Skip to main content

Existence results for classes of infinite semipositone problems

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, c0, 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.

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, c0, Δu=div(u) is the Laplacian of u and h: Ω ¯ R is a C 1 ( Ω ¯ ) function satisfying h(x)0 for xΩ, h(x)0, max x Ω ¯ h(x)=1 and h(x)=0 for xΩ. 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)=asb s 2 ch(x) satisfies f(0,x)<0 for some xΩ. 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, c0, 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, c0, 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 ) KC([ r 0 ,),(0,)) and satisfies K(r)< 1 r n + θ for r1, 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 hC([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)|ξ0 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 ψZ in Ω. Then (2) has a solution u such that ψuZ 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 ={xΩ|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 2b k γ p + β p 1 λ 1 k α a. Define c 1 =min{ k p 1 + α β p 1 (β1)(p1) 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 (β1)(p1) | 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 )2b 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 (β1)(p1) 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 pβ(p1+α)>0,

k p 1 β p 1 (β1)(p1) | 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 ¯ eψ. 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(1t) for t(0,1). Also, let ϵ< ϵ 1 and m,μ>0 be such that | ϕ 1 |m in (0,ϵ][1ϵ,1) and ϕ 1 μ in (ϵ,1ϵ). Let β(1, p ρ p 1 + α ) be fixed and choose k>0 such that 2b 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 (β1)(p1) | 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 )2b 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,1ϵ) 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 pβ(p1)>αβ+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 (β1)(p1) | ϕ 1 | p ϕ 1 p β ( p 1 ) c h ( t ) k α ϕ 1 α β .
(21)

Proving (18) holds in [1ϵ,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 ¯ eψ. 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 (β1)(p1) | 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, ψZ. 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, u0 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, c0 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)dt 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)dt 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 .

References

  1. Oruganti S, Shi J, Shivaji R: Diffusive logistic equation with constant harvesting, I: steady states. Trans. Am. Math. Soc. 2002, 354(9):3601-3619. 10.1090/S0002-9947-02-03005-2

    Article  MATH  MathSciNet  Google Scholar 

  2. Oruganti S, Shi J, Shivaji R: Logistic equation with the p -Laplacian and constant yield harvesting. Abstr. Appl. Anal. 2004, 9: 723-727.

    Article  MathSciNet  Google Scholar 

  3. Ambrosetti A, Arcoya D, Biffoni B: Positive solutions for some semipositone problems via bifurcation theory. Differ. Integral Equ. 1994, 7: 655-663.

    MATH  Google Scholar 

  4. Anuradha V, Hai DD, Shivaji R: Existence results for superlinear semipositone boundary value problems. Proc. Am. Math. Soc. 1996, 124(3):757-763. 10.1090/S0002-9939-96-03256-X

    Article  MATH  MathSciNet  Google Scholar 

  5. Arcoya D, Zertiti A: Existence and nonexistence of radially symmetric nonnegative solutions for a class of semipositone problems in an annulus. Rend. Mat. Appl. 1994, 14: 625-646.

    MATH  MathSciNet  Google Scholar 

  6. Castro A, Garner JB, Shivaji R: Existence results for classes of sublinear semipositone problems. Results Math. 1993, 23: 214-220. 10.1007/BF03322297

    Article  MATH  MathSciNet  Google Scholar 

  7. Castro A, Shivaji R: Nonnegative solutions for a class of nonpositone problems. Proc. R. Soc. Edinb. 1998, 108(A):291-302.

    MathSciNet  Google Scholar 

  8. Castro A, Shivaji R: Nonnegative solutions for a class of radially symmetric nonpositone problems. Proc. Am. Math. Soc. 1989, 106(3):735-740.

    MATH  MathSciNet  Google Scholar 

  9. Castro A, Shivaji R: Positive solutions for a concave semipositone Dirichlet problem. Nonlinear Anal. 1998, 31: 91-98. 10.1016/S0362-546X(96)00189-7

    Article  MATH  MathSciNet  Google Scholar 

  10. Ghergu M, Radulescu V: Sublinear singular elliptic problems with two parameters. J. Differ. Equ. 2003, 195: 520-536. 10.1016/S0022-0396(03)00105-0

    Article  MATH  MathSciNet  Google Scholar 

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

    MATH  MathSciNet  Google Scholar 

  12. Hernandez J, Mancebo FJ, Vega JM: Positive solutions for singular nonlinear elliptic equations. Proc. R. Soc. Edinb. 2007, 137A: 41-62.

    MathSciNet  Google Scholar 

  13. Lee E, Shivaji R, Ye J: Classes of infinite semipositone systems. Proc. R. Soc. Edinb. 2009, 139(A):853-865.

    Article  MATH  MathSciNet  Google Scholar 

  14. Lee E, Shivaji R, Ye J: Positive solutions for elliptic equations involving nonlinearities with falling zeros. Appl. Math. Lett. 2009, 22: 846-851. 10.1016/j.aml.2008.08.020

    Article  MATH  MathSciNet  Google Scholar 

  15. Ramaswamy M, Shivaji R, Ye J: Positive solutions for a class of infinite semipositone problems. Differ. Integral Equ. 2007, 20(11):1423-1433.

    MATH  MathSciNet  Google Scholar 

  16. Shi J, Yao M: On a singular nonlinear semilinear elliptic problem. Proc. R. Soc. Edinb. 1998, 128A: 1389-1401.

    Article  MathSciNet  Google Scholar 

  17. Zhang Z: On a Dirichlet problem with a singular nonlinearity. J. Math. Anal. Appl. 1995, 194: 103-113. 10.1006/jmaa.1995.1288

    Article  MATH  MathSciNet  Google Scholar 

  18. Cui S: Existence and nonexistence of positive solutions for singular semilinear elliptic boundary value problems. Nonlinear Anal. 2000, 41: 149-176. 10.1016/S0362-546X(98)00271-5

    Article  MATH  MathSciNet  Google Scholar 

  19. Laetsch T: The number of solutions of a nonlinear two point boundary value problem. Indiana Univ. Math. J. 1970, 20: 1-13. 10.1512/iumj.1970.20.20001

    Article  MATH  MathSciNet  Google Scholar 

Download references

Acknowledgements

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

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to R Shivaji.

Additional information

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

Equal contributions from all authors.

Authors’ original submitted files for images

Rights and permissions

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

Reprints and permissions

About this article

Cite this article

Goddard, J., Lee, E.K., Sankar, L. et al. Existence results for classes of infinite semipositone problems. Bound Value Probl 2013, 97 (2013). https://doi.org/10.1186/1687-2770-2013-97

Download citation

  • Received:

  • Accepted:

  • Published:

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

Keywords