Special Issue in honor of Alan C. Lazer

Electronic Journal of Differential Equations, Special Issue 01 (2021), pp. 293–300.

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

EXISTENCE AND MULTIPLICITY RESULTS FOR

p-q-LAPLACIAN BOUNDARY VALUE PROBLEMS

ANANTA ACHARYA, UJJAL DAS, RATNASINGHAM SHIVAJI

Dedicated to the living memory of Alan C. Lazer

Abstract. We study positive solutions to the boundary value problem

−∆pu−∆qu = λf(u) in Ω,

u = 0 on ∂Ω,

where q ∈ (1, p) and Ω is a bounded domain in RN , N > 1 with smooth bound-

ary, λ is a positive parameter, and f : [0,∞) → (0,∞) is C1, nondecreasing,
and p-sublinear at infinity i.e. limt→∞ f(t)/tp−1 = 0. We discuss existence

and multiplicity results for classes of such f . Further, when N = 1, we discuss

an example which exhibits S-shaped bifurcation curves.

1. Introduction

In [12], the authors studied the the p-Laplacian boundary value problem

−∆pu = λf(u) in Ω,

u = 0 on ∂Ω
(1.1)

where p > 1, Ω is a bounded domain in RN , N > 1 with smooth boundary, λ is a
positive parameter, ∆pu = div(|∇u|p−2∇u), p > 1 and f : [0,∞)→ (0,∞) satisfies

(H1) f is a C1 non-decreasing, p-sublinear function at infinity, i.e.

lim
t→∞

f(t)

tp−1
= 0.

In particular, they established the existence of a positive solution for each λ, and
when there exists 0 < a < b such that

Q(a, b) :=
ap−1/f(a)

bp−1/f(b)
� 1,

they obtained multiple positive solutions for certain range of λ. When p = 2, these
results were discussed in [4]. Their study was motivated by the applications in
chemical reaction theory (see [2]) and in combustion theory (see [11, 14]).

2010 Mathematics Subject Classification. 35J70, 35J55.

Key words and phrases. Multiple positive solutions; p-q Laplacian; sub-super solution.
©2021 This work is licensed under a CC BY 4.0 license.
Published January 18, 2022, 2021.

293



294 A. ACHARYA, U. DAS, R. SHIVAJI EJDE/SI/01

In this article, we extend this study to the p-q-Laplacian boundary value problem

−∆pu−∆qu = λf(u) in Ω,

u = 0 on ∂Ω,
(1.2)

for q ∈ (1, p). Namely, we establish

Theorem 1.1. Assume (H1). Then the following results hold:

(1) Equation (1.2) has a positive solution for each λ > 0.
(2) There exists positive constants C0 = C0(Ω), C1 = C1(p,Ω, N), and C2 =

C2(Ω) such that if b > C0 and

Q(a, b) =
a/(f(a))p−1

b/(f(b))p−1
>
C1

C2
,

for some points a and b, a < b, then (1.2) has at least two positive solutions

for λ ∈
(
bp−1

f(b) C1,
ap−1

f(a) C2

]
.

As in [4] and [12] we use the method of sub-super solutions to establish our
results. In [4], the availability of Green’s function played an important role in the
construction of a positive strict sub-solution that was used to establish a multiplicity
result. In [12], the authors had to develop another idea to construct this special
sub-solution because of the lack of a Green’s function for the p-Laplacian operator
for p 6= 2. Here we adapt and extend the ideas used in [12] to establish our results.
However, unlike in [12], our multiplicity result is restricted to only two solutions.
In [12], the authors used results in [1, 15] to guarantee three solutions. If the
results in [1, 15] can be extended to the p-q Laplacian case, our construction of
sub-super solutions will also yield at least three positive solutions for the range of
λ in Theorem 1.1 part (2)

A time-dependent version of an operator such as in (1.2) often occurs in the
mathematical modeling of chemical reactions and plasma physics. In recent years,
a lot of attention has been given to study the boundary value problem involving
p-q Laplacian, see for instance [3, 5, 10, 13] and the references therein.

The rest of this article is organized as follows. In Section 2, we recall some
important results that are required for the development of this article. In Section
3 we prove of Theorem 1.1. In Section 4 we provide an application of our results.
Finally in Section 5, we obtain exact bifurcation diagrams for the case when Ω =
(0, 1), p = 4 and q = 2, namely to the two-point boundary value problem

−[(u′)3]′ − µ[(u′)]′ = λf(u); (0, 1)

u(0) = 0 = u(1)
(1.3)

where f(s) = exp ( γs
γ+s ), γ > 0, and µ is a non-negative parameter.

2. Preliminaries

In this section, we recall some results concerning a sub-super solution method
for p-q-Laplacian boundary value problem. First, by a weak solution of (1.2) we

mean a function u ∈W 1,p
0 (Ω) which satisfies∫

Ω

|∇u|p−2∇u · ∇φ+

∫
Ω

|∇u|q−2∇u · ∇φ = λ

∫
Ω

f(u)φ, ∀φ ∈ C∞0 (Ω).



EJDE-2021/SI/01 p-q-LAPLACIAN BOUNDARY VALUE PROBLEMS 295

However, in this article, we in fact study C1(Ω) solution. Next, by a sub-solution
(super solution) of (1.2) we mean a function v ∈W 1,p(Ω)∩C(Ω) such that v ≤ (≥)0
on ∂Ω and satisfies∫

Ω

|∇v|p−2∇v · ∇φ+

∫
Ω

|∇v|q−2∇v · ∇φ ≤ (≥)λ

∫
Ω

f(v)φ,

for all φ ∈ C∞0 (Ω) and φ ≥ 0 in Ω. Then the following sub-super solution result
holds.

Lemma 2.1. Let ψ, z be sub and super solutions of (1.2) respectively such that
ψ ≤ z in Ω. Then (1.2) has a solution u ∈ C1(Ω) such that ψ ≤ u ≤ z.

For a proof of the above lemma see [7, Corollary 1].

3. Proof of Theorem 1.1

In this section we use sub-super solution method to prove Theorem 1.1. At first
we prove the results when Ω = BR, a ball of radius R and centered at origin in RN .
We adopt and extend the ideas presented in [12] to construct a crucial sub-solution
on BR.

Construction of two sub-solutions on BR. Clearly φ1 = 0 is a sub-solution to
the problem (1.2). Now we construct another sub-solution. For that we consider
the function

v(r) =

{
1, r ≤ ε

1− [1− (R−rR−ε )β ]α, ε ≤ r ≤ R,
where ε ∈ (0, R), α > 1 and β > 1. Let us denote µ1(r) = R−r

R−ε and µ2(r) =

1 − (µ1(r))β . Taking ṽ(r) = bv(r) we note that |ṽ′(r)| ≤ bαβ/(R − ε). Now let ψ
be a radially symmetric solution of

−∆pψ −∆qψ = λf(ṽ(|x|)) in BR,

ψ = 0 on ∂BR.
(3.1)

Then ψ satisfies

−(rN−1Gp(ψ
′(r)))′ − (rN−1Gqψ

′(r))′ = λrN−1f(ṽ(r))

ψ′(0) = ψ(R) = 0,
(3.2)

where Gs(t) = |t|s−2t, s = p, q. Integrating the above equation over 0 < r < R we
obtain

−Gp(ψ′(r))−Gqψ′(r) =
λ

rN−1

∫ r

0

sN−1f(ṽ(s)) ds. (3.3)

Notice that G̃(t) := Gp(t) +Gq(t) is a continuous, monotone function. Hence, G̃−1

exists and is also continuous. Therefore, (3.3) yields

ψ′(r) = G̃−1
( λ

rN−1

∫ r

0

sN−1f(ṽ(s)) ds
)
. (3.4)

Next, we claim that ṽ(r) ≤ ψ(r), for 0 ≤ r ≤ R. If this claim is true, then ψ is a
sub-solution as f is nondecreasing. Now, since ψ(R) = v(R) = 0, it is sufficient to
show that ψ′(R) ≤ v′(R) for all 0 ≤ r ≤ R. Observe that ψ′(r) = 0 for 0 ≤ r ≤ R
and ṽ′(r) = 0 for 0 ≤ r ≤ ε. Where as for r ≥ ε we have∫ r

0

sN−1f(ṽ(s)) ds ≥
∫ ε

0

sN−1f(ṽ(s)) ds ≥ f(b)
εN

N
.



296 A. ACHARYA, U. DAS, R. SHIVAJI EJDE/SI/01

It follows from (3.4) that

−ψ′(r) ≥ G̃−1
(
f(b)

λεN

NRN−1

)
.

Recall that |ṽ′(r)| ≤ bαβ
(R−ε) . Thus, ψ′(r) ≤ v′(r) if G̃−1(f(b) λεN

NRN−1 ) ≥ bαβ
R−ε i.e. if

f(b)
λεN

NRN−1
≥ G̃(

bαβ

R− ε
). (3.5)

Note that (3.5) will be satisfied if

f(b)
λεN

NRN−1
≥ max

{
2, 2Gp(

bαβ

R− ε
)
}
,

i.e. if

λ ≥ 2NRN−1

f(b)εN
max

{
1,
( bαβ
R− ε

)p−1}
. (3.6)

Let b > R. Then we can choose α ≈ 1, β ≈ 1 so that ( bαβR−ε ) > 1 and (3.6) will be
satisfied if

λ ≥ bp−1

f(b)
C1(αβ)p−1, (3.7)

where C1 = infε
2NRN−1

εN (R−ε)p−1 . In fact, this infimum is achieved at ε = ε0 = NR
N+p−1 ,

which will be our choice of ε. Assume that

λ >
bp−1

f(b)
C1 . (3.8)

Then clearly we can fix α(> 1) ≈ 1, β(> 1) ≈ 1 so that (3.7) holds. Hence, for
these choice of α, β, ε, ψ will be a sub-solution, when (3.8) is satisfied. Further,
since ψ ≥ ṽ, ‖ψ‖∞ ≥ b.
Construction of super solutions on BR. Let σ(r) = (1 − ( rR )p

′
)/p′ on BR,

where 1
p + 1

p′ = 1. Notice that 0 ≤ σ ≤ 1. Also that for 0 ≤ r ≤ R,

σ′(r) = −r
p′−1

Rp′
,

−(rN−1Gs(σ
′(r)))′ = −(rN−1|σ′(r)|s−2σ′(r))′ =

(rN−1r(p′−1)(s−1)

Rp′(s−1)

)′
≥ 0,

(3.9)

In particular, −(rN−1Gp(σ
′(r)))′ = NrN−1

Rp . Now let ξa = N
1

1−p aσ, where a as in
the assumption of Theorem 1.1. Then, since f is nondecreasing,

−(rN−1Gp(ξ
′
a(r)))′ − (rN−1Gq(ξ

′
a(r)))′ ≥ rN−1ap−1

Rp
≥ λrN−1f(ξa(r))

if λ ≤ ap−1

f(a)Rp . Thus, ξ̃a := ξa(|x|) satisfies

−∆pξ̃a −∆q ξ̃a ≥
ap−1

Rp
≥ λf(ξ̃a) if λ ≤ ap−1

f(a)Rp
. (3.10)

Hence, ξ̃a is a super solution when λ ≤ ap−1

f(a)Rp . Next for a given λ > 0, let

ξ̃λ = N
1

1−pM(λ)σ(|x|), where M(λ) >> 1 so that [m(λ)]p−1

f(M(λ)) ≥ λRp. Then, again

using f is nondecreasing we have

−∆pξ̃λ −∆q ξ̃λ ≥
M(λ)p−1

Rp
≥ λf(ξ̃λ), (3.11)



EJDE-2021/SI/01 p-q-LAPLACIAN BOUNDARY VALUE PROBLEMS 297

hence ξ̃λ is a super solution on BR.

Comparison of Sub-Sup solutions on BR. For any λ > 0, ψ1 ≡ 0 is a strict sub-

solution (as f(0) > 0) and z2 = ξ̃λ = N
1

1−pM(λ)σ(|x|) with M(λ) >> 1 is a super
solution. Hence, by Lemma 2.1, (1.2) has a positive solution for each λ > 0. Next

let λ ∈ ( b
p−1

f(b) C1,
ap−1

f(a) C2], where C2 = 1
Rp and 0 < a < b such that b > R = C0(Ω)

and Q(a, b) ≥ C1

C2
. For such λ, ψ1 ≡ 0 is a strict sub-solution, ψ2 = ψ (ψ as in

(3.1)) is a sub-solution, z1 = ξ̃a = N
1

1−p aσ(|x|) is a super solution (see (3.10)) and

z2 = ξ̃λ = N
1

1−pM(λ)σ(|x|) is a super solution. Hence, by Lemma 2.1, (1.2) has
two solutions u1, u2 for such λ > 0, where u1 ∈ [ψ1, z1] and u2 ∈ [ψ2, z2]. Note
that, u1 and u2 are distinct since ‖z1‖∞ ≤ a, ‖ψ2‖∞ ≥ b and a < b.

Now we proceed to prove our result for any open, bounded subset Ω of RN .

Proof of Theorem 1.1. Note that ψ̃1 = 0 still remains a sub-solution on Ω to (1.2)
on Ω. Now let BR be the largest inscribed ball inside Ω and we define

ψ̃2(x) =

{
ψ(x), if x ∈ BR
0, if x ∈ Ω \BR

where ψ is as in (3.1). Clearly, ψ̃2 ∈W 1,p
0 (Ω) and when λ > C1b

p−1/f(b) we have

−∆pψ̃2(x) = −∆pψ2(x) ≤ λf(ψ2(x)) = λf(ψ2(x)) on BR.

Also −∆pψ̃2(x) = 0 < λf(0) = λf(ψ2(x)) in Ω \ BR. Hence, ψ̃2 is a strict sub-

solution when λ > C1b
p−1/f(b). Also ‖φ̃2‖∞ ≥ b. Next, we consider a ball BR

containing Ω. By taking z1 = ξ̃a = N
1

1−p aσ(|x|) and z2 = ξ̃λ = N
1

1−pM(λ)σ(|x|) as
earlier (but now in ball BR) and taking their restrictions on Ω, it is easy to see that

z1 is a strict super solution (1.2) on Ω if λ ≤ ap−1

Rpf(a) , while z2 = ξ̃λ with M(λ) >> 1

is a super solution to (1.2) on Ω for any λ > 0. Noticing again ‖z1‖∞ ≤ a and
using Lemma 2.1, the proof of Theorem 1.1 follows in the general region Ω. �

Remark 3.1. Under assumption (H1), if f(s)/sq−1 is strictly decreasing for s > 0,
then (1.2) has a unique positive solution [8, Theorem 2.2].

4. Application in combustion theory

For 1 < q < p, we consider

−∆pu−∆qu = λ exp
( γu

γ + u

)
in Ω,

u = 0 on ∂Ω.
(4.1)

The reaction term f(s) = exp
(
γs
γ+s

)
, γ > 0 occurs in the theory of combustion and it

has been discussed in [4] (Laplacian case), and in [12] (p-Laplacian case). In [4], the
authors obtained that γ > 4 is a necessary condition for multiple positive solutions
for the Laplacian case; while in [12] the authors obtained the same for γ > 4(p−1) in
the p-Laplacian case. Here we present analogous result for p-q Laplacian. Towards

this we first notice that f̃(u) := f(u)
uq−1 is decreasing if γ ≤ 4(q − 1). Thus, Remark

3.1 ensures that γ > 4(q − 1) is a necessary condition for (4.1) to have multiple
solutions. Further, taking a = 1 and b = γ, we have

Q(a, b) :=
ap−1/f(a)

bp−1/f(b)
= γ(1−p) exp

(γ
2
− γ

γ + 1

)
.



298 A. ACHARYA, U. DAS, R. SHIVAJI EJDE/SI/01

Thus, for any C0, C1 and C2, we can choose γ large enough such that b > C0

Q(1, γ) > C1

C2
and hence, by Theorem 1.1, (4.1) admits at least two solutions at

least for certain range of λ.

5. Bifurcation diagram for positive solutions to (5.1)

Here we study the two-point boundary value problem

−[(u′)3]′ − µ[(u′)]′ = λf(u); (0, 1)

u(0) = 0 = u(1)
(5.1)

where f(s) = exp ( γs
γ+s ); γ > 0, and µ is a non-negative parameter. We will provide

the exact bifurcation diagram via a quadrature method and Mathematica com-
putations. We will also study how this bifurcation curve evolves when γ and µ
vary.

Here we use the quadrature method described in [6] which was obtained by
extending the method initially introduced in [9]. First we note that since (5.1) is
autonomous, any positive solution u must be symmetric about x = 1/2, increasing
on (0, 1/2), and decreasing on (1/2, 1). Assume u is a positive solution of (5.1) and
let u(1/2) = ρ.

Figure 1. Shape of a positive solution to (5.1)

Multiplying (5.1) by u′ and integrating we obtain

−3

4
[(u′)4]′ − µ

2
[(u′)2]′ = λ(F (u))′ in (0, 1),

where F (s) =
∫ s

0
f(z)dz. Further integrating we obtain

3[u′(x)]4 + 2µ[u′(x)]2 = 4λ[F (ρ)− F (u(x))] in [0,
1

2
],

and hence

u′(x) =

√
[µ2 + 12λ(F (ρ)− F (u(x)))]

1
2 − µ

√
3

in [0,
1

2
]. (5.2)

Integrating (5.2), we obtain∫ u(x)

0

ds√
[µ2 + 12λ(F (ρ)− F (s))]

1
2 − µ

=
x√
3

in [0,
1

2
), (5.3)



EJDE-2021/SI/01 p-q-LAPLACIAN BOUNDARY VALUE PROBLEMS 299

and setting x→
(

1
2

)−
, we obtain

G(λ, ρ) =

∫ ρ

0

ds√
[µ2 + 12λ(F (ρ)− F (s))]

1
2 − µ

=
1

2
√

3
. (5.4)

Inversely, if λ, ρ are such that (5.4) is satisfied, u(x) is defined via (5.3) for
x ∈ [0, 1

2 ), u(1/2) = ρ, and u(x) = u(1 − x) for x ∈ (1/2, 1], it follows that u will
be a positive solution of (5.1). Hence (5.4) determines the bifurcation diagram
of positive solutions for (5.1). Now, for f(s) = exp ( γs

γ+s ), we use Mathematica

computations to obtain the bifurcation diagram using (5.4).

Observations. For a given µ ≥ 0, there exists γ0(µ) such that for γ < γ0(µ), we
obtain a unique solution of (5.1) for all λ > 0, and for γ > γ0(µ) the bifurcation
curve is S−shaped with multiplicity in the region (λ1, λ2). Further, γ0(µ) decreases
in µ and λ1 decreases in γ (see Figure 2). Furthermore, strength of the multiplicity
range (i.e. the length (λ2 − λ1)) increases in γ (See Figure 3).

𝛾 = 10
𝛾 = 11
𝛾 = 12
𝛾 = 13
𝛾 = 14
𝛾 = 30

𝜇 = 100

µ = 0 µ = 100

Figure 2. Bifurcation diagrams of (5.1) for different values of γ
for given µ.

Strength of multiplicity range

Figure 3. Heat map showing the strength of multiplicity range
w.r.t γ and µ.



300 A. ACHARYA, U. DAS, R. SHIVAJI EJDE/SI/01

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Ananta Acharya
Department of Mathematics and Statistics, University of North Carolina at Greens-

boro, Greensboro, NC 27412, USA

Email address: a achary@uncg.edu

Ujjal Das

Technion-Israel Institute of Technology, Haifa-32000, Israel
Email address: ujjaldas@campus.technion.ac.il, ujjal.rupam.das@gmail.com

Ratnasingham Shivaji
Department of Mathematics and Statistics, University of North Carolina at Greens-

boro, Greensboro, NC 27412, USA

Email address: r shivaj@uncg.edu


	1. Introduction
	2. Preliminaries
	3. Proof of Theorem 1.1
	4. Application in combustion theory
	5. Bifurcation diagram for positive solutions to (5.1)
	Observations

	References