Abstract
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.
1 Introduction
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 [1]. 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 [2], these results were extended to the case of the pLaplacian operator, , where , . Boundary value problems of the form (1) are known as semipositone problems since the nonlinearity satisfies for some . See [39] 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 [1014], and [1517] for some recent existence results of infinite semipositone problems. We establish the following theorem.
Theorem 1.1Given, , and, there exists a constantsuch 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:
() 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 .
We then establish the following theorem.
Theorem 1.2Given, , , and assume () holds. Then there exists a constantsuch 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 1.3The boundary value problem (6) has a branch of positive solutions bifurcating from the trivial branch of solutionsat (as shown in Figure 1).
Figure 1. Bifurcation diagram,avs.for (6).
Our results are obtained via the method of subsuper 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 [13].
Letψbe a subsolution of (2) andZbe a supersolution of (2) such thatin Ω. Then (2) has a solutionusuch thatin 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,
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,
Next, we prove (18) holds in . Since , 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 [19]), 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.
Figure 2. Bifurcation diagrams,cvs.ρfor (23) with,.
Bifurcation diagrams of positive solutions of (23) when () and for different values of α is shown in Figure 3.
Figure 3. Bifurcation diagrams,cvs.ρfor (23) with,.
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 .
Figure 4. Bifurcation diagram,avs.ρfor (25) with,.
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
Equal contributions from all authors.
Acknowledgements
EK Lee was supported by 2year Research Grant of Pusan National University.
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