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# Positive solutions for a class of superlinear semipositone systems on exterior domains

## Abstract

We study the existence of a positive radial solution to the nonlinear eigenvalue problem $−Δu=λ K 1 (|x|)f(v)$ in $Ω e$, $−Δv=λ K 2 (|x|)g(u)$ in $Ω e$, $u(x)=v(x)=0$ if $|x|= r 0$ (>0), $u(x)→0$, $v(x)→0$ as $|x|→∞$, where $λ>0$ is a parameter, $Δu=div(∇u)$ is the Laplace operator, $Ω e ={x∈ R n ∣|x|> r 0 ,n>2}$, and $K i ∈ C 1 ([ r 0 ,∞),(0,∞))$; $i=1,2$ are such that $K i (|x|)→0$ as $|x|→∞$. Here $f,g:[0,∞)→R$ are $C 1$ functions such that they are negative at the origin (semipositone) and superlinear at infinity. We establish the existence of a positive solution for λ small via degree theory and rescaling arguments. We also discuss a non-existence result for $λ≫1$ for the single equations case.

MSC: 34B16, 34B18.

## 1 Introduction

We consider the nonlinear elliptic boundary value problem

(1.1)

where $λ>0$ is a parameter, $Δu=div(∇u)$ is the Laplace operator, and $Ω e ={x∈ R n ∣|x|> r 0 ,n>2}$ is an exterior domain. Here the nonlinearities $f,g:[0,∞)→R$ are $C 1$ functions which satisfy:

(H1):$f(0)<0$ and $g(0)<0$ (semipositone).

(H2): For $i=1,2$ there exist $b i >0$ and $q i >1$ such that $lim s → ∞ f ( s ) s q 1 = b 1$, and $lim s → ∞ g ( s ) s q 2 = b 2$.

Further, for $i=1,2$, the weight functions $K i ∈ C 1 ([ r 0 ,∞),(0,∞))$ are such that $K i (|x|)→0$ as $|x|→∞$. In particular, we are interested in the challenging case, where $K i$ do not decay too fast. Namely, we assume

(H3): There exist $d 1 ˜ >0$, $d 2 ˜ >0$, $ρ∈(0,n−2)$ such that for $i=1,2$

We then establish the following.

### Theorem 1.1

Let (H1)-(H3) hold. Then (1.1) has a positive radial solution$(u,v)$ ($u>0$, $v>0$in$Ω e$) when λ is small, and$∥ u ∥ ∞ →∞$, $∥ v ∥ ∞ →∞$as$λ→0$.

We prove this result via the Leray-Schauder degree theory, by arguments similar to those used in  and . The study of such eigenvalue problems with semipositone structure has been documented to be mathematically challenging (see , ), yet a rich history is developing starting from the 1980s (see –) until recently (see –). In ,  the authors studied such superlinear semipositone problems on bounded domains. In particular, in  the authors studied the system

where Ω is a bounded domain in $R n$, $n≥1$, and establish an existence result when λ is small. The main motivation of this paper is to extend this study in the case of exterior domains (see Theorem 1.1).

We also discuss a non-existence result for the single equation model:

(1.2)

for large values of λ, when $f ˜$, $K 1$ satisfy the following hypotheses:

(H4):$f ˜ ∈ C 1 ([0,∞),R)$, $f ˜ ′ (z)>0$ for all $z>0$, $f ˜ (0)<0$, and there exists $m 0 >0$ such that $lim z → ∞ f ˜ ( z ) z ≥ m 0$.

(H5): The weight function $K 1 ∈ C 1 ([ r 0 ,∞),(0,∞))$ is such that $s − 2 ( n − 1 ) n − 2 K 1 ( r 0 s 1 2 − n )$ is decreasing for $s∈(0,1]$.

We establish the following.

### Theorem 1.2

Let (H3)-(H5) hold. Then (1.2) has no nonnegative radial solution for$λ≫1$.

We establish Theorem 1.2 by recalling various useful properties of solutions established in , where the authors prove a uniqueness result for $λ≫1$ for such an equation in the case when $f ˜$ is sublinear at ∞. However, the properties we recall from  are independent of the growth behavior of $f ˜$ at ∞. Non-existence results for such superlinear semipositone problems on bounded domain also have a considerable history starting from the work in the 1980s in  leading to the recent work in . Here we discuss such a result for the first time on exterior domains.

Finally, we note that the study of radial solutions $(u(r),v(r))$ (with $r=|x|$) of (1.1) corresponds to studying

which can be reduced to the study of solutions $(u(s),v(s))$; $s∈[0,1]$ to the singular system:

$− u ″ ( s ) = λ h 1 ( s ) f ( v ( s ) ) , 0 < s < 1 , − v ″ ( s ) = λ h 2 ( s ) g ( u ( s ) ) , 0 < s < 1 , u ( 0 ) = u ( 1 ) = 0 , v ( 0 ) = v ( 1 ) = 0 , }$
(1.3)

via the Kelvin transformation $s= ( r r 0 ) 2 − n$, where $h i (s)= r 0 2 ( n − 2 ) 2 s − 2 ( n − 1 ) ( n − 2 ) K i ( r 0 s 1 2 − n )$, $i=1,2$ (see ).

### Remark 1.3

The assumption (H3) implies that $lim s → 0 + h i (s)=∞$, for $i=1,2$, $h ˆ = inf t ∈ ( 0 , 1 ) { h 1 (t), h 2 (t)}>0$, and there exist $d>0$, $η∈(0,1)$ such that $h i (s)≤ d s η$ for $s∈(0,1]$, and for $i=1,2$. When in addition (H5) is satisfied, $h 1$ is decreasing in $(0,1]$.

We will prove Theorem 1.1 in Section 2 by studying the singular system (1.3), and Theorem 1.2 in Section 3 by studying the corresponding single equation

$− u ″ ( s ) = λ h 1 ( s ) f ˜ ( u ( s ) ) , 0 < s < 1 , u ( 0 ) = u ( 1 ) = 0 . }$
(1.4)

## 2 Existence result

We first establish some useful results for solutions to the system

$− u ″ ( s ) = b 1 h 1 ( s ) | v ( s ) + l | q 1 , 0 < s < 1 , − v ″ ( s ) = b 2 h 2 ( s ) | u ( s ) + l | q 2 , 0 < s < 1 , u ( 0 ) = u ( 1 ) = 0 , v ( 0 ) = v ( 1 ) = 0 , }$
(2.1)

where $l≥0$ is a parameter. (Clearly, any solution $( u l , v l )$ of (2.1) for $l>0$ must satisfy $u l (s)>0$, $v l (s)>0$ for $s∈(0,1)$. This is also true for any nontrivial solution when $l=0$.) We prove the following.

### Lemma 2.1

1. (i)

There exists $l 0 >0$ such that 2.1 has no solution if $l≥ l 0$.

2. (ii)

For each $l∈[0, l 0 )$, there exists $M>0$ (independent of l) such that if $( u l , v l )$ is a solution of (2.1), then $max{ ∥ u l ∥ ∞ , ∥ v l ∥ ∞ }≤M$.

### Proof of (i)

Let $λ 1 := π 2$, $ϕ 1 :=sin(πs)$. Here $λ 1$ is the principal eigenvalue and $ϕ 1$ a corresponding eigenfunction of $− ϕ ″ (s)=λϕ(s)$ in $(0,1)$ with $ϕ(0)=0=ϕ(1)$. Let $a> λ 1 b 1 b 2 h ˆ$, $c>0$ be such that $( s + l ) q i ≥as−c$ for all $s≥0$ and for $i=1,2$. Now let $( u l , v l )$ be a solution of (2.1). Multiplying (2.1) by $ϕ 1$ and integrating, we obtain

$λ 1 ∫ 0 1 u l ϕ 1 ds= b 1 ∫ 0 1 h 1 (s) ( v l + l ) q 1 ϕ 1 ds≥ b 1 ∫ 0 1 h 1 (s)(a v l −c) ϕ 1 ds$

and

$λ 1 ∫ 0 1 v l ϕ 1 ds= b 2 ∫ 0 1 h 2 (s) ( u l + l ) q 2 ϕ 1 ds≥ b 2 ∫ 0 1 h 2 (s)(a u l −c) ϕ 1 ds.$

By Remark 1.3, $h ˆ = inf t ∈ ( 0 , 1 ) { h 1 (t), h 2 (t)}>0$, and $∥ h i ∥ 1 := ∫ 0 1 h i (s)ds<∞$ for $i=1,2$. Then from the above inequalities we obtain

$∫ 0 1 v l ϕ 1 ds≤ 1 a b 1 h ˆ ( λ 1 ∫ 0 1 u l ϕ 1 d s + b 1 c ∥ h 1 ∥ 1 )$

and

$∫ 0 1 u l ϕ 1 ds≤ 1 a b 2 h ˆ ( λ 1 ∫ 0 1 v l ϕ 1 d s + b 2 c ∥ h 2 ∥ 1 ) .$

Hence we deduce that

$∫ 0 1 u l ϕ 1 ds≤ m 1 m := m 2 ,$

where $m:=(a b 2 h ˆ − λ 1 2 a b 1 h ˆ )$, and $m 1 := λ 1 c ∥ h 1 ∥ 1 a h ˆ + b 2 c ∥ h 2 ∥ 1$. This implies

$∫ 0 1 ( v l + l ) q 1 ϕ 1 ds≤ λ 1 m 2 b 1 h ˆ := m 3 .$

In particular, this implies $∫ 1 4 3 4 l q 1 ds≤ m 3 inf [ 1 4 , 3 4 ] ϕ 1$. Since $m 3$ is independent of l, clearly this is a contradiction for $l≫1$, and hence there must exists an $l 0 >0$ such that for $l≥ l 0$, (2.1) has no solution.

### Proof of (ii)

Assume the contrary. Then without loss of generality we can assume there exists ${ l n }⊂(0, l 0 )$ such that $∥ u l n ∥ ∞ →∞$ as $n→∞$. Clearly $u l n ″ (s)<0$, and $v l n ″ (s)<0$ for all $s∈(0,1)$. Let $s 1 ( l n ) ∈(0,1)$, $s 2 ( l n ) ∈(0,1)$ be the points at which $u l n$ and $v l n$ attain their maximums. Now since $u l n ″ (s)<0$ for all $s∈(0,1)$, we have

Hence $u l n (s)≥min{ s ∥ u l n ∥ ∞ s 1 ( l n ) , ( 1 − s ) ∥ u l n ∥ ∞ 1 − s 1 ( l n ) }$, and in particular, for $s∈[ 1 4 , 3 4 ]$,

$u l n (s)≥min { 1 4 ∥ u l n ∥ ∞ , 1 4 ∥ u l n ∥ ∞ } = 1 4 ∥ u l n ∥ ∞ .$

Let $s l n ˜ , s l n ¯ ∈[ 1 4 , 3 4 ]$ be such that $min [ 1 4 , 3 4 ] u l n (s)= u l n ( s l n ˜ )$, and $min [ 1 4 , 3 4 ] v l n (s)= v l n ( s l n ¯ )$. Now for $s∈[ 1 4 , 3 4 ]$,

$v l n (s)≥ b 2 h ˆ m ˜ ∫ 1 4 3 4 | u l n ( t ) + l | q 2 dt,$

where $m ˜ := min [ 1 4 , 3 4 ] × [ 1 4 , 3 4 ] G(s,t)$ (>0), and G is the Green’s function of $− Z ″$ with $Z(0)=0=Z(1)$. In particular, $v l n ( s l n ¯ )≥ b 2 h ˆ m ˜ 2 ( u l n ( s l n ˜ ) ) q 2$. Similarly $u l n ( s l n ˜ )≥ b 1 h ˆ m 2 ( v l n ( s l n ¯ ) ) q 1$. Hence, there exists a constant $A>0$ such that

$u l n ( s l n ˜ )≥A ( u l n ( s l n ˜ ) ) q 1 q 2 .$

This is a contradiction since $q 1 q 2 >1$ and $u l n ( s l n ˜ )≥ 1 4 ∥ u l n ∥ ∞ →∞$ as $n→∞$. Thus (ii) holds. □

### Proof of Theorem 1.1

We first extend f and g as even functions on by setting $f(−s)=f(s)$ and $g(−s)=g(s)$. Then we use the rescaling, $λ= γ δ$, $w 1 =γu$, and $w 2 = γ θ v$ with $γ>0$, $θ= q 2 + 1 q 1 + 1$, and $δ= q 1 q 2 − 1 q 1 + 1$. With this rescaling, (1.3) reduces to

$− w 1 ″ ( s ) = F ( s , γ , w 2 ) , 0 < s < 1 , − w 2 ″ ( s ) = G ( s , γ , w 1 ) , 0 < s < 1 , w 1 ( 0 ) = w 1 ( 1 ) = 0 , w 2 ( 0 ) = w 2 ( 1 ) = 0 , }$
(2.2)

where

$F ( s , γ , w 2 ) : = γ 1 + δ h 1 ( s ) ( f ( w 2 γ θ ) − b 1 | w 2 γ θ | q 1 ) + b 1 | w 2 | q 1 h 1 ( s ) , and G ( s , γ , w 1 ) : = γ θ + δ h 2 ( s ) ( g ( w 1 γ ) − b 2 | w 1 γ | q 2 ) + b 2 | w 1 | q 2 h 2 ( s ) .$

Note that by our hypothesis (H2), $F(s,γ, w 2 )→ b 1 | w 2 | q 1 h 1 (s)$ and $G(s,γ, w 1 )→ b 2 | w 1 | q 2 h 2 (s)$ as $γ→0$. Hence we can continuously extend $F(s,γ, w 2 )$ and $G(s,γ, w 1 )$ to $F(s,0, w 2 )= b 1 | w 2 | q 1 h 1 (s)$ and $G(s,0, w 1 )= b 2 | w 1 | q 2 h 2 (s)$, respectively. Note that proving (1.3) has a positive solution for λ small is equivalent to proving (2.2) has a solution $( w 1 , w 2 )$ with $w 1 >0$, $w 2 >0$ in $(0,1)$ for small $γ>0$. We will achieve this by establishing that the limiting equation (when $γ=0$)

$− w 1 ″ ( s ) = F ( s , 0 , w 2 ) = b 1 h 1 ( s ) | w 2 | q 1 , 0 < s < 1 , − w 2 ″ ( s ) = G ( s , 0 , w 1 ) = b 2 h 2 ( s ) | w 1 | q 2 , 0 < s < 1 , w 1 ( 0 ) = w 1 ( 1 ) = 0 , w 2 ( 0 ) = w 2 ( 1 ) = 0 }$
(2.3)

(which is the same as (2.1) with $l=0$) has a positive solution $w 1 >0$, $w 2 >0$ in $(0,1)$ that persists for small $γ>0$.

Let $X= C 0 [0,1]× C 0 [0,1]$ be the Banach space equipped with $∥ w ̲ ∥ X = ∥ ( w 1 , w 2 ) ∥ X =max{ ∥ w 1 ∥ ∞ , ∥ w 2 ∥ ∞ }$, where $∥ ⋅ ∥ ∞$ denotes the usual supremum norm in $C 0 ([0,1])$. Then for fixed $γ≥0$, we define the map $S(γ,⋅):X→X$ by

$S(γ, w ̲ ):= w ̲ − ( K ( F ( s , γ , w 2 ) ) , K ( G ( s , γ , w 1 ) ) ) ,$

where $K(H(s,γ,Z(s)))= ∫ 0 1 G(t,s)H(t,γ,Z(t))dt$. Note that $F(s,γ,⋅),G(s,γ,⋅): C 0 ([0,1])→ L 1 (0,1)$ are continuous and $K: L 1 (0,1)→ C 0 1 ([0,1])$ is compact. Hence $S(γ,⋅)$ is a compact perturbation of the identity. Clearly for $γ>0$, if $S(γ, w ̲ )= 0 ̲$, then $w ̲ =( w 1 , w 2 )$ is a solution of (2.2), and if $S(0, w ̲ )= 0 ̲$, then $w ̲ =( w 1 , w 2 )$ is a solution of (2.3).

We first establish the following.

### Lemma 2.2

There exists$R>0$such that$S(0, w ̲ )≠ 0 ̲$for all$w ̲ =( w 1 , w 2 )∈X$with$∥ w ̲ ∥ X =R$and$deg(S(0,⋅), B R ( 0 ̲ ), 0 ̲ )=0$.

### Proof

Define $S l (0, w ̲ ):X→X$ by

$S l (0, w ̲ ):= w ̲ − ( K ( b 1 h 1 ( s ) | w 2 + l | q 1 ) , K ( b 2 h 2 ( s ) | w 1 + l | q 2 ) )$

for $l≥0$. (Note $S 0 (0, w ̲ )=S(0, w ̲ )$.) By Lemma 2.1, if $l≥ l 0$ then $S l (0, w ̲ )≠ 0 ̲$ and if $S l (0, w ̲ )= 0 ̲$ for $l∈[0, l 0 )$, then $∥ w ∥ X ≤M$. This implies that there exists $R≫1$ such that $S l (0, w ̲ )≠ 0 ̲$ for $w ̲ ∈∂ B R ( 0 ̲ )$ for any $l≥0$. Also, since (2.1) has no solution for $l≥ l 0$, $deg( S l 0 (0,⋅), B R ( 0 ̲ ), 0 ̲ )=0$. Hence, using the homotopy invariance of degree with the parameter $l∈[0, l 0 ]$ we get

$deg ( S ( 0 , ⋅ ) , B R ( 0 ̲ ) , 0 ̲ ) =deg ( S l 0 ( 0 , ⋅ ) , B R ( 0 ̲ ) , 0 ̲ ) =0.$

□

Next we establish the following.

### Lemma 2.3

There exists$r∈(0,R)$small enough such that$S(0, w ̲ )≠ 0 ̲$for all$w ̲ =( w 1 , w 2 )∈X$with$∥ w ̲ ∥ X =r$and$deg(S(0,⋅), B r ( 0 ̲ ), 0 ̲ )=1$.

### Proof

Define $T τ (0, w ̲ ):X→X$ by

$T τ (0, w ̲ ):= w ̲ − ( K ( τ b 1 h 1 ( s ) | w 2 | q 1 ) , K ( τ b 2 h 2 ( s ) | w 1 | q 2 ) )$

for $τ∈[0,1]$. Clearly $T 1 (0, w ̲ )=S(0, w ̲ )$, and $T 0 (0, w ̲ )=I$ is the identity operator. Note that $T τ (0, w ̲ )=0$ if $w ̲ =( w 1 , w 2 )$ is a solution of

$− w 1 ″ ( s ) = τ b 1 h 1 ( s ) | w 2 | q 1 , 0 < s < 1 , − w 2 ″ ( s ) = τ b 2 h 2 ( s ) | w 1 | q 2 , 0 < s < 1 , w 1 ( 0 ) = w 1 ( 1 ) = 0 , w 2 ( 0 ) = w 2 ( 1 ) = 0 , }$
(2.4)

and for $τ=1$, (2.4) coincides with (2.3). Assume to the contrary that (2.4) has a solution $w ̲ =( w 1 , w 2 )$ with $∥ w ̲ ∥ X = r ˜ >0$. Without loss of generality assume $∥ w 1 ∥ ∞ = r ˜$. Now,

$w 1 (s)=τ ∫ 0 1 G(s,t) b 1 h 1 (s) | w 2 | q 1 ds.$

Then $∥ w 1 ∥ ∞ ≤ C ˜ ∥ w 2 ∥ ∞ q 1$ for some constant $C ˜ >0$ independent of $τ∈[0,1]$. Similarly $∥ w 2 ∥ ∞ ≤ C ˆ ∥ w 1 ∥ ∞ q 2$ for some constant $C ˆ >0$. This implies that

$r ˜ = ∥ w 1 ∥ ∞ ≤C ∥ w 1 ∥ ∞ q 1 q 2 =C r ˜ q 1 q 2$

for some constant $C>0$. But $q 1 q 2 >1$, and hence this is a contradiction if $r ˜ >0$ is small. Thus there exists small $r>0$ such that (2.4) has no solution $w ̲$ with $∥ w ̲ ∥ X =r$ for all $τ∈[0,1]$. Now using the homotopy invariance of degree with the parameter $τ∈[0,1]$, in particular using the values $τ=1$ and $τ=0$, we obtain

$deg ( S ( 0 , ⋅ ) , B r ( 0 ̲ ) , 0 ̲ ) =deg ( T 1 ( 0 , ⋅ ) , B r ( 0 ̲ ) , 0 ̲ ) =deg ( T 0 ( 0 , ⋅ ) , B r ( 0 ̲ ) , 0 ̲ ) =1.$

□

By Lemma 2.2 and Lemma 2.3, with $0, we conclude that

$deg ( S ( 0 , ⋅ ) , B R ( 0 ̲ ) ∖ B r ¯ ( 0 ̲ ) , 0 ̲ ) =−1,$

and hence (2.3) has a solution $w ̲ =( w 1 , w 2 )$ with $w 1 >0$, $w 2 >0$ in $(0,1)$, and $r< ∥ w ∥ X . Now we show that the solution obtained above (when $γ=0$) persists for small $γ>0$ and remains positive componentwise.

### Lemma 2.4

Let R, r be as in Lemmas 2.2, 2.3, respectively. Then there exists$γ 0 >0$such that:

1. (i)

$deg(S(γ,⋅), B R ( 0 ̲ )∖ B r ¯ ( 0 ̲ ), 0 ̲ )=−1$ for all $γ∈[0, γ 0 ]$.

2. (ii)

If $S(γ, w ̲ )= 0 ̲$ for $γ∈[0, γ 0 ]$ with $r< ∥ w ̲ ∥ X , then $w 1 >0$, $w 2 >0$ in $(0,1)$.

### Proof of (i)

We first show that there exists $γ 0 >0$ such that $S(γ, w ̲ )≠ 0 ̲$ for all $w ̲ =( w 1 , w 2 )∈X$ with $∥ w ̲ ∥ X ∈{R,r}$, for all $γ∈[0, γ 0 ]$. Suppose to the contrary that there exists ${ γ n }$ with $γ n →0$, $S( γ n , w n ̲ )= 0 ̲$ and $∥ w n ̲ ∥ X ∈{r,R}$. Since $K ̲ =(K,K): L 1 (0,1)× L 1 (0,1)→ C 0 1 ([0,1])× C 0 1 ([0,1])$ is compact, and ${F(s, γ n , w 2 n ),G(s, γ n , w 1 n )}$ are bounded in $L 1 (0,1)× L 1 (0,1)$, $w ̲ n → Z ̲ =( Z 1 , Z 2 )∈ C 0 1 ([0,1])× C 0 1 ([0,1])$ (up to a subsequence) with $∥ Z ̲ ∥ X =R$ or r and $S(0, Z ̲ )= 0 ̲$. This is a contradiction to Lemma 2.2 or 2.3 and hence there exists a small $γ 0 >0$ satisfying the assertions. Now, by the homotopy invariance of degree with respect to $γ∈[0, γ 0 ]$,

$deg ( S ( γ , ⋅ ) , B R ( 0 ̲ ) ∖ B r ¯ ( 0 ̲ ) , 0 ̲ ) =deg ( S ( 0 , ⋅ ) , B R ( 0 ̲ ) ∖ B r ¯ ( 0 ̲ ) , 0 ̲ ) =−1$

for all $γ∈[0, γ 0 ]$.

### Proof of (ii)

Assume to the contrary that there exists $γ n →0$ and a corresponding solution $w n ̲ =( w 1 n , w 2 n )$ such that $r< ∥ w n ̲ ∥ X and

Arguing as before, $w n ̲ → Z ̲ ∈ C 0 1 ([0,1])× C 0 1 ([0,1])$ with $S(0, Z ̲ )= 0 ̲$ (up to a subsequence). Note that $Z ̲ ≢ 0 ̲$ since $∥ Z ̲ ∥ X ≥r>0$. By the strong maximum principle $Z 1 >0$, $Z 2 >0$, $Z 1 ′ (0)>0$, $Z 2 ′ (0)>0$, $Z 1 ′ (1)<0$ and $Z 2 ′ (1)<0$. Now suppose there exists ${ x n }∈(0,1)$ with ${ x n }∈ Ω n$ and $w 1 n ( x n )≤0$. Then ${ x n }$ must have a subsequence (renamed as ${ x n }$ itself) such that $x n → x ˜ ∈[0,1]$. But $Z 1 >0$ in $(0,1)$ implies that $x ˜ ∈{0,1}$. Suppose $x ˜ =0$. Since $w 1 n ( x n )≤0$ and $w 1 n (0)=0$, there exists $y n ∈(0, x n )$ such that $w 1 n ′ ( y n )≤0$, and hence taking the limit as $n→∞$ we will have $Z 1 ′ (0)≤0$, which is a contradiction since $Z 1 ′ (0)>0$. A similar contradiction follows if $x ˜ =1$, using the fact that $Z 1 ′ (1)<0$. Further, contradictions can be achieved if there exists ${ x n }∈Ω$ with ${ x n }∈ Ω n$ and $w 2 n ( x n )≤0$ using the facts that $Z 2 ′ (0)>0$ and $Z 2 ′ (1)<0$. This completes the proof of the lemma. □

We now easily conclude the proof of Theorem 1.1. From Lemma 2.4, since $w ̲ =( w 1 , w 2 )$ is a positive solution of (2.2) for γ small, $(u,v)=( γ − 1 w 1 , γ − θ w 2 )$ with $θ= q 2 + 1 q 1 + 1$ is a positive solution of (1.3) for $λ= γ δ$ where $δ= q 1 q 2 − 1 q 1 + 1$. Further, since $w 1 >0$ and $w 2 >0$ in $(0,1)$ for $γ∈[0, γ 0 ]$, $∥ u ∥ ∞ →∞$ and $∥ v ∥ ∞ →∞$ as $λ(= γ δ )→0$. This completes the proof of Theorem 1.1.  □

## 3 Non-existence result

We first recall from  that, when (H5) is satisfied, one can prove via an energy analysis that a nonnegative solution u of (1.4) must be positive in $(0,1)$ and have a unique interior maximum with maximum value greater than θ, where θ is the unique positive zero of $F ˜ (s)= ∫ 0 s f ˜ (y)dy$. Further, for $λ≫1$ and $s 1 , s 1 ˆ ∈(0,1)$ such that $s 1 ˆ > s 1$, $u( s 1 )=u( s 1 ˆ )=β$ (see Figure 1), where $β>0$ is the unique zero of $f ˜$, there exists a constant C such that $s 1 ≤C λ − 1 2$ and $(1− s 1 ˆ )≤C λ − 1 2$. Hence we can assume $( s 1 ˆ − s 1 )> 1 2$ for $λ≫1$. Now we provide the proof of Theorem 1.2. Figure 1

### Proof of Theorem 1.2

Let $v:=u−β$. Then $v>0$ in $( s 1 , s 1 ˆ )$ and satisfies

$− v ″ = λ h 1 ( s ) f ˜ ( u ) u − β v , s 1 < s < s 1 ˆ , v ( s 1 ) = v ( s 1 ˆ ) = 0 . }$

Note that $ϕ(s)=−(sin( π ( s − s 1 ) ( s 1 ˆ − s 1 ) ))>0$ in $( s 1 , s 1 ˆ )$, $ϕ( s 1 )=ϕ( s 1 ˆ )=0$, and it satisfies $− ϕ ″ = π 2 ( s 1 ˆ − s 1 ) 2 ϕ$ in $( s 1 , s 1 ˆ )$. Hence using the fact that $∫ s 1 s 1 ˆ (−ϕ v ″ +v ϕ ″ )ds=0$, we obtain

$∫ s 1 s 1 ˆ ( λ f ˜ ( u ) u − β h 1 ( s ) − π 2 ( s 1 ˆ − s 1 ) 2 ) vϕds=0.$

In particular,

(3.1)

But $h ˆ = inf ( 0 , 1 ) h 1 (s)>0$, and $( s 1 ˆ − s 1 )> 1 2$ for $λ≫1$. Thus clearly (3.1) can hold when $λ→∞$, only if $Z=u( s λ )→∞$ with $f ˜ ( u ( s λ ) ) u ( s λ ) − β →0$. But by (H4), this is not possible since $lim Z → ∞ f ˜ ( Z ) Z ≥ m 0 >0$. Hence the nonnegative solution cannot exist for $λ≫1$. □

## References

1. 1.

Ambrosetti A, Arcoya D, Buffoni B: Positive solutions for some semi-positone problems via bifurcation theory. Differ. Integral Equ. 1994, 7(3-4):655-663.

2. 2.

Maya C, Girg P: Existence and nonexistence of positive solutions for a class of superlinear semipositone systems. Nonlinear Anal. 2009, 71(10):4984-4996. 10.1016/j.na.2009.03.070

3. 3.

Berestycki H, Caffarelli LA, Nirenberg L: Inequalities for second-order elliptic equations with applications to unbounded domains. I. Duke Math. J. 1996, 81(2):467-494. A celebration of John F. Nash, Jr 10.1215/S0012-7094-96-08117-X

4. 4.

Lions P-L: On the existence of positive solutions of semilinear elliptic equations. SIAM Rev. 1982, 24(4):441-467. 10.1137/1024101

5. 5.

Brown KJ, Shivaji R: Simple proofs of some results in perturbed bifurcation theory. Proc. R. Soc. Edinb., Sect. A 1982/1983, 93(1-2):71-82. 10.1017/S030821050003167X

6. 6.

Castro A, Shivaji R: Nonnegative solutions for a class of nonpositone problems. Proc. R. Soc. Edinb., Sect. A 1988, 108(3-4):291-302. 10.1017/S0308210500014670

7. 7.

Castro A, Shivaji R: Nonnegative solutions to a semilinear Dirichlet problem in a ball are positive and radially symmetric. Commun. Partial Differ. Equ. 1989, 14(8-9):1091-1100. 10.1080/03605308908820645

8. 8.

Hai DD, Sankar L, Shivaji R: Infinite semipositone problems with asymptotically linear growth forcing terms. Differ. Integral Equ. 2012, 25(11-12):1175-1188.

9. 9.

Lee EK, Sankar L, Shivaji R: Positive solutions for infinite semipositone problems on exterior domains. Differ. Integral Equ. 2011, 24(9-10):861-875.

10. 10.

Lee EK, Shivaji R, Ye J: Classes of infinite semipositone systems. Proc. R. Soc. Edinb., Sect. A 2009, 139(4):853-865. 10.1017/S0308210508000255

11. 11.

Sankar L, Sasi S, Shivaji R: Semipositone problems with falling zeros on exterior domains. J. Math. Anal. Appl. 2013, 401(1):146-153. 10.1016/j.jmaa.2012.11.031

12. 12.

Maya C, Girg P: Existence of positive solutions for a class of superlinear semipositone systems. J. Math. Anal. Appl. 2013, 408(2):781-788. 10.1016/j.jmaa.2013.06.041

13. 13.

Castro A, Sankar L, Shivaji R: Uniqueness of nonnegative solutions for semipositone problems on exterior domains. J. Math. Anal. Appl. 2012, 394(1):432-437. 10.1016/j.jmaa.2012.04.005

14. 14.

Brown KJ, Castro A, Shivaji R: Nonexistence of radially symmetric nonnegative solutions for a class of semi-positone problems. Differ. Integral Equ. 1989, 2(4):541-545.

15. 15.

Shivaji R, Ye J:Nonexistence results for classes of $3×3$ elliptic systems. Nonlinear Anal. 2011, 74(4):1485-1494. 10.1016/j.na.2010.10.021

16. 16.

Ko E, Lee EK, Shivaji R: Multiplicity results for classes of singular problems on an exterior domain. Discrete Contin. Dyn. Syst. 2013, 33(11-12):5153-5166. 10.3934/dcds.2013.33.5153

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## Acknowledgements

The third author is funded by the project EXLIZ - CZ.1.07/2.3.00/30.0013, which is co-financed by the European Social Fund and the state budget of the Czech Republic.

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