In this paper we are interested in the study of existence of positive solutions for a semilinear elliptic equation with logistic type nonlinearity and harvesting under the so-called strong growth assumption. More precisely, we consider the following equation arising in modeling population biology of one species:
Here Ω is a bounded smooth domain in , , and u represents the population density of the species whose growth follows a logistics model. The positive constant a is the so-called linear growth rate and the nonnegative crowding coefficient is assumed to be spatially dependent due to the heterogeneity of the environment. Finally, the last term on the right-hand side models the presence of a constant yield harvesting pattern (see  for more details). The existence and the structure of the positive solution set of (1.1) has been extensively studied under various assumptions on and h (see [9, 15, 12, 13, 14]). Here we are mainly interested in the so-called degenerate logistic case, where , and the zero set of b is the closure of some suitably regular sub-domain , that is,
so that our model is a mix of logistic and Malthusian models.
To properly set up our problem and give a review of the state of affairs regarding this equation we start with a few words about the notation. For a bounded smooth domain O in we let denote the i-th eigenvalue of over the region O with Dirichlet boundary condition. We omit the potential ϕ and write if . Furthermore, for a solution u of equation (1.1), we let denote the i-th eigenvalue of the linearization of (1.1) at u, that is, . Following the classical terminology, u will be called stable if , and unstable if . We also recall the well-known fact that
F̱or the case , that is, in the absence of harvesting, the equation
Assume in Ω.
If , then for every , equation ( 1.2 ) has a unique positive solution .
Furthermore, in either case the curve is continuous and increasing and the positive solution is stable, i.e., .
This result indicates that equation (1.2) behaves similar to the logistic model for , but a dramatic change occurs as the linear growth rate a crosses the threshold value . In fact, as a approaches the critical value , the degeneracy of the crowding coefficient in causes the solution to blow up in . More precisely, we have the following result.
Theorem 1.2 ([8, Theorem 3.6]).
Let . Then the following hold:
uniformly on as .
uniformly as on any compact subset of , where is the minimal positive solution of the boundary blow-up equation
Following the terminology of , we call and , the weak and strong growth rate case, respectively. As for the existence of positive solutions in the presence of harvesting , Oruganti et al. in  considered the case of . Their results were then extended to the degenerate logistic case considered here in the weak growth rate regime in . The following two theorems summarize the main results in this case.
Theorem 1.3 ([18, Theorem 2.6]).
Suppose that , and . Then there exists such that the following hold:
Theorem 1.4 ([18, Theorem 2.8]).
Under the assumptions of Theorem 1.3, there exists such that for , the following hold:
Equation ( 1.1 ) has exactly two positive solutions and for , exactly one positive solution with , and no positive solution for .
The Morse index is 1 for when , and is degenerate with .
All solutions lie on a smooth curve Σ that, on space, starts from , continues to the right, reaches the unique turning point at where it turns back, then continues to the left without any turnings until it reaches , where is the unique positive solution of ( 1.1 ) with .
At this point it is worth making a few comments. Firstly, we note that under the weak growth rate assumption, any positive solution of equation (1.1) is a sub-solution of (1.2), and therefore, by a classical comparison result, is point wise bounded by the unique solution of equation (1.2), that is, for . This observation plays a crucial role in the proof of both existence and nonexistence results of Theorem 1.3 above. In particular, in the light of this uniform (with respect to c) point wise bound, the nonexistence result for positive solutions and c large, that is, the fact that equation (1.1) does not have a positive solution as c crosses the critical value is rather obvious. In fact, as positive solutions remain uniformly bounded, heuristically one does not expect survival of the species (i.e., existence of a positive density distribution u) as the harvesting rate c approaches infinity.
However, we note that if , then such a uniform point wise bound is not available. In fact, as mentioned before, as a increases toward , the solution blows up in impeding existence of a positive solution for the pure logistic equation (1.2) for . On the other hand, it seems reasonable to inquire whether one may be able to offset the absence of crowding effect in through the presence of a strong harvesting term, and therefore prove the existence of a positive solution in this case. To the best of our knowledge, the question of existence of positive solutions to equation (1.1) in the strong growth rate regime has not been considered before, and the above observations were our initial motivation for taking up this study here.
In this work we provide some results in this direction. Since a simple application of the implicit function theorem (see Proposition 2.8) provides the necessary and sufficient condition for the existence of positive solutions for c small, our concentration here is on the question of existence of positive solutions for c large. In particular, we show that under the strong growth rate assumption, in is a necessary condition for existence of positive solutions as . Furthermore, under the same condition, we will establish an existence (and somewhat surprisingly) uniqueness result for positive solutions of (1.1) for all c large in a right neighborhood of the threshold growth rate (see Theorem 3.11).
Our approach is based on variational and topological arguments and makes extensive use of classical elliptic estimates and comparison principles. In Section 2 we provide some background and preliminary results and consider the basic setup of our problem. In Section 3 we consider the case in and then finish the proof of our main existence and uniqueness result.
We start with the consideration of equation (1.1), by stating our main hypotheses. Here Ω is a bounded smooth domain in , . The constant c is nonnegative and throughout we assume the following:
and () are nonnegative functions.
There exists a smooth region such that and for , and on .
In this section we first gather some useful background material and then present some preliminary results. Throughout this paper, we will repeatedly use the saddle-node bifurcation result of Crandall and Rabinowitz , which we recall below.
Theorem 2.1 (Saddle-node bifurcation at a turning point ).
Let X and Y be Banach spaces and assume that . Let F be a continuously differentiable mapping of an open neighborhood V of into Y. Suppose that the following hold:
If Z is a complement of in X, then the solutions of near form a curve , where is a continuously differentiable function near and , . Moreover, if F is k times continuously differentiable, then so are and .
Furthermore, by [19, Theorem 2.4], we have
where satisfies .
The following two results provide additional useful information for the pure logistic equation set in a bounded domain with :
Lemma 2.2 ().
Suppose that and are, respectively, sub and super solutions of (2.1), that is,
Furthermore, assume that
Then in Ω.
Next we recall a result of Du and Huang (see ) on the existence of boundary blow-up solutions for
Theorem 2.3 ([8, Theorem 2.4]).
We are finally ready to state our first result. In the following section we will have occasions where knowing the structure of the set of all (not just positive) solutions of equation (1.1) in the weak growth rate case is of great value. The following provides a picture, similar to one obtained in Theorem 1.3 above, for the set of all solutions.
Suppose that and (A1) and (A2) hold. Then there exists such that the following hold:
The curve is decreasing with respect to the parameter c for and is stable, that is, . Furthermore, is the unique stable solution of ( 1.1 ).
For , there exists a degenerate solution , i.e., , and equation ( 1.1 ) has another unstable solution for c near and to the left of .
As arguments similar to the ones used in the proof of this result will be needed repeatedly later on, we will provide all the details here so as to be able to refer to them later on. First note that if equation (1.1) has a solution , then either or the comparison lemma above applied on the set implies that , where is the unique positive solution of equation (1.2). Multiplying equation (1.1) with , the first eigenfunction of in Ω with Dirichlet boundary condition, and then integrating over Ω, we obtain
implying that is well defined. Next for let and . We define by . Note that if , that is, if is a solution of (1.1) with harvesting rate , and if , then by applying the implicit function theorem, we can continue the curve of solutions forward (i.e., for ) with respect to c. Moreover, denoting the curve of solutions by , it will be decreasing as solves the equation
Hence, is decreasing with respect to c as well. Therefore, the curve of solutions starting at , can be continued to the right until a point with . Next applying the saddle-node bifurcation (Theorem 2.1), one easily sees that the curve turns back at the degenerate solution , therefore generating a second unstable solution in a left neighborhood of . Furthermore, the curve of solutions obtained above is indeed the curve of maximal solutions. To see this, first note that if equation (1.1) has a solution at , then (1.1) will have a maximal solution with . Indeed, is a pair of sub-super solutions of (2.1), therefore the comparison lemma above applied on the set implies that . Next, considering as an ordered pair of sub-super solutions of (1.1) for , the standard iteration process starting at the super solution will provide the maximal solution of (1.1) for , which, by construction, will have (see ). Also, as is a sub-solution of (1.1) with , if , it is clear that is decreasing in c and therefore left continuous. In addition, for , since , we have . This implies that the implicit function theorem applies at every point , from which one easily concludes that is right continuous as well. Hence, the curve of maximal solutions is continuous and decreasing for . Finally, as is the unique positive (and therefore maximal) solution of equation (1.1) for , we have , which together with the fact that both curves and are continuous, decreasing and constitute of stable solutions yield for .
Next we will show that equation (1.1) does not have a solution for . Otherwise, if is a solution of (1.1) for some , then as was shown above, (1.1) has a maximal solution at with . Now depending on whether or , we can continue the curve of solutions from backwards (i.e., for ) in an increasing fashion, initially using either the implicit function theorem or the saddle-node bifurcation theorem, and then (as becomes positive) through the implicit function theorem all the way to . Thus, equation (1.1) will have a solution u at with . Since , we get , which is a contradiction. Finally, a similar argument yields uniqueness of stable solutions. The proof of the theorem is now complete. ∎
It is obvious that . In the remarks on page 3614 of , Oruganti, Shi and Shivaji claim, in passing, that . Although the validity of this claim for arbitrary is not clear to us, we are able to show that if in , then for a sufficiently close to (see Lemma 3.5).
The next result, covering the strong growth rate case, is crucial in the proof of our main result in Section 3.
Assume . Then any solution u (not necessary positive) of (1.1) is unstable, that is, .
Multiplying the above equation by , the first eigenfunction of in with Dirichlet boundary condition, and integrating over , we obtain
which readily yields . ∎
Recalling the classical result (see ) that a solution u obtained between an ordered pair of sub and super solutions through the classical iteration procedure (starting at the sub or the super solution) will automatically satisfy , the above result, in particular, indicates that in our search for positive solutions of (1.1) in the strong growth rate case, we cannot utilize the technique of sub and super solutions.
We end this section by providing a necessary and sufficient condition for existence of positive solutions of (1.1) for c small. In fact, since for , is the unique nonnegative solution of (1.1) with , a simple application of implicit function theorem yields the following result.
Assume (A1) and (A2), and suppose that and for all . Then equation (1.1) has a positive solution for (with some ) if and only if the linear equation
has a positive solution.
In regards to the existence of a positive solution to (2.3), we may write , where is the first eigenfunction of in Ω with Dirichlet boundary condition and . Then clearly v, the unique solution of equation (2.3), is given by
Therefore, there exists , depending on a and , such that if , then .
3 Existence and nonexistence results in the strong growth rate case
In this section we will consider the existence of positive solutions of equation (1.1) for and c large. Our first result provides a necessary condition for existence of positive solutions for c large.
Assume that and in . Then there exists such that equation (1.1) does not have a nonnegative solution for .
Let u be a nonnegative solution of (1.1). Using the comparison Lemma 2.2, we have in , where is the minimal boundary blow-up solution given in Theorem 2.3. Let . Clearly, in for all n large. Fix such an n and let be the first eigenfunction of in with Dirichlet boundary condition. Multiplying equation (1.1) with and then integrating over , we obtain
where ν is the outer normal vector of . Therefore, we have
if , and
if . Therefore, c has a bound independent of u. ∎
In the rest of this section we will assume that in . In order to prove the existence of positive solutions of (1.1) for and c large, we first show that (1.1) has an unstable solution for large c as a approaches from bellow. Using this and the fact that unstable solutions can not blow up as , we prove the existence of a positive solution for by considering the limit of these unstable solutions. Next, using the degree theory on cones we are able to establish the same result for a in a right neighborhood of . We start this procedure by first considering the behavior of as .
Assume that in . Then as .
Fix arbitrary large. We will show that (1.1) has a positive solution for a sufficiently close by constructing an ordered pair of sub and super solutions. Clearly, is a super solution of (1.1). To construct a sub-solution, we consider
where m and n are large positive constants (given below), denotes the first eigenfunction of in with Dirichlet boundary condition, and is the positive solution of the equation
Now we assume that for some small. First we take n large enough such that . Next, using the fact that , we choose m sufficiently large so that on . Finally, we choose a close enough to such that in and . Then is a weak sub-solution and, in addition, (note that on ). So, equation (1.1) has a positive solution between and . This completes the proof. ∎
Before stating our next result, we recall that by the anti-maximum principle (see ), there exists such that for , the following equation has a unique positive solution :
There exists such that if is a sequence of nonnegative solutions of equation (1.1), with , such that , and . Then uniformly on .
We note that this lemma is in fact valid for arbitrary nonnegative .
Up to a subsequence, with . We divide the proof into several steps. Step 1: uniformly in every compact subsets of . Note that . Thus, up to a subsequence, . Let . From equation (1.1) we have
Since and , there exists such that weakly in and strongly in for . Multiplying (3.2) by and then integrating over Ω, we get
Thus, , and so in .
Next, multiplying (3.2) by and integrating over , we have
Passing n to infinity, we obtain
Therefore, is a nonnegative weak solution of the equation
If , then, thanks to the Fredholm alternative, we have , since and . Therefore, or in (recall that is the first eigenfunction of with Dirichlet boundary condition in ).
We claim that . In fact, if , then, by equation (3.2), , so that
uniformly in Ω, as in for . This contradicts .
Next if , then assuming that , we have , as . But this will again yield a contradiction as above. Thus, and , the unique positive solution of equation (3.1).
Now standard elliptic estimates (see ) imply that uniformly in every compact subset of , completing the proof of step 1.
Next we let and show that . This will be done in the next two steps. First note that since by assumption is , it satisfies a uniform interior ball assumption, i.e., there exists such that for every , there exists a ball with radius R and center y such that and . Let K be a compact subset of such that for all .
Step 2: Let . If is bounded, then for all n large. Using some of the ideas of the proof of [8, Lemma 3.3], we argue by contradiction by assuming that for a subsequence, still denoted by , we have . Then step 1 implies . Hence, there exists such that . Next we consider the auxiliary functions
and show that there exists a sequence such that
To start with, we fix a suitably chosen large , so that satisfy the following properties on :
Note that the choice of σ depends only on R and the function . Let . The rest of the proof proceeds by considering the two cases and separately.
If , since in and , then . Hence, we can choose so that . Therefore, for , we have
and, in addition,
In the second case, that is, , by step 1 we have and . Thus,
Hence, by further decreasing δ, we can guarantee that
Now we may proceed as in the previous case, obtaining (3.3) and a contradiction as before. The proof of step 2 is now complete.
Step 3: Let . Then . To argue by contradiction, we assume that is bounded, i.e., for some independent of n. Then, by step 2, we have . We follow the argument in step 2 where now , and conclude
for some sequence . Lemma 2.3 in  implies that the equation
has a unique positive solution and by the comparison lemma, in . Similarly, w, the unique positive solution of
satisfies in . Thus, is bounded, and therefore standard elliptic estimates imply that is bounded in , and so, in particular, remains uniformly bounded. Since and in , we have
contradicting (3.7). This completes the proof of step 3, and therefore the proof of the lemma. ∎
At this point for , we define
where is the boundary of the cone
and denotes the subspace of functions in which are zero on the boundary of Ω. Note that is a nonnegative solution of (1.1) with , so is well defined and . In the following lemma we show that does not go to infinity as .
Assume on the contrary that there exists a sequence such that and . Let be a sequence of nonnegative solutions of equation (1.1) with and for some . Since , by Lemma 3.3, we have uniformly in . Thus, for large n, we have on . Now clearly we can choose large so that
Hence, by the maximum principle and the Hopf boundary lemma in Ω and on , contradicting . ∎
Our next result shows that for equation (1.1) has an unstable positive solution.
There exists such that if and , then (1.1) has an unstable positive solution.
Since as a approaches , Lemma 3.4 implies the existence of such that if . Next, given such an a, we fix and . For the positive constants σ, and , we define
where is the first eigenfunction of in Ω with Dirichlet boundary condition and is such that
Firstly, taking into account the definition of , a simple limiting argument implies that for σ sufficiently small, equation (1.1) has no solution on for (where denotes the relative boundary of in P). Fixing such a σ, we then take sufficiently large so that maps into P. Now for , we have
Thus, (3.8) and (3.9) imply that for , equation (1.1) has another solution in which, by the uniqueness of stable solutions, satisfies . We claim that in fact . Indeed, if , then applying the saddle-node bifurcation theorem, there exist and an open neighborhood of , such that the solution set of (1.1) in is an -shaped curve. Moreover, the upper part of the curve consists of stable solutions. However, by the uniqueness of the curve of stable solutions, this can only happen if . This completes the proof. ∎
Assume that . There exists such that for all , equation (1.1) has a positive solution.
First fix . Lemma 3.5 implies the existence of a sequence of unstable positive solutions of equation (1.1) with and . We claim that is bounded. Otherwise, , and therefore by Lemma 3.3, uniformly in . Hence, there exist and such that in and . Now an application of the comparison lemma in easily implies in all of Ω (note that is a super solution and a solution of the logistic equation on ). Also, as , we have that is a sub-solution of (1.1) with . Therefore, equation (1.1) for has a minimal solution between and with (see [17, Theorem 4.1]). Hence, , contradicting the fact that is unstable.
Therefore, is bounded and weakly in and strongly in for some . Obviously, is a nonnegative solution of (1.1) with .
Therefore, for all , equation (1.1) has a nonnegative solution . Since , by Lemma 3.3, we get uniformly in as . Therefore, there exists such that in for , and a further application of the maximum principle in (see the proof of Lemma 3.4) yields in Ω. ∎
Fix an a. Assume on the contrary that is a sequence of solutions of (1.1) corresponding to and for some . From Lemma 3.3, we have uniformly in . Thus, for a subsequence in and therefore (by an application of the comparison lemma, as in the proof of Theorem 3.6) in all of Ω. Since , we have for all and . On the other hand, since , we have . Therefore, there exists a fixed such that for all . This contradicts the fact that the sequence is strictly increasing, and therefore so is . ∎
Fix an a. Assume on the contrary that and are two sequences of positive solutions of equation (1.1) corresponding to . If we subtract the equations for and , then we obtain that for some . By Lemma 3.3, we have uniformly in . Thus, there exists a subsequence such that in , and therefore by the comparison lemma in all of Ω. Therefore, for all and . On the other hand, since , we have . We can now continue as in the proof of Lemma 3.7 and reach a contradiction as before. ∎
The next two results prepare the ground for the application of degree theory arguments in order to prove our main existence result on positive solutions of (1.1) for all and c large. In what follows, we let
Note that the above results imply that equation (1.1) for and has a unique positive solution which, in addition, is nondegenerate.
Assume on the contrary that there exists a sequence of nonnegative solutions of (1.1) with and , where , and . Now, by Lemma 3.3, we have in . Denoting the unique positive solution of (1.1) for and by , we have for large, in , and then by the comparison lemma in all of Ω (note that is a super solution and a solution of the logistic equation on ). Hence, is an ordered pair of sub-super solution of (1.1) with and , and therefore (1.1) has a solution (achieved with the iteration process starting at ) between and with . This contradicts Theorem 2.6. ∎
Assume on the contrary that is a sequence of solutions of (1.1) with and with and . By Lemma 3.3, uniformly in , and therefore, by the maximum principle (as in the proof of Lemma 3.4), in Ω for large n, contradicting . ∎
We are now ready to state the main result of this paper.
The only statement that requires a proof is the existence of a positive solution for . The proof uses similar arguments as in the proof of Theorem 3.6, so we shall brief them here. Fixing a , we define
where e and K are positive constants. First notice that by Lemmas 3.9 and 3.10, we can choose e sufficiently small and K sufficiently large so that (1.1) has no solution on (the relative boundary of in P) for . Next we define by
By taking sufficiently large, we may assume that is increasing in for all and that maps into P. Indeed, if and , then we have
for all . The proof is now complete. ∎
Finally, it is an interesting open problem to study existence of positive solutions of (1.1) for all c large when a is large and away from . In particular, for any , it is easily seen (through a familiar limiting argument) that the existence of a positive solution to the equation
is a necessary condition for existence of a positive solutions to (1.1) as . Although the techniques used in this paper seem inadequate to deal with the question of sufficiency of this necessary condition, we do believe that some of the ideas used here should be of value in dealing with this problem.
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About the article
Published Online: 2017-04-19
Citation Information: Advances in Nonlinear Analysis, Volume 8, Issue 1, Pages 455–467, ISSN (Online) 2191-950X, ISSN (Print) 2191-9496, DOI: https://doi.org/10.1515/anona-2016-0208.
© 2019 Walter de Gruyter GmbH, Berlin/Boston. This work is licensed under the Creative Commons Attribution 4.0 Public License. BY 4.0