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Existence results for classes of infinite semipositone problems

  • Jerome GoddardII1,
  • Eun Kyoung Lee2,
  • Lakshmi Sankar3 and
  • R Shivaji4Email author
Boundary Value Problems20132013:97

https://doi.org/10.1186/1687-2770-2013-97

Received: 23 October 2012

Accepted: 5 April 2013

Published: 19 April 2013

Abstract

We consider the problem

{ Δ p u = a u p 1 b u γ 1 c u α , x Ω , u = 0 , x Ω ,

where Δ p u = div ( | u | p 2 u ) , p > 1 , Ω is a smooth bounded domain in R n , a > 0 , b > 0 , c 0 , γ > p and α ( 0 , 1 ) . Given a, b, γ and α, we establish the existence of a positive solution for small values of c. These results are also extended to corresponding exterior domain problems. Also, a bifurcation result for the case c = 0 is presented.

Keywords

Bifurcation DiagramExistence ResultExterior DomainPrincipal EigenvalueSingular Boundary

1 Introduction

Consider the nonsingular boundary value problem:
{ Δ u = a u b u 2 c h ( x ) , x Ω , u = 0 , x Ω ,
(1)

where Ω is a smooth bounded domain in R n , a > 0 , b > 0 , c 0 , Δ u = div ( u ) is the Laplacian of u and h : Ω ¯ R is a C 1 ( Ω ¯ ) function satisfying h ( x ) 0 for x Ω , h ( x ) 0 , max x Ω ¯ h ( x ) = 1 and h ( x ) = 0 for x Ω . Existence of positive solutions of problem (1) was studied in [1]. In particular, it was proved that given an a > λ 1 and b > 0 there exists a c ( a , b , Ω ) > 0 such that for c < c (1) has positive solutions. Here, λ 1 is the first eigenvalue of −Δ with Dirichlet boundary conditions. Nonexistence of a positive solution was also proved when a λ 1 . Later in [2], these results were extended to the case of the p-Laplacian operator, Δ p , where Δ p u = div ( | u | p 2 u ) , p > 1 . Boundary value problems of the form (1) are known as semipositone problems since the nonlinearity f ( s , x ) = a s b s 2 c h ( x ) satisfies f ( 0 , x ) < 0 for some x Ω . See [39] for some existence results for semipositone problems.

In this paper, we study positive solutions to the singular boundary value problem:
{ Δ p u = a u p 1 b u γ 1 c u α , x Ω , u = 0 , x Ω ,
(2)

where Δ p u = div ( | u | p 2 u ) , p > 1 , Ω is a smooth bounded domain in R n , a > 0 , b > 0 , c 0 , α ( 0 , 1 ) , p > 1 , and γ > p . In the literature, problems of the form (2) are referred to as infinite semipositone problems as the nonlinearity f ( s ) = a s p 1 b s γ 1 c s α satisfies lim s 0 + f ( s ) = . One can refer to [1014], and [1517] for some recent existence results of infinite semipositone problems. We establish the following theorem.

Theorem 1.1 Given a , b > 0 , γ > p , and α ( 0 , 1 ) , there exists a constant c 1 = c 1 ( a , b , α , p , γ , Ω ) > 0 such that for c < c 1 , (2) has a positive solution.

Remark 1.1 In the nonsingular case ( α = 0 ), positive solutions exist only when a > λ 1 (the principal eigenvalue) (see [1, 2]). But in the singular case, we establish the existence of a positive solution for any given a > 0 .

Next, we study positive radial solutions to the problem:
{ Δ p u = K ( | x | ) ( a u p 1 b u γ 1 c u α ) , x Ω , u = 0 , if  | x | = r 0 , u 0 , as  | x | ,
(3)
where Ω = { x R n | | x | > r 0 } is an exterior domain, n > p , a > 0 , b > 0 , c 0 , α ( 0 , 1 ) , p > 1 , γ > p and K : [ r 0 , ) ( 0 , ) belongs to a class of continuous functions such that lim r K ( r ) = 0 . By using the transformation: r = | x | and s = ( r r 0 ) n + p p 1 , we reduce (3) to the following boundary value problem:
{ ( | u | p 2 u ) = h ( s ) ( a u p 1 b u γ 1 c u α ) , 0 < s < 1 , u ( 0 ) = u ( 1 ) = 0 ,
(4)

where h ( s ) = ( p 1 n p ) p r 0 p s p ( n 1 ) n p K ( r 0 s ( p 1 ) n p ) . We assume:

( H 1 ) K C ( [ r 0 , ) , ( 0 , ) ) and satisfies K ( r ) < 1 r n + θ for r 1 , and for some θ such that ( n p p 1 ) α < θ < n p p 1 .

With the condition ( H 1 ), h satisfies:
(5)

We note that if θ n p p 1 then h ( s ) is nonsingular at 0 and h C ( [ 0 , 1 ] , ( 0 , ) ) . In this case, problem (4) can be studied using ideas in the proof of Theorem 1.1. Hence, our focus is on the case when θ < n p p 1 in which, h may be singular at 0. Note that in this case h ˆ = inf s ( 0 , 1 ) h ( s ) > 0 .

Remark 1.2 Note that ρ + α < 1 since θ > ( n p p 1 ) α .

We then establish the following theorem.

Theorem 1.2 Given a , b > 0 , γ > p , α ( 0 , 1 ) , and assume ( H 1 ) holds. Then there exists a constant c 2 = c 2 ( a , b , α , p , γ ) > 0 such that for c < c 2 , (3) has a positive radial solution.

Finally, we prove a bifurcation result for the problem
{ Δ p u = a u p 1 b u γ 1 u α , x Ω , u = 0 , on  Ω ,
(6)

where Ω is a smooth bounded domain in R n , a is a positive parameter, b , α > 0 , p > 1 + α and γ > p . We prove the following.

Theorem 1.3 The boundary value problem (6) has a branch of positive solutions bifurcating from the trivial branch of solutions ( a , 0 ) at ( 0 , 0 ) (as shown in Figure 1).
Figure 1
Figure 1

Bifurcation diagram, a vs. u for ( 6 ).

Our results are obtained via the method of sub-super solutions. By a subsolution of (2), we mean a function ψ W 1 , p ( Ω ) C ( Ω ¯ ) that satisfies
{ Ω | ψ | p 2 ψ w d x Ω a ψ p 1 b ψ γ 1 c ψ α w d x , for every  w W , ψ > 0 , in  Ω , ψ = 0 , on  Ω ,
and by a supersolution we mean a function Z W 1 , p ( Ω ) C ( Ω ¯ ) that satisfies:
{ Ω | Z | p 2 Z w d x Ω a Z p 1 b Z γ 1 c Z α w d x , for every  w W , Z > 0 , in  Ω , Z = 0 , on  Ω ,

where W = { ξ C 0 ( Ω ) | ξ 0  in  Ω } . The following lemma was established in [13].

Lemma 1.4 (see [13, 18])

Let ψ be a subsolution of (2) and Z be a supersolution of (2) such that ψ Z in Ω. Then (2) has a solution u such that ψ u Z in Ω.

Finding a positive subsolution, ψ, for such infinite semipositone problems is quite challenging since we need to construct ψ in such a way that lim x Ω Δ p ψ = and Δ p ψ > 0 in a large part of the interior. In this paper, we achieve this by constructing subsolutions of the form ψ = k ϕ 1 β , where k is an appropriate positive constant, β ( 1 , p p 1 ) and ϕ 1 is the eigenfunction corresponding to the first eigenvalue of Δ p ϕ = λ | ϕ | p 2 ϕ in Ω, ϕ = 0 on Ω.

In Sections 2, 3, and 4, we provide proofs of our results. Section 5 is concerned with providing some exact bifurcation diagrams of positive solutions of (2) when Ω = ( 0 , 1 ) and p = 2 .

2 Proof of Theorem 1.1

We first construct a subsolution. Consider the eigenvalue problem Δ p ϕ = λ | ϕ | p 2 ϕ in Ω, ϕ = 0 on Ω. Let ϕ 1 be an eigenfunction corresponding to the first eigenvalue λ 1 such that ϕ 1 > 0 and ϕ 1 = 1 . Also, let δ , m , μ > 0 be such that | ϕ 1 | m in Ω δ and ϕ 1 μ in Ω Ω δ , where Ω δ = { x Ω | d ( x , Ω ) δ } . Let β ( 1 , p p 1 + α ) be fixed. Here, note that since α ( 0 , 1 ) , p p 1 + α > 1 . Choose a k > 0 such that 2 b k γ p + β p 1 λ 1 k α a . Define c 1 = min { k p 1 + α β p 1 ( β 1 ) ( p 1 ) m p , 1 2 k p 1 μ β ( p 1 ) ( a β p 1 λ 1 k α ) } . Note that c 1 > 0 by the choice of k and β. Let ψ = k ϕ 1 β . Then
Δ p ψ = k p 1 β p 1 λ 1 ϕ 1 β ( p 1 ) k p 1 β p 1 ( β 1 ) ( p 1 ) | ϕ 1 | p ϕ 1 p β ( p 1 ) .
To prove ψ is a subsolution, we need to establish:
(7)
in Ω if c < c 1 . To achieve this, we split the term k p 1 β p 1 λ 1 ϕ 1 β ( p 1 ) into three, namely,
k p 1 β p 1 λ 1 ϕ 1 β ( p 1 ) = a k p 1 α ϕ 1 β ( p 1 α ) 1 2 k p 1 α ϕ 1 β ( p 1 α ) ( a k α ϕ 1 α β β p 1 λ 1 ) 1 2 k p 1 α ϕ 1 β ( p 1 α ) ( a k α ϕ 1 α β β p 1 λ 1 ) .
Now to prove (7) holds in Ω, it is enough to show the following three inequalities:
(8)
(9)
(10)
From the choice of k, ( a β p 1 λ 1 k α ) 2 b k γ p , hence,
1 2 k p 1 α ϕ 1 β ( p 1 α ) ( a k α ϕ 1 α β β p 1 λ 1 ) b k γ 1 α ϕ 1 β ( p 1 α ) b k γ 1 α ϕ 1 β ( γ 1 α ) .
(11)
Using ϕ 1 μ in Ω Ω δ and c < 1 2 k p 1 μ β ( p 1 ) ( a β p 1 λ 1 k α )
1 2 k p 1 α ϕ 1 β ( p 1 α ) ( a k α ϕ 1 α β β p 1 λ 1 ) k p 1 ϕ 1 β ( p 1 ) ( a k α λ 1 β p 1 ) 2 k α ϕ 1 α β c k α ϕ 1 α β .
(12)
Finally, since | ϕ 1 | m , in Ω δ , and c < k p 1 + α β p 1 ( β 1 ) ( p 1 ) m p ,
k p 1 β p 1 ( β 1 ) ( p 1 ) | ϕ 1 | p ϕ 1 p β ( p 1 ) k p 1 + α β p 1 ( β 1 ) ( p 1 ) m p k α ϕ 1 α β ϕ 1 p β ( p 1 ) α β c k α ϕ 1 α β ϕ 1 p β ( p 1 + α ) .
Since p β ( p 1 + α ) > 0 ,
k p 1 β p 1 ( β 1 ) ( p 1 ) | ϕ 1 | p ϕ 1 p β ( p 1 ) c k α ϕ 1 α β .
(13)

From (11), (12) and (13) we see that equation (7) holds in Ω, if c < c 1 . Next, we construct a supersolution. Let e be the solution of Δ p e = 1 in Ω , e = 0 on Ω. Choose M ¯ > 0 such that a u p 1 b u γ 1 c u α M ¯ p 1 u > 0 and M ¯ e ψ . Define Z = M ¯ e . Then Z is a supersolution of (2). Thus, Theorem 1.1 is proven.

3 Proof of Theorem 1.2

We begin the proof by constructing a subsolution. Consider
( | ϕ | p 2 ϕ ) = λ | ϕ | p 2 ϕ , t ( 0 , 1 ) , ϕ ( 0 ) = ϕ ( 1 ) = 0 .
(14)
Let ϕ 1 be an eigenfunction corresponding to the first eigenvalue of (14) such that ϕ 1 > 0 and ϕ 1 = 1 . Then there exist d 1 > 0 such that 0 < ϕ 1 ( t ) d 1 t ( 1 t ) for t ( 0 , 1 ) . Also, let ϵ < ϵ 1 and m , μ > 0 be such that | ϕ 1 | m in ( 0 , ϵ ] [ 1 ϵ , 1 ) and ϕ 1 μ in ( ϵ , 1 ϵ ) . Let β ( 1 , p ρ p 1 + α ) be fixed and choose k > 0 such that 2 b k γ p + β p 1 λ 1 k α h ˆ a . Define c 2 = min { k p 1 + α β p 1 ( β 1 ) ( p 1 ) m p d 1 ρ , 1 2 k p 1 μ β ( p 1 ) ( a β p 1 λ 1 k α h ˆ ) } . Then c 2 > 0 by the choice of k and β. Let ψ = k ϕ 1 β . This implies that:
( | ψ | p 2 ψ ) = k p 1 β p 1 λ 1 ϕ 1 β ( p 1 ) k p 1 β p 1 ( β 1 ) ( p 1 ) | ϕ 1 | p ϕ 1 p β ( p 1 ) .
To prove ψ is a subsolution, we need to establish:
(15)
Here, we note that the term k p 1 β p 1 λ 1 ϕ 1 β ( p 1 ) = h ˆ k p 1 β p 1 λ 1 ϕ 1 β ( p 1 ) h ˆ h ( t ) ( a k p 1 α ϕ 1 β ( p 1 α ) 1 2 k p 1 α ϕ 1 β ( p 1 α ) ( a k α ϕ 1 α β β p 1 λ 1 h ˆ ) 1 2 k p 1 α ϕ 1 β ( p 1 α ) ( a k α ϕ 1 α β β p 1 λ 1 h ˆ ) ) , where h ˆ = inf s ( 0 , 1 ) h ( s ) . Now to prove (15) holds in ( 0 , 1 ) , it is enough to show the following three inequalities:
(16)
(17)
(18)
From the choice of k, ( a β p 1 λ 1 k α h ˆ ) 2 b k γ p , hence,
1 2 k p 1 α ϕ 1 β ( p 1 α ) ( a k α ϕ 1 α β β p 1 λ 1 h ˆ ) b k γ 1 α ϕ 1 β ( p 1 α ) b k γ 1 α ϕ 1 β ( γ 1 α ) .
(19)
Using ϕ 1 μ in ( ϵ , 1 ϵ ) and c < 1 2 k p 1 μ β ( p 1 ) ( a β p 1 λ 1 k α h ˆ )
1 2 k p 1 α ϕ 1 β ( p 1 α ) ( a k α ϕ 1 α β β p 1 λ 1 h ˆ ) k p 1 ϕ 1 β ( p 1 ) ( a k α λ 1 β p 1 h ˆ ) 2 k α ϕ 1 α β c k α ϕ 1 α β .
(20)
Next, we prove (18) holds in ( 0 , ϵ ] . Since | ϕ 1 | m , and p β ( p 1 ) > α β + ρ
k p 1 β p 1 ( β 1 ) ( p 1 ) | ϕ 1 | p ϕ 1 p β ( p 1 ) k p 1 + α β p 1 ( β 1 ) ( p 1 ) m p k α ϕ 1 α β ϕ 1 ρ k p 1 + α β p 1 ( β 1 ) ( p 1 ) m p k α ϕ 1 α β d 1 ρ t ρ .
Since h ( t ) 1 t ρ in ( 0 , ϵ ] , and c < k p 1 + α β p 1 ( β 1 ) ( p 1 ) m p d 1 ρ ,
k p 1 β p 1 ( β 1 ) ( p 1 ) | ϕ 1 | p ϕ 1 p β ( p 1 ) c h ( t ) k α ϕ 1 α β .
(21)

Proving (18) holds in [ 1 ϵ , 1 ) is straightforward since h is not singular at t = 1 . Thus, from equations (19), (20) and (21), we see that (15) holds in ( 0 , 1 ) . Hence, ψ is a subsolution. Let Z = M ¯ e where e satisfies ( | e | p 2 e ) = h ( t ) in ( 0 , 1 ) , e ( 0 ) = e ( 1 ) = 0 and M ¯ is such that a u p 1 b u γ 1 c u α M ¯ p 1 u > 0 and M ¯ e ψ . Then Z is a supersolution of (4) and there exists a solution u of (4) such that u [ ψ , Z ] . Thus, Theorem 1.2 is proven.

4 Proof of Theorem 1.3

We first prove (6) has a positive solution for every a > 0 . We begin by constructing a subsolution. Let ϕ 1 be as in the proof of Theorem 1.1 (see Section 2). Let β ( 1 , p p 1 ) , and choose a k > 0 such that b k γ p + β p 1 λ 1 k α a . Let ψ = k ϕ 1 β . Then
Δ p ψ = k p 1 β p 1 λ 1 ϕ 1 β ( p 1 ) k p 1 β p 1 ( β 1 ) ( p 1 ) | ϕ 1 | p ϕ 1 p β ( p 1 ) .
To prove ψ is a subsolution, we will establish:
k p 1 β p 1 λ 1 ϕ 1 β ( p 1 ) a k p 1 α ϕ 1 β ( p 1 α ) b k γ 1 α ϕ 1 β ( γ 1 α )
(22)
in Ω. To achieve this, we rewrite the term k p 1 β p 1 λ 1 ϕ 1 β ( p 1 ) as k p 1 β p 1 λ 1 ϕ 1 β ( p 1 ) = a k p 1 α ϕ 1 β ( p 1 α ) k p 1 α ϕ 1 β ( p 1 α ) ( a k α ϕ 1 α β β p 1 λ 1 ) . Now to prove (22) holds in Ω, it is enough to show k p 1 α ϕ 1 β ( p 1 α ) ( a k α ϕ 1 α β β p 1 λ 1 ) b k γ 1 α ϕ 1 β ( γ 1 α ) . From the choice of k, ( a β p 1 λ 1 k α ) b k γ p , hence,
k p 1 α ϕ 1 β ( p 1 α ) ( a k α ϕ 1 α β β p 1 λ 1 ) b k γ 1 α ϕ 1 β ( p 1 α ) b k γ 1 α ϕ 1 β ( γ 1 α ) .

Thus, ψ is a subsolution. It is easy to see that Z = ( a b ) 1 γ p is a supersolution of (6). Since k, can be chosen small enough, ψ Z . Thus, (6) has a positive solution for every a > 0 . Also, all positive solutions are bounded above by Z. Hence, when a is close to 0, every positive solution of (6) approaches 0. Also, u 0 is a solution for every a. This implies we have a branch of positive solutions bifurcating from the trivial branch of solutions ( a , 0 ) at ( 0 , 0 ) .

5 Numerical results

Consider the boundary value problem
{ u ( x ) = a u b u 2 c u α , x ( 0 , 1 ) , u ( 0 ) = 0 = u ( 1 ) ,
(23)
where a , b > 0 , c 0 and α ( 0 , 1 ) . Using the quadrature method (see [19]), the bifurcation diagram of positive solutions of (23) is given by
G ( ρ , c ) = 0 ρ d s [ 2 ( F ( ρ ) F ( s ) ) ] = 1 2 ,
(24)
where F ( s ) : = 0 s f ( t ) d t where f ( t ) = a t b t 2 c t α and ρ = u ( 1 2 ) = u . We plot the exact bifurcation diagram of positive solutions of (23) using Mathematica. Figure 2 shows bifurcation diagrams of positive solutions of (23) when a = 8 ( < λ 1 ) and b = 1 for different values of α.
Figure 2
Figure 2

Bifurcation diagrams, c vs. ρ for ( 23 ) with a = 8 , b = 1 .

Bifurcation diagrams of positive solutions of (23) when a = 15 ( > λ 1 ) and b = 1 for different values of α is shown in Figure 3.
Figure 3
Figure 3

Bifurcation diagrams, c vs. ρ for ( 23 ) with a = 15 , b = 1 .

Finally, we provide the exact bifurcation diagram for (6) when p = 2 , and Ω = ( 0 , 1 ) . Consider
{ u ( x ) = a u b u 2 u α , x ( 0 , 1 ) , u ( 0 ) = 0 = u ( 1 ) ,
(25)
where a , b , α > 0 . The bifurcation diagram of positive solutions of (25) is given by
G ˜ ( ρ , a ) = 0 ρ d s [ 2 ( F ˜ ( ρ ) F ˜ ( s ) ) ] = 1 2 ,
(26)
where F ˜ ( s ) : = 0 s f ˜ ( t ) d t where f ˜ ( t ) = a t b t 2 t α and ρ = u ( 1 2 ) = u . The bifurcation diagram of positive solutions of (25) as well as the trivial solution branch are shown in Figure 4 when α = 0.5 and b = 1 .
Figure 4
Figure 4

Bifurcation diagram, a vs. ρ for ( 25 ) with α = 0.5 , b = 1 .

Declarations

Acknowledgements

EK Lee was supported by 2-year Research Grant of Pusan National University.

Authors’ Affiliations

(1)
Department of Mathematics, Auburn University Montgomery, Montgomery, USA
(2)
Department of Mathematics Education, Pusan National University, Busan, Korea
(3)
Department of Mathematics & Statistics, Mississippi State University, Mississippi State, USA
(4)
Department of Mathematics & Statistics, University of North Carolina at Greensboro, Greensboro, USA

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© Goddard II et al.; licensee Springer. 2013

This article is published under license to BioMed Central Ltd. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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