'}{pq'}} \biggr] \\ &\leq C(p,q,N)\sup_{s\in[\omega_{*}(\theta),+\infty]}z^{\frac{p'}{p}}(s) \frac{1}{\theta^{(1-\frac{1}{N})p'}}\bigl[\omega_{*}(0)\theta^{\frac {p'}{pq'}}+\theta^{\frac{p'}{pq'}} \bigr]. \end{aligned}$$
(3.8)
Since
\(q>\frac{N}{p}\), we have
\(q_{0}=\frac{p'}{pq'}+\frac {p'}{N}-p'+1>0\). From (
3.5), we deduce that there exists
\(t_{0}>0\) such that
$$C(p,q,N)z^{\frac{p'}{p}}(s)|\Omega|^{q_{0}}\leq\frac{1}{2}\quad \mbox{for all }s\geq t_{0}. $$
Hence, upon integration over
\([0,\mu_{\omega}(t_{0})]\), inequality (
3.8) gives
$$ \omega_{*}(0) \leq1+2t_{0}, $$
which implies that
\(\|u\|_{L^{\infty}(\Omega)}\leq\phi^{-1}(1+2t_{0})\). We observe that
\(t_{0}\) only depends on
p,
q,
N,
\(|\Omega|\),
α,
β, thus the proof of Lemma
3.1 is finished. □