Existence of positive radial solutions of general quasilinear elliptic systems

Daniel Devine

doi:10.1515/anona-2025-0083

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Existence of positive radial solutions of general quasilinear elliptic systems

Daniel Devine

Published/Copyright: July 29, 2025

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From the journal Advances in Nonlinear Analysis Volume 14 Issue 1

Abstract

Let Ω ⊂ R n ( n ≥ 2 ) be either an open ball B R centred at the origin or the whole space. We study the existence of positive, radial solutions of quasilinear elliptic systems of the form

Δ p u = f 1 ( ∣ x ∣ ) g 1 ( v ) ∣ ∇ u ∣ α in Ω , Δ p v = f 2 ( ∣ x ∣ ) g 2 ( v ) h ( ∣ ∇ u ∣ ) in Ω ,

where α ≥ 0 , Δ p is the p -Laplace operator, p > 1 , and for i , j = 1 , 2, we assume f i , g j , and h are continuous, non-negative and non-decreasing functions. For functions g j which grow polynomially, we prove sharp conditions for the existence of positive radial solutions which blow up at ∂ B R , and for the existence of global solutions.

Keywords: radial solutions; elliptic systems; coercive systems; p-Laplace operator; nonlinear gradient terms

MSC 2010: 35B44; 35J47; 35J92

1 Introduction

Let Ω ⊂ R n ( n ≥ 2 ) be either an open ball B R centred at the origin or the whole space, and for i , j = 1 , 2 suppose f i , g j , and h are continuous, non-negative and non-decreasing functions. We study the existence of positive, radial solutions of quasilinear elliptic systems of the form

(1.1) Δ p u = f 1 ( ∣ x ∣ ) g 1 ( v ) ∣ ∇ u ∣ α in Ω , Δ p v = f 2 ( ∣ x ∣ ) g 2 ( v ) h ( ∣ ∇ u ∣ ) in Ω ,

where Δ p u = div ( ∣ ∇ u ∣ p − 2 ∇ u ) is the p -Laplace operator, p > 1 and α ≥ 0 . The functions g j are assumed to satisfy growth conditions which are stated below. For Ω = B R , the following boundary behaviours are possible:

u and v are bounded in B R ;
u is bounded in B R and lim ∣ x ∣ → R − v ( x ) = ∞ ;
lim ∣ x ∣ → R − u ( x ) = lim ∣ x ∣ → R − v ( x ) = ∞ .

The first equation of (1.1) prevents a positive radial solution satisfying lim ∣ x ∣ → R − u ( x ) = ∞ and v is bounded in B R from existing (see Lemma 2.4 below for details). We prove optimal Keller-Osserman type conditions for the existence of positive radial solutions of (1.1) with boundary behaviours (B1)–(B3). This solves a problem left open in [22] by Filippucci and Vinti, who studied a special case of system (1.1), namely, to prove a sufficient condition for the blow up of positive radial solutions near ∂ B R . The key result needed to prove this is Lemma 2.5, which was previously only known for the particular case p = 2 . We then turn our attention to the case Ω = R n and prove a sharp condition for the existence of global positive radial solutions of (1.1).

There is by now a vast literature on elliptic problems containing nonlinear gradient terms. In the scalar case, these terms arise in models which investigate the influence of nonlinear diffusion on the elliptic Hamilton-Jacobi equation:

(1.2) − Δ p u = H ( x , u , ∇ u ) in Ω ,

where Ω is either bounded or the whole space R n . See [4] for a study of the case H ( x , u , ∇ u ) = u q + h ( x , u , ∇ u ) with Ω bounded and smooth, and [5,21] for the case H ( x , u , ∇ u ) = u m ∣ ∇ u ∣ q and Ω = R n , to mention but a few recent works. When it comes to the coercive analogue of equation (1.2), namely, Δ p u = H ( x , u , ∇ u ) , solutions which blow up at ∂ Ω are the subject of much attention, which began in the 1950s with the seminal papers by Keller [25] and Osserman [29]. The authors independently studied the semilinear problem:

Δ u = h ( u ) in Ω , u → ∞ as x → ∂ Ω ,

where h ∈ C 1 ( [ 0 , ∞ ) ) and Ω ⊂ R n is bounded and smooth and found that the aforementioned problem has C 2 ( Ω ) solutions if and only if

∫ 1 ∞ d s ∫ 0 s h ( t ) d t < ∞ ,

which is the well-known Keller-Osserman condition. Their methods have since been adapted and generalised to a very broad range of problems, see [1,11,17,20] and the references therein.

The study of coercive elliptic systems without the presence of gradient terms also has a rich history. As with the scalar case, particular interest has been devoted to deriving Keller-Osserman type conditions for the existence of solutions which blow up at ∂ Ω , and also to the existence of unbounded entire solutions when Ω = R n . For semilinear elliptic systems, we refer to, for instance, [10,13,23,26–28], and for the case of quasilinear elliptic systems, we refer to [2,7–10,12,18]. More recently, coercive systems with nonlinear gradient terms such as (1.1) have been investigated [3,6,14–16,19,22,24,30]. Their study began with Díaz et al. [16] who focused on the case

Δ u = v in B R , Δ v = ∣ ∇ u ∣ 2 in B R .

The solutions of this system are steady states of a related parabolic system which models the unidirectional flow of a viscous, heat-conducting fluid. See the introduction of [24] and the references therein for further discussion on the physical motivation of such systems. Singh [30] later considered more general semilinear systems of the following form:

Δ u = v m in Ω , Δ v = h ( ∣ ∇ u ∣ ) in Ω ,

where Ω = B R or R n , m > 0 and h ∈ C 1 ( [ 0 , ∞ ) ) is non-negative and non-decreasing. He found optimal Keller-Osserman type conditions for the existence of positive radial solutions with boundary behaviours (B1)–(B3). Singh’s results were then partially extended to the p -Laplace operator by Filippucci and Vinti [22]. For the system

(1.3) Δ p u = v m in B R , Δ p v = h ( ∣ ∇ u ∣ ) in B R ,

with the same assumptions as mentioned earlier, in the case p ≥ 2 , the authors show that a necessary condition for system (1.3) to admit a positive radial solution ( u , v ) which blows up at ∂ B R is given by

(1.4) ∫ 1 ∞ s ( p − 2 ) ( p − 1 ) ⁄ ( m p + p − 1 ) d s ∫ 0 s h ( t ) p d t m p ⁄ ( m p + p − 1 ) < ∞ .

As mentioned earlier, however, the authors were unable to show condition (1.4) is optimal. In the particular case h ( t ) = t q for q > 0 , this problem was later solved by Ghergu et al. [24]. The authors fully classified the positive radial solutions of the system

(1.5) Δ p u = v m ∣ ∇ u ∣ α in B R , Δ p v = v β ∣ ∇ u ∣ q in B R ,

according to their behaviour at ∂ B R . The exponents in (1.5) are assumed to satisfy m , q > 0 , α ≥ 0 , 0 ≤ β ≤ m and ( p − 1 − α ) ( p − 1 − β ) − q m ≠ 0 , and optimal conditions on the exponents for the existence of positive radial solutions with boundary behaviours (B1)–(B3) are found. In the case that solutions exist in the whole space R n , the precise behaviour at infinity of solutions of (1.5) is also determined. For a discussion on the exact rate of blow-up of solutions of (1.5), we refer to [3], and for the behaviour at infinity of solutions of more general systems such as (1.1), we refer to recent work by Devine and Karageorgis [14].

Throughout this article, we assume for each i , j that

f i , g j , h are continuous, non-decreasing on [ 0 , ∞ ) and positive on ( 0 , ∞ ) .

When it comes to the functions g j , we make the following growth assumption:

there exist constants k 1 > 0 and 0 ≤ k 2 ≤ k 1 such that g j ( t ) ⁄ t k j is non-increasing in ( 0 , ∞ ) and
lim t → ∞ g j ( t ) t k j > 0 .

Systems (1.3) and (1.5) clearly satisfy (A1) and (A2), but our assumption (A2) allows a much more general class of functions g j to be considered. For example, each g j ( t ) could be any sum of non-negative powers of t with non-negative coefficients. Moreover, thanks to Lemma 2.5, the results are shown to be optimal for functions h only assumed to satisfy (A1).

Our first goal in this article is to give optimal conditions for the existence of non-constant positive radial solutions of (1.1) in Ω = B R which satisfy (B1)–(B3). According to [22, Theorem 1.2], condition (1.4) is necessary for positive radial solutions of system (1.3) to satisfy (B2) or (B3). We extend this result in three directions. Namely, we extend it to the much more general system (1.1), we extend it to all p > 1 and we also show that our conditions are sharp. In fact, our result also extends [22, Theorem 1.1] and [24, Theorem 2.1].

Our second goal is to then give sharp conditions for the existence of global non-constant positive radial solutions of (1.1). By using a simple comparison argument, we show that global positive radial solutions exist if and only if all positive radial solutions satisfy (B1). In other words, either all positive radial solutions of (1.1) are global or none are.

Before stating the main results of this article, we again clarify that we are only interested in positive radial solutions, by which we shall always mean a pair ( u ( r ) , v ( r ) ) , r = ∣ x ∣ , such that

u , v ∈ C 2 ( Ω ) are positive and solve (1.1);
u and v are non-constant in any neighbourhood of the origin.

In the case Ω = R n , we call solutions of (1.1) global solutions. To make our main existence result easier to state, we define the function

(1.6) ℋ ( s ) ≔ ∫ 0 s h t 1 p − 1 − α 1 p d t − k 1 p k 1 p + p − 1 − k 2 .

We note that k 1 p + p − 1 − k 2 > 0 by assumption, and remark that p − 1 − α > 0 is a necessary condition for solutions to exist (Lemma 2.2).

Theorem 1.1

Assume (A1) and (A2), 0 ≤ α < p − 1 , and let θ = 1 p − 1 − α . Let ( u , v ) be a non-constant positive radial solution of (1.1). Then:

( u , v ) satisfies (B1) if and only if
(1.7) ∫ 1 ∞ ℋ ( s ) d s = ∞
( u , v ) satisfies (B2) if and only if
(1.8) ∫ 1 ∞ s θ ℋ ( s ) d s < ∞
( u , v ) satisfies (B3) if and only if
(1.9) ∫ 1 ∞ ℋ ( s ) d s < ∞ and ∫ 1 ∞ s θ ℋ ( s ) d s = ∞ .

Theorem 1.2

Assume (A1), (A2), Ω = R n and α ≥ 0 . Then (1.1) admits non-constant global positive radial solutions if and only if α < p − 1 and (1.7) holds.

Remark 1.3

Although not explicitly treated here, the results in this article actually hold for several generalisations of system (1.1). For example, in the first equation, we could replace g 1 ( v ) by g 1 ( v ) ⋅ g 3 ( u ) for a continuous, non-decreasing and bounded function g 3 which is positive on ( 0 , ∞ ) . When it comes to the second equation, we could similarly replace g 2 ( v ) by g 2 ( v ) ⋅ g 4 ( u ) , with g 4 subject to the same conditions as g 3 . Such additional factors would not affect the arguments whatsoever. For the sake of simplicity, though, such variations are not treated explicitly here.

In the case h ( t ) also grows polynomially, we have the following.

Corollary 1.4

With the same assumptions as Theorem 1.2, if there exists k 3 > 0 such that h ( t ) ⁄ t k 3 is non-increasing for t > 0 and lim t → ∞ h ( t ) ⁄ t k 3 > 0 , then (1.1) admits non-constant global positive radial solutions if and only if α < p − 1 and ( p − 1 − α ) ( p − 1 − k 2 ) > k 1 k 3 .

The remaining sections are organized as follows. In Section 2, we gather some preliminary information about solutions of system (1.1). In particular, we show all positive radial solutions of (1.1) are increasing and convex. In Section 3, we prove Theorem 1.1, and finally, the proof of Theorem 1.2 is given in Section 4.

2 Properties of solutions

In this section, we gather some properties which all non-constant positive radial solutions of (1.1) must have. First, if ( u , v ) is a positive radial solution, then both u and v are increasing and convex. In particular, solutions which exist locally, but not globally, must blow up around ∂ Ω . Moreover, if (1.1) admits any positive radial solutions, then we necessarily have 0 ≤ α < p − 1 . The proofs of these results can be found in [14], so we omit them. We conclude the section with a comparison result for integrals which will allow us to prove the optimality of conditions (1.7)–(1.9).

Lemma 2.1

Assume (A1), Ω = B R , α ≥ 0 , and suppose ( u , v ) is a non-constant radial solution of (1.1) with u ( 0 ) > 0 and v ( 0 ) > 0 . Then u ′ ( r ) , v ′ ( r ) > 0 for all 0 < r < R .

Lemma 2.2

Assume (A1), Ω = B R and α ≥ p − 1 . Then (1.1) does not admit any non-constant positive radial solutions for any R > 0 .

Lemma 2.3

Assume (A1), Ω = B R , 0 ≤ α < p − 1 , and suppose ( u , v ) is a non-constant positive radial solution of (1.1). Then

(2.1) p − 1 − α n ( p − 1 − α ) + α f 1 ( r ) g 1 ( v ) ≤ [ u ′ ( r ) p − 1 − α ] ′ ≤ p − 1 − α p − 1 f 1 ( r ) g 1 ( v ) ,

(2.2) 1 n f 2 ( r ) g 2 ( v ) h ( u ′ ) ≤ [ v ′ ( r ) p − 1 ] ′ ≤ f 2 ( r ) g 2 ( v ) h ( u ′ ) ,

for all 0 < r < R . In particular, both u ( r ) and v ( r ) are convex for all 0 < r < R .

Lemma 2.4

Assume (A1), Ω = B R and 0 ≤ α < p − 1 . Then (1.1) does not admit any positive radial solutions ( u , v ) satisfying

(2.3) lim r → R − u ( r ) = ∞ and lim r → R − v ( r ) < ∞ .

Proof

By Lemma 2.1, we may rewrite system (1.1) as follows:

(2.4) [ r δ u ′ ( r ) p − 1 − α ] ′ = δ n − 1 r δ f 1 ( r ) g 1 ( v ( r ) ) for all 0 < r < R , [ r n − 1 v ′ ( r ) p − 1 ] ′ = r n − 1 f 2 ( r ) g 2 ( v ( r ) ) h ( u ′ ( r ) ) for all 0 < r < R , u ′ ( 0 ) = v ′ ( 0 ) = 0 , u ( 0 ) > 0 , v ( 0 ) > 0 ,

where δ is defined as follows:

(2.5) δ = ( n − 1 ) ( p − 1 − α ) p − 1 > 0 .

Suppose ( u , v ) is a solution satisfying (2.3). For each 0 ≤ r < R , we have

u ( r ) = u ( 0 ) + ∫ 0 r u ′ ( t ) d t ,

so from (2.3), we see that lim r → R − u ′ ( r ) = ∞ . Now, since v ( r ) is increasing for 0 < r < R , recalling our assumption (A1) the first equation in (2.4) tells us

r δ u ′ ( r ) p − 1 − α = δ n − 1 ∫ 0 r t δ f 1 ( t ) g 1 ( v ( t ) ) d t ≤ δ n − 1 f 1 ( r ) g 1 ( v ( r ) ) ∫ 0 r t δ d t

for all 0 < r < R . So we have the estimate

1 r u ′ ( r ) p − 1 − α ≤ δ ( δ + 1 ) ( n − 1 ) f 1 ( r ) g 1 ( v ( r ) )

for all 0 < r < R . From (2.3) and (A1), we thus have

u ′ ( r ) p − 1 − α ≤ δ r ( δ + 1 ) ( n − 1 ) f 1 ( r ) g 1 ( v ( r ) ) ≤ C

for all 0 < r < R , and some constant C > 0 . Letting r → R − , we clearly reach a contradiction since α < p − 1 .□

Lemma 2.5

Let h ( t ) satisfy (A1), and H ( t ) = ∫ 0 t h ( s ) d s . Then for s > 0 , we have

( p − 1 ) 2 p − 1 ∫ 0 s H ( t ) 1 p − 1 d t p − 1 ≤ ( p − 1 ) p − 1 ∫ 0 p s h ( t ) 1 p d t p ≤ ∫ 0 p 2 s H ( t ) 1 p − 1 d t p − 1 ,

and consequently, for each ν > 0 ,

∫ 1 ∞ d s ∫ 0 s H ( t ) 1 p − 1 d t ν ( p − 1 ) < ∞ i f a n d o n l y i f ∫ 1 ∞ d s ∫ 0 s h ( t ) 1 p d t ν p < ∞ .

Proof

We start by noting that since p > 1 , the function x ↦ x p is convex. If we fix s > 0 , we have

∫ 0 s h ( t ) 1 p d t p = s p 1 s ∫ 0 s h ( t ) 1 p d t p ≤ s p − 1 ∫ 0 s h ( t ) d t by Jensen’s inequality = ( p − 1 ) 1 − p H ( s ) 1 p − 1 ∫ s p s d σ p − 1 ≤ ( p − 1 ) 1 − p ∫ s p s H ( σ ) 1 p − 1 d σ p − 1 ≤ ( p − 1 ) 1 − p ∫ 0 p s H ( σ ) 1 p − 1 d σ p − 1

and the second inequality follows. When it comes to the first inequality, since both h and H are non-decreasing, we have

∫ 0 s H ( t ) 1 p − 1 d t p − 1 ≤ H ( s ) 1 p − 1 ∫ 0 s d t p − 1 = s p − 1 ∫ 0 s h ( t ) d t ≤ s p h ( s ) = 1 p − 1 h ( s ) 1 p ∫ s p s d t p ≤ 1 p − 1 ∫ 0 p s h ( t ) 1 p d t p ,

and the first inequality follows.□

Lemma 2.5 is an extension of [30, Lemma 4.1], and is key to proving the sharpness of the conditions in Theorem 1.1. If we define for each θ > 0 ,

(2.6) H θ ( t ) ≔ ∫ 0 t h ( s θ ) d s , t > 0 ,

then Lemma 2.5 tells us that for each ν > 0 we have

∫ 1 ∞ d s ∫ 0 s H θ ( t ) 1 p − 1 d t ν ( p − 1 ) < ∞ if and only if ∫ 1 ∞ d s ∫ 0 s h ( t θ ) 1 p d t ν p < ∞ .

It is this formulation, with θ = 1 p − 1 − α and ν = k 1 k 1 p + p − 1 − k 2 , that we shall use in the subsequent sections.

3 Proof of Theorem 1.1

Before the main proof, we first argue that system (1.1) has a non-constant positive radial solution in Ω = B R for any R > 0 . To see this, we note the existence of a non-constant positive radial solution of (1.1) in a small ball B ρ follows from a standard fixed point argument (see [22, Proposition A1] for details). In particular, the map

T : C 1 [ 0 , ρ ] × C 1 [ 0 , ρ ] → C 1 [ 0 , ρ ] × C 1 [ 0 , ρ ]

given by

T [ u , v ] ( r ) = T 1 [ u , v ] ( r ) T 2 [ u , v ] ( r ) ,

where

T 1 [ u , v ] ( r ) = u ( 0 ) + ∫ 0 r δ n − 1 s − δ ∫ 0 s τ δ f 1 ( τ ) g 1 ( v ( τ ) ) d τ 1 p − 1 − α d s T 2 [ u , v ] ( r ) = v ( 0 ) + ∫ 0 r s 1 − n ∫ 0 s τ n − 1 f 2 ( τ ) g 2 ( v ( τ ) ) h ( ∣ u ′ ( τ ) ∣ ) d τ 1 p − 1 d s

and δ is defined as in (2.5), has a fixed point ( u , v ) for sufficiently small ρ > 0 . Note from Lemma 2.2 that p − 1 − α > 0 . Now, let R > 0 be arbitrary. We claim that (1.1) has a positive radial solution in B R . By the aforementioned the following system:

Δ p u ˜ = f ˜ 1 ( ∣ x ∣ ) g ˜ 1 ( v ˜ ) ∣ ∇ u ˜ ∣ α in B ρ , Δ p v ˜ = f ˜ 2 ( ∣ x ∣ ) g ˜ 2 ( v ˜ ) h ˜ ( ∣ ∇ u ˜ ∣ ) in B ρ ,

has a positive radial solution ( u ˜ , v ˜ ) for sufficiently small ρ , where

f ˜ 1 ( r ) = λ p − 1 − α f 1 ( λ r ) f ˜ 2 ( r ) = λ p − 1 f 2 ( λ r ) g ˜ 1 ( t ) = g 1 ( t ) g ˜ 2 ( t ) = g 2 ( t ) h ˜ ( t ) = h t λ ,

for λ > 0 . We then have that

u ( r ) = u ˜ r λ v ( r ) = v ˜ r λ

is a solution of (1.1) in B λ ρ , and the claim follows by letting λ = R ρ . We are thus guaranteed the existence of non-constant positive radial solutions of (1.1) in B R , and are now in a position to classify them according to their behaviour as r → R − . In what follows, C will be used to denote a positive constant which may vary on each occurrence. In general, C = C ( n , p , α , R , u , v ) , but its exact value will not be important.

When it comes to part 1, we first assume that (1.1) admits a solution ( u , v ) satisfying

(3.1) lim r → R − v ( r ) = ∞ .

From assumption (A2), for some constant C > 0 , we have the estimates

C v ( r ) k 1 ≤ g 1 ( v ( r ) ) ≤ g 1 ( v ( 0 ) ) v ( 0 ) k 1 v ( r ) k 1 C v ( r ) k 2 ≤ g 2 ( v ( r ) ) ≤ g 2 ( v ( 0 ) ) v ( 0 ) k 2 v ( r ) k 2

for all r > 0 . For simplicity, we shall let w = u ′ , and we note that w is positive and increasing on ( 0 , R ) by Lemmas 2.1 and 2.3. Multiplying the right inequality in (2.2) by v ′ , which is positive by Lemma 2.1, and integrating over [ 0 , r ] for r < R , we find

∫ 0 r [ v ′ ( t ) p − 1 ] ′ v ′ ( t ) d t ≤ ∫ 0 r f 2 ( t ) g 2 ( v ( t ) ) h ( w ( t ) ) v ′ ( t ) d t .

Now, by Lemmas 2.1 and 2.3, we know that v ( r ) and w ( r ) are increasing in ( 0 , R ) , so by our assumptions (A1)–(A2), the aforementioned inequality becomes

p − 1 p ∫ 0 r ( v ′ ( t ) p ) ′ d t ≤ C h ( w ( r ) ) ∫ 0 r v ( t ) k 2 v ′ ( t ) d t

from which we have, recalling v ′ ( 0 ) = 0 ,

v ′ ( r ) p ≤ C h ( w ( r ) ) ∫ v ( 0 ) v ( r ) t k 2 d t ≤ C v ( r ) k 2 + 1 h ( w ( r ) )

for all r ∈ ( 0 , R ) . In other words,

(3.2) v ′ ( r ) v ( r ) − k 2 + 1 p ≤ C h ( w ( r ) ) 1 p

for all r ∈ ( 0 , R ) . Fix now some ρ ∈ ( 0 , R ) . Inequality (2.1) and our assumptions (A1) and (A2) tell us that for all r ∈ [ ρ , R ) , we have

C v ( r ) k 1 ≤ f 1 ( ρ ) g 1 ( v ( r ) ) ≤ f 1 ( r ) g 1 ( v ( r ) ) ≤ n ( p − 1 − α ) + α p − 1 − α [ w ( r ) p − 1 − α ] ′ .

Combining this with (3.2) yields

v ′ ( r ) v ( r ) k 1 − k 2 + 1 p ≤ C h ( w ( r ) ) 1 p [ w ( r ) p − 1 − α ] ′ ,

which we can integrate over [ ρ , r ] for any r ∈ ( ρ , R ) to obtain, after changing variables,

(3.3) ∫ v ( ρ ) v ( r ) t k 1 − k 2 + 1 p d t ≤ C ∫ w ( ρ ) p − 1 − α w ( r ) p − 1 − α h ( t 1 p − 1 − α ) 1 p d t ≤ C ∫ 0 w ( r ) p − 1 − α h ( t 1 p − 1 − α ) 1 p d t

for all r ∈ ( ρ , R ) . Noting that k 1 p > k 1 ≥ k 2 by our assumption (A2), we may rewrite (3.3) as follows:

v ( r ) k 1 p + p − 1 − k 2 p ≤ v ( ρ ) k 1 p + p − 1 − k 2 p + C ∫ 0 w ( r ) p − 1 − α h ( t 1 p − 1 − α ) 1 p d t = v ( ρ ) k 1 p + p − 1 − k 2 p ∫ 0 w ( r ) p − 1 − α h ( t 1 p − 1 − α ) 1 p d t + C ∫ 0 w ( r ) p − 1 − α h ( t 1 p − 1 − α ) 1 p d t ≤ v ( ρ ) k 1 p + p − 1 − k 2 p ∫ 0 w ( ρ ) p − 1 − α h ( t 1 p − 1 − α ) 1 p d t + C ∫ 0 w ( r ) p − 1 − α h ( t 1 p − 1 − α ) 1 p d t ,

and since ρ > 0 is fixed, we can express the aforementioned as follows:

v ( r ) k 1 p + p − 1 − k 2 p ≤ C ∫ 0 w ( r ) p − 1 − α h ( t 1 p − 1 − α ) 1 p d t ,

for all r ∈ [ ρ , R ) . In other words,

v ( r ) ≤ C ∫ 0 w ( r ) p − 1 − α h ( t 1 p − 1 − α ) 1 p d t p k 1 p + p − 1 − k 2

for all r ∈ [ ρ , R ) . By combining this with (2.1), we have

(3.4) [ w ( r ) p − 1 − α ] ′ ≤ p − 1 − α p − 1 f 1 ( r ) g 1 ( v ( r ) ) ≤ C v ( r ) k 1 ≤ C ∫ 0 w ( r ) p − 1 − α h ( t 1 p − 1 − α ) 1 p d t k 1 p k 1 p + p − 1 − k 2 = C ℋ ( w ( r ) p − 1 − α ) − 1

for all r ∈ [ ρ , R ) , recalling definition (1.6). It follows from (3.4) that for some constant C > 0 and all r ∈ [ ρ , R )

ℋ ( w ( r ) p − 1 − α ) [ w ( r ) p − 1 − α ] ′ ≤ C .

We can integrate both sides of this expression over [ ρ , r ] for r ∈ ( ρ , R ) to find

∫ ρ r ℋ ( w ( s ) p − 1 − α ) [ w ( s ) p − 1 − α ] ′ d s ≤ C ∫ ρ r d s

or, after a change of variables

(3.5) ∫ w ( ρ ) p − 1 − α w ( r ) p − 1 − α ℋ ( s ) d s ≤ C ( r − ρ ) ≤ C r .

Letting r → R − , we see, recalling (2.1) and (3.1)

∫ w ( ρ ) p − 1 − α ∞ ℋ ( s ) d s ≤ C R < ∞ .

Hence,

∫ 1 ∞ ℋ ( s ) d s < ∞ .

By recalling Lemma 2.4, we thus have that all positive radial solutions of (1.1) satisfy (B1) if (1.7) holds. To prove the converse statement, we first note that system (1.1) has a positive radial solution ( u , v ) defined on some maximal interval [ 0 , R 0 ) . Suppose then that (1.7) does not hold, and fix ρ ∈ ( 0 , R 0 ) . Recall from (2.1) and (2.2) and our assumptions (A1) and (A2) that we have

(3.6) [ w ( r ) p − 1 − α ] ′ ≤ C v ( r ) k 1

(3.7) v ( r ) k 2 h ( w ( r ) ) ≤ C [ v ′ ( r ) p − 1 ] ′

for all r ∈ [ ρ , R 0 ) . Multiplying (3.6) and (3.7) and rearranging gives

h ( w ( r ) ) [ w ( r ) p − 1 − α ] ′ ≤ C v ( r ) k 1 − k 2 [ v ′ ( r ) p − 1 ] ′

for all r ∈ [ ρ , R 0 ) . By integrating the aforementioned inequality over [ ρ , r ] for any r ∈ ( ρ , R 0 ) , we obtain

∫ w ( ρ ) p − 1 − α w ( r ) p − 1 − α h ( t 1 p − 1 − α ) d t ≤ C ∫ ρ r v ( t ) k 1 − k 2 [ v ′ ( t ) p − 1 ] ′ d t ≤ C ∫ 0 r v ( t ) k 1 − k 2 [ v ′ ( t ) p − 1 ] ′ d t ≤ C v ( r ) k 1 − k 2 ∫ 0 r [ v ′ ( t ) p − 1 ] ′ d t ≤ C v ( r ) k 1 − k 2 v ′ ( r ) p − 1 ,

recalling that k 1 ≥ k 2 by (A2) and v ′ ( 0 ) = 0 . Taking the extreme left and right above, one has

(3.8) ∫ 0 w ( r ) p − 1 − α h ( t 1 p − 1 − α ) d t ≤ ∫ 0 w ( ρ ) p − 1 − α h ( t 1 p − 1 − α ) d t + C v ( r ) k 1 − k 2 v ′ ( r ) p − 1 = ∫ 0 w ( ρ ) p − 1 − α h ( t 1 p − 1 − α ) d t v ( r ) k 1 − k 2 v ′ ( r ) p − 1 + C v ( r ) k 1 − k 2 v ′ ( r ) p − 1 ≤ ∫ 0 w ( ρ ) p − 1 − α h ( t 1 p − 1 − α ) d t v ( ρ ) k 1 − k 2 v ′ ( ρ ) p − 1 + C v ( r ) k 1 − k 2 v ′ ( r ) p − 1 ≤ C v ( r ) k 1 − k 2 v ′ ( r ) p − 1

for all r ∈ [ ρ , R 0 ) , recalling that v ′ ( r ) is positive for r > 0 by Lemma 2.1. We now let θ = 1 p − 1 − α , and recall from (2.6) that we define

H θ ( t ) = ∫ 0 t h ( s θ ) d s , t > 0 .

Multiplying inequalities (3.6) and (3.8) thus gives

H θ ( w ( r ) p − 1 − α ) 1 p − 1 [ w ( r ) p − 1 − α ] ′ ≤ C v ( r ) k 1 p − k 2 p − 1 v ′ ( r )

for all r ∈ [ ρ , R 0 ) . A further integration over [ ρ , r ] yields

∫ w ( ρ ) p − 1 − α w ( r ) p − 1 − α H θ ( t ) 1 p − 1 d t ≤ C v ( r ) k 1 p + p − 1 − k 2 p − 1 ,

for all r ∈ ( ρ , R 0 ) . Similar to that mentioned earlier, since k 1 p + p − 1 − k 2 > 0 , there exists C > 0 such that

∫ 0 w ( r ) p − 1 − α H θ ( t ) 1 p − 1 d t ≤ C v ( r ) k 1 p + p − 1 − k 2 p − 1

for all r ∈ ( ρ , R 0 ) . By using (2.1), and our assumption (A2), the aforementioned inequality becomes

∫ 0 w ( r ) p − 1 − α H θ ( t ) 1 p − 1 d t ≤ C [ v ( r ) k 1 ] k 1 p + p − 1 − k 2 k 1 ( p − 1 ) ≤ C ( [ w ( r ) p − 1 − α ] ′ ) k 1 p + p − 1 − k 2 k 1 ( p − 1 )

for all r ∈ ( ρ , R 0 ) , yielding

(3.9) C ≤ [ w ( r ) p − 1 − α ] ′ ∫ 0 w ( r ) p − 1 − α H θ ( t ) 1 p − 1 d t k 1 ( p − 1 ) k 1 p + p − 1 − k 2 .

By Integrating (3.9) over [ ρ , r ] and letting r → R 0 − , we find

(3.10) C ( R 0 − ρ ) ≤ ∫ w ( ρ ) p − 1 − α w ( R 0 − ) p − 1 − α d s ∫ 0 s H θ ( t ) 1 p − 1 d t k 1 ( p − 1 ) k 1 p + p − 1 − k 2 ≤ ∫ w ( ρ ) p − 1 − α ∞ d s ∫ 0 s H θ ( t ) 1 p − 1 d t k 1 ( p − 1 ) k 1 p + p − 1 − k 2 ,

where w ( R 0 − ) ∈ R + ∪ { ∞ } is understood to mean lim r → R 0 − w ( r ) . If (1.7) does not hold, then Lemma 2.5 together with (3.10) implies the maximal interval of existence [ 0 , R 0 ) is finite. By Lemmas 2.1 and 2.4, this implies v ( R 0 − ) = ∞ , proving part 1.

Turning now to part 2, assume that (3.1) holds, and let Φ : ( 0 , ∞ ) → ( 0 , ∞ ) be defined as follows:

Φ ( t ) ≔ ∫ t ∞ ℋ ( s ) d s .

We note that Φ is decreasing and by part 1, we have lim t → ∞ Φ ( t ) = 0 . From (3.5) and (3.10), we see that there exists ρ ∈ ( 0 , R ) such that

C 1 ( R − r ) ≤ Φ ( w ( r ) p − 1 − α ) ≤ C 2 ( R − r )

for all r ∈ [ ρ , R ) and some constants C 2 > C 1 > 0 . Since Φ is decreasing, this implies

Φ − 1 ( C 2 ( R − r ) ) θ ≤ w ( r ) ≤ Φ − 1 ( C 1 ( R − r ) ) θ ,

where again θ = 1 p − 1 − α . Now, recalling that for all r ∈ [ ρ , R )

u ( r ) = u ( ρ ) + ∫ ρ r w ( t ) d t ,

we see that lim r → R − u ( r ) < ∞ if and only if

∫ ρ R w ( t ) d t < ∞ ,

which holds if and only if

∫ ρ R Φ − 1 ( C ( R − t ) ) θ d t < ∞

for some C > 0 . Hence, if lim r → R − u ( r ) < ∞ , after a change of variables, we see

∫ 0 C ( R − ρ ) Φ − 1 ( s ) θ d s < ∞ ,

which yields

∫ 0 1 Φ − 1 ( s ) θ d s < ∞ .

The change of variables t = Φ − 1 ( s ) gives us that lim r → R − u ( r ) < ∞ if and only if

∫ 1 ∞ s θ ℋ ( s ) d s < ∞ ,

finishing the proof. □

Remark 3.1

Some possible extensions of the aforementioned results would be to consider nonlinear terms which are not separable, that is systems of the form

Δ p u = f ( ∣ x ∣ , v ) in Ω , Δ p v = g ( ∣ x ∣ , ∣ ∇ u ∣ ) in Ω ,

or to somehow relax the growth assumption (A2). Even in the semilinear case p = 2 , the analysis becomes quite technical, and the results obtained are no longer sharp. For this reason, we do not discuss such extensions here.

4 Proof of Theorem 1.2

First assume that (1.7) holds and 0 ≤ α < p − 1 . From the discussion at the beginning of Section 3, we know that when 0 ≤ α < p − 1 we are able to construct a non-constant positive radial solution in a maximal ball. By Lemma 2.1, both u and v are increasing, and by Lemma 2.4 and Theorem 1.1 part 1, we know that u and v are bounded when (1.7) holds. Hence, the domain of existence must be all of R n .

Conversely, we know from Lemma 2.2 that if α ≥ p − 1 , then no positive radial solution of (1.1) exists in B R for any R > 0 . So suppose now that 0 ≤ α < p − 1 , (1.7) does not hold, and ( U , V ) is a global positive radial solution of (1.1). We consider the related system

(4.1) Δ p u = f ˜ 1 ( ∣ x ∣ ) g ˜ 1 ( v ) ∣ ∇ u ∣ α in Ω , Δ p v = f ˜ 2 ( ∣ x ∣ ) g ˜ 2 ( v ) h ˜ ( ∣ ∇ u ∣ ) in Ω ,

where

f ˜ 1 ( r ) = λ p − 1 − α f 1 ( λ r ) f ˜ 2 ( r ) = λ ( p − 1 ) ( 1 − c ) f 2 ( λ r ) g ˜ 1 ( t ) = g 1 ( λ c t ) g ˜ 2 ( t ) = g 2 ( λ c t ) h ˜ ( t ) = h t λ

for some 0 < c < 1 and λ > 0 . Then ( U λ ( r ) , V λ ( r ) ) = ( U ( λ r ) , λ − c V ( λ r ) ) is a global positive radial solution of (4.1). The system (4.1) also satisfies the hypotheses (A1) and (A2), so by Theorem 1.1, since (1.7) does not hold, there exists a positive radial solution ( u , v ) of (4.1) satisfying, without loss of generality, lim r → 1 − v ( r ) = ∞ . By letting λ > 0 be sufficiently small, we may assume V λ ( 0 ) > v ( 0 ) > 0 . Let ( 0 , R 0 ) ⊂ ( 0 , 1 ) be the maximal interval on which

(4.2) V λ ( r ) > v ( r ) for all 0 < r < R 0 .

Now, since both ( U λ , V λ ) and ( u , v ) solve system (4.1), if we rewrite system (4.1) in an analogous manner to (2.4), we see that ( U λ , V λ ) and ( u , v ) also satisfy

[ r δ ( U ′ ( r ) p − 1 − α − u ′ ( r ) p − 1 − α ) ] ′ = δ n − 1 r δ f ˜ 1 ( r ) ⋅ [ g ˜ 1 ( V λ ) − g ˜ 1 ( v ) ] ,

with δ as defined in (2.5). Along the interval ( 0 , R 0 ) , we have V λ > v > 0 by (4.2), so g ˜ 1 ( V λ ) > g ˜ 1 ( v ) by (A2). Since f ˜ 1 ( r ) is positive by (A1), the right-hand side above is clearly positive, so it follows that U ′ ( r ) > u ′ ( r ) for all 0 < r < R 0 , recalling that p − 1 − α > 0 .

When it comes to the functions V λ , v , by using the second equation in (4.1), we obtain

(4.3) [ r n − 1 ( V ′ ( r ) p − 1 − v ′ ( r ) p − 1 ) ] ′ = r n − 1 f ˜ 2 ( r ) ⋅ [ g ˜ 2 ( V λ ) h ˜ ( U ′ ) − g ˜ 2 ( v ) h ˜ ( u ′ ) ] .

Since U ′ > u ′ on the interval ( 0 , R 0 ) by the aforementioned equation, we have h ˜ ( U ′ ) ≥ h ˜ ( u ′ ) on ( 0 , R 0 ) by (A1). In addition, V λ > v on ( 0 , R 0 ) by (4.2), and thus, g ˜ 2 ( V λ ) ≥ g ˜ 2 ( v ) on ( 0 , R 0 ) by (A1), showing that the right-hand side of (4.3) is non-negative. If the right-hand side of (4.3) is identically zero, then since V ′ ( 0 ) = v ′ ( 0 ) = 0 , we in fact have V ′ ( r ) = v ′ ( r ) on ( 0 , R 0 ) . If the right-hand side of (4.3) becomes positive on ( 0 , R 0 ) , we may argue similarly to above to conclude V ′ ( r ) ≥ v ′ ( r ) for all 0 < r < R 0 . So in both cases, we have V ′ ( r ) ≥ v ′ ( r ) for all 0 < r < R 0 . But since V λ ( r ) > v ( r ) on ( 0 , R 0 ) , it cannot possibly be the case that v ( r ) ≥ V λ ( r ) at r = R 0 . In other words, the maximal interval on which (4.2) holds coincides with the maximal interval on which v ( r ) is defined, implying lim r → 1 − V λ ( r ) ≥ lim r → 1 − v ( r ) = ∞ , which contradicts the fact that ( U λ , V λ ) is a global solution of (4.1).□

Acknowledgments

The author thanks Paschalis Karageorgis and Gurpreet Singh for their helpful discussions and comments during the preparation of this article.

Funding information: The research conducted in this publication was funded by the Irish Research Council, under grant number GOIPG/2022/469.
Author contribution: The author confirms the sole responsibility for the conception of the study, presented results and manuscript preparation.
Conflict of interest: The author states no conflict of interest.

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Received: 2024-08-05

Revised: 2025-01-07

Accepted: 2025-03-24

Published Online: 2025-07-29

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Keywords for this article

radial solutions; elliptic systems; coercive systems; p-Laplace operator; nonlinear gradient terms

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