1 Introduction
Let Ω be a bounded domain in ℝN(N≥2), with C2 boundary ∂Ω, and let T>0 with QT:=Ω×(0,T) and ΓT:=∂Ω×(0,T). In this paper, we are interested in the study of following (p,q)-parabolic equation involving singular nonlinearity,
($\mathrm{P}_{t}$)
{
u
t
-
Δ
p
u
-
Δ
q
u
=
ϑ
u
-
δ
+
f
(
x
,
u
)
,
u
>
0
in
Ω
×
(
0
,
T
)
,
u
=
0
on
∂
Ω
×
(
0
,
T
)
,
u
(
x
,
0
)
=
u
0
(
x
)
in
Ω
,
where 1<q<p<∞, δ>0 and ϑ>0 is a parameter. Furthermore, we assume that f:Ω×[0,∞)→ℝ is a bounded below Carathéodory function, the map s↦f(x,s) is locally Lipschitz uniformly in x∈Ω and u0∈L∞(Ω)∩W01,p(Ω) is a nonnegative function. In what follows, Δp is the p-Laplace operator, defined as Δpu=∇⋅(|∇u|p-2∇u). The operator Ap,q:=-Δp-Δq is known as (p,q)-Laplacian.
The motivation to study problem (($\mathrm{P}_{t}$)) comes from its large variety of applications in the physical world. For instance, when p=q=2, it has applications in non-Newtonian fluids, in particular pseudoplastic fluids, in boundary layer phenomena for viscous fluids, in the Langmuir–Hinshelwood model of chemical heterogeneous catalyst kinetics, in enzymatic kinetic models, as well as in the theory of heat conduction in electrically conducting materials and in the study of guided modes of an electromagnetic field in the nonlinear medium. For p≠2, it arises from the study of turbulent flow of gas in porous media, see [3] for a brief introduction to this topic. For further details, we refer to the survey of Hernández and Mancebo [26] and the book of Ghergu and Rǎdulescu [22]. For p≠q, that is, the equations of kind (($\mathrm{P}_{t}$)) arise in the form of general reaction-diffusion equation:
(1.1)
u
t
=
div
[
A
(
u
)
∇
u
]
+
r
(
x
,
u
)
,
where A(u)=|∇u|p-2+|∇u|q-2. Problem (1.1) has wide range of important applications in biophysics, plasma physics and chemical reactions, where the function u corresponds to the concentration term, the first term on the right-hand side represents diffusion with a diffusion coefficient A(u) and the second term is the reaction which relates to sources and loss processes. For more details, readers are referred to [16, 30] and references therein.
The evolution equations of type (($\mathrm{P}_{t}$)) and corresponding stationary problems have extensively been explored by researchers in recent decades when the differential operator is homogeneous in nature, that is, p=q. In particular, when p=q=2, the seminal work of Crandall, Rabinowitz and Tartar in [18] is considered as the beginning of vast research on stationary problems involving singular nonlinearity. Interested reader may refer to the work of Boccardo and Orsina [7] and the bibliography of Hernández and Mancebo [26], and for the case p=q≠2 to the work of Giacomoni et al. [9, 24]. The case of stationary problems with singular nonlinearity and non-homogeneous elliptic operators have very recent history, for instance see the contributions of Rǎdulescu et al. [27, 32] for the case δ<1 and of Giacomoni, Kumar and Sreenadh [23] for all δ>0.
Turning to the parabolic problems involving singular nonlinearity, for p=q=2, Takáč in [34] studied the stabilization results for the semilinear parabolic problem of type (($\mathrm{P}_{t}$)), that is, the solution u(t) converges to the solution of steady state problem in C1(Ω¯), as t→∞. For the case p≠2, Badra, Bal and Giacomoni in [3], De Bonis and Giachetti in [19] and Bougherara, Giacomoni and Takáč in [10] studied parabolic equation involving p-Laplacian and singular nonlinearity, almost simultaneously. In [3], authors studied problem (($\mathrm{P}_{t}$)) with p=q and the term f(x,u) exhibiting the subhomogeneous growth with respect to p in the variable u. In this work, among other results, authors proved the existence of unique solution to (($\mathrm{P}_{t}$)) when δ<2+1p-1 by semi-discretization in time and using implicit Euler method. Furthermore, they proved the stabilization result under some additional assumptions on f and the initial condition u0. While authors in [19] considered the nonlinear term in (($\mathrm{P}_{t}$)) as f(x,t)(u-δ+1) with p=q, δ>0 and 0≤f∈Lr(0,T;Lm(Ω)), where 1r+Npm<1. Here, authors proved the existence of a positive solution. Subsequently, authors in [10] studied problem (($\mathrm{P}_{t}$)) when the function f has a form of f(x,u,∇u) and it exhibit superhomogeneous growth with respect to p in u variable. Assuming u0∈Lr(Ω), for some large r, authors proved the existence of a solution u in Cloc0,α(QT) such that ur-2+pp∈Lp(0,T;W01,p(Ω)), for all δ>0, as the limit of solution to some auxiliary problem.
For the case when the differential operator in problem (($\mathrm{P}_{t}$)) is non-homogeneous in nature, we mention the contributions of Bögelein, Duzaar and Marcellini in [8], Cai and Zhou in [14], Baroni and Lindfors in [6], and references therein. In [6], authors considered the following kind of general quasilinear parabolic problem:
(1.2)
{
u
t
-
div
A
(
∇
u
)
=
0
on
Ω
T
,
u
=
ψ
on
∂
p
Ω
T
,
where A:ℝN→ℝN is a C1 vector field exhibiting Orlicz-type growth conditions and ψ is a continuous function on the parabolic boundary ∂pΩT. Here, authors proved the existence of a solution to (1.2), which is continuous up to the boundary and local boundedness of its gradient. While in [8, 14], authors have proved the existence of solution for similar equations as in (1.2) with differential operator exhibiting Orlicz-type growth. Diening, Scharle and Schwarzacher in [21] discussed the local regularity results for solution of problem of type (1.2). The study of parabolic problems involving non-homogeneous operator and singular nonlinearity was still open after these works.
Coming back to our paper, here we obtain the existence, uniqueness, regularity results and asymptotic behavior of the solution u to problem (($\mathrm{P}_{t}$)) depending on the growth of the nonlinear term f. For the existence part in the subhomogeneous case, we employ the technique of time discretization and implicit Euler method together with existence and regularity results of associated stationary problems (see (($\mathrm{S}_{\lambda}$)) in Section 2). This also generalizes the work of [3] to non-homogeneous differential operators case. The novelty of the paper is the study of equations with combined effect of the singular nonlinear terms and the non-homogeneous nature of the leading differential operator. One of the main contribution of the paper is the construction of appropriate sub- and supersolutions of stationary problems corresponding to problem (($\mathrm{P}_{t}$)) which are comparable to the initial condition u0. In case of homogeneous operators, e.g., the operator -Δp (the p-Laplacian), it is well known that the scalar multiple of ϕ1, the first eigenfunction of -Δp with zero Dirichlet boundary condition, can be used to construct sub- and supersolutions. Since the operator is not homogeneous in our case, we introduce a parameter in front of the singular term in the problem of type (PS) (Section 2) and study the behavior of the solution as the parameter goes to 0 or ∞ by using the recently developed regularity results for purely singular problems in [23]. Precisely, in Propositions 3.3 and 3.4, we prove that the solution uρ converges uniformly to 0, as ρ→0 and to ∞, as ρ→∞. These results help us to construct small subsolutions and large supersolutions (see (3.6), (3.8)) of stationary problems of kinds (($\mathrm{S}_{\lambda}$)) and (P), that are comparable with initial condition u0. Once the sub- and supersolutions with aforementioned properties are obtained, the method of time discretization and implicit Euler method similar to [3] applies to our case too (with some modifications). To establish the uniqueness result, we prove a comparison principle as in Theorem 3.6 using Díaz–Sáa’s inequality and consequently, exploiting the local Lipschitz nature of f, we have the uniqueness of weak solution of (($\mathrm{P}_{t}$)). Furthermore, using semigroup theory, we prove the stabilization result under some assumptions on the initial data, that is, the solution u(t) converges to u∞, the solution of steady state problem (P), in L∞(Ω) as t→∞.
At the end of Section 3, we prove the higher Sobolev integrability for the solution of purely singular problem (PS), as in Theorem 2.8, which is of independent interest. Precisely, we prove the (unique) solution to problem (PS) belongs to W01,m(Ω) for m<p-1+δδ-1 for all δ>1 in the range (p-2)δ<2(p-1). That is, for the case 1<p≤2, the result is true for all δ>1 and for p>2, it is true for δ<2+2p-2. The significance of the result can be understood as follows. In [23], it is proved that the solution u of problem (PS) is in C1,α(Ω¯), for some α∈(0,1) and thus in all W01,m(Ω) when 0<δ<1. For the case δ≥1, u is only C0,α(Ω¯) regular with u∈W01,p(Ω) if and only if δ<2+1p-1 (see [23, Theorems 1.4, 1.7, 1.8]). Thus, the above-mentioned regularity result of Theorem 2.8 improves the integrability condition of the gradient as u∈W01,m(Ω) for m<p-1+δδ-1, clearly which is bigger than p in the case of δ<2+1p-1, whereas for the case 2+1p-1≤δ<2+2p-2, it was not known in the previous works whether u is in some Sobolev space. To prove the theorem, using the approach of DiBenedetto and Manfredi [20] (see also [13]), we first prove the higher integrability result (Calderón–Zygmund-type result) for solution of some non-homogeneous equation involving a more general quasilinear operator with the right-hand side term as the divergence of some vector field (see (3.24)). Similar Calderón–Zygmund-type results for the equations involving the double phase feature can be seen in [17] whereas for the parabolic equations involving the p-Laplacian, we refer to [1]. Then, taking help of the solution to the problem -Δw=u-δ+b(x), for b∈L∞(Ω), via Green function approach, we complete the proof of Theorem 2.8. Subsequently, we use the higher Sobolev integrability result of Theorem 2.8 to improve the regularity of solution to (($\mathrm{P}_{t}$)) as in Corollary 2.14. Precisely, under some additional assumption on u0, we prove that the solution u∈C([0,T];W01,m(Ω)) for all p≤m<p-1+δδ-1. To the best of our knowledge, these regularity results are new even for the equations involving p-Laplacian operator with singular nonlinearities.
Concerning the case of superhomogeneous growth, we apply nonlinear semigroup theory and Picard’s iteration process to obtain first a solution to problem (($\mathrm{P}_{t}$)) where the nonlinear term is replaced by f(v), for v∈L∞(QT). Then, applying the Banach Fixed Point Theorem, we get the local existence result as in Theorem 2.16. For the blow-up result, we refer the readers to [12, 15] for the case of semilinear parabolic equations and to [2] for the quasilinear parabolic problems involving variable exponent. In this regard, the available literature deals with the equations involving only the superhomogeneous nonlinearity but not the singularity on the nonlinear terms. Thus, this work is an extension of the previous contributions in the field concerning the finite time blow-up behavior. The main difficulty in this regard is presence of the singular term in the equation, which causes non-differentiability of the energy functional. We overcome this difficulty by establishing a suitable comparison of the solution to a sufficiently regular function (see Lemma 5.1). Additional difficulty is that, for the case δ<1, the singular nonlinearity makes the energy functional to exhibit concave-convex-type growth which forces us to prove the blow-up behavior only for a small parameter. Using the classical energy method and concavity method, we prove that the solution u of problem (($\mathrm{P}_{t}$)) blows up in the sense that ∥u(t)∥L∞(Ω)→∞ as t→∞, for a small parameter ϑ>0 (its threshold value depends only on the initial data) in the case δ≤1 while for all ϑ>0 in the case δ>1. To the best of our knowledge, there is no work dealing with the blow-up phenomenon for equation involving singular as well as superhomogeneous nonlinearity even for the case p=q (that is, equations involving only homogeneous operator -Δp).
2 Main Results
In this section, we state our main results concerning the existence, uniqueness, regularity and asymptotic behavior of weak solutions of problem (($\mathrm{P}_{t}$)). Here we also discuss several results regarding the stationary problems which are used to prove the aforementioned properties of solution to (($\mathrm{P}_{t}$)). Based on the growth rate of the nonlinear term f, we distinguish the two cases as subhomogeneous problem and superhomogeneous problem.
For the subhomogeneous growth condition, we assume that the function f satisfies the following:
0
≤
lim
s
→
∞
f
(
x
,
s
)
s
q
-
1
:=
α
f
<
∞
, and
the map s↦f(x,s)sq-1 is nonincreasing in ℝ+ for a.e. x∈Ω.
We first study the following general form of problem (($\mathrm{P}_{t}$)):
($\mathrm{G}_{t}$)
{
u
t
-
Δ
p
u
-
Δ
q
u
=
ϑ
u
-
δ
+
g
(
x
,
t
)
,
u
>
0
in
Q
T
,
u
=
0
on
Γ
T
,
u
(
x
,
0
)
=
u
0
(
x
)
in
Ω
,
where g∈L∞(QT).
Definition 2.1.
Let 𝒱(QT):={u∈L∞(QT):ut∈L2(QT) and u∈L∞(0,T;W01,p(Ω))} and for u∈𝒱(QT) we say that u>0 in QT if for any compact set K⊂QT, essinfKu>0.
We define the notion of weak solution to problem (($\mathrm{G}_{t}$)) as follows.
Definition 2.2.
A function u∈𝒱(QT) is said to be a weak solution of (($\mathrm{G}_{t}$)) if u>0 in QT, u(x,0)=u0(x) a.e. in Ω and it satisfies
∫
Q
T
(
ϕ
∂
u
∂
t
+
(
|
∇
u
|
p
-
2
+
|
∇
u
|
q
-
2
)
∇
u
∇
ϕ
-
(
ϑ
u
-
δ
+
g
(
x
,
t
)
)
ϕ
)
𝑑
x
𝑑
t
=
0
for all ϕ∈𝒱(QT).
Definition 2.3.
We define the conical shell 𝒞δ⊂L∞(Ω) as the set of functions u satisfying the following in Ω:
c
1
φ
δ
(
d
(
x
)
)
≤
u
(
x
)
≤
c
2
φ
δ
(
d
(
x
)
)
,
where d(x):=dist(x,∂Ω), c1,c2>0 are constants and for A>0 large enough, the function
φ
δ
:
[
0
,
∞
)
→
[
0
,
∞
)
is defined as
φ
δ
(
s
)
:=
{
s
if
δ
<
1
,
s
log
1
p
(
A
s
)
if
δ
=
1
,
s
p
p
-
1
+
δ
if
δ
>
1
.
We follow the approach similar to [3], to obtain the existence result for (($\mathrm{G}_{t}$)). In this regard, we consider the following stationary problem: for h∈L∞(Ω) and λ>0,
($\mathrm{S}_{\lambda}$)
{
u
-
λ
(
Δ
p
u
+
Δ
q
u
+
ϑ
u
-
δ
)
=
h
,
u
>
0
in
Ω
,
u
=
0
on
∂
Ω
.
Theorem 2.4.
Let h∈L∞(Ω) and let 0<δ<2+1p-1. Then, for any λ,ϑ>0, problem (($\mathrm{S}_{\lambda}$)) admits a unique solution uλ∈W01,p(Ω)∩Cδ∩C0(Ω¯).
Definition 2.5.
For the case δ≥2+1p-1, u∈Wloc1,p(Ω) is said to be a weak solution of problem (($\mathrm{S}_{\lambda}$)) if
∫
Ω
(
u
ϕ
+
λ
|
∇
u
|
p
-
2
∇
u
∇
ϕ
+
λ
|
∇
u
|
q
-
2
∇
u
∇
ϕ
)
𝑑
x
=
∫
Ω
(
λ
ϑ
u
-
δ
+
h
)
ϕ
𝑑
x
for all ϕ∈Cc∞(Ω) and the Dirichlet datum is understood in the sense that there exists ν≥1 such that uν∈W01,p(Ω).
Theorem 2.6.
Let δ≥2+1p-1 and let h∈L∞(Ω). Then, for any positive λ and ϑ, there exists a weak solution u∈Wloc1,p(Ω)∩Cδ∩C0(Ω¯) of problem (($\mathrm{S}_{\lambda}$)) such that u∉W01,p(Ω).
For the stabilization result, we study the following stationary problem associated to (($\mathrm{P}_{t}$)):
(P)
{
-
Δ
p
u
-
Δ
q
u
=
ϑ
u
-
δ
+
f
(
x
,
u
)
,
u
>
0
in
Ω
,
u
=
0
on
∂
Ω
.
Theorem 2.7.
Let 0<δ<2+1p-1 and assume that (f1) and (f2) are satisfied. Then, for all ϑ>0, there exists a unique solution u∞ of problem (P) in W01,p(Ω)∩Cδ∩C0(Ω¯).
Now, we state our result concerning higher Sobolev integrability.
Theorem 2.8.
Let δ>1 and u be the solution to the following problem:
(PS)
{
-
Δ
p
u
-
Δ
q
u
=
u
-
δ
+
b
(
x
)
,
u
>
0
in
Ω
,
u
=
0
on
∂
Ω
,
where b∈L∞(Ω). If δ satisfies (p-2)δ<2(p-1), then u∈W01,m(Ω) for all m<p-1+δδ-1.
Remark 2.9.
We remark that the result of Theorem 2.8 holds for equations involving a more general class of operators with singular nonlinearity as in (PS), for example:
the operator -div(|∇u|p-2∇u+a(x)|∇u|q-2∇u), where 0≤a(x)∈W1,∞(Ω)∩C(Ω¯) with 1<q<p<∞,
the operator -div(a(x)|∇u|p-2∇u+|∇u|q-2∇u), where 0<infΩ¯a(x)≤a(x)∈C1(Ω¯) with 1<q<p.
Concerning the parabolic case, our first existence theorem in this regard is stated below.
Theorem 2.10.
Let 0<δ<2+1p-1, g∈L∞(QT), u0∈W01,p(Ω)∩Cδ. Then, for all ϑ>0, there exists a unique solution u to problem (($\mathrm{G}_{t}$)) such that u(⋅,t)∈Cδ uniformly for t∈[0,T]. Moreover, u∈C([0,T];W01,p(Ω)) and the following holds for δ≠1, and for all t∈[0,T]:
(2.1)
∫
0
t
∫
Ω
(
∂
u
∂
t
)
2
𝑑
x
𝑑
τ
+
1
p
∫
Ω
|
∇
u
|
p
𝑑
x
+
1
q
∫
Ω
|
∇
u
|
q
𝑑
x
-
ϑ
1
-
δ
∫
Ω
u
1
-
δ
𝑑
x
=
∫
0
t
∫
Ω
g
∂
u
∂
t
𝑑
x
𝑑
τ
+
1
p
∫
Ω
|
∇
u
0
|
p
𝑑
x
+
1
q
∫
Ω
|
∇
u
0
|
q
𝑑
x
-
ϑ
1
-
δ
∫
Ω
u
0
1
-
δ
𝑑
x
.
(For δ=1, terms of the form v1-δ1-δ are replaced by logv in the above expression.)
For problem (($\mathrm{P}_{t}$)), we have existence result as follows.
Theorem 2.11.
Let 0<δ<2+1p-1 and u0∈W01,p(Ω)∩Cδ. Suppose (f1) and (f2) are satisfied. Then, for any T>0 and ϑ>0, there exists a unique weak solution u of (($\mathrm{P}_{t}$)) such that u(⋅,t)∈Cδ uniformly for t∈[0,T], u∈C([0,T];W01,p(Ω)) and the following holds for δ≠1 and for any t∈[0,T]:
∫
0
t
∫
Ω
(
∂
u
∂
t
)
2
𝑑
x
𝑑
τ
+
1
p
∫
Ω
|
∇
u
|
p
𝑑
x
+
1
q
∫
Ω
|
∇
u
|
q
𝑑
x
-
ϑ
1
-
δ
∫
Ω
u
1
-
δ
𝑑
x
=
∫
Ω
F
(
x
,
u
(
t
)
)
𝑑
x
+
1
p
∫
Ω
|
∇
u
0
|
p
𝑑
x
+
1
q
∫
Ω
|
∇
u
0
|
q
𝑑
x
-
ϑ
1
-
δ
∫
Ω
u
0
1
-
δ
𝑑
x
-
∫
Ω
F
(
x
,
u
0
)
𝑑
x
,
where F(x,w):=∫0wf(x,s)𝑑s. (For δ=1, the terms of the form v1-δ1-δ are replaced by logv in the above expression).
Remark 2.12.
If δ≥2+1p-1, using Theorem 2.6 together with the comparison principle of [23, Theorem 1.5] and the uniform behavior of solution to (($\mathrm{S}_{\lambda}$)) with respect to the distance function, one can generalize the existence result of Theorem 2.11 for problem (($\mathrm{P}_{t}$)) such that the solution up-1+νp∈Lp(0,T;W01,p(Ω)), for some ν≥2, whenever u0∈W01,p(Ω)∩𝒞δ.
Furthermore, we prove the following regularity result for solution of problem (($\mathrm{P}_{t}$)). Set
𝒟
(
𝒜
)
:=
{
v
∈
𝒞
δ
∩
W
0
1
,
p
(
Ω
)
:
𝒜
v
:=
-
Δ
p
v
-
Δ
q
v
-
ϑ
v
-
δ
∈
L
∞
(
Ω
)
}
.
Proposition 2.13.
Assume the hypotheses of Theorem 2.11 are satisfied and u0∈D(A)¯L∞(Ω). Then the solution u of problem (($\mathrm{P}_{t}$)) belongs to C([0,T];C0(Ω¯)) and:
if
v
is a solution of (
($\mathrm{P}_{t}$)
) with initial datum
v
0
∈
𝒟
(
𝒜
)
¯
L
∞
(
Ω
)
, then
∥
u
(
⋅
,
t
)
-
v
(
⋅
,
t
)
∥
L
∞
(
Ω
)
≤
e
ω
t
∥
u
0
-
v
0
∥
L
∞
(
Ω
)
,
0
≤
t
≤
T
,
if
u
0
∈
𝒟
(
𝒜
)
, then
u
∈
W
1
,
∞
(
0
,
T
;
L
∞
(
Ω
)
)
with
Δ
p
u
+
Δ
q
u
+
ϑ
u
-
δ
∈
L
∞
(
Q
T
)
and
∥
d
u
(
⋅
,
t
)
d
t
∥
L
∞
(
Ω
)
≤
e
ω
t
∥
Δ
p
u
0
+
Δ
q
u
0
+
ϑ
u
0
-
δ
+
f
(
x
,
u
0
)
∥
L
∞
(
Ω
)
,
where ω>0 is the Lipschitz constant for f in [u¯,u¯] with u¯ and u¯ as sub- and supersolution to (P), respectively, constructed as in the proof of Theorem 2.7.
Corollary 2.14.
Let the hypotheses of Theorem 2.11 be true and assume u0∈D(A). Then the solution u of problem (($\mathrm{P}_{t}$)) belongs to C([0,T];W01,m(Ω)) for all m<p-1+δδ-1, in the case of 1<δ<2+1p-1. (Note that the result is true for all m>1 in the case of δ<1 from the C1,α(Ω¯) regularity, α∈(0,1), of solutions to (PS).)
Next, we prove the following stabilization result.
Theorem 2.15.
Under the hypotheses of Theorem 2.11, the solution u of problem (($\mathrm{P}_{t}$)) is defined in Ω×(0,∞) and it satisfies
u
(
⋅
,
t
)
→
u
∞
(
⋅
)
in
L
∞
(
Ω
)
as
t
→
∞
,
where u∞ is the solution to stationary problem (P).
Now, we turn to the superhomogeneous case. We assume f(x,u) is of the form f(u) only and f∈C(ℝ) together with
there exists r∈(q,p*) and cr>0 such that cr|s|r≤rF(s)≤sf(s) for all s∈ℝ, where F(u)=∫0uf(t)𝑑t, and p*:=NpN-p if p<N and p*<∞ be an arbitrary large number if p≥N.
We have the local existence result as below.
Theorem 2.16.
Let u0∈W01,p(Ω)∩Cδ. Then, for all ϑ>0 and δ<2+1p-1, there exists T^>0 small enough, such that problem (($\mathrm{P}_{t}$)) has a solution u∈C([0,T];L∞(Ω)), which is defined in QT for all T<T^.
Remark 2.17.
We note that the solution u obtained in Theorem 2.16 belongs to the space C([0,T];W01,p(Ω)) and the energy estimate of Theorem 2.11 holds in this case too. Moreover, the results of Proposition 2.13 and Corollary 2.14 also hold for small T>0 (possibly less than T^). Proofs of the aforementioned results are essentially same as in the subhomogeneous case by noticing the fact that the solution is in C([0,T];L∞(Ω)) and the local Lipschitz nature of f makes problem (($\mathrm{P}_{t}$)) of similar kind to problem (($\mathrm{G}_{t}$)). Moreover, as a consequence of the integral representation of the solution as in (5.6), we see that the solution exists globally or there exists T*<∞ such that ∥u(⋅,t)∥L∞(Ω)→∞ as t→T*.
To obtain the blow-up result, for u∈W01,p(Ω), we define the following,
J
ϑ
(
u
)
:=
1
p
∫
Ω
|
∇
u
|
p
+
1
q
∫
Ω
|
∇
u
|
q
-
ϑ
1
-
δ
∫
Ω
|
u
|
1
-
δ
-
∫
Ω
F
(
u
)
,
I
ϑ
(
u
)
:=
∫
Ω
|
∇
u
|
p
+
∫
Ω
|
∇
u
|
q
-
ϑ
∫
Ω
|
u
|
1
-
δ
-
∫
Ω
f
(
u
)
u
.
For δ=1, the term 11-δ|u|1-δ in the first equation is replaced by log|u|. The corresponding Nehari manifold is defined as
𝒩
ϑ
:=
{
u
∈
W
0
1
,
p
(
Ω
)
:
I
ϑ
(
u
)
=
0
}
with
Θ
ϑ
=
inf
u
∈
𝒩
ϑ
J
ϑ
(
u
)
.
We note that Θϑ<0, for the case δ<1. Indeed, proceeding similar to [27, Lemma 4.4], we can prove that the infimum over 𝒩ϑ+∪𝒩ϑ0⊂𝒩ϑ is negative for ϑ<λ*.
Theorem 2.18.
Let the hypotheses in Theorem 2.16 be satisfied and the function f satisfies (f3) with r∈[p,p*) and r>2. Then the solution u of problem (($\mathrm{P}_{t}$)) blows up in finite time in the sense that, there exists T*<∞ such that
lim
t
→
T
*
∥
u
(
⋅
,
t
)
∥
L
∞
(
Ω
)
=
∞
,
under the following conditions:
For
δ
≤
1
, 2NN-2≤q and for all ϑ<ϑ*, where ϑ*>0 is a constant, provided Iϑ(u0)<0 and
J
ϑ
(
u
0
)
≤
{
Θ
ϑ
for
δ
<
1
,
min
{
Θ
ϑ
,
0
}
for
δ
=
1
.
For
δ
∈
(
1
,
2
+
1
p
-
1
)
and for all
ϑ
>
0
, provided
J
ϑ
(
u
0
)
≤
0
.
3 Existence and Regularity Results for Stationary Problems
In this section, we prove the existence and uniqueness of solution to stationary problems associated to (($\mathrm{P}_{t}$)) using the method of sub- and supersolution. Furthermore, using Díaz–Sáa’s inequality, we obtain the comparison principle as in Theorem 3.6. Moreover, we prove higher Sobolev integrability of solution to the purely singular problem, see Theorem 2.8.
To construct a suitable subsolution for the case δ<1, we recall the following proposition proved in [32]. For this purpose, we define the following set:
int
C
+
:=
{
u
∈
C
1
(
Ω
¯
)
:
u
>
0
in
Ω
,
u
=
0
on
∂
Ω
,
∂
u
∂
ν
|
∂
Ω
<
0
}
.
Lemma 3.1 ([32, Proposition 10]).
For all ρ>0, there exists a unique solution u~ρ∈intC+ to the problem
(3.1)
{
-
Δ
p
u
-
Δ
q
u
=
ρ
in
Ω
,
u
=
0
on
∂
Ω
.
Furthermore, the map ρ↦u~ρ is increasing from (0,1] to C01(Ω¯) and u~ρ→0 in C01(Ω¯) as ρ→0+.
Corollary 3.2.
We have u~ρd→0 uniformly in Ω, as ρ→0.
Below we mention results which will be used in the sequel to construct supersolutions.
Proposition 3.3.
Let M≥1, L,l≥0 and δ<1. Then there exists vM∈W01,p(Ω)∩Cδ, solution to the problem
($\mathcal{M}$)
{
-
Δ
p
u
-
Δ
q
u
=
M
u
-
δ
+
l
u
q
-
1
+
L
,
u
>
0
in
Ω
,
u
=
0
on
∂
Ω
.
Moreover, vMd→∞ uniformly in Ω, as M→∞.
Proof.
We first consider the following problem:
{
-
Δ
p
v
-
Δ
q
v
=
v
-
δ
in
Ω
,
v
=
0
on
∂
Ω
.
Then, by [23, Theorem 1.4] with β=0, there exists a unique positive solution v¯∈W01,p(Ω)∩𝒞δ. Next, for s∈ℝ, set
h
~
(
x
,
s
)
:=
{
s
-
δ
if
s
≥
v
¯
(
x
)
,
v
¯
(
x
)
-
δ
otherwise
,
and H~(x,t):=∫0th~(x,s)𝑑s for t∈ℝ. Define E:W01,p(Ω)→ℝ as
E
(
u
)
=
1
p
∫
Ω
|
∇
u
|
p
𝑑
x
+
1
q
∫
Ω
(
|
∇
u
|
q
-
l
|
u
|
q
)
𝑑
x
-
M
∫
Ω
H
~
(
x
,
u
)
𝑑
x
-
L
∫
Ω
u
𝑑
x
.
Due to the fact that δ<1 and q<p, E is coercive and weakly lower semicontinuous in W01,p(Ω), and hence bounded below. Let {vn}∈W01,p(Ω) be a minimizing sequence for E. Since infu∈W01,p(Ω)E(u)<0, it is easy to prove that the sequence {vn} is bounded in W01,p(Ω). Therefore, up to a subsequence, we have the following:
v
n
⇀
v
M
weakly in
W
0
1
,
p
(
Ω
)
and
v
n
→
v
M
in
L
m
(
Ω
)
for
1
≤
m
<
p
*
,
where p*:=NpN-p for p<N and p*<∞ otherwise. Using weak lower semicontinuity of norms and the aforementioned compactness results, we get that vM is a global minimizer for E in W01,p(Ω). By [24, Lemma A2], we know that E is Gâteaux differentiable and thus vM satisfies
{
-
Δ
p
v
M
-
Δ
q
v
M
=
M
h
~
(
x
,
v
M
)
+
l
v
M
q
-
1
+
L
in
Ω
,
v
M
=
0
on
∂
Ω
.
Employing comparison principle, we obtain vM≥v¯, therefore vM is a weak solution of (($\mathcal{M}$)). Next, since δ<1, from [27, Lemma 3.2], we obtain vM∈L∞(Ω). Consequently, using [23, Remark 2.8], we get that vM∈𝒞δ. For the second part, we divide the equation in (($\mathcal{M}$)) by M and substitute wM=M-1p-1+δvM, then
(3.2)
{
-
Δ
p
w
M
-
M
q
-
p
p
-
1
+
δ
Δ
q
w
M
=
w
M
-
δ
+
l
M
q
-
p
p
-
1
+
δ
w
M
q
-
1
+
L
M
1
-
p
p
-
1
+
δ
in
Ω
,
w
M
=
0
on
∂
Ω
.
Due to the fact M≥1, we note that the coefficients of terms on the right-hand side of equation (3.2) are bounded above by quantities which are independent of M. Therefore, careful reading of [25, Theorem 2, p. 361] and [27, Lemma 3.2] implies that ∥wM∥L∞(Ω)≤C1, where C1>0 is a constant independent of M. Now, using [23, Remark 2.8 and Proposition 2.7], we get wM≤C2d in Ω, where C2>0 is a constant independent of M. Thus, invoking [23, Theorem 1.7], we obtain ∥wM∥C1,γ(Ω¯)≤C3, where γ∈(0,1) and C3>0 is a constant independent of M. Therefore, there exists w∈C1(Ω¯) such that wM→w as M→∞. By uniqueness of solution (see [24, Lemma 3.1]) to the problem
(3.3)
{
-
Δ
p
u
=
u
-
δ
,
u
>
0
in
Ω
,
u
=
0
on
∂
Ω
,
we get that w≢0 in Ω and it satisfies the above equation. Due to the convergence of wM to w in C1(Ω¯), we conclude that
w
M
d
→
w
d
uniformly in
Ω
,
as
M
→
∞
.
Since w∈C1(Ω¯) and it satisfies (3.3), by [24, Lemma 3.1], there exists a positive constant c such that w(x)≥cd(x) in Ω, which implies that infΩwd>0. Therefore, we get the required result of the proposition by using the definition of wM. ∎
Next, we consider the case δ≥1. In what follows, we fix the constants c1,c2 appearing in the definition of 𝒞δ for u0 as k1 and k2, respectively. That is, for the case δ>1,
(3.4)
k
1
d
(
x
)
p
p
-
1
+
δ
≤
u
0
(
x
)
≤
k
2
d
(
x
)
p
p
-
1
+
δ
in
Ω
.
Proposition 3.4.
Let M≥1, L≥0, ρ>0 and 1≤δ<2+1p-1. Then:
There exists
v
ρ
∈
W
0
1
,
p
(
Ω
)
∩
𝒞
δ
, solution of the following equation:
(3.5)
{
-
Δ
p
v
-
Δ
q
v
=
ρ
v
-
δ
,
v
>
0
in
Ω
,
v
=
0
on
∂
Ω
,
such that the sequence
v
ρ
→
0
uniformly in
K
, as
ρ
→
0
, for every compact subset
K
of Ω . Moreover, for sufficiently small ρ>0, the following holds:
(3.6)
v
ρ
(
x
)
≤
{
k
1
d
(
x
)
p
p
-
1
+
δ
if
δ
>
1
,
k
1
d
(
x
)
log
1
p
(
A
d
(
x
)
)
if
δ
=
1
,
in
Ω
,
where
k
1
is as in (
3.4
) and
A
is a large enough constant.
Consider the problem
(3.7)
{
-
Δ
p
u
-
Δ
q
u
=
M
u
-
δ
,
u
>
0
in
Ω
,
u
=
0
on
∂
Ω
.
Then there exists
u
M
∈
W
0
1
,
p
(
Ω
)
∩
𝒞
δ
, solution of (
3.7
), such that the sequence
u
M
→
∞
uniformly in
K
, as
M
→
∞
, for every compact subset
K
of Ω . Furthermore, for sufficiently large M, the following holds:
(3.8)
u
M
(
x
)
≥
{
k
2
d
(
x
)
p
p
-
1
+
δ
if
δ
>
1
,
k
2
d
(
x
)
log
1
p
(
A
d
(
x
)
)
if
δ
=
1
,
in
Ω
,
where the constant
k
2
appears in (
3.4
) and
A
is large enough.
Proof.
(a) Proceed similar to the proof of [23, Theorem 1.4] for β=0 to get the existence of vρ∈Wloc1,p(Ω)∩𝒞δ. Moreover, since δ<2+1p-1, we have vρ∈W01,p(Ω) and the regularity result of [23, Theorem 1.8] implies vρ∈C0,α(Ω¯), for some α∈(0,1) (which may depend on ρ). By the weak comparison principle, we have vρ≤v1 for all ρ∈(0,1). Since v1∈𝒞δ, there exists c1>0 such that
0
<
v
ρ
(
x
)
≤
v
1
(
x
)
≤
c
1
d
(
x
)
p
p
-
1
+
δ
≤
c
1
diam
(
Ω
)
p
p
-
1
+
δ
in
Ω
.
That is,
v
ρ
≤
c
1
d
(
x
)
p
p
-
1
+
δ
and
∥
v
ρ
∥
L
∞
(
Ω
)
≤
c
2
,
where c2 is independent of ρ. Therefore, from [23, proof of claim (i), Theorem 1.8], we see that
∫
B
r
|
∇
v
ρ
|
p
≤
C
3
r
N
-
p
,
here C3>0 is a constant independent of ρ. Thus, completing the proof similarly as in the theorem, we get that vρ∈C0,α(Ω¯) and the bound on the norm is independent of ρ. Therefore, by the Arzela–Ascoli Theorem, up to a subsequence, vρ→v uniformly in K, for every compact subset K of Ω. Moreover, from the equation (3.5), it is clear that v=0 in Ω, because the equation
{
-
Δ
p
v
-
Δ
q
v
=
0
in
Ω
,
v
=
0
on
∂
Ω
,
has only trivial solution. Therefore, vρ→0 uniformly in K, as ρ→0, for every compact subset K of Ω.
Now, we will show that vρ(x)≤k1d(x)pp-1+δ, for sufficiently small ρ>0. Set w¯(x)=k1d(x)τ with τ=pp-1+δ. Since the boundary ∂Ω is of class C2, it follows that d∈C2(Ωμ) with |∇d|=1 and |Δd|≤D in Ωμ, where Ωμ:={x∈Ω¯:d(x)<μ}, for μ>0. Choose 0<ϱ≤min{(q-1)(1-τ)2D,μ,1}. Then, for ψ∈Cc∞(Ωϱ) and r∈{p,q}, we deduce that
∫
Ω
ϱ
(
-
Δ
r
w
¯
)
ψ
=
(
k
1
τ
)
r
-
1
∫
Ω
ϱ
d
(
τ
-
1
)
(
r
-
1
)
∇
d
∇
ψ
=
(
k
1
τ
)
r
-
1
∫
Ω
ϱ
(
(
-
Δ
d
)
d
(
τ
-
1
)
(
r
-
1
)
ψ
+
(
r
-
1
)
(
1
-
τ
)
d
(
x
)
(
τ
-
1
)
(
r
-
1
)
-
1
ψ
)
,
that is,
-
Δ
r
w
¯
=
d
(
τ
-
1
)
(
r
-
1
)
-
1
(
k
1
τ
)
r
-
1
[
(
-
Δ
d
)
d
+
(
1
-
τ
)
(
r
-
1
)
]
weakly in
Ω
ϱ
.
Recalling the definition of τ, we see that
(
τ
-
1
)
(
p
-
1
)
-
1
=
-
τ
δ
and
(
τ
-
1
)
(
q
-
1
)
-
1
=
-
τ
δ
+
(
1
-
τ
)
(
p
-
q
)
:=
-
τ
δ
+
η
.
Therefore, the following holds, in the weak sense in Ωϱ:
-
Δ
p
w
¯
-
Δ
q
w
¯
=
k
1
δ
w
¯
δ
[
(
k
1
τ
)
p
-
1
(
-
Δ
d
d
+
(
p
-
1
)
(
1
-
τ
)
)
+
(
k
1
τ
)
q
-
1
d
η
(
-
Δ
d
d
+
(
q
-
1
)
(
1
-
τ
)
)
]
.
By the choice of ϱ, we see that
(
-
Δ
d
d
+
(
q
-
1
)
(
1
-
τ
)
)
≥
(
-
D
ϱ
+
(
q
-
1
)
(
1
-
τ
)
)
≥
(
q
-
1
)
(
1
-
τ
)
2
≥
0
.
Therefore, we have
(3.9)
-
Δ
p
w
¯
-
Δ
q
w
¯
≥
k
1
δ
w
¯
δ
(
k
1
τ
)
p
-
1
(
q
-
1
)
(
1
-
τ
)
2
=:
c
(
k
1
)
w
¯
δ
weakly in
Ω
ϱ
.
Since vρ→0 uniformly in K, for every compact subset K of Ω, there exists ρ1>0 such that ρ1≤c(k1) and for all ρ≤ρ1,
v
ρ
≤
k
1
d
(
x
)
τ
in
Ω
∖
Ω
ϱ
.
Therefore, taking into account (3.5) and (3.9), for all ρ≤ρ1<c(k1), we have
-
Δ
p
w
¯
-
Δ
q
w
¯
-
c
(
k
1
)
w
¯
-
δ
≥
0
≥
-
Δ
p
v
ρ
-
Δ
q
v
ρ
-
c
(
k
1
)
v
ρ
-
δ
in
Ω
ϱ
,
which, by using the weak comparison principle, implies that vρ≤k1d(x)τ in Ωϱ. Hence, for ρ>0 small enough, we have vρ satisfies (3.5) and vρ(x)≤k1d(x)τ≤u0(x) in Ω.
(b) Proceeding similar to case (a), we get the existence of solution uM∈W01,p(Ω)∩𝒞δ of (3.7). Next, we prove the convergence result for sequence uM in the interior of Ω in three steps.
Step (i). Making the substitution vM=M-1p-1+δuM, we see that (3.7) transforms into
(3.10)
{
-
Δ
p
v
M
-
M
q
-
p
p
-
1
+
δ
Δ
q
v
M
=
v
M
-
δ
in
Ω
,
v
M
=
0
on
∂
Ω
.
Since uM∈W01,p(Ω)∩𝒞δ for all M≥1 (here coefficients of the function dτ may depend on M), we see that uMγ∈W01,p(Ω) for all γ≥1. Therefore, for fixed γ>1 (to be specified later), we set wM=vMγ and then (3.10) implies
-
Δ
p
w
M
-
M
q
-
p
p
-
1
+
δ
γ
p
-
q
w
M
(
γ
-
1
)
(
p
-
q
)
/
γ
Δ
q
w
M
+
(
γ
-
1
)
(
p
-
1
)
γ
|
∇
w
M
|
p
w
M
(3.11)
+
M
q
-
p
p
-
1
+
δ
γ
p
-
q
-
1
(
γ
-
1
)
(
q
-
1
)
|
∇
w
M
|
q
w
M
1
+
(
γ
-
1
)
(
q
-
p
)
/
γ
=
γ
p
-
1
w
M
(
-
δ
+
(
γ
-
1
)
(
p
-
1
)
)
/
γ
.
We choose γ>1 such that -δ+(γ-1)(p-1)>0, this implies that -δ+(γ-1)(p-1)γ∈(0,p-1). Therefore, from the equation, it is clear that the sequence {wM} is bounded in W01,p(Ω) with respect to M. Noticing the fact that the right-hand side of (3.11) is bounded from above by 2γp-1(1+|wM|p*-1) and the terms involving gradient of wM on the left are nonnegative, a standard Moser iteration procedure, e.g., see Ladyzhenskaya and Uraĺtseva [28], implies that there exists a constant C1>0, independent of M, such that
(3.12)
∥
w
M
∥
L
∞
(
Ω
)
≤
C
1
.
Step (ii). We will prove that there exists Γ>0, independent of M, such that vM≤Γdpp-1+δ. For this, we set w¯M(x)=M1p-1+δΓd(x)pp-1+δ, for Γ>1. Then a computation similar to (3.9) gives us
-
Δ
p
w
¯
M
-
Δ
q
w
¯
M
≥
Γ
p
-
1
+
δ
M
(
w
¯
M
)
-
δ
≥
M
(
w
¯
M
)
-
δ
in
Ω
ϱ
,
and taking into account (3.12), we can choose Γ>1, independent of M, such that
u
M
≤
M
1
p
-
1
+
δ
C
1
1
γ
≤
M
1
p
-
1
+
δ
Γ
d
p
p
-
1
+
δ
in
Ω
∖
Ω
ϱ
.
Then the weak comparison principle gives us uM≤w¯M=M1p-1+δΓdpp-1+δ, i.e., vM≤Γdpp-1+δ.
Step (iii). Arguing as in case (a) and using [23, Theorem 1.8], we get that {vM} is uniformly bounded in C0,α(Ω¯) with respect to M. Therefore, up to a subsequence, vM→v uniformly in K, as M→∞, for every compact subset K of Ω. Moreover, we see that v is the solution of
{
-
Δ
p
v
=
v
-
δ
,
v
>
0
in
Ω
,
v
=
0
on
∂
Ω
,
since the above equation possesses unique solution. This implies that uM→∞ uniformly in K, as M→∞, for every compact subset K of Ω.
To complete the proof, we set w¯(x)=k2d(x)τ with τ=pp-1+δ. Then a computation similar to the one used to derive (3.9) gives us
-
Δ
p
w
¯
-
Δ
q
w
¯
=
k
2
δ
w
¯
δ
[
(
k
2
τ
)
p
-
1
(
-
Δ
d
d
+
(
p
-
1
)
(
1
-
τ
)
)
+
(
k
2
τ
)
q
-
1
d
η
(
-
Δ
d
d
+
(
q
-
1
)
(
1
-
τ
)
)
]
≤
C
k
2
δ
w
¯
δ
:=
c
(
k
2
)
w
¯
δ
in
Ω
ϱ
,
where we have used |Δd|≤D in Ωϱ and η=(τ-1)(q-p). Since uM→∞ uniformly in K, as M→∞, for every compact subset K of Ω, there exists M1>0 such that M1≥c(k3) and for all M≥M1
u
M
≥
k
2
d
(
x
)
τ
in
Ω
∖
Ω
ϱ
.
Now, proceeding as in case (a), we complete the proof for the case δ>1.
For the case δ=1, the proof differs only in proving counterparts of the inequalities (3.6) and (3.8). For these, we take w¯=k1dlog1p(Ad) and w¯=k2dlog1p(Ad), where A is a large enough constant satisfying A≫diam(Ω). Then, proceeding as above and using the bounds as in [23, Lemma 2.3], we complete the prove of (3.6) and (3.8). ∎
Corollary 3.5.
For every M≥1 and l,L≥0, there exists u~M∈W01,p(Ω)∩Cδ, solution to the following equation:
(3.13)
{
-
Δ
p
u
-
Δ
q
u
=
M
u
-
δ
+
l
u
q
-
1
+
L
in
Ω
,
u
>
0
in
Ω
,
u
=
0
on
∂
Ω
.
Moreover, for sufficiently large M, the following holds:
(3.14)
u
~
M
(
x
)
≥
{
k
2
d
(
x
)
p
p
-
1
+
δ
if
δ
>
1
k
2
d
(
x
)
log
1
p
(
A
d
(
x
)
)
if
δ
=
1
in
Ω
,
where the constant k2 appears in (3.4) and A is large enough.
Proof.
Since 1<q<p and M>0, using the standard minimization technique and employing [23, Lemma 2.2] together with the proof of [23, Theorem 1.4], we get the existence of unique solution u~M∈W01,p(Ω)∩𝒞δ of (3.13) (uniqueness follows from Theorem 3.6 below). Furthermore, it is easy to observe that u~M is a supersolution of (3.7), hence by weak comparison principle (Theorem 3.6), we get u~M≥uM for all M≥1, where uM is the unique solution of (3.7). Therefore, the required result follows from part (b) of Proposition 3.4. ∎
Proof of Theorem 2.4.
We distinguish two cases.
Case (i): When δ<1. For λ>0, we define the energy functional associated to problem (($\mathrm{S}_{\lambda}$)) as follows:
I
λ
(
u
)
:=
1
2
∫
Ω
u
2
𝑑
x
+
λ
p
∫
Ω
|
∇
u
|
p
𝑑
x
+
λ
q
∫
Ω
|
∇
u
|
q
𝑑
x
-
λ
ϑ
1
-
δ
∫
Ω
u
+
1
-
δ
𝑑
x
-
∫
Ω
h
u
+
𝑑
x
.
We see that the first term in Iλ is well defined for all u in W01,p(Ω) for the case p≥2NN+2 and in W01,p(Ω)∩L2(Ω) for 1<p<2NN+2. Hence, functional Iλ is well defined in X, where
X
=
{
W
0
1
,
p
(
Ω
)
if
p
≥
2
N
N
+
2
,
W
0
1
,
p
(
Ω
)
∩
L
2
(
Ω
)
if
1
<
p
<
2
N
N
+
2
.
Furthermore, due to the fact δ<1, it is easy to prove that Iλ is continuous and coercive in X, and strictly convex on the positive cone of X. Therefore, there exists a unique minimizer uλ∈X of Iλ. Moreover, since Iλ(u+)≤Iλ(u) for all u∈X, we may assume uλ≥0 a.e. in Ω. Next, we will show that this uλ is in fact a weak solution of (($\mathrm{S}_{\lambda}$)).
We first construct a subsolution to problem (($\mathrm{S}_{\lambda}$)). As a consequence of Lemma 3.1, for all ρ∈(0,1), we have ∥u~ρ∥C(Ω¯)≤C(ρ), where C(ρ) is a positive constant which goes to 0 as ρ→0. Therefore, for given λ>0, there exists ρλ>0 (sufficiently small) such that
(3.15)
u
~
ρ
λ
-
λ
(
Δ
p
u
~
ρ
λ
+
Δ
q
u
~
ρ
λ
+
ϑ
u
~
ρ
λ
-
δ
)
=
u
~
ρ
λ
-
λ
(
-
ρ
λ
+
ϑ
u
~
ρ
λ
-
δ
)
<
-
∥
h
∥
L
∞
(
Ω
)
≤
h
in
Ω
,
where u~ρλ∈C01(Ω¯) is the unique solution of (3.1). Set vλ:=(u~ρλ-uλ)+ and Θ(t):=Iλ(uλ+tvλ) for t≥0. Then, applying the arguments of [3, p. 5049] and using (3.15), we have cρλd≤u~ρλ≤uλ which, on account of [24, Lemma A2], implies that Iλ is Gâteaux differentiable. Consequently, uλ is a weak solution of (($\mathrm{S}_{\lambda}$)). Next, choosing C>0 large enough so that ∥h∥L∞(Ω)<C-λϑC-δ and using the weak comparison principle, we get uλ≤C, that is, uλ∈L∞(Ω). To complete the proof in this case, we need to prove uλ∈𝒞δ. For this purpose, we consider uM∈W01,p(Ω)∩𝒞δ, the unique solution to (($\mathcal{M}$)) given by Proposition 3.3 for M>max{1,ϑ}, l=0 and λL>∥h∥L∞(Ω). Then
u
M
-
λ
(
Δ
p
u
M
+
Δ
q
u
M
+
ϑ
u
M
-
δ
)
=
u
M
-
λ
(
-
M
u
M
-
δ
-
L
+
ϑ
u
M
-
δ
)
≥
λ
(
M
-
ϑ
)
u
M
-
δ
+
λ
L
≥
∥
h
∥
L
∞
(
Ω
)
,
that is, uM is a supersolution to problem (($\mathrm{S}_{\lambda}$)). Therefore, by comparison principle, we get uλ≤uM in Ω, consequently uλ∈𝒞δ. This completes the proof of theorem for the case δ<1.
Case (ii): When 1≤δ<2+1p-1. For the supersolution, we choose the numbers M,L,l in (3.13) as in case (i) above. Then, proceeding similarly, it is easy to observe that u¯:=u~M∈W01,p(Ω)∩𝒞δ, satisfying (3.13), is a supersolution of (($\mathrm{S}_{\lambda}$)). Furthermore, from the proof of Proposition 3.4, it is clear that the choice of the constant k1 in (3.6) is arbitrary, that is, for given η>0, there exists ρ>0 (sufficiently small) such that
(3.16)
v
ρ
(
x
)
≤
{
η
d
(
x
)
p
p
-
1
+
δ
if
δ
>
1
η
d
(
x
)
log
1
p
(
A
d
(
x
)
)
if
δ
=
1
in
Ω
.
Therefore, for δ>1 and λ,ϑ>0, we fix η>0 satisfying
η
δ
≤
min
{
λ
ϑ
2
diam
(
Ω
)
α
δ
δ
(
diam
(
Ω
)
α
δ
+
∥
h
∥
L
∞
(
Ω
)
)
,
1
}
,
where αδ=pp-1+δ. Hence, by above discussion, we can choose ρ∈(0,ϑ2) (sufficiently small) such that (3.16) holds and
v
ρ
-
λ
(
Δ
p
v
ρ
+
Δ
q
v
ρ
+
ϑ
v
ρ
-
δ
)
=
v
ρ
+
λ
(
ρ
-
ϑ
)
v
ρ
-
δ
≤
∥
v
ρ
∥
L
∞
(
Ω
)
-
λ
ϑ
2
∥
v
ρ
∥
L
∞
(
Ω
)
-
δ
≤
η
diam
(
Ω
)
α
δ
-
λ
ϑ
2
η
-
δ
diam
(
Ω
)
-
α
δ
δ
≤
-
∥
h
∥
L
∞
(
Ω
)
,
where vρ∈W01,p(Ω)∩𝒞δ is the solution of (3.5). For the case δ=1, we observe that
d
(
x
)
log
1
p
(
A
d
(
x
)
)
≤
A
1
p
d
(
x
)
1
-
1
p
.
Therefore, rest of the calculations follows similarly with αδ=1-1p. Thus, the above constructed vρ is a subsolution to problem (($\mathrm{S}_{\lambda}$)).
By noticing δ<2+1p-1, a truncation argument similar to Proposition 3.3 and the weak comparison principle of [23, Theorem 1.5] proves the existence of a solution u∈W01,p(Ω)∩𝒞δ of (($\mathrm{S}_{\lambda}$)) for all ϑ>0 and δ∈(1,2+1p-1). The strict monotonicity of the operator in (($\mathrm{S}_{\lambda}$)) implies the uniqueness of u. ∎
Proof of Theorem 2.6.
The standard minimization technique (see [23, Lemma 2.1]) shows that there exists a unique solution uϵ∈C1,σ(Ω¯), for some σ∈(0,1), to the following auxiliary problem:
($\mathrm{S}_{\epsilon}$)
{
u
ϵ
-
λ
(
Δ
p
u
ϵ
+
Δ
q
u
ϵ
+
ϑ
(
u
ϵ
+
ϵ
)
-
δ
)
=
h
in
Ω
,
u
ϵ
>
0
in
Ω
,
u
ϵ
=
0
on
∂
Ω
.
Moreover, following [23, Lemma 2.2], with
u
¯
ϵ
(
x
)
=
η
(
(
d
(
x
)
+
ϵ
p
-
1
+
δ
p
)
p
p
-
1
+
δ
-
ϵ
)
and
u
¯
ϵ
(
x
)
=
Γ
(
(
d
(
x
)
+
ϵ
p
-
1
+
δ
p
)
p
p
-
1
+
δ
-
ϵ
)
for η>0 small and Γ large enough, we can prove u¯ϵ≤uϵ≤u¯ϵ. Next, we take uϵγ as a test function in the weak formulation of problem (($\mathrm{S}_{\epsilon}$)) for some γ>0,
(3.17)
1
λ
∫
Ω
u
ϵ
1
+
γ
+
∫
Ω
|
∇
u
ϵ
|
p
-
2
∇
u
ϵ
∇
u
ϵ
γ
+
∫
Ω
|
∇
u
ϵ
|
q
-
2
∇
u
ϵ
∇
u
ϵ
γ
=
ϑ
∫
Ω
u
ϵ
γ
(
u
ϵ
+
ϵ
)
-
δ
+
1
λ
∫
Ω
h
u
ϵ
γ
.
We then observe that
∫
Ω
|
∇
u
ϵ
|
p
-
2
∇
u
ϵ
∇
u
ϵ
γ
=
γ
(
p
p
+
γ
-
1
)
p
∫
Ω
|
∇
u
ϵ
p
+
γ
-
1
p
|
p
and similar result holds for the third term on the left of (3.17). Therefore, using the relation uϵ≤u¯ϵ, for γ≥δ, from (3.17), we infer that
γ
(
p
p
+
γ
-
1
)
p
∫
Ω
|
∇
u
ϵ
p
+
γ
-
1
p
|
p
≤
C
∫
Ω
u
¯
ϵ
(
γ
-
δ
)
𝑑
x
+
λ
-
1
∥
h
∥
L
∞
(
Ω
)
∫
Ω
u
¯
ϵ
γ
𝑑
x
.
Furthermore, noticing u¯ϵ≤Γdpp-1+δ, we get that the right-side quantity is finite if and only if γ>δ-p-1+δp. Thus, uϵρ∈W01,p(Ω) is uniformly bounded for all ρ>(p-1)(p-1+δ)p2. Therefore, on account of embedding results of W01,p(Ω), we can extract a subsequence, still denoted by uϵ, such that uϵ(x)→u(x) a.e. in Ω, for some u∈Wloc1,p(Ω). By the local Hölder regularity result of Lieberman [29, Theorem 1.7], we infer that the sequence uϵ converges to u in Cloc1(Ω). Therefore, u satisfies equation (P) in the sense of distribution. Moreover, from the relation u¯ϵ≤uϵ≤u¯ϵ and passing to limit ϵ→0, we deduce that
(3.18)
η
d
(
x
)
p
p
-
1
+
δ
≤
u
(
x
)
≤
Γ
d
(
x
)
p
p
-
1
+
δ
in
Ω
.
Repeating the proof of boundedness of the sequence {uϵρ} and using the comparison estimate above, we see that u(p+δ-1)/p∈W01,p(Ω). Taking limx→x0∈∂Ωu(x), we get u∈C0(Ω¯), consequently u∈𝒞δ. Next, we will show that u∉W01,p(Ω). On the contrary, we assume u∈W01,p(Ω), then from the weak formulation, it is clear that ∫Ωu1-δ<∞. Using (3.18), we obtain a contradiction. This completes proof of the theorem. ∎
Next, we prove the following weak comparison principle for singular problems.
Theorem 3.6.
Let 1<q<p<∞ and let g:Ω×R+→R be a Carathéodory function bounded from below such that it satisfies condition (f2) with g in place of f. Let u,v∈L∞(Ω)∩W01,p(Ω) be such that u,v>0 in Ω, ∫Ωu1-δ𝑑x<∞, ∫Ωv1-δ𝑑x<∞ and the following hold, in weak sense, in Ω:
-
Δ
p
u
-
Δ
q
u
≤
u
-
δ
+
g
(
x
,
u
)
𝑎𝑛𝑑
-
Δ
p
v
-
Δ
q
v
≥
v
-
δ
+
g
(
x
,
v
)
.
Furthermore, suppose that there exists a positive function w∈L∞(Ω) and constants c1,c2>0 such that c1w≤u,v≤c2w with
(3.19)
∫
Ω
|
g
(
x
,
c
1
w
)
|
w
𝑑
x
<
∞
𝑎𝑛𝑑
∫
Ω
|
g
(
x
,
c
2
w
)
|
w
𝑑
x
<
∞
.
Then the comparison principle holds, that is, u≤v in Ω.
Proof.
For ϵ>0, set uϵ=u+ϵ and vϵ=v+ϵ. Let
ϕ
:=
u
ϵ
q
-
v
ϵ
q
u
ϵ
q
-
1
and
ψ
:=
v
ϵ
q
-
u
ϵ
q
v
ϵ
q
-
1
.
We recall the following Díaz–Saá inequality [35, Theorem 2.5] (see also [11, Remark 2.10]):
(3.20)
∫
Ω
(
-
Δ
p
w
1
w
1
r
-
1
+
Δ
p
w
2
w
2
r
-
1
)
(
w
1
r
-
w
2
r
)
𝑑
x
≥
0
for all 1<r≤p, and the equality holds if and only if w1=kw2, for some constant k>0. Set
Ω
+
=
{
x
∈
Ω
:
u
(
x
)
>
v
(
x
)
}
.
Then we have ϕ≥0 and ψ≤0 in Ω+. Now testing the first equation by ϕ and the second by ψ, we obtain
∫
Ω
+
(
-
Δ
p
u
-
Δ
q
u
)
ϕ
+
∫
Ω
+
(
-
Δ
p
v
-
Δ
q
v
)
ψ
≤
∫
Ω
+
(
u
-
δ
+
g
(
x
,
u
)
)
ϕ
+
∫
Ω
+
(
v
-
δ
+
g
(
x
,
v
)
)
ψ
.
Rearranging the terms and using the Díaz–Saá inequality (3.20), we get
(3.21)
0
≤
∫
Ω
+
(
-
Δ
p
u
ϵ
u
ϵ
q
-
1
+
Δ
p
v
ϵ
v
ϵ
q
-
1
)
(
u
ϵ
q
-
v
ϵ
q
)
+
∫
Ω
+
(
-
Δ
q
u
ϵ
u
ϵ
q
-
1
+
Δ
q
v
ϵ
v
ϵ
q
-
1
)
(
u
ϵ
q
-
v
ϵ
q
)
≤
∫
Ω
+
(
u
-
δ
+
g
(
x
,
u
)
)
ϕ
+
∫
Ω
+
(
v
-
δ
+
g
(
x
,
v
)
)
ψ
.
Noticing the fact that u-δϕ+v-δψ≤0 in Ω+, the right side quantity of above equation simplifies to
∫
Ω
+
(
u
-
δ
+
g
(
x
,
u
)
)
ϕ
+
∫
Ω
+
(
v
-
δ
+
g
(
x
,
v
)
)
ψ
≤
∫
Ω
+
[
g
(
x
,
u
)
u
q
-
1
u
q
-
1
u
ϵ
q
-
1
-
g
(
x
,
v
)
v
q
-
1
v
q
-
1
v
ϵ
q
-
1
]
(
u
ϵ
q
-
v
ϵ
q
)
𝑑
x
.
Since uuϵ→1 and vvϵ→1 as ϵ→0+ a.e. in Ω, using (3.19), (f2) with g in place of f, and the Dominated Convergence Theorem, we obtain
(3.22)
lim
ϵ
→
0
+
∫
Ω
+
(
g
(
x
,
u
)
ϕ
+
g
(
x
,
v
)
ψ
)
≤
0
.
Taking into account (3.20), (3.21), (3.22) and using Fatou’s lemma, we deduce that
∫
Ω
+
(
-
Δ
q
u
u
q
-
1
+
Δ
q
v
v
q
-
1
)
(
u
q
-
v
q
)
=
0
,
therefore, there exists k∈ℝ such that u=kv in Ω+. Now, we will prove that k≤1. On the contrary assume k>1. Then the following hold:
k
q
∫
Ω
+
(
|
∇
v
|
p
+
|
∇
v
|
q
)
≤
∫
Ω
+
(
|
∇
u
|
p
+
|
∇
u
|
q
)
≤
∫
Ω
+
(
u
1
-
δ
+
g
(
x
,
u
)
u
)
=
∫
Ω
+
(
k
1
-
δ
v
1
-
δ
+
g
(
x
,
k
v
)
k
v
)
,
k
q
∫
Ω
+
(
|
∇
v
|
p
+
|
∇
v
|
q
)
≥
k
q
∫
Ω
+
(
v
1
-
δ
+
g
(
x
,
v
)
v
)
≥
∫
Ω
+
(
k
q
v
1
-
δ
+
g
(
x
,
k
v
)
k
v
)
,
where in the last inequality we have used the fact that k>1, v>0 and g(x,s)sq-1 is decreasing in s. Therefore, we get kq+δ-1≤1, which yields a contradiction, since δ>0,q>1. This implies that u≤v in Ω+ and from the definition of Ω+, we get u≤v in Ω. ∎
Proof of Theorem 2.7.
By using (f1) and the assumptions that f is bounded from below and uniformly locally Lipschitz, there exist 0<αf<m and L>0 such that -L≤f(x,s)≤msq-1+L. We first consider the case δ<1. Fix ϵ>0 small enough so that the unique positive solution u~ϵ of (3.1) satisfies the following inequality in weak sense:
-
Δ
p
u
~
ϵ
-
Δ
q
u
~
ϵ
-
ϑ
u
~
ϵ
-
δ
≤
-
L
in
Ω
.
Furthermore, we take l>m and M>max{1,ϑ} in Proposition 3.3 and denote v∈W01,p(Ω)∩𝒞δ as the unique positive solution of problem (($\mathcal{M}$)) in (($\mathcal{M}$)). Then it follows that
-
Δ
p
v
-
Δ
q
v
-
ϑ
v
-
δ
≥
m
v
q
-
1
+
L
in
Ω
.
By the weak comparison principle, we have u~ϵ≤v in Ω. Next, for s∈ℝ, we set the following:
h
(
x
,
s
)
=
{
s
-
δ
if
s
≥
u
~
ϵ
(
x
)
,
u
~
ϵ
(
x
)
-
δ
otherwise
,
g
(
x
,
s
)
=
{
f
(
x
,
u
~
ϵ
(
x
)
)
if
u
~
ϵ
(
x
)
≤
s
,
f
(
x
,
s
)
if
u
~
ϵ
(
x
)
≤
s
≤
v
(
x
)
,
f
(
x
,
v
(
x
)
)
if
s
≤
v
(
x
)
.
Let
H
(
x
,
t
)
:=
∫
0
t
h
(
x
,
s
)
𝑑
s
and
G
(
x
,
t
)
:=
∫
0
t
g
(
x
,
s
)
𝑑
s
for t∈ℝ. We define the energy functional 𝒥:W01,p(Ω)→ℝ as follows:
𝒥
(
u
)
=
1
p
∫
Ω
|
∇
u
|
p
𝑑
x
+
1
q
∫
Ω
|
∇
u
|
q
𝑑
x
-
ϑ
∫
Ω
H
(
x
,
u
)
𝑑
x
-
∫
Ω
G
(
x
,
u
)
𝑑
x
.
It is easy to verify that 𝒥 is coercive and weakly lower semicontinuous in W01,p(Ω). Similar to the proof of Proposition 3.3, we obtain a global minimizer u of 𝒥 in W01,p(Ω). Again by using [24, Lemma A2], we get that 𝒥 is Gâteaux differentiable and thus u satisfies
{
-
Δ
p
u
-
Δ
q
u
=
ϑ
h
(
x
,
u
)
+
g
(
x
,
u
)
in
Ω
,
u
=
0
on
∂
Ω
.
Then, by repeated application of weak comparison principle, first we can conclude that u~ϵ≤u, consequently h(x,u)=u-δ, and then u≤v. Therefore, u satisfies (P) in the weak sense. From the above procedure it is clear that u∈𝒞δ.
Next, we consider the case δ≥1. Since f is a uniformly local Lipschitz function, there exists K>0 such that the map t↦f(x,t)+Kt is nondecreasing in [0,∥u¯∥L∞(Ω)], where u¯ is specified below. Consider the following iterative scheme:
(3.23)
{
-
Δ
p
u
n
-
Δ
q
u
n
-
ϑ
u
n
-
δ
+
K
u
n
=
f
(
x
,
u
n
-
1
)
+
K
u
n
-
1
in
Ω
,
u
n
=
0
on
∂
Ω
,
with u0:=u¯. Now, we specify the choices for u¯ and u¯. Since f(x,s)≥-L, we construct a subsolution, as in the proof of Theorem 2.4 (with L in place of ∥h∥L∞(Ω)), for the case δ≥1. Therefore, for sufficiently small ρ>0, we set u¯=vρ. For the supersolution, we again follow the similar steps as in Theorem 2.4 (with M>max{1,ϑ} and l=m in (3.13)) and set u¯=u~M. By the choice of K and Theorem 2.4, we see that the scheme is well defined and produces a sequence {un}⊂W01,p(Ω)∩𝒞δ∩C0(Ω¯) as solution to problem (3.23). Using weak comparison principle (Theorem 3.6), we obtain u¯≤un≤u¯ and again using this together with the monotonicity of the map t↦f(x,t)+Kt, we get that the sequence {un} is monotone increasing. Moreover, as a consequence of [23, Theorem 1.8], un∈C0,σ(Ω¯) and from the relation u¯≤un≤u¯, we get that the sequence {un} is uniformly bounded in 𝒞δ∩C0(Ω¯) with respect to n. Therefore, by the Arzela–Ascoli Theorem, we get that un→u in 𝒞δ∩C0(Ω¯), as n→∞. This, together with the weak formulation of (3.23), implies that the sequence {un} is Cauchy in W01,p(Ω) and therefore converges to u in W01,p(Ω). Then, passing to the limit as n→∞ in the equation (3.23) and using Lebesgue Dominated Convergence Theorem, we get that u is a solution of problem (P). Furthermore, the uniqueness of the solution follows by Theorem 3.6. ∎
Now, we prove some higher Sobolev integrability result for the following equation:
(3.24)
div
A
(
x
,
∇
u
)
=
div
f
in
Ω
,
where f∈Lpp-1(Ω;ℝN), Ω is a bounded domain and A:Ω¯×ℝN→ℝN satisfies the following:
|
A
(
x
,
z
)
|
+
|
∂
z
A
(
x
,
z
)
|
|
z
|
≤
Λ
(
1
+
|
z
|
p
-
1
)
,
〈
z
,
A
(
x
,
z
)
〉
ℝ
N
≥
ν
|
z
|
p
,
∑
i
=
1
N
|
A
i
(
x
,
z
)
-
A
i
(
y
,
z
)
|
≤
Λ
(
1
+
|
z
|
p
-
1
)
|
x
-
y
|
ω
,
where 0<ν≤Λ are constants. We have the following notation:
D
m
(
Ω
)
:=
{
g
∈
L
m
(
Ω
;
ℝ
N
)
:
there exists
u
∈
L
loc
1
(
Ω
)
with
g
=
∇
u
}
.
We state our theorem in this regard as follows.
Theorem 3.7.
Let f∈Lpp-1(Ω;RN) and let u be a weak solution of (3.24) such that ∇u∈Dp(Ω). Suppose f∈Lmp-1(Ω;RN) for some m≥p. Then ∇u belongs to Lm(Ω;RN) and
(3.25)
∥
∇
u
∥
L
m
p
-
1
≤
C
m
(
∥
f
∥
L
m
p
-
1
+
1
)
,
where Cm>0 is a constant which depends only on N,m,p,Ω and Λ,ν,ω.
Proof.
The proof of the theorem is essentially the same as the one of [20, Theorem 1.1](see also [13]). Indeed, let w∈W1,p(BR) be the unique solution to the problem
{
div
A
(
0
,
∇
w
)
=
0
in
B
R
,
w
=
u
on
∂
B
R
.
Then, proceeding similar to [20, Lemma 2.1], with |F|=|f|1p-1, one can prove the following: for η∈(0,1), there exists a constant Cη>0 such that
(3.26)
⨍
B
ρ
|
∇
(
u
-
w
)
|
p
≤
η
(
R
ρ
)
N
⨍
B
R
|
∇
w
|
p
+
C
η
(
R
ρ
)
N
⨍
B
R
|
f
|
p
′
for
0
<
ρ
≤
R
.
Moreover, from the proof of [20, Proposition 2.1], it follows that
(3.27)
⨍
B
ρ
|
|
∇
u
|
p
-
(
|
∇
u
|
p
)
ρ
|
≤
γ
⨍
B
ρ
|
∇
(
u
-
w
)
|
p
+
γ
⨍
B
ρ
|
∇
w
-
(
∇
w
)
ρ
|
p
for all
ρ
∈
(
0
,
R
)
,
where (v)ρ=⨍Bρv𝑑x and γ is a positive constant depending only on p. Then, for all ρ∈(0,R7) and η∈(0,1), using the estimates of [23, Lemma 4.1] (which holds for the interior balls too) and taking into account (3.26), we deduce from (3.27) that
⨍
B
ρ
|
|
∇
u
|
p
-
(
|
∇
u
|
p
)
ρ
|
≤
(
R
ρ
)
N
[
η
⨍
B
R
|
∇
w
|
p
+
C
η
⨍
B
R
|
f
|
p
′
]
+
C
(
ρ
R
)
ς
p
(
R
-
N
∫
B
R
|
∇
w
|
p
+
1
)
≤
C
η
(
R
ρ
)
N
⨍
B
R
|
f
|
p
′
+
[
η
(
R
ρ
)
N
+
C
(
ρ
R
)
ς
p
]
⨍
B
R
|
∇
w
|
p
+
C
(
ρ
R
)
ς
p
,
where Cη>0 is a constant and ς∈(0,1) as appeared in [23, Lemma 4.1]. Now, proof of (3.25) is a standard procedure by making the use of maximal operators and can be completed as in [20, proof of Theorem 1.1, pp. 1113–1116]. ∎
Corollary 3.8.
Let 1<q<p and let u be a solution to the problem:
{
-
Δ
p
u
-
Δ
q
u
=
g
(
x
)
+
b
(
x
)
in
Ω
,
u
=
0
on
∂
Ω
,
where b∈L∞(Ω), c1d(x)-α≤g(x)≤c2d(x)-α with α∈(1,2) and c1,c2>0 are constants. Then |∇u|∈Lm(Ω) for m<p-1α-1.
Proof.
We first consider the following problem:
{
-
Δ
w
=
g
(
x
)
+
b
(
x
)
in
Ω
,
w
=
0
on
∂
Ω
.
Since the domain is C2 regular, there exists Green function G(x,y) for -Δ in Ω. Furthermore, there exists a positive constant c such that the following holds for x,y∈Ω (see [36]):
1
c
|
x
-
y
|
N
min
{
|
x
-
y
|
2
,
d
(
x
)
d
(
y
)
}
≤
G
(
x
,
y
)
≤
c
|
x
-
y
|
N
min
{
|
x
-
y
|
2
,
d
(
x
)
d
(
y
)
}
.
The solution w is represented by
w
(
x
)
:=
∫
Ω
G
(
x
,
y
)
(
g
(
y
)
+
b
(
y
)
)
𝑑
y
.
Then, using the behavior of g, it is not difficult to prove that
{
c
3
d
(
x
)
≤
w
(
x
)
≤
c
4
d
(
x
)
if
α
<
1
,
c
7
d
(
x
)
2
-
α
≤
w
(
x
)
≤
c
8
d
(
x
)
2
-
α
if
α
∈
(
1
,
2
)
.
Proceeding similar to the proof of [18, Theorem 2.5], we can prove that |∇w|≤c9d(x)1-α, for some positive constant c9. Therefore, we see that ∇w∈Lmp-1(Ω;ℝN) for all m<p-1α-1. Thus, for given functions g and b, satisfying the assumptions of the Corollary, there exists a function w such that
-
Δ
p
u
-
Δ
q
u
=
g
(
x
)
+
b
(
x
)
=
-
Δ
w
=
div
(
-
∇
w
)
in
Ω
.
Therefore, taking A(x,ξ):=|ξ|p-2ξ+|ξ|q-2ξ, where x∈Ω and ξ∈ℝN, and f=-∇w in (3.25), we get the required result of the corollary. ∎
Proof of Theorem 2.8.
Let u∈Wloc1,p(Ω) be the solution of problem (PS), whose existence and uniqueness is guaranteed by similar steps of Theorems 1.4 and 1.5, respectively, of [23]. Moreover, u satisfies the following:
c
1
d
(
x
)
p
p
-
1
+
δ
≤
u
(
x
)
≤
c
2
d
(
x
)
p
p
-
1
+
δ
,
for some positive constants c1,c2. Indeed, using the estimates of [23, Lemma 2.2], one can construct a sub- and supersolution u¯ and u¯ (which are in W01,p(Ω)∩𝒞δ) of (PS), respectively, as below:
-
Δ
p
u
¯
-
Δ
q
u
¯
≤
u
¯
-
δ
-
∥
b
∥
L
∞
(
Ω
)
,
-
Δ
p
u
¯
-
Δ
q
u
¯
≥
u
¯
-
δ
+
∥
b
∥
L
∞
(
Ω
)
.
Then we observe that u-δ behaves like d(x)-pδp-1+δ with pδp-1+δ<2. Let v be the solution of the following problem, as obtained in Corollary 3.8:
(3.28)
{
-
Δ
p
v
-
Δ
q
v
=
u
-
δ
+
b
(
x
)
in
Ω
,
v
=
0
on
∂
Ω
.
Employing the result of Corollary 3.8, we get that |∇v|∈Lm(Ω) for all m<p-1+δδ-1. Since u satisfies (3.28) and u∈C0(Ω¯), we get the required result of the theorem. ∎
4 Existence and Regularity of Solution to (($\mathrm{G}_{t}$))
In this section, we obtain the existence result for problem (($\mathrm{G}_{t}$)) with the help of Theorem 2.4 and using semi-discretization in time with implicit Euler method.
Proof of Theorem 2.10.
Fix N0∈ℕ and set Δt=TN0. For 0≤n≤N0, define tn=nΔt and
g
n
(
x
)
=
1
Δ
t
∫
t
n
-
1
t
n
g
(
x
,
τ
)
𝑑
τ
.
Since g∈L∞(QT), we have gn∈L∞(Ω). Set
g
Δ
t
(
x
,
t
)
=
g
n
(
x
)
if
t
∈
[
t
n
-
1
,
t
n
)
for
1
≤
n
≤
N
0
.
It follows from Jensen’s inequality that, for 1<r<∞,
(4.1)
∥
g
Δ
t
∥
L
r
(
Q
T
)
≤
(
T
|
Ω
|
)
1
r
∥
g
∥
L
∞
(
Q
T
)
and gΔt→g in Lr(QT) as Δt→0. We take λ=Δt,h=Δtgn+un-1∈L∞(Ω) in (($\mathrm{S}_{\lambda}$)) and define iteratively un∈W01,p(Ω)∩𝒞δ with the following implicit Euler scheme:
(4.2)
{
u
n
-
u
n
-
1
Δ
t
-
Δ
p
u
n
-
Δ
q
u
n
-
ϑ
(
u
n
)
-
δ
=
g
n
in
Ω
,
u
n
=
0
on
∂
Ω
,
where u0=u0∈W01,p(Ω)∩𝒞δ. Then, for all n∈{1,…,N0} and t∈[tn-1,tn), we set
(4.3)
u
Δ
t
(
⋅
,
t
)
:=
u
n
(
⋅
)
and
u
~
Δ
t
(
⋅
,
t
)
:=
(
t
-
t
n
-
1
)
Δ
t
(
u
n
(
⋅
)
-
u
n
-
1
(
⋅
)
)
+
u
n
-
1
(
⋅
)
.
From the above definition, it is easy to observe that
(4.4)
∂
u
~
Δ
t
∂
t
-
Δ
p
u
Δ
t
-
Δ
q
u
Δ
t
-
ϑ
u
Δ
t
-
δ
=
g
Δ
t
∈
L
∞
(
Q
T
)
.
Now, multiplying (4.2) by Δtun and summing from n=1 to N1≤N0, for ϵ>0, using Young’s inequality, we get
∑
n
=
1
N
1
∫
Ω
(
u
n
-
u
n
-
1
)
u
n
+
Δ
t
(
∑
n
=
1
N
1
∥
u
n
∥
W
0
1
,
p
(
Ω
)
p
+
∑
n
=
1
N
1
∥
u
n
∥
W
0
1
,
q
(
Ω
)
q
-
ϑ
∑
n
=
1
N
1
∫
Ω
(
u
n
)
1
-
δ
)
≤
C
(
ϵ
)
∥
g
∥
L
∞
(
Ω
)
p
′
+
ϵ
Δ
t
∑
n
=
1
N
1
∥
u
n
∥
W
0
1
,
p
(
Ω
)
p
,
where C(ϵ)>0 is a constant and p′=pp-1. An easy manipulation yields
∑
n
=
1
N
1
∫
Ω
(
u
n
-
u
n
-
1
)
u
n
=
1
2
(
∑
n
=
1
N
1
∫
Ω
|
u
n
-
u
n
-
1
|
2
+
∫
Ω
(
u
N
1
)
2
-
∫
Ω
u
0
2
)
.
To control the integrals involving singular term, we will construct a suitable sub- and supersolution, u¯ and u¯ in W01,p(Ω)∩𝒞δ such that u¯≤u0≤u¯ and the following hold in the weak sense in Ω:
(4.5)
-
Δ
p
u
¯
-
Δ
q
u
¯
-
ϑ
u
¯
-
δ
≤
-
∥
g
∥
L
∞
(
Q
T
)
,
-
Δ
p
u
¯
-
Δ
q
u
¯
-
ϑ
u
¯
-
δ
≥
∥
g
∥
L
∞
(
Q
T
)
.
Indeed, since u0∈𝒞δ, there exist positive constants k1,k2 such that k1d(x)≤u0(x)≤k2d(x) in Ω, for δ<1. On account of Corollary 3.2, we can choose ρ>0 sufficiently small in Lemma 3.1 such that u~ρ≤u0 and the first equation of (4.5) holds with u¯=u~ρ, where u~ρ is the unique positive solution of (3.1). To obtain a suitable supersolution, we invoke Proposition 3.3. Let uM∈W01,p(Ω)∩𝒞δ be the unique solution of problem (($\mathcal{M}$)) in (($\mathcal{M}$)) with l=0, L≥∥g∥L∞(QT) and M>max{1,ϑ}. Since uMd→∞ uniformly in Ω as M→∞, we can choose u¯=uM, for M sufficiently large, such that the second equation in (4.5) holds and u0≤u¯=uM in Ω (for details in the case δ<1, see case (i) of the proof of Theorem 2.4). Whereas for the case δ≥1, we construct subsolution by following similar procedure to the proof of Theorem 2.4 with L=∥g∥L∞(QT) and an additional assumption that ρ>0 is sufficiently small such that (3.6) holds. That is, u¯=vρ, where vρ is the solution of (3.5) and ρ>0 is sufficiently small such that u¯≤u0 (a consequence of (3.6)). While for the supersolution, we take l=0, L=∥g∥L∞(QT) and M large enough in (3.13) such that the solution u~M:=u¯ satisfies (3.14), that is, u¯≥u0. Then repeated application of the weak comparison principle gives us u¯≤un≤u¯ and thus
(4.6)
u
¯
≤
u
Δ
t
,
u
~
Δ
t
≤
u
¯
.
Therefore,
Δ
t
∑
n
=
1
N
1
∫
Ω
(
u
n
)
1
-
δ
≤
{
T
∫
Ω
u
¯
1
-
δ
<
∞
if
δ
≤
1
,
T
∫
Ω
u
¯
1
-
δ
<
∞
if
δ
>
1
.
Resuming all the information, we conclude that the sequences uΔt,u~Δt∈𝒞δ uniformly and are bounded in Lp(0,T;W01,p(Ω))∩L∞(0,T;L∞(Ω)). Proceeding similar to [3, (2.10)–(2.12), p. 5054], we obtain
(4.7)
Δ
t
2
∑
n
=
1
N
1
∫
Ω
(
u
n
-
u
n
-
1
Δ
t
)
2
+
1
p
∫
Ω
|
∇
u
N
1
|
p
-
1
p
∫
Ω
|
∇
u
0
|
p
+
1
q
∫
Ω
|
∇
u
N
1
|
q
-
1
q
∫
Ω
|
∇
u
0
|
q
-
ϑ
1
-
δ
(
∫
Ω
|
u
N
1
|
1
-
δ
-
∫
Ω
|
u
0
|
1
-
δ
)
≤
|
Ω
|
T
2
∥
g
∥
L
∞
(
Q
T
)
2
.
From the above expression together with the fact ∫Ω(un)1-δ≤max{∫Ωu¯1-δ,∫Ωu¯1-δ}, we infer that
(4.8)
∂
u
~
Δ
t
∂
t
is bounded in
L
2
(
Q
T
)
uniformly in
Δ
t
,
(4.9)
u
Δ
t
,
u
~
Δ
t
are bounded in
L
∞
(
0
,
T
;
W
0
1
,
p
(
Ω
)
)
uniformly in
Δ
t
.
On account of the preceding observation and (4.7), an easy computation yields
(4.10)
∥
u
Δ
t
-
u
~
Δ
t
∥
L
∞
(
0
,
T
;
L
2
(
Ω
)
)
≤
max
1
≤
n
≤
N
0
∥
u
n
-
u
n
-
1
∥
L
2
(
Ω
)
≤
C
(
Δ
t
)
1
2
.
Indeed, for the last inequality, from (4.7) (which holds for N1=N0 also) and the uniform bound on un with respect to n, we have
1
2
Δ
t
∫
Ω
(
u
n
-
u
n
-
1
)
2
≤
|
Ω
|
T
2
∥
g
∥
L
∞
(
Q
T
)
2
+
1
p
∫
Ω
|
∇
u
0
|
p
+
1
q
∫
Ω
|
∇
u
0
|
q
+
ϑ
1
-
δ
∫
Ω
|
u
N
0
|
1
-
δ
for all n∈{1,…,N0}. Furthermore, taking N0→∞, that is, Δt→0+, from (4.6), (4.8) and (4.9), up to a subsequence, we obtain
(4.11)
{
u
~
Δ
t
⇀
∗
u
in
L
∞
(
0
,
T
;
W
0
1
,
p
(
Ω
)
∩
L
∞
(
Ω
)
)
,
u
Δ
t
⇀
∗
v
in
L
∞
(
0
,
T
;
W
0
1
,
p
(
Ω
)
∩
L
∞
(
Ω
)
)
,
∂
u
~
Δ
t
∂
t
⇀
∂
u
∂
t
in
L
2
(
Q
T
)
,
as Δt→0+, for some u,v∈L∞(0,T;W01,p(Ω)∩L∞(Ω)) with u,v∈𝒞δ uniformly and ∂u∂t∈L2(QT). With the help of (4.10) and (4.6), we conclude that u≡v and u¯≤u≤u¯. Therefore, u∈𝒱(QT). Now we will prove that u is a solution of (($\mathrm{G}_{t}$)). For this, set
S
=
{
u
∈
L
∞
(
0
,
T
;
W
0
1
,
p
(
Ω
)
)
:
∂
u
∂
t
∈
L
2
(
Q
T
)
}
.
By the Aubin–Lions–Simon Lemma, we get that S is compactly embedded into C([0,T];L2(Ω)). Therefore, by interpolation identity, for all r>1, up to a subsequence, we have
(4.12)
u
~
Δ
t
→
u
in
C
(
[
0
,
T
]
;
L
r
(
Ω
)
)
as
Δ
t
→
0
.
Using (4.10), for all r>1, we deduce that
u
Δ
t
→
u
in
L
∞
(
0
,
T
;
L
r
(
Ω
)
)
as
Δ
t
→
0
.
Multiplying (4.4) by (uΔt-u) and using the above convergence results, we obtain
∫
Q
T
(
∂
u
~
Δ
t
∂
t
-
∂
u
∂
t
)
(
u
~
Δ
t
-
u
)
𝑑
x
𝑑
t
+
∫
0
T
〈
-
Δ
p
u
Δ
t
,
u
Δ
t
-
u
〉
𝑑
t
+
∫
0
T
〈
-
Δ
q
u
Δ
t
,
u
Δ
t
-
u
〉
𝑑
t
-
ϑ
∫
Q
T
u
Δ
t
-
δ
(
u
Δ
t
-
u
)
𝑑
x
𝑑
t
=
∫
Q
T
g
Δ
t
(
u
Δ
t
-
u
)
𝑑
x
𝑑
t
+
o
Δ
t
(
1
)
.
Since u¯≤uΔt,u~Δt≤u¯, by the Dominated Convergence Theorem, we have
∫
Q
T
u
Δ
t
-
δ
(
u
Δ
t
-
u
)
𝑑
x
𝑑
t
=
o
Δ
t
(
1
)
,
∫
Q
T
g
Δ
t
(
u
Δ
t
-
u
)
𝑑
x
𝑑
t
=
o
Δ
t
(
1
)
.
Moreover, performing integration by parts and taking into account the convergence results of uΔt and u~Δt, we deduce that
(4.13)
1
2
∫
Ω
|
u
~
Δ
t
-
u
|
2
(
T
)
𝑑
x
+
∫
0
T
(
〈
-
Δ
p
u
Δ
t
,
u
Δ
t
-
u
〉
+
〈
-
Δ
q
u
Δ
t
,
u
Δ
t
-
u
〉
)
𝑑
t
=
o
Δ
t
(
1
)
.
Now, we recall that the (p,q)-Laplacian operator satisfies the (S+) property (see [31, proof of Lemma 3.2]), that is, for a sequence {un} in W01,p(Ω) satisfying
u
n
⇀
u
weakly in
W
0
1
,
p
(
Ω
)
,
lim sup
n
→
∞
〈
-
Δ
p
u
n
-
Δ
q
u
n
,
u
n
-
u
〉
≤
0
,
one has un→u in W01,p(Ω). Therefore, by using the (S+) property of the (p,q)-Laplacian operator for the sequence uΔt, and the Dominated Convergence Theorem (on account of (4.9)), we deduce from (4.13) that
∫
0
T
∫
Ω
|
∇
(
u
Δ
t
-
u
)
|
p
𝑑
x
𝑑
t
=
o
Δ
t
(
1
)
and
∫
0
T
∫
Ω
|
∇
(
u
Δ
t
-
u
)
|
q
𝑑
x
𝑑
t
=
o
Δ
t
(
1
)
.
Furthermore, from the above discussion, it is clear that
(4.14)
-
Δ
p
u
Δ
t
→
-
Δ
p
u
in
L
p
′
(
0
,
T
;
W
-
1
,
p
′
(
Ω
)
)
,
-
Δ
q
u
Δ
t
→
-
Δ
q
u
in
L
q
′
(
0
,
T
;
W
-
1
,
p
′
(
Ω
)
)
,
where p′ and q′ are the Hölder conjugate of p and q, respectively. Moreover, from [3, (2.23), p. 5056], we have
(4.15)
u
Δ
t
-
δ
→
u
-
δ
in
L
∞
(
0
,
T
;
W
-
1
,
p
′
(
Ω
)
)
.
Therefore, from (4.1), (4.11), (4.14), (4.15) and using the Dominated Convergence Theorem, we get that u is a weak solution of (($\mathrm{G}_{t}$)). Next, we will show that such a weak solution u∈𝒞δ is unique. On the contrary, assume v∈𝒞δ is another weak solution of (($\mathrm{G}_{t}$)). Then, from the weak formulation of (($\mathrm{G}_{t}$)), we have
∫
Ω
∂
(
u
-
v
)
∂
t
(
u
-
v
)
-
〈
Δ
p
u
-
Δ
p
v
,
u
-
v
〉
-
〈
Δ
q
u
-
Δ
q
v
,
u
-
v
〉
=
ϑ
∫
Ω
(
u
-
δ
-
v
-
δ
)
(
u
-
v
)
.
We recall the following inequality (see [33, Lemma A.0.5]):
(4.16)
〈
|
x
|
p
-
2
x
-
|
y
|
p
-
2
y
,
x
-
y
〉
ℝ
N
≥
{
c
p
|
x
-
y
|
p
if
p
≥
2
c
p
|
x
-
y
|
2
(
|
x
|
+
|
y
|
)
2
-
p
if
1
<
p
<
2
,
for all
x
,
y
∈
ℝ
N
.
Consequently,
∂
∂
t
∫
Ω
1
2
(
u
-
v
)
2
𝑑
x
≤
0
.
Therefore, the function H:[0,T]→ℝ, defined by H(t)=∫Ω12(u-v)2𝑑x is decreasing. Since u and v are distinct and u(⋅,0)=v(⋅,0), we have
0
<
H
(
t
)
≤
H
(
0
)
=
0
for
t
∈
[
0
,
T
]
,
which is a contradiction. Thus H(t)=0 for all t∈[0,T] and hence u≡v.
To complete the proof of the theorem, we need to show that u∈C([0,T];W01,p(Ω)) and (2.1) holds. From (4.11) and (4.12), it is clear that u∈L∞(0,T;W01,p(Ω)) and u∈C([0,T];L2(Ω)). Therefore, it follows that u:t∈[0,T]→W01,p(Ω) is weakly continuous and u(⋅,t0)∈W01,p(Ω) together with
∥
u
(
⋅
,
t
0
)
∥
W
0
1
,
p
(
Ω
)
≤
lim inf
t
→
t
0
∥
u
(
⋅
,
t
)
∥
W
0
1
,
p
(
Ω
)
for all t0∈[0,T] (and analogous result holds for W01,q(Ω) also). Multiplying (4.2) by un-un-1 and integrating over Ω then summing from N2≤N1, we get
Δ
t
∑
n
=
N
2
N
1
∫
Ω
[
u
n
-
u
n
-
1
Δ
t
]
2
+
1
p
[
∫
Ω
|
∇
u
N
1
|
p
-
∫
Ω
|
∇
u
N
2
-
1
|
p
]
+
1
q
[
∫
Ω
|
∇
u
N
1
|
q
-
∫
Ω
|
∇
u
N
2
-
1
|
q
]
-
ϑ
1
-
δ
[
∫
Ω
|
u
N
1
|
1
-
δ
-
∫
Ω
|
u
N
2
-
1
|
1
-
δ
]
≤
∑
n
=
N
2
N
1
∫
Ω
g
Δ
t
(
u
n
-
u
n
-
1
)
𝑑
x
.
For t1∈[t0,T], taking N2 and N1 such that N2Δt→t0 and N1Δt→t1 as Δt→0+, we obtain
∫
t
0
t
1
∫
Ω
(
∂
u
∂
t
)
2
𝑑
x
𝑑
t
+
1
p
∥
u
(
⋅
,
t
1
)
∥
W
0
1
,
p
(
Ω
)
p
+
1
q
∥
u
(
⋅
,
t
1
)
∥
W
0
1
,
q
(
Ω
)
q
-
ϑ
1
-
δ
∫
Ω
|
u
(
x
,
t
1
)
|
1
-
δ
𝑑
x
(4.17)
≤
∫
t
0
t
1
∫
Ω
g
∂
u
∂
t
𝑑
x
𝑑
t
+
1
p
∥
u
(
⋅
,
t
0
)
∥
W
0
1
,
p
(
Ω
)
p
+
1
q
∥
u
(
⋅
,
t
0
)
∥
W
0
1
,
q
(
Ω
)
q
-
ϑ
1
-
δ
∫
Ω
|
u
(
x
,
t
0
)
|
1
-
δ
𝑑
x
.
Assume that ∥u(⋅,t1)∥W01,p(Ω)→l1 and ∥u(⋅,t1)∥W01,q(Ω)→l2 as t1→t0+. Then, using that u∈L∞(0,T;Lr(Ω)) for all r>1 and the Dominated Convergence Theorem, passing on the limit t1→t0+, we get
(4.18)
1
p
l
1
p
+
1
q
l
2
q
≤
1
p
∥
u
(
⋅
,
t
0
)
∥
W
0
1
,
p
(
Ω
)
p
+
1
q
∥
u
(
⋅
,
t
0
)
∥
W
0
1
,
q
(
Ω
)
q
.
Since
l
1
≥
∥
u
(
⋅
,
t
0
)
∥
W
0
1
,
p
(
Ω
)
and
l
2
≥
∥
u
(
⋅
,
t
0
)
∥
W
0
1
,
q
(
Ω
)
(these are consequence of the fact that the map u:t∈[0,T]→W01,p(Ω) is weakly continuous), from (4.18) we obtain
∥
u
(
⋅
,
t
1
)
∥
W
0
1
,
p
(
Ω
)
p
→
∥
u
(
⋅
,
t
0
)
∥
W
0
1
,
p
(
Ω
)
p
as
t
1
→
t
0
+
(and a similar result in W01,q(Ω)). Therefore, u(⋅,t)→u(⋅,t0) in W01,p(Ω) as t→t0+, which implies that u is right continuous on the interval [0,T]. To prove the left continuity, we take t1>t0 and 0<k<t1-t0. Set ρk(u)(s):=u(s+k)-u(s)k. Then taking it as a test function in the weak formulation of (($\mathrm{G}_{t}$)) and using convexity argument, we deduce that
∫
t
0
t
1
∫
Ω
∂
u
∂
t
ρ
k
(
u
)
𝑑
x
𝑑
t
+
1
k
p
∫
t
0
t
1
∫
Ω
(
|
∇
u
(
t
+
k
)
|
p
-
|
∇
u
(
t
)
|
p
)
+
1
k
q
∫
t
0
t
1
∫
Ω
(
|
∇
u
(
t
+
k
)
|
q
-
|
∇
u
(
t
)
|
q
)
-
ϑ
k
(
1
-
δ
)
∫
t
0
t
1
∫
Ω
(
u
1
-
δ
(
t
+
k
)
-
u
1
-
δ
(
t
)
)
≥
∫
t
0
t
1
∫
Ω
g
ρ
k
(
u
)
𝑑
x
𝑑
t
.
Now proceeding similar to [3, p. 5058], as k→0+, we get
∫
t
0
t
1
∫
Ω
(
∂
u
∂
t
)
2
𝑑
x
𝑑
t
+
1
p
∥
u
(
⋅
,
t
1
)
∥
W
0
1
,
p
(
Ω
)
p
+
1
q
∥
u
(
⋅
,
t
1
)
∥
W
0
1
,
q
(
Ω
)
q
-
ϑ
1
-
δ
∫
Ω
|
u
(
x
,
t
1
)
|
1
-
δ
𝑑
x
≥
∫
t
0
t
1
∫
Ω
g
∂
u
∂
t
𝑑
x
𝑑
t
+
1
p
∥
u
(
⋅
,
t
0
)
∥
W
0
1
,
p
(
Ω
)
p
+
1
q
∥
u
(
⋅
,
t
0
)
∥
W
0
1
,
q
(
Ω
)
q
-
ϑ
1
-
δ
∫
Ω
|
u
(
x
,
t
0
)
|
1
-
δ
𝑑
x
.
From (4), it is clear that the above inequality is in fact an equality. Therefore, using the fact that the map t↦∫Ωu(x,t)1-δ𝑑x is continuous, we obtain u∈C([0,T];W01,p(Ω)). Moreover, (2.1) follows by taking t0=0 in the above equality. ∎
,
Deepak Kumar
