Electronic Journal of Differential Equations, Vol. 2022 (2022), No. 40, pp. 1–16.

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

EXISTENCE OF SOLUTIONS TO NONLOCAL ELLIPTIC

PROBLEMS WITH SINGULAR AND COMBINED

NONLINEARITIES

JESUS ALBERTO LEON TORDECILLA

Abstract. We use an approximation scheme together with a variation of the
fixed point theorem to show the existence of a positive solution to a nonlo-

cal boundary value problem. This problem has a smooth bounded domain in

RN , a singular term, and combined nonlinearities. We also study the symmet-
ric, monotonicity, and asymptotic behavior of the solutions with respect to a

parameter involved in the problem.

1. Introduction

Let Ω ⊂ RN (N ≥ 2) be a smooth bounded domain. We prove the existence of
a positive solution to the nonlocal boundary value problem

−M
(∫

Ω

|∇u|2
)

∆u = λ(a(x)u−γ + uq) + f(u), x ∈ Ω,

u = 0, x ∈ ∂Ω,
(1.1)

where 0 < q < 1, 0 < γ < 1, λ > 0 is a parameter, a ∈ L∞(Ω) with a ≥ 0,
M : R+ → R+ is a continuous and positive function in [0, 1] and f : R → R is a
continuous function satisfying

0 ≤ tf(t) ≤ C|t|p, (1.2)

with 1 < p ≤ N+2
N−2 if N ≥ 3 or 1 < p if N = 2. Two typical examples are

M(t) = ct+ d with c > 0 and d ≥ 0, and f(u) = up.
By a solution of (1.1) we mean a function u ∈ H1

0 (Ω) such that u > 0 in Ω and

−M
(∫

Ω

|∇u|2
)∫

Ω

∇u∇φ = λ

∫
Ω

(a(x)u−γ + uq)φ+

∫
Ω

f(u)φ = 0

for all φ ∈ H1
0 (Ω).

Problem (1.1) is called nonlocal because of the presence of the term M , which
implies that the equation in (1.1) is no longer a point-wise function. This phe-
nomenon provokes some mathematical difficulties, which makes the study of such
problems particularly interesting. This problem has a physical motivation. In fact,

2020 Mathematics Subject Classification. 35J75, 35B40, 35J60, 35B25, 58K55.
Key words and phrases. Nonlocal problem; singular equation; Galerkin method;

positive solution; asymptotic behavior.
©2022. This work is licensed under a CC BY 4.0 license.

Submitted October 9, 2021. Published June 27, 2022.

1



2 JESUS A. L. TORDECILLA EJDE-2022/40

the operator M(
∫

Ω
|∇u|2)∆u appears in the Kirchhoff equation, which arises in

nonlinear vibrations, namely

utt −M
(∫

Ω

|∇u|2
)

∆u = g(u), x ∈ Ω,

u = 0, x ∈ ∂Ω× [0, T ],

u(x, 0) = u0(x), ut(x, 0) = u1(x).

(1.3)

Such a hyperbolic equation is a general version of the Kirchhoff equation

ρ
∂2u

∂t2
−
(P0

h
+

E

2L

∫ L

0

|∂u
∂x
|dx
)∂2u

∂t2
= 0 (1.4)

presented by Kirchhoff in [20]. This equation extends the classical d’Alembert’s
wave equation by considering the effects of the changes in the length of the strings
during the vibrations. The parameters in equation (1.4) have the following mean-
ings: L is the length of the string, h is the area of cross-section, E is the Young
modulus of the material, ρ is the mass density and P0 is the initial tension.

Equations with singularities have attracted a great attention due to the rela-
tionship with models of non-Newtonian fluids, in applications to heat conduction
in electrically conducting materials, boundary layer phenomena for viscous fluids,
and chemical heterogeneous catalysts (see, e.g., [6, 7, 24, 28] and the references
therein). One of the first studies appeared in [10, 12]. There, the authors consider
an approximation of the singular equation by a regular problem, where monotonic-
ity methods can be applied and then passing to the limit to obtain the solution
of the original equation. Recently, the existence of a solution to the problem (1.1)
was investigated, for the cases f = 0 or f(u) = up, M(t) = ct+ d, and Ω ⊂ R3 is a
bounded domain, see [3, 8, 13, 21]. Furthermore, for the cases a = 0 and f(u) = up

where the nonlinearity is convex-concave, the existence of solutions to (1.1) has
been extensively researched in [1, 11, 22].

WhenM = 1, equation (1.1) is reduced to the singular semilinear elliptic problem

−∆u = λ(a(x)u−γ + uq) + f(u), x ∈ Ω,

u = 0, x ∈ ∂Ω.
(1.5)

Many authors have considered problems related with (1.5). In [14], it was studied
the existence, nonexistence, and uniqueness of positive solution to the problem
−∆u + a(x)g(u) = µf(x, u) + λh(x) in Ω and u = 0 on ∂Ω, where f is a positive
function with sublinear growth, a, h ∈ C0,α(Ω) with a > 0 and h > 0, and g is a
singular nonlinearity. The same hypotheses were used to address similar questions
by [9] for a different equation −∆u = a(x)g(u) + λf(u). In [15], the existence
of multiple positive solutions was studied for the singular, critical elliptic problem
−∆u = λ(u−δ + uq + ρ(u)) in Ω and u = 0 on ∂Ω, where δ > 0, 1 < q ≤ 2∗− 1 and
ρ is a smooth function with subcritical asymptotic behavior at infinite. Moreover,
in [16], in addition to studying the existence of a positive solution, the authors
investigated the asymptotic behavior of the solutions when the exponent p → 1.
Note that in the problem (1.1) we establish the asymptotic behavior of the solutions
regarding the parameter λ. Our main results read as follows.

Theorem 1.1. Suppose that f : R → R is a continuous function satisfying (1.2)
and M : R+ → R+ is a continuous and positive function satisfying (1.6). Then



EJDE-2022/40 NONLOCAL ELLIPTIC PROBLEMS 3

there exists λ∗ > 0 such that for every λ ∈ (0, λ∗), the problem (1.1) has a positive
solution u ∈ H1

0 (Ω).

Proposition 1.2. Suppose that, under the conditions in Theorem 1.1, a(x) = 0
and Ω = B% is an open ball in RN with radius % and center x = 0. If u ∈ H1

0 (Ω) is a
positive solution to (1.1), give by Theorem 1.1, then u is symmetric with respect to
the hyperplane x1 = 0 and decreasing in the direction x1 > 0, where x = (x1, x

′) ∈
B%.

Remark 1.3. If a(x) = 0, then we can show using well-known Bootstrap arguments
that the positive solutions u ∈ H1

0 (Ω) to (1.1) are classical, that is, u ∈ C2,α(Ω) for
some α ∈ (0, 1).

Proposition 1.4. Suppose that uλ ∈ H1
0 (Ω) is a positive solution to (1.1), given

by Theorem 1.1, then
‖uλ‖H1

0 (Ω) → 0 as λ→ 0+.

Notice that in this article we do not impose any extra hypotheses on M be-
yond continuity and positivity in [0, 1]. In comparison with problems that found
in the literature, the novelty in problem (1.1) is that our results hold for a new
nonlocal problem, that is, when M(t) = d − ct with d > c > 0. Recently, non-
singular problems related to this operator were studied in [25, 27]. The results
obtained in Proposition 1.2 are apparently new in the study of nonlocal problems
with singularity.

Remark 1.5. If M is a continuous and positive function in the compact set [0, 1],
then there exist m0 > 0 and m∞ > 0 such that

m0 ≤M(t) ≤ m∞ for every t ∈ [0, 1]. (1.6)

This article is organized as follows: in Section 2 we give some auxiliary results
that will be used throughout the paper. We approximate f by a sequence (fn)
of Lipschitz functions. Then, in Section 3 we prove the existence of solutions
vn for an approximate problem (3.1) in finite dimension. In Section 4 we prove
Theorem 1.1 where we show that the solutions vn of (3.1) are bounded and converge
to a positive solution of (1.1). Finally, in Section 5, we investigated the symmetry,
monotonicity and asymptotic behavior of solutions to the problem (1.1), that is,
we prove Propositions 1.2 and 1.4.

2. Auxiliary results

In this section, we present some preliminary results that will be used throughout
the paper. Initially, we approximate the function f give in (1.1) by a sequence of
Lipschitz functions fk : R→ R defined by

fk(t) =



−k[G(−k − 1
k )−G(−k)], if t ≤ −k

−k[G(t− 1
k )−G(t)], if − k < t ≤ −1/k

k2t[G(− 2
k )−G(− 1

k )], if − 1/k < t ≤ 0

k2t[G( 2
k )−G( 1

k )], if 0 < t ≤ 1/k

k[G(t+ 1
k )−G(t)], if 1/k < t ≤ k

k[G(k + 1
k )−G(k)], if t > k

(2.1)

where G(t) =
∫ t

0
f(τ)dτ .



4 JESUS A. L. TORDECILLA EJDE-2022/40

The following approximation result was proved in [26] and it uses an explicit
expression of the sequence defined in (2.1).

Lemma 2.1. Let f : R→ R be a continuous function such that tf(t) ≥ 0 for every
t ∈ R. Then there exists a sequence fk : R→ R of continuous functions satisfying

(i) tfk(t) ≥ 0 for every t ∈ R;
(ii) ∀k ∈ N, ∃ck > 0 such that |fk(ξ)− fk(η)| ≤ ck|ξ − η| for every ξ, η ∈ R;

(iii) fk → f uniformly in bounded subsets of R.

The sequence (fk) in Lemma 2.1 has some additional properties that are deduced
from (1.2).

Lemma 2.2. Let f : R → R be a continuous function satisfying (1.2) for every
t ∈ R. Then the sequence fk of Lemma 2.1 satisfies

(i) ∀k ∈ N, 0 ≤ tfk(t) ≤ C1|t|p for every |t| ≥ 1/k;
(ii) ∀k ∈ N, 0 ≤ tfk(t) ≤ C2|t|2 for every |t| ≤ 1/k,

where C1 and C2 are positive constants independent of k.

Proof. The proof consist of four steps and it is basically deduced using the mean
value theorem. Everywhere in this proof, the constant C is the one of (1.2).

Step 1. Suppose of −k ≤ t ≤ −1/k. By the mean value theorem, there exists
η ∈ (t− 1

k , t) such that

fk(t) = −k
[
G
(
t− 1

k

)
−G(t)

]
= −kG′(η)

(
t− 1

k
− t
)

= f(η)

and so, tfk(t) = tf(η). Since t − 1
k < η < t < 0 and f(η) < 0, we obtain

tfk(t) ≤ ηf(η). Therefore,

tfk(t) ≤ ηf(η) ≤ C|η|p ≤ C|t− 1

k
|p ≤ C

(
|t|+ 1

k

)p ≤ C(2|t|)p ≤ C2p|t|p.

Step 2. Assume 1
k ≤ t ≤ k. By the mean value theorem, there exists η ∈ (t, t+ 1

k )
such that

fk(t) = k
[
G
(
t+

1

k

)
−G(t)

]
= kG′(η)

(
t+

1

k
− t
)

= f(η)

and thus fk(t) = tf(η). Since 0 < t < η < t + 1
k and f(η) > 0, we have tfk(t) ≤

ηf(η). Therefore

tfk(t) ≤ ηf(η) ≤ C|η|p ≤ C|t+
1

k
|p ≤ C(2|t|)p ≤ C2p|t|p.

Step 3. Suppose that |t| ≥ k, then

fk(t) =

{
−k[G(−k − 1

k )−G(−k)], if t ≤ −k
k[G(k + 1

k )−G(k)], if t ≥ k.

If t ≤ −k, by the mean value theorem, there exists η ∈ (−k − 1
k ,−k) such that

fk(t) = k
[
G
(
− k − 1

k

)
−G(−k)

]
= kG′(η)

(
− k +

1

k
+ k
)

= f(η)



EJDE-2022/40 NONLOCAL ELLIPTIC PROBLEMS 5

and tfk(t) = tf(η). Since −k − 1
k < η < −k < 0 and k < |η| < k + 1

k , we conclude
that

tfk(t) =
s

η
ηf(η) ≤ C |t|

|η|
|η|p

≤ C|t|
(
k +

1

k

)p−1

≤ C|t|
(
|t|+ 1

k

)p−1 ≤ C2p|t|p.

(2.2)

If t ≥ k, by the mean value theorem, there exists η ∈ (k, k + 1
k ) such that

fk(t) = k
[
G
(
k, k +

1

k

)
−G(k)

]
= kG′(η)

(
k +

1

k
− k
)

= f(η).

By computations similar to those for (2.2) one has

tfk(t) =
s

η
ηf(η) ≤ C |t|

|η|
|η|p ≤ C2p|t|p.

Step 4. Suppose that |t| ≤ 1
k , then

fk(t) =

{
k2t[G(− 2

k )−G(− 1
k )], if − 1

k ≤ t ≤ 0

k2t[G( 2
k )−G( 1

k )], if 0 ≤ t ≤ 1
k .

If −1/k ≤ t ≤ 0, by the mean value theorem, there exists η ∈ (− 1
k ,−

2
k ) such that

fk(t) = k2t
[
G
(
− 2

k

)
−G

(
− 1

k

)]
= k2tG′(η)

(
− 2

k
+

1

k
)
)

= −ktf(η).

Therefore,

tfk(t) = −kt2f(η) = −k t
2

η
ηf(η)

≤ k t
2

η
ηf(η) ≤ C|t|2|η|p−1

≤ Ck|t|2(
2

k
)p−1 ≤ C2p−1|t|2

(2.3)

If 0 ≤ t ≤ 1/k, by the mean value theorem, there exists η ∈ ( 1
k ,

2
k ) such that

fk(t) = k
[
G
(2

k

)
−G

(1

k

)]
= k2tG′(η)

(2

k
− 1

k

)
= ktf(η).

By computations similar to those for (2.3) one obtains

tfk(t) = kt2f(η) = k
t2

η
ηf(η) ≤ k t

2

η
ηf(η) ≤ C2p−1|t|2.

The proof of lemma follows by talking C1 = C2p and C2 = C2p−1. �

The next lemma will be used to show the symmetry and monotonicity of the
positive solutions to (1.1).

Lemma 2.3. Let Ω ⊂ RN be a bounded open set, convex in the direction of x1 and
symmetric with respect the hyperplane x1 = 0. Let u ∈ C2(Ω)∩C0(Ω) be a positive
solution to the problem −∆u = g(u) in Ω and u = 0 in ∂Ω, where g : R → R is a
locally Lipschitz function. Then u(x1, x

′) ≤ u(−x1, x
′) for every x = (x1, x

′) ∈ Ω
such that x1 > 0. Furthermore, ∂u

∂x1
< 0 for every x ∈ Ω, x1 > 0. See [4, Theorem

1.2].



6 JESUS A. L. TORDECILLA EJDE-2022/40

Now, we recall the Hardy-Sobolev inequality, which will play a key role in the
proof of our main result:

Lemma 2.4 (Hardy-Sobolev inequality [18]). If u ∈ W 1,p
0 (Ω) with 1 < p ≤ N ,

then u
dτ ∈ L

σ(Ω), for 1
σ = 1

p −
1−τ
N , 0 < τ ≤ 1 and

‖ u
dτ
‖Lσ(Ω) ≤ C‖∇u‖Lp(Ω),

where d(x) = dist(x, ∂Ω) and C > 0 is a constant which does not depend on x.

The following lemma will be used for showing that the solutions of an approxi-
mate problem discussed in Section 3 converges to a solution to the problem (1.1).

Lemma 2.5 ([26, Theorem 1.1]). Let Ω be a bounded open set in RN , uk : R→ R
be a sequence of functions, and gk : R → R be a sequence of functions such that
gk(uk) are mensurable in Ω for every k ∈ N. Assume that gk(uk) → v a.e. in
Ω and

∫
Ω
gk(uk)uk ≤ C for a constant C independent of k. And suppose that for

every B ⊂ R, B bounded, there is a constant CB depending only on B such that
|gk(x)| ≤ CB, for all x ∈ B and k ∈ N. Then v ∈ L1(Ω) and gk(uk)→ v in L1(Ω).

We conclude this section by presenting a lemma, which is a consequence of
Brouwer’s Fixed Point Theorem. However, our statement is a subtle (but very
useful) generalization by comparing it with the literature.

Lemma 2.6. Let F : Rd → Rd be a continuous function such that 〈F (ξ), ξ〉 ≥ 0 for
every ξ ∈ Rd with |ξ| = r for some r > 0. Then there exists z0 in the closed ball
Br(0) such that F (z0) = 0.

3. Approximate problem in a finite dimensional space

For each n ∈ N, consider the sequence (fn) of Lipschitz functions given by the
Lemmas 2.1 and 2.2. We will show the existence of a solution to the approximate
problem

−M
(∫

Ω

|∇v|2
)

∆v = λ
(
a(x)vs(v + 1/

√
n)−(γ+s) + vq

)
+ fn(v) +

φ

n
, x ∈ Ω,

u = 0, x ∈ ∂Ω,

(3.1)
where 0 < γ < s < 1, 0 < q < 1, λ > 0 is a parameter, φ(x) is a positive function
such that φ ∈ C0,α(Ω) for some α ∈ (0, 1]. Notice that for us to show the existence
of a solution to the approximate problem (3.1), we will use the Galerkin method
together with the fixed point theorem given in Lemma 2.6. The main result in this
section is the following.

Lemma 3.1. For each n ∈ N, there exists λ∗ > 0 and n∗ ∈ N such that (3.1) admits
a positive solution vn ∈ H1

0 (Ω) for every λ ∈ (0, λ∗) and n ≥ n∗. Furthermore,

‖vn‖H1
0 (Ω) ≤ r, ∀n ∈ N,

where r does not depend on n.

In the following lemma we prove that nonnegative solutions for the approximate
problem (3.1) are in fact regular.

Lemma 3.2. Let v ∈ H1
0 (Ω) be a nonegative solution to (3.1). Then v ∈ C2,α(Ω)

for some α ∈ (0, 1).



EJDE-2022/40 NONLOCAL ELLIPTIC PROBLEMS 7

In the next corollary, we will emphasize the significance of considering the se-
quence (fn) of Lipchitz functions in the approximate problem (3.1).

Corollary 3.3. Suppose that a(x) = 0 and Ω = B%. Then, for each n ∈ N, the

solution vn ∈ C2,α(Ω) to (3.1) satisfies

vn(x1, x
′) ≤ vn(−x1, x

′), (3.2)

for every x = (x1, x
′) ∈ B% such that x1 > 0. Furthermore,

∂u

∂x1
< 0 for every x ∈ Ω, x1 > 0. (3.3)

Proof Lemma 3.1. Let B = {w1, w2, . . . , wm, . . .} be an orthonormal basis ofH1
0 (Ω).

For each m ∈ N, we define

Wm = [w1, w2, . . . , wm],

to be the m-dimensional space generated by {w1, w2, . . . , wm}. Define the function
F : Rm → Rm such that

F (ξ) = (F1(ξ), F2(ξ), . . . , Fm(ξ)),

where ξ = (ξ1, ξ2, . . . , ξm) ∈ Rm,

Fj(ξ)

= M
(∫

Ω

|∇v|2
)∫

Ω

∇v∇wj − λ
∫

Ω

(
a(x)(v+)s(v+ + 1/

√
n)−(γ+s) + (v+)q

)
wj

−
∫

Ω

fn(v+)wj −
1

n

∫
Ω

φwj

for j = 1, 2, . . . ,m and v =
∑m
i=1 ξiwi belongs to Wm. Therefore,

〈F (ξ), ξ〉

= M
(∫

Ω

|∇v|2
)∫

Ω

|∇v|2 − λ
∫

Ω

(
a(x)(v+)s+1(v+ + 1/

√
n)−(γ+s) + (v+)q

)
−
∫

Ω

fn(v+)v+ −
1

n

∫
Ω

φv,

(3.4)
where v+ = max{0, v} and v− = v+ − v.

Notice that F is continuous by Sobolev embedding and dominated convergence
theorem. Given v ∈Wm, we define

Ω+
n = {x ∈ Ω : |v| ≥ 1

n
}, Ω−n = {x ∈ Ω : |v| < 1

n
}.

Now we rewrite (3.4) as 〈F (ξ), ξ〉 = 〈F (ξ), ξ〉+ + 〈F (ξ), ξ〉− where

〈F (ξ), ξ〉+

= M
(∫

Ω

|∇v|2
)∫

Ω+
n

|∇v|2 − λ
∫

Ω+
n

(
a(x)(v+)s+1(v+ + 1/

√
n)−(γ+s) + (v+)q

)
−
∫

Ω+
n

fn(v+)v+ −
1

n

∫
Ω+
n

φv

and

〈F (ξ), ξ〉−



8 JESUS A. L. TORDECILLA EJDE-2022/40

= M
(∫

Ω

|∇v|2
)∫

Ω−n

|∇v|2 − λ
∫

Ω−n

(
a(x)(v+)s+1(v+ + 1/

√
n)−(γ+s) + (v+)q

)
−
∫

Ω−n

fn(v+)v+ −
1

n

∫
Ω−n

φv.

Let K1 = ‖a‖L∞(Ω) and K2 = max{|φ(x)| : x ∈ Ω}. In the next two steps, we

estimate 〈F (ξ), ξ〉+ and 〈F (ξ), ξ〉−.

Step 1. Since 0 < q < 1 and 0 < γ < s < 1, using Holder inequality and by the
Sobolev embedding we obtain∫

Ω+
n

a(x)(v+)s+1(v+ + 1/
√
n)−(γ+s) ≤ K1

∫
Ω

|v|1−γ ≤ C2‖v‖1−γH1
0 (Ω)

, (3.5)∫
Ω+
n

(v+)q+1 ≤
∫

Ω

|v|q+1 = ‖v‖q+1
Lq+1(Ω) ≤ C3‖v‖q+1

H1
0 (Ω)

. (3.6)

Furthermore, ∫
Ω

φv ≤ K2|Ω|‖v‖L2(Ω) ≤ C4‖v‖H1
0 (Ω) . (3.7)

By Lemma 2.2(i) and Sobolev embedding, we deduce∫
Ω+
n

fn(v+)v+ ≤ C
∫

Ω

|v+|p+1 ≤ C‖v‖p+1
Lp+1(Ω) ≤ C1‖v‖p+1

H1
0 (Ω)

. (3.8)

Thus, it follows from (3.5)–(3.8) that

〈F (ξ), ξ〉+ ≥M(‖v‖2H1
0 (Ω))

∫
Ω+
n

|∇v|2 − λ(C2‖v‖1−γH1
0 (Ω)

+ C3‖v‖q+1
H1

0 (Ω)
) (3.9)

− C1‖v‖p+1
H1

0 (Ω)
− C4

n
‖v‖H1

0 (Ω) , (3.10)

where C1, C2, C3, and C4 are positive constants that do not depend on n nor m.

Step 2. Since 0 < q < 1 and 0 < s < 1, we deduce∫
Ω−n

a(x)(v+)s+1(v+ + 1/
√
n)−(γ+s) ≤ K1n

γ+s
2

∫
Ω−n

|v|s+1 ≤ K1|Ω|n
γ−s
2 , (3.11)∫

Ω−n

(v+)q+1 ≤ |Ω| 1

nq+1
. (3.12)

By Lemma 2.2(ii) we find that∫
Ω−n

fn(v+)v+ ≤ C2

∫
Ω

|v+|2 ≤ C5|Ω|
1

n2
. (3.13)

It follows from (3.11)–(3.13) that

〈F (ξ), ξ〉−

≥M(‖v‖2H1
0 (Ω))

∫
Ω−n

|∇v|2 − λ(K1|Ω|n
γ−s
2 + |Ω| 1

nq+1
)− C5|Ω|

1

n2
.

(3.14)

As a direct consequence of estimates (3.9) and (3.14) we obtain

〈F (ξ), ξ〉

≥M(‖v‖2H1
0 (Ω))

∫
Ω

|∇v|2 − λ(C2‖v‖1−γH1
0 (Ω)

+ C3‖v‖q+1
H1

0 (Ω)
)

− C1‖v‖p+1
H1

0 (Ω)
− C4

n
‖v‖H1

0 (Ω) − λ(K1|Ω|n
γ−s
2 + |Ω| 1

nq+1
)− C5|Ω|

1

n2
.

(3.15)



EJDE-2022/40 NONLOCAL ELLIPTIC PROBLEMS 9

Assume that ‖v‖H1
0 (Ω) = r ≤ 1 for some 0 < r ≤ 1 to be fixed later. It follows from

(1.5) that M(‖v‖2
H1

0 (Ω)
) = M(r2) ≥ m0, and so we deduce

〈F (ξ), ξ〉 ≥ m0r
2 − λ(C2r

1−γ + C3r
q+1)− C1r

p+1 − C4

n
r

− λ(K1|Ω|n
γ−s
2 + |Ω| 1

nq+1
)− C2|Ω|

1

n2
.

(3.16)

Note that if

r ≤
( m0

2C1

)1/(p−1)

,

then m0r
2 − C1r

p+1 ≥ m0r
2/2. Thus, by considering

r =
1

2
min

{
1,
( m0

2C1

)1/(p−1)}
,

we obtain

〈F (ξ), ξ〉 ≥ m0r
2

2
− λ(C2r

1−γ + C3r
q+1)− C4

n
r

− λ(K1|Ω|n
γ−s
2 + |Ω| 1

nq+1
)− C5|Ω|

1

n2
.

Now, defining ρ = m0r
2

2 − λ(C2r
1−γ + C3r

q+1), we choose λ∗ > 0 such that ρ > 0
for every λ < λ∗. Therefore, we may take

λ∗ = min
{m0r

1+γ

4C2
,
m0r

1−q

4C3

}
> 0.

Moreover, since γ < s we may choose n∗ ∈ N such that

C4

n
r + λ(K1|Ω|n

γ−s
2 + |Ω| 1

nq+1
) + C5|Ω|

1

n2
<
ρ

2
, ∀n ≥ n∗.

Let ξ ∈ Rm, such that |ξ| = r, then for λ < λ∗ and n ≥ n∗ we have

〈F (ξ), ξ〉 ≥ ρ

2
> 0.

For every n ∈ N, fn is a Lipschitz function, and then by Lemma 2.6, for every
m ∈ N there exists y ∈ Rm with |y|m ≤ r such that F (y) = 0, that is, there exists
vm ∈Wm satisfying

‖vm‖H1
0 (Ω) ≤ r for every m ∈ N,

and such that

M
(∫

Ω

|∇vm|2
)∫

Ω

∇vm∇w

= λ

∫
Ω

(
a(x)(vm+)s(vm+ + 1/

√
n)−(γ+s) + (vm+)q

)
w

−
∫

Ω

fn(vm+)w − 1

n

∫
Ω

φw, ∀w ∈Wm.

(3.17)

Since Wm ⊂ H1
0 (Ω) for every m ∈ N and r does not depend on m, it follows that

(vm) is a bounded sequence in H1
0 (Ω). Thus, for some subsequence, there exist

0 ≤ t0 ≤ 1 and vn ∈ H1
0 (Ω) (denoting vn by v) such that

‖vm‖2H1
0 (Ω) → t0, (3.18)

vm ⇀ v weakly in H1
0 (Ω). (3.19)



10 JESUS A. L. TORDECILLA EJDE-2022/40

From (3.19) and Sobolev compact embedding, we have

vm → v in L2(Ω) and a.e. in Ω. (3.20)

Let k ∈ N, then for every k ≥ m we obtain that Wk ⊂Wm and

M
(∫

Ω

|∇vm|2
)∫

Ω

∇vm∇wk

= λ

∫
Ω

(
a(x)(vm+)s(vm+ + 1/

√
n)−(γ+s) + (vm+)q

)
wk

−
∫

Ω

fn(vm+)wk −
1

n

∫
Ω

φwk, ∀wk ∈Wk.

(3.21)

It follows from (3.19) that∫
Ω

∇vm∇wk →
∫

Ω

∇v∇wk as m→∞. (3.22)

Also, using (3.18) and continuity of M we deduce that

M
(∫

Ω

|∇vm|2
)
→M(t0) as m→∞. (3.23)

By (3.20),

a(x)(vm+)s(vm+ + 1/
√
n)−(γ+s) → a(x)(v+)s(v+ + 1/

√
n)−(γ+s) a.e. in Ω

and (vm+)s|wk| → (v+)s|wk| a.e. in Ω. Furthermore,

a(x)(vm+)s(vm+ + 1/
√
n)−(γ+s) ≤ K1Cn(vm+)s|wk|, ∀m ∈ N,

and by the Sobolev compact embedding one obtains∫
Ω

(vm+)s|wk| →
∫

Ω

(v+)s|wk| as m→∞.

Therefore, from generalized dominate convergence theorem,∫
Ω

a(x)(vm+)s(vm+ + 1/
√
n)−(γ+s) →

∫
Ω

a(x)(v+)s(v+ + 1/
√
n)−(γ+s), (3.24)

as m→∞. Also, thanks to (3.20) and Lemma 2.1 (ii) we obtain∫
Ω

fn(vm+)wk →
∫

Ω

fn(v+)wk as m→∞. (3.25)

Thus, by (3.22), (3.24), and (3.25), letting m→∞ we deduce that

λ

∫
Ω

(
a(x)(vm+)s(vm+ + 1/

√
n)−(γ+s) + (vm+)q

)
wk +

∫
Ω

fn(vm+)wk

+
1

n

∫
Ω

φwk

→ λ

∫
Ω

(
a(x)(v+)s(v+ + 1/

√
n)−(γ+s) + (v+)q

)
wk

+

∫
Ω

fn(v+)wk +
1

n

∫
Ω

φwk.

(3.26)



EJDE-2022/40 NONLOCAL ELLIPTIC PROBLEMS 11

It follows from (3.21), (3.22), (3.23), and (3.26) that

M(t0)

∫
Ω

∇v∇wk = λ

∫
Ω

(
a(x)(v+)s(v+ + 1/

√
n)−(γ+s) + (v+)q

)
wk

+

∫
Ω

fn(v+)wk +
1

n

∫
Ω

φwk, ∀wk ∈Wk.

(3.27)

Since [Wk]k∈N is dense in W 1,N
0 (Ω), we derive that

M(t0)

∫
Ω

∇v∇w = λ

∫
Ω

(
a(x)(v+)s(v+ + 1/

√
n)−(γ+s) + (v+)q

)
w

+

∫
Ω

fn(v+)w +
1

n

∫
Ω

φw, ∀w ∈ H1
0 (Ω).

(3.28)

We claim that t0 = ‖vm‖2H1
0 (Ω)

. Indeed, taking w = vm in (3.17) we obtain

M
(∫

Ω

|∇vm|2
)∫

Ω

|∇vm|2

= λ

∫
Ω

(
a(x)(vm+)s(vm+ + 1/

√
n)−(γ+s) + (vm+)q

)
vm

+

∫
Ω

fn(vm+)vm +
1

n

∫
Ω

φvm, ∀w ∈Wm.

(3.29)

So that, passing to the limit as m→∞ in (3.29), we find

M(t0)t0 = λ

∫
Ω

(
a(x)(v+)s(v+ + 1/

√
n)−(γ+s) + (v+)q

)
v

+

∫
Ω

fn(v+)v +
1

n

∫
Ω

φv.

(3.30)

Therefore, taking w = v in (3.28), it follows from (3.30) that t0 = ‖v‖2
H1

0 (Ω)
, since

M(t0) ≥ m0 > 0. The claim is proved.
Moreover, v ≥ 0 a.e. in Ω. In fact since v− ∈ H1

0 (Ω), by (3.28), we obtain

M

(∫
Ω

|∇v|2
)∫

Ω

∇v∇v− = λ

∫
Ω

(
a(x)(v+)s(v+ + 1/

√
n)−(γ+s) + (v+)q

)
v−

−
∫

Ω

fn(v+)v− −
1

n

∫
Ω

φv−.

Hence,

−M
(
‖v‖2H1

0 (Ω)

)
‖v−‖2H1

0 (Ω) = −M
(
‖v‖2H1

0 (Ω)

)∫
Ω

∇v∇v−

=

∫
Ω

fn(v+)v− +
1

n

∫
Ω

φv− ≥ 0,

then v− = 0 a.e. in Ω, since M(‖v‖2
H1

0 (Ω)
) ≥ M(t0) > 0. This completes the

proof. �

Proof of Lemma 3.2. Define

g(x) =
1

M(t0)

[
λ
(
a(x)v(x)s(v(x) + 1/

√
n)−(γ+s) + v(x)q

)
+ fn(v(x)) +

φ(x)

n

]
.



12 JESUS A. L. TORDECILLA EJDE-2022/40

Clearly,

|g| ≤ 1

M(t0)

[
K1n

γ+s
2 |v|s + |v|q + |fn(v)|+ K2

n

]
, (3.31)

where K1 = ‖a‖L∞(Ω) and K2 = max{φ(x) : x ∈ Ω}. Notice that

|v|s ≤ 1 + |v|β−1, |v|q ≤ 1 + |v|β−1, (3.32)

where 2 ≤ β ≤ 2N
N−2 . Moreover, since fn is a Lipschitz function and fn(0) = 0, we

obtain |fn(v)| ≤ Cn|v|, and so

|fn(v)| ≤ Cn(1 + |v|β−1). (3.33)

It follows from (3.31), (3.32), and (3.33) that

|g| ≤ C1 + C2|v|β−1, (3.34)

where

C1 =
1

M(t0)

[
K1n

γ+s
2 + λ+ Cn +

K2

n

]
, C2 =

1

M(t0)

[
K1n

γ+s
2 + λ+ Cn

]
.

Therefore, by applying bootstrap arguments and using (3.34), similar to those found
in [19], we conclude that v ∈ C2,α(Ω) for some α ∈ (0, 1). The proof is complete. �

Remark 3.4. Since φ/n 6= 0, we deduce that v 6= 0 in Ω. Therefore, it follows
from maximum principle that v > 0 in Ω, since v ≥ 0 in Ω.

Proof of Lemma 3.3. Let g : R → R be a function defined by g(t) = 1
M(t0) (λtq +

fn(t)). Notice that g is a locally Lipschitz function because tq and fn are Lipschitz
functions. According to Lemma 2.3 the solutions vn ∈ C2,α(Ω) of problem (3.1)
satisfy (3.2) and (3.3). �

4. Proof of Theorem 1.1

We will use the existence of a unique solution z to the problem

−∆z = zq, x ∈ Ω,

z > 0, x ∈ Ω,

z = 0, x ∈ ∂Ω,

(4.1)

where 0 < q < 1, to show that vn ≥ az in Ω, with a a constant independent of
n. This implies that the limit of the sequence vn of solutions to the approximate
problem (3.1) is positive. See for instance [5] for the details of problem (4.1).

Also, we use Lemma 2.5 together with the Hardy-Sobolev inequality, see Lemma
2.4, to show that vn converges to a positive solution v to (1.1).

Proof of Theorem 1.1. By Lemmas 3.1 and 3.2, equation (3.1) has a solution vn ∈
C2,α(Ω), for some 0 < α < 1, for each n ∈ N. From (3.19) we know that

vm ⇀ vn weakly in H1
0 (Ω) as m→∞. (4.2)

Thus

‖vn‖H1
0 (Ω) ≤ lim inf

m→∞
‖vm‖H1

0 (Ω) ≤ r ≤ 1 for every n ∈ N,

and r does not depend on n. Therefore, up to a subsequence, there exist 0 ≤ t̃0 ≤ 1
and v ∈ H1

0 (Ω) such that

‖vn‖2H1
0 (Ω) → t̃0 as n→∞, (4.3)



EJDE-2022/40 NONLOCAL ELLIPTIC PROBLEMS 13

vn ⇀ v weakly in H1
0 (Ω), (4.4)

and, by the Sobolev embedding for 1 ≤ σ < +∞,

vn → v in Lσ(Ω) and a.e. in Ω. (4.5)

Moreover, since M is continuous we obtain, by (4.3), that

M(‖vn‖2H1
0 (Ω))→M(t̃0). (4.6)

Now, it follows from (1.5) that 0 < m0 ≤M(‖vn‖2H1
0 (Ω)

) ≤ m∞ and since vn > 0,

then by taking µ = λm−1
∞ we find that

−∆vn ≥ µvqn, x ∈ Ω,

vn > 0, x ∈ Ω,

vn = 0, x ∈ ∂Ω.

Thus, by defining zn = µ
1

1−q vn we deduce that

−∆
( zn

µ
1

1−q

)
= µ

( zn

µ
1

1−q

)q
,

that is, −∆zn ≥ zqn. By [2, Lemma 3.3], it follows that zn ≥ z for every n ∈ N,
implying

vn ≥ µ
1

1−q z, ∀n ∈ N. (4.7)

Letting n→∞ in (4.7), we have v ≥ µ
1

1−q z in Ω. Therefore, v > 0 a.e. in Ω.
We claim that v is a solution of (1.1). Since vn → v a.e. in Ω we have

fn(vn(x))→ f(v(x)) a.e. in Ω (4.8)

by the uniform convergence of Lemma 2.1(iii).
Observe that ∫

Ω

|fn(vn)vn| ≤ C, ∀n ∈ N, (4.9)

where C > 0 is a constant independent of n. Indeed, let z be a solution to (4.1).
In view of the maximum principle we have

∂z

∂ν
< 0 in ∂Ω.

So that, by (4.7) and using [23, Lemma 2.6], we find

vn(x) ≥ λ
1

1−q z(x) ≥ Cd(x) > 0, (4.10)

where C is a positive constant. Furthermore, using the Hardy-Sobolev inequality,
Lemma 2.4, we deduce that vn/d

γ ∈ Lσ(Ω) with 1
σ = 1

2 −
1−γ
N , and

‖vn
dγ
‖Lσ(Ω) ≤ C‖∇vn‖L2(Ω).

Using the estimate ‖vn‖H1
0 (Ω) ≤ r, we obtain ‖vndγ ‖Lσ(Ω) ≤ Cr and so, by (4.10)

and Hölder’s inequality, we have∫
Ω

a(x)(vn)s(vn + 1/
√
n)−(γ+s) ≤ K1

∫
Ω

vn
(vn)γ

≤
∫

Ω

vn
Cdγ

≤ C1

(∫
Ω

( vn
Cdγ

)σ )1/σ

≤ C,
(4.11)

where C = C1r is a constant independent of n.



14 JESUS A. L. TORDECILLA EJDE-2022/40

Recall from (3.28) that

M(‖vn‖2H1
0 (Ω))

∫
Ω

∇vn∇w

= λ

∫
Ω

(
a(x)(vn)s(vn + 1/

√
n)−(γ+s) + (vn)q

)
w

+

∫
Ω

fn(vn)w +
1

n

∫
Ω

φw, ∀w ∈ H1
0 (Ω).

(4.12)

Talking w = vn in (4.12) and since vn is bounded in H1
0 (Ω), by the Sobolev compact

embedding we obtain (4.9).
By (4.8), (4.9), and the expression of fn in (2.1), the assumptions of Lemma 2.5

are satisfied, implying f(v) ∈ L1(Ω) and

fn(vn)→ f(v) in L1(Ω).

Furthermore, since vn → v a.e. in Ω, from (4.11) and the dominated converge
theorem we have∫

Ω

a(x)(vn)s(vn + 1/
√
n)−(γ+s)w →

∫
Ω

a(x)v−γw, ∀w ∈ H1
0 (Ω).

Note that, by (4.8), we have v(x) ≥ Cd(x) a.e. in Ω and in addition it follows from
the Hardy-Sobolev inequality that v−γw ∈ L1(Ω), since 0 < γ < 1.

Finally, letting n→ 1 in (4.12), we have

M(t̃0)

∫
Ω

∇v∇w = λ

∫
Ω

(a(x)v−γ + vq)w +

∫
Ω

fn(v)w, ∀w ∈ H1
0 (Ω). (4.13)

Observe that, similarly to Lemma 3.1 we can show that t̃0 = ‖v‖2
H1

0 (Ω)
. Thus, we

conclude from (4.13) that v ∈ H1
0 (Ω) is a positive solution to problem (1.1). This

completes the proof. �

5. Symmetric, monotonicity, and asymptotic behavior of the
solutions

In this section we show Propositions 1.2 and 1.4.

Proof of Proposition 3.3. Since vn → v a.e. in Ω, where v ∈ H1
0 (Ω) is a solution of

(1.1), by Lemma 3.3, letting n → ∞, we have v(x1, x
′) ≤ v(−x1, x

′) for every x =
(x1, x

′) ∈ B% such that x1 > 0. Similarly, we may show that v(−x1, x
′) ≤ v(x1, x

′).
Furthermore,

∂u

∂x1
< 0 for every x ∈ B% with x1 > 0.

Therefore, v is symmetric with respect to the hyperplane x1 = 0 and decreasing in
the direction x1 with x1 > 0, where x = (x1, x

′) ∈ B%. �

Remark 5.1. If we consider f instead of fn, we cannot apply Theorem 2.3 because
the function g(t) = λtq + f(t) is not necessarily Lipschitz continuous.

Proof of Proposition 1.4. It follows from definition of solution to problem (1.1),
considering ϕ = u as a test function, that

M(‖u‖H1
0 (Ω))

∫
Ω

|∇u|2 = λ

∫
Ω

a(x)(u1−γ + uq+1) +

∫
Ω

fn(u)u

≤ λ(C3‖u‖1−γH1
0 (Ω)

+ C4‖u‖q+1
H1

0 (Ω)
) + C1‖u‖p+1

H1
0 (Ω)

,

(5.1)



EJDE-2022/40 NONLOCAL ELLIPTIC PROBLEMS 15

where C1 is give by (3.8).
Since u 6= 0, it follows from (5.1) that

‖u‖1+γ
H1

0 (Ω)

[
m0 − C1‖u‖p−2

H1
0 (Ω)

]
≤ λ(C3 + C4‖u‖q+γH1

0 (Ω)
). (5.2)

On the other hand, by the choice of r in Lemma 3.1 we deduce that

m0 − C1‖u‖p−2
H1

0 (Ω)
≥ m0 − C1r

p−2 ≥ 2.

Then, from (5.2) we find that ‖u‖H1
0 (Ω) ≤ Cλ

1
1+γ . Therefore ‖u‖H1

0 (Ω) → 0 as

λ→ 0+. The proof is complete. �

Acknowledgments. The author express their gratitude to the referees for their
helpful suggestions.

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Jesus Alberto Leon Tordecilla

Universidade Estadual de Campinas, IMECC, Departamento de Matematica, Rua Sergio

Buarque de Holanda, 651, Campinas, SP, Brazil, CEP 13083-970
Email address: jleontordecilla@gmail.com


	1. Introduction
	2. Auxiliary results
	3. Approximate problem in a finite dimensional space
	4. Proof of Theorem 1.1
	5. Symmetric, monotonicity, and asymptotic behavior of the solutions
	Acknowledgments

	References