Electronic Journal of Differential Equations, Vol. 2022 (2022), No. 55, pp. 1–10.

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

DOUBLE PHASE EQUATIONS WITH AN INDEFINITE

CONCAVE TERM

ZHENHAI LIU, NIKOLAOS S. PAPAGEORGIOU

Abstract. We consider a Dirichlet problem having a double phase differential
operator with unbalanced growth and reaction involving the combined effects

of a concave (sublinear) and of a convex (superlinear) terms. We allow the

coefficient E ∈ L∞(Ω) of the concave term to be sign changing. We show that
when ‖E‖∞ is small the problem has at least two bounded positive solutions.

1. Introduction

Let Ω ⊆ RN be a bounded domain with a Lipschitz boundary ∂Ω. In this paper
we study the double phase problem

−∆a
pu(z)−∆qu(z) = E(z)u(z)τ−1 + f(z, u(z)) in Ω,

u|∂Ω = 0, 1 < τ < q < p < N, ];u ≥ 0.
(1.1)

By ∆a
p we denote the weighted p-Laplace differential operator with weight a(·),

defined by

∆a
pu = div(a(z)|Du|p−2Du).

By ∆q we denote the standard q-Laplace differential operator

∆qu = div(|Du|q−2Du).

Problem (1.1) has the sum of these two operators. So, the differential operator
in (1.1), is not homogeneous. In the reaction tern (right hand side) of (1.1), we
have the combined effects of two nonlinearities of different nature. One is the func-
tion x → E(z)xτ−1, x ≥ 0 with E ∈ L∞(Ω). Since τ < q < p, this is a “concave”
((q− 1)-sublinear) term, while the perturbation f(z, x) is a Caratheodory function
(that is, for all x ∈ R, z → f(z, x) is measurable and for a.a. z ∈ Ω, x → f(z, x)
is continuous) which is assumed to be (p − 1)-superlinear as x → ∞ but without
satisfying the Ambrosetti-Rabinowitz condition (the AR-condition for short). So,
we have a “concave-convex” reaction, with the distinguishing feature that the coef-
ficient E ∈ L∞(Ω) of the concave term, is in general sign changing. In the standard
concave-convex problem E(·) ≡ λ > 0 and we can prove existence and multiplicity
of positive solutions and the result is global with respect to the parameter λ > 0
(a bifurcation-type result). We refer to the works of Ambrosetti-Brezis-Cerami

2020 Mathematics Subject Classification. 35J75, 35J20, 35J60.
Key words and phrases. Unbalanced growth; generalized Orlicz spaces; positive solution;

concave-convex problem; mountain pass theorem.
©2022. This work is licensed under a CC BY 4.0 license.

Submitted January 12, 2022. Published July 28, 2022.

1



2 Z. LIU, N. S. PAPAGEORGIOU EJDE-2022/55

[1] and Anello [2] (equations driven by the Laplacian), Garcia Azoreio-Manfredi-
Peral Alonso [4] (equations driven by the p-Laplacian) and Liu-Papageorgiou [9]
(anisotropic (p, q)-equations).

We do not assume that the weight function a ∈ L∞(Ω) is bounded away from
zero, that is, we do not require that 0 < ess infΩ a(·). Therefore the integrand in
the energy functional corresponding to the differential operator

η(z, t) = a(z)tp + tq for all z ∈ Ω, all t ≥ 0

is a Caratheodory function which exhibits unbalanced growth with respect to t ≥ 0,
namely we have

tq ≤ η(z, t) ≤ c0[1 + tp] for a.a. z ∈ Ω, all t ≥ 0, some c0 > 0.

Such functionals were first investigated in the context of problems of the calculus
of variations and nonlinear elasticity theory by Marcellini [10] and Zhikov [16]. For
unbalanced elliptic problems there is no global regularity theory (up to the bound-
ary), analogous to the one existing for balanced problems (see, for example, [5]).
There are only local regularity results, see Baroni-Colombo-Mingione [3], Marcellini
[11] and the nice survey paper of Mingione-Rădulescu [12]. The unbalanced growth
of η(z, ·) leads to a functional framework for double phase problems which is based
on generalized Orlicz spaces. In the next section we discuss this family of spaces.
Details can be found in the book of Harjulehto-Hästo [7].

Using variational tools in the framework of such spaces, we show that when
‖E‖∞ is small problem (1.1) has at least two bounded positive solutions. The lack
of a global regularity theory for double phase unbalanced growth problems and the
fact that the coefficient function of the concave term is nodal (sign-changing), are
features that make problem 1.1 more difficult to handle and also more interesting.

2. Mathematical background

As we already mentioned in the Introduction, the functional framework for the
analysis of problem (1.1) is provided by generalized Orlicz spaces. Recall that by
C0,1(Ω) we denote the space of Lipschitz continuous functions. Our hypotheses on
the weight a(·) and the exponents τ, q, p are the following:

(H0) a ∈ C0,1(Ω), a 6≡ 0, a(z) ≥ 0 for all z ∈ Ω, 1 < τ < q < p < N, pq < 1 + 1
N .

Remark 2.1. The above inequality restricting the exponents q p is common in
Dirichlet double phase problems and it implies that p < q∗ = Nq

N−q . This leads to

some useful compact embeddings of spaces. Moreover, since a ∈ C0,1(Ω), we know
that the Poincare inequality is valid on the relevant Orlicz-Sobolev space (see [7,
p.138]).

Let M(Ω) be the linear space of all measurable functions u : Ω → R. As usual
we identify two such functions which differ only on a Lebesgue-null set. Recall that

η(z, t) = a(z)tp + tq.

Then the Orlicz-Lebesghe space is

Lη(Ω) = {u ∈M(Ω) : ρη(u) <∞},
where ρη(·) is the modular function

ρη(u) =

∫
Ω

η(z, |u|)dz.



EJDE-2022/55 DOUBLE PHASE EQUATIONS 3

We equip Lη(Ω) with the so called “Luxemburg norm”

‖u‖η = inf
[
λ > 0 : ρη(

u

λ
) ≤ 1

]
.

Then Lη(Ω) becomes a Banach space which is separable and reflexive. In fact
the uniform convexity of η(z, ·) implies the uniform convexity of Lη(Ω). Moreover,
Lη(Ω)∗ = Lη

∗
(Ω) with η∗(z, y) being the convex conjugate of η(z, ·) and we have

the following version of the Hölder’s inequality∫
Ω

|hg|dz ≤ 2‖h‖η‖g‖η∗ for all h ∈ Lη(Ω), all g ∈ Lη
∗
(Ω).

Using Lη(Ω) we can define the corresponding Orlicz-Sobolev space

W 1,η(Ω) = {u ∈ Lη(Ω) : |Du| ∈ Lη(Ω)}

with Du denoting the weak gradient of u. We equip this space with the norm

‖u‖1,η = ‖u‖η + ‖Du‖η

where ‖Du‖η = ‖|Du|‖η. Let W 1,η
0 (Ω) = C∞0 (Ω)

‖·‖1,η
. For this space the Poincaré

inequality is true and so on W 1,η
0 (Ω) we can consider the equivalent norm

‖u‖ = ‖Du‖η for all u ∈W 1,η
0 (Ω).

We have the following useful embeddings.

Proposition 2.2. If (H0) holds, then

(a) Lη(Ω) ↪→ Lr(Ω) and W 1,η
0 (Ω) ↪→W 1,r

0 (Ω) continuously for all r ∈ [1, q];

(b) W 1,η
0 (Ω) ↪→ Lr(Ω) continuously (resp. compactly) for all r ∈ [1, q∗] (resp.

r ∈ [1, q∗);
(c) Lp(Ω) ↪→ Lη(Ω) continuously.

The norm ‖ · ‖ and the modular function ρη(·) are closely related.

Proposition 2.3. If hypotheses (H0) holds, then

(a) for u ∈ Lη(Ω) \ {0} we have ‖u‖η = λ⇔ ρη(uλ ) = 1;
(b) ‖u‖η < 1(resp. = 1, > 1)⇔ ρη(u) < 1(resp. = 1, > 1);
(c) ‖u‖η < 1⇒ ‖u‖pη ≤ ρη(u) ≤ ‖u‖qη;
(d) ‖u‖η > 1⇒ ‖u‖qη ≤ ρη(u) ≤ ‖u‖pη;
(e) ‖u‖η → 0(resp. → +∞)⇔ ρη(u)→ 0(resp. → +∞).

We consider the nonlinear operator V : W 1,η
0 (Ω)→W 1,η

0 (Ω)∗ defined by

〈V (u), h〉 =

∫
Ω

[a(z)|Du|p−2 + |Du|q−2](Du,Dh)RNdz, for all u, h ∈W 1,η
0 (Ω).

This operator has the following properties (see Liu-Dai [8]).

Proposition 2.4. If hypotheses (H0) holds, then V (·) is bounded (that is, maps
bounded sets to bounded sets), continuous, strictly monotone (thus maximal mono-

tone too) and of type (S)+, that is, “un
w−→ u in W 1,η

0 (Ω) and lim supn→∞〈V (un), un−
u〉 ≤ 0 imply that un → u in W 1,η

0 (Ω).′′



4 Z. LIU, N. S. PAPAGEORGIOU EJDE-2022/55

If u ∈M(Ω), then u± = max{±u(z), 0}, for all z ∈ Ω and we have u = u+−u−,

|u| = u+ + u− and if u ∈W 1,η
0 (Ω), then u± ∈W 1,η

0 (Ω).
If X is a Banach space and ϕ ∈ C1(X), then Kϕ = {u ∈ X : ϕ′(u) = 0} (the

critical set of ϕ). Also, we say that ϕ(·) satisfies the C-condition, if the following
property hold:

every sequence {un}n∈N ⊆ X such that {ϕ(un)}n∈N ⊆ R is bounded
and (1 + ‖un‖X)ϕ′(un) → 0 in X∗ as n → +∞ admits a strongly
convergent subsequence.

This is a compactness-type condition on the functional ϕ(·) which compensates
for the fact the ambient space need not be locally compact (being in general infinite
dimensional). It leads to a deformation lemma from which follow the minimax
theorems for the critical values of ϕ(·) (see Papageorgiou-Rădulescu-Repovš [4,
Chaper 5]).

By λ̂1(q) we denote the principal eigenvalue of (−∆q,W
1,q
0 (Ω)). So, we consider

the following nonlinear eigenvalue problem

−∆qu(z) = λ̂|u(z)|q−2u(z) in Ω, u|∂Ω = 0. (2.1)

We say λ̂ is an “eigenvalue” of (−∆q,W
1,q
0 (Ω)) if problem (2.1) has a nontrivial

solution known as an “eigenfunction” corresponding to the eigenvalue λ̂. We know

that there is a smallest eigenvalue λ̂1(q) > 0 given by

λ̂1(q) = inf
[‖Du‖qq
‖u‖qq

: u ∈W 1,q
0 (Ω), u 6= 0

]
. (2.2)

The eigenvalue is isolated and simple (that is, if u, v are eigenfunctions corre-

sponding to λ̂1(q) > 0, then u = θv for some θ ∈ R \ {0}). The infimum in (2.2)
is realized on the corresponding one-dimensional eigenspace the elements of which
have constant sign and belong in C1(Ω). So, if û is an eigenfunction corresponding

to λ̂1(q) > 0, then û(z) > 0 or û(z) < 0 for all z ∈ Ω (by the nonlinear maximum
principle). For details we refer to Gasiński-Papageorgiou [5, Chapter 6]. Using the

aforementioned properties of λ̂1(q) > 0 and of its corresponding eigenfunctions, we
show the following proposition (see Mugnai-Papageorgiou [13, Lemma 4.11]).

Proposition 2.5. If θ ∈ L∞(Ω), θ(z) ≤ λ̂1(q) for a.a. z ∈ Ω and θ 6≡ λ̂1(q), then
there exists c0 > 0 such that

c0‖Du‖qq ≤ ‖Du‖qq −
∫

Ω

θ(z)|u|qdz for all u ∈W 1,q
0 (Ω).

Next we introduce our hypotheses on the rest of the data of (1.1).

(H1) E ∈ L∞(Ω), E+ 6≡ 0 and there exists ũ ∈ W 1,η
0 (Ω) with ũ(z) > 0, for a.a.

z ∈ Ω and
∫

Ω
E(z)ũτdz > 0.

Remark 2.6. If E(·) ≡ λ > 0 (that is, we have the standard concave-convex

problem), then hypotheses (H1) is satisfied with any u ∈W 1,η
0 (Ω), u 6≡ 0, u ≥ 0.

(H2) f : Ω × R → R is a Carathéodory function such that f(z, 0) = 0 for a.a.
z ∈ Ω and

(i) |f(z, x)| ≤ â(z)[1 + xr−1] for a.a. z ∈ Ω, all x ≥ 0, with â ∈ L∞(Ω),
p < r < q∗;



EJDE-2022/55 DOUBLE PHASE EQUATIONS 5

(ii) If F (z, x) =
∫ x

0
f(z, s)ds, then limx→+∞

F (z,x)
xp = +∞ uniformly for

a.a. z ∈ Ω and there exists µ ∈
(
(r − q)Nq , q

∗) such that τ < µ and

0 < β0 ≤ lim inf
x→+∞

f(z, x)x− pF (z, x)

xµ

uniformly for a.a. z ∈ Ω;
(iii) there exist δ > 0 and θ ∈ L∞(Ω) such that

θ(z) ≤ λ̂1(q) for a.a. z ∈ Ω, θ 6≡ λ̂1(q),

f(z.x) ≥ 0 for a.a. z ∈ Ω, all 0 ≤ x ≤ δ,

lim sup
x→0+

qF (z, x)

xq
≤ θ(z) uniformly for a.a. z ∈ Ω.

Since we look for positive solutions and the above hypotheses concern the pos-
itive semiaxis R+ = [0,+∞), without any loss of generality, we may assume that
f(z, x) = 0 for a.a. z ∈ Ω and all x ≤ 0.

3. Positive solutions

In the section we show that when ‖E‖∞ is small, then problem (1.1) has at least
two bounded positive solutions.

Let ϕ : W 1,η
0 (Ω)→ R be the energy functional for problem (1.1) defined by

ϕ(u) =
1

p
ρa(Du) +

1

q
‖Du‖qq −

1

τ

∫
Ω

E(z)(u+)τdz −
∫

Ω

f(z, u+)dz,

for all u ∈W 1,η
0 (Ω), with ρa(Du) =

∫
Ω
a(z)|Du|pdz. We know that ϕ ∈ C1(W 1,η

0 (Ω)).

Proposition 3.1. If hypotheses (H0)–(H2) hold, then ϕ(·) satisfies the C-condition.

Proof. We consider a sequence {un}n∈N ⊆W 1,η
0 (Ω) such that

|ϕ(un)| ≤ c1 for some c1 > 0, and all n ∈ N, (3.1)

(1 + ‖un‖)ϕ′(un)→ 0 in W 1,η
0 (Ω)∗ as n→ +∞. (3.2)

From (3.1) we have

ρa(Dun) +
p

q
‖Dun‖qq −

p

τ

∫
Ω

E(z)(u+
n )τdz −

∫
Ω

pF (z, u+
n )dz ≤ pc1 (3.3)

for all n ∈ N. Also from (3.2) we have

|〈V (un), h〉 −
∫

Ω

E(z)(u+
n )τ−1hdz −

∫
Ω

f(z, u+
n )hdz| ≤ εn‖h‖

1 + ‖un‖
(3.4)

for all h ∈W 1,η
0 (Ω) and all εn → 0+.

In (3.4) we choose h = un ∈W 1,η
0 (Ω) and obtain

− ρa(Dun)− ‖Dun‖qq +

∫
Ω

E(z)(u+
n )τdz +

∫
Ω

f(z, u+
n )u+

n dz ≤ εn (3.5)

for all n ∈ N. When we add (3.3) and (3.5), and recall that τ < q < p, we obtain∫
Ω

[f(z, u+
n )u+

n − pF (z, u+
n )]dz ≤ [

p

τ
− 1]‖E‖∞‖u+

n ‖ττ + c2 (3.6)

for some c2 > 0 and all n ∈ N.



6 Z. LIU, N. S. PAPAGEORGIOU EJDE-2022/55

Hypotheses (H2)(i),(ii) imply that we can find β̂0 ∈ (0, β0) and c3 > 0 such that

β̂0x
µ − c3 ≤ f(z, x)x− pF (z, x) for a.a. z ∈ Ω, all x ≥ 0. (3.7)

We return to (3.6) and use (3.7) to obtain

‖u+
n ‖µµ ≤ c4

[
1 + ‖u+

n ‖τµ
]

for some c4 > 0 all n ∈ N (recall that τ < µ),

⇒ {u+
n }n∈N ⊆ Lµ(Ω) is bounded.

(3.8)

From hypothesis (H2)(ii) we see that we can always assume that µ < r < q∗. Hence
we can find t ∈ (0, 1) such that

1

r
=

1− t
µ

+
t

q∗
(3.9)

Invoking the interpolation inequality (Papageorgiou-Winkert [15, p. 116]), we have

‖u+
n ‖r ≤ ‖u+

n ‖1−tµ ‖u+
n ‖tq∗ for all n ∈ N,

⇒ ‖u+
n ‖rr ≤ c5‖u+

n ‖tr for some c5 > 0, all n ∈ N
(3.10)

(recall that W 1,η
0 (Ω) ↪→ Lq

∗
(Ω) continuously, see Proposition 2.2). In (3.4) we use

the test function h = u+
n ∈W

1,η
0 (Ω). Then

ρa(Du+
n ) ≤ εn +

∫
Ω

f(z, u+
n )u+

n dz + ‖E‖∞‖u+
n ‖ττ (3.11)

for all n ∈ N, see hypotheses (H1).

Since our aim is to show that {u+
n }n∈N ⊆ W 1,η

0 (Ω) is bounded, we may assume
without any loss of generality that ‖u+

n ‖ ≥ 1 for all n ∈ N. Then from (3.11) and
using Proposition 2.3, we have

‖u+
n ‖q ≤ c6[1 + ‖un‖tr] for some c6 > 0, all n ∈ N (3.12)

(see hypothesis (H2)(i),(ii), (3.10) and recall that τ < µ).) Since q∗ = Nq
N−q , from

(3.9) we have

tr =
q∗(r − µ)

q∗ − µ
=

Nq(r − µ)

Nq −Nµ+ qµ
< q (see hypothesis (H2)(ii)). (3.13)

From (3.13) and (3.12) it follows that

{u+
n }n∈N ⊆W

1,η
0 (Ω) is bounded. (3.14)

Next in (3.4) we choose h = u−n ∈W
1,η
0 (Ω). Then

ρa(Du−n ) + ‖Du−n ‖qq ≤ εn for all n ∈ N

⇒ u−n → 0 in W 1,η
0 (Ω) as n→ +∞.

(3.15)

(see Proposition 2.3 and use Poincaré inequality). From (3.14) and (3.15), we infer
that

{un}n∈N ⊆W 1,η
0 (Ω) is bounded.

So, by passing to a suitable subsequence if necessary, we assume that

un
w−→ u in W 1,η

0 (Ω) and un → u in Lr(Ω). (3.16)

In (3.4) we choose h = un − u ∈ W 1,η
0 (Ω), pass to the limit as n → +∞ and use

(3.16). We obtain

lim
n→+∞

〈V (un), un − u〉 = 0;



EJDE-2022/55 DOUBLE PHASE EQUATIONS 7

⇒ un → u in W 1,η
0 (Ω) (see Proposition 2.4).

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

Next we show that when ‖E‖∞ is small, then the function ϕ(·) satisfied the
mountain pass geometry.

Proposition 3.2. If hypotheses (H0)—(H2) hold, then there exists λ∗ > 0 such
that if ‖E‖∞ < λ∗, we can find ρ̂ > 0 such that ϕ(u) ≥ m̂ > 0 for all ‖u‖ = ρ̂.

Proof. On account of hypotheses (H2)(i),(iii), given ε > 0, we can find c7 = c7(ε)
such that

F (z, x) ≤ 1

q
[θ(z) + ε]xp + c7x

r for a.a. z ∈ Ω, all x ≥ 0. (3.17)

Then for u ∈W 1,η
0 (Ω) we have

ϕ(u) ≥ 1

p
ρa(Du) +

1

q

[
‖Du‖qq −

∫
Ω

θ(z)|u|qdz − εc8‖Du‖qq
]

− c9[‖E‖∞‖u‖τ + ‖u‖r]
(3.18)

for some c8, c9 > 0 (see (3.17), (2.2)). From Proposition 2.5, we know that

‖Du‖qq −
∫

Ω

θ(z)|u|qdz ≥ c0‖Du‖qq for all u ∈W 1,η
0 (Ω).

So, choosing ε ∈ (0, c0/c8) we see that

‖Du‖qq −
∫

Ω

θ(z)|u|qdz − εc8‖Du‖qq ≥ 0. (3.19)

Assume that ‖u‖ ≤ 1. Then using Proposition 2.3 and (3.19), from (3.18) we have

ϕ(u) ≥ 1

p
‖u‖p − c9‖E‖∞‖u‖τ − c9‖u‖r

≥ [
1

p
− c9(‖E‖∞‖u‖τ−p + ‖u‖r−p)]‖u‖p.

(3.20)

Let γ(t) = ‖E‖∞tτ−p + tr−p for all t > 0. Evidently γ ∈ C1(0,∞) and since
τ < p < r, we have γ(t)→ +∞ as t→ 0+ and as t→ +∞. Therefore we can find
t0 > 0 such that

γ(t0) = min
t>0

γ(t),

⇒ γ′(t0) = 0,

⇒ (p− τ)‖E‖∞ = (r − p)tr−τ0 ,

⇒ t0 = [
(p− τ)‖E‖∞

r − p
]

1
r−τ .

So, we have

γ(t0) = ‖E‖∞[
r − p

(p− τ)‖E‖∞
]
p−τ
r−τ + [

(p− τ)‖E‖∞
r − p

]
r−p
r−τ .

Since p−τ
r−τ < 1, we see that ‖E‖∞ → 0+ ⇒ γ(t0) → 0+. Therefore we can find

λ∗ > 0 such that

‖E‖∞ < λ∗ ⇒ γ(t0) <
1

c9p
.



8 Z. LIU, N. S. PAPAGEORGIOU EJDE-2022/55

Using this in (3.20) we see that

ϕ(u) ≥ m̂ > 0 for all ‖u‖ = ρ̂ = t0(‖E‖∞).

If ‖u‖ > 1, then the same argument works with p replaced by q (since now ρa(Du) ≥
‖u‖q, see Proposition 2.3). �

Proposition 3.3. If hypotheses (H0)–(H2) hold, then for t > 0 small we have

ϕ(tũ) < 0 with ũ ∈W 1,η
0 (Ω) as postulated by hypotheses (H1).

Proof. On account of hypotheses (H2)(i), (iii), we can find c10 > 0 such that

F (z, x) ≥ −c10x
r for a.a. z ∈ Ω, all x ≥ 0. (3.21)

Then for t > 0, we have

ϕ(tũ) ≤ tp

p
ρa(Dũ) +

tq

q
‖Dũ‖qq + c11t

r‖ũ‖r − tτ

τ

∫
Ω

E(z)ũτdz

for some c11 > 0, see (3.1). For t ∈ (0, 1) we have

ϕ(tũ) ≤ c12t
q − tτ

τ

∫
Ω

E(z)ũτdz for some c12 > 0.

Using hypotheses (H1) and since τ < q, for t ∈ (0, 1) small, we have ϕ(tũ) < 0. �

Now we are ready to prove the multiplicity theorem when ‖E‖∞ is small. One
solution will be produced using the mountain pass theorem and the other will be a
local minimizer of ϕ(·).

Theorem 3.4. If hypotheses (H0)–(H2) hold, then for ‖E‖∞ small problem (1.1)

has at least two nontrivial solutions u0, û ∈W 1,η
0 (Ω) ∩ L∞(Ω) such that

u0(z), û(z) ≥ 0 for a.a. z ∈ Ω.

Proof. From Proposition 3.2 we know that if ‖E‖∞ is small, then we can find ρ̂ > 0
such that

0 < m̂ ≤ ϕ(u) for all u ∈W 1,η
0 (Ω), ‖u‖ = ρ̂.

Consider the closed ball Bρ̂ = {u ∈W 1,η
0 (Ω) : ‖u‖ ≤ ρ̂}. The reflexivity of W 1,η

0 (Ω)

implies that Bρ̂ is weakly compact and then by the Eberlein-Smulian theorem, we

have that Bρ̂ is sequentially weakly compact. Using Proposition 2.2, we see that
ϕ(·) is sequentially weakly lower semicontinuous. So, by the Weierstrass-Tonelli

theorem we know that there exists u0 ∈W 1,η
0 (Ω) such that

ϕ(u0) = inf[ϕ(u) : u ∈ Bρ̂]. (3.22)

From Propositions 3.2 and 3.3, we have that

0 < ‖u0‖ < ρ̂,

⇒ ϕ′(u0) = 0 (see(25))

⇒ |〈V (u0), h〉 =

∫
Ω

E(z)(u+
0 )τ−1hdz +

∫
Ω

f(z, u+
0 )hdz

(3.23)

for all h ∈W 1,η
0 (Ω).

In (3.18) we use the test function h = −u−0 ∈ W
1,η
0 (Ω) and obtain that u0 ≥

0, u0 6= 0. Hence u0 is a nontrivial positive solution of problem (1.1). Invoking [6,

Theorem 3.1] we have that u0 ∈W 1,η
0 (Ω) ∩ L∞(Ω).



EJDE-2022/55 DOUBLE PHASE EQUATIONS 9

Consider u ∈W 1,η
0 (Ω) with u(z) > 0 for a.a. z ∈ Ω. Hypothesis (H2)(ii) implies

that

ϕ(tu)→ −∞ as t→ +∞.
This fact and Propositions 3.1 and 3.2, permit the use the mountain pass theorem.
So, we can find û ∈W 1,η

0 (Ω) such that

û ∈ Kϕ and ϕ(u0) < 0 = ϕ(0) < m̂ ≤ ϕ(û).

Then û ≥ 0, û 6= 0, is a solution of (1.1) and as before û ∈ L∞(Ω). �

Acknowledgments. The authors wish to thank the referees for their helpful re-
marks. This research was partially supported by the NNSF of China Grant No.
12071413.

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Zhenhai Liu

Guangxi Colleges and Universities, Key Laboratory of Complex System Optimization
and Big Data Processing, Yulin Normal University, Yulin 537000, China.

Guangxi Colleges and Universities, Key Laboratory of Optimization Control and En-

gineering Calculation, Guangxi Minzu University, Nanning, Guangxi, 530006, China
Email address: zhhliu@hotmail.com



10 Z. LIU, N. S. PAPAGEORGIOU EJDE-2022/55

Nikolaos S. Papageorgiou

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

Athens, Greece
Email address: npapg@math.ntua.gr


	1. Introduction
	2. Mathematical background
	3. Positive solutions
	Acknowledgments

	References