In this case, we obtain a priori estimates on \(u_{n}\) in \(H_{0}^{1}( \Omega)\) only if f is more regular than \(L^{1}(\Omega)\). We have the following results.
Lemma 3.1
([20])
Let
\(u_{n}\)
be the solution of (2.2) with
\(0<\alpha^{-}\leqslant\alpha(x)\leqslant\alpha^{+}<1\), and suppose that
\(f\in L^{m}(\Omega)\)
with
\(m=\frac{Np}{Np-N+p+(N-p)\alpha^{-}}= (\frac{p^{*}}{1-\alpha^{-}} )^{\prime}\). Then the sequence
\(\{u_{n}\}\)
is bounded in
\(W_{0}^{1,p}(\Omega)\).
Once we have the boundedness of \(u_{n}\), we can prove an existence result for (1.1).
Theorem 3.1
([20])
Suppose that
f
is a nonnegative function in
\(L^{m}(\Omega)\) (\(f\not\equiv0\)), with
\(m=\frac{Np}{Np-N+p+(N-p)\alpha ^{-}}= (\frac{p^{*}}{1-\alpha^{-}} )'\), \(f\not\equiv0\), and let
\(0<\alpha^{-}\leqslant\alpha(x)\leqslant\alpha^{+}<1\). Then problem (1.1) has a solution
\(u\in W_{0}^{1,p}(\Omega)\)
satisfying (2.1).
The summability of u depends on the summability of f, which is proved in the next lemma.
Lemma 3.2
Suppose that
\(f\in L^{m}(\Omega)\), \(m\geqslant\frac{Np}{Np-N+p+(N-p) \alpha^{-}}\), and let
\(0<\alpha^{-}\leqslant\alpha(x)\leqslant \alpha^{+}<1\). Then the solution
u
of (1.1) given by Theorem
3.1
is such that:
-
(i)
if
\(m>\frac{N}{p}\), then
\(u\in L^{\infty}(\Omega)\);
-
(ii)
if
\(\frac{Np}{Np-N+p+(N-p)\alpha^{-}}\leqslant m<\frac {N}{p}\), then
\(u\in L^{s}(\Omega)\), \(s=\frac{Nm(\alpha^{-}+p-1)}{N-pm}\).
Proof
To prove (i), let \(k>1\) and define \(G_{k}(s)=(s-k)_{+}\). Taking \(G_{k}(u_{n})\) as a test function in (2.2), we obtain
$$ \int_{\Omega}\bigl\vert \nabla G_{k}(u_{n}) \bigr\vert ^{p}\,\mathrm{dx}\leqslant \int_{ \Omega}\bigl(\bigl\vert \nabla G_{k}(u_{n}) \bigr\vert ^{p-2}\nabla G_{k}(u_{n})\bigr)\cdot \nabla G_{k}(u_{n})\,\mathrm{dx}= \int_{\Omega}\frac{f_{n}G_{k}(u_{n})}{(u _{n}+\frac{1}{n})^{\alpha(x)}}\,\mathrm{dx}. $$
Since \(G_{k}(u_{n})\neq0\), it implies that
$$ \int_{\Omega}\bigl\vert \nabla G_{k}(u_{n}) \bigr\vert ^{p}\,\mathrm{dx}\leqslant \int_{ \Omega}fG_{k}(u_{n})\,\mathrm{dx}. $$
(3.1)
Starting from inequality (3.1), Theorem 4.2 in [21] shows that there exists a constant C (independent of n) such that
$$ \Vert u_{n}\Vert _{L^{\infty}(\Omega)}\leqslant C\Vert f\Vert _{L^{m}(\Omega)}, $$
which implies that u belongs to \(L^{\infty}(\Omega)\).
To prove (ii), noting that if \(m=\frac{Np}{Np-N+p+(N-p)\alpha^{-}}\), \(s=\frac{Np}{N-p}=p^{*}\), since \(u\in W_{0}^{1,p}(\Omega)\), the result when \(m=\frac{Np}{Np-N+p+(N-p)\alpha^{-}}\) is true by the Sobolev embedding theorem. If \(\frac{Np}{Np-N+p+(N-p)\alpha^{-}}< m< \frac{N}{p}\), letting \(\delta>1\) and choosing \(u_{n}^{p\delta-p+1}\) as a test function in (2.2), using Hölder’s inequality, we get that
$$ \begin{aligned}[b] &(p\delta-p+1) \int_{\Omega} \vert \nabla u_{n}\vert ^{p}u_{n}^{p\delta-p} \,\mathrm{dx}\\ &\quad \leqslant \int_{\{x\in\Omega, u_{n}\geqslant1\}}\frac{fu_{n}^{p \delta-p+1}}{u_{n}^{\alpha^{-}}}\,\mathrm{dx}+ \int_{\{x\in\Omega, u_{n}< 1\}}\frac{fu_{n}^{p\delta-p+1}}{u_{n} ^{\alpha^{+}}}\,\mathrm{dx}\\ &\quad = \Vert f\Vert _{L^{m}(\Omega)} \biggl( \biggl( \int_{\Omega }u_{n}^{(p\delta-p+1- \alpha^{-})m'}\,\mathrm{dx}\biggr)^{\frac{1}{m'}} \\ &\quad\quad{} + \vert \Omega \vert ^{ \frac{\alpha^{+}-\alpha^{-}}{(p\delta-p+1-\alpha^{-})m'}} \biggl( \int_{\Omega}u_{n}^{(p\delta-p+1-\alpha^{-})m'}\,\mathrm{dx}\biggr)^{\frac{p \delta-p+1-\alpha^{+}}{(p\delta-p+1-\alpha^{-})m'}} \biggr). \end{aligned} $$
(3.2)
By the Sobolev inequality (on the left-hand side), we have that
$$ \int_{\Omega} \vert \nabla u_{n}\vert ^{p}u_{n}^{p\delta -p}\,\mathrm{dx}=\frac{1}{ \delta^{p}} \int_{\Omega}\bigl\vert \nabla u_{n}^{\delta} \bigr\vert ^{p}\,\mathrm{dx}\geqslant\frac{S}{\delta^{p}} \biggl( \int_{\Omega}u_{n}^{p^{*}\delta} \,\mathrm{dx}\biggr)^{\frac{p}{p^{*}}}, $$
(3.3)
where S is the constant of the Sobolev embedding theorem. Combining with (3.2) and (3.3), we have that
$$ \begin{aligned}[b] & \frac{S(p\delta-p+1)}{\delta^{p}} \biggl( \int_{\Omega}u_{n}^{p^{*} \delta} \,\mathrm{dx}\biggr)^{\frac{p}{p^{*}}} \\ & \quad \leqslant \Vert f\Vert _{L^{m}( \Omega)} \biggl( \biggl( \int_{\Omega}u_{n}^{(p\delta-p+1-\alpha^{-})m'} \,\mathrm{dx}\biggr)^{\frac{1}{m'}} \\ &\quad\quad {} + \vert \Omega \vert ^{ \frac{\alpha^{+}-\alpha^{-}}{(p\delta-p+1-\alpha^{-})m'}} \biggl( \int_{\Omega}u_{n}^{(p\delta-p+1-\alpha^{-})m'}\,\mathrm{dx}\biggr)^{\frac{p \delta-p+1-\alpha^{+}}{(p\delta-p+1-\alpha^{-})m'}} \biggr). \end{aligned} $$
(3.4)
We choose δ in such a way that \(p^{*}\delta=(p\delta-p+1- \alpha^{-})m'\), i.e.,
$$ \delta=\frac{(\alpha^{-}+p-1)m(N-p)}{p(N-mp)}, $$
which yields that \(\delta>1\) if and only if \(\frac{Np}{Np-N+p+(N-p) \alpha^{-}}< m<\frac{N}{p}\), and that \(p^{*}\delta=\frac{Nm(\alpha ^{-}+p-1)}{N-pm}=s\). Therefore, (3.4) becomes
$$ \begin{aligned} \biggl( \int_{\Omega}u_{n}^{s}\,\mathrm{dx}\biggr)^{\frac{p}{p^{*}}} &\leqslant \frac{\delta^{p}}{S(p\delta-p+1)}\Vert f\Vert _{L^{m}(\Omega)} \biggl( \biggl( \int_{\Omega}u_{n}^{s}\,\mathrm{dx}\biggr)^{\frac{1}{m'}} \\ &\quad{} +\vert \Omega \vert ^{ \frac{\alpha^{+}-\alpha^{-}}{(p\delta-p+1-\alpha^{-})m'}} \biggl( \int_{\Omega}u_{n}^{s}\,\mathrm{dx}\biggr)^{\frac{p\delta-p+1-\alpha ^{+}}{(p\delta-p+1-\alpha^{-})m'}} \biggr), \end{aligned} $$
which implies that
$$ \biggl( \int_{\Omega}u_{n}^{s}\,\mathrm{dx}\biggr)^{\frac{p-1+\alpha^{+}}{p ^{*}\delta}} \leqslant\frac{\delta^{p}}{S(p\delta-p+1)}\Vert f\Vert _{L ^{m}(\Omega)} \biggl( \biggl( \int_{\Omega}u_{n}^{s}\,\mathrm{dx}\biggr)^{\frac{ \alpha^{+}-\alpha^{-}}{p^{*}\delta}} +\vert \Omega \vert ^{\frac{\alpha^{+}- \alpha^{-}}{p^{*}\delta}} \biggr). $$
(3.5)
Using Young’s inequality on the right-hand side in (3.5), we have that
$$ \biggl( \int_{\Omega}u_{n}^{s}\,\mathrm{dx}\biggr)^{\frac{p-1+\alpha^{+}}{p ^{*}\delta}} \leqslant\frac{\delta^{p}}{S(p\delta-p+1)}\Vert f\Vert _{L ^{m}(\Omega)} \biggl(\varepsilon \biggl( \int_{\Omega}u_{n}^{s} \,\mathrm{dx}\biggr)^{\frac{p-1+\alpha^{+}}{p^{*}\delta}} + \varepsilon^{-\frac{\alpha^{+}-\alpha^{-}}{p-1+\alpha^{-}}}+\vert \Omega \vert ^{\frac{ \alpha^{+}-\alpha^{-}}{p^{*}\delta}} \biggr), $$
where \(\varepsilon=\frac{S(p\delta-p+1)}{2\delta^{p}\Vert f\Vert _{L^{m}( \Omega)}}\). Thus, we get that
$$ \biggl( \int_{\Omega}u_{n}^{s}\,\mathrm{dx}\biggr)^{\frac{p-1+\alpha^{+}}{p ^{*}\delta}} \leqslant\frac{2\delta^{p}\Vert f\Vert _{L^{m}(\Omega)}}{S(p \delta-p+1)} \biggl( \biggl( \frac{2\delta^{p}\Vert f\Vert _{L^{m}(\Omega)}}{S(p \delta-p+1)} \biggr)^{\frac{\alpha^{+}-\alpha^{-}}{p-1+\alpha^{-}}} +\vert \Omega \vert ^{\frac{\alpha^{+}-\alpha^{-}}{p^{*}\delta}} \biggr). $$
(3.6)
Therefore, we know that \(u_{n}\) is bounded in \(L^{s}(\Omega)\), so is \(u\in L^{s}(\Omega)\). □
Theorem 3.2
Suppose that
\(f\in L^{m}(\Omega)\), \(\frac{(p-1+\alpha^{+})N}{(\alpha ^{-}+p-1)(N-p)+p(p-1+\alpha^{+})}\leqslant m<\frac{Np}{Np-N+p+(N-p) \alpha^{-}}\), and
\(0<\alpha^{-}\leqslant\alpha(x)\leqslant\alpha ^{+}<1\). Then problem (1.1) has a solution
u
in
\(W_{0}^{1,q}( \Omega)\), \(q=\frac{Nm(\alpha^{-}+p-1)}{N-m(1-\alpha^{-})}\).
Proof
The lines of our proof are that if we can prove that \(u_{n}\) is bounded in \(W_{0}^{1,q}(\Omega)\) (with q as in the statement), the existence of a solution u in \(W_{0}^{1,q}(\Omega)\) of (1.1) will be proved by passing to the limit in (2.2) as in the proof of Theorem 3.1. To prove that \(u_{n}\) is bounded in \(W_{0}^{1,q}( \Omega)\), we begin by proving that it is bounded in \(L^{s}(\Omega)\), with \(s=\frac{Nm(\alpha^{-}+p-1)}{N-pm}\). To attain this goal, we choose \(u_{n}^{p\delta-p+1}\) as a test function in (2.2) as in the statement of Lemma 3.2, where \(\frac{p-1+\alpha^{+}}{p} \leqslant\delta<1\); however, \(\nabla u_{n}^{p\delta-p+1}\) will be singular at \(u_{n}=0\). Therefore, we choose \((u_{n}+\varepsilon)^{p \delta-p+1}-\varepsilon^{p\delta-p+1}\) as a test function in (2.2), where \(\varepsilon<\frac{1}{n}\) for n fixed. We have that
$$ (p\delta-p+1) \int_{\Omega} \vert \nabla u_{n}\vert ^{p}(u_{n}+\varepsilon)^{p \delta-p}\,\mathrm{dx}\leqslant \int_{\Omega}\frac{f_{n}(u_{n}+ \varepsilon)^{p\delta-p+1}}{(u_{n}+\varepsilon)^{\alpha(x)}} \,\mathrm{dx}. $$
Since \(f_{n}\leqslant f\), we have that
$$ \begin{aligned}[b] &(p\delta-p+1) \int_{\Omega} \vert \nabla u_{n}\vert ^{p}(u_{n}+\varepsilon)^{p \delta-p}\,\mathrm{dx}\\ &\quad \leqslant \int_{\Omega}f(u_{n}+\varepsilon)^{p\delta-p+1-\alpha ^{-}} \,\mathrm{dx}+ \int_{\Omega}f(u_{n}+\varepsilon)^{p\delta-p+1- \alpha^{+}} \,\mathrm{dx}. \end{aligned} $$
(3.7)
By the Sobolev embedding theorem (\(W_{0}^{1,p}(\Omega)\hookrightarrow L^{p^{*}}(\Omega)\)) on the left-hand side, it follows that
$$ \begin{aligned}[b] \int_{\Omega} \vert \nabla u_{n}\vert ^{p}(u_{n}+\varepsilon)^{p\delta-p} \,\mathrm{dx}&= \int_{\Omega}\frac{\vert \nabla ((u_{n}+\varepsilon)^{ \delta}-\varepsilon^{\delta} )\vert ^{p}}{\delta^{p}}\,\mathrm{dx}\\ &\geqslant \frac{S}{\delta^{p}} \biggl( \int_{\Omega} \bigl((u_{n}+\varepsilon )^{\delta}-\varepsilon^{\delta} \bigr)^{p^{*}}\,\mathrm{dx}\biggr)^{\frac{p}{p ^{*}}}, \end{aligned} $$
(3.8)
where S is the best constant of the Sobolev embedding theorem. Combining (3.7) with (3.8), we have that
$$ \begin{aligned}[b] &\frac{S(p\delta-p+1)}{\delta^{p}} \biggl( \int_{\Omega} \bigl((u_{n}+ \varepsilon)^{\delta}- \varepsilon^{\delta} \bigr)^{p^{*}}\,\mathrm{dx}\biggr)^{\frac{p}{p^{*}}} \\ &\quad \leqslant \int_{\Omega}f(u_{n}+\varepsilon)^{p\delta-p+1-\alpha ^{-}} \,\mathrm{dx}+ \int_{\Omega}f(u_{n}+\varepsilon)^{p\delta-p+1- \alpha^{+}} \,\mathrm{dx}. \end{aligned} $$
(3.9)
Using Hölder’s inequality on the right-hand side, we get
$$ \begin{aligned} &\frac{S(p\delta-p+1)}{\delta^{p}} \biggl( \int_{\Omega} \bigl((u_{n}+ \varepsilon)^{\delta}- \varepsilon^{\delta} \bigr)^{p^{*}}\,\mathrm{dx}\biggr)^{\frac{p}{p^{*}}} \\ &\quad \leqslant \Vert f\Vert _{L^{m}(\Omega)} \biggl( \int_{\Omega}(u_{n}+\varepsilon)^{(p\delta-p+1-\alpha^{-})m'} \,\mathrm{dx}\biggr)^{\frac{1}{m'}} \\ &\quad\quad {} + \vert \Omega \vert ^{ \frac{\alpha^{+}-\alpha^{-}}{(p\delta-p+1-\alpha^{-})m'}}\Vert f\Vert _{L ^{m}(\Omega)} \biggl( \int_{\Omega}(u_{n}+\varepsilon)^{(p\delta-p+1- \alpha^{-})m'}\,\mathrm{dx}\biggr) ^{\frac{p\delta-p+1-\alpha^{+}}{(p \delta-p+1-\alpha^{-})m'}}. \end{aligned} $$
Letting \(\varepsilon\rightarrow0\), we get (3.4), i.e.,
$$ \begin{aligned} \biggl( \int_{\Omega}u_{n}^{p^{*}\delta} \,\mathrm{dx}\biggr)^{\frac{p}{p ^{*}}} &\leqslant \frac{\delta^{p}}{S(p\delta-p+1)}\Vert f\Vert _{L^{m}( \Omega)} \biggl( \biggl( \int_{\Omega}u_{n}^{(p\delta-p+1-\alpha^{-})m'} \,\mathrm{dx}\biggr)^{\frac{1}{m'}} \\ &\quad{} + \vert \Omega \vert ^{ \frac{\alpha^{+}-\alpha^{-}}{(p\delta-p+1-\alpha^{-})m'}} \biggl( \int_{\Omega}u_{n}^{(p\delta-p+1-\alpha^{-})m'}\,\mathrm{dx}\biggr)^{\frac{p \delta-p+1-\alpha^{+}}{(p\delta-p+1-\alpha^{-})m'}} \biggr), \end{aligned} $$
where δ is chosen in such a way that \(p^{*}\delta=(p\delta-p+1- \alpha^{-})m'\), i.e.,
$$ \delta=\frac{(\alpha^{-}+p-1)(N-p)m}{p(N-mp)}. $$
If \(m= \frac{(p-1+\alpha^{+})N}{(\alpha^{-}+p-1)(N-p)+p(p-1+\alpha^{+})}\), we choose \(\delta=\frac{p-1+\alpha^{+}}{p}\) in (3.9), and letting \(\varepsilon\rightarrow0\), we have that
$$ \biggl( \int_{\Omega}u_{n}^{p^{*}\delta} \,\mathrm{dx}\biggr)^{\frac{p}{p ^{*}}}\leqslant\frac{\delta^{p}}{S(p\delta-p+1)} \biggl( \int_{\Omega }fu_{n}^{p\delta-p+1-\alpha^{-}}\,\mathrm{dx}+ \int_{\Omega}f \,\mathrm{dx}\biggr). $$
Using Hölder’s inequality and Young’s inequality, we get that
$$ \biggl( \int_{\Omega}u_{n}^{p^{*}\delta} \,\mathrm{dx}\biggr)^{\frac{p}{p ^{*}}} \leqslant \frac{\delta^{p}\Vert f\Vert _{L^{m}(\Omega)}}{S(p\delta -p+1)} \biggl(\varepsilon \biggl( \int_{\Omega}u_{n}^{p^{*}\delta} \,\mathrm{dx}\biggr)^{\frac{p}{p^{*}}}+ \varepsilon^{- \frac{p^{*}}{pm'-p^{*}}}+\vert \Omega \vert ^{\frac{1}{m'}} \biggr), $$
where \(\varepsilon=\frac{S(p\delta-p+1)}{2\delta^{p}\Vert f\Vert _{L^{m}( \Omega)}}\). Thus we have that
$$ \biggl( \int_{\Omega}u_{n}^{p^{*}\delta} \,\mathrm{dx}\biggr)^{\frac{p}{p ^{*}}} \leqslant \frac{2\delta^{p}\Vert f\Vert _{L^{m}(\Omega)}}{S(p\delta -p+1)} \biggl( \biggl( \frac{2 \delta^{p}\Vert f\Vert _{L^{m}(\Omega)}}{S(p\delta -p+1)} \biggr)^{\frac{p^{*}}{pm'-p ^{*}}}+\vert \Omega \vert ^{\frac{1}{m'}} \biggr). $$
Therefore we obtain that \(u_{n}\) is bounded in \(L^{\frac{N(p-1+\alpha ^{+})}{N-p}}(\Omega)\), where \(\frac{N(p-1+\alpha^{+})}{N-p}\) is the value of s for \(m=\frac{(p-1+\alpha^{+})N}{(\alpha^{-}+p-1)(N-p)+p(p-1+ \alpha^{+})}\).
If \(\frac{(p-1+\alpha^{+})N}{(\alpha^{-}+p-1)(N-p)+p(p-1+\alpha ^{+})}< m<\frac{Np}{Np-N+p+(N-p) \alpha^{-}}\), it is clear that the inequality on m holds true if and only if \(\frac{p-1+\alpha^{+}}{p}<\delta<1\), starting from (3.4) and arguing as in the proof of Lemma 3.2, we also get that \(u_{n}\) is bounded in \(L^{s}(\Omega)\) with \(s=\frac{Nm( \alpha^{-}+p-1)}{N-pm}\).
The right-hand side of (3.7) is bounded with respect to n (and ε, which we take smaller than 1) by using the estimate on \(u_{n}\) in \(L^{s}(\Omega)\) and the choice of δ.
Since \(\delta<1\),
$$ \int_{\Omega}\frac{\vert \nabla u_{n}\vert ^{p}}{(u_{n}+\varepsilon)^{p-p \delta}} \,\mathrm{dx}= \int_{\Omega} \vert \nabla u_{n}\vert ^{p}(u_{n}+\varepsilon )^{p\delta-p}\,\mathrm{dx}\leqslant C. $$
If \(q=\frac{Nm(\alpha^{-}+p-1)}{N-m(1-\alpha^{-})}< p\), by Hölder’s inequality, we have that
$$ \begin{aligned}[b] \int_{\Omega} \vert \nabla u_{n}\vert ^{q}\,\mathrm{dx}&= \int _{\Omega}\frac{\vert \nabla u_{n}\vert ^{q}}{(u_{n}+\varepsilon)^{(1-\delta )q}}(u_{n}+\varepsilon )^{(1-\delta)q}\,\mathrm{dx}\\ &\leqslant \biggl( \int_{\Omega}\frac{\vert \nabla u_{n}\vert ^{p}}{(u_{n}+ \varepsilon)^{p(1-\delta)}}\,\mathrm{dx}\biggr)^{\frac{q}{p}} \biggl( \int_{\Omega}(u_{n}+\varepsilon)^{\frac{pq(1-\delta)}{p-q}} \,\mathrm{dx}\biggr)^{1-\frac{q}{p}} \\ &\leqslant C \biggl( \int_{\Omega}(u_{n}+\varepsilon)^{\frac{pq(1- \delta)}{p-q}}\,\mathrm{dx}\biggr)^{1-\frac{q}{p}}. \end{aligned} $$
(3.10)
The choice of δ and the value of q are such that \(\frac{pq(1- \delta)}{p-q}=s\), so that the right-hand side of (3.10) is bounded with respect to n and ε. Hence, \(u_{n}\) is bounded in \(W_{0}^{1,q}(\Omega)\). □
Theorem 3.3
Suppose that
\(f\in L^{m}(\Omega)\), \(\frac{1}{2-p-\alpha^{+}+p\delta }< m<\frac{(p-1+ \alpha^{+})N}{(\alpha^{-}+p-1)(N-p)+p(p-1+\alpha^{+})}\) (\(\frac{p-1+ \alpha^{-}}{p}<\delta<\frac{p-1+\alpha^{+}}{p}\)), and
\(0<\alpha ^{-}\leqslant\alpha(x)\leqslant\alpha^{+}<1\). Then problem (1.1) has a solution
u
in
\(W_{0}^{1,q}(\Omega)\), \(q=\frac{Nm( \alpha^{-}+p-1)}{N-m(1-\alpha^{-})}\).
Proof
The lines of our proof are similar to those in the proof of Theorem 3.2. We also begin by proving that \(u_{n}\) is bounded in \(L^{s}(\Omega)\), with \(s=\frac{Nm(\alpha^{-}+p-1)}{N-pm}\). To this aim, we also choose \((u_{n}+\varepsilon)^{p\delta-p+1}- \varepsilon^{p\delta-p+1}\) as a test function in (2.2), where \(\frac{p-1+\alpha^{-}}{p}<\delta<\frac{p-1+\alpha^{+}}{p}\), \(\varepsilon<\frac{1}{n}\) for n fixed. Since \(f_{n}\leqslant f\), using the Sobolev embedding theorem (\(W_{0}^{1,p}(\Omega )\hookrightarrow L^{p^{*}}(\Omega)\)) on the left-hand side again, we have that
$$ \begin{aligned} &\frac{S(p\delta-p+1)}{\delta^{p}} \biggl( \int_{\Omega} \bigl((u_{n}+ \varepsilon)^{\delta}- \varepsilon^{\delta} \bigr)^{p^{*}}\,\mathrm{dx}\biggr)^{\frac{p}{p^{*}}} \\ &\quad \leqslant \int_{\Omega}f(u_{n}+\varepsilon)^{p\delta-p+1-\alpha ^{-}} \,\mathrm{dx}+ \int_{\Omega}f(u_{n}+\varepsilon)^{p\delta-p+1- \alpha^{+}} \,\mathrm{dx}, \end{aligned} $$
where S is the best constant of the Sobolev embedding theorem.
Using Hölder’s inequality and Lemma 2.3 on the right-hand side, we get that
$$ \begin{aligned} &\frac{S(p\delta-p+1)}{\delta^{p}} \biggl( \int_{\Omega} \bigl((u_{n}+ \varepsilon)^{\delta}- \varepsilon^{\delta} \bigr)^{p^{*}}\,\mathrm{dx}\biggr)^{\frac{p}{p^{*}}} \\ &\quad \leqslant \int_{\Omega}f(u_{n}+\varepsilon)^{p\delta-p+1-\alpha ^{-}} \,\mathrm{dx}+ \int_{\Omega}\frac{f}{u_{1}^{p-1+\alpha^{+}-p\delta }}\,\mathrm{dx}\\ &\quad \leqslant \Vert f\Vert _{L^{m}(\Omega)} \biggl( \int_{\Omega }(u_{n}+\varepsilon )^{(p\delta-p+1-\alpha^{-})m'} \,\mathrm{dx}\biggr)^{\frac{1}{m'}}+C\Vert f\Vert _{L^{m}(\Omega)}. \end{aligned} $$
Letting \(\varepsilon\rightarrow0\), we have that
$$ \biggl( \int_{\Omega}u_{n}^{p^{*}\delta} \,\mathrm{dx}\biggr)^{\frac{p}{p ^{*}}} \leqslant\frac{\delta^{p}}{S(p\delta-p+1)}\Vert f\Vert _{L^{m}( \Omega)} \biggl( \biggl( \int_{\Omega}u_{n}^{(p\delta-p+1-\alpha^{-})m'} \,\mathrm{dx}\biggr)^{\frac{1}{m'}}+C \biggr), $$
(3.11)
where δ is chosen in such a way that \(p^{*}\delta=(p\delta-p+1- \alpha^{-})m'\), i.e.,
$$ \delta=\frac{(\alpha^{-}+p-1)(N-p)m}{p(N-mp)}. $$
If \(1< m<\frac{(p-1+\alpha^{+})N}{(\alpha^{-}+p-1)(N-p)+p(p-1+\alpha ^{+})}\), it is clear that the inequality on m holds true if and only if \(\frac{p-1+\alpha^{-}}{p}<\delta<\frac{p-1+\alpha^{+}}{p}\), and arguing as to the case \(m=\frac{(p-1+\alpha^{+})N}{(\alpha^{-}+p-1)(N-p)+p(p-1+ \alpha^{+})}\) in the proof of Theorem 3.2, we also obtain that \(u_{n}\) is bounded in \(L^{s}(\Omega)\), with \(s=\frac{Nm(\alpha ^{-}+p-1)}{N-pm}\).
Since \(\delta<1\),
$$ \int_{\Omega}\frac{\vert \nabla u_{n}\vert ^{p}}{(u_{n}+\varepsilon)^{p-p \delta}} \,\mathrm{dx}= \int_{\Omega} \vert \nabla u_{n}\vert ^{p}(u_{n}+\varepsilon )^{p\delta-p}\,\mathrm{dx}\leqslant C. $$
If \(q=\frac{Nm(\alpha^{-}+p-1)}{N-m(1-\alpha^{-})}< p\), similarly to the proof of Theorem 3.2, we have by Hölder’s inequality that
$$ \int_{\Omega} \vert \nabla u_{n}\vert ^{q}\,\mathrm{dx}\leqslant C \biggl( \int_{ \Omega}(u_{n}+\varepsilon)^{\frac{pq(1-\delta)}{p-q}}\,\mathrm{dx}\biggr)^{1-\frac{q}{p}}. $$
Due to the choice of δ and the value of q, the right-hand side of the above inequality is bounded with respect to n and ε. Hence, \(u_{n}\) is bounded in \(W_{0}^{1,q}(\Omega)\). □