CUBO A Mathematical Journal
Vol.15, No¯ 02, (21–31). June 2013

Nonnegative solutions of quasilinear elliptic problems with
sublinear indefinite nonlinearity1

Weihui Wang
a
and Zuodong Yang

a,b

a Institute of Mathematics, School of Mathematical Sciences,

Nanjing Normal University,

Jiangsu Nanjing 210046, China.

b College of Zhongbei,

Nanjing Normal University,

Jiangsu Nanjing 210046, China.

335348332@qq.com zdyang jin@263.net

ABSTRACT

We study the existence, nonexistence and multiplicity of nonnegative solutions for the

quasilinear elliptic problem

{
− △p u = a(x)u

q + λb(x)ur, in Ω

u = 0, on ∂Ω

where Ω is a bounded domain in RN, λ > 0 is a parameter, △p = div(|∇u|
p−2∇u)

is the p−Laplace operator of u, 1 < p < N, 0 < q < p − 1 < r ≤ p∗ − 1, a(x), b(x)

are bounded functions, the coefficient b(x) is assumed to be nonnegative and a(x) is

allowed to change sign. The results of the semilinear equations are extended to the

quasilinear problem.

1Project Supported by the National Natural Science Foundation of China(Grant No.11171092). Project
Supported by the Natural Science Foundation of the Jiangsu Higher Education Institutions of China (Grant
No.08KJB110005)



22 Weihui Wang and Zuodong Yang CUBO
15, 2 (2013)

RESUMEN

Estudiamos la existencia, no existencia y multiplicidad de soluciones no negativas del

problema eĺıptico cuasi-lineal

{
− △p u = a(x)u

q + λb(x)ur, in Ω

u = 0, on ∂Ω

donde Ω es un dominio acotado en RN, λ > 0 es un parámetro, △p = div(|∇u|
p−2∇u)

es el operador p−Laplaciano de u , 1 < p < N, 0 < q < p − 1 < r ≤ p∗ − 1, a(x), b(x)

son funciones acotadas, el coeficiente b(x) se supone que es no negativo y a(x) se le

permite cambiar de signo. Los resultados de las ecuaciones semilineales se extienden a

el problema cuasi-lineal.

Keywords and Phrases: Nonnegative solutions; quasilinear elliptic problems; sublinear indefi-

nite nonlinearity; Existence and nonexistence.

2010 AMS Mathematics Subject Classification: 35J50, 35J55, 35J60.



CUBO
15, 2 (2013)

Nonnegative solutions of quasilinear elliptic problems ... 23

1 Introduction

Let us consider the problem
{

− △p u = a(x)u
q + λb(x)ur, in Ω

u = 0, on ∂Ω
(Pλ)

where Ω ⊂ RN is a smooth bounded domain, λ > 0, 1 < p < N, 0 < q < p − 1 < r ≤ p∗ − 1,

p∗ =
Np
N−p

, b(x) ≥ 0, a(x) change its sign, △p = div(|∇u|
p−2∇u) is the p−Laplace operator of u.

Equations of the above form are mathematical models occuring in studies of the p-Laplace equation,

generalized reaction-diffusion theory([7]), non-Newtonian fluid theory, and the turbulent flow of

a gas in porous medium([8]). In the non-Newtonian fluid theory, the quantity p is characteristic

of the medium. Media with p > 2 are called dilatant fluids and those with p < 2 are called

pseudoplastics. If p = 2, they are Newtonian fluids.

Recently, A.V.Lair and A.Mohammed in [11] considered the existence and nonexistence of

positive entire large solutions of the semilinear elliptic equation

△u = p(x)uα + q(x)uβ, 0 < α ≤ β.

Francisco in [1] considered a sublinear indefinite nonlinearity problem of the form

{
−△u = a(x)uq + λb(x)up, in Ω

u = 0, on ∂Ω

where Ω is a smooth bounded domain in RN, λ ∈ R, 0 < q < 1 < p < r ≤ 2∗ − 1, b(x) ≥ 0,

a(x) change its sign. For more results we refer the reader to the works [12-15] and the references

therein.

In recent years, the existence and uniqueness of the positive solutions for the single quasilinear

elliptic equation with eigenvalue problems
{

div(|∇u|p−2∇u) + λf(u) = 0 in Ω,

u(x) = 0 ∂Ω,
(1.1)

with λ > 0, p > 1, Ω ⊂ RN, N ≥ 2 have been studied by many authors, see [16-23] and the

references therein. When f is strictly increasing on R+, f(0) = 0, lims→0+ f(s)/s
p−1 = 0 and

f(s) ≤ α1 + α2s
µ, 0 < µ < p − 1, α1, α2 > 0, it was shown in [16] that there exist at least two

positive solutions for Eqs (1.1) when λ is sufficiently large. If lims→0+ inf f(s)/s
p−1 > 0, f(0) = 0

and the monotonicity hypothesis (f(s)/sp−1)′ < 0 holds for all s > 0. It was also shown in [17]

that problem (1.1) has a unique positive large solution and at least one positive small solution

when λ is large if f is nondecreasing; there exist α1, α2 > 0 such that f(s) ≤ α1 + α2s
β, 0 < β <

p − 1; lims→0+
f(s)

sp−1
= 0, and there exist T, Y > 0 with Y ≥ T such that

(f(s)/sp−1)′ > 0 for s ∈ (0, T)



24 Weihui Wang and Zuodong Yang CUBO
15, 2 (2013)

and

(f(s)/sp−1)′ < 0 for s > Y.

Yang and Xu in [10] established the existence for quasilinear elliptic equation






−△pu = a(x)(u
m + λun), x ∈ RN

u > 0, x ∈ RN

u → 0, |x| → ∞

(1.2)

where 0 < m < p − 1 < n, they proved there exists a λ∗ > 0 such that (1.2) has a positive solution

for 0 < λ < λ∗.

The quasilinear elliptic equations when a(x) ≡ b(x) ≡ 1 was considered in [2], although here

under some restrictions on the p, q in the critical case r = p∗−1. Problems of local ”superlinearrity”

and ”sublinearity” for the p− Laplace problem was considered in [3]. A class of quasilinear elliptic

equations are study in [4]. For more results we refer the reader to the works [5-6] and the references

therein.

Motivated by the results of the above papers. In this paper, we consider the quasilinear elliptic

equations (Pλ). We modify the method developed Francisco Odair de Paiva in [1] and extend the

results a quasilinear elliptic equation (Pλ), and complement results in [2-4, 10].

The paper is organized as follows. In section 2, we recall some facts that will be needed in the

paper, and give the main results. In section 3, we give the proofs of the main results in this paper.

2 Main results and Preliminary

Let us first consider the following parameterized elliptic problems






−△pu = a(x)u
q + λb(x)ur, in Ω

u ≥ 0, in Ω

u = 0, on ∂Ω

(Qλ)

where Ω is a bounded domain in RN, λ > 0 is a parameter, 1 < p < N, 0 < q < p − 1 < r ≤

p∗ − 1, a(x), b(x) are bounded functions, the coefficient b(x) is assumed to be nonnegative and

a(x) is allowed to change sign. Because that a(x) changes sign in Ω, so the Maximum principal is

not applicable. Then, define

Fλ(u) =
1

p

∫

Ω

|∇u|p −
1

q + 1

∫

Ω

a(x)(u+)q+1 −
λ

r + 1

∫

Ω

b(x)(u+)r+1, u ∈ W
1,p
0

(Ω)

We know that Fλ(u) is well define in W
1,p
0

(Ω) and is of C10(Ω)



CUBO
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Nonnegative solutions of quasilinear elliptic problems ... 25

Definition 2.1. We call u ∈ W
1,p
0

(Ω) is a weak solution of (Qλ), if u is a critical points of

Fλ(u).

Throughout this paper, we always suppose that

(H1) There exist λ > 0,a smooth subdomain B1 ∈ Ω
+
a , m(x) ∈ L

∞(B1) with m(x) ≥ 0, m(x) 6≡

0, µ > λ1(B1, m(x)) such that

a(x)sq + λb(x)sr ≥ µm(x)sp−1

for a.e.x ∈ B1 and all s ≥ 0; here λ1(B1, m(x)) denotes the principal eigenvalue of −△p on W
1,p
0

(B1)

for the weight m(x).

(H2) For any λ > 0, there exists a smooth subdomain B2 ⊂ Ω
+
a ,s1 > 0 and θ1 > λ1(B2),

such that

a(x)sq + λb(x)sr ≥ θ1s
p−1

for a.e. x ∈ B2, and all s ∈ [0, s1]; here λ1(B2) denotes the principal eigenvalue of −△p on W
1,p
0

(Ω)

(F1) a(x), b(x) ∈ L
∞(Ω), and

Ωa = {x ∈ Ω : a(x) ≥ 0}, Ω
+
a = {x ∈ Ω : a(x) > 0}

Ω−a = {x ∈ Ω : a(x) < 0}, Ω
+
b = {x ∈ Ω : b(x) > 0}

are nonempty;

(F2) Ω
+
a is open, |Ω

−
a | > 0 and Ω

+
a

⋂
Ω−a = ∅;

(F3) int(Ω
+
b
) 6= ∅ and b ≥ 0;

(F4) Ω
+
a ⊂ Ω

+
b
and Ω+a ⊂ Ω;

(F5) int(Ωa) =
⋃k

1
Ui, Ui connected, and Ui

⋂
Ω+a 6= ∅.

As a consequence of assumption (F5), by the Maximum principle, if u is a solution of (Qλ)

such that u is nontrivial in the components of Ωa, then u > 0 in int(Ωa) ⊃ Ω
+
a .

Definition 2.2. If u is a weak solution of (Qλ) and u(x) > 0, a.e.x ∈ Ω
+
a , then u ∈ W

1,p
0

(Ω)

is a solution of (1.1).

Let

λ∗ = sup{λ > 0; (1.1) has a solution}.

By a modification of the method given in [1], we obtain the following main results.

Theorem 2.1. Let 0 < q < p − 1 < r ≤ p∗ − 1. Assume that (F1) − (F5) hold, then there

exists λ∗ ∈ (0, ∞) such that

(1) for all λ ∈ (0, λ∗), problem (Pλ) has at least one weak solutions;

(2) for λ = λ∗, problem (Pλ) has at least one solution;



26 Weihui Wang and Zuodong Yang CUBO
15, 2 (2013)

(3) for all λ > λ∗, problem (Pλ) has no solution.

Theorem 2.2. Let 0 < q < p − 1 < r < p∗ − 1. Assume that (F1) − (F5) hold, then problem

(Pλ) has at least two solutions for 0 < λ < λ
∗.

3 The proof of main results

Lemma 3.1. There is λ0 > 0 such that for 0 < λ ≤ λ0, problem (Pλ) has a solution.

Proof. Let e be the unique positive solution of

{
−△pe = 1, in Ω;

e = 0, on ∂Ω.

Since 0 < q < p−1 < r, we can find λ0 > 0 such that for all 0 < λ ≤ λ0 there exists M = M(λ) > 0

satisfying

Mp−1 ≥ Mq‖a‖∞‖e‖
q
∞

+ λMr‖b‖∞‖e‖
r
∞
.

As a consequence, the function Me satisfies

−△p(Me) = M
p−1 ≥ Mq‖a‖∞‖e‖

q
∞

+ λMr‖b‖∞‖e‖
r
∞
.

Hence Me is a supersolution of (Pλ). Then let u = Me, we have that u is a supersolution

for (Qλ). Moreover 0 is a solution of (Qλ), so let u = 0 is a subsolution for (Qλ). It follows

form the sub-supersolution argument as in [5] or [6] that (Qλ) has a nonnegative solution in

A = {u ∈ W
1,p
0

: 0 ≤ u(x) ≤ Me a.e.x ∈ Ω}. Then let c = infA Fλ,

Fλ(u) =
1

p

∫

Ω

|∇u|p −
1

q + 1

∫

Ω

a(x)(u+)q+1 −
λ

r + 1

∫

Ω

b(x)(u+)r+1, u ∈ w
1,p
0

(Ω),

there exist uλ ∈ A such that c = infA Fλ(uλ) and uλ is a solution of (Qλ). Also uλ solves (Pλ) if

uλ > 0 a.e.x ∈ Ω
+
a .

By contradiction, suppose that uλ ≡ 0 a.e.x ∈ Ω
+
a , let ϕ ∈ C

∞

c (Ω
+
a ) be nonnegative and

nontrivial, then for sufficiently small s > 0, uλ + sϕ ∈ A

Fλ(uλ + sϕ) = Fλ(uλ) + Fλ(sϕ)

= Fλ(uλ) +
sp

p
‖ϕ‖p −

sq+1

q + 1

∫

Ω

a(x)ϕq+1 −
λsr+1

r + 1

∫

Ω

b(x)ϕr+1

Then we have Fλ(uλ + sϕ) < Fλ(uλ), if s > 0 is small enough, however this contradicts that the

infimum c = inf Fλ is achieve at uλ . So uλ > 0 a.e.x ∈ Ω
+
a and is a solution of (Pλ).

Lemma 3.2. (Pλ) has a solution for all λ ∈ (0, λ
∗).



CUBO
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Nonnegative solutions of quasilinear elliptic problems ... 27

Proof. Given λ < λ∗, let u
λ
be a solution of (P

λ
), with λ < λ < λ∗. Then

− △p uλ = a(x)u
q

λ
+ λb(x)ur

λ
≥ a(x)u

q

λ
+ λb(x)ur

λ
,

which u
λ
is a supersolution for (Pλ).

Consider A = {u ∈ W
1,p
0

: 0 ≤ u ≤ u
λ
}, there exist uλ ∈ A such that Fλ(uλ) = infA Fλ, and

uλ is a solution of (Qλ), as the proof of Lemma 3.1, uλ is also the solution of (Pλ).

Lemma 3.3. Let λ∗ = sup{λ > 0 : (Pλ) has a solution}, then 0 < λ
∗ < ∞.

Proof. Under the assume (H1), suppose that when λ > 0, (Pλ) has a solution uλ ∈

W
1,p
0

(Ω)
⋂

L∞(Ω). Consider the eigenvalue problem with weight
{

− △p v = µm(x)|v|
p−2, in B1;

v = 0, on ∂B1.

Since by (H1), we have
∫

B1

| ▽ uλ|
p−2 ▽ uλ ▽ ϕ =

∫

B1

(a(x)u
q
λ
+ λb(x)urλ)ϕ ≥ µ

∫

B1

m(x)u
p−1
λ

ϕ

for all ϕ ∈ C∞c (Ω), ϕ ≥ 0. This show that uλ is an supersolution of (Eµ). Furthermore, ǫϕ1 is a

subsolution of (Eµ), and ǫϕ1 ≤ uλ for ǫ small enough.
∫

B1

| ▽ (ǫϕ1)|
p−2 ▽ (ǫϕ1) ▽ ϕ = λ1

∫

B1

m(x)(ǫϕ1)
p−1ϕ < µ

∫

B1

m(x)(ǫϕ1)
p−1ϕ

for ϕ ∈ C∞c (Ω), ϕ ≥ 0; ϕ1 is a positive eigenfunction associated to λ1(B1, m(x)). Then (Eλ) has a

solution v with ǫϕ1 ≤ v ≤ uλ,in particular v ≥ 0, v 6≡ 0. For above that µ is a principal eigenvalue

of −△p u on B for the weight m(x). This is contradiction with µ > λ1(B1, m(x)), and consequently

λ∗ < +∞, moreover we can also obtain λ∗ > 0 to the Lemma 4.1, so, λ∗ ∈ (0, ∞). Hence, when

λ > λ∗, problem (Pλ) has no solution.

Lemma 3.4. For λ = λ∗, problem (Pλ) has at least one solution.

Proof. For the definition of λ∗, let λn be a sequence such that λn −→ λ
∗ with 0 < λn < λ

∗,λn
increasing, let un be a solution of Pλn with Fλn(un) < 0 and F

′

λn
(un) = 0. We obtain

Fλn(un) + F
′

λn
(un) · un ≤ C‖un‖,

where

Fλn(un) =
1

p

∫

Ω

|∇un|
p −

1

q + 1

∫

Ω

a(x)(u+n)
q+1 −

λn

r + 1

∫

Ω

b(x)(u+n)
r+1,

F
′

λn
(un) · un =

∫

Ω

|∇un|
p −

∫

Ω

a(x)(u+n)
q+1 − λn

∫

Ω

b(x)(u+n)
r+1

so by Theorem 1.2.1 of [9], we have

(
1

p
+ 1)‖un‖

p ≤ C‖un‖
q+1 + c.



28 Weihui Wang and Zuodong Yang CUBO
15, 2 (2013)

It shows that un is bounded in W
1,p
0

, we have, for a subsequence, un −→ u
∗ in C1(Ω), hence u∗

solves (Qλ) in Ω. u
∗ is a solution of (Pλ)) if u

∗ 6≡ 0 in Ω+a . Assume by contradiction u
∗ ≡ 0 in

Ω+a . Under the assume (H2), we have

∫

B2

| ▽ un|
p−2 ▽ un ▽ ϕ =

∫

B2

(a(x)uqn + λnb(x)u
r)ϕ ≥ θ1

∫

B2

up−1n ϕ

for n sufficiently large(so that 0 ≤ un(x) ≤ s1 on B2,which is possible since un −→ 0 uniformly).

So that un is a supersolution for the problem

{
− △p v = θ1|v|

p−2v, in B2;

v = 0, on ∂B2.

Moreover, since θ1 > λ1, let uε = εϕ1. We have

− △p (uε) = λ1u
p−1
ε < θ1u

p−1
ε

and εϕ1 ≤ un on B2, for (ε > 0 sufficiently small). It shows that the existence of a solution v of

(Eθ1) with εϕ1 ≤ v ≤ un. This is a contradiction with θ1 > λ1 in assume (H2). So, u
∗ 6≡ 0 in Ω+a

and is a solution of (Pλ).

Proof of Theorem 2.2. From the Lemma 3.2, we have obtained uλ is a local minimizer of

Fλ(u) and is a solution of (Pλ). In this section, we hope to find the second solution of the form

v = uλ + u, by the moutnain pass theorem,where u is a nonnegative solution of

{
− △p (uλ + u) = a(x)(uλ + u

+)q + λb(x)(uλ + u
+)r, in Ω;

u = 0, on ∂Ω.

u ∈ W
1,p
0

(Ω), and u ≥ 0. Then, uλ + u is a second solution of (Pλ). Define the associated

functional

Iλ(u) =
1

p

∫

Ω

|∇(uλ + u)|
p −

∫

Ω

Hλ(x, u)

Hλ(x, u) = Gλ(x, uλ + u
+) − Gλ(x, uλ) − gλ(x, uλ)u

+;

Gλ(x, u) =

∫

Ω

gλ(x, u)du; gλ(x, u) = a(x)u
q + λb(x)ur.

Then, it follows that

Iλ(u) =
1

p

∫

Ω

|∇(uλ + u)|
p −

1

q + 1

∫

Ω

a(x)[(uλ + u
+)q+1 − u

q+1
λ

− (q + 1)u
q
λ
u+]

−
λ

r + 1

∫

Ω

b(x)[(uλ + u
+)r+1 − ur+1λ − (r + 1)u

r
λu

+]

(i) let u+ ∈ W
1,p
0

(Ω+a ), and for ‖u
+‖ sufficiently small, we have

Iλ(u) ≥
1

p

∫

Ω

|∇(uλ + u)|
p −

1

p

∫

|Ω∇(uλ + u
+)|p +

1

p

∫

Ω

|∇uλ|
p +

∫

Ω

gλ(x, uλ)u
+



CUBO
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Nonnegative solutions of quasilinear elliptic problems ... 29

then,

Iλ(u) ≥
1

p

∫

Ω

|∇uλ|
p +

∫

Ω

gλ(x, uλ)u
+ ≥

1

p

∫

Ω

|∇uλ|
p = Iλ(0)

(ii) let v1 ∈ W
1,p
0

(Ω+
b
), v1 ≥ 0, v1 6≡ 0,such that

∫
Ω
b(x)vr+1

1
> 0. We have, for large s

Iλ(sv1) =
1

p

∫

Ω

|∇(uλ + sv1)|
p −

1

q + 1

∫

Ω

a(x)[(uλ + sv1)
q+1 − u

q+1
λ

− (q + 1)u
q
λ
sv1]

−
λ

r + 1

∫

Ω

b(x)[(uλ + sv1)
r+1 − ur+1λ − (r + 1)u

r
λsv1]

=
sp

p

∫

Ω

|∇(
uλ

s
+ v1)|

p −
sq+1

q + 1

∫

Ω

a(x)[(
uλ

s
+ v1)

q+1 − (
uλ

s
)q+1 −

(q + 1)u
q
λ
v1

sq
]

−
λsr+1

r + 1

∫

Ω

b(x)[(
uλ

s
+ v1)

r+1 − (
uλ

s
)r+1 −

(r + 1)urλv1

sr
]

= O(sp) −
λsr+1

r + 1

∫

Ω

b(x)vr+11 −→ −∞

as s −→ ∞.

(iii) We now prove Iλ(u) satisfies the (PS) condition in W
1,p
0

(Ω). Indeed, if uk is a (PS)

sequence, i.e. Iλ(uk) −→ c, I
′

λ(uk) −→ 0. Then, for p < θ < r + 1,εk −→ 0, and some constant c,

we have,

θIλ(uk) − I
′

λ(uk) · uk ≤ c + εk‖uk‖

where ‖uk‖ denotes the W
1,p
0

(Ω) norm (
∫
Ω
|∇u|p)

1
p .

(
θ

p
− 1)‖uk‖

p ≤ (
θ

q + 1
− 1)

∫

Ω

a(x)u
q+1
k

+ λ(
θ

r + 1
− 1)

∫

Ω

b(x)ur+1k + c + εk‖uk‖

(
θ

p
− 1)‖uk‖

p + λ(1 −
θ

r + 1
)

∫

Ω

b(x)ur+1k ≤ (
θ

q + 1
− 1)

∫

Ω

a(x)u
q+1
k

+ c + εk‖uk‖

By a(x), b(x) is bounded in Ω, we obtain,

(
θ

p
− 1)‖uk‖

p + c2λ(1 −
θ

r + 1
)‖uk‖

r+1 ≤ c1(
θ

q + 1
− 1)‖uk‖

q+1 + c + εk‖uk‖

since q + 1 < p < r + 1, this implies that the sequence (uk) be bounded in W
1,p
0

(Ω). Thus, from

(i)-(iii), Iλ satisfies the assumptions of the mountain pass theorem,i.e.Iλ has a nontrivial critical

point. This concludes the proof of Theorem 2.2.

Received: February 2012. Accepted: October 2012.



30 Weihui Wang and Zuodong Yang CUBO
15, 2 (2013)

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