Special Issue in honor of Alan C. Lazer

Electronic Journal of Differential Equations, Special Issue 01 (2021), pp. 169–182.

ISSN: 1072-6691. URL: https://ejde.math.txstate.edu or https://ejde.math.unt.edu

TWIN POSITIVE SOLUTIONS FOR RESONANT SINGULAR

(p, q)-EQUATIONS

FLORIN-IULIAN ONETE, NIKOLAOS S. PAPAGEORGIOU,

VICENŢIU D. RĂDULESCU

Abstract. We consider a Dirichlet (p, q)-equation with a reaction having the

combined effects of a singular term and of a resonant perturbation. Using an
auxiliary problem to bypass the singularity and variational tools from critical

point theory, with truncation and comparison techniques, we show that the

problem has two positive smooth solutions.

1. Introduction

Let Ω ⊆ RN be a bounded domain with a C2-boundary ∂Ω. In this paper
we study the existence of positive solutions for the Dirichlet (p, q)-equation with
singular reaction

−∆pu(z)−∆qu(z) = a(z)u(z)−η + f(z, u(z)) in Ω,

u
∣∣
∂Ω

= 0, 1 < q < p, 0 < η < 1, u > 0.
(1.1)

For every r ∈ (1,∞), we denote by ∆r the r-Laplace differential operator defined
by

∆ru = div(|Du|r−2Du) for all u ∈W 1,r
0 (Ω).

In the reaction there are both a singular term a(z)x−η and a Carathéodory
perturbation f(z, x) (that is, for all x ∈ R the mapping z 7→ f(z, x) is measurable
and for a.a. z ∈ Ω the function x 7→ f(z, x) is continuous). We assume that the
Carathéodory perturbation f(z, ·) exhibits (p−1) linear growth as x→ +∞ and, in
fact, asymptotically we can have a resonance with respect to the principal eigenvalue
of the Dirichlet p-Laplacian. Actually the resonance occurs from the right of the
principal eigenvalue, making the energy functional of the problem indefinite, that
is, noncoercive. So, in problem (1.1), the reaction exhibits the combined effects of
a singular term and of a resonant perturbation.

We point out that our problem is nonparametric. Usually singular problems
involve a parameter, see for example the works of Sun, Wu and Long [25], Gia-
comoni, Schindler and Takač [8], Lü and Xie [13], Papageorgiou, Rădulescu and
Repovš [17], Papageorgiou, Vetro and Vetro [21], Papageorgiou and Winkert [22].
When the equation is parametric, by varying the parameter, we can achieve certain

2010 Mathematics Subject Classification. 35J75, 35J92.
Key words and phrases. Resonance; nonlinear regularity; comparison principle;
maximum principle; positive solutions.
c©2021 This work is licensed under a CC BY 4.0 license.

Published October 6, 2021.

169



170 F. I. ONETE, N. S. PAPAGEORGIOU, V. D. RĂDULESCU EJDE/SI/01

geometric configurations, which permit the use of minimax theorems from critical
point theory. This is also the case of nodal solutions of nonsingular (p, 2)-equations
with competing nonlinearities in the reaction (see Papageorgiou and Scapellato [19]
and Papageorgiou and Zhang [23]).

Nonlinear nonparametric singular equations were investigated by Bai, Gasiński
and Papageorgiou [2], Papageorgiou, Rădulescu and Repovš [16], Papageorgiou,
Vetro and Vetro [20]. The first two papers consider equations driven by the p-
Laplacian, while the third one deals with (p, 2)-equations and has a perturbation
term f(z, ·) which is (p− 1)-superlinear.

We mention that the exponent of the singular term satisfies η ∈ (0, 1). The more
difficult case η ≥ 1 (strong singularity), was examined by Lazer and McKenna
[11] in the context of semilinear equations driven by the Dirichlet Laplacian. An
overview of singular problems with a rich bibliography, can be found in the book
of Ghergu and Rădulescu [7].

2. Mathematical background and hypotheses

The main spaces in the study of problem (1.1) are the Sobolev space W 1,p
0 (Ω)

and the ordered Banach space C1
0 (Ω) = {u ∈ C1(Ω) : u

∣∣
∂Ω

= 0}. By ‖ · ‖ we denote

the norm of W 1,p
0 (Ω). On account of the Poincaré inequality, we can have

‖u‖ = ‖Du‖p for all u ∈W 1,p
0 (Ω).

The positive (order) cone for the space C1
0 (Ω) is

C+ = {u ∈ C1
0 (Ω) : u(z) ≥ 0 for all z ∈ Ω}.

This cone has a nonempty interior given by

intC+ = {u ∈ C+ : u(z) > 0 for all z ∈ Ω,
∂u

∂n

∣∣∣
∂Ω

< 0}

with n(·) being the outward unit normal on ∂Ω.

Given r ∈ (1,∞), we denote by Ar : W 1,r
0 (Ω) → W−1,r′(Ω) = W 1,r

0 (Ω)∗,
(

1
r +

1
r′ = 1

)
the nonlinear operator defined by

〈Ar(u), h〉 =

∫
Ω

|Du|r−2(Du,Dh)RNdz for all u, h ∈W 1,r
0 (Ω).

The next proposition summarizes the well-known properties of this map. We
refer to Problem 2.192 of Gasiński and Papageorgiou [5, p.279] for a more general
result.

Proposition 2.1. The operator Ar(·) is bounded (that is, maps bounded sets to
bounded sets) continuous, strictly monotone (hence maximal monotone, too) and
of type (S)+, that is,

if un
w→ u in W 1,r

0 (Ω) and lim supn→∞〈Ar(un), un − u〉 ≤ 0, then

un → u in W 1,r
0 (Ω).

If x ∈ R, then we set x± = max{±x, 0}. For u ∈ W 1,p
0 (Ω) we define u±(z) =

u(z)± for all z ∈ Ω. We know that

u± ∈W 1,p
0 (Ω), u = u+ − u−, |u| = u+ + u−.



EJDE-2021/SI/01 RESONANT SINGULAR (p, q)-EQUATIONS 171

Given u, v : Ω → R two measurable functions, with u(z) ≤ v(z) for a.a. z ∈ Ω,
we define

[u, v] =
{
h ∈W 1,p

0 (Ω) : u(z) ≤ h(z) ≤ v(z) for a.a. z ∈ Ω
}
,

[u) =
{
h ∈W 1,p

0 (Ω) : u(z) ≤ h(z) for a.a. z ∈ Ω
}
,

intC1
0 (Ω)[u, v] is the interior in C1

0 (Ω) of [u, v] ∩ C1
0 (Ω).

Also we write u � u if and only if for every compact K ⊆ Ω we have 0 < cK ≤
v(z)− u(z) for a.a. z ∈ K. Evidently, if u, v ∈ C(Ω) and u(z) < v(z) for all z ∈ Ω,
then u � v.

By λ̂1(p) we denote the first eigenvalue of
(
− ∆p,W

1,p
0 (Ω)

)
. We know that

λ̂1(p) > 0, it is simple, isolated and admits the following variational characterization

λ̂1(p) = inf
{‖Du‖pp
‖u‖pp

: u ∈W 1,p
0 (Ω), u 6= 0

}
. (2.1)

The infimum in (2.1) is realized on the corresponding one-dimensional eigenspace,
the elements of which have fixed sign. By û1(p) we denote the positive, Lp-

normalized (that is, ‖û1(p)‖p = 1) eigenfunction corresponding to λ̂1(p). We know
that û1(p) ∈ intC+ (by the nonlinear maximum principle, see Pucci and Serrin
[24]). We will also consider a weighted version of the eigenvalue problem. So, let
m ∈ L∞(Ω), m(z) ≥ 0 for a.a. z ∈ Ω, m 6= 0 and consider the nonlinear eigenvalue
problem

−∆pu(z) = λ̃m(z)|u(z)|p(z)−2u(z) in Ω, u
∣∣∣
∂Ω

= 0.

This problem too has a smallest eigenvalue λ̃1(m, p) > 0 which has the same

properties as λ̂(p). Note that if m ≡ 1, then λ̃1(m, p) = λ̂1(p). Moreover, the map

m 7→ λ̂1(m, p) has the following monotonicity property. We refer to Proposition
9.47(d) of Motreanu, Motreanu and Papageorgiou [14, p.250] for details and a
complete proof.

Proposition 2.2. If m,m′ ∈ L∞(Ω), 0 ≤ m(z) ≤ m′(z) for a.a. z ∈ Ω, m 6≡ 0,

m 6≡ m′, then λ̃(m′, p) < λ̃1(m, p).

Finally, we mention that λ̃1(m, p) > 0 is the only eigenvalue with eigenfunctions
of constant sign. All the other eigenvalues have eigenfunctions which are nodal
(that is, sign changing).

Let X be a Banach space and ϕ ∈ C1(X). We say that ϕ(·) satisfies the “C-
condition”, if it has the following property:

Every sequence {un}n∈N ⊆ X such that {ϕ(un)}n∈N ⊆ R is bounded,
and (1+‖un‖)ϕ′(un)→ 0 in X∗, admits a strongly convergent sub-
sequence.

By Kϕ we denote the critical set of ϕ, that is,

Kϕ = {u ∈ X : ϕ′(u) = 0}.
Now we introduce the hypotheses on the data of problem (1.1).

(H0) a ∈ C1
0 (Ω), a(z) > 0 for all z ∈ Ω.

(H1) f : Ω × R → R is a Carathéodory function such that f(z, 0) = 0 for a.a.
z ∈ Ω and

(i) |f(z, x)| ≤ a(z)(1 + xp−1) for a.a. z ∈ Ω, all x ≥ 0, with a ∈ L∞(Ω);



172 F. I. ONETE, N. S. PAPAGEORGIOU, V. D. RĂDULESCU EJDE/SI/01

(ii) λ̂1(p) ≤ lim infx→+∞
f(z,x)
xp−1 uniformly for a.a. z ∈ Ω;

(iii) if F (z, x) =
∫ x

0
f(z, x)ds then there exists τ ∈ (q, p) such that

0 < β0 ≤ lim inf
x→+∞

pF (z, x)− f(z, x)x

xτ

uniformly for a.a. z ∈ Ω;
(iv) there exist µ ∈ (1, q) and δ, ϑ > 0 such that C0x

µ ≤ f(z, x)x ≤
µF (z, x) for a.a. z ∈ Ω, all 0 ≤ x ≤ δ, some C0 > 0, and

a(z)ϑ−η + f(z, ϑ) ≤ −Ĉ < 0 for a.a. z ∈ Ω;

(v) for every ρ > 0, there exists ξ̂ρ > 0 such that for a.a. z ∈ Ω, the
function

x 7→ f(z, x) + ξ̂ρx
p−1

is nondecreasing on [0, ρ].

We point out that we assume a ∈ C1
0 (Ω) since we use the Hardy inequality in

the proof of Proposition 3.1 and because we need a(·)u0(·)−η ∈ L∞(Ω) and 0 �
a(·)u0(·)−η in the proof of Proposition 4.1 in order to apply the strong comparison
principle (see Gasiński and Papageorgiou [6]). We also mention that in hypothesis

(H1)(iv) we simply say that there exists Ĉ > 0 such that

a(z)ϑ−η + f(z, ϑ) ≤ −Ĉ < 0 for a.a. z ∈ Ω.

In other words, the mapping a(·)ϑ−η +f(·, ϑ) is bounded away from zero uniformly
for a.a. z ∈ Ω.

Remark 2.3. Since we look for positive solutions and the hypotheses concern the
positive semiaxis R+ = [0,+∞) only, without any loss of generality we may assume
that f(z, x) = 0 for a.a. z ∈ Ω, all x ≤ 0. HypothesisH1(ii) implies that as x→ +∞
we can have resonance with respect to the principal eigenvalue. Hypothesis (H1)(iv)
implies the presence of a “concave” nonlinearity near 0+. Indeed integrating the
first inequality in (H1)(iv), we obtain C1x

µ ≤ F (z, x) for a.a. z ∈ Ω, all 0 ≤ x ≤ δ,
some C1 > 0. The second inequality in (H1)(iv) implies an oscillatory behavior for
the reaction near 0+.

By a solution of problem (1.1), we mean a function u ∈ W 1,p
0 (Ω) such that

u−ηh ∈ L1(Ω) for all h ∈W 1,p
0 (Ω) and

〈Ap(u), h〉+ 〈Aq(u), h〉 =

∫
Ω

[a(z)u−η + f(z, u)]h for all u ∈W 1,p
0 (Ω).

The presence of the singular term implies that ϕ(·) is not C1 and so we cannot
use the results of critical point theory directly on this functional. We need to find
ways to bypass the singularity and deal with a C1-functional. For this reason, in
the next section we consider an auxiliary problem, the solution of which will be
used to bypass the singularity as indicated above.

3. An auxiliary problem

Let r ∈ (p, p∗). On account of hypotheses (H1)(i) and (H1)(iv), we have

f(z, x) ≥ C0x
µ−1 − C2x

r−1 for a.a. z ∈ Ω, all x ≥ 0, some C2 > 0. (3.1)



EJDE-2021/SI/01 RESONANT SINGULAR (p, q)-EQUATIONS 173

Then we introduce the Carathéodory function

k(z, x) =

{
C0(x+)µ−1 − C2(x+)r−1, if x ≤ ϑ
C0ϑ

µ−1 − C2ϑ
r−1, if ϑ < x.

(3.2)

We consider the Dirichlet (p, q)-equation

−∆pu(z)−∆qu(z) = k(z, u(z)) in Ω,

u
∣∣
∂Ω

= 0, 1 < q < p, u > 0.
(3.3)

Proposition 3.1. Problem (3.3) admits a unique positive solution u ∈ intC+,
u(z) ≤ ϑ for all z ∈ Ω and a(·)u−η ∈ L∞(Ω).

Proof. First we show the existence of a positive solution. To this end, let K(z, x) =∫ x
0
k(z, s)ds and consider the C1-functional τ : W 1,p

0 (Ω)→ R defined by

τ(u) =
1

p
‖Du‖pp +

1

q
‖Du‖qq −

∫
Ω

K(z, u)dz for all u ∈W 1,p
0 (Ω).

It is clear from (3.2) that τ(·) is coercive. Also using the Sobolev embedding
theorem, we see that τ(·) is sequentially weakly lower semicontinuous. So, by the

Weierstrass-Tonelli theorem, we can find u ∈W 1,p
0 (Ω) such that

τ(u) = min
{
τ(u) : u ∈W 1,p

0 (Ω)
}
. (3.4)

Let u ∈ intC+ and choose t ∈ (0, 1) small such that 0 ≤ tu(z) ≤ ϑ for all z ∈ Ω.
Then on account of (3.2) we have

τ(tu) =
tp

p
‖Du‖pp +

tq

q
‖Du‖qq +

trC2

r
‖u‖rr −

tµC0

µ
‖u‖µµ.

Since, by hypothesis 1 < µ < q < p < r, then choosing t ∈ (0, 1) even smaller if
necessary, we have

τ(tu) < 0⇒ τ(u) < 0 = τ(0) (see (3.4)),

⇒ u 6= 0.

From (3.4) we have τ ′(u) = 0 which implies

〈Ap(u), h〉+ 〈Aq(u), h〉 =

∫
Ω

k(z, u)h dz for all h ∈W 1,p
0 (Ω). (3.5)

In (3.5) first we choose h = −u− ∈W 1,p
0 (Ω). We obtain

‖Du−‖pp ≤ 0 (see (3.2)),

⇒ u ≥ 0, u 6= 0.

Next, in (3.5) we choose h = [u− ϑ]+ ∈W 1,p
0 (Ω). Then we have

〈Ap(u), (u− ϑ)+〉+ 〈Aq(u), (u− ϑ)+〉

=

∫
Ω

[
C0ϑ

µ−1 − C2ϑ
r−1
]

(u− ϑ)+dz (see (3.2))

≤
∫

Ω

f(z, ϑ)(u− ϑ)+dz (see (3.1))

≤ 0 (see (H1)(iv))

= 〈Ap(ϑ), (u− ϑ)+〉+ 〈Aq(ϑ), (u− ϑ)+〉,
⇒ u ≤ ϑ (see Proposition 2.1).



174 F. I. ONETE, N. S. PAPAGEORGIOU, V. D. RĂDULESCU EJDE/SI/01

So, we have proved that

u ∈ [0, ϑ], u 6= 0. (3.6)

From (3.6), (3.2), (3.5) it follows that u ∈ W 1,p
0 (Ω) is a solution of problem

(3.3). By Theorem 7.1 of Ladyzhenskaya and Uraltseva [10, p. 286], we have that
u ∈ L∞(Ω). Then the nonlinear regularity theory of Lieberman [12] implies that
u ∈ C+ \ {0}. We have

∆pu+ ∆qu ≤ C2‖u‖r−p∞ up−1 in Ω

⇒ u ∈ intC+ (see Pucci and Serrin [24, pp. 111, 120]).

Next, we show that this positive solution is unique. To this end, we consider the
integral functional j : L1(Ω)→ R = R ∪ {+∞} defined by

j(u) =

{
1
p‖Du

1/µ‖pp + 1
q‖Du

1/µ‖qq, if u ≥ 0, u1/µ ∈W 1,p
0 (Ω)

∞, otherwise.

From Lemma 1 (and its proof) of Diaz and Saa [4], we have that the functional

j(·) is convex. Let ũ ∈ W 1,p
0 (Ω) be another positive solution of (3.3). Again we

show that ũ ∈ [0, ϑ]∩ intC+. Using Proposition 4.1.22 of Papageorgiou, Rădulescu
and Repovš [18, p. 274], we have

u

ũ
∈ L∞(Ω) and

ũ

u
∈ L∞(Ω).

If dom j = {u ∈ L1(Ω) : j(u) < ∞} (the effective domain of j(·)) and h =

uµ − ũµ ∈W 1,p
0 (Ω), then for |t| < 1 small we have

uµ + th ∈ dom j and ũµ + th ∈ dom j.

So, from the convexity of j(·), we obtain that it is Gâteaux differentiable at uµ

and ũµ in the direction h. Using the nonlinear Green identity (see Corollary 1.5.17
of Papageorgiou, Rădulescu and Repovš [18, p. 35]), we have

j′(uµ)(h) =
1

µ

∫
Ω

−∆pu−∆qu

uµ−1
(uµ − ũµ)dz

=
1

µ

∫
Ω

[C0 − C2u
r−µ](uµ − ũµ)dz,

j′(ũµ)(h) =
1

µ

∫
Ω

−∆pũ−∆qũ

ũµ−1
(uµ − ũµ)dz

=
1

µ

∫
Ω

[C0 − C2ũ
r−µ](uµ − ũµ)dz.

The convexity of j(·) implies the monotonicity of j′(·). Therefore

0 ≤
∫

Ω

C2[ũr−µ − ur−µ](uµ − ũµ)dz ≤ 0, ⇒ u = ũ.

This proves the uniqueness of the positive solution u ∈ intC+ of (3.3).

Let d̂(z) = d(z, ∂Ω) for all z ∈ Ω. It follows by Lemma 14.16 of Gilbarg and

Trudinger [9, p. 355] that we can find δ0 > 0 small such that d̂ ∈ C2(Ωδ0), where

Ωδ0 = {z ∈ Ω : d(z, ∂Ω) < δ0}.



EJDE-2021/SI/01 RESONANT SINGULAR (p, q)-EQUATIONS 175

Then it follows that d̂ ∈ intC+ and so using Proposition 4.1.22 of Papageorgiou,

Rădulescu and Repovš [18, p. 274], we can find C3, C4 > 0 such that C3d̂ ≤ u ≤
C4d̂.

Let s > 1. Then we have∫
Ω

(
a(z)u−η

)s
dz =

∫
Ω

(u1−η)s
(a(z)

u

)s
dz

≤ C4

∫
Ω

(a(z)

u

)s
dz for some C4 > 0, since u ∈ intC+.

≤ C5

∫
Ω

(a(z)

d̂

)s
dz for some C5 > 0

≤ C5‖Da‖ss (by Hardy’s inequality, see Brezis [3, p. 313]),

which implies
‖au−η‖s ≤ C6 for some C6 > 0, all s > 1,

Therefore, a(·)u(·)−η ∈ L∞(Ω). The proof is now complete. �

4. Positive solutions

In this section, using u ∈ intC+ from Proposition 3.1, we are able to bypass the
singularity and deal with C1-functionals on which we can use the minimax theorems
of critical point theory.

So, with u ∈ intC+ from Proposition 3.1, we introduce the Carathéodory func-
tions g, ĝ : Ω× R→ R defined by

g(z, x) =

{
a(z)u(z)−η + f(z, u(z)), if x ≤ u(z)

a(z)x−η + f(z, x), if u(z) < x,
(4.1)

ĝ(z, x) =

{
g(z, x), if x ≤ ϑ
g(z, ϑ), if ϑ < x,

(4.2)

recall that u ≤ ϑ.
We set G(z, x) =

∫ x
0
g(z, s)ds, Ĝ(z, x) =

∫ x
0
ĝ(z, s)ds and consider the C1-

functionals Ψ, Ψ̂ : W 1,p
0 (Ω)→ R defined by

Ψ(u) =
1

p
‖Du‖pp +

1

q
‖Du‖qq −

∫
Ω

G(z, u)dz,

Ψ̂(u) =
1

p
‖Du‖pp +

1

q
‖Du‖qq −

∫
Ω

Ĝ(z, u)dz for all u ∈W 1,p
0 (Ω).

Proposition 4.1. If hypotheses (H0), (H1) hold, then problem (1.1) admits a pos-
itive solution

u0 ∈ intC1
0 (Ω)[u, ϑ].

Proof. From (4.2) it is clear that Ψ̂ is coercive. Also by the Sobolev embedding

theorem Ψ̂(·) is sequentially weakly lower semicontinuous. So, we can find u0 ∈
W 1,p

0 (Ω) such that

Ψ̂(u0) = inf{Ψ̂(u) : u ∈W 1,p
0 (Ω)},

which implies Ψ̂′(u) = 0, and

〈Ap(u0), h〉+ 〈Aq(u0), h〉 =

∫
Ω

ĝ(z, u0)h dz for all h ∈W 1,p
0 (Ω). (4.3)



176 F. I. ONETE, N. S. PAPAGEORGIOU, V. D. RĂDULESCU EJDE/SI/01

In (4.3) first we choose h = [u− u0]+ ∈W 1,p
0 (Ω). We have

〈Ap(u0), (u− u0)+〉+ 〈Aq(u0), (u− u0)+〉

=

∫
Ω

[a(z)u−η + f(z, u)](u− u0)+dz (see (4.1), (4.2))

≥
∫

Ω

f(z, u)(u− u0)+dz (see hypothesis (H0))

≥
∫

Ω

[C0u
µ−1 − C2u

r−1](u− u0)+dz (see (3.1))

= 〈Ap(u), (u− u0)+〉+ 〈Aq(u), (u− u0)+〉 (see Proposition 3.1),

⇒ u ≤ u0.

Next, in (4.3) we choose h = [u0 − ϑ]+ ∈W 1,p
0 (Ω). We have

〈Ap(u0), (u0 − ϑ)+〉+ 〈Aq(u0), (u0 − ϑ)+〉

=

∫
Ω

[a(z)ϑ−η + f(z, ϑ)](u0 − ϑ)+dz (see (4.1), (4.2))

≤ 0 (from (H1)(iv))

= 〈Ap(ϑ), (u0 − ϑ)+〉+ 〈Aq(ϑ), (u0 − ϑ)+〉,
⇒ u0 ≤ ϑ.

So, we have proved that

u0 ∈ [u, ϑ]. (4.4)

From (4.4), (4.1), (4.2) and (4.3), it follows that u0 is a positive solution of (1.1)
and, as before, the nonlinear regularity theory of Lieberman [12] implies that

u0 ∈ [u, ϑ] ∩ intC+. (4.5)

Since 1 < µ < q < p < r, we can find ξ̃ϑ > 0 such that the function x 7→
C0x

µ−1 − C2x
r−1 + ξ̃ϑx

p−1 is nondecreasing on [0, ϑ]. So, we have

−∆pu0 −∆qu0 + ξ̃ϑu
p−1
0

= a(z)u−η0 + f(z, u0) + ξ̃ϑu
p−1
0

≥ C0u
µ−1
0 − C2u

r−1
0 + ξ̃ϑu

p−1
0 (see (3.1) and (H0))

≥ C0u
µ−1 − C2u

r−1 + ξ̃ϑu
p−1 (see (4.5))

= −∆pu−∆qu+ ξ̃ϑu
p−1 (see Proposition 3.1).

(4.6)

Since u0 ∈ intC+ and a(z) > 0 for all z ∈ Ω (see hypothesis (H0)), it follows that
0 � a(·)u0(·)−η. Hence from (4.6) and Proposition 3.2 of Gasiński and Papageorgiou
[6], we infer that

u0 − u ∈ intC+. (4.7)

On the other hand, let ξ̂ϑ > 0 be as postulated by hypothesis H1(v). We have

−∆pu0 −∆qu0 + ξ̂ϑu
p−1
0 − a(z)u−η0

= f(z, u0) + ξ̂ϑu
p−1
0

≤ f(z, ϑ) + ξ̂ϑϑ
p−1 (see (4.5) and (H1)(v))

≤ −∆pϑ−∆qϑ+ ξ̂ϑϑ
p−1 − a(z)ϑ−η (see hypothesis (H1)(iv)).

(4.8)



EJDE-2021/SI/01 RESONANT SINGULAR (p, q)-EQUATIONS 177

We know that

ϑ−η + f(z, ϑ) ≤ −Ĉ < 0 for a.a. z ∈ Ω.

So, from (4.8) and Proposition 6 of Papageorgiou, Rădulescu and Repovš [17] we
have

u0(z) < ϑ for all z ∈ Ω. (4.9)

From (4.7) and (4.9), we conclude that u0 ∈ intC1
0 (Ω)[u, ϑ]. The proof is complete.

�

Proposition 4.2. If hypotheses (H0), (H1) hold, then u0 ∈ intC+ is a local mini-
mizer of the functional Ψ(·).

Proof. From (4.1) and (4.2) it is clear that

Ψ
∣∣
[u,ϑ]

= Ψ̂
∣∣
[u,ϑ]

. (4.10)

From the proof of Proposition 4.1 we know that u0 is a minimizer of Ψ̂. Since
u0 ∈ intC1

0 (Ω)[u, ϑ], it follows from (4.10) that

u0 is a local C1
0 (Ω)-minimizer of Ψ(·)

⇒ u0 is a local W 1,p
0 (Ω)-minimizer of Ψ(·)

(see Papageorgiou and Rădulescu [15, Proposition 2.12]). This completes the proof.
�

Using (4.1) and the nonlinear regularity theory, we have

KΨ ⊆ [u) ∩ intC+. (4.11)

From (4.1) and (4.11), we see that we may assume that

KΨ is finite. (4.12)

Otherwise we already have an infinity of positive smooth solutions of (1.1) and so
we are done.

Combining Proposition 5, relation (4.12) and Theorem 5.7.6 of Papageorgiou,
Rădulescu and Repovš [18, p. 449], we deduce that we can find ρ ∈ (0, 1) small
such

Ψ(u0) < inf{Ψ(u) : ‖u− u0‖ = ρ} = m. (4.13)

Proposition 4.3. If hypotheses (H0), (H1) hold, then Ψ(tû1(p)) → −∞ as t →
+∞.

Proof. We have

d

dx
[
F (z, x)

xp
] =

f(z, x)xp − pF (z, x)xp−1

x2p

=
f(z, x)x− pF (z, x)

xp+1
for a.a. z ∈ Ω, all x > 0.

(4.14)

On account of hypothesis (H1)(iii), we can find M > 0 and β1 ∈ (0, β0) such that

f(z, x)− pF (z, x) ≤ −β1x
τ for a.a. z ∈ Ω, all x ≥M. (4.15)

We return to (4.14) and use (4.15) to obtain

d

dx
[
F (z, x)

xp
] ≤ −β1x

τ

xp+1
=
−β1

xp−τ+1
for a.a. z ∈ Ω, all x ≥M,



178 F. I. ONETE, N. S. PAPAGEORGIOU, V. D. RĂDULESCU EJDE/SI/01

which implies

F (z, v)

vp
− F (z, x)

xp
≤ β1

p− τ
[

1

vp−τ
− 1

xp−τ
] for a.a. z ∈ Ω, all v ≥ x ≥M.

We let v → +∞ and using (H1)(ii) we obtain

λ̂1(p)

p
− F (z, x)

xp
≤ − β1

p− τ
1

xp−τ
,

⇒ λ̂1(p)xp − pF (z, x)

xτ
≤ − pβ1

p− τ
= −β2 for a.a. z ∈ Ω, all x ≥M ,

⇒ lim sup
x→+∞

λ̂1(p)xp − pF (z, x)

xτ
≤ −β2 < 0 uniformly for a.a. z ∈ Ω

. (4.16)

Then we have

Ψ(tû1(p)) =
tp

p
λ̂1(p) +

tq

q
‖Dû1(p)‖qq −

∫
Ω

G(z, tû1(p))dz

=
tp

p
λ̂1(p) +

tq

q
‖Dû1(p)‖qq −

∫
{tû1(p)≤u}

[a(z)u−η + f(z, u)](tû1(p))dz

− 1

1− η

∫
{tû1(p)>u}

a(z)1−η[(tû1(p))1−η − u1−η]dz

−
∫
{tû1(p)>u}

[F (z, tû1(p))− F (z, u)]dz

≤ 1

p

∫
Ω

λ̂1(p)(tû1(p))p − pF (z, tû1(p))

tτ
tτdz + C7 for some C7 > 0.

Passing to the limit as t → −∞ and using (4.16), we obtain Ψ(tû1(p)) → −∞ as
t→ +∞. The proof is now complete. �

Remark 4.4. From the above proof we see that

pF (z, x)− λ̂1(p)xp → +∞

uniformly for a.a. z ∈ Ω, as x→ +∞. Hence the resonance is from the right of the
principal eigenvalue, making our problem noncoercive. This means that the direct
method of the calculus of variations cannot be used and we need to appeal to the
minimax theorems of critical point theory.

Proposition 4.5. If hypotheses (H0), (H1), hold, then the functional Ψ(·) satisfies
the C-condition.

Proof. Let {un}n∈N ⊆W 1,p
0 (Ω) be a sequence such that

|Ψ(un)| ≤ M̂ for some M̂ > 0, all n ∈ N, (4.17)

(1 + ‖un‖)Ψ′(un)→ 0 in W−1,p′(Ω) as n→∞. (4.18)

From (4.18) we have

|〈Ap(un), h〉+ 〈Aq(un), h〉| −
∫

Ω

g(z, un)h dz| ≤ εn‖h‖
1 + ‖un‖

(4.19)



EJDE-2021/SI/01 RESONANT SINGULAR (p, q)-EQUATIONS 179

for all h ∈W 1,p
0 (Ω), with εn → 0+. In (4.19) we let h = −u−n ∈W

1,p
0 (Ω). Then

‖Du−n ‖p ≤ C8 for some C8 > 0, all n ∈ N,

⇒ {u−n }n∈N ⊆W
1,p
0 (Ω) is bounded.

(4.20)

Next we show that {u+
n }n∈N ⊆ W 1,p

0 (Ω) is bounded. Arguing by contradiction,
assume that at least for a subsequence, we have

‖u+
n ‖ → ∞ as n→∞. (4.21)

We set yn = u+
n /‖u+

n ‖ for all n ∈ N. We have that ‖yn‖ = 1 and yn ≥ 0 for all
n ∈ N. So, we may assume that

yn
w→ y in W 1,p

0 (Ω) and yn → y in Lp(Ω) as n→∞, y ≥ 0. (4.22)

From (4.19) and (4.20) we have∣∣〈Ap(u+
n ), h〉+ 〈Aq(u+

n ), h〉 −
∫

Ω

g(z, u+
n )h dz

∣∣ ≤ C9‖h‖

for some C9 > 0, all h ∈W 1,p
0 (Ω), all n ∈ N. Then∣∣〈Ap(yn), h〉+

1

‖u+
n ‖p−q

〈Aq(yn), h〉 −
∫

Ω

g(z, u+
n )

‖u+
n ‖p−1

h dz
∣∣ ≤ C9‖h‖
‖u+

n ‖p−1
(4.23)

for all h ∈W 1,p
0 (Ω), all n ∈ N.

From (4.1), Proposition 3.1 and hypothesis (H1)(i), we see that{g(·, u+
n (·))

‖u+
n ‖p−1

}
n∈N
⊆ Lp

′
(Ω) is bounded. (4.24)

So, if in (4.23) we choose h = yn − y ∈ W 1,p
0 (Ω), pass to the limit as n → ∞ and

use (4.22), (4.21), (4.24), then we obtain

lim
n→∞

〈Ap(yn), yn − y〉 = 0,

⇒ yn → y in W 1,p
0 (Ω), hence ‖y‖ = 1, y ≥ 0 (see Proposition 2.1).

(4.25)

From (4.24), (4.1) and (H1)(ii), we see that at least for a subsequence, we have

g(·, u+
n (·))

‖u+
n ‖p−1

w→ η̂(·)y(·)p−1 in Lp
′
(Ω) (4.26)

with η̂ ∈ L∞(Ω), λ̂1(p) ≤ η̂(z) for a.a. z ∈ Ω (see Aizicovici, Papageorgiou and
Staicu [1], proof of Proposition 16).

So, if in (4.23) we pass the limit as n→∞ and use (4.25), (4.21) and (4.26), we
obtain

〈Ap(y), h〉 =

∫
Ω

η̂(z)yp−1h dz for all h ∈W 1,p
0 (Ω),

⇒ −∆py = η̂(z)yp−1 in Ω, y
∣∣
∂Ω

= 0.

(4.27)

If η̂ 6≡ λ̂1(p) (see (4.26)), then by Proposition 2.2 we have

λ̃1(η̂, p) < λ̃1(λ̂1(p), p) = 1.

Then form (4.27) we infer that y must be nodal, which contradicts (4.25).

Now suppose that η̂(z) = λ̂1(p) for a.a. z ∈ Ω. Then from (4.27) it follows that

y = µû1(p) ∈ intC+, with µ > 0.



180 F. I. ONETE, N. S. PAPAGEORGIOU, V. D. RĂDULESCU EJDE/SI/01

This means that u+
n (z)→ +∞ for a.a. z ∈ Ω, which implies

lim inf
n→∞

∫
Ω

pF (z, u+
n )− f(z, u+

n )u+
n

(u+
n )τ

dz ≥ β̂ > 0 (4.28)

(by Fatou’s lemma and hypothesis (H1)(iii)). From (4.17) and (4.20) we have

− ‖Du+
n ‖pp −

p

q
‖Du+

n ‖qq +

∫
Ω

pF (z, u+
n )dz ≤M1 (4.29)

for some M1 > 0, all n ∈ N. Also from (4.19) with h = u+
n ∈W

1,p
0 (Ω), we have

‖Du+
n ‖pp + ‖Du+

n ‖qq −
∫

Ω

f(z, u+
n )u+

n dz ≤M2 (4.30)

for some M2 > 0, all n ∈ N.
We add (4.29) and (4.30) to obtain∫

Ω

[pF (z, u+
n )− f(z, u+

n )u+
n ]dz ≤

(p
q
− 1
)
‖Du+

n ‖qq +M3

with M3 = M1 +M2 > 0, for all n ∈ N. Then∫
Ω

pF (z, u+
n )− f(z, u+

n )u+
n

(u+
n )τ

yτndz ≤
(p
q
− 1
) ‖Dyn‖qq
‖u+

n ‖τ−q
+

M3

‖u+
n ‖τ

⇒ lim sup
n→∞

∫
Ω

pF (z, u+
n )− f(z, u+

n )u+
n

(u+
n )τ

yτndz ≤ 0

(4.31)

(since q < τ and use (4.21)).
Comparing (4.31) and (4.28), we have a contradiction. This proves that {u+

n }n∈N ⊆
W 1,p

0 (Ω) is bounded. It follows that

{un}n∈N ⊆W 1,p
0 (Ω) is bounded (see (4.20)).

So, we may assume that

un
w→ u in W 1,p

0 (Ω) and un → u ∈ Lp(Ω). (4.32)

Now we return to (4.19), choose h = un − u ∈ W 1,p
0 (Ω), pass to the limit as

n→∞ and use (4.32). Then

lim
n→∞

[〈Ap(un), un − u〉+ 〈Aq(un), un − u〉] = 0,

⇒ lim sup
n→∞

[〈Ap(un), un − u〉+ 〈Aq(u), un − u〉] ≤ 0 (since Aq(·) is monotone)

⇒ lim sup
n→∞

〈Ap(un), un − u〉 ≤ 0,

⇒ un → u in W 1,p
0 (Ω) (see Proposition 1).

This proves that Ψ(·) satisfies the C-condition. �

Proposition 4.6. If hypotheses (H0), (H1) hold, then problem (1.1) admits a sec-
ond positive solution û ∈ intC+, with û 6= u0.

Proof. Propositions 4.3, 4.5 and relation (4.13), permit the use of the mountain

pass theorem. So, we can find û ∈W 1,p
0 (Ω) such that

û ∈ KΨ ⊆ [u) ∩ intC+ (see (4.11)),

Ψ(u0) < m ≤ Ψ(û) (see (4.13)).

It follows that û ∈ intC+ is a positive solution of (1.1) (see (4.1)) and û 6= u0. �



EJDE-2021/SI/01 RESONANT SINGULAR (p, q)-EQUATIONS 181

Summarizing our findings, we can state the following multiplicity theorem for
problem (1.1).

Theorem 4.7. If hypotheses (H0), (H1) hold, then problem (1.1) admits at least
two positive solutions

u0, û ∈ intC+, u0 6= û, u0(z) < ϑ for all z ∈ Ω.

Acknowledgements. N. S. Papageorgiou and V. D. Rădulescu were supported by
the Slovenian Research Agency program P1-0292. F.-I. Onete and V. D. Rădulescu
were supported by a grant of the Romanian Ministry of Research, Innovation and
Digitization, CNCS/CCCDI-UEFISCDI, project number PCE 137/2021, within
PNCDI III.

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Florin I. Onete

Department of Mathematics, University of Craiova, 200585 Craiova, Romania

Email address: oneteflorin@yahoo.com

Nikolaos S. Papageorgiou

National Technical University, Department of Mathematics, Zografou Campus, Athens
15780, Greece

Email address: npapg@math.ntua.gr

Vicenţiu D. Rădulescu
Faculty of Applied Mathematics, AGH University of Science and Technology, al. Mick-

iewicza 30, 30-059 Kraków, Poland.

Department of Mathematics, University of Craiova, 200585 Craiova, Romania
Email address: radulescu@inf.ucv.ro


	1. Introduction
	2. Mathematical background and hypotheses
	3. An auxiliary problem
	4. Positive solutions
	Acknowledgements

	References