Electronic Journal of Differential Equations, Vol. 2025 (2025), No. 47, pp. 1–13.

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

DOI: 10.58997/ejde.2025.47

EXISTENCE OF THREE POSITIVE SOLUTIONS FOR A p-SUBLINEAR

PROBLEM INVOLVING A SCHRÖDINGER p-LAPLACIAN TYPE

OPERATOR

SIGIFREDO HERRÓN, EMER LOPERA, DIANA SÁNCHEZ

Abstract. We prove the existence of three positive solutions for the problem

−∆pu+ V (x)φp(u) = λf(u), x ∈ Ω,

u(x) = 0, x ∈ ∂Ω,

where λ > 0, ∆p is the p-Laplacian operator, N > p > 1, φp(s) := |s|p−2s, s ∈ R, Ω is a

bounded domain in RN with connected and smooth boundary. In our study, V ∈ L∞(Ω) and

f : [0,∞) → R is a C1 function. The reaction term, f , is increasing and p-sublinear at infinity.

Our method relies on sub-super solution techniques and the use of a theorem on the existence
of multiple fixed points. We extend some results known in the literature.

1. Introduction

The purpose of this article is to prove the existence of three positive solutions for the problem

−∆pu+ V (x)φp(u) = λf
(
u
)
, x ∈ Ω,

u(x) = 0, x ∈ ∂Ω,
(1.1)

where ∆p stands for the p-Laplacian operator, N > p > 1, φp(s) := |s|p−2s, s ∈ R, Ω is a bounded
domain in RN with connected and smooth boundary. Furthermore, we assume that V ∈ L∞(Ω),
λ > 0 and f : [0,∞) → R is a C1 function.

Throughout this article, W 1,p
0 (Ω) denotes the Sobolev space with the norm

∥u∥ :=
(∫

Ω

|∇u|p dx
)1/p

.

Also, ∥u∥q will denote the usual norm in Lq(Ω), for 1 ⩽ q ⩽ ∞.
Let R > 0 be the largest number such that BR ⊆ Ω, where BR is the ball with radius R centered

at the origin in RN . Consider the positive number

M1 := inf
0<ε<R

NRN−1

εN (R− ε)p−1

and B > 0 such that

0 <
1−B∥V ∥∞

B
< M1. (1.2)

We shall use the following assumptions:

(A1) f ∈ C1([0,∞)) is increasing and f(0) > 0.
(A2) limu→∞ f(u)/up−1 = 0.

2020 Mathematics Subject Classification. 35B09, 35B50, 35B51, 35D30, 35G30, 35J10, 35J92, 47H10.

Key words and phrases. Subsolution; supersolution; multiple solutions; p-Laplacian; Schrödinger type operator.
©2025. This work is licensed under a CC BY 4.0 license.
Submitted March 2, 2025. Published May 8, 2025.

1



2 S. HERRÓN, E. LOPERA, D. SÁNCHEZ EJDE-2025/47

(A3) There exists 0 < cV < λ1 such that −cV < V (x) a.e. x ∈ Ω, where

λ1 := inf
{∫

Ω

|∇u|p dx : u ∈ W 1,p
0 (Ω), ∥u∥p = 1

}
,

i.e., λ1 is the principal eigenvalue of (−∆p,W
1,p
0 (Ω)).

Remark 1.1. Let us observe that under hypothesis (A3),

µ1 := inf
{∫

Ω

(|∇u|p + V (x)|u|p) dx : u ∈ W 1,p
0 (Ω), ∥u∥p = 1

}
> 0,

which is the first eigenvalue of the problem −∆pu + V (x)φp(u) = µφp(u) with homogeneous
boundary condition. This fact is fundamental to our approach.

For our analysis we shall use the properties of the solution of the e-problem

−∆pe+ V (x)φp(e) = 1, in Ω,

e = 0, on ∂Ω.
(1.3)

Indeed, since µ1 > 0, there exists e ∈ W 1,p
0 (Ω) such that e(x) > 0 a.e. x ∈ Ω, it satisfies (1.3), [12,

Theorem 6.4.6]. Moreover, e ∈ L∞(Ω) [12, Theorem 6.2.6] and by [12, Theorem 6.2.7] there exists

0 < β < 1 such that e ∈ C1,β
0 (Ω). Furthermore, ∂e

∂η < 0 on ∂Ω, where for x0 ∈ ∂Ω, η := η(x0)

denotes the outward unit normal to ∂Ω at x0 [12, Theorem 6.2.8]. Also, we assume that

(A4) There exist positive numbers a < b < d such that

Q(a, b) :=
φp(a)f(b)

f(a)φp(b)
>

M1∥e∥p−1
∞

1−B∥V ∥∞
, (1.4)

dp−1 >
Rp(1−B∥V ∥∞)f(b)

(p′)p−1∥e∥p−1
∞ f(a)

ap−1 (1.5)

and the function f̃(s) := f(s)− f(b)
bp−1B∥V ∥∞sp−1 is positive for all s ∈ [0, d] and is increasing over

[a, d] (see Figure 2). Our main theorems read as follows.

Theorem 1.2. Let f be a continuous, non-negative and non-decreasing function and λ > 0.
Assume also that problem (1.1) admits a subsolution w1, a strict supersolution w1, a strict subso-
lution w2 and a supersolution w2, such that w1 < w1 < w2, w1 < w2 < w2 and w2 ⩽̸ w1. Then
problem (1.1) has at least three distinct solutions ui, i = 1, 2, 3 such that w1 ⩽ u1 < u2 < u3 ⩽ w2.

As applications of this theorem we obtain the following results.

Theorem 1.3. Let Ω := BR the ball of radius R centered at the origin in RN . Assume that
hypotheses (A1)–(A4) hold. Then, for each λ ∈ [λ∗, λ

∗], problem (1.1) admits at least three
positive solutions, where

λ∗ := M1
φp(b)

f̃(b)
and λ∗ :=

φp(a)

f(a)∥e∥p−1
∞

.

Observe that (1.4) implies that λ∗ < λ∗.

Theorem 1.4. Let Ω be a bounded domain in RN containing the origin with connected boundary
of class C2. Assume that the hypotheses (A1)–(A4) hold. Then, for each λ ∈ [λ∗, λ

∗], problem
(1.1) admits at least three positive solutions.

The solutions of problem (1.1) will be understood in the weak sense. Similar results to Theorem
1.2 have been established in several contexts, some of them based on Lemma 2.5 below (see
[7, 9, 10, 17]). Nevertheless, to the best of our knowledge, Theorem 1.2 has not been proven yet,
which is one of the contributions of this work. Furthermore, our hypothesis on V , (A3), permits
considering a diverse range of potential forms, encompassing positive, negative and sign-changing.
The proof of Theorem 1.2 essentially depends on the properties of the corresponding solution
operator A for problem (1.1). As in our case, this kind of theorem has been used as the main tool
in establishing a multiplicity of solutions for problems like (1.1). One of the main difficulties in



EJDE-2025/47 POSITIVE SOLUTIONS FOR A SCHRÖDINGER p-LAPLACIAN 3

applying this theorem is the construction of a suitable strict subsolution, w2. To our knowledge,
the existence of three positive solutions has never been established for problem (1.1); so Theorems
1.3 and 1.4 extend the results in [18] where the authors studied the multiplicity of positive solutions
for problem (1.1) in the case V ≡ 0.

Multiplicity results applying fixed point techniques have blossomed in recent years (see [13, 17,
18]). For instance, in [18] Ramaswamy and Shivaji established the existence of three solutions for
problem (1.1) in the case V ≡ 0. In [9], the authors established a three-solution theorem for a
singular problem with p = 2. Recently in [13], Ko, Lee and Shivaji proved the existence of three
solutions for a Schrödinger type operator with p = 2 and with a singular reaction term at the
origin. In contrast, we obtain multiple positive solutions for problem (1.1) when V ̸= 0 and p ̸= 2.
Other works on multiplicity of positive solutions in the singular case with V ≡ 0 are, for instance,
[4, 14, 15, 20]. In the case of λ = 0, considering a suitable function g which is perturbed for an
exactly once sign-changing potential V , authors in [5, 6] obtained infinitely many sign-changing
radial solutions.

There are many papers that investigated problems similar to (1.1). These researches involve
N = 1, V ≡ 0, λ = 0, non-linearities having a singularity and/or N < p. Only a few works are
known in the literature considering exactly the problem (1.1). To illustrate, in [2], the authors
investigated problem (1.1) for p > N , where V represents a positive potential. They demonstrated
the existence of at least three weak solutions, each with a bounded norm. Indeed, we have extended
the result obtained in [2] because our potential can assume negative values. Furthermore, we have
augmented the range of values for p. See also [16, 19].

This paper is organized in the following manner. In Section 2, we will present some relevant
preliminary results on sub and super solutions in the context of problem (1.1), which are necessary
for the proofs of Theorems 1.3 and 1.4. To prove Theorem 1.2 we use some results related to
completely continuous maps defined on retracting Banach spaces, as well as strong comparison
principles and maximum principles. Section 3 is devoted to the proof of Theorem 1.2. In section
4 we construct a strict subsolution to problem (1.1) and then, applying Theorem 1.2, we prove
Theorem 1.3. Finally, in Section 5, we prove Theorem 1.4.

2. Preliminary results

Definition 2.1. By a subsolution of (1.1) we mean a function u ∈ W 1,p
0 (Ω) ∩ C(Ω̄) such that∫

Ω

|∇u|p−2∇u∇v dx+

∫
Ω

V (x)|u|p−1uv dx ⩽ λ

∫
Ω

f(u)v dx,

for all v ∈ W 1,p
0 (Ω), v ⩾ 0. If u is not a solution of this problem then we call it a strict subsolution.

Similarly, we say that u ∈ W 1,p
0 (Ω) ∩ C(Ω̄) is a supersolution of (1.1) if∫

Ω

|∇u|p−2∇u∇v dx+

∫
Ω

V (x)|u|p−1uv dx ⩾ λ

∫
Ω

f(u)v dx,

for all v ∈ W 1,p
0 (Ω), v ⩾ 0. Similarly we define the concept of strict supersolution.

Remark 2.2. If g is an appropriate function defined on [0, R], the radial version of the problem

−∆pu = g(|x|), x ∈ BR,

u(x) = 0, |x| = R,

is

−
(
rN−1φp(v

′)
)′

= rN−1g(r), 0 < r < R,

v(R) = 0, v′(0) = 0,
(2.1)

where v(r) := u(x) and r = |x|. In addition, every solution of (2.1) satisfies

−v′(r) = φ−1
p

(
r1−N

∫ r

0

tN−1g(v) dt
)
, 0 < r ⩽ R.



4 S. HERRÓN, E. LOPERA, D. SÁNCHEZ EJDE-2025/47

Given z ∈ C0(Ω) and by extending f(t) = f(0) for all t < 0, we see that λf ◦ z ∈ L∞(Ω) and

λf ◦ z ⩾ 0. From [12, Theorem 6.4.6], we know that there exists a unique w ∈ W 1,p
0 (Ω), w > 0 in

Ω such that

−∆p(w) + V φp(w) = λf(z). (2.2)

Also, from the regularity theory [12, Theorems 6.2.6 and 6.2.7], w ∈ C1,α
0 (Ω) for some 0 < α < 1.

Therefore, we can define the operator

A : C0(Ω) → C1
0 (Ω)

as follows: A(z) = w if and only if w is a weak solution of (2.2). Now, for δ > 0, let

Ωδ = {x ∈ Ω: dist(x,Ω) < δ}.

The following proposition is a standard result (see, for example [11, Theorem 6.1]).

Proposition 2.3 (Strong Comparison Principle). For i = 1, 2, suppose that fi ∈ L∞(Ω) and that

ui ∈ W 1,p
0 (Ω) is a weak solution to

−∆pui + V (x)φp(ui) = fi(x),

where 0 ⩽ f1 ⩽ f2 but f1 ̸= f2. Then

0 ≤ u1 < u2 in Ω and
∂u2

∂η
<

∂u1

∂η
on ∂Ω.

The following lemma was proven in [12, Theorem 6.4.6].

Lemma 2.4 (Maximum Principle). Let us assume (A3) and let u ∈ W 1,p
0 (Ω), u ⩾ 0 be a super

solution of

−∆p(u) + V (x)φp(u) = 0.

Then either u ≡ 0 or u(x) > 0 for all x ∈ Ω.

Now, we consider the space

Ce(Ω) := {u ∈ C0(Ω) : −te ⩽ u ⩽ te for some t > 0},

where e is the solution of (1.3), equipped with the norm

∥u∥e := inf{t > 0 : −te ⩽ u ⩽ te}.

A standard procedure shows that (Ce(Ω), ∥ · ∥e) is a Banach space.
We define the positive cone in Ce(Ω) as Pe := {u ∈ Ce(Ω): u(x) ⩾ 0} whose interior is

P̊e = {u ∈ Ce(Ω): t1e ⩽ u(x) ⩽ t2e, for some t1, t2 > 0}.

An operator  : Ce(Ω) → Ce(Ω) is said strongly increasing if u1 < u2 implies Â(u2)− Â(u1) ∈ P̊e.
The following lemma is proved in [3, Lemma 14.1]. We recall that a nonempty subset Y of a
topological space X is called a retract if there exists a continuous map r : X → Y such that
r|Y = idY .

Lemma 2.5. Let X be a retract of some Banach space and F : X → X be a completely continuous
map. Suppose that X1 and X2 are disjoint retracts of X and let Uk, k = 1, 2 be open subsets of
X such that Uk ⊂ Xk, k = 1, 2. Moreover, suppose that F (Xk) ⊂ Xk and that F has no fixed
points on Xk ∖Uk, k = 1, 2. Then F has at least three distinct fixed points x, x1, x2 with xk ∈ Xk,
k = 1, 2 and x ∈ X ∖ (X1 ∪X2).

We want to recall a compactness result for Hölder spaces which is based on the theorem of
Arzéla-Ascoli (see [1, Theorems 1.30, 1.31]).

Proposition 2.6. Suppose Ω is a relatively compact domain in RN and let m ∈ N and 0 ⩽ α <
β ⩽ 1. Then Cm,β(Ω) ↪→ Cm,α(Ω) compactly.



EJDE-2025/47 POSITIVE SOLUTIONS FOR A SCHRÖDINGER p-LAPLACIAN 5

3. A sub-super solution theorem

The purpose of this section is to prove Theorem 1.2. To achieve this, we need to explore some
important properties of the solution operator A and related spaces. Therefore, we start this section
with statements and proofs of some lemmas.

Lemma 3.1. The following chain of continuous embeddings holds

C1
0 (Ω) ↪→ Ce(Ω) ↪→ C0(Ω). (3.1)

Proof. First we prove that C1
0 (Ω) ⊆ Ce(Ω). Let u ∈ C1

0 (Ω). For x0 ∈ ∂Ω denote by L(x0) the
straight line parallel to η crossing x0. Since

∂e
∂η < 0, ∂e

∂η is continuous and ∂Ω is compact, we can

choose and ε > 0 such that for all x ∈ ∂Ω, ∂e
∂η (x) ⩽ −ε. Moreover, due to the continuity of ∇e and

η, there exist ε0 and δ > 0 such that ∂e
∂η (z) ⩽ −ε0, for all x0 ∈ ∂Ω and all z ∈ Ωδ ∩ L(x0). Now,

for all x ∈ Ωδ, take x0 ∈ ∂Ω the closest point to x. Then, η = x0−x
|x0−x| (see Figure 1). Observe that

there exists ξ(x) ∈ Ωδ ∩ L(x0) such that e(x)− e(x0) = ∇e(ξ(x)) · (x− x0). Taking into account
that e vanishes on the boundary of Ω we see that

e(x)

|x− x0|
=

∣∣∇e(ξ(x)) · x− x0

|x− x0|
∣∣ = ∣∣∇e(ξ(x)) · η

∣∣ ⩾ ε0.

Also, we have that for all x ∈ Ωδ,∣∣u(x)
e(x)

∣∣ ⩽ |u(x)− u(x0)|
ε0|x− x0|

⩽
1

ε0
|∇u(h(x))| ⩽ t1, (3.2)

where h(x) is a point in the segment [x, x0] and t1 is a constant that depends on δ. This proves that
C1

0 (Ω) ⊆ Ce(Ω). We shall now proceed to prove the continuity of the inclusion. Let zn, z ∈ C1
0 (Ω)

such that zn → z in C1
0 (Ω). We need to see that zn → z in Ce(Ω). In fact, applying (3.2) to

u := zn − z, we obtain for all x ∈ Ωδ,∣∣zn(x)− z(x)

e(x)

∣∣ ⩽ 1

ε0
|∇(zn − z)(h(x))| ⩽

∥zn − z∥C1
0 (Ω)

ε0
.

The same result applies to all x ∈ Ω ∖ Ωδ. In this way, we have the first embedding in (3.1).To
establish the second one, let zn, z ∈ Ce(Ω) such that zn → z in Ce(Ω). Let ε > 0. Then there
exists n0 such that for all n ⩾ n0, ∥zn − z∥e < ε/∥e∥∞. For all n ⩾ n0 there is Tn such that
∥zn − z∥e < Tn < ε/∥e∥∞. Thus for all x ∈ Ω,

|zn(x)− z(x)| ⩽ Tne(x) ⩽ Tn∥e∥∞ < ε.

As a consequence, ∥zn − z∥∞ < ε. Which proves the second embedding in (3.1). □

Lemma 3.2. For any λ > 0, A : Ce(Ω) → Ce(Ω) is strongly increasing.

Proof. Let u1 < u2, wi := A(ui) (i = 1, 2) and w̃ := w2 −w1. By the strong comparison principle
(Proposition 2.3) we obtain that w̃ > 0 in Ω. We claim that there exists t1 > 0 such that
t1e < w2 − w1. For any t > 0 consider the function gt(x) := w̃(x)− te(x), x ∈ Ω. We claim that
there exits t1 > 0 such that gt1(x) > 0 for all x ∈ Ω. In fact, by Proposition 2.3 we have that
∂w̃
∂η < 0, on ∂Ω. Let

2t0 := min
{∂w̃
∂η

(x)/
∂e

∂η
(x) : x ∈ ∂Ω

}
> 0

which is well defined since ∂Ω is compact. Thence,
∂gt0
∂η (x) < 0, x ∈ ∂Ω. By the continuity of ∇gt0 ,

there exists r > 0 such that ∇gt0(x) ̸= 0 for all x ∈ Ωr. We claim that for all x ∈ Ωr, gt0(x) ⩾ 0.
If there exists x ∈ Ωr with gt0(x) < 0, then gt0 would attain a minimum at a point x0 ∈ Ωr. Thus
∇gt0(x0) = 0, which is a contradiction. From this, for all x ∈ Ωr, t0e(x) ⩽ w̃(x). On the other
hand, for all x ∈ Ω ∖ Ωr, w̃(x)/e(x) > 0. Since Ω ∖ Ωr is compact, then there exist t > 0 such
that w̃(x)/e(x) ⩾ t and thus w̃(x) ⩾ te(x). Setting t1 = 2−1 min{t0, t}, we see that for all x ∈ Ω,
t1e(x) < w̃(x). On the other hand, since w̃ ∈ C1

0 (Ω), then by Lemma 3.1, w̃ ∈ Ce(Ω). Therefore,

there exists t2 > 0 such that w̃ ⩽ t2e; which implies that Au2 −Au1 ∈ P̊ , as desired. □



6 S. HERRÓN, E. LOPERA, D. SÁNCHEZ EJDE-2025/47

Figure 1. Outward unit normal

Lemma 3.3. For any λ > 0, A : C0(Ω) → C1
0 (Ω) is completely continuous.

Proof. To show that A is continuous, let {un} be a sequence in C0(Ω) and u ∈ C0(Ω) such that
un → u. In particular {un} is bounded in C0(Ω), i.e. there exists C > 0 such that ∥un∥∞ ⩽ C for

all n. Set wn := A(un) and w := A(u). Let us see that {wn} is bounded in W 1,p
0 (Ω). From the

growth behavior of f we see that ∥f(un)∥∞ ⩽ C1 := f(C), for all n. On the other hand, since wn

is a weak solution of

−∆p(wn) + V φp(wn) = λf(un) in Ω; wn = 0 on ∂Ω,

we have that for all ϕ ∈ W 1,p
0 (Ω),∫

Ω

|∇wn|p−2∇wn · ∇ϕdx+

∫
Ω

V (x)|wn|p−2wnϕdx = λ

∫
Ω

f(un)ϕdx.

Then, using wn as a test function

∥wn∥p = λ

∫
Ω

f(un)wn dx−
∫
Ω

V (x)|wn|p dx

⩽ λC1∥wn∥1 + cV ∥wn∥pp
⩽ Cλ∥wn∥+

cV
λ1

∥wn∥p.

Thus (
1− cV

λ1

)
∥wn∥p − Cλ∥wn∥ ⩽ 0. (3.3)

Since 1 − cV
λ1

> 0, we have that {wn} is bounded in W 1,p
0 (Ω). Therefore, up to a subsequence,

wn ⇀ ŵ in W 1,p
0 (Ω) and wn → ŵ in Lp(Ω) and in L1(Ω), for some ŵ. From the definition of a

weak solution, we have that∫
Ω

|∇wn|p−2∇wn(∇wn −∇ŵ) dx = −
∫
Ω

V (x)|wn|p−2wn(wn − ŵ) dx+ λ

∫
Ω

f(un)(wn − ŵ) dx.

(3.4)
Now, from Hölder inequality we see that∣∣ ∫

Ω

f(un)(wn − ŵ) dx
∣∣ ⩽ C1∥wn − ŵ∥1

and ∣∣ ∫
Ω

V (x)|wn|p−2wn(wn − ŵ) dx
∣∣ ⩽ ∥V ∥∞

(∫
Ω

|wn|(p−1)p′
dx

)1/p′(∫
Ω

|wn − ŵ|p dx
)1/p



EJDE-2025/47 POSITIVE SOLUTIONS FOR A SCHRÖDINGER p-LAPLACIAN 7

⩽ ∥V ∥∞∥wn∥p−1
p ∥wn − ŵ∥p.

Then from (3.4) we obtain

lim
n→∞

∫
Ω

|∇wn|p−2∇wn(∇wn −∇ŵ) dx = 0. (3.5)

On the other hand, since wn ⇀ ŵ in W 1,p
0 (Ω), it follows that

lim
n→∞

∫
Ω

|∇ŵ|p−2∇ŵ(∇wn −∇ŵ) dx = 0. (3.6)

Observe also that using the Hölder inequality we reach∫
Ω

(|∇wn|p−2∇wn−|∇ŵ|p−2∇ŵ)(∇wn−∇ŵ) dx ⩾ (∥wn∥p−1−∥ŵ∥p−1)(∥wn∥−∥ŵ∥) ⩾ 0. (3.7)

From (3.5), (3.6) and (3.7) we see that limn→∞ ∥wn∥ = ∥ŵ∥. Since W 1,p
0 (Ω) is reflexive and

wn ⇀ ŵ, it follows that wn → ŵ strongly in W 1,p
0 (Ω). Consequently, from the Lebesgue dominated

convergence theorem we have that for any test function ϕ ∈ W 1,p
0 (Ω),

lim
n→∞

∫
Ω

|∇wn|p−2∇wn · ∇ϕdx+

∫
Ω

V (x)|wn|p−2wnϕdx

=

∫
Ω

|∇ŵ|p−2∇ŵ · ∇ϕdx+

∫
Ω

V (x)|ŵ|p−2ŵϕ dx.

(3.8)

On the other hand, since f is continuous and un → u uniformly in Ω, then f(un) → f(u) uniformly

in Ω. Therefore, for any ϕ ∈ W 1,p
0 (Ω), we have∫

Ω

f(un)ϕdx →
∫
Ω

f(u)ϕdx as n → ∞. (3.9)

From (3.8) and (3.9) we have that ŵ is weak solution of −∆pŵ + V φp(ŵ) = λf(u) in Ω, ŵ = 0
on ∂Ω. That is, ŵ = A(u) = w. Now let us see that since ∥un∥∞ ⩽ C, then there exists
0 < β < 1 such that ∥wn∥C1,β

0 (Ω) ⩽ C2, for some C2 > 0 independent of n. First we claim that∫
E
|wn|p

∗
dx → 0 uniformly in n, as |E| → 0. Indeed, from [12, Theorem 6.2.6 ], we have that

∥wn∥∞ ⩽ C∥wn∥p∗, for some constant C independent of n. Furthermore, from (3.3) and the

Sobolev inequalities we obtain that ∥wn∥∞ ⩽ Ĉ for some Ĉ. Thus, there exists C > 0 such that
for all n ∫

E

|wn|p
∗
dx ⩽ ∥wn∥p

∗

∞|E| ⩽ C|E|.

This proves the claim. On the other hand, taking

hn(x, t) := λf(un(x))− V (x)φp(t),

then, taking into account that f is continuous and V ∈ L∞(Ω), we have

|hn(x, t)| ⩽ C1 + C2|t|p
∗−1.

Therefore, from [8, Proposition 3.7] it follows that {wn} remains bounded in C1,β
0 (Ω) for some

0 < β < 1.

Because of the compact embeddings C1,β
0 (Ω) ⊂⊂ C1

0 (Ω), up to a subsequence, wn → w0 in

C1
0 (Ω) for some w0. Hence, wn → w0 in Lp(Ω). By the uniqueness of the limit, w0 = w. Therefore

wn → w in C1
0 (Ω), which proves the continuity of A.

Now, let us prove that A is compact. Let us assume that {un} is a bounded sequence in C0(Ω).

Arguing as above we see that {wn} remains bounded in C1,α
0 (Ω) for some 0 < α < 1. Now, due to

the compact embedding C1,α
0 (Ω) ↪→ C1

0 (Ω) (see Proposition 2.6), up to subsequences, wn → w′ in

C1
0 (Ω), for some w′. This proves that A is compact, which completes the proof of the lemma. □

From Lemmas 3.1 and 3.3 we obtain the following result.

Corollary 3.4. For any λ > 0, A : Ce(Ω) → Ce(Ω) is completely continuous.

The proof of Theorem 1.2 is inspired by [9] and relies strongly in [3, Lemma 14.1].



8 S. HERRÓN, E. LOPERA, D. SÁNCHEZ EJDE-2025/47

Proof of the Theorem 1.2. Let us consider the subsets X := [w1, w2], X1 := [w1, w1], and X2 :=
[w2, w2] of the Banach space Ce(Ω). Each Xi, i = 1, 2, is a nonempty, closed and convex subset of
X and, in consequence, is a retract of X. Clearly X1 ∩X2 = ∅. From Lemma 3.2 and Corollary
3.4 we have that A : Ce(Ω) → Ce(Ω) is strongly increasing and completely continuous. Also,
A|X : X → X is well defined. Indeed, since

−∆pw1 + V φp(w1) ⩽ λf(w1) = −∆p(A(w1)) + V φp(A(w1)),

then, by the strong comparison principle (Proposition 2.3) we obtain w1 ⩽ A(w1). Also we have
A(w2) ⩽ w2. Therefore, if w ∈ Ce(Ω) is such that w1 ⩽ w ⩽ w2, then

w1 ⩽ A(w1) ⩽ A(w) ⩽ A(w2) ⩽ w2.

Hence, A(X) ⊆ X. Moreover, A|X is completely continuous and strongly increasing, which is
inherited from A. Observe that we also have A(Xi) ⊆ Xi, i = 1, 2. On the other hand, since w1

is a strict supersolution of problem (1.1), then by the strong comparison principle A(w1) < w1.
From [3, Corollary 6.2 ] A has a maximal fixed point u1 ∈ X1 and w1 ⩽ u1 < w1. Likewise, A has
a minimal fixed point u2 ∈ X2 with w2 < u2 ⩽ w1. Now, since 0 ⩽ u1 < w1 and f is increasing,

it follows that λf(u1) ⩽ λf(w1) and therefore, by the strong comparison principle, ∂(w1−u1)
∂η < 0

on ∂Ω. Thus there exists t1 > 0 such that w1 − u1 > t1e. Similarly there exists t2 > 0 such that
u2 − w2 > t2e. In this way, the open sets

Bi := X ∩ {z ∈ Ce(Ω): ∥z − ui∥e < ti},
i = 1, 2, satisfies that Bi ⊆ Xi. In fact, if z ∈ Bi, then z ∈ X which implies that w1 ⩽ z ⩽ w2.
Moreover, from the definition of the norm ∥ ·∥e, there exists t̂i < ti such that |z−ui| < t̂ie. Hence,
z < t̂1e+ u1 < t1e+ u1w1 and −z < t̂2e− u2 < t2e− u2 < −w2 and then w2 < z. So that, in any
case, z ∈ Xi. We claim that there exists a set U1, open in X1, such that A has no fixed points
in X1 ∖ U1. Arguing by contradiction, let us assume that for any open U ⊆ X1, A has a fixed
point in X1 ∖ U . In particular, there exists u3 ∈ X1 ∖ int(X1) such that Au3 = u3. Since u1 is
a maximal fixed point of A in X1 then we have w1 ⩽ u3 ⩽ u1. Due to the fact that u3 ̸= u1 the
strong comparison principle implies that u3 < u1. Notice that X1 ∩ [w1, u1) is open in X1 (since
[w1, u1) is open in X) and u3 ∈ X1 ∩ [w1, u1). This contradicts the assumption that u3 /∈ int(X1).
A similar argument shows that there exists a set U2, open in X2 such that A has no fixed points in
X2∖U2. Therefore, Lemma 2.5 leads us to the existence of at least three solutions to the problem
(1.1), u1 ∈ X1, u2 ∈ X2 and u3 ∈ X ∖ (X1 ∩X2). This concludes the proof of the theorem. □

4. The case Ω = BR

In this section we prove Theorem 1.3. We assume that Ω is the ball in RN centered at the
origin with radius R. Let a∗ ∈ [0, a] such that f̃(a∗) = min0⩽s⩽a f̃(s) (see (A4)). There exists a
function h ∈ C([0,∞)) satisfying

h(u) =

{
f̃(a∗), u ⩽ a∗,

f̃(u), a ⩽ u,

which is non-decreasing in [0, d] and h(u) ⩽ f̃(u) for all u > 0 (see Figure 2).
Observe that 0 ⩽ h(u) for all 0 ⩽ u ⩽ d. We consider the problem

−∆pu = λh(u), in Ω,

u = 0, on ∂Ω.

Let us define, for some α, β > 1 and ε > 0,

v(r) =

{
1, 0 ⩽ r ⩽ ε,

1−
(
1− (R−r

R−ε )
β
)α

, ε < r ⩽ R,

and v̂(r) = bv(r) (see Figure 3). Note that for ε < r < R we have

−v̂′(r) = |v̂′(r)| ⩽ b
αβ

R− ε
. (4.1)



EJDE-2025/47 POSITIVE SOLUTIONS FOR A SCHRÖDINGER p-LAPLACIAN 9

Figure 2. Graphs of f̃ and h

Figure 3. Graphs of v̂ and w

The proof of the following lemma is inspired by ideas from [13], where the authors choose
appropriate values of α, β and ε, such that, after the natural extension to the ball BR, the solution
of (4.2) leads us to a subsolution of problem (1.1).

Lemma 4.1. Let M1 > 0 be the number defined in (A4). Then for any λ such that

M1b
p−1

f̃(b)
⩽ λ ⩽

(p′)p−1dp−1

Rpf̃(b)
,

problem (1.1) has a positive subsolution w2 with b ⩽ ∥w2∥∞ ⩽ d.



10 S. HERRÓN, E. LOPERA, D. SÁNCHEZ EJDE-2025/47

Proof. Let w be a positive solution of(
rN−1φp(w

′)
)′

= −λrN−1h(v̂(r)), 0 < r < R,

w′(0) = 0, w(R) = 0.
(4.2)

which exists by [12, Theorem 6.4.6]. We claim that w satisfies

−
(
rN−1φp(w

′)
)′

⩽ rN−1λh(w(r)), for all 0 < r < R.

First, we prove that w′(r) ⩽ v̂′(r), for all r ∈ [0, R]. Integrating over [0, r], r > 0, the differential
equation in (4.2) and taking into account the initial condition w′(0) = 0 we see that

−w′(r) = φ−1
p

(
r1−N

∫ r

0

λtN−1h(v̂(t)) dt
)
, 0 < r ⩽ R. (4.3)

From this and the fact that h(r) ⩾ 0 for all r ∈ [0, b], we have that w′(r) ⩽ 0. In particular,

w′(r) ⩽ 0 = v̂′(r), for all r ∈ [0, ε] (4.4)

On the other hand, for all r ∈ (ε,R], from (4.3),

−φp(w
′(r)) = r1−N

∫ r

0

λtN−1h(v̂(t)) dt ⩾
λ

rN−1

∫ ε

0

tN−1h(v̂(t)) dt ⩾
λh(b)εN

NRN−1
.

Therefore,

−w′(r) ⩾ φ−1
p

(λh(b)εN
NRN−1

)
. (4.5)

Taking into account that (see (A4)) M1 = inf0<ε<R
NRN−1

εN (R−ε)p−1 , then for any λ > M1
bp−1

h(b) , there

exists ε1 > 0 such that

h(b)

bp−1
λ >

NRN−1

εN1 (R− ε1)p−1
.

Then there exist α, β > 1 such that

h(b)

bp−1
λ >

NRN−1

εN1 (R− ε1)p−1
(αβ)p−1.

Thus,

h(b)εN1
NRN−1

λ > φp

( bαβ

R− ε1

)
. (4.6)

From this inequality, (4.1) and (4.5) we see that

−w′(r) ⩾
bαβ

R− ε1
⩾ −v̂′(r). (4.7)

Then from (4.4) and (4.7) we have

w′(r) ⩽ v̂′(r), for all 0 ⩽ r ⩽ R. (4.8)

Now, integrating both sides of (4.8) over the interval [r,R] and using the conditions w(R) =
v̂(R) = 0, we obtain

v̂(r) ⩽ w(r), for all 0 ⩽ r ⩽ R. (4.9)



EJDE-2025/47 POSITIVE SOLUTIONS FOR A SCHRÖDINGER p-LAPLACIAN 11

Moreover, integrating (4.3) in [t, R], 0 ⩽ t ⩽ R, from (4.9) and taking into account that w(R) = 0
and that h is increasing on [0, d] we obtain

w(t) =

∫ R

t

φ−1
p

(
r1−N

∫ r

0

λsN−1h(v̂(s)) ds
)
dr

⩽
∫ R

t

φ−1
p

(
r1−Nλh(b)

∫ r

0

sN−1 ds
)
dr

⩽
∫ R

t

φ−1
p

(
rλh(b)

)
dr

⩽ φ−1
p

(
λh(b)

) ∫ R

0

rp
′−1 dr

= φ−1
p (λ)

φ−1
p

(
h(b)

)
Rp′

p′
.

(4.10)

Now, the hypothesis implies that φ−1
p (λ) < p′d/[Rp′

φ−1
p (h(b))]. Then from (4.10) we obtain

w(t) ⩽ d. Set w2(x) := w(|x|), x ∈ BR. Taking into account that w ⩽ d and (4.9) we obtain,
b ⩽ ∥w2∥∞ ⩽ d. On the other hand, since w is solution of (4.2) then from Remark 2.2,

−∆pw2(x) = λh(v̂(|x|)).

Furthermore, since h is nondecreasing over [0, d], h(u) ⩽ f̃(u) for all u > 0 and (4.9), then

−∆pw2(x) ⩽ λh(w(|x|)) ⩽ λf̃(w(|x|)) = λf̃(w2(x)). (4.11)

Since M1b
p−1/f̃(b) ⩽ λ, then because of (1.2) we obtain bp−1

Bf(b) < λ. Thus

V (x) ⩽ ∥V ∥∞ ⩽ λ
f(b)

bp−1
B∥V ∥∞ .

From (4.11) and the definition of f̃ (see (A4)), we see that

−∆pw2 + V (x)φp(w2) ⩽ −∆pw2 + λ
f(b)

bp−1
B∥V ∥∞φp(w2) ⩽ λf(w2).

That is, w2 is a positive subsolution of (1.1), which completes the proof of the Lemma. □

Proof of Theorem 1.3. We will construct appropriate sub and super solutions of (1.1) so that we
can apply the Theorem 1.2. Since f(0) > 0, then we see immediately that w1 := 0 is a subsolution
of (1.1) for every λ > 0. Now, since f is increasing, the function w1 := ae/∥e∥∞, where e is
the solution of (1.3), is a supersolution of (1.1) whenever λ ⩽ φp(a)/

(
f(a)∥e∥p−1

∞
)
= λ∗. Notice

that ∥w1∥∞ = a and from (1.5), λ ⩽ (p′)p−1dp−1/(Rpf̃(b)). Therefore, according to Lemma 4.1
there exists, w2, a positive subsolution of (1.1), if λ ⩾ M1φp(b)/h(b) = λ∗. We have ∥w2∥∞ ⩾ b.
Remember that from (1.4), we have λ∗ < λ∗. From hypothesis (A2), for every λ ∈ [λ∗, λ

∗], there
exists M = M(λ) > 0 such that

Mp−1

f(M)
⩾ λ∥e∥p−1

∞ . (4.12)

Therefore, from (4.12) and the fact that f is increasing we have that for any λ ∈ [λ∗, λ
∗], w2 :=

Me/∥e∥∞ is a supersolution of (1.1). Furthermore, since ∂e
∂η < 0 on ∂Ω, we can choose M large

enough such that w2 > w2 and w2 > w1. From Theorem 1.2, for every λ ∈ [λ∗, λ
∗], problem (1.1)

has at least three solutions, ui, i = 1, 2, 3, such that w1 = 0 ⩽ u1 < u2 < u3 ⩽ w2. Finally, by the
maximum principle (Lemma 2.4) we have 0 < u1. This completes the proof of the theorem. □

5. General case: Ω smooth and bounded

It is worth mentioning that the functions w1 = 0, w1 and w2 can be constructed in any bounded
domain Ω with connected smooth boundary. Next, we can prove our main result of this section.
Namely, we extend the previous result when Ω is a smooth bounded domain containing the origin.



12 S. HERRÓN, E. LOPERA, D. SÁNCHEZ EJDE-2025/47

Proof of the Theorem 1.4. Let R > 0 be the largest number such that BR ⊆ Ω and λ∗ ⩽ λ ⩽ λ∗.
It is clear that w1 = 0 is a subsolution of (1.1). Arguing as in the proof of Theorem 1.3 we obtain
supersolutions w1 and w2 of problem (1.1). Take w2 defined in BR as in Lemma 4.1. Then we
define w∗(x) = w2(x) for x ∈ BR and w∗(x) = 0 if x ∈ Ω∖BR. Observe that for x ∈ Ω∖BR,

−∆p(w∗(x)) + V (x)φp(w∗(x)) = 0 < λf
(
0
)
= λf

(
w∗(x)

)
.

On the other hand, for x ∈ BR, it follows from definition of w∗ that

−∆p(w∗(x)) + V (x)φp(w∗(x)) ⩽ λf
(
w∗(x)

)
.

This proves that w∗ is a subsolution of (1.1). Arguing as in the proof of Theorem 1.3 we obtain
the result. □

Acknowledgments. We would like to express our gratitude to Ratnasingham Shivaji for en-
couraging us to address these kind of problems and for providing us with some references that
were essential in the development of this work. Diana Sánchez would like to express her gratitude
to Maya Chhetri for her guidance and support. We really want to thank the referee for their
thoughtful comments, suggestions and indications. We believe that, thanks to their remarks, the
paper has been improved.

This research was partially supported by Facultad de Ciencias, Universidad Nacional de Colom-
bia, Sede Medelĺın, - Facultad de Ciencias – Departamento de Matemáticas – Grupo de investi-
gación en Matemáticas de la Universidad Nacional de Colombia Sede Medelĺın. Proyecto de
facultad: Análisis no lineal aplicado a problemas mixtos en ecuaciones diferenciales parciales,
HERMES code 60827. This research was also supported by Facultad de Ciencias Exactas y Natu-
rales, Universidad Nacional de Colombia, Sede Manizales – Facultad de Ciencias – Departamento
de Matemáticas – Grupo de investigación: Análisis Matemático AM de la Universidad Nacional
de Colombia-Sede Manizales. Projects: Problemas en ecuaciones eĺıpticas no lineales, HERMES
code 63271, Ciencia y Tecnoloǵıa para la Calidad en la Industria Licorera de Caldas (CalCIL): Un
Enfoque desde la Facultad de Ciencias Exactas y Naturales HERMES code 63945 and, Acerca de
la Topoloǵıa Algebraica y sus aplicaciones a las Ecuaciones Diferenciales HERMES code 63636.

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Sigifredo Herrón

Universidad Nacional de Colombia, Medelĺın, Colombia
Email address: sherron@unal.edu.co

Emer Lopera
Universidad Nacional de Colombia, Manizales, Colombia

Email address: edloperar@unal.edu.co

Diana Sánchez
Universidad Nacional de Colombia, Manizales, Colombia

Email address: dmsanchezm@unal.edu.co


	1. Introduction
	2. Preliminary results
	3. A sub-super solution theorem
	4. The case =BR
	5. General case:  smooth and bounded
	Acknowledgments

	References