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

Boundary Value Problems20142014:198

https://doi.org/10.1186/s13661-014-0198-z

Received: 8 May 2014

Accepted: 5 August 2014

Published: 25 September 2014

Abstract

We study the existence of a positive radial solution to the nonlinear eigenvalue problem Δ u = λ K 1 ( | x | ) f ( v ) in Ω e , Δ v = 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, Δ u = 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.

Keywords

superlinearsemipositonepositive solutionsexistencenon-existenceexterior domains

1 Introduction

We consider the nonlinear elliptic boundary value problem
Δ u = λ K 1 ( | x | ) f ( v ) in  Ω e , Δ v = 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 | , }
(1.1)

where λ > 0 is a parameter, Δ u = 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
d 1 ˜ | x | n + ρ K i ( | x | ) d 2 ˜ | x | n + ρ for  | x | 1 .

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 [1] and [2]. The study of such eigenvalue problems with semipositone structure has been documented to be mathematically challenging (see [3], [4]), yet a rich history is developing starting from the 1980s (see [5]–[7]) until recently (see [8]–[12]). In [1], [2] the authors studied such superlinear semipositone problems on bounded domains. In particular, in [12] the authors studied the system
Δ u = λ f ( v ) in  Ω , Δ v = λ g ( u ) in  Ω , u = v = 0 on  Ω , }

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:
Δ u = λ K 1 ( | x | ) f ˜ ( u ) in  Ω e , u ( x ) = 0 if  | x | = r 0 ( > 0 ) , u ( x ) 0 as  | x | , }
(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 [13], 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 [13] 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 [14] leading to the recent work in [15]. 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
( r n 1 u ( r ) ) = λ r n 1 K 1 ( r ) f ( v ( r ) ) for  r > r 0 , ( r n 1 v ( r ) ) = λ r n 1 K 2 ( r ) g ( u ( r ) ) for  r > r 0 , u ( r ) = v ( r ) = 0 if  r = r 0 ( > 0 ) , u ( r ) 0 , v ( r ) 0 as  r , }
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 [16]).

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 ( π s ) . Here λ 1 is the principal eigenvalue and ϕ 1 a corresponding eigenfunction of ϕ ( s ) = λ ϕ ( 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 a s 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 d s = b 1 0 1 h 1 ( s ) ( v l + l ) q 1 ϕ 1 d s b 1 0 1 h 1 ( s ) ( a v l c ) ϕ 1 d s
and
λ 1 0 1 v l ϕ 1 d s = b 2 0 1 h 2 ( s ) ( u l + l ) q 2 ϕ 1 d s b 2 0 1 h 2 ( s ) ( a u l c ) ϕ 1 d s .
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 ) d s < for i = 1 , 2 . Then from the above inequalities we obtain
0 1 v l ϕ 1 d s 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 d s 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 d s 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 d s λ 1 m 2 b 1 h ˆ : = m 3 .

In particular, this implies 1 4 3 4 l q 1 d s 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
u l n ( s ) { s u l n ( s 1 ( l n ) ) s 1 ( l n ) for  s ( 0 , s 1 ( l n ) ) , ( 1 s ) u l n ( s 1 ( l n ) ) 1 s 1 ( l n ) for  s ( s 1 ( l n ) , 1 ) .
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 d t ,
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 = γ u , 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 ) ) d t . 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 d s .
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 < r < R , 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 < R . 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 < R , 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 < R and
Ω n : = { x ( 0 , 1 ) w 1 n ( x ) 0  or  w 2 n ( x ) 0 } .

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 [13] 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 ) d y . 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
Figure 1

Graph of u .

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 ) d s = 0 , we obtain
s 1 s 1 ˆ ( λ f ˜ ( u ) u β h 1 ( s ) π 2 ( s 1 ˆ s 1 ) 2 ) v ϕ d s = 0 .
In particular,
λ f ˜ ( u ( s λ ) ) u ( s λ ) β h 1 ( s λ ) = π 2 ( s 1 ˆ s 1 ) 2 , for some  s λ ( s 1 , s 1 ˆ ) .
(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 . □

Declarations

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.

Authors’ Affiliations

(1)
Department of Mathematics and Statistics, The University of North Carolina at Greensboro, Greensboro, USA
(2)
Department of Mathematics & NTIS, University of West Bohemia, Plzeň, Czech Republic

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© Abebe et al.; licensee Springer 2014

This article is published under license to BioMed Central Ltd.Open Access This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly credited.

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