Electronic Journal of Differential Equations, Vol. 2021 (2021), No. 40, pp. 1–13.

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

EXISTENCE OF SIGN-CHANGING SOLUTIONS FOR RADIALLY

SYMMETRIC p-LAPLACIAN EQUATIONS WITH VARIOUS

POTENTIALS

WEI-CHUAN WANG

Abstract. In this article, we study the nonlinear equation(
rn−1|u′(r)|p−2u′(r)

)′
+ rn−1w(r)|u(r)|q−2u(r) = 0,

where q > p > 1. For positive potentials (w > 0), we investigate the existence

of sign-changing solutions with prescribed number of zeros depending on the

increasing initial parameters. For negative potentials, we deduce a finite in-
terval in which the positive solution will tend to infinity. The main methods

using in this work are the scaling argument, Prüfer-type substitutions, and

some integrals involving the p-Laplacian.

1. Introduction

The purpose of this article is to investigate some properties related to the radially
symmetric problem for

−∆pu = g(|x|, u), on Ω ⊆ Rn, (1.1)

where ∆pu = div(|∇u|p−2∇u), p > 1 and n ≥ 1. The p-Laplacian operator ∆pu
itself has the originally physical meaning, and also can be treated as a generalization
of the Laplacian operator. The quantity p is a characteristic of the medium of non-
Newtonian fluids or nonlinear diffusion problems. Media with p > 2 are called
dilatant fluids and those with p < 2 are called pseudoplastics. If p = 2, they are
Newtonian fluids. For the above, we refer the readers to [7, 8, 18, 22, 26, 31, 32, 33]
and their references. Also, some results for radial solutions related to (1.1) have
been obtained in [5, 6, 12, 13, 14, 15, 29, 25, 36, 37]. In [5], the authors extended
the eigenvalue theory to the radially symmetric p-Laplcian in Rn corresponding to
Weyl’s limit point and Weyl’s limit circle theories in the case p = 2. In [12], the
authors determined the structure of positive radial solutions related to (1.1). In
particular, Kabeya et al. [14] deduced the existence of radially fast-decay solutions
of (1.1) with prescribed number of zeros in (0,∞). They also considered further
boundary problems with similar results. More recently, the authors [6] derived the
existence of radial solutions having prescribed number of sign changes on (0,∞)
for n ≥ p > 1. Another direction is to deduce the existence of blow-up solutions.
A solution u of (1.1) is called boundary blow-up if limk→∞ u(xk) = ∞ for each

2010 Mathematics Subject Classification. 34A12, 34B15, 34A55.
Key words and phrases. Nonlinear p-Laplacian equation; sign-changing solution;

blow-up solution.
c©2021 Texas State University.

Submitted September 8, 2020. Published May 7, 2021.

1



2 W.-C. WANG EJDE-2021/40

sequence in Ω which converges to a point on ∂Ω. For p = 2 this type of problem
has a long history. Bieberbach [3] and Rademacher [27] started to study this theme.
Bieberbach was motivated by problems in geometry, Rademacher by a problem in
mathematical physics. Later, Keller [16] derived a well-known result. The author
gave necessary and sufficient conditions on the the growth of g(u) at infinity to
guarantee that such solutions exist. Under more restrictive assumptions, but for
general N -dimensional domains, the blow-up problem has been studied by Bandle
and Marcus [1, 2], Lazer and McKenna [19, 20] for p = 2 and by Diaz and Letelier
[9] for general p > 1. For a nonlinear radial p-Laplacian, Reichel and Walter [29]
employed a strong comparison principle to develop some properties of boundary
blow-up solutions. Also, McKenna et al. [25] have treated the radial case for
g(u) = |u|q and general p > 1. We mention a part of work in [25], where the
authors developed the existence of blow-up solutions for the one-dimensional case
(n = 1) first and applied a crucial inequality to obtain the existence of blow-up
solutions to the general case (n ≥ 1).

By considering radially symmetric solutions to (1.1), we are led to study the
nonlinear problem(

rn−1|u′(r)|p−2u′(r)
)′

+ rn−1w(r)|u(r)|q−2u(r) = 0, (1.2)

u(0) = α > 0, u′(0) = 0, (1.3)

where r = |x| and ′ = d
dr . Motivated by the previous results [6, 14, 17, 25, 35, 37],

we study two issues related to (1.2)-(1.3). When w > 0, we investigate the existence
of sign-changing solutions with the prescribed number of zeros in a finite interval.
For this issue, we consider a right endpoint condition

u(1) = 0 (1.4)

for the sake of simplicity. We denote the solution of (1.2)-(1.3) by u(r;α). The
following results (Theorem 1.1 and 1.3) can be treated as the Sturmian theory
which is related to the existence of solutions having prescribed number of zeros.
Some results closely relevant to this issue can be referred to [6, 14, 15, 35, 36, 37]
and their bibliographies. In this article we to consider a wide class of potential
functions and employ the interesting methods, scaling arguments and Prüfer-type
substitutions, to achieve the goal. The method seems to be classical and can be
found in [14, 35, 39]. However, this extension is not trivial and need more subtle
arguments in the analysis of generalized polar coordinates. Throughout this paper
we assume the following conditions:

(A1) q > p;
(A2) w ∈ C1(R), and |w| ≥ δ1 on [0,∞) for some δ1 > 0;

(A3) κ := max
{ |w′(r)|

w(r) : r ∈ [0, 1]
}

;

(A4) p > n, w1 := max{|w(r)| : r ∈ [0, 1]} and K := w1

(w(0)
δ1

eκ
) q−p

q .

Note that (A1)–(A3) hold for the existence of solutions with prescribed number of
zeros to the two-endpoint boundary condition case (Theorem 1.1). (A4) is added to
the case of multi-point boundary conditions (Theorem 1.3). Here is our first result.

Theorem 1.1. Assume that w > 0 in (1.2) and (A1)–(A3) hold. Then there exists
a strictly increasing sequence of positive numbers {αn}∞n=1, such that the solution
u(r;αn) is a solution to the BVP (1.2)-(1.4). Moreover, u(r;αn) has exactly n− 1
zeros in (0, 1) for n ∈ N.



EJDE-2021/40 RADIALLY SYMMETRIC p-LAPLACIAN EQUATIONS 3

Remark 1.2. The initial parameter corresponding to the solution with the pre-
scribed number of zeros is not unique usually. For the case of n = 1, the author
[34] showed that such a sequence is unique, provided w ∈ C2(R), ([w(r)]−1/p)′′ ≤ 0
on R and 1 < p ≤ 2.

Under the derivation of the existence of sign-changing solutions to the two-point
BVP (Theorems 1.1), we observe an application to the case of multi-point boundary
conditions. The existence of solutions, especially positive solutions, of boundary
value problems with multi-point boundary conditions have been studied extensively,
see, for example, [11, 23, 24, 28, 38]. Moreover, to the best of the author’s knowl-
edge, there is few work done so far on the existence of nodal solutions to problems
with the multi-point boundary conditions related to (1.2). Motivated by the idea in
[17, 37] and as a byproduct from the derivation of Theorem 1.1, we intend to extend
the Sturmian theory to the case of (1.2)-(1.3) coupled with the multi-point bound-
ary conditions. Here we consider a more general situation that the multi-point
conditions are dependent on the initial parameter. It is similar to Sturm-Liouville
problems coupled with eigendependent boundary conditions. We mention that the
Prüfer-type substitution is significant to derive the Sturmian theory. The Prüfer
phase is efficient to study the number of sign changes of solutions, and the Prüfer
radius estimate leads to satisfy the multi-point boundary condition. The details of
Prüfer-type substitutions will be discussed in Section 2. Now we impose

u(1)−
d∑
i=1

τie
− (α

q−p
p K)
p−1 u(ri) = 0, (1.5)

where K is defined in (A4), τi ∈ R and ri ∈ (0, 1) for i = 1, 2, 3, · · · , d with d ∈ N.
The following is our second result.

Theorem 1.3. Assume that w > 0 in (1.2) and (A1)–(A4) hold. Also assume that

1−
d∑
i=1

|τi| > 0. (1.6)

Then there exists a strictly increasing sequence of positive initial values {αn}∞n=1,
such that the solution u(r;αn) is a solution to the multi-point boundary value prob-
lem (1.2)-(1.3) and (1.5). Moreover, u(r;αn) has exactly n or n+ 1 zeros in (0, 1)
for n ∈ N.

Next, the counterpart of this paper is to investigate the negative potential case
(w < 0). We intend to discuss the blow-up solutions of (1.2)-(1.3). Motivated by
the interesting idea as in [25] and under some minor assumption of w, we plan to
discuss the existence of blow-up solutions in a finite interval. We also derive such
an interval associated with the initial parameter α precisely by analyzing some
integrals involving the p-Laplcian.

Theorem 1.4. For nonincreasing w with w < 0, assume that (A1) and (A2) hold.
The nonlinear problem (1.2)-(1.3) has at least one positive blow-up solution u(r;α)
in (0, Rα), where

Rα := p

√
nq(p− 1)

pδ1αq−p

( 1

p− 1
(2p − 1)

p−1
p +

2p

q − p

)
. (1.7)



4 W.-C. WANG EJDE-2021/40

That is, the positive solution u(r;α) tends to infinity as r tends to R ≤ Rα. More-
over, such a positive blow-up solution can not occur when the problem is considered
in a finite interval as q ≤ p.

The article is organized as follows. Some elementary properties related to (1.2)-
(1.3) and the proofs of Theorem 1.1 and 1.3 will be given in Section 2. The existence
of blow-up solutions (Theorem 1.4) will be represented in Section 3.

2. Preliminaries and Sturmian theory

First, the existence of solutions to (1.2)-(1.3) is valid and can be found in [12, 14,
15, 25, 29, 36, 37]. Here we quote the following to coincide with our setting. Also
the regularity requirements for a solution u are u ∈ C1 and rn−1|u′|p−2u′ ∈ C1.

Theorem 2.1 ([25, Theorem EUCD],[29, Theorems 1 and 4],[36, Corollary 2.3]).
Assume conditions (A1) and (A2) hold. For the positive potential function (w > 0),
there exists a unique local solution u(r;α) of (1.2)-(1.3). Moreover, the solution
u(r;α) can be extended to the whole real axis.

Now, we introduce a Prüfer-type substitution for the solution u(r;α) of (1.2)-
(1.3) by using the generalized sine function Sp(r). The generalized sine function Sp
has been well studied in the literature (see Lindqvist [21] or [4, 10, 30] with a minor
difference in setting). Here we outline some properties for the reader’s convenience.
The function Sp satisfies

|S′p(r)|p +
|Sp(r)|p

p− 1
= 1, (2.1)

(|S′p|p−2S′p)′ + |Sp|p−2Sp = 0. (2.2)

Moreover,

πp ≡ 2

∫ (p−1)1/p

0

dt

(1− tp

p−1 )1/p
=

2(p− 1)1/pπ

p sin(π/p)

is the first zero of Sp in the positive real axis. Similarly, one has Sp(
πp
2 + nπp) =

(−1)n, S′p(nπp) = (−1)n, Sp(nπp) = 0 and S′p(
πp
2 + nπp) = 0 for n ∈ Z. With

the help of the generalized sine function, we introduce the phase-plane coordinates
ρ > 0 and θ for the solution u(r;α) of (1.2)-(1.3) as follows:

|u(r;α)|p−2u(r;α) = ρ(r;α)|Sp(mθ(r;α))|p−2Sp(mθ(r;α)),

rn−1|u′(r;α)|p−2u′(r;α) = gρ(r;α)|S′p(mθ(r;α))|p−2S′p(mθ(r;α)),
(2.3)

with

mθ(0;α) =
πp
2

and ρ(0;α) =
( α

p− 1

)p−1
, (2.4)

where m and g are some positive constants that will be specified later. Then

(gρ(r;α))
p
p−1 =

g
p
p−1

p− 1
|u(r;α)|p + r

p(n−1)
p−1 |u′(r;α)|p, (2.5)

rn−1|u′|p−2u′

|u|p−2u
=
g|S′p|p−2S′p
|Sp|p−2Sp

. (2.6)



EJDE-2021/40 RADIALLY SYMMETRIC p-LAPLACIAN EQUATIONS 5

Differentiating both sides of (2.6) with respect to r and employing (1.2) and (2.1)-
(2.3), one can obtain

mgθ′(r;α)

=
rn−1

p− 1
w(r)|u(r;α)|q−p|Sp(mθ(r;α))|p + g

p
p−1

(
r

1−n
p−1 |S′p(mθ(r;α))|p

)
,

(2.7)

and
ρ′(r;α)

ρ(r;α)
=
[
r

1−n
p−1 g

1
p−1 − rn−1g−1w(r)|u(r;α)|q−p

]
× |Sp(mθ(r;α))|p−2Sp(mθ(r;α))S′p(mθ(r;α)).

(2.8)

Employing the above, one can conclude that u(r;α) is the solution of (1.2)-(1.3) if
and only if {θ(r;α), ρ(r;α)} satisfies (2.7)-(2.8) and (2.4).

Motivated by a similar idea in [35] (or [14]), we introduce the scaling argument.
Assume that {αi} is a positively and strictly increasing sequence which tends to
infinity, and define the sequence {µi} to satisfy the following relation:

µi = max{x > 0 : xpw(x) = αp−qi } (2.9)

for i ∈ N. Note that tpw(t) = O(tp) and q > p. Hence, if {αi} is a positively
increasing sequence which tends to infinity, then the corresponding sequence {µi}
satisfying (2.9) decreases to zero. Then, the scaled function vi is defined by

vi(r) =
u(µir;αi)

αi
. (2.10)

By (1.2) and (2.9)-(2.10), a direct calculation yields that vi satisfies(
rn−1|v′i(r)|p−2v′i(r)

)′
+ rn−1

w(µir)

w(µi)
|vi(r)|q−2vi(r) = 0, (2.11)

vi(0) = 1, v′i(0) = 0. (2.12)

From Theorem 2.1 and for each fixed i, the function vi which solves (2.11)-(2.12)

exists on [0, µ−1i ]. By the assumption on w, w(µir)
w(µi)

→ 1 as µi → 0 uniformly on any

bounded interval in [0,∞). Thus vi converges to a function V uniformly on any
bounded interval in [0,∞), where V solves(

rn−1|V ′|p−2V ′
)′

+ rn−1|V |q−2V = 0,

V (0) = 1, V ′(0) = 0.
(2.13)

Next we define an energy functional for the scaled function vi(r) and deduce an
a priori estimate for this energy. Let a functional E[vi](r, α) be defined by

E[vi](r, α) ≡ |v
′
i(r)|p

p
+

w(µir)

q(p− 1)w(µi)
|vi(r)|q (2.14)

with

E[vi](0, r) =
w(0)

q(p− 1)w(µi)
. (2.15)

Note that from (2.11), one can obtain the following equation by multiplying v′i(r),

(n− 1)rn−2|v′i(r)|p + (p− 1)rn−1|v′i(r)|p−2v′i(r)v′′i (r)

+ rn−1
w(µir)

w(µi)
|vi|q−2vi(r)v′i(r) = 0.



6 W.-C. WANG EJDE-2021/40

i.e., for r 6= 0,

−|v′i(r)|p

r
=

(p− 1)

(n− 1)
|v′i(r)|p−2v′i(r)v′′i (r)+

w(µir)

(n− 1)w(µi)
|vi(r)|q−2vi(r)v′i(r). (2.16)

Since there exists a unique solution in [0, µ−1i ], by Theorem 2.1 and (2.10), all the

terms on the right-hand side of (2.16) are bounded in [0, µ−1i ] and v′i tends to zero
as r vanishes by the initial condition. This implies that

lim
r→0+

|v′i(r)|p

r
= 0 (2.17)

and the term
|v′i(r)|

p

r is bounded in [0, µ−1i ]. Then, it follows from (2.11) and (2.16)

that for r ∈ (0, µ−1i ],

d

dr
E[vi](r, α) = E[vi]

′(r, α)

= v′i(r)
[
|v′i(r)|p−2v′′i (r) +

w(µir)

(p− 1)w(µi)
|vi(r)|q−2vi(r)

]
+

µiw
′(µir)

q(p− 1)w(µi)
|vi(r)|q

= − (n− 1)

(p− 1)
· |v
′
i(r)|p

r
+

µiw
′(µir)

q(p− 1)w(µi)
|vi(r)|q

≤ µiκ
|u′(r)|p

p
+ µiκ

w(µir)

q(p− 1)w(µi)
|vi(r)|q

= µiκE[vi](r, α),

(2.18)

where κ = max
{ |w′(µir)|

w(µir)
: r ∈ [0, µ−1i ]

}
= max

{ |w′(r)|
w(r) : r ∈ [0, 1]

}
. In particular,

the above inequality holds for the whole interval [0, µ−1i ] by (2.17). Hence, for any

r ∈ [0, µ−1i ]

E[vi](r, α) ≤ E[vi](0, α)eµiκr =
w(0)eµiκr

q(p− 1)w(µi)
(2.19)

by (2.18) and (2.15). This means that both vi(r;α) and v′i(r;α) are bounded as
long as the solution exists.

Proposition 2.2. Assume the conditions (A1) and (A2) hold. Let µi be defined as
in (2.9). Then, vi satisfies

|vi(r)| ≤
( w(0)

w(µir)
eκ
)1/q

≤
(w(0)

δ1
eκ
)1/q

(2.20)

for r ∈ [0, µ−1i ], where κ = max
{ |w′(r)|

w(r) : r ∈ [0, 1]
}

. Moreover, the function V

solving (2.13) satisfies the uniform boundedness,

|V (r)| ≤ eκ/q (2.21)

on any bounded interval in [0,∞).

Now we prove the result related to the Sturmian theory.



EJDE-2021/40 RADIALLY SYMMETRIC p-LAPLACIAN EQUATIONS 7

Proof of Theorem 1.1. From the Prüfer angular equation (2.7), one has

mgθ′(r;α)

=
rn−1

p− 1
w(r)|u(r;α)|q−p|Sp(mθ(r;α))|p + g

p
p−1 r

1−n
p−1 |S′p(mθ(r;α))|p.

(2.22)

Applying the scaling argument (2.10) and choosing m = g
1
p−1 = α

q−p
p , the phase

equation (2.22) can be rewritten as

θ′(r;α) =
rn−1

p− 1
w(r)|v(µ−1r)|q−p|Sp(α

q−p
p θ(r;α))|p

+ r
1−n
p−1 |S′p(α

q−p
p θ(r;α))|p.

(2.23)

Note that |Sp(mθ(r;α))|p and |S′p(mθ(r;α))|p will not vanish at the same point
by (2.1). And if Sp(mθ(r;α)) tends to zero, |S′p(mθ(r;α))| will approach to one.

Furthermore, v(µ−1r) and Sp(mθ(r;α)) vanish at the same point by (2.3) and
(2.10). Integrating the phase equation (2.23) over [0, r] for r ∈ (0, 1], one can
obtain

mθ(r;α) =
πp
2

+m

∫ r

0

( sn−1
p− 1

w(s)|v(µ−1s)|q−p|Sp(mθ(s;α))|p

+ s
1−n
p−1 |S′p(mθ(s;α))|p

)
ds,

(2.24)

where m = α
q−p
p . A detailed analysis similar as in [30, Lemma 3] (or [5]) shows

that mθ(r;α)− πp
2 = O(rn) as r → 0+. Hence, we can observe that for any α > 0

the integral term in (2.24) is bounded and never vanishes by the above explanation.
And the Prüfer phase θ is continuous dependence on α obviously. Then, one can
conclude that for r ∈ (0, 1],

lim
α→0

α
q−p
p θ(r;α) =

πp
2
, lim

α→∞
α
q−p
p θ(r;α) =∞. (2.25)

Now by (2.4) and (2.25), there exists an increasing sequence of of positive numbers
{αn}∞n=1 such that

α
q−p
p

n θ(0;αn) =
πp
2

and α
q−p
p

n θ(1;αn) = nπp.

This means that u(r;αn) is a solution of (1.2)-(1.4) which has exactly n− 1 zeros
in (0, 1). The proof is complete. �

Next, we deal with multi-point boundary conditions.

Proof of Theorem 1.3. Recall that m = g
1
p−1 = α

q−p
p are as in the proof of Theorem

1.1. From (2.25) and the continuity of θ(r;α) in α, there exist a maximal αn and
a minimal αn+1 such that

α
q−p
p

n θ(1;αn) = (n+
1

2
)πp, α

q−p
p

n θ(1;αn+1) = (n+
3

2
)πp, (2.26)

(n+
1

2
)πp < α

q−p
p θ(1;α) < (n+

3

2
)πp for αn < α < αn+1. (2.27)

Now by (2.8),(2.10), (A3), (A4), and (2.20), for r ∈ (0, 1] and j = n, n+ 1 one can
obtain

ρ′(r;αj)

ρ(r;αj)
≥ −α

q−p
p

j

(
r

1−n
p−1 + rn−1w(r)|v(µ−1j r)|q−p

)



8 W.-C. WANG EJDE-2021/40

≥ −α
q−p
p

j

(
r

1−n
p−1 + w1

(w(0)

δ1
eκ
) q−p

q
)

= −α
q−p
p

j

(
r

1−n
p−1 + K

)
.

Integrating the above inequality over [ri, 1] (1 ≤ i ≤ d), by (A4) one can get

ln
ρ(1;αj)

ρ(ri;αj)
≥ −α

q−p
p

j

( p− 1

p− n
(1− r

p−n
p−1

i ) + K(1− ri)
)

≥ −α
q−p
p

j

( p− 1

p− n
+ K

)
> −α

q−p
p

j K

for j = n, n+ 1. Then,

ρ(ri;αj) < e

(
α
q−p
p

j K
)
ρ(1;αj), i = 1, 2, 3, . . . , d and j = n, n+ 1. (2.28)

By the Prüfer-type substitution (2.5) and (2.26), one can observe that

|u(ri;αj)| ≤
p−1

√
(p− 1)

p−1
p ρ(ri;αj),

|u(1;αj)| =
p−1

√
(p− 1)

p−1
p ρ(1;αj).

(2.29)

for 1 ≤ i ≤ d and j = n, n+ 1. Now define

Γ(α) = u(1;α)−
d∑
i=1

τie
−

(
α

q−p
p K
)

p−1 u(ri;α).

Assume that n = 2k − 1 for k ∈ N. Note that

u(1;α2k−1) = u(1;αn) < 0 and u(1;α2k) = u(1;αn+1) > 0 (2.30)

from (2.26) and (2.3). By applying (2.28)-(2.30) and (1.6), one can obtain that

Γ(α2k−1)

= u(1;α2k−1)−
d∑
i=1

τie
−

(
α

q−p
p

2k−1
K
)

p−1 u(ri;α2k−1)

≤ − p−1

√
(p− 1)

p−1
p ρ(1;α2k−1) +

d∑
i=1

|τi|e−
(
α

q−p
p

2k−1
K
)

p−1
p−1

√
(p− 1)

p−1
p ρ(ri;α2k−1)

< − p−1

√
(p− 1)

p−1
p ρ(1;α2k−1)

+

d∑
i=1

|τi|e−
(
α

q−p
p

2k−1
K
)

p−1

p−1

√
(p− 1)

p−1
p e

(
α
q−p
p

2k−1K
)
ρ(1;α2k−1)

=
p−1

√
(p− 1)

p−1
p ρ(1;α2k−1)

(
− 1 +

d∑
i=1

|τi|
)
< 0,

and

Γ(α2k)

= u(1;α2k)−
d∑
i=1

τie
−

(
α

q−p
p

2k
K
)

p−1 u(ri;α2k)



EJDE-2021/40 RADIALLY SYMMETRIC p-LAPLACIAN EQUATIONS 9

≥ p−1

√
(p− 1)

p−1
p ρ(1;α2k)−

d∑
i=1

|τi|e−
(
α

q−p
p

2k
K
)

p−1
p−1

√
(p− 1)

p−1
p ρ(ri;α2k)

>
p−1

√
(p− 1)

p−1
p ρ(1;α2k)

−
d∑
i=1

|τi|e−
(
α

q−p
p

2k
K
)

p−1

p−1

√
(p− 1)

p−1
p e

(
α
q−p
p

2k K
)
ρ(1;α2k)

=
p−1

√
(p− 1)

p−1
p ρ(1;α2k)

(
1−

d∑
i=1

|τi|
)
> 0.

By the continuity of Γ(α), there exists ᾱ ∈ (α2k−1, α2k) such that Γ(ᾱ) = 0. It is
similar to the case of n = 2k with k ∈ N. Now in both cases, from (2.27) one has

(n+
1

2
)πp < ᾱ

q−p
p θ(1; ᾱ) < (n+

3

2
)πp.

Hence, the above implies that u(r; ᾱ) has n or n+ 1 zeros in (0, 1) and satisfies the
multi-point boundary condition (1.5). The proof is complete. �

3. Blow-up solutions in finite intervals

In this section, we consider the negative potential (w < 0) and focus on the issue
related to the existence of unbounded solutions in a finite interval. Motivated by
the interesting idea raised in [25], we first study the one-dimensional case.

Theorem 3.1. Let w < 0 and assume that (A1) and (A2) hold. Then, the one-
dimensional problem (|u′|p−2u′)′+w(r)|u|q−2u = 0 has at least one blow-up solution
in a finite interval. That is, for u(0) = α > 0 one positive solution will tend to

infinity in the finite interval (0,
p
√
n−1Rα), where Rα is defined as in (1.7). For

q ≤ p, such a blow-up solution satisfying this problem can not exist in any finite
interval.

Proof. For w < 0, a positive unique local solution u(r) in J with u(0) = α > 0 and
u′(0) = 0 satisfies u′ > 0, and then∫ r

0

(|u′|p−2u′)′u′ds = −
∫ r

0

w(s)|u|q−2uu′ds ≥ δ1
∫ r

0

|u|q−2uu′ds

by (A2). The above implies that

u′p −
∫ r

0

(|u′|p−2u′)u′′ds =
p− 1

p
u′p ≥ δ1

q
(uq − αq) .

i.e.,

u′

p
√
uq − αq

≥ p

√
pδ1

q(p− 1)
.

Then, ∫ u(r)

α

du
p
√
uq − αq

≥ p

√
pδ1

q(p− 1)
r.



10 W.-C. WANG EJDE-2021/40

Hence, if the solution becomes infinite at r = `, then

` ≤ p

√
q(p− 1)

pδ1

∫ ∞
α

du
p
√
uq − αq

= p

√
q(p− 1)

pδ1αq−p

∫ ∞
1

ds
p
√
sq − 1

= C
(∫ 2

1

+

∫ ∞
2

) ds
p
√
sq − 1

< C
(∫ 2

1

ds
p
√
sp − 1

+

∫ ∞
2

ds

p

√
sq − 1

2s
q

)
(letting sp − 1 = t)

= C
(1

p

∫ 2p−1

0

t
−1
p (1 + t)

1−p
p dt+ 2

q
p

∫ ∞
2

s
−q
p ds

)
≤ C

(1

p

∫ 2p−1

0

t
−1
p dt+

2p

q − p

)
= C

( 1

p− 1
(2p − 1)

p−1
p +

2p

q − p

)
=

p
√
n−1Rα,

(3.1)

where C = p

√
q(p−1)
pδ1αq−p

and Rα is defined as in (1.7). This shows that there is at

least one positive blow-up solution in (0,
p
√
n−1Rα).

For q ≤ p and a positive solution u(r) with u(0) = β > 0 and u′(0) = 0, assume
this unique local solution exists in J = [0, a) and let |w| ≤ δa in J . Then∫ r

0

(|u′|p−2u′)′u′ds = −
∫ r

0

w(s)|u|q−2uu′ds ≤ δa
∫ r

0

|u|q−2uu′ds.

Apply the similar argument as in the above case (q > p) and let the solution become
infinite at r = R(β), where

R(β) ≥ p

√
q

(p− 1)

pδa

∫ ∞
β

du
p
√
uq − βq

= C̃

∫ ∞
1

ds
p
√
sq − 1

=∞,

where C̃ = p

√
q(p−1)
pδaβq−p

. This shows that the blow-up solution can not occur when

the problem is considered in a finite interval as q ≤ p. �

The following is a technical and crucial lemma whose main concept is quoted
from [25, Lemma 1]. It represents some elementary properties for solutions of (1.2)
and the significant relationship between (1.2) and its corresponding one-dimensional
problem (n = 1). Here we give the details for the reader’s convenience and make it
coincide with our setting.

Lemma 3.2. For ` > 0, assume that u ∈ C1(0, `) with |u′|p−2u′ ∈ C1(0, `) is a
solution of (

rn−1|u′|p−2u′
)′

+ rn−1g(u) = 0 in (0, `), (3.2)

where g ∈ C(R) and u′ is bounded near zero. Then u(0) := limr→0 u(r) exists, and,
with this definition,

(i) u′(0) = 0, u ∈ C1[0, `) and |u′|p−2u′ ∈ C1[0, `);
(ii) if the function g is negative and nonincreasing, then u′(r) ≥ 0 for r > 0

and

(|u′|p−2u′)′ ≤ r−n+1
(
rn−1|u′|p−2u′

)′ ≤ n(|u′|p−2u′)′;



EJDE-2021/40 RADIALLY SYMMETRIC p-LAPLACIAN EQUATIONS 11

(iii) the function v(r) = v(r; c;µ) = cu(µr) (c, µ > 0) satisfies

r−n+1
(
rn−1|v′|p−2v′

)′
+ cp−1µpg(v/c) = 0 in (0, `/µ).

Proof. From (3.2) and the boundedness of u′, one has

rn−1|u′|p−2u′ = −
∫ r

0

sn−1g(u(s))ds = −rn
∫ 1

0

tn−1g(u(rt))dt. (3.3)

The boundedness of u′ near zero implies that limr−→o+ u(r) := α exists. Letting

α = u(0), one can obtain that |u
′|p−2u′

r is bounded near zero, hence u′(0) = 0. The
other properties of (i) are valid by (3.3). For (ii), by the assumption of g one has
u′(r) ≥ 0 from (3.3) obviously. Besides,

r−n+1
(
rn−1|u′|p−2u′

)′
= (n− 1)

|u′|p−2u′

r
+ (|u′|p−2u′)′ ≥ (|u′|p−2u′)′. (3.4)

Employing (3.3) and g′ ≤ 0, one can obtain that

(|u′|p−2u′)′ = −
∫ 1

0

tn−1g(u(rt))dt− r
[ ∫ 1

0

tng′(u(rt)u′(rt)dt
]

≥ −
∫ 1

0

tn−1g(u(rt)dt =
|u′|p−2u′

r
.

(3.5)

Hence, by (3.4)-(3.5)

(|u′|p−2u′)′ ≤ r−n+1
(
rn−1|u′|p−2u′

)′
≤ (n− 1)(|u′|p−2u′)′ + (|u′|p−2u′)′ = n(|u′|p−2u′)′.

This completes the proof of (ii). Finally, (iii) is valid by a direct substitution. �

The following is a version of the comparison lemma for the radial p-Laplacian
and can be found as a consequence of [25, 29].

Lemma 3.3 (Comparison). Let 0 ≤ a < b. Assume that u, v ∈ C2[a, b] satisfy

r−n+1
(
rn−1|u′|p−2u′

)′ ≤ g(r, u) and r−n+1
(
rn−1|v′|p−2v′

)′ ≥ g(r, v)

in [a, b], u(a) ≤ v(a) and u′(a) ≤ v′(a), where g(r, s) is increasing in s. Then
u′ ≤ v′ in [a, b], which implies u ≤ v in [a, b]. In addition, if u(a+) < v(a+), it
follows that u′ ≤ v′ and u < v in (a, b].

Proof of Theorem 1.4. By Theorem 3.1 we assume that v is a positive blow-up
solution of

(|v′(r)|p−2v′(r))′ = n−1w(µ−1r)|v(r)|q−2v(r) in (0, Rv)

satisfying v(0) = 1 and v′(0) = 0, where µ > 0 and

R1 = p

√
q(p− 1)n

pδ1

( 1

p− 1
(2p − 1)

p−1
p +

2p

q − p

)
.

That is, v becomes infinite as r tends to ` ≤ R1. Now let u1 be the solution of

r−n+1
(
rn−1|u′1(r)|p−2u′1(r)

)′
= w(µ−1r)|u1(r)|q−2u1(r), u1(0) = 1, u′1(0) = 0.

By Lemma 3.2 (ii), (|u′1(r)|p−2u′1(r))′ ≥ n−1w(µ−1r)|u1(r)|q−2u1(r). Then the
comparison lemma gives u1 ≥ v. That is, u1 tends to infinity as r → `1 with

`1 ≤ R1. Now we define uα(r) = αu1(µr) with µ = α
q−p
p . Applying Lemma 3.2



12 W.-C. WANG EJDE-2021/40

(iii), one can obtain that uα(r) solves (1.2)-(1.3). Hence, uα has the asymptote

R(α) ≤ α
p−q
p R1 = Rα (as in (1.7)), which implies that there is at least one positive

blow-up solution in (0, Rα). For q ≤ p, such a blow-up solution of (1.2)-(1.3) can
not occur when the problem is considered in a finite interval by applying Theorem
3.1 and Lemma 3.2 (ii) directly. The proof is complete. �

Acknowledgments. The author was partially supported by MOST 108-2115-M-
507-001.

References

[1] C. Bandle, M. Marcus; Large solutions of semilinear elliptic equations: existence, uniqueness
and asymptotic behavior, J. d’Analyse Math., 58 (1992), 9-24.

[2] C. Bandle, M. Marcus; Asymptotic behaviour of solutions and their derivatives, for semilinear

elliptic problems with blowup on the boundary, Ann. Inst. Henri Poincar, 12 (1995), 155-171.
[3] L. Bieberbach; ∆u = exp(u) und die automorphen Funktionen, Math. Annln, 77 (1916),

173-212.

[4] P. Binding, P. Drabek; Sturm–Liouville theory for the p-Laplacian, Studia Scientiarum Math-
ematicarum Hungarica, 40 (2003), 373-396.

[5] B. M. Brown, W. Reichel; Eigenvalues of the radial symmetric p-Laplacian in Rn, J. London

Math. Soc., 69 (2004), 657-675.
[6] C. Cortazar, M. Garcia-Huidobro, C. S. Yarur; On the existence of sign changing bound state

solutions of a quasilinear equation, J. Differential Equations, 254 (2013), 2603-2625.

[7] J. I. Diaz; Nonlinear Partial Differential Equations and Free Boundaries, vol. I. Elliptic
Equations, Res. Notes Math., vol. 106, Pitman, Boston, MA, 1985.

[8] J. I. Diaz; Qualitative study of nonlinear parabolic equations: an introduction, Extracta

Math., 16 (2001), 303-341.
[9] G. Diaz, R. Letelier; Explosive solutions of quasilinear elliptic equations: existence and

uniqueness, Nonlinear Analysis, T.M.A., 20 (1993), 97-125.
[10] A. Elbert; A half-linear second order differential equation, Colloqia mathematica Societatis

Jónos Bolyai, 30., Qualitiative Theory of Differential Equations, Szeged (Hungary) (1979),

153-180.
[11] H. Feng, W. Ge, M. Jiang; Multiple positive solutions for m-point boundary-value problems

with a one-dimensional p-Laplacian, Nonlinear Analysis, 68 (2008), 2269-2279.

[12] M. Garcia-Huidobro, R. Manasevich, C. S. Yarur; On the structure of positive radial solutions
to an equation containing a p-Laplacian with weight, J. Differential Equations, 223 (2006),

51-95.

[13] Chunhua Jin, Jingxue Yin, Zejia Wang; Positive radial solutions of p-Laplacian equation with
sign changing nonlinear sources, Math. Meth. Appl. Sci., 30 (2007), 1-14.

[14] Y. Kabeya, E. Yanagida, S. Yotsutani; Existence of nodal fast-decay solutions to

div(|∇u|m−2∇u) + K(|x|)|u|q−1u = 0 in Rn, Differential and Integral Equations, 9 (1996),
981-1004.

[15] Y. Kabeya, E. Yanagida, S. Yotsutani; Number of zeros of solutions to singular initial value

problems, Tohoku Math. J., 50 (1998), 1-22.
[16] J. B. Keller; On solutions of ∆u = f(u), Communs pure appl. Math., 10 (1957), 503-510.

[17] L. Kong, Q. Kong; Existence of nodal solutions of multi-point boundary value problems,
Discrete Contin. Dyn. Syst. (Suppl.), (2009), 457-465.

[18] O. A. Ladyzhenskaya; The Mathematical Theory of Viscous Incompressible Flow, 2nd ed.,
Gordon and Breach (New York), 1969.

[19] A. Lazer, P. J. McKenna; Asymptotic behaviour of solutions of boundary blow-up problems,
Differential and Integral Equations, 7 (1994), 1001-1019.

[20] A. Lazer, P. J. McKenna; A singular elliptic boundary value problem, Applied Mathematics
and Computation, 65 (1994), 183-194.

[21] P. Lindqvist; Some remarkable sine and cosine functions, Ricerche Mat., 44 (1995), 269-290.

[22] J. L. Lions; Quelques methodes de resolution des problemes aux limites non lineaires, Dunod
(Paris) 1969.



EJDE-2021/40 RADIALLY SYMMETRIC p-LAPLACIAN EQUATIONS 13

[23] B. Liu; Positive solutions of singular three-point boundary value problems for the one-

dimensional p-Laplacian, Comput. Math. Appl., 48 (2004), 913-925.

[24] R. Ma; Positive solutions for multipoint boundary value problem with a one-dimensional
p-Laplacian, Comput. Math. Appl., 42 (2001), 755-765.

[25] P. J. Mckenna, W. Reichel, W. Walter; Symmetry and multiplicity for nonlinear elliptic

differential equations with boundary blow-up, Nonlinear Analysis : Theory, Methods & Ap-
plications, 28 (1997), 1213-1225.

[26] M. C. Pelissier, M. L. Reynaud; Etude d’un modele mathematique d’ecoulement de glacier,

C. R. Acad. Sci. Paris Ser. I Math., 279 (1974), 531-534.
[27] H. Rademcher; Einige besondere Probleme partieller Differential Gleichungen, in Die differ-

ential und Integral Gleichungen der Mechanik und Physik I, 2nd edition, 838-845. Rosenberg,

New York 1943.
[28] M. Ramaswamy, R. Shivaji; Multiple positive solutions for classes of p-Laplacian equations,

Differential and Integral Equations, 17 (11-12) (2004), 1255-1261.
[29] W. Reichel, W. Walter; Radial solutions of equations and inequalities involving the p-

Laplacian, J. Inequalities Appl., 1 (1997), 47-71.

[30] W. Reichel, W. Walter; Sturm-Liouville type problems for the p-Laplacian under asymptotic
non-resonance conditions, J. Differential Equations, 156 (1999), 50-70.

[31] Gelson C. G. dos Santos, Giovany M. Figueiredo, Leandro S. Tavares; Sub-super solution

method for nonlocal systems involving the p(x)-Laplacian operator, Electron. J. Differential
Equations, Vol. 2020 (2020), No. 25, 1-19.

[32] R. E. Showalter, N. J. Walkington; Diffusion of fluid in a fissured medium with microstructure,

SIAM J. Math. Anal., 22 (1991), 1702-1722.
[33] S. Takeuchi; Generalized Jacobian elliptic functions and their application to bifurcation prob-

lems associated with p-Laplacian, J. Math. Anal. Appl., 385 (2012), 24-35.

[34] S. Tanaka; An identity for a quasilinear ODE and its applications to the uniqueness of
solutions of BVPs, J. Math. Anal. Appl., 351 (2009), 206-217.

[35] W. C. Wang; Some nodal properties for a quasilinear differential equation involving the p-
Laplacian, J. Math. Anal. Appl., 472 (2019), 1093-1105.

[36] W. C. Wang, Y. H. Cheng; On the existence of sign-changing radial solutions to nonlinear

p-Laplacian equations in Rn, Nonlinear Analysis, 102 (2014), 14-22.
[37] W. C. Wang, Y. H. Cheng; Sign-changing solutions to a class of nonlinear equations involving

p-Laplacian, Rocky Mountain J. Math., 47 (2017), 971-996.

[38] Y. Wang, C. Hou; Existence of multiple positive solutions for one-dimensional p-Laplacian,
J. Math. Anal. Appl., 315 (2006), 144-153.

[39] E. Yanagida; Sturmian theory for a class of nonlinear second-order differential equations, J.

Math. Anal. Appl., 187 (1994), 650-662.

Wei-Chuan Wang
Department of Civil Engineering and Engineering Management, Center for General

Education, National Quemoy University, Kinmen, Taiwan 892, ROC
Email address: wangwc@nqu.edu.tw, wangwc72@gmail.com


	1. Introduction
	2. Preliminaries and Sturmian theory
	3. Blow-up solutions in finite intervals
	Acknowledgments

	References