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Existence results for classes of infinite semipositone problems
Boundary Value Problems volume 2013, Article number: 97 (2013)
We consider the problem
where , , Ω is a smooth bounded domain in , , , , and . 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 is presented.
Consider the nonsingular boundary value problem:
where Ω is a smooth bounded domain in , , , , is the Laplacian of u and is a function satisfying for , , and for . Existence of positive solutions of problem (1) was studied in . In particular, it was proved that given an and there exists a such that for (1) has positive solutions. Here, is the first eigenvalue of −Δ with Dirichlet boundary conditions. Nonexistence of a positive solution was also proved when . Later in , these results were extended to the case of the p-Laplacian operator, , where , . Boundary value problems of the form (1) are known as semipositone problems since the nonlinearity satisfies for some . See [3–9] for some existence results for semipositone problems.
In this paper, we study positive solutions to the singular boundary value problem:
where , , Ω is a smooth bounded domain in , , , , , , and . In the literature, problems of the form (2) are referred to as infinite semipositone problems as the nonlinearity satisfies . One can refer to [10–14], and [15–17] for some recent existence results of infinite semipositone problems. We establish the following theorem.
Theorem 1.1 Given , , and , there exists a constant such that for , (2) has a positive solution.
Remark 1.1 In the nonsingular case (), positive solutions exist only when (the principal eigenvalue) (see [1, 2]). But in the singular case, we establish the existence of a positive solution for any given .
Next, we study positive radial solutions to the problem:
where is an exterior domain, , , , , , , and belongs to a class of continuous functions such that . By using the transformation: and , we reduce (3) to the following boundary value problem:
where . We assume:
() and satisfies for , and for some θ such that .
With the condition (), h satisfies:
We note that if then is nonsingular at 0 and . 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 in which, h may be singular at 0. Note that in this case .
Remark 1.2 Note that since .
We then establish the following theorem.
Theorem 1.2 Given , , , and assume () holds. Then there exists a constant such that for , (3) has a positive radial solution.
Finally, we prove a bifurcation result for the problem
where Ω is a smooth bounded domain in , a is a positive parameter, , and . 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 at (as shown in Figure 1).
Our results are obtained via the method of sub-super solutions. By a subsolution of (2), we mean a function that satisfies
and by a supersolution we mean a function that satisfies:
where . The following lemma was established in .
Let ψ be a subsolution of (2) and Z be a supersolution of (2) such that in Ω. Then (2) has a solution u such that 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 and in a large part of the interior. In this paper, we achieve this by constructing subsolutions of the form , where k is an appropriate positive constant, and is the eigenfunction corresponding to the first eigenvalue of in Ω, 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 and .
2 Proof of Theorem 1.1
We first construct a subsolution. Consider the eigenvalue problem in Ω, on ∂ Ω. Let be an eigenfunction corresponding to the first eigenvalue such that and . Also, let be such that in and in , where . Let be fixed. Here, note that since , . Choose a such that . Define . Note that by the choice of k and β. Let . Then
To prove ψ is a subsolution, we need to establish:
in Ω if . To achieve this, we split the term into three, namely,
Now to prove (7) holds in Ω, it is enough to show the following three inequalities:
From the choice of k, , hence,
Using in and
Finally, since , in , and ,
From (11), (12) and (13) we see that equation (7) holds in Ω, if . Next, we construct a supersolution. Let e be the solution of in on ∂ Ω. Choose such that and . Define . 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
Let be an eigenfunction corresponding to the first eigenvalue of (14) such that and . Then there exist such that for . Also, let and be such that in and in . Let be fixed and choose such that . Define . Then by the choice of k and β. Let . This implies that:
To prove ψ is a subsolution, we need to establish:
Here, we note that the term ≤ , where . Now to prove (15) holds in , it is enough to show the following three inequalities:
From the choice of k, , hence,
Using in and
Next, we prove (18) holds in . Since , and
Since in , and ,
Proving (18) holds in is straightforward since h is not singular at . Thus, from equations (19), (20) and (21), we see that (15) holds in . Hence, ψ is a subsolution. Let where e satisfies in , and is such that and . Then Z is a supersolution of (4) and there exists a solution u of (4) such that . Thus, Theorem 1.2 is proven.
4 Proof of Theorem 1.3
We first prove (6) has a positive solution for every . We begin by constructing a subsolution. Let be as in the proof of Theorem 1.1 (see Section 2). Let , and choose a such that . Let . Then
To prove ψ is a subsolution, we will establish:
in Ω. To achieve this, we rewrite the term as . Now to prove (22) holds in Ω, it is enough to show . From the choice of k, , hence,
Thus, ψ is a subsolution. It is easy to see that is a supersolution of (6). Since k, can be chosen small enough, . Thus, (6) has a positive solution for every . Also, all positive solutions are bounded above by Z. Hence, when a is close to 0, every positive solution of (6) approaches 0. Also, is a solution for every a. This implies we have a branch of positive solutions bifurcating from the trivial branch of solutions at .
5 Numerical results
Consider the boundary value problem
where , and . Using the quadrature method (see ), the bifurcation diagram of positive solutions of (23) is given by
where where and . We plot the exact bifurcation diagram of positive solutions of (23) using Mathematica. Figure 2 shows bifurcation diagrams of positive solutions of (23) when () and for different values of α.
Bifurcation diagrams of positive solutions of (23) when () and for different values of α is shown in Figure 3.
Finally, we provide the exact bifurcation diagram for (6) when , and . Consider
where . The bifurcation diagram of positive solutions of (25) is given by
where where and . The bifurcation diagram of positive solutions of (25) as well as the trivial solution branch are shown in Figure 4 when and .
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EK Lee was supported by 2-year Research Grant of Pusan National University.
The authors declare that they have no competing interests.
Equal contributions from all authors.
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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
- Bifurcation Diagram
- Existence Result
- Exterior Domain
- Principal Eigenvalue
- Singular Boundary