Results Math (2021) 76:169
c
2021 The Author(s),
corrected publication 2021
https://doi.org/10.1007/s00025-021-01484-5 Results in Mathematics
Existence and Nonexistence of Positive
Solutions for Singular (p, q)-Equations with
Superdiffusive Perturbation
Nikolaos S. Papageorgiou and Patrick Winkert
Abstract. We consider a nonlinear Dirichlet problem driven by the (p, q)-
Laplacian and with a reaction which is parametric and exhibits the com-
bined effects of a singular term and of a superdiffusive one. We prove an
existence and nonexistence result for positive solutions depending on the
value of the parameter λ∈
◦
R+=(0,+∞).
Mathematics Subject Classification. 35J75, 35J92.
Keywords. (p, q)-Laplaciana, superdiffusive perturbation, positive solu-
tions, nonlinear regularity, truncation and comparison methods.
1. Introduction
Let Ω ⊆RNbe a bounded domain with a C2-boundary ∂Ω. In this paper, we
study the following singular (p, q)-equation with logistic perturbation
−Δpu−Δqu=λu−η+uθ−1−f(x, u)inΩ,
u=0 on∂Ω,
u>0,λ>0,0<η<1,1<q<p<θ.
(Pλ)
For r∈(1,∞) we denote the r-Laplace differential operator defined by
Δru=div|∇u|r−2∇ufor all u∈W1,r
0(Ω).
In problem (Pλ) we have the sum of two such operators with different
exponents which implies that the differential operator on the left-hand side
is not homogeneous. The right-hand side of (Pλ) has the combined effects of
a singular term s→λs−ηfor s>0 with 0 <η<1 and of a perturbation
169 Page 2 of 20 N. S. Papageorgiou and P. Winkert Results Math
which is of logistic type, namely the function s→λsθ−1−f(x, s) for almost
all (a. a.) x∈Ω. The function f:Ω×R→Ris a Carath´eodory function,
that is, x→ f(x, s) is measurable for all s∈Rand s→ f(x, s) is continuous
for a. a. x∈Ω. We assume that f(x, ·)is(θ−1)-superlinear as s→+∞for
a. a. x∈Ω. So, the logistic perturbation is of the superdiffusive type. We are
interested in positive solutions whenever the parameter λis positive.
Parametric superdiffusive logistic equations with no singular term
present, were investigated by Afrouzi–Brown [1] (for semilinear Dirichlet
problems), Takeuchi [23,24] (for nonlinear Dirichlet problems driven by
the p-Laplacian), Gasi´nski–O’Regan–Papageorgiou [3] (for nonlinear Dirich-
let problems driven by a nonhomogeneous differential operator), Cardinali–
Papageorgiou–Rubbioni [2], Gasi´nski–Papageorgiou [7] (both dealing with
nonlinear problems driven by the p-Laplacian) and Papageorgiou–R˘adulescu-
Repovˇs[16] (for semilinear mixed problems). These works reveal that the
superdiffusive logistic equations exhbit a kind of “bifurcation” for large values
of the parameter λ>0. More precisely, there is a critical parameter value
λ∗>0 such that the problem has at least two positive solutions for all λ>λ
∗,
the problem has at least one positive solution for λ=λ∗and there are no
positive solutions for λ∈(0,λ
∗). This is in contrast to subdiffusive and equid-
iffusive logistic equations for which we do not have multiplicity of positive
solutions, see Papageorgiou–Winkert [19].
When we introduce a singular term in the reaction, the geometry of the
problem changes since u= 0 is no longer a local minimizer of the energy
functional and so we cannot have a multiplicity result. In addition, the singular
term generates an energy functional which is not C1and so we have difficulties
in using the results of critical point theory. Therefore, we need to find a way to
bypass the singular term and deal with a C1-functional to which we can apply
the results of the critical point theory. Nonlinear singular problems but with a
different kind of perturbation were studied recently by Papageorgiou-Winkert
[20] (equations driven by the p-Laplacian) and by Papageorgiou–R˘adulescu-
Repovˇs[15] (equations driven by a nonhomogeneous differential operator).
The main result of our work here establishes the existence of a critical
parameter λ∗such that
•problem (Pλ) has at least one positive smooth solution for all λ≥λ∗;
•problem (Pλ) has no positive solutions for all λ<λ
∗.
Finally we mention that equations driven by the sum of two differential
operators of different nature (such as (p, q)-equations) arise in many mathe-
matical models of physical processes. We refer to the survey papers of Marano–
Mosconi [12]andR˘adulescu [22].
Vol. 76 (2021) Existence and Nonexistence of Positive Solutions Page 3 of 20 169
2. Preliminaries and Hypotheses
In this section we present some preliminaries which are needed in the sequel
and also the hypotheses on the data of problem (Pλ).
For every 1 ≤r<∞we consider the usual Lebesgue spaces Lr(Ω) and
Lr(Ω; RN) equipped with the norm ·
r. When 1 <r<∞we denote by
W1,r(Ω) and W1,r
0(Ω) the corresponding Sobolev spaces equipped with the
norms ·
1,r and ·
1,r,0, respectively. Because of the Poincar´e inequality we
can equip the space W1,r
0(Ω) with the following norm
u=∇urfor all u∈W1,r
0(Ω),
The Banach space
C1
0(Ω) = u∈C1(Ω) : u∂Ω=0
is an ordered Banach space with positive cone
C1
0(Ω)+=u∈C1
0(Ω) : u(x)≥0 for all x∈Ω.
This cone has a nonempty interior given by
int C1
0(Ω)+=u∈C1
0(Ω)+:u(x)>0 for all x∈Ω, ∂u
∂n(x)<0 for all x∈∂Ω,
where n(·) stands for the outward unit normal on ∂Ω.
Let r∈(1,+∞) and recall that W1,r
0(Ω)∗=W−1,r(Ω) with 1
r+1
r=1.
By ·,·1,r we denote the duality brackets of the pair (W1,r
0(Ω),W−1,r(Ω)).
For notational simplicity when r=p, we simply write ·,·.
For r∈(1,+∞), let Ar:W1,r
0(Ω) →W−1,r(Ω) = W1,r
0(Ω)∗with 1
r+
1
r= 1 be the nonlinear map defined by
Ar(u),h1,r =Ω
|∇u|r−2∇u·∇hdx for all u, h ∈W1,r
0(Ω).(2.1)
From Gasi´nski–Papageorgiou [5, Problem 2.192, p. 279] we have the fol-
lowing properties of Ar.
Proposition 2.1. The map Ar:W1,r
0(Ω) →W−1,r(Ω) defined in (2.1)is
bounded, that is, it maps bounded sets to bounded sets, continuous, strictly
monotone, hence maximal monotone and it is of type (S)+, that is,
un
w
→uin W1,r
0(Ω) and lim sup
n→∞
Ar(un),u
n−u≤0,
imply un→uin W1,r
0(Ω).
For s∈R,wesets±= max{±s, 0}and for u∈W1,p
0(Ω) we define
u±(·)=u(·)±. It is well known that
u±∈W1,p
0(Ω),|u|=u++u−,u=u+−u−.
169 Page 4 of 20 N. S. Papageorgiou and P. Winkert Results Math
Furthermore, given a measurable function g:Ω×R→R, we denote by
Ngthe corresponding Nemytskii (superposition) operator defined by
Ng(u)(·)=g(·,u(·)) for all measurable u:Ω→R.
It is clear that x→g(x, u(x)) is measurable. Recall that if g:Ω×R→R
is a Carath´eodory function, then gis measurable in both arguments, see, for
example, Papageorgiou–Winkert [18, Proposition 2.2.31, p. 106].
If h1,h
2:Ω→Rare two measurable functions, then we write h1≺h2
if and only if for every compact K⊆Ωwehave0<c
K≤h2(x)−h1(x)for
a. a. x∈K. Note that if h1,h
2∈C(Ω) and h1(x)<h
2(x) for all x∈Ω, then
h1≺h2.
For u, v ∈W1,p
0(Ω) with u(x)≤v(x) for a. a. x∈Ω we define
[u, v]=h∈W1,p
0(Ω) : u(x)≤h(x)≤v(x) for a. a. x∈Ω,
[u)=h∈W1,p
0(Ω) : u(x)≤h(x) for a. a. x∈Ω.
Now we are ready to introduce the hypotheses on the perturbation f:Ω×
R→R.
H: f:Ω×R→Ris a Carath´eodory function such that, for a. a. x∈Ω,
f(x, 0) = 0, f(x, ·) is nondecreasing and
(i)
f(x, s)≤a(x)1+sr−1
for a.a. x∈Ω, for all s≥0, with a∈L∞(Ω) and θ<r<p
∗, where
p∗denotes the critical Sobolev exponent with respect to pgiven by
p∗=Np
N−pif p<N,
+∞if N≤p;
(ii)
lim
s→+∞
f(x, s)
sθ−1=+∞uniformly for a. a. x∈Ω;
(iii) there exist 0 <ˆη1≤ˆη2and δ0>0 such that
ˆη1sq−1≤f(x, s) for a. a. x∈Ω and for all s∈[0,δ
0]
and
lim sup
s→0+
f(x, s)
sq−1≤ˆη2uniformly for a. a. x∈Ω.
Remark 2.2. With view to our problem it is clear that we are looking for
positive solutions and the hypotheses above only concern the positive semiaxis
R+=[0,+∞). Therefore, without any loss generality, we may assume that
f(x, s) = 0 for a.a. x∈Ω and for all s≤0.
Vol. 76 (2021) Existence and Nonexistence of Positive Solutions Page 5 of 20 169
Hypothesis H(ii) implies that f(x, ·)is(θ−1)-superlinear as s→+∞for
a. a. x∈Ω. Dropping the x-dependence for simplicity, the following functions
satisfy hypotheses H
f1(x)=(s+)q−1if s≤1,
sθ−1[ln(x) + 1] if 1 <s, with 1 <q<p<θ<p
∗,
f2(x)=μ(s+)q−1−(s+)τ−1if s≤1,
(μ−1)sr−1if 1 <s with 1 <q<p<r<p
∗,
and τ>qas well as μ≥p−1
q−1.
As we already mentioned in the Introduction, the presence of the singular
term leads to an energy functional which is not C1. This creates problems in the
usage of variational tools. In the next section we examine an auxiliary singular
problem and the solution of them will help us in order to avoid difficulties of
having to do with a nonsmooth energy functional.
3. An Auxiliary Singular Problem
In this section we deal with the following parametric singular Dirichlet (p, q)-
equation
−Δpu−Δqu=λu−η−f(x, u)inΩ,
u=0 on∂Ω,
u>0,λ>0,0<η<1,1<q<p.
(Qλ)
For this problem we have the following existence and uniqueness result.
Proposition 3.1. If hypotheses Hhold, then for every λ>0, problem (Qλ)
has a unique positive solution uλ∈int C1
0(Ω)+and the map λ→uλis
nondecreasing from ◦
R+=(0,+∞)into C1
0(Ω).
Proof. First we show the existence of a positive solution for problem (Qλ)for
every λ>0.
To this end, let g∈Lp(Ω) and ε>0. We consider the following Dirichlet
problem
−Δpu−Δqu+f(x, u)= λ
[|g|+ε]ηin Ω,
u=0 on∂Ω,
Moreover, we consider the nonlinear operator V:W1,p
0(Ω) →W−1,p(Ω)
defined by
V(u)=Ap(u)+Aq(u)+Nf(u) for all u∈W1,p
0(Ω).
169 Page 6 of 20 N. S. Papageorgiou and P. Winkert Results Math
Recall that W1,p
0(Ω) →W1,q
0(Ω) continuously and densely implies that
W−1,q(Ω) →W−1,p(Ω) continuously and densely as well, see Gasi´nski–
Papageorgiou [6, Lemma 2.2.27, p. 141].
By Proposition 2.1 and the fact that f(x, ·) is nondecreasing, we know
that V:W1,p
0(Ω) →W−1,p(Ω) is continuous and strictly monotone, hence,
maximal monotone as well. In addition we have
V(u),u≥Ap(u),u=∇up
p=upfor all u∈W1,p
0(Ω),
which implies that V:W1,p
0(Ω) →W−1,p(Ω) is also coercive. Therefore, it is
surjective, see Papageorgiou–R˘adulescu–Repovˇs[14, Corollary 2.8.7, p. 135].
Note that
λ
[|g(·)|+ε]η∈L∞(Ω) →W−1,p(Ω).
Hence, there exists vε∈W1,p
0(Ω) such that
V(vε)= λ
[|g|+ε]η.
The strict monotonicity of Vimplies that this solution vεis unique. Since
W1,p
0(Ω) →Lp(Ω) by the Sobolev embedding theorem, we can define the
solution map kε:Lp(Ω) →Lp(Ω) by kε(g)=vε. Note that
Ap(vε)+Aq(vε)+Nf(vε)= λ
[|g|+ε]ηin W−1,p(Ω).(3.1)
On (3.1) we take the test function vε∈W1,p
0(Ω) and obtain
∇vεp
p=vεp≤λ
εη(3.2)
because f(x, vε)vε≥0. From the compactness of W1,p
0(Ω) →Lp(Ω) it follows
that
kε(Lp(Ω))·p⊆Lp(Ω) is compact.
Suppose that gn→gin Lp(Ω). From (3.2) we see that
{vn
ε}n∈N={kε(gn)}n∈N⊆W1,p
0(Ω) is bounded.
Hence, by passing to a suitable subsequence if necessary, we may assume that
vn
ε
w
→v∗
εin W1,p
0(Ω) and vn
ε→v∗
nin Lp(Ω).(3.3)
We have
Ap(vn
ε)+Aq(vn
ε)+Nf(vn
ε)= λ
[|gn|+ε]ηin W−1,p(Ω) (3.4)
for all n∈N. Applying vn
ε−v∗
ε∈W1,p
0(Ω) on (3.4), passing to the limit as
n→∞and using (3.3), we obtain
lim
n→∞ [Ap(vn
ε),vn
ε−v∗
ε+Aq(vn
ε),vn
ε−v∗
ε]=0.
Vol. 76 (2021) Existence and Nonexistence of Positive Solutions Page 7 of 20 169
Since Aqis monotone, we derive
lim sup
n→∞
[Ap(vn
ε),vn
ε−v∗
ε+Aq(v∗
ε),vn
ε−v∗
ε]≤0
and due to (3.3), we get
lim sup
n→∞
Ap(vn
ε),vn
ε−v∗
ε≤0.
Then, by Proposition 2.1, it follows that
vn
ε→v∗
εin W1,p
0(Ω).(3.5)
So,ifwepassin(3.4) to the limit as n→∞and use (3.5)aswellasthe
fact that |gn|→|g|in Lp(Ω), we obtain
Ap(v∗
ε)+Aq(v∗
ε)+Nf(v∗
ε)= λ
[|g|+ε]ηin W−1,p(Ω).
Hence, v∗
ε=kε(g).
By the Urysohn’s criterion for the convergence of sequences we have
for the initial sequence kε(gn)→kε(g)inLp(Ω), see Gasi´nski–Papageorgiou
[4, p. 33]. This proves that the solution map kεis continuous. Therefore, we
can apply the Schauder–Tychonov fixed point theorem, see Papageorgiou–
R˘adulescu–Repovˇs[14, Theorem 4.3.21, p. 298], which gives the existence of
ˆvε∈W1,p
0(Ω) such that
kε(ˆvε)=ˆvε,ˆvε≥0,ˆvε=0.
We have
−Δpˆvε−Δqˆvε=λ
[ˆvε+ε]η−f(x, ˆvε)inΩ,
ˆvε=0 on∂Ω.
Theorem 7.1 of Ladyzhenskaya–Ural’tseva [10, p. 286] implies that ˆvε∈
L∞(Ω). Then, from the nonlinear regularity theory of Lieberman [11]wehave
that ˆvε∈C1
0(Ω)+\{0}. Hypotheses H(i), (iii) imply that if ρε=ˆvε∞, then
there exists ˆ
ξρε>0 such that ˆ
ξρεsp−1−f(x, s)≥0 for a. a. x∈Ω and for all
s∈[0,ρ
ε]. Using this we obtain
−Δpˆvε−Δqˆvε+ˆ
ξρεˆvp−1≥ˆ
ξρεˆvp−1−f(x, ˆvε)≥0inΩ.
Hence, we have
Δpˆvε+Δ
qˆvε≤ˆ
ξρεˆvp−1,
which implies that ˆvε∈int C1
0(Ω)+, see Pucci–Serrin [21, pp. 111 and 120].
Therefore, we produced a solution ˆvε∈int C1
0(Ω)+for the following
approximation of problem (Qλ)
−Δpu−Δqu=λ
[|u|+ε]η−f(x, u)inΩ,
u∂Ω=0,u≥0.(3.6)
169 Page 8 of 20 N. S. Papageorgiou and P. Winkert Results Math
In fact this solutions is unique. Indeed, if ˜vε∈W1,p
0(Ω) is another positive
solution of (3.6), then we have
0≤Ap(ˆvε)−Ap(˜vε),ˆvε−˜vε+Aq(ˆvε)−Aq(˜vε),ˆvε−˜vε
+Ω
[f(x, ˆvε)−f(x, ˜vε)] (ˆvε−˜vε)dx
=Ω
λ1
(ˆvε+ε)η−1
(˜vε+ε)η(ˆvε−˜vε)dx ≤0.
Since u→Ap(u)+Aq(u) is strictly monotone, see Proposition 2.1, it follows
that ˆvε=˜vε. This proves the uniqueness of the solution ˆvε∈int C1
0(Ω)+of
(3.6).
claim. If 0 <ε
<ε≤1, then ˆvε≤ˆvε.
We have
−Δpˆvε−Δqˆvε+f(x, ˆvε)= λ
[ˆvε+ε]η≥λ
[ˆvε+ε]ηin Ω.(3.7)
Now we introduce the Carath´eodory function eε:Ω×R→Rdefined by
eε(x, s)=λ
[s++ε]ηif s≤ˆvε(x),
λ
[ˆvε(x)+ε]ηif ˆvε(x)<s. (3.8)
We set Eε(x, s)=s
0eε(x, t)dt and consider the C1-functional σε:W1,p
0(Ω) →
Rdefined by
σε(u)=1
p∇up
p+1
q∇uq
q+Ω
Fx, u+dx −Ω
Eε(x, u)dx
for all u∈W1,p
0(Ω). From (3.8) and since F≥0 we see that σε:W1,p
0(Ω) →R
is coercive and because of the Sobolev embedding theorem it is also sequentially
weakly lower semicontinuous. Therefore, by the Weierstraß-Tonelli theorem
there exists ˜vε∈W1,p
0(Ω) such that’
σε(˜vε) = min σε(v):v∈W1,p
0(Ω).
This implies that σ
ε(˜vε) = 0, that is,
Ap(˜vε),h+Aq(˜vε),h+Ω
f(x, ˜vε)hdx=Ω
eε(x, ˜vε)hdx (3.9)
for all h∈W1,p
0(Ω). Taking h=−˜v−
ε∈W1,p
0(Ω) as test function in (3.9)
and applying (3.8) we obtain that ˜vε≥0. Moreover, we can choose h=
(˜vε−ˆvε)+∈W1,p
0(Ω). Then, using once again (3.8) and also (3.7) we infer
that ˜vε≤ˆvε. So, we have proved that
˜vε∈[0,ˆvε].(3.10)
Vol. 76 (2021) Existence and Nonexistence of Positive Solutions Page 9 of 20 169
From (3.10), (3.8) and (3.9) it follows that
−Δp˜vε−Δq˜vε+f(x, ˜vε)=[˜vε+ε]−ηin Ω,
˜vε∂Ω=0,˜vε≥0.
It is clear that ˜vε= 0 and so from the first part of the proof we have ˜vε=ˆvε∈
int C1
0(Ω)+. Then, due to (3.10), we obtain ˜vε≤˜vε. This proves the Claim.
Now we are ready to send ε→0+in order to produce a solution for
problem (Qλ). So, we consider a sequence εn→0+and set ˆvn=ˆvεnfor all
n∈N.Wehave
Ap(ˆvn),h+Aq(ˆvn),h+Ω
f(x, ˆvn)hdx=Ω
λh
[ˆvn+εn]ηdx (3.11)
for all h∈W1,p
0(Ω). Testing (3.11) with h=ˆvn∈W1,p
0(Ω) and applying the
Claim gives
ˆvnp=∇ˆvnp
p≤Ω
λˆvn
[ˆvn+εn]ηdx ≤Ω
λˆvnˆv−η
1dx (3.12)
for all n∈N.
Let ˆ
d(x)=d(x, ∂Ω) for x∈Ω. We know that ˆ
d∈int C1
0(Ω)+,see
Gilbarg–Trudinger [9, p. 355]. Since ˆv1∈int C1
0(Ω)+,wehave
Ω
λˆvnˆv−η
1dx =Ω
λˆv1−η
1
ˆvn
ˆv1
dx ≤λc1Ω
ˆvn
ˆv1
dx ≤λc2Ω
ˆvn
ˆ
ddx
≤λc3
ˆvn
ˆ
d
p
≤λc4ˆvn
(3.13)
for some c1,c
2,c
3,c
4>0.
From (3.12) and (3.13) it follows that {ˆvn}⊆W1,p
0(Ω) is bounded. There-
fore we may assume that
ˆvn
w
→uλin W1,p
0(Ω) and ˆvn→uλin Lr(Ω).(3.14)
Now we choose h=ˆvn−uλ∈W1,p
0(Ω) in (3.11). This yields
Ap(ˆvn),ˆvn−uλ+Aq(ˆvn),ˆvn−uλ+Ω
f(x, ˆvn)(ˆvn−uλ)dx
=Ω
λ[ˆvn−uλ]
[ˆvn+εn]ηdx
≤Ω
λ[ˆvn−uλ]1−ηdx
≤λc6ˆvn−uλrfor some c6>0 and for all n∈N,
since uλ≥0. Then, from the convergence properties in (3.14), we conclude
that
lim sup
n→∞
[Ap(ˆvn),ˆvn−uλ+Aq(ˆvn),ˆvn−uλ]≤0.
169 Page 10 of 20 N. S. Papageorgiou and P. Winkert Results Math
By the monotonicity of Aqwe obtain
lim sup
n→∞
[Ap(ˆvn),ˆvn−uλ+Aq(uλ),ˆvn−uλ]≤0.
Therefore,
lim sup
n→∞
Ap(ˆvn),ˆvn−uλ≤0,
which by Proposition 2.1 implies that
ˆvn→uλin W1,p
0(Ω).(3.15)
From the Claim we know that ˆv1≤ˆvnfor all n∈Nand so, ˆv1≤uλ.
Thus, uλ=0.
For every h∈W1,p
0(Ω), since ˆv1∈int C1
0(Ω)+, by Hardy’s inequality,
we have
0≤|h(x)|
[ˆvn+εn]η≤|h(x)|ˆv−η
1∈L1(Ω) for all n∈N.
Moreover, we have
h(x)
[ˆvn(x)+εn]η→h(x)
uλ(x)ηfor a. a. x∈Ω
due to (3.14). Therefore, we can apply the Dominated Convergence Theorem
and obtain
Ω
h
[ˆvn+εn]ηdx →Ω
h
uη
λ
as n→∞.(3.16)
We return to (3.11), pass to the limit as n→∞and use (3.15) as well
as (3.16). We obtain
Ap(uλ),h+Aq(uλ),h+Ω
f(x, uλ)hdx=Ω
λh
uη
λ
dx
for all h∈W1,p
0(Ω). Hence, uλis a positive solution of (Qλ)forλ>0.
From Marino–Winkert [13]wehavethat
ˆvn∈L∞(Ω) and ˆvn∞≤c7
for some c7>0 and for all n∈N. Then, by hypothesis H(i) we know that
{Nf(ˆvn)}n∈N⊆L∞(Ω) is bounded.
We have
−Δpˆvn−Δqˆvn=λ
[ˆvn+εn]η−f(x, ˆvn)inΩ,ˆvn∂Ω=0
for all n∈N.
Using the nonlinear regularity theory of Lieberman [11], we have that
{ˆvn}n∈N⊆C1
0(Ω) is relatively compact.
Hence, due to (3.15), we obtain ˆvn→uλin C1
0(Ω). Since ˆv1≤uλ, we then
conclude that uλ∈int C1
0(Ω)+.
Vol. 76 (2021) Existence and Nonexistence of Positive Solutions Page 11 of 20 169
So, we have proved that for every λ>0, problem (Qλ) has a solution
uλ∈int C1
0(Ω)+.
We need to show that this is the unique positive solution of (Qλ). To this
end, let vλ∈W1,p
0(Ω) be another positive solution of (Qλ). Since Apand Aq
are strictly monotone and f(x, ·) is nondecreasing, we have
0≤Ap(uλ)−Ap(vλ), uλ−vλ+Ap(uλ)−Aq(vλ), uλ−vλ
+Ω
[f(x, uλ)−f(x, vλ)] (uλ−vλ)dx
=Ω
λ1
uη
λ
−1
vη
λ(uλ−vλ)dx ≤0.
Therefore, uλ=vλ.
Finally, we are going to show the monotonicity of λ→uλ. So, let λ<μ.
We consider the Carath´eodory function dμ:Ω×R→Rdefined by
dμ(x, s)=μuλ(x)−η−f(x, uλ(x)) if s≤uλ(x),
μs−η−f(x, s)ifuλ(x)<s. (3.17)
We set Dμ(x, s)=s
0dμ(x, t)dt and consider the C1-functional τμ:W1,p
0(Ω) →
Rdefined by
τμ(u)=1
p∇up
p+1
q∇uq
q−Ω
Dμ(x, u)dx for all u∈W1,p
0(Ω).
Since τμ:W1,p
0(Ω) →Ris coercive, the direct method of the calculus of vari-
ations produces ˜uμ∈W1,p
0(Ω) such that
τμ(˜uμ) = min τμ(u):u∈W1,p
0(Ω).
From (3.17) we see that
˜uμ∈Kτμ=u∈W1,p
0(Ω) : τ
μ(u)=0
⊆[uλ)∩int C1
0(Ω)+
and
˜uμ=uμ∈int C1
0(Ω)+.
Hence, uλ≤uμ.
4. Positive Solutions
In this section we prove the existence and nonexistence of positive solutions
for problem (Pλ)asλmoves in ◦
R+=(0,+∞).
We introduce the following two sets
L={λ>0 : problem (Pλ) has a positive solution},
Sλ={u:uis a positive solution of problem (Pλ)}.
169 Page 12 of 20 N. S. Papageorgiou and P. Winkert Results Math
Proposition 4.1. If hypotheses H hold, then uλ≤ufor all u∈S
λ.
Proof. Let u∈S
λ. We introduce the Carath´eodory function kλ:Ω×◦
R+→R
defined by
kλ(x, s)=λs−η−f(x, s)if0<s≤u(x),
λu(x)−η−f(x, u(x)) if u(x)<s. (4.1)
We consider the following Dirichlet singular (p, q)-equation
−Δpu−Δqu=kλ(x, u)inΩ,u
∂Ω=0,u>0.(4.2)
Reasoning as in the proof of Proposition 3.1, see also Papageorgiou–R˘adulescu–
Repovˇs[15, Proposition 10], we show that (4.2) has a positive solution ˜uλ∈
int C1
0(Ω)+. The weak formulation of (4.2) is given by
Ap(˜uλ),h+Aq(˜uλ),h=Ω
kλ(x, ˜uλ)hdx for all u∈W1,p
0(Ω).(4.3)
Now, we choose h=(˜uλ−u)+∈W1,p
0(Ω) as test function in (4.3). Then, by
applying (4.1), u≥0andu∈S
λ, we obtain
Ap(˜uλ),(˜uλ−u)++Aq(˜uλ),(˜uλ−u)+
=Ωλu−η−f(x, u)(˜uλ−u)+dx
≤λu−u+uθ−1−f(x, u)(˜uλ−u)+dx
=Ap(u),(˜uλ−u)++Aq(u),(˜uλ−u)+.
Therefore, ˜uλ≤ubecause of the monotonicity of Apand Aq.
Then, from (4.1) and Proposition 3.1, it follows that ˜uλ=uλ∈
int C1
0(Ω)+and so, uλ≤ufor all u∈S
λ.
Next we determine the regularity of the elements of the solution set Sλ.
Proposition 4.2. If hypotheses H hold, then Sλ⊆int C1
0(Ω)+for all λ>0.
Proof. The result is trivially true if Sλ=∅. So, suppose that Sλ=∅and let
u∈S
λ. From Proposition 4.1 we know that uλ≤uand so u−η≤u−η
λ∈L1(Ω).
Recall that ˆv1≤uλand ˆv−η
1∈L1(Ω), see the proof of Proposition 3.1. There-
fore, using Theorem B.1 of Giacomoni–Saoudi [8], we see that u∈C1
0(Ω)+\{0}.
On account of hypotheses H(i), (ii), if ρ=u∞, then we can find ˆ
ξρ>0
such that
ˆ
ξρsp−1−f(x, s)≥0 for a.a. x∈Ω and for all 0 ≤s≤ρ.
Using this, we have
Δpu+Δ
qu≤ˆ
ξρup−1in Ω.
Then, by Pucci–Serrin [21, pp. 111 and 120], we derive u∈int C1
0(Ω)+.
Hence, Sλ⊆int C1
0(Ω)+.
Vol. 76 (2021) Existence and Nonexistence of Positive Solutions Page 13 of 20 169
Next, we are going to prove the nonemptiness of L.
Proposition 4.3. If hypotheses H hold, then L =∅.
Proof. Let uλ∈int C1
0(Ω)+be the unique positive solution of (Qλ), see
Proposition 3.1. We introduce the Carath´eodory function eλ:Ω×R→R
defined by
eλ(x, s)=λuλ(x)−η−f(x, uλ(x)) + λ(s+)θ−1if s≤uλ(x),
λs−η−f(x, s)+λsθ−1if uλ(x)<s. (4.4)
We set Eλ(x, s)=s
0eλ(x, t)dt and consider the functional γλ:W1,p
0(Ω) →R
defined by
γλ(u)=1
p∇up
p+1
q∇uq
q−Ω
Eλ(x, u)dx for all u∈W1,p
0(Ω).
Since u−η
λ∈L1(Ω), see the proof of Proposition 3.1,wehavethatγλ∈
C1(W1,p
0(Ω)), see also Proposition 3 of Papageorgiou–Smyrlis [17].
From (4.4) and hypothesis H(ii), we infer that γλis coercive. Moreover, it
is also sequentially weakly lower semicontinuous. Hence, there exists a global
minimizer u◦
λ∈W1,p
0(Ω) of γλ,thatis,
γλ(u◦
λ) = min γλ(u):u∈W1,p
0(Ω).(4.5)
Let u∈int C1
0(Ω)+and choose t∈(0,1) small so that tu ≤uλ.
Recall that uλ∈int C1
0(Ω)+and use Proposition 4.1.22 of Papageorgiou–
R˘adulescu–Repovˇs[14, p. 274].
We have
γλ(tu)≤tp
p∇up
p+tq
q∇uq
q−tΩλu−η
λ−f(x, uλ)udx. (4.6)
Let λ0=uη
λf(x, uλ)∞, see hypothesis H(i), and let λ>λ
0. Then
Ωλu−η
λ−f(x, uλ)dx =μ>0.
So,from(4.6)wehave
γλ(tu)≤c10tq−μt for some c10 >0,
since t∈(0,1) and q<p.
Since q>1, by taking t∈(0,1) even smaller if necessary, we see that
γλ(tu)<0. Taking (4.5) into account we know that
γλ(u◦
λ)<0=γλ(0) for all λ>λ
0.
Thus, u◦
λ=0.
From (4.5)wehaveγ
λ(u◦
λ) = 0, that is,
Ap(u◦
λ),h+Aq(u◦
λ),h=Ω
eλ(x, u◦
λ)hdx for all h∈W1,p
0(Ω).(4.7)
169 Page 14 of 20 N. S. Papageorgiou and P. Winkert Results Math
We choose h=(uλ−u◦
λ)+∈W1,p
0(Ω) as test function in (4.7). Applying (4.4)
and Proposition 3.1 gives
Ap(u◦
λ),(uλ−u◦
λ)++Aq(u◦
λ),(uλ−u◦
λ)+
=Ωλu−η
λ−f(x, uλ)+λ(u◦
λ)+θ−1(uλ−u◦
λ)+dx
≥Ωλu−η
λ−f(x, uλ)(uλ−u◦
λ)+dx
=Ap(uλ),(uλ−u◦
λ)++Aq(uλ),(uλ−u◦
λ)+.
As before, by the monotonicity of Apand Aqwe conclude that uλ≤u◦
λ.Using
this fact along with (4.4) and (4.7) we infer that
u◦
λ∈S
λ⊆int C1
0(Ω)+,
see Proposition 4.2. Therefore, λ∈Land so (λ0,+∞)⊆L=∅.
The next proposition establishes a structural property for L, namely that
Lis an upper half-line.
Proposition 4.4. If hypotheses H hold, λ∈Land μ>λ, then μ∈L.
Proof. Since λ∈Lthere exists uλ∈S
λ⊆int C1
0(Ω)+, see Proposition 4.2.
From Proposition 4.1 we have uλ≤uλ. Therefore,
u−η
λ∈L1(Ω).(4.8)
We now introduce the Carath´eodory function gμ:Ω×R→Rdefined by
gμ(x, s)=μuλ(x)−η+uλ(x)θ−1−f(x, uλ(x)) if s≤uλ(x),
μs−η+sθ−1−f(x, s)ifuλ(x)<s. (4.9)
We set Gμ(x, s)=s
0gμ(x, t)dt and consider the C1-functional ϕμ:W1,p
0(Ω) →
Rdefined by
ϕμ(u)=1
p∇up
p+1
q∇uq
q−Ω
Gμ(x, u)dx for all u∈W1,p
0(Ω),
see (4.8).
From (4.8) and hypothesis H(ii) we see that ϕμis coercive and we know
it is also sequentially weakly lower semicontinuous. Hence, we can find uμ∈
W1,p
0(Ω) such that
ϕμ(uμ) = min ϕμ(u):u∈W1,p
0(Ω).
This implies that ϕ
μ(uμ) = 0, that is,
Ap(uμ),h+Aq(uμ),h=Ω
gμ(x, uμ)hdx for all h∈W1,p
0(Ω).
(4.10)
Vol. 76 (2021) Existence and Nonexistence of Positive Solutions Page 15 of 20 169
We choose h=(uλ−uμ)+∈W1,p
0(Ω) as test function in (4.10). Applying
(4.9), λ<μand uλ∈S
λ, we obtain
Ap(uμ),(uλ−uμ)++Aq(uμ),(uλ−uμ)+
=Ωμu−η
λ+uθ−1
λ−f(x, uλ)(uλ−uμ)+dx
≥Ωλu−η
λ+uθ−1
λ−f(x, uλ)(uλ−uμ)+dx
=Ap(uλ),(uλ−uμ)++Aq(uλ),(uλ−uμ)+.
Again, from the monotonicity of Apand Aq, we deduce that uλ≤uμ. This
along with (4.9)aswellas(4.10) implies that uμ∈S
μ⊆int C1
0(Ω)+. Hence,
μ∈L.
So, according to Proposition 4.4,Lis an upper half-line. Moreover, a
byproduct of the proof of Proposition 4.4 is the following corollary.
Corollary 4.5. If hypotheses H hold, λ∈L,uλ∈S
λand μ>λ, then μ∈L
and there exists uμ∈S
μsuch that uλ≤uμ.
If we strengthen a little the conditions on f(x, ·), we can improve the
assertion of this corollary.
H’: f:Ω×R→Ris a Carath´eodory function such that f(x, 0) = 0 for
a. a. x∈Ω, f(x, ·) is nondecreasing, hypotheses H’(i), (ii), (iii) are the
same as the corresponding hypotheses H(i), (ii), (iii) and
(iv) for every >0 there exists ˆ
ξ>0 such that the function
s→ˆ
ξsp−1−f(x, s)
is nondecreasing on [0,] for a. a. x∈Ω.
Remark 4.6. The examples in Sect. 2satisfy this extra condition.
Proposition 4.7. If hypotheses H’ hold, λ∈L,uλ∈S
λand μ>λ, then μ∈L
and there exists uμ∈S
μsuch that uμ−uλ∈int C1
0(Ω)+.
Proof. From Corollary 4.5 we already know that μ∈Land we can find uμ∈
Sμ⊆int C1
0(Ω)+such that
uλ≤uμ.(4.11)
169 Page 16 of 20 N. S. Papageorgiou and P. Winkert Results Math
Let =uμ∞and let ˆ
ξ>0 be as postulated by hypothesis H’(iv). Since
λ<μ,uλ∈S
λand due to (4.11) as well as hypothesis H’(iv) we obtain
−Δpuλ−Δquλ+ˆ
ξup−1
λ−μu−η
λ
≤−Δpuλ−Δquλ+ˆ
ξup−1
λ−λu−η
λ
=λuθ−1
λ+ˆ
ξup−1
λ−f(x, uλ)
≤μuθ−1
μ+ˆ
ξup−1
μ−f(x, uμ)
=−Δpuμ−Δquμ+ˆ
ξup−1
μ−μu−η
μ.
(4.12)
Note that since uλint C1
0(Ω)+we have
0≺[μ−λ]uθ−1
λ.
So, from (4.12) and Proposition 7 of Papageorgiou–R˘adulescu–Repovˇs[15], we
conclude that uμ−uλ∈int C1
0(Ω)+.
Let λ∗=infL.
Proposition 4.8. If hypotheses H’ hold, then λ∗>0.
Proof. On account of hypotheses H’(ii), (iii) we can find ˆ
λ>0 such that
ˆ
λsθ−1−f(x, s)≤0 for a.a. x∈Ω and for all s≥0.(4.13)
Consider λ∈(0,ˆ
λ) and suppose that λ∈L. Then we can find uλ∈S
λ⊆
int C1
0(Ω)+.Wesetλ= maxΩuλ. Then, for δ∈(0,
λ) small enough, we
set δ
λ=λ−δ>0. For ˆ
ξλ=ˆ
ξλ>0 as postulated by hypothesis H’(iv) along
with (4.13), ˆ
λ>λ,uλ∈S
λand δ>0 small enough, we obtain
−Δpδ
λ−Δqδ
λ+ˆ
ξλδ
λp−1−λδ
λ−η
≥ˆ
ξλp−1
λ−χ(δ) with χ(δ)→0+as δ→0+
≥ˆ
λθ−1
λ−f(x, λ)+ˆ
ξλp−1
λ−χ(δ)
=λθ−1
λ−f(x, λ)+ˆ
ξλp−1
λ+ˆ
λ−λθ−1
λ−χ(δ)
≥λθ−1
λ−f(x, λ)+ˆ
ξλp−1
λ
≥λuθ−1
λ−f(x, uλ)+ˆ
ξλup−1
λ
=−Δpuλ−Δquλ+ˆ
ξλup−1
λ−λu−η
λ.
Invoking Proposition 6 of Papageorgiou–R˘adulescu–Repovˇs[15], we have that
δ
λ>u
λ(x) for all x∈Ω and for all small δ∈(0,
λ),
Vol. 76 (2021) Existence and Nonexistence of Positive Solutions Page 17 of 20 169
a contradiction to the definition of λ. Therefore
0<ˆ
λ≤λ∗=infL.
Next, we show that λ∗is admissible, that is, λ∗>0.
Proposition 4.9. If hypotheses H’ hold, then λ∗∈L.
Proof. Let {λn}n∈N⊆Lbe such that λnλ∗. For every n∈N,letun∈
Sλn⊆int C1
0(Ω)+. From Proposition 3.1 we know that
uλ∗≤unfor all n∈N.(4.14)
Moreover we have
Ap(un),h+Aq(un),h=Ωλnu−η
n+uθ−1
n−f(x, un)hdx (4.15)
for all h∈W1,p
0(Ω) and for all n∈N.
On account of hypotheses H’(i), (ii), (iii) there exists c11 >0 such that
λnsθ−1−f(x, s)≤c11 (4.16)
for a. a. x∈Ω, for all s≥0 and for all n∈N.
Choosing h=un∈W1,p
0(Ω) in (4.15) and using (4.14) and (4.16), results
in
unp≤c12 unfor some c12 >0 and for all n∈N.
Therefore, {un}n∈N⊆W1,p
0(Ω) is bounded.
So, we may assume that
un
w
→u∗in W1,p
0(Ω) and un→u∗in Lr(Ω).(4.17)
Taking h=un−u∗∈W1,p
0(Ω) as test function in (4.15), passing to the limit
as n→∞and using (4.17) yields
lim sup
n→∞
Ap(un),u
n−u∗≤0,
see the proof of Proposition 3.1. Then, from Proposition 2.1 we conclude that
un→u∗in W1,p
0(Ω).(4.18)
Now we can apply (4.18) along with (4.14)aswellas(4.15), as in the proof of
Proposition 3.1, in the limit as n→∞, we obtain
uλ∗≤u∗
and
Ap(u∗),h+Aq(u∗),h=Ωλ∗u−η
∗+uθ−1
∗f(x, u∗)hdx
for all h∈W1,p
0(Ω). Finally, we reach u∗∈S
λ∗⊆int C1
0(Ω)+and so λ∗∈L.
169 Page 18 of 20 N. S. Papageorgiou and P. Winkert Results Math
So, we have L=[λ∗,+∞) and we can state the following theorem for the
positive solutions of problem (Pλ).
Theorem 4.10. If hypotheses H’ hold, then there exists λ∗>0such that
(1) for every λ≥λ∗, problem (Pλ) has a positive solution uλ∈int C1
0(Ω)+;
(2) for every λ∈(0,λ
∗), problem (Pλ) has no positive solutions.
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Nikolaos S. Papageorgiou
Department of Mathematics, Zografou Campus
National Technical University
15780 Athens
Greece
e-mail: [email protected]
Patrick Winkert
Institut f¨ur Mathematik
Technische Universit¨at Berlin
Straße des 17. Juni 136
10623 Berlin
Germany
e-mail: [email protected]
