Consider the following truncation-perturbation of the convection term \(f(z,\cdot,y)\):
$$ \widehat {f}(z,x,y)= \textstyle\begin{cases} f(z,x,y)+\xi_{\eta}(x^{+})^{p-1} & \mbox{if } x\leqslant\eta,\\ f(z,\eta,y)+\xi_{\eta}\eta^{p-1} & \mbox{if } \eta< x. \end{cases} $$
(3.1)
Evidently f̂ is a Carathéodoty function.
Given \(v\in C^{1}(\overline{\Omega})\), we consider the following auxiliary Robin problem:
$$ \textstyle\begin{cases} -\operatorname{div}a(Du(z))+\xi_{\eta}u(z)^{p-1}=\widehat {f}(z,u(z),Dv(z))&\mbox{in } \Omega,\\ \frac{\partial u}{\partial n_{a}}+\beta(z)u(z)^{p-1}=0&\mbox{on } \partial\Omega , u\geqslant0. \end{cases} $$
(3.2)
Proposition 3.1
If hypotheses
\(H(a)\), \(H(\beta)\)
and
\(H(f)\)
hold, then problem (3.2) admits a positive solution
\(u_{v}\in[0,\eta]\cap D_{+}\).
Proof
Let
$$\widehat {F}_{v}(z,x)= \int_{0}^{x} \widehat {f}\bigl(z,s,Dv(z)\bigr)\,ds $$
and consider the \(C^{1}\)-functional \(\widehat {\varphi }_{v}\colon W^{1,p}(\Omega)\longrightarrow \mathbb {R}\) defined by
$$\widehat {\varphi }_{v}(u)= \int_{\Omega}G(Du)\,dz+\frac{\xi_{\eta }}{p} \Vert u \Vert _{p}^{p}+\frac {1}{p} \int_{\partial\Omega }\beta(z) \vert u \vert ^{p}\,d\sigma- \int_{\Omega }\widehat {F}_{v}(z,u)\,dz $$
for all \(u\in W^{1,p}(\Omega)\). From (3.1), Corollary 2.3 and hypothesis \(H(\beta)\), we see that \(\widehat {\varphi }_{v}\) is coercive. Also, using the Sobolev embedding theorem, the compactness of the trace map and the convexity of G, we see that \(\widehat {\varphi }_{v}\) is sequentially weakly lower semicontinuous. So, by the Weierstrass–Tonelli theorem, we can find \(u_{v}\in W^{1,p}(\Omega)\) such that
$$ \widehat {\varphi }_{v}(u_{v})=\inf _{u\in W^{1,p}(\Omega)}\widehat {\varphi }_{v}(u). $$
(3.3)
Let \(M> \Vert v \Vert _{C^{1}(\overline{\Omega})}\). Hypothesis \(H(f)\)(ii) implies that given \(\varepsilon >0\), we can find \(\delta\in(0,\eta]\) such that
$$f(z,x,y)\geqslant\bigl(\eta_{M}(z)-\varepsilon \bigr)x^{q-1} \quad\mbox{for a.a. } z\in\Omega, \mbox{ all } 0\leqslant x\leqslant\delta,\mbox{ all } \vert y \vert \leqslant M, $$
so
$$\widehat {f}\bigl(z,x,Dv(z)\bigr)\geqslant\bigl(\eta_{M}(z)- \varepsilon \bigr)x^{q-1}+\xi _{\eta }x^{p-1}\quad \mbox{for a.a. } z\in\Omega, \mbox{ all } 0\leqslant x\leqslant\delta $$
(see (3.1)) and thus
$$ \widehat {F}_{v}(z,x)\geqslant\frac{1}{q}\bigl( \eta_{M}(z)-\varepsilon \bigr)x^{q}+\frac{\xi _{\eta}}{p}x^{p} \quad\textrm{for a.a. } z\in\Omega, \mbox{ all } 0\leqslant x\leqslant\delta. $$
(3.4)
Hypothesis \(H(a)\)(iv) implies that
$$ G(y)\leqslant\frac{c^{*}+\varepsilon }{q} \vert y \vert ^{q} \quad\mbox{for all } \vert y \vert \leqslant\delta. $$
(3.5)
Since \(\widehat {u}_{1}(q)\in D_{+}\), we can find \(t\in(0,1)\) small such that
$$ t \widehat {u}_{1}(q) (z)\in(0,\delta],\qquad t \bigl\vert D\widehat {u}_{1}(q) (z) \bigr\vert \leqslant\delta\quad \forall z\in \overline{\Omega}. $$
(3.6)
Then we have
$$\begin{aligned} \widehat {\varphi }_{v}\bigl(t\widehat {u}_{1}(q)\bigr) \leqslant& \frac{c^{*}+\varepsilon }{q}t^{q}\widehat {\lambda}_{1}(q) -\frac{t^{q}}{q} \int_{\Omega}\bigl(\eta_{M}(z)-\varepsilon \bigr) \widehat {u}_{1}(q)^{q}\,dz \\ \leqslant& \frac{t^{q}}{q} \biggl( \int_{\Omega}\bigl(c^{*}\widehat {\lambda}_{1}(q)-\eta _{M}(z)\bigr)\widehat {u}_{1}(q)^{q}\,dz+ \varepsilon \widehat {\lambda}_{1}(q) \biggr) \end{aligned}$$
(3.7)
(recall that \(\Vert \widehat {u}_{1}(q) \Vert _{q}=1\)). Using hypothesis \(H(f)\)(ii) and the fact that \(\widehat {u}_{1}(q)\in D_{+}\), we have
$$r_{0}= \int_{\Omega}\bigl(\eta_{M}(z)-c^{*}\widehat { \lambda}_{1}(q)\bigr)\widehat {u}_{1}(q)^{q}\,dz>0. $$
Then from (3.7) we have
$$\widehat {\varphi }_{v}\bigl(t\widehat {u}_{1}(q)\bigr) \leqslant\frac {t^{q}}{q}\bigl(-r_{0}+\varepsilon \widehat { \lambda}_{1}(q)\bigr). $$
Choosing \(\varepsilon \in(0,\frac{r_{0}}{\widehat {\lambda}_{1}(q)})\), we see that
$$\widehat {\varphi }_{v}\bigl(t\widehat {u}_{1}(q)\bigr)< 0, $$
so
$$\widehat {\varphi }_{v}(u_{v})< 0=\widehat {\varphi }_{v}(0), $$
thus
From (3.3) we have
$$\widehat {\varphi }_{v}'(u_{v})=0, $$
so
$$\begin{aligned} & \bigl\langle A(u_{v}),h\bigr\rangle + \xi_{\eta} \int_{\Omega} \vert u_{v} \vert ^{p-2}u_{v} h\,dz + \int_{\partial\Omega }\beta(z) \vert u_{v} \vert ^{p-2}u_{v} h \,d\sigma \\ &\quad = \int_{\Omega}\widehat {f}(z,u_{v},Dv)h\,dz \quad\forall h\in W^{1,p}(\Omega). \end{aligned}$$
(3.8)
In (3.8) we choose \(h=-u_{v}^{-}\in W^{1,p}(\Omega)\). Using Lemma 2.2 and (3.1), we have
$$\frac{c_{1}}{p-1} \bigl\Vert Du_{v}^{-} \bigr\Vert _{p}^{p}+\xi_{\eta} \bigl\Vert u_{v}^{-} \bigr\Vert _{p}^{p}\leqslant0, $$
so
$$u_{v}\geqslant0,\qquad u_{v}\ne0. $$
Next in (3.8) we choose \(h=(u_{v}-\eta)^{+}\in W^{1,p}(\Omega)\). Then
$$\begin{aligned} & \bigl\langle A(u_{v}),(u_{v}-\eta)^{+}\bigr\rangle + \xi_{\eta} \int_{\Omega}u_{v}^{p-1}(u_{v}- \eta)^{+}\,dz + \int_{\partial\Omega }\beta(z)u_{v}^{p-1}(u_{v}- \eta)^{+}\,d\sigma \\ &\quad = \int_{\Omega} \bigl(f(z,\eta,Dv)+\xi_{\eta} \eta^{p-1} \bigr) (u_{v}-\eta)^{+}\,dz + \int_{\Omega}\xi_{\eta}\eta^{p-1}(u_{v}- \eta)^{+}\,dz \end{aligned}$$
(see (3.1) and hypothesis \(H(f)\)(i)), so
$$\bigl\langle A(u_{v})-A(\eta),(u_{v}-\eta)^{+}\bigr\rangle + \xi_{\eta} \int_{\Omega}\bigl(u_{v}^{p-1}- \eta^{p-1}\bigr) (u_{v}-\eta)^{+}\, dz\leqslant0 $$
(see hypothesis \(H(\beta)\) and note that \(A(\eta)=0\)), thus
So, we have proved that
$$ u_{v}\in[0,\eta]. $$
(3.9)
Then, from (3.1), (3.8) and (3.9), we have
$$\bigl\langle A(u_{v}),h\bigr\rangle + \int_{\partial\Omega }\beta(z)u_{v}^{p-1}h\,d\sigma = \int_{\Omega}f(z,u_{v}, Dv)h\,dz \quad\forall h\in W^{1,p}(\Omega), $$
so
$$ \textstyle\begin{cases} -\operatorname{div}a(Du_{v}(z))=f(z,u_{v}(z),Dv(z))&\mbox{for a.a. } z\in \Omega,\\ \frac{\partial u_{v}}{\partial n_{a}}+\beta(z)u_{v}(z)^{p-1}=0&\mbox{on } \partial\Omega \end{cases} $$
(3.10)
(see Papageorgiou and Rădulescu [21]). From (3.10) and Papageorgiou and Rădulescu [17], we have
$$u_{v}\in L^{\infty}(\Omega). $$
Then from Lieberman [3] (see also Fukagai and Narukawa [23]), we have
$$u_{v}\in C_{+}\setminus\{0\}. $$
Hypothesis \(H(f)\)(iii) implies that
$$f(z,x,y)+\xi_{\eta}x^{p-1}\geqslant0\quad\mbox{for a.a. } z\in \Omega, \mbox{ all } 0\leqslant x\leqslant\eta,\mbox{ all } y\in \mathbb{R}^{N}. $$
Then from (3.10) we have
$$ \operatorname{div}a\bigl(Du_{v}(z)\bigr)\leqslant \xi_{\eta}u_{v}(z)^{p-1}\quad \mbox{for a.a. } z\in \Omega. $$
(3.11)
From (3.11), the strong maximum principle (see Pucci and Serrin [14, p. 111]) and the boundary point lemma (see Pucci and Serrin [14, p. 120]), we have \(u_{v}\in D_{+}\). □
Next we show that problem (3.2) has a smallest positive solution in the order interval \([0,\eta]\). So, let
$$S_{v}=\bigl\{ u\in W^{1,p}(\Omega): u\ne0, u\in[0,\eta] \mbox{ is a solution of (3.2)}\bigr\} . $$
From Proposition 3.1 we know that
$$\emptyset\ne S_{v}\subseteq[0,\eta]\cap D_{+}. $$
Given \(\varepsilon >0\) and \(r\in(p,p^{*})\), where
$$p^{*}= \textstyle\begin{cases} \frac{Np}{N-p} & \mbox{if } p< N,\\ +\infty& \mbox{if } N\leqslant p \end{cases} $$
(the critical Sobolev exponent corresponding to p), hypotheses \(H(f)\)(i) and (ii) imply that we can find \(c_{6}=c_{6}(\varepsilon ,r,M)>0\) (recall that \(M> \Vert v \Vert _{C^{1}(\overline{\Omega})}\)) such that
$$ f\bigl(z,x,Dv(z)\bigr) \geqslant \bigl(\eta_{M}(z)- \varepsilon \bigr)x^{q-1}-c_{6} x^{r-1}\quad \mbox{for a.a. } z\in\Omega, \mbox{ all } 0\leqslant x\leqslant \eta. $$
(3.12)
This unilateral growth restriction on \(f(z,\cdot,Dv(z))\) leads to the following auxiliary Robin problem:
$$ \textstyle\begin{cases} -\operatorname{div}a(Du(z))=(\eta_{M}(z)-\varepsilon )u(z)^{q-1}-c_{6} u(z)^{r-1}& \mbox{in } \Omega,\\ \frac{\partial u}{\partial n_{a}}+\beta(z)u(z)^{p-1}=0& \mbox{on } \partial\Omega , u\geqslant0. \end{cases} $$
(3.13)
Proposition 3.2
If hypotheses
\(H(a)\)
and
\(H(\beta)\)
hold, then for all
\(\varepsilon >0\)
small problem (3.13) admits a unique positive solution
\(u^{*}\in D_{+}\).
Proof
First we show the existence of a positive solution for problem (3.13). To this end, let \(\psi\colon W^{1,p}(\Omega)\longrightarrow \mathbb{R}\) be the \(C^{1}\)-functional defined by
$$\begin{aligned} \psi(u) = & \int_{\Omega}G(Du)\,dz +\frac{1}{p} \bigl\Vert u^{-} \bigr\Vert _{p}^{p} +\frac{1}{p} \int_{\partial\Omega }\beta(z) \vert u \vert ^{p}\,d\sigma \\ &{} -\frac{1}{q} \int_{\Omega}\bigl(\eta_{M}(z)-\varepsilon \bigr) \bigl(u^{+}\bigr)^{q}\,dz +\frac{c_{6}}{r} \bigl\Vert u^{+} \bigr\Vert _{p}^{p} \quad\forall u\in W^{1,p}(\Omega). \end{aligned}$$
Using Corollary 2.3, we obtain
$$\begin{aligned} \psi(u) \geqslant& \frac{c_{1}}{p(p-1)} \bigl\Vert Du^{+} \bigr\Vert _{p}^{p} +\frac{c_{6}}{r} \bigl\Vert u^{+} \bigr\Vert _{r}^{r} +\frac{c_{1}}{p(p-1)} \bigl\Vert Du^{-} \bigr\Vert _{p}^{p} +\frac{1}{p} \bigl\Vert u^{-} \bigr\Vert _{p}^{p} \\ &{} -\frac{1}{q} \int_{\Omega}\bigl(\eta_{M}(z)-\varepsilon \bigr) \bigl(u^{+}\bigr)^{q}\,dz, \end{aligned}$$
so
$$\psi(u)\geqslant c_{7} \Vert u \Vert ^{p}-c_{8} \bigl( \Vert u \Vert ^{q}+1\bigr) $$
for some \(c_{7},c_{8}>0\). Since \(q< p\), it follows that ψ is coercive. Also, from the Sobolev embedding theorem, the compactness of the trace map and the convexity of G, we have that ψ is sequentially weakly lower semicontinuous. Invoking the Weierstrass–Tonelli theorem, we can find \(u^{*}\in W^{1,p}(\Omega)\) such that
$$ \psi\bigl(u^{*}\bigr)=\inf_{u\in W^{1,p}(\Omega)}\psi(u). $$
(3.14)
As in the proof of Proposition 3.1, using the condition on \(\eta_{M}\) (see hypothesis \(H(f)\)(ii)), we show that, for \(t\in(0,1)\) and \(\varepsilon >0\) small, we have
$$\psi\bigl(t\widehat {u}_{1}(q)\bigr)< 0, $$
so
$$\psi\bigl(u^{*}\bigr)< 0=\psi(0) $$
(see (3.14)), thus
From (3.14) we have
$$\psi'\bigl(u^{*}\bigr)=0, $$
so, for all \(h\in W^{1,p}(\Omega)\), we have
$$\begin{aligned} & \bigl\langle A\bigl(u^{*}\bigr),h\bigr\rangle - \int_{\Omega}\bigl(\bigl(u^{*}\bigr)^{-}\bigr)^{p-1}h\,dz + \int_{\partial\Omega }\beta(z) \bigl\vert u^{*} \bigr\vert ^{p-2}u^{*} h\,d\sigma \\ &\quad = \int_{\Omega}\bigl(\eta_{M}(z)-\varepsilon \bigr) \bigl( \bigl(u^{*}\bigr)^{+}\bigr)^{q-1}h\,dz -c_{6} \int_{\Omega}\bigl(\bigl(u^{*}\bigr)^{+}\bigr)^{r-1}h\,dz. \end{aligned}$$
(3.15)
In (3.15) we choose \(h=-(u^{*})^{-}\in W^{1,p}(\Omega)\). Then
$$\frac{c_{1}}{p-1} \bigl\Vert D\bigl(u^{*}\bigr)^{-} \bigr\Vert _{p}^{p}+ \bigl\Vert \bigl(u^{*}\bigr)^{-} \bigr\Vert _{p}^{p}\leqslant0 $$
(see Lemma 2.2 and hypothesis \(H(\beta)\)), so
$$u^{*}\geqslant0,\qquad u^{*}\ne0. $$
Hence (3.15) becomes
$$\bigl\langle A\bigl(u^{*}\bigr),h\bigr\rangle + \int_{\partial\Omega }\beta(z) \bigl(u^{*}\bigr)^{p-1}h\,d\sigma = \int_{\Omega}\bigl(\eta_{M}(z)-\varepsilon \bigr) \bigl(u^{*}\bigr)^{q-1}h\,dz -c_{6} \int_{\Omega}\bigl(u^{*}\bigr)^{r-1}h\,dz $$
for all \(h\in W^{1,p}(\Omega)\), thus
$$ \textstyle\begin{cases} -\operatorname{div}a(Du^{*}(z))=(\eta_{M}-\varepsilon )(u^{*})(z)^{q-1}-c_{6}(u^{*})(z)^{r-1}& \mbox{for a.a. } z\in\Omega,\\ \frac{\partial u^{*}}{\partial n_{a}}+\beta(z)(u^{*})^{p-1}=0& \mbox{on } \partial\Omega , u\geqslant 0 \end{cases} $$
(3.16)
(see Papageorgiou and Rădulescu [21]). As before, via the nonlinear regularity theory, we have
$$u^{*}\in C_{+}\setminus\{0\}. $$
From (3.16) we have
$$\operatorname{div}a\bigl(Du^{*}(z)\bigr)\leqslant c_{6} \bigl\Vert u^{*} \bigr\Vert _{\infty }^{r-p}u^{*}(z)^{p-1}\quad \mbox{for a.a. } z\in\Omega $$
(recall \(r>p\)), so
(see Pucci and Serrin [14, pp. 111, 120]).
Next we show that this positive solution is unique. For this purpose, we introduce the integral functional \(j\colon L^{1}(\Omega)\longrightarrow \overline {\mathbb{R}}=\mathbb {R}\cup\{+\infty\}\) defined by
$$j(u)= \textstyle\begin{cases} \int_{\Omega}G(Du^{\frac{1}{q}})\,dz+\frac{1}{p}\int_{\partial \Omega }\beta(z)u^{\frac {p}{q}}\,d\sigma & \mbox{if } u\geqslant0, u^{\frac{1}{q}}\in W^{1,p}(\Omega),\\ +\infty& \mbox{otherwise}. \end{cases} $$
Let \(\operatorname{dom}j=\{u\in L^{1}(\Omega): j(u)<+\infty\}\) (the effective domain of the functional j) and consider \(u_{1},u_{2}\in\operatorname{dom}j\). We set \(u=(1-t)u_{1}+tu_{2}\) with \(t\in[0,1]\). Using Lemma 1 of Díaz and Saá [24], we have
$$\bigl\vert Du(z)^{\frac{1}{q}} \bigr\vert \leqslant \bigl( (1-t) \bigl\vert Du_{1}(z)^{\frac{1}{q}} \bigr\vert ^{q}+t \bigl\vert Du_{2}(z)^{\frac {1}{q}} \bigr\vert ^{q} \bigr)^{\frac{1}{q}}\quad\mbox{for a.a. } z\in\Omega. $$
Recalling that \(G_{0}\) is increasing, we have
$$\begin{aligned} G_{0}\bigl( \bigl\vert Du(z)^{\frac{1}{q}} \bigr\vert \bigr) \leqslant& G_{0} \bigl( \bigl( (1-t) \bigl\vert Du_{1}(z)^{\frac{1}{q}} \bigr\vert ^{q}+t \bigl\vert Du_{2}(z)^{\frac {1}{q}} \bigr\vert ^{q} \bigr)^{\frac{1}{q}} \bigr) \\ \leqslant& (1-t) G_{0}\bigl( \bigl\vert Du_{1}(z)^{\frac{1}{q}} \bigr\vert \bigr)+t G_{0}\bigl( \bigl\vert Du_{2}(z)^{\frac{1}{q}} \bigr\vert \bigr) \end{aligned}$$
(see hypothesis \(H(a)\)(iv)), so
$$G\bigl(Du(z)^{\frac{1}{q}}\bigr)\leqslant(1-t)G\bigl(Du_{1}(z)^{\frac{1}{q}} \bigr)+t G\bigl(Du_{2}(z)^{\frac {1}{q}}\bigr)\quad\mbox{for a.a. } z\in \Omega, $$
thus the map \(\operatorname{dom}j\ni u\longmapsto \int_{\Omega}G(Du^{\frac {1}{q}})\,dz\) is convex.
Since \(q< p\) and \(\beta\geqslant0\), it follows that the map \(\operatorname{dom}j\ni u\longmapsto \frac{1}{p}\int_{\partial \Omega }\beta(z)u^{\frac{p}{q}}\, d\sigma\) is convex.
Therefore the integral functional j is convex.
Suppose that \(\widetilde {u}^{*}\) is another positive solution of (3.13). As we did for \(u^{*}\), we can show that
$$\widetilde {u}^{*}\in D_{+}. $$
Hence, given \(h\in C^{1}(\overline{\Omega})\) for \(|t|\) small, we have
$$u^{*}+th\in\operatorname{dom}j\quad\mbox{and}\quad \widetilde {u}^{*}+th\in \operatorname{dom}j. $$
Using the convexity of j, we can easily see that j is Gâteaux differentiable at \(u^{*}\) and at \(\widetilde {u}^{*}\) in the direction h. Using the chain rule and the nonlinear Green’s identity (see Gasiński and Papageorgiou [1, p. 210]), we have
$$j'\bigl(u^{*}\bigr) (h)=\frac{1}{q} \int_{\Omega}\frac{-\operatorname {div}a(Du^{*})}{(u^{*})^{q-1}}h\,dz\quad\forall h\in C^{1}(\overline{\Omega}) $$
and
$$j'\bigl(\widetilde {u}^{*}\bigr) (h)=\frac{1}{q} \int_{\Omega}\frac {-\operatorname{div}a(D\widetilde {u}^{*})}{(\widetilde {u}^{*})^{q-1}}h\,dz\quad\forall h\in C^{1}(\overline{\Omega}). $$
The convexity of j implies the monotonicity of \(j'\). Therefore
$$\begin{aligned} 0 \leqslant& \frac{1}{q} \int_{\Omega} \biggl(\frac{-\operatorname {div}(Du^{*})}{(u^{*})^{q-1}}-\frac{-\operatorname{div} a(D\widetilde {u}^{*})}{(\widetilde {u}^{*})^{q-1}} \biggr) \bigl(\bigl(u^{*}\bigr)^{q}-\bigl(\widetilde {u}^{*}\bigr)^{q} \bigr)\,dz \\ = & \frac{c_{6}}{q} \int_{\Omega} \bigl(\bigl(\widetilde {u}^{*}\bigr)^{r-q}- \bigl(u^{*}\bigr)^{r-q} \bigr) \bigl(\bigl(u^{*}\bigr)^{q}-\bigl( \widetilde {u}^{*}\bigr)^{q} \bigr)\,dz \end{aligned}$$
(see (3.13)), so
$$u^{*}=\widetilde {u}^{*} $$
(since \(q< p< r\)). This proves the uniqueness of the positive solution \(u^{*}\in D_{+}\). □
Proposition 3.3
If hypotheses
\(H(a)\), \(H(\beta)\), \(H(f)\)
hold and
\(u\in S_{v}\), then
\(u^{*}\leqslant u\).
Proof
We consider the Carathéodory function \(e\colon\Omega\times\mathbb {R}\longrightarrow \mathbb{R}\) defined by
$$ e(z,x)= \textstyle\begin{cases} (\eta_{M}(z)-\varepsilon )(x^{+})^{q-1}-c_{6}(x^{+})^{r-1}+\xi_{\eta }(x^{+})^{p-1} & \mbox{if } x\leqslant u(z),\\ (\eta_{M}(z)-\varepsilon )u(z)^{q-1}-c_{6}u(z)^{r-1}+\xi_{\eta }u(z)^{p-1} & \mbox{if } u(z)< x. \end{cases} $$
(3.17)
We set
$$E(z,x)= \int_{0}^{x} e(z,s)\,ds $$
and consider the \(C^{1}\)-functional \(\tau\colon W^{1,p}(\Omega)\longrightarrow \mathbb{R}\) defined by
$$\tau(u)= \int_{\Omega}G(Du)\,dz +\frac{\xi_{\eta}}{p} \Vert u \Vert _{p}^{p} +\frac{1}{p} \int_{\partial\Omega }\beta(z) \vert u \vert ^{p}\,d\sigma - \int_{\Omega}E(z,u)\,dz \quad\forall u\in W^{1,p}(\Omega). $$
From (3.17) it is clear that τ is coercive. Also, it is sequentially weakly lower semicontinuous. So, we can find \(\widetilde {u}^{*}\in W^{1,p}(\Omega)\) such that
$$ \tau\bigl(\widetilde {u}^{*}\bigr)=\inf_{h\in W^{1,p}(\Omega)} \tau(h). $$
(3.18)
As before, since \(q< p< r\), we have
$$\tau\bigl(\widetilde {u}^{*}\bigr)< 0=\tau(0), $$
so
$$\widetilde {u}^{*}\ne0. $$
From (3.18) we have
$$\tau'\bigl(\widetilde {u}^{*}\bigr)=0, $$
so
$$\begin{aligned} & \bigl\langle A\bigl(\widetilde {u}^{*}\bigr),h\bigr\rangle + \xi_{\eta} \int_{\Omega} \bigl\vert \widetilde {u}^{*} \bigr\vert ^{p-2}\widetilde {u}^{*}h\,dz + \int_{\partial\Omega }\beta(z) \bigl\vert \widetilde {u}^{*} \bigr\vert ^{p-2}\widetilde {u}^{*}h\,d\sigma \\ &\quad = \int_{\Omega}e\bigl(z,\widetilde {u}^{*}\bigr)h\,dz \quad\forall h\in W^{1,p}(\Omega). \end{aligned}$$
(3.19)
In (3.19) first we choose \(h=-(\widetilde {u}^{*})^{-}\in W^{1,p}(\Omega)\). Then
$$\frac{c_{1}}{p-1} \bigl\Vert D\bigl(\widetilde {u}^{*}\bigr)^{-} \bigr\Vert _{p}^{p} +\xi_{\eta} \bigl\Vert \bigl(\widetilde {u}^{*}\bigr)^{-} \bigr\Vert _{p}^{p} + \int_{\partial\Omega }\beta(z) \bigl(\bigl(\widetilde {u}^{*}\bigr)^{-} \bigr)^{p}\,d\sigma=0 $$
(see (3.17)), so
$$\widetilde {u}^{*}\geqslant0,\qquad \widetilde {u}^{*}\ne0 $$
(see hypothesis \(H(\beta)\)).
Next in (3.19) we choose \(h=(\widetilde {u}^{*}-u)^{+}\in W^{1,p}(\Omega)\). Then
$$\begin{aligned} & \bigl\langle A\bigl(\widetilde {u}^{*}\bigr),\bigl(\widetilde {u}^{*}-u\bigr)^{+} \bigr\rangle +\xi_{\eta} \int_{\Omega}\bigl(\widetilde {u}^{*}\bigr)^{p-1}(\widetilde {u}-u)^{+}\,dz\\ &\qquad {} + \int_{\partial\Omega }\beta(z) \bigl(\widetilde {u}^{*}\bigr)^{p-1} \bigl(\widetilde {u}^{*}-u\bigr)^{+}\,d\sigma \\ &\quad = \int_{\Omega} \bigl( \bigl(\eta_{M}(z)-\varepsilon \bigr)u^{q-1}-c_{6} u^{r-1} \bigr) \bigl(\widetilde {u}^{*}-u\bigr)^{+}\, dz \\ &\quad \leqslant \int_{\Omega}f(z,u,Dv) \bigl(\widetilde {u}^{*}-u\bigr)^{+}\,dz \\ &\quad = \bigl\langle A(u),\bigl(\widetilde {u}^{*}-u\bigr)^{+}\bigr\rangle + \xi_{\eta} \int_{\Omega}u^{p-1}\bigl(\widetilde {u}^{*}-u\bigr)^{+}\,dz\\ &\qquad {} + \int_{\partial\Omega }\beta(z)u^{p-1}\bigl(\widetilde {u}^{*}-u \bigr)^{+}\, d\sigma \end{aligned}$$
(see (3.17), (3.12) and recall that \(u\in S_{v}\)), so
$$\bigl\langle A\bigl(\widetilde {u}^{*}\bigr)-A(u),\bigl(\widetilde {u}^{*}-u\bigr)^{+} \bigr\rangle +\xi_{\eta} \int_{\Omega} \bigl(\bigl(\widetilde {u}^{*}\bigr)^{p-1}-u^{p-1} \bigr) \bigl(\widetilde {u}^{*}-u\bigr)\, dz\leqslant0 $$
(see hypothesis \(H(\beta)\)), thus
$$\widetilde {u}^{*}\leqslant u. $$
We have proved that
$$ \widetilde {u}^{*}\in[0,u]\setminus\{0\}. $$
(3.20)
Then, from (3.17) and (3.20), equation (3.19) becomes
$$\begin{aligned} &\bigl\langle A\bigl(\widetilde {u}^{*}\bigr),h\bigr\rangle + \int_{\partial\Omega }\beta(z) \bigl(\widetilde {u}^{*}\bigr)^{p-1}h \,d\sigma \\ &\quad = \int_{\Omega} \bigl(\bigl(\eta_{M}(z)-\varepsilon \bigr) \bigl(\widetilde {u}^{*}\bigr)^{q-1}-c_{6}\bigl(\widetilde {u}^{*}\bigr)^{r-1} \bigr)h\,dz \quad\forall h\in W^{1,p}( \Omega), \end{aligned}$$
so \(\widetilde {u}^{*}=u^{*}\) (see Proposition 3.2), thus
□
Using this proposition, we can show that problem (3.2) admits a smallest positive solution \(\widehat {u}_{v}\in D_{+}\) on \([0,\eta]\).
Proposition 3.4
If hypotheses
\(H(a)\), \(H(\beta)\), \(H(f)\)
hold, then problem (3.2) admits a smallest positive solution
\(\widehat {u}_{v}\in D_{+}\).
Proof
Invoking Lemma 3.10 of Hu and Papageorgiou [22, p. 178], we can find a decreasing sequence \(\{u_{n}\}_{n\geqslant1}\subseteq S_{v}\) such that
$$ \inf S_{v}=\inf_{n\geqslant1}u_{n}. $$
(3.21)
For all \(n\geqslant1\), we have
$$ \bigl\langle A(u_{n}),h\bigr\rangle + \int_{\partial\Omega }\beta(z)u_{n}^{p-1}h\,d\sigma = \int_{\Omega}f(z,u_{n},Dv)h\,dz \quad\forall h\in W^{1,p}(\Omega), $$
(3.22)
so
$$ u^{*}\leqslant u_{n}\leqslant\eta. $$
(3.23)
Then, on account of hypotheses \(H(f)\)(i), \(H(\beta)\) and Lemma 2.2, we have that the sequence \(\{u_{n}\}_{n\geqslant1}\subseteq W^{1,p}(\Omega)\) is bounded. Passing to a subsequence, we may assume that
$$ u_{n}\stackrel{w}{\longrightarrow}\widehat {u}_{v}\quad\mbox{in }W^{1,p}(\Omega) \quad\mbox{and}\quad u_{n}\longrightarrow \widehat {u}_{v}\quad\mbox{in }L^{p}(\Omega) \mbox{ and in } L^{p}(\partial\Omega ). $$
(3.24)
In (3.22) we choose \(h=u_{n}-\widehat {u}_{v}\in W^{1,p}(\Omega )\), pass to the limit as \(n\to\infty\) and use (3.24). Then
$$\lim_{n\rightarrow+\infty}\bigl\langle A(u_{n}),u_{n}- \widehat {u}_{v}\bigr\rangle =0, $$
so
$$ u_{n}\longrightarrow \widehat {u}_{v}\quad \mbox{in }W^{1,p}(\Omega) $$
(3.25)
(see Proposition 2.5). If in (3.22) we pass to the limit as \(n\to+\infty\) and use (3.25), then
$$\bigl\langle A(\widehat {u}_{v}),h\bigr\rangle + \int_{\partial\Omega }\beta(z)\widehat {u}_{v}^{p-1}h\,d \sigma = \int_{\Omega}f(z,\widehat {u}_{v},Dv)h\,dz \quad\forall h\in W^{1,p}(\Omega), $$
so \(u^{*}\leqslant\widehat {u}_{v}\) (see (3.23)).
From the above it follows that
$$\widehat {u}_{v}\in S_{v}\quad\mbox{and}\quad \widehat {u}_{v}=\inf S_{v}. $$
□
Let
$$C=\bigl\{ u\in C^{1}(\overline{\Omega}): 0\leqslant u(z)\leqslant\eta \mbox{ for all } z\in\overline{\Omega}\bigr\} , $$
and let \(\vartheta \colon C\longrightarrow C\) be the map defined by
$$\vartheta (v)=\widehat {u}_{v}. $$
A fixed point of this map is clearly a positive solution of problem (1.1). We will produce a fixed point for ϑ using the Leray–Schauder alternative principle (see Theorem 2.9). To this end, we will need the following lemma.
Lemma 3.5
If hypotheses
\(H(a)\), \(H(\beta)\), \(H(f)\)
hold, \(\{v_{n}\}_{n\geqslant1}\subseteq C\), \(v_{n}\to v\)
in
\(C^{1}(\overline{\Omega})\)
and
\(u\in S_{v}\), then we can find
\(u_{n}\in S_{v_{n}}\)
for
\(n\geqslant1\)
such that
\(u_{n}\longrightarrow u\)
in
\(C^{1}(\overline{\Omega})\).
Proof
Consider the following nonlinear Robin problem:
$$ \textstyle\begin{cases} -\operatorname{div}a(Dw(z))+\xi_{\eta} \vert w(z) \vert ^{p-2}w(z)=\widehat {f}(z,u(z),Dv_{n}(z))&\mbox{in } \Omega,\\ \frac{\partial w}{\partial n_{a}}+\beta(z) \vert w \vert ^{p-2}w=0&\mbox{on } \partial\Omega , n\geqslant1. \end{cases} $$
(3.26)
Since \(u\in S_{v}\subseteq[0,\eta]\cap D_{+}\), we see that
$$\widehat {f}\bigl(\cdot,u(\cdot),Dv_{n}(\cdot)\bigr)\not\equiv0\quad \forall n\geqslant1 $$
(see (3.1)) and
$$\widehat {f}\bigl(z,u(z),D v_{n}(z)\bigr)\geqslant0\quad\mbox{for a.a. } z\in \Omega, \mbox{ all } n\geqslant1 $$
(see hypothesis \(H(f)\)(i)). Therefore problem (3.26) has a unique nontrivial solution \(u_{n}^{0}\in D_{+}\). Also we have
$$\begin{aligned} & \bigl\langle A\bigl(u_{n}^{0}\bigr), \bigl(u_{n}^{0}-\eta\bigr)^{+}\bigr\rangle +\xi_{\eta} \int_{\Omega}\bigl(u_{n}^{0} \bigr)^{p-1}\bigl(u_{n}^{0}-\eta\bigr)^{+}\,dz \\ &\qquad {} + \int_{\partial\Omega }\beta(z) \bigl(u_{n}^{0} \bigr)^{p-1}\bigl(u_{n}^{0}-\eta\bigr)^{+}\, d\sigma \\ &\quad = \int_{\Omega}\bigl(f(z,u,Dv_{n})+\xi_{\eta} u^{p-1}\bigr) \bigl(u_{n}^{0}-\eta\bigr)^{+}\,dz \\ &\quad \leqslant \int_{\Omega}\bigl(f(z,\eta,D v_{n})+\xi_{\eta} \eta^{p-1}\bigr) \bigl(u_{n}^{0}-\eta \bigr)^{+}\,dz \\ &\quad = \int_{\Omega}\xi_{\eta}\eta^{p-1} \bigl(u_{n}^{0}-\eta\bigr)^{+}\,dz \end{aligned}$$
(see (3.1), hypotheses \(H(f)\)(iii) and (i) and recall that \(u\in S_{v}\subseteq[0,\eta]\cap D_{+}\)), so
$$\bigl\langle A\bigl(u_{n}^{0}\bigr)-A(\eta), \bigl(u_{n}^{0}-\eta\bigr)^{+}\bigr\rangle +\xi_{\eta} \int_{\Omega}\bigl(\bigl(u_{n}^{0} \bigr)^{p-1}-\eta^{p-1}\bigr) \bigl(u_{n}^{0}- \eta\bigr)^{+}\, dz\leqslant0 $$
(see hypothesis \(H(\beta)\) and note that \(A(\eta)=0\)), thus
$$u_{n}^{0}\leqslant\eta. $$
So, we have that
$$u_{n}^{0}\in[0,\eta]\setminus\{0\}\quad\forall n\geqslant1. $$
Moreover, the nonlinear regularity theory (see Lieberman [3]) and the nonlinear maximum principle (see Pucci and Serrin [14]) imply that
$$ u_{n}^{0}\in[0,\eta]\cap D_{+}\quad\forall n \geqslant1. $$
(3.27)
We have
$$ \textstyle\begin{cases} -\operatorname{div}a(Du_{n}^{0}(z))=f(z,u(z),Dv_{n}(z))&\mbox{for a.a. } z\in \Omega,\\ \frac{\partial u_{n}^{0}}{\partial n_{a}}+\beta(z)(u_{n}^{0})^{p-1}=0& \mbox{on } \partial\Omega . \end{cases} $$
(3.28)
Then \(\{u_{n}^{0}\}_{n\geqslant1}\subseteq W^{1,p}(\Omega)\) is bounded (see (3.27), (3.28), Lemma 2.2 and hypothesis \(H(f)\)(i)). So, on account of the nonlinear regularity theory of Lieberman [3], we can find \(\mu\in(0,1)\) and \(c_{9}>0\) such that
$$u_{n}^{0}\in C^{1,\mu}(\overline{\Omega})\quad \mbox{and}\quad \bigl\Vert u_{n}^{0} \bigr\Vert _{C^{1,\mu}(\overline{\Omega})}\leqslant c_{9}\quad\forall n\geqslant1. $$
The compactness of the embedding \(C^{1,\mu}(\overline{\Omega })\subseteq C^{1}(\overline{\Omega})\) implies that we can find a subsequence \(\{u_{n_{k}}^{0}\}_{k\geqslant1}\) of the sequence \(\{u_{n}^{0}\}_{n\geqslant 1}\) such that
$$u_{n_{k}}^{0}\longrightarrow \widetilde {u}_{0}\quad \mbox{in }C^{1}(\overline{\Omega})\mbox{ as }k\to+\infty. $$
Note that
$$ \textstyle\begin{cases} -\operatorname{div}a(D\widetilde {u}^{0}(z))=f(z,u(z),Dv(z))& \mbox{for a.a. } z\in \Omega,\\ \frac{\partial\widetilde {u}^{0}}{\partial n_{a}}+\beta(z)(\widetilde {u}^{0})^{p-1}=0&\mbox{on } \partial\Omega . \end{cases} $$
(3.29)
Since \(u\in S_{v}\) solves (3.29) which has a unique solution, we infer that
$$\widetilde {u}^{0}=u\in S_{v}. $$
Hence, for the original sequence \(\{u_{n}^{0}\}_{n\geqslant1}\), we have
$$u_{n}^{0}\longrightarrow u\quad\mbox{in }C^{1}( \overline{\Omega})\mbox{ as }n\to+\infty. $$
Next consider the following nonlinear Robin problem:
$$\textstyle\begin{cases} -\operatorname{div}a(D w(z))+\xi_{\eta} \vert w(z) \vert ^{p-2}w(z)=\widehat {f}(z,u_{n}^{0}(z),Dv_{n}(z))&\mbox{in } \Omega,\\ \frac{\partial w}{\partial n_{a}}+\beta(z) \vert w \vert ^{p-2}w=0&\mbox{on } \partial\Omega , n\geqslant1. \end{cases} $$
As above, we establish that this problem has a unique solution
$$u_{n}^{1}\in[0,\eta]\cap D_{+}\quad\forall n\geqslant1. $$
Again we have
$$u_{n}^{1}\longrightarrow u\quad\mbox{in }C^{1}( \overline{\Omega})\mbox{ as }n\to+\infty. $$
Continuing this way, we generate a sequence \(\{u_{n}^{k}\}_{k,n\geqslant1}\) such that
$$\begin{aligned} &\textstyle\begin{cases} -\operatorname{div}a(D u_{n}^{k}(z))+\xi_{\eta}u_{n}^{k}(z)^{p-1}=\widehat {f}(z,u_{n}^{k-1}(z),Dv_{n}(z))&\mbox{in } \Omega,\\ \frac{\partial u_{n}^{k}}{\partial n_{a}}+\beta(z)(u_{n}^{k})^{p-1}=0& \mbox{on } \partial\Omega , n,k\geqslant1, \end{cases}\displaystyle \end{aligned}$$
(3.30)
$$\begin{aligned} &u_{n}^{k}\in[0,\eta]\cap D_{+}\quad\forall n,k\geqslant1 \end{aligned}$$
(3.31)
and
$$ u_{n}^{k}\longrightarrow u\quad\mbox{in }C^{1}(\overline{\Omega}) \mbox{ as }n\to+\infty\ \forall k\geqslant1. $$
(3.32)
Fix \(n\geqslant1\). As before we have that the sequence \(\{u_{n}^{k}\}_{k\geqslant1}\subseteq C^{1}(\overline{\Omega })\) is relatively compact. So, we can find a subsequence \(\{u_{n}^{k_{m}}\}_{m\geqslant1}\) of the sequence \(\{u_{n}^{k}\}_{k\geqslant1}\) such that
$$u_{n}^{k_{m}}\longrightarrow \widetilde {u}_{n} \quad\mbox{in }C^{1}(\overline{\Omega})\mbox{ as }m\to+\infty, $$
so
$$ \textstyle\begin{cases} -\operatorname{div}a(D \widetilde {u}_{n}(z))+\xi_{\eta}\widetilde {u}_{n}(z)^{p-1}=\widehat {f}(z,\widetilde {u}_{n}(z),Dv_{n}(z))&\mbox{for a.a. } z\in\Omega,\\ \frac{\partial\widetilde {u}_{n}}{\partial n_{a}}+\beta(z)\widetilde {u}_{n}^{p-1}=0&\mbox{on } \partial\Omega , n\geqslant1 \end{cases} $$
(3.33)
(see (3.30)). Using the nonlinear regularity theory of Lieberman [3], (3.32) and the double limit lemma (see Aubin and Ekeland [25] and Gasiński and Papageorgiou [26, p. 61]), we have
$$\widetilde {u}_{n}\longrightarrow u\quad\mbox{in } C^{1}( \overline{\Omega}), $$
so
$$\widetilde {u}_{n}\in[0,\eta]\cap D_{+}\quad\forall n\geqslant n_{0}, $$
and thus
$$\widetilde {u}_{n}\in S_{v_{n}}\quad\forall n\geqslant n_{0}\quad\mbox{and}\quad \widetilde {u}_{n}\longrightarrow u\quad\mbox{in }C^{1}(\overline {\Omega}). $$
□
Using this lemma, we can show that the map \(\vartheta \colon C\longrightarrow C\) defined earlier is compact.
Proposition 3.6
If hypotheses
\(H(a)\), \(H(\beta)\), \(H(f)\)
hold, then the map
\(\vartheta \colon C\longrightarrow C\)
is compact.
Proof
First we show that ϑ is continuous.
So, suppose that \(v_{n}\longrightarrow v\) in \(C^{1}(\overline{\Omega})\), \(\{v_{n}\}_{n\geqslant 1}\subseteq C\), \(v\in C\), and let \(\widehat {u}_{n}=\vartheta (v_{n})\) for \(n\geqslant1\). We have
$$ \textstyle\begin{cases} -\operatorname{div}a(D\widehat {u}_{n}(z))=f(z,\widehat {u}_{n}(z),Dv_{n}(z))&\mbox{for a.a. } z\in\Omega,\\ \frac{\partial\widehat {u}_{n}}{\partial n_{a}}+\beta(z)\widehat {u}_{n}(z)^{p-1}=0&\mbox{on } \partial\Omega , \widehat{u}_{n}\in[0,\eta], n\geqslant1. \end{cases} $$
(3.34)
From (3.34) we see that \(\{\widehat {u}_{n}\}_{n\geqslant1}\subseteq W^{1,p}(\Omega)\) is bounded and so, according to Lieberman [3], we can find \(\tau\in(0,1)\) and \(c_{10}>0\) such that
$$\widehat {u}_{n}\in C^{1,\tau}(\overline{\Omega})\quad\mbox{and}\quad \Vert \widehat {u}_{n} \Vert _{C^{1,\tau}(\overline{\Omega})}\leqslant c_{10}\quad\forall n\geqslant1. $$
So, we may assume that
$$ \widehat {u}_{n}\longrightarrow \widehat {u}\quad \mbox{in }C^{1}(\overline{\Omega})\mbox{ as }n\to+\infty. $$
(3.35)
In (3.34) we pass to the limit as \(n\to\infty\) and use (3.35). Then
$$ \textstyle\begin{cases} -\operatorname{div}a(D\widehat {u}(z))=f(z,\widehat {u}(z),Dv(z))&\mbox{for a.a. } z\in\Omega,\\ \frac{\partial\widehat {u}}{\partial n_{a}}+\beta(z)\widehat {u}(z)^{p-1}=0&\mbox{on } \partial\Omega . \end{cases} $$
(3.36)
From Proposition 3.3 we have
$$u^{*}\leqslant\widehat {u}_{n}\quad\forall n\geqslant1 $$
(in this case \(M>\sup_{n\geqslant1} \Vert v_{n} \Vert _{C^{1}(\overline {\Omega})}\)), so
$$u^{*}\leqslant\widehat {u} $$
(see (3.35)), thus
$$ \widehat {u}\in S_{v}. $$
(3.37)
We claim that \(\widehat {u}=\vartheta (v)\). According to Lemma 3.5, we can find \(u_{n}\in S_{v_{n}}\), \(n\geqslant1\), such that
$$ u_{n}\longrightarrow \vartheta (v)\quad\mbox{in }C^{1}(\overline {\Omega})\mbox{ as }n\to+\infty. $$
(3.38)
We have
$$\widehat {u}_{n}=\vartheta (v_{n})\leqslant u_{n}\quad\forall n\geqslant1, $$
so
$$\widehat {u}\leqslant\vartheta (v) $$
(see (3.35) and (3.38)), thus
$$\widehat {u}=\vartheta (v) $$
(see (3.37)), and hence ϑ is continuous.
Next we show that ϑ maps bounded sets in C to relatively compact subsets of C. So, let \(B\subseteq C\) be bounded in \(C^{1}(\overline{\Omega})\). As above, we have that the set \(\vartheta (B)\subseteq W^{1,p}(\Omega)\) is bounded. But then the nonlinear regularity theory of Lieberman [3] and the compactness of the embedding \(C^{1,s}(\overline{\Omega})\subseteq C^{1}(\overline{\Omega})\) (with \(0< s<1\)) imply that the set \(\vartheta (B)\subseteq C^{1}(\overline{\Omega})\) is relatively compact, thus ϑ is compact. □
Now we are ready for the existence theorem.
Theorem 3.7
If hypotheses
\(H(a)\), \(H(\beta)\), \(H(f)\)
hold, then problem (1.1) admits a solution
\(\widehat {u}\in[0,\eta ]\cap D_{+}\).
Proof
We consider the set
$$S(\vartheta )=\bigl\{ u\in C: u=\lambda\vartheta (u), 0< \lambda< 1\bigr\} . $$
If \(u\in S(\vartheta )\), then
$$\frac{1}{\lambda} u=\vartheta (u), $$
so
$$ \biggl\langle A\biggl(\frac{1}{\lambda} u\biggr),h\biggr\rangle + \int_{\partial\Omega }\beta(z) \biggl(\frac{u}{\lambda} \biggr)^{p-1}h\, d\sigma = \int_{\Omega}f\biggl(z,\frac{u}{\lambda},Du\biggr)h\,dz \quad \forall h\in W^{1,p}(\Omega). $$
(3.39)
In (3.39) we choose \(h=\frac{u}{\lambda}\in W^{1,p}(\Omega)\). Using Lemma 2.2 and hypothesis \(H(\beta)\), we have
$$\begin{aligned} \frac{c_{1}}{p-1} \biggl\Vert D\biggl(\frac{u}{\lambda}\biggr) \biggr\Vert _{p}^{p} \leqslant& \int_{\Omega}f\biggl(z,\frac{u}{\lambda},Du\biggr) \frac{u}{\lambda} \,dz \leqslant \int_{\Omega}f(z,u,Du)\frac{u}{\lambda^{p}}\,dz \\ \leqslant& \int_{\Omega}f\biggl(z,u,D\biggl(\frac{u}{\lambda}\biggr)\biggr)u \,dz \leqslant \int_{\Omega} \biggl(\widetilde {c}_{1}+\widetilde {c}_{2} \biggl\vert D\biggl(\frac {u}{\lambda}\biggr) \biggr\vert ^{p} \biggr)\,dz \end{aligned}$$
(see (2.4), hypotheses \(H(f)\)(iii) and (i)). Recalling that \(\widetilde {c}_{2}<\frac{\widetilde {c}_{1}}{p-1}\) (see hypothesis \(H(f)\)(i)), we have
$$\biggl\Vert D\biggl(\frac{u}{\lambda}\biggr) \biggr\Vert _{p} \leqslant c_{11}\quad\forall \lambda\in(0,1), $$
for some \(c_{11}>0\), thus
$$ \biggl\{ D\biggl(\frac{u}{\lambda}\biggr)\biggr\} _{u\in S(\vartheta )} \subseteq L^{p}\bigl(\Omega; \mathbb{R}^{N}\bigr)\quad \mbox{is bounded}. $$
(3.40)
As above, from (3.39) with \(h=\frac{u}{\lambda} \in W^{1,p}(\Omega)\), using hypotheses \(H(f)\)(i), (iii) and (3.40), we obtain
$$\frac{c_{1}}{p-1} \biggl\Vert D\biggl(\frac{u}{\lambda}\biggr) \biggr\Vert _{p}^{p}+ \int_{\partial\Omega }\beta (z) \biggl(\frac{u}{\lambda} \biggr)^{p}\,dz\leqslant c_{12}\quad\forall \lambda \in(0,1), $$
for some \(c_{12}>0\), so
$$\frac{c_{1}}{p-1}\widehat {\lambda}_{1}(p,\widehat {\beta}) \biggl\Vert \frac{u}{\lambda} \biggr\Vert _{p}^{p}\leqslant c_{12}, $$
where \(\widehat {\beta}=\frac{p-1}{c_{1}}\beta\) (see (2.3)), thus
$$\biggl\{ \frac{u}{\lambda} \biggr\} _{u\in S(\vartheta )}\subseteq L^{p}( \Omega)\quad \mbox{is bounded}, $$
hence
$$ \biggl\{ \frac{u}{\lambda} \biggr\} _{u\in S(\vartheta )}\subseteq W^{1,p}(\Omega)\quad \mbox{is bounded} $$
(3.41)
(see (3.40)). From (3.39) we have
$$ \textstyle\begin{cases} -\operatorname{div}a(D(\frac{u}{\lambda})(z))=f(z,\frac{u}{\lambda }(z),Du(z))&\mbox{for a.a. } z\in\Omega,\\ \frac{\partial(\frac{u}{\lambda})}{\partial n_{a}}+\beta(z)(\frac {u}{\lambda})^{p-1}=0&\mbox{on } \partial\Omega . \end{cases} $$
(3.42)
Hypothesis \(H(f)\)(iii) implies that
$$ f\biggl(z,\frac{u}{\lambda},Du\biggr)\leqslant \lambda^{p} f\biggl(z,\frac{u}{\lambda },D\biggl(\frac{u}{\lambda} \biggr)\biggr) \quad\mbox{for a.a. } z\in\Omega. $$
(3.43)
Then, from (3.41), (3.42), (3.43) and the nonlinear regularity theory of Lieberman [3], we have
$$\biggl\Vert \frac{u}{\lambda} \biggr\Vert _{C^{1}(\overline{\Omega})}\leqslant c_{13}\quad \forall u\in S(\vartheta ), $$
for some \(c_{13}>0\), thus \(S(\vartheta )\subseteq C^{1}(\overline{\Omega})\) is bounded.
Since ϑ is compact (see Proposition 3.6), we can use the Leray–Schauder alternative theorem (see Theorem 2.9) and find \(\widehat {u}\in C\) such that
$$\widehat {u}=\vartheta (\widehat {u}), $$
so \(\widehat {u}\in[0,\eta]\cap D_{+}\) is a solution of (1.1). □