Proof of Theorem 2.6
We consider
$$ textstylebegin{cases} -Delta _{Phi}u= H(u, x, h_{1}(zeta (u,x)), h_{2}(zeta (u,x))), quad xin Omega , \ u=0, quad xin partial Omega , end{cases} $$
(3.1)
where
$$ H(u, x, s, t)= J_{1}(u)s+ J_{2}(u)t-gamma (u,x). $$
We have the following claims:
Claim 1. Problem (3.1) has a solution in (W^{1,Phi}_{0}(Omega )cap L^{infty}(Omega )).
Define (B:W_{0}^{1, Phi}(Omega ): to W^{-1, Phi}(Omega )) as
$$ begin{aligned} bigl(B(u), wbigr)&= int _{Omega}-Delta _{Phi}uw- int _{Omega}bigl[h_{1}bigl(zeta (u,x) bigr)J_{1}(u)+h_{2}bigl(zeta (u,x)bigr)J_{2}(u) – gamma (u,x)bigr]w \ &= int _{Omega}rho bigl( vert nabla u vert bigr) (nabla u cdot nabla w)- int _{Omega}bigl[h_{1}bigl(zeta (u,x) bigr)J_{1}(u)+h_{2}bigl(zeta (u,x)bigr)J_{2}(u) \ &quad {}-gamma (u,x)bigr]w,quad forall u, win W_{0}^{1, Phi}( Omega ), end{aligned} $$
where ρ satisfies ((rho _{1})), ((rho _{2})), and ((rho _{3})).
First, we want to show that B is continuous, bounded, and coercive.
It is easy to see that the conditions on ρ and the continuity of (h_{1}) and (h_{2}) guarantees that B is bounded and continuous.
According to ((rho _{3})’), there exist (kappa , sin (1,N)) such that
$$ kappa le frac{rho (t)t^{2}}{Phi (t)}le s,quad forall t>0, $$
which implies that
$$ begin{aligned} frac{(B(u), u)}{ Vert u Vert _{1, Phi}}&= frac{int _{Omega}rho ( vert nabla u vert ) vert nabla u vert ^{2}-int _{Omega}[h_{1}(zeta (u,x))J_{1}(u)+h_{2}(zeta (u,x))J_{2}(u)-gamma (u,x)]u}{ Vert u Vert _{1, Phi}} \ &ge frac{kappa int _{Omega}Phi ( vert nabla u vert )-int _{Omega}[h_{1}(zeta (u,x))J_{1}(u)+h_{2}(zeta (u,x))J_{2}(u)-gamma (u,x)]u}{ Vert u Vert _{1, Phi}}. end{aligned} $$
From the Lemma 2.3 and Lemma 2.1 in [12], we have
$$ min bigl{ Vert nabla u Vert _{L^{Phi}}^{kappa}, Vert nabla u Vert _{L^{Phi}}^{s} bigr} =xi _{0}bigl( Vert nabla u Vert _{L^{Phi}}bigr)le int _{Omega}Phi bigl( vert nabla u vert bigr) $$
and
$$ int _{Omega}Phi bigl( vert nabla u vert bigr)ge int _{Omega}Phi biggl(frac{ vert u vert }{d}biggr), $$
then we deduce
$$ begin{aligned} frac{(B(u), u)}{ Vert u Vert _{1, Phi}}&ge frac{frac{kappa}{2}min { Vert nabla u Vert _{L^{Phi}}^{kappa}, Vert nabla u Vert _{L^{Phi}}^{s}}+frac{kappa}{2}min { Vert frac{u}{d} Vert _{L^{Phi}}^{kappa}, Vert frac{u}{d} Vert _{L^{Phi}}^{s}}}{ Vert u Vert _{1, Phi}} \ &quad {}- frac{int _{Omega}[h_{1}(zeta (u,x))J_{1}(u)+h_{2}(zeta (u,x))J_{2}(u)-gamma (u,x)]u}{ Vert u Vert _{1, Phi}}. end{aligned} $$
It follows that
$$ begin{aligned} frac{kappa int _{Omega}Phi ( vert nabla u vert )}{ Vert u Vert _{1, Phi}} &= frac{kappa int _{Omega}Phi ( vert nabla u vert )}{ vert nabla u vert _{L^{Phi}}+ vert u vert _{L^{Phi}}} \ &ge frac{frac{kappa}{2}min { Vert nabla u Vert _{L^{Phi}}^{kappa}, Vert nabla u Vert _{L^{Phi}}^{s}}+frac{kappa}{2}min { Vert frac{u}{d} Vert _{L^{Phi}}^{kappa}, Vert frac{u}{d} Vert _{L^{Phi}}^{s}}}{ vert nabla u vert _{L^{Phi}}+ vert u vert _{L^{Phi}}} \ &=frac{kappa}{2} frac{min { Vert nabla u Vert _{L^{Phi}}^{kappa}, Vert nabla u Vert _{L^{Phi}}^{s}}+min { Vert frac{u}{d} Vert _{L^{Phi}}^{kappa}, Vert frac{u}{d} Vert _{L^{Phi}}^{s}}}{ vert nabla u vert _{L^{Phi}}+ vert u vert _{L^{Phi}}} to infty end{aligned} $$
if (|u|_{1, Phi}to infty ). Then we have
$$ begin{aligned} frac{(B(u), u)}{ Vert u Vert _{1, Phi}}to infty quad bigl( Vert u Vert _{1, Phi}to infty bigr). end{aligned} $$
Hence we can conclude that the operator B is coercive.
In the end, we will prove that operator B is pseudomonotone, i.e., if
$$ u_{n}rightharpoonup u quad text{in } W_{0}^{1,Phi}( Omega )cap L^{ infty}(Omega )$$
and
$$ lim_{nto infty}sup bigl(B(u_{n}), (u_{n}-u) bigr)le 0,$$
then
$$ lim_{nto infty}inf bigl(B(u_{n}), (u_{n}-w)bigr)ge bigl(B(u), (u-w)bigr), quad forall wtext{ in } W_{0}^{1,Phi}(Omega )cap L^{infty}( Omega ). $$
(3.2)
From
$$ int _{Omega}bigl[h_{1}bigl(zeta (u_{n},x)bigr)J_{1}(u_{n})+gbigl(zeta (u_{n},x)bigr)J_{2}(u_{n})- gamma (u_{n},x)bigr](u_{n}-u)to 0 $$
and
$$ limsup_{nto infty}bigl(B(u_{n}), (u_{n}-u) bigr)le 0, $$
we obtain
$$ limsup_{nto infty} int _{Omega}rho bigl( vert nabla u_{n} vert bigr) bigl( nabla u_{n}cdot nabla (u_{n}-u)bigr)le 0. $$
(3.3)
From Lemma 3.1 in [12], we infer
$$ begin{aligned} biglVert nabla (u_{n}-u) bigrVert _{L^{Phi}}le int _{Omega}Phi bigl( biglvert nabla (u_{n}-u) bigrvert bigr). end{aligned} $$
(3.4)
From Lemma 2.5, we can obtain a (k_{0}>0) such that
$$ begin{aligned} &Phi bigl( biglvert nabla (u_{n}-u) bigrvert bigr) \ &quadle frac{[Phi ( vert nabla u_{n} vert )+Phi ( vert nabla u vert )]^{frac{1}{kappa +1}}}{k_{0}^{frac{kappa}{kappa +1}}} \ &quadquad {}times bigl[rho bigl( vert nabla u_{n} vert bigr) bigl(nabla u_{n}cdot nabla (u_{n}-u)bigr)- rho bigl( vert nabla u vert bigr) bigl(nabla ucdot nabla (u_{n}-u) bigr)bigr]^{ frac{kappa}{kappa +1}}, end{aligned} $$
(3.5)
that is,
$$ begin{aligned} & int _{Omega}Phi bigl( biglvert nabla (u_{n}-u) bigrvert bigr) \ &quadle int _{Omega} biggl{ frac{[Phi ( vert nabla u_{n} vert )+Phi ( vert nabla u vert )]^{frac{1}{kappa +1}}}{k_{0}^{frac{kappa}{kappa +1}}} \ &quad quad {}times bigl[rho bigl( vert nabla u_{n} vert bigr) bigl( nabla u_{n}cdot nabla (u_{n}-u)bigr)- rho bigl( vert nabla u vert bigr) bigl(nabla ucdot nabla (u_{n}-u) bigr)bigr]^{ frac{kappa}{kappa +1}} biggr} \ &quadle biggl{ int _{Omega} biggl[ frac{[Phi ( vert nabla u_{n} vert )+Phi ( vert nabla u vert )]^{frac{1}{kappa +1}}}{k_{0}^{frac{kappa}{kappa +1}}} biggr]^{kappa +1} biggr} ^{frac{1}{kappa +1}} \ &quadquad {}times biggl{ int _{Omega}bigl[rho bigl( vert nabla u_{n} vert bigr) bigl(nabla u_{n} cdot nabla (u_{n}-u) bigr)-rho bigl( vert nabla u vert bigr) bigl(nabla ucdot nabla (u_{n}-u)bigr)bigr] biggr} ^{frac{kappa}{kappa +1}}. end{aligned} $$
(3.6)
Since (u_{n}rightharpoonup u), we have
$$ int _{Omega}rho bigl( vert nabla u vert bigr) bigl( nabla ucdot nabla (u_{n}-u)bigr) to 0, $$
which, together with (3.3), guarantees that
$$ int _{Omega}rho bigl( vert nabla u_{n} vert bigr) bigl(nabla u_{n}cdot nabla (u_{n}-u)bigr)- rho bigl( vert nabla u vert bigr) bigl(nabla ucdot nabla (u_{n}-u)bigr)to 0quad text{as } n to +infty . $$
(3.7)
From (3.5), (3.6), and (3.7), we have
$$ int _{Omega}Phi bigl( biglvert nabla (u_{n}-u) bigrvert bigr)to 0, $$
that is,
$$ biglVert nabla (u_{n}-u) bigrVert _{L^{Phi}}to 0. $$
Therefore,
$$ begin{aligned} Vert u_{n}-u Vert _{1, Phi}= Vert u_{n}-u Vert _{L^{Phi}}+ biglVert nabla (u_{n}-u) bigrVert _{L^{ Phi}}to 0, end{aligned} $$
which implies that (3.2) is true.
According to Lemma 2.2.2 in [21], there is a (uin W_{0}^{1,Phi}(Omega )cap L^{infty}(Omega )) such that for (forall win W_{0}^{1,Phi}(Omega )),
$$ bigl(B(u), wbigr)=0. $$
Therefore, we know that u is a (weak) solution of problem (3.1).
Claim 2. We show that the solution u of problem (3.1) obtained above is a solution of (1.1).
We shall prove that
$$ begin{aligned} underline{w}_{*}le ule overline{w}^{*} quad text{in } Omega . end{aligned} $$
(3.8)
Choosing (w=(u-overline{w}^{*})_{+}) as a test function, we have
$$ begin{aligned} int _{Omega}-Delta _{Phi}ubigl(u-overline{w}^{*} bigr)_{+} &= int _{Omega}bigl[Hbigl(x, u, h_{1}bigl(zeta (u,x) bigr), h_{2}bigl(zeta (u,x)bigr)bigr)- gamma (u,x)bigr]bigl(u- overline{w}^{*}bigr)_{+} \ &= int _{Omega}bigl[h_{1}bigl(zeta (u,x) bigr)J_{1}(u)+ h_{2}bigl(zeta (u,x)bigr)J_{2}(u)- gamma (u,x)bigr]bigl(u-overline{w}^{*}bigr)_{+}. end{aligned} $$
(3.9)
Define
$$ Omega _{1}:=bigl{ xin Omega mid u>overline{w}^{*} bigr} . $$
Then
$$ begin{aligned} & int _{Omega}bigl[h_{1}bigl(zeta (u,x) bigr)J_{1}(u)+ h_{2}bigl(zeta (u,x)bigr)J_{2}(u)- gamma (u,x)bigr]bigl(u-overline{w}^{*}bigr)_{+} \ &quad= int _{Omega _{1}}+ int _{Omega -Omega _{1}}bigl[h_{1}bigl(zeta (u,x) bigr)J_{1}(u)+ h_{2}bigl(zeta (u,x)bigr)J_{2}(u)- gamma (u,x)bigr]bigl(u-overline{w}^{*}bigr)_{+} \ &quad= int _{Omega _{1}}bigl[h_{1}bigl(zeta (u,x) bigr)J_{1}(u)+ h_{2}bigl(zeta (u,x)bigr)J_{2}(u)- gamma (u,x)bigr]bigl(u-overline{w}^{*}bigr)_{+}+0 \ &quad= int _{Omega _{1}}bigl[h_{1}bigl(overline{w}^{*} bigr)J_{1}(u)+ h_{2}bigl( overline{w}^{*} bigr)J_{2}(u)-bigl(u-overline{w}^{*}bigr)_{+}^{nu} bigr]bigl(u- overline{w}^{*}bigr)_{+}. end{aligned} $$
(3.10)
Since Ψ and Λ are increasing, from Lemma 2.1 and (|zeta (u,x)|leq overline{w}^{*}), we have
$$ biggl{ varsigma >0Bigm| int _{Omega} Psi biggl( frac{ vert zeta (u,x) vert }{varsigma} biggr) le 1 biggr} supseteq biggl{ varsigma >0Bigm| int _{Omega} Psi biggl( frac{overline{w}^{*}}{varsigma} biggr) le 1 biggr} $$
and
$$ biggl{ varsigma >0Bigm| int _{Omega} Lambda biggl( frac{ vert zeta (u,x) vert }{varsigma} biggr) le 1 biggr} supseteq biggl{ varsigma >0Bigm| int _{Omega} Lambda biggl( frac{overline{w}^{*}}{varsigma} biggr) le 1 biggr} , $$
which implies that
$$ J_{1}bigl(zeta (u,x)bigr)leq J_{1} bigl(overline{w}^{*}bigr),qquad J_{2}bigl(zeta (u,x)bigr) leq J_{2}bigl(overline{w}^{*}bigr). $$
(3.11)
From (3.9), (3.10), and (3.11), we have
$$ int _{Omega}-Delta _{Phi}ubigl(u-overline{w}^{*} bigr)_{+}le int _{ Omega}bigl[h_{1}bigl(overline{w}^{*} bigr)J_{1}bigl(overline{w}^{*}bigr)+h_{2} bigl( overline{w}^{*}bigr)J_{2}bigl(overline{w}^{*} bigr)-bigl(u-overline{w}^{*}bigr)_{+}^{ nu} bigr]bigl(u-overline{w}^{*}bigr)_{+}. $$
By Definition 2.2, we have
$$ int _{Omega}-Delta _{Phi}ubigl(u-overline{w}^{*} bigr)_{+}le int _{ Omega}bigl[-Delta _{Phi}overline{w}^{*}- bigl(u-overline{w}^{*}bigr)_{+}^{nu}bigr] bigl(u- overline{w}^{*}bigr)_{+}. $$
Hence
$$ int _{Omega}-Delta _{Phi}ubigl(u-overline{w}^{*} bigr)_{+}+ int _{Omega} Delta _{Phi}overline{w}^{*} bigl(u-overline{w}^{*}bigr)_{+}le int _{ Omega}bigl[-bigl(u-overline{w}^{*} bigr)_{+}^{nu +1}bigr]le 0, $$
i.e.,
$$ int _{Omega}bigl(rho bigl( vert nabla u vert bigr) nabla u-rho bigl( biglvert nabla overline{w}^{*} bigrvert bigr) nabla overline{w}^{*}bigr)cdot nabla bigl(u- overline{w}^{*}bigr)_{+}le int _{Omega}bigl[-bigl(u-overline{w}^{*} bigr)_{+}^{nu +1}bigr]le 0. $$
(3.12)
From Lemma 2.5, there exists a (k_{0}>0) such that
$$ begin{aligned} & int _{Omega}bigl(rho bigl( vert nabla u vert bigr) nabla u-rho bigl( biglvert nabla overline{w}^{*} bigrvert bigr)nabla overline{w}^{*}bigr)cdot nabla bigl(u- overline{w}^{*}bigr)_{+} \ &quad ge int _{Omega}k_{0} frac{Phi ( vert nabla u-nabla overline{w}^{*} vert )^{frac{kappa +1}{kappa}}}{(Phi ( vert nabla u vert )+Phi ( vert nabla overline{w}^{*} vert ))^{frac{1}{kappa}}} frac{nabla (u-overline{w}^{*})_{+}}{nabla (u-overline{w}^{*})}. end{aligned} $$
(3.13)
Since
$$ int _{Omega}k_{0} frac{Phi ( vert nabla u-nabla overline{w}^{*} vert )^{frac{kappa +1}{kappa}}}{(Phi ( vert nabla u vert )+Phi ( vert nabla overline{w}^{*} vert ))^{frac{1}{kappa}}} frac{nabla (u-overline{w}^{*})_{+}}{nabla (u-overline{w}^{*})}= int _{Omega _{1}}k_{0} frac{Phi ( vert nabla u-nabla overline{w}^{*} vert )^{frac{kappa +1}{kappa}}}{(Phi ( vert nabla u vert )+Phi ( vert nabla overline{w}^{*} vert ))^{frac{1}{kappa}}} $$
and Φ is continuous, we obtain that there is an (M_{1}>0) such that
$$ int _{Omega _{1}}k_{0} frac{Phi ( vert nabla u-nabla overline{w}^{*} vert )^{frac{kappa +1}{kappa}}}{(Phi ( vert nabla u vert )+Phi ( vert nabla overline{w}^{*} vert ))^{frac{1}{kappa}}}= frac{k_{0}}{M_{1}} int _{{u>overline{w}^{*}}}Phi bigl( biglvert nabla u- nabla overline{w}^{*} bigrvert bigr)^{frac{kappa +1}{kappa}}. $$
(3.14)
From (3.12), (3.13), and (3.14), we have
$$ int _{{u>overline{w}^{*}}}Phi bigl( biglvert nabla u-nabla overline{w}^{*} bigrvert bigr)^{ frac{kappa +1}{kappa}}le 0. $$
From Lemma 2.2 in [11] and [14], we obtain
$$ int _{{u>overline{w}^{*}}}Phi biggl(frac{ vert u-overline{w}^{*} vert }{d} biggr)^{frac{kappa +1}{kappa}} le int _{{u>overline{w}^{*}}} Phi bigl( biglvert nabla u-nabla overline{w}^{*} bigrvert bigr)^{frac{kappa +1}{kappa}} le 0, $$
where (d=mathrm{diam}(Omega )). Therefore, we can conclude that
$$ biglvert bigl{ u>overline{w}^{*}bigr} bigrvert =0, $$
and then (ule overline{w}^{*}).
A similar argument shows that (ugeq underline{w}_{*}).
Therefore, (3.8) is true and thus u is a solution of problem (1.1).
The proof is completed. □
Proof of Theorem 2.7
In order to get positive solutions of problem (1.3), we study the following problem:
$$ textstylebegin{cases} -Delta _{Phi}u=(u+frac{1}{n})^{beta} Vert u Vert _{L^{Psi}}^{alpha}, quad xin Omega , \ u=0, quad xin partial Omega , end{cases} $$
(3.15)
for (ngeq 1). We will use Theorem 2.6 to discuss problem (3.15).
First, we will construct a supersolution u̅ of problem (3.15).
From Lemma 2.4, problem (2.1) has a unique positive (z_{lambda}in W_{0}^{1, Psi}(Omega )) which satisfies
$$ begin{aligned} 0< z_{lambda}(x)le Klambda ^{frac{1}{kappa -1}}, quad xin Omega end{aligned} $$
(3.16)
for (lambda >0) big enough, where K is independent of λ.
Let (M=Klambda ^{frac{1}{kappa -1}}). Then
$$ begin{aligned} Klambda ^{frac{1}{kappa -1}}< z_{lambda}(x)+Mle 2K lambda ^{ frac{1}{kappa -1}}, quad xin Omega . end{aligned} $$
The condition (0<alpha <kappa -1) implies that there is a (lambda >1) big enough such that
$$ lambda ^{frac{alpha}{kappa -1}} Vert 2K Vert _{L^{Psi}}^{alpha}le lambda ,qquad M=Klambda ^{frac{1}{kappa -1}}>1$$
and (3.16) holds. Hence
$$ biggl(z_{lambda}+M+frac{1}{n}biggr)^{beta} Vert z_{lambda}+M Vert _{L^{Psi}}^{ alpha}le Vert z_{lambda}+M Vert _{L^{Psi}}^{alpha}le lambda ^{ frac{alpha}{kappa -1}} Vert 2K Vert _{L^{Psi}}^{alpha}le lambda $$
and
$$ -Delta _{Phi}(z_{lambda}+M)= -Delta _{Phi}z_{lambda}= lambda geq biggl(z_{lambda}+M+frac{1}{n} biggr)^{beta} Vert z_{lambda}+M Vert _{L^{ Psi}}^{alpha}. $$
Therefore, (z_{lambda}+M) is a supersolution of (3.15).
Second, we will construct a positive subsolution (underline{u}_{*}) of problem (3.15).
Define (d(x):=mathrm{dist}(x,partial Omega )), then by a direct calculation one can deduce that (|nabla d(x)|=1). Because ∂Ω is (C^{2}), we can get a constant (tau >0) such that (din C^{2}(overline{Omega _{3tau}})) with (overline{Omega _{3tau}}:={xin overline{Omega}:d(x)le 3tau } ) (see [9, 10]). Let (varpi in (0, tau )). Define
$$ eta (x):= textstylebegin{cases} e^{vartheta d(x)}-1, &text{for } d(x)< varpi , \ e^{vartheta varpi}-1+int _{varpi}^{d(x)}vartheta e^{vartheta d(x)}( frac{2tau -t}{2tau -varpi})^{frac{s}{kappa -1}},dt,&text{for } varpi le d(x)le 2tau , \ e^{vartheta varpi}-1+int _{varpi}^{2tau}ke^{vartheta d(x)}( frac{2tau -t}{2tau -varpi})^{frac{s}{kappa -1}},dt,&text{for } 2tau < d(x), end{cases} $$
where (vartheta >0) is an arbitrary number. Direct computations imply that
$$ -Delta _{Phi}(mu eta ) = textstylebegin{cases} -vartheta Theta (x)frac{d}{dt}(rho (t)t)|_{t=Theta (x)} – rho (Theta (x))Theta (x)Delta d ,&text{for } d(x)< varpi , \ frac{Theta _{0}(frac{s}{kappa -1})chi (x)^{frac{s}{kappa -1}-1}}{2tau -varpi} frac{d}{dt}(rho (t)t)|_{t=Theta _{0}chi (x)^{ frac{s}{kappa -1}}} \ quad {}-rho (Theta _{0}chi (x)^{frac{s}{kappa -1}})Theta _{0}chi (x)^{ frac{s}{kappa -1}}Delta d,&text{for } varpi le d(x) le 2tau , \ 0, &text{for } 2tau < d(x), end{cases} $$
with (Theta (x)=mu vartheta e^{vartheta d(x)}), (Theta _{0}=mu vartheta e^{vartheta varpi}), and (chi (x)=frac{2tau -d(x)}{2tau -varpi}) for all (mu >0).
There are three cases: (1) (d(x)<varpi ); (2) (varpi < d(x)<2tau ); and (3) (d(x)>2tau ).
(1) We consider the case (d(x)<varpi ).
Since Δd is a bounded function near ∂Ω and (kappa >1), there is a ϑ large enough such that
$$ begin{aligned} -Delta _{Phi}(mu eta )&= – mu vartheta ^{2}e^{vartheta d(x)} frac{d}{dt}bigl(rho (t)t bigr)bigg|_{t=mu vartheta e^{vartheta d(x)}}- rho bigl(mu vartheta e^{vartheta d(x)}bigr)mu vartheta e^{vartheta d(x)} Delta d \ &le -vartheta ^{2}mu e^{vartheta d(x)}(kappa -1)rho bigl(mu vartheta e^{mu vartheta e^{vartheta d(x)}}bigr)-rho bigl(mu vartheta e^{ vartheta d(x)}bigr) mu vartheta e^{vartheta d(x)}Delta d \ &=mu vartheta e^{vartheta d(x)}rho bigl(mu vartheta e^{vartheta d(x)}bigr) bigl(- vartheta (kappa -1)-Delta dbigr) \ &le 0, end{aligned} $$
which implies that
$$ begin{aligned} -Delta _{Phi}(mu eta )le 0le (mu eta )^{beta} vert mu eta vert _{L^{ Psi}}^{alpha}, end{aligned} $$
when (d(x)<varpi ) and ϑ is large enough.
(2) We consider the case (varpi < d(x)<2delta ).
From the condition ((rho _{3})) and Lemma 2.3, we have
$$ begin{aligned} &mu vartheta e^{vartheta varpi} biggl(frac{s}{kappa -1} biggr) biggl( frac{2tau -d(x)}{2tau -varpi} biggr)^{frac{s}{kappa -1}-1} biggl( frac{1}{2tau -varpi} biggr)frac{d}{dt}bigl(rho (t)t bigr)bigg|_{t=mu vartheta e^{vartheta varpi} (frac{2tau -d(x)}{2tau -varpi} )^{frac{s}{kappa -1}}} \ &quadle mu vartheta e^{vartheta varpi} biggl(frac{s}{kappa -1} biggr) biggl( frac{2tau -d(x)}{2tau -varpi} biggr)^{frac{s}{kappa -1}-1} biggl(frac{s-1}{2tau -varpi} biggr)rho biggl(mu vartheta e^{ vartheta varpi} biggl(frac{2tau -d(x)}{2tau -varpi} biggr)^{ frac{s}{kappa -1}} biggr) \ &quadle biggl(frac{s}{kappa -1} biggr) biggl(frac{s-1}{2tau -varpi} biggr) frac{sPhi (mu vartheta e^{vartheta varpi} (frac{2tau -d(x)}{2tau -varpi} )^{frac{s}{kappa -1}} )}{mu vartheta e^{vartheta varpi} (frac{2delta -d(x)}{2tau -varpi} )^{frac{s}{kappa -1}}} frac{1}{frac{2tau -d(x)}{2tau -varpi}} \ &quadle biggl(frac{s^{2}}{kappa -1} biggr) biggl(frac{s-1}{2tau -varpi} biggr)max biggl{ bigl(mu vartheta e^{vartheta varpi}bigr)^{s-1} biggl( frac{2tau -d(x)}{2tau -varpi} biggr)^{s(frac{s}{kappa -1})-( frac{s}{kappa -1}+1)}, \ &quadquad bigl(mu vartheta e^{ vartheta varpi}bigr)^{kappa -1} biggl( frac{2tau -d(x)}{2tau -varpi} biggr)^{kappa (frac{s}{kappa -1})-(frac{s}{kappa -1}+1)} biggr} Phi (1). end{aligned} $$
(3.17)
Now (s, kappa >1) implies (kappa (frac{s}{kappa -1} )-s (frac{s}{kappa -1}+1 ), s (frac{s}{kappa -1} )-s (frac{s}{kappa -1}+1 )>0), which, together with (0le frac{2tau -d(x)}{2tau -varpi}le 1) and (3.17), guarantees that
$$begin{aligned} &mu vartheta e^{vartheta varpi} biggl(frac{s}{kappa -1} biggr) biggl( frac{2tau -d(x)}{2tau -varpi} biggr)^{frac{s}{kappa -1}-1} biggl( frac{1}{2tau -varpi} biggr)frac{d}{dt}bigl(rho (t)t bigr)bigg|_{t=mu vartheta e^{vartheta varpi} ( frac{2delta -d(x)}{2tau -varpi} )^{frac{s}{kappa -1}}} \ &quad le biggl(frac{s^{2}}{kappa -1} biggr) biggl(frac{s-1}{2tau -varpi} biggr)Phi (1) max bigl{ bigl(mu vartheta e^{vartheta varpi}bigr)^{s-1}, bigl(mu vartheta e^{vartheta varpi}bigr)^{kappa -1}bigr} \ &quad =C_{1} biggl(frac{1}{2tau -varpi} biggr)max bigl{ bigl(mu vartheta e^{ vartheta varpi}bigr)^{s-1}, bigl(mu vartheta e^{vartheta varpi}bigr)^{ kappa -1}bigr} , end{aligned}$$
(3.18)
where (C_{1}=frac{s^{2}(s-1)Phi (1)}{kappa -1}) is a constant independent of μ and ϑ. Similarly, one has
$$ begin{aligned} & bigglvert rho biggl(mu vartheta e^{vartheta varpi} biggl( frac{2tau -d(x)}{2tau -varpi} biggr)^{frac{s}{kappa -1}} biggr)mu vartheta e^{vartheta varpi} frac{(2tau -d(x))^{frac{s}{kappa -1}}}{(2tau -varpi )^{frac{s}{kappa -1}}} Delta d biggrvert \ &quadle rho biggl(mu vartheta e^{vartheta varpi} biggl( frac{2tau -d(x)}{2tau -varpi} biggr)^{frac{s}{r-1}}biggr)mu vartheta e^{ vartheta varpi} biggl( frac{2tau -d(x)}{2tau -varpi} biggr)^{ frac{s}{kappa -1}}sup_{overline{Omega _{3tau}}} vert Delta d vert \ &quadle C frac{Phi (mu vartheta e^{vartheta varpi} (frac{2tau -d(x)}{2tau -varpi} )^{frac{s}{kappa -1}} )}{mu vartheta e^{vartheta varpi} (frac{2tau -d(x)}{2tau -varpi} )^{frac{s}{kappa -1}}} \ &quadle Cmax biggl{ bigl(mu vartheta e^{vartheta varpi}bigr)^{s-1} biggl( frac{2tau -d(x)}{2tau -varpi} biggr)^{s(frac{s}{kappa -1})-( frac{s}{kappa -1}+1)}, \ &quadquad bigl(mu vartheta e^{vartheta varpi}bigr)^{kappa -1} biggl( frac{2tau -d(x)}{2tau -varpi} biggr)^{kappa ( frac{s}{kappa -1})-(frac{s}{kappa -1}+1)} biggr} \ &quadle C_{2}max bigl{ bigl(mu vartheta e^{vartheta varpi} bigr)^{s-1}, bigl( mu vartheta e^{vartheta varpi}bigr)^{kappa -1} bigr} , end{aligned} $$
(3.19)
where (C_{2}) is a constant independent of ϖ, ϑ, and μ. Thus from (3.18) and (3.19) we have
$$ begin{aligned} -Delta _{Phi}ule max biggl{ frac{C_{1}}{2tau -varpi}, C_{2} biggr} max bigl{ bigl(mu vartheta e^{vartheta varpi}bigr)^{s-1}, bigl(mu vartheta e^{vartheta varpi} bigr)^{kappa -1} bigr} , end{aligned} $$
when (varpi < d(x)<2tau ).
Let (varpi =frac{ln2}{vartheta }) and (mu =e^{-vartheta }), then (e^{vartheta varpi}=2). Since
$$ begin{aligned} eta (x)&=e^{vartheta varpi}-1+ int _{varpi}^{d(x)}vartheta e^{ vartheta d(x)} biggl( frac{2tau -t}{2tau -varpi} biggr)^{ frac{s}{kappa -1}},dt \ &>2-1+2vartheta int _{varpi}^{d(x)} biggl( frac{2tau -t}{2tau -varpi} biggr)^{frac{s}{kappa -1}},dt \ &=1+vartheta C_{3} \ &ge 1, end{aligned} $$
where (C_{3}>0) is a constant, we have that when μ is small enough and n is large enough,
$$begin{aligned} biggl(mu eta +frac{1}{n} biggr)^{beta} vert mu eta vert _{L^{Psi}}^{alpha}& ge vert mu eta vert _{L^{Psi}}^{alpha} \ &={inf}^{alpha} biggl{ varsigma >0: int _{Omega}Psi biggl( frac{ vert mu eta vert }{varsigma} biggr)< 1 biggr} \ &={inf}^{alpha} biggl{ tau mu >0: int _{Omega}Psi biggl( frac{ vert mu eta vert }{tau mu} biggr)< 1 biggr} \ &=mu ^{alpha}{inf}^{alpha} biggl{ tau >0: int _{Omega}Psi biggl( frac{ vert eta vert }{tau} biggr)< 1 biggr} \ &ge mu ^{alpha}C_{4}, end{aligned}$$
where (C_{4}>0) is a constant independent of (vartheta >0).
Since (0<alpha <kappa -1), we have the result
$$ begin{aligned} lim_{vartheta to +infty} frac{vartheta ^{kappa -1}}{e^{vartheta (kappa -1-alpha )}}=0. end{aligned} $$
In view of
$$ begin{aligned} -Delta _{Phi}(mu eta )le max biggl{ frac{C_{1}}{2tau -varpi}, C_{2} biggr} max bigl{ 2^{s-1}, 2^{kappa -1}bigr} biggl( frac{vartheta }{e^{vartheta }} biggr)^{kappa -1}, end{aligned} $$
choose a (vartheta _{0}>0) large enough such that
$$ begin{aligned} C_{4}ge max biggl{ frac{C_{1}}{2tau -frac{ln 2}{vartheta }}, C_{2} biggr} max bigl{ 2^{s-1}, 2^{kappa -1}bigr} biggl( frac{vartheta ^{kappa -1}}{e^{vartheta (kappa -1-alpha )}} biggr) end{aligned} $$
for all (vartheta ge vartheta _{0}).
Thus,
$$ begin{aligned} -Delta _{Phi}(mu eta )le biggl(mu eta +frac{1}{n} biggr)^{beta} vert mu eta vert _{L^{Psi}}^{alpha} end{aligned} $$
in the case (varpi < d(x)<2tau ) for (vartheta >0) large enough.
(3) We consider the case (d(x)>2tau ).
Obviously,
$$ begin{aligned} -Delta _{Phi}(mu eta )=0le biggl(mu eta +frac{1}{n} biggr)^{ beta} vert mu eta vert _{L^{Psi}}^{alpha}. end{aligned} $$
It is obvious that (underline{w}_{*}leq overline{w}^{*}) if M is large enough and μ is small enough. And ((underline{w}_{*},overline{w}^{*})) is a sub-supersolution pair of problem (3.15). Now Theorem 2.6 guarantees that problem (3.15) has a solution (u_{n}) which satisfies (0<mu eta le u_{n}le z_{lambda}+M).
Now we consider the set ({u_{n}}).
From Lemma 2.2 in [12], one has that (|u|_{1, Phi}) and (|!|!|nabla u|!|!|_{L^{Phi}}) defined on (W_{0}^{1, Phi}) are equivalent. And from the proof of the coercivity of the operator B, we know that if (|!|!|nabla u|!|!|_{L^{Phi}}>1), then
$$ int _{Omega}Phi bigl( vert nabla u vert bigr)ge |!|!|nabla u|!|!|_{L^{Phi}}, $$
that is,
$$ int _{Omega}Phi bigl( vert nabla u vert bigr)ge Vert u Vert _{1, Phi}, $$
when (|u|_{1, Phi}>1).
If (|u_{n}|_{1, Phi}le 1), then ({u_{n}}) is bounded in (W_{0}^{1, Phi}(Omega )) naturally.
If (|u_{n}|_{1, Phi}>1), then
$$ Vert u_{n} Vert _{1, Phi}le int _{Omega}Phi bigl( vert nabla u_{n} vert bigr). $$
By the condition ((rho _{3})’) and due to
$$ int _{Omega}-Delta _{Phi}u_{n}u_{n}= int _{Omega}u_{n} biggl(u_{n}+ frac{1}{n} biggr)^{beta} Vert u_{n} Vert _{L^{Psi}}^{alpha}, $$
we have
$$ kappa int _{Omega}Phi bigl( vert nabla u_{n} vert bigr)le int _{Omega}phi bigl( vert nabla u_{n} vert bigr) vert nabla u_{n} vert ^{2}= int _{Omega}u_{n} biggl(u_{n}+ frac{1}{n} biggr)^{beta} Vert u_{n} Vert _{L^{Psi}}^{alpha}, $$
which, together with (alpha ge 0), (-1<beta <0 ), gives
$$ int _{Omega}Phi bigl( vert nabla u_{n} vert bigr)le frac{1}{kappa} int _{Omega} overline{w}^{* beta +1} biglVert overline{w}^{*} bigrVert _{L^{Psi}}^{alpha}, $$
that is,
$$ Vert u_{n} Vert _{1, Phi}le frac{1}{kappa} int _{Omega}overline{w}^{* beta +1} biglVert overline{w}^{*} bigrVert _{L^{Psi}}^{alpha}. $$
Therefore, ({u_{n}}) is bounded in (W_{0}^{1, Phi}(Omega )).
Since (W_{0}^{1, Phi}(Omega )) is reflexive, ({u_{n}}) has weakly convergent subsequences in (W_{0}^{1,Phi}(Omega )cap L^{infty}(Omega )), and we still use (u_{n}) to denote its subsequence. From the analysis in [3], we have
$$ begin{aligned} u_{n}rightharpoonup uquad text{in } W_{0}^{1,Phi}(Omega )cap L^{ infty}(Omega ) end{aligned} $$
and
$$ begin{aligned} u_{n}(x)stackrel{text{a.e.}}{to} u(x), quad xin Omega . end{aligned} $$
Since
$$ underline{w}_{*}le u_{n}le overline{w}^{*}, quad xin Omega ,$$
Lebesgue theorem implies
$$ u_{n} to u quad text{in } L^{q}(Omega ) forall qin [1, + infty ).$$
(3.20)
Since (u_{n}) is a (weak) solution of (3.15) for all , we have
$$ begin{aligned} int _{Omega}-Delta _{Phi}u_{n}w= int _{Omega} biggl(u_{n}+ frac{1}{n} biggr)^{beta} Vert u_{n} Vert _{L^{Psi}}^{alpha}w, end{aligned} $$
for all (win W_{0}^{1,Phi}(Omega )).
Denoting (w=u_{n}-u), we have
$$ begin{aligned} int _{Omega}-Delta _{Phi}u_{n}(u_{n}-u)= int _{Omega} biggl(u_{n}+ frac{1}{n} biggr)^{beta} Vert u_{n} Vert _{L^{Psi}}^{alpha}(u_{n}-u). end{aligned} $$
Since
$$ biggl(u_{n}+frac{1}{n} biggr)^{beta}le underline{w}_{*}^{beta}, quad xin Omega , $$
one has
$$ begin{aligned} int _{Omega} biggl(u_{n}+frac{1}{n} biggr)^{beta} Vert u_{n} Vert _{L^{Psi}}^{ alpha} vert u_{n}-u vert &le int _{Omega}underline{w}_{*}^{beta} vert u_{n}-u vert Vert u_{n} Vert _{L^{ Psi}}^{alpha} \ &le biggl[ int _{Omega} bigl(underline{w}_{*}^{beta} Vert u_{n} Vert _{L^{ Psi}}^{alpha} bigr)^{p} biggr]^{frac{1}{p}} biggl[ int _{Omega} vert u_{n}-u vert ^{q} biggr]^{frac{1}{q}}, end{aligned} $$
where (p, q>1), (frac{1}{p}+frac{1}{q}=1), and (beta in (-1,0)). From (3.20), we have
$$ biggl[ int _{Omega} bigl(underline{w}_{*}^{beta} Vert u_{n} Vert _{L^{Psi}}^{ alpha} bigr)^{p} biggr]^{frac{1}{p}} biggl[ int _{Omega} vert u_{n}-u vert ^{q} biggr]^{frac{1}{q}}to 0,$$
and so
$$ begin{aligned} int _{Omega} biggl(u_{n}+frac{1}{n} biggr)^{beta} Vert u_{n} Vert _{L^{Psi}}^{ alpha} vert u_{n}-u vert to 0quad text{as }nto +infty , end{aligned} $$
which implies
$$ begin{aligned} int _{Omega}-Delta _{Phi}u_{n}(u_{n}-u) to 0. end{aligned} $$
Obviously,
$$ begin{aligned} int _{Omega}-Delta _{Phi}u (u_{n}-u)to 0. end{aligned} $$
(3.21)
Similar to the previous proof, from (3.4), (3.6), and (3.21), we have
$$ u_{n}to uquad text{in } W_{0}^{1,Phi}(Omega ) cap L^{infty}( Omega ),$$
and so
$$ Vert u_{n} Vert _{L^{Psi}}^{alpha}to Vert u Vert _{L^{Psi}}^{alpha}.$$
Therefore, taking the limit as (nto infty ) in (3.15), we have
$$ -Delta _{Phi}u=u^{beta} Vert u Vert _{L^{Psi}}^{alpha}.$$
The limit value u is just the solution which we are looking for, and it satisfies (underline{w}_{*}le ule overline{w}^{*}), obviously. Therefore, the proof is finished. □
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