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Advances in Nonlinear Analysis

Editor-in-Chief: Radulescu, Vicentiu / Squassina, Marco


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Diffusive logistic equations with harvesting and heterogeneity under strong growth rate

Saeed Shabani Rokn-e-vafa / Hossein T. Tehrani
Published Online: 2017-04-19 | DOI: https://doi.org/10.1515/anona-2016-0208

Abstract

We consider the equation

-Δu=au-b(x)u2-ch(x)in Ω,u=0on Ω,

where Ω is a smooth bounded domain in N, b(x) and h(x) are nonnegative functions, and there exists Ω0Ω such that {x:b(x)=0}=Ω¯0. We investigate the existence of positive solutions of this equation for c large under the strong growth rate assumption aλ1(Ω0), where λ1(Ω0) is the first eigenvalue of the -Δ in Ω0 with Dirichlet boundary condition. We show that if h0 in ΩΩ¯0, then our equation has a unique positive solution for all c large, provided that a is in a right neighborhood of λ1(Ω0). For this purpose, we prove and utilize some new results on the positive solution set of this equation in the weak growth rate case.

Keywords: Logistic equation; harvesting; heterogeneity; strong growth rate; comparison principles, stable solutions

MSC 2010: 35J25; 35J61; 92D25

1 Introduction

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:

-Δu=au-b(x)u2-ch(x)in Ω,u=0on Ω.(1.1)

Here Ω is a bounded smooth domain in N, N3, 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 b(x)C(Ω¯) 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 [15] for more details). The existence and the structure of the positive solution set of (1.1) has been extensively studied under various assumptions on a,b and h (see [9, 15, 12, 13, 14]). Here we are mainly interested in the so-called degenerate logistic case, where b(x)0, b0 and the zero set of b is the closure of some suitably regular sub-domain Ω0, that is,

Ω¯0:={x:b(x)=0},Ω0Ω,

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 N we let λi(ϕ,O) denote the i-th eigenvalue of -Δ+ϕ over the region O with Dirichlet boundary condition. We omit the potential ϕ and write λi(O) if ϕ=0. Furthermore, for a solution u of equation (1.1), we let μi(u) denote the i-th eigenvalue of the linearization of (1.1) at u, that is, μi(u)=λi(-a+2b(x)u,Ω). Following the classical terminology, u will be called stable if μ1(u)>0, and unstable if μ1(u)<0. We also recall the well-known fact that

λi(ϕ1,O)<λi(ϕ2,O)if ϕ1ϕ2 and ϕ1ϕ2.

F̱or the case c=0, that is, in the absence of harvesting, the equation

-Δu=au-b(x)u2in Ω,u=0on Ω,(1.2)

has been investigated by a number of authors (cf, [1, 6, 2, 7, 10, 16]) and a complete picture of the structure of the positive solution set is available. Indeed, we have the following theorem.

Theorem 1.1.

Assume b(x)0 in Ω.

  • (i)

    If Ω0= , then for every a>λ1(Ω) , equation ( 1.2 ) has a unique positive solution ua.

  • (ii)

    If Ω0 , then for any a(λ1(Ω),λ1(Ω0)) , equation ( 1.2 ) has a unique positive solution ua . In addition, if aλ1(Ω0) , then ( 1.2 ) has no nonnegative solution except zero.

Furthermore, in either case the curve aua is continuous and increasing and the positive solution ua is stable, i.e., μ1(ua)>0.

This result indicates that equation (1.2) behaves similar to the logistic model for a(λ1(Ω),λ1(Ω0)), but a dramatic change occurs as the linear growth rate a crosses the threshold value λ1(Ω0). In fact, as a approaches the critical value λ1(Ω0), the degeneracy of the crowding coefficient b(x) in Ω0 causes the solution ua to blow up in Ω0. More precisely, we have the following result.

Theorem 1.2 ([8, Theorem 3.6]).

Let a0=λ1(Ω0). Then the following hold:

  • (i)

    ua+ uniformly on Ω¯0 as aa0.

  • (ii)

    uaU¯a0 uniformly as aa0 on any compact subset of Ω¯Ω¯0 , where U¯a0 is the minimal positive solution of the boundary blow-up equation

    -Δu=au-b(x)u2in ΩΩ¯0,u=0on Ω,u=+on Ω0.

Following the terminology of [9], we call a<λ1(Ω0) and aλ1(Ω0), the weak and strong growth rate case, respectively. As for the existence of positive solutions in the presence of harvesting , Oruganti et al. in [15] considered the case of b(x)=b>0. Their results were then extended to the degenerate logistic case considered here in the weak growth rate regime in [18]. The following two theorems summarize the main results in this case.

Theorem 1.3 ([18, Theorem 2.6]).

Suppose that λ1(Ω)<a<λ1(Ω0), b(x)0 and h(x)0. Then there exists c^a>0 such that the following hold:

  • (i)

    If 0c<c^a , then equation ( 1.1 ) has a maximal positive solution u¯a,c . If c>c^a , then no solution of ( 1.1 ) stays positive in Ω.

  • (ii)

    The curve cu¯a,c is decreasing with respect to the parameter c for c[0,c^a) and u¯a,c is stable, that is, μ1(u¯a,c)>0 . Furthermore, u¯a,c is the unique positive stable solution of ( 1.1 ).

Theorem 1.4 ([18, Theorem 2.8]).

Under the assumptions of Theorem 1.3, there exists ϵ>0 such that for a(λ1(Ω),λ1(Ω)+ϵ), the following hold:

  • (i)

    Equation ( 1.1 ) has exactly two positive solutions u¯a,c and u¯a,c for c(0,c^a) , exactly one positive solution u^a with c=c^a , and no positive solution for c>c^a.

  • (ii)

    The Morse index M(u) is 1 for u=u¯a,c when c[0,c^a) , and u^a is degenerate with μ1(u^a)=0.

  • (iii)

    All solutions lie on a smooth curve Σ that, on (c,u) space, starts from (0,0) , continues to the right, reaches the unique turning point at c=c^a where it turns back, then continues to the left without any turnings until it reaches (0,ua) , where ua is the unique positive solution of ( 1.1 ) with c=0.

At this point it is worth making a few comments. Firstly, we note that under the weak growth rate assumption, any positive solution ua,c 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 ua of equation (1.2), that is, ua,c(x)ua(x) for xΩ. 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 c^a 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 aλ1(Ω0), then such a uniform point wise bound is not available. In fact, as mentioned before, as a increases toward λ1(Ω0), the solution ua blows up in Ω¯0 impeding existence of a positive solution for the pure logistic equation (1.2) for aλ1(Ω0). On the other hand, it seems reasonable to inquire whether one may be able to offset the absence of crowding effect in Ω0 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, h(x)0 in ΩΩ¯0 is a necessary condition for existence of positive solutions as c. 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 a=λ1(Ω0) (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 h(x)0 in ΩΩ¯0 and then finish the proof of our main existence and uniqueness result.

2 Preliminaries

We start with the consideration of equation (1.1), by stating our main hypotheses. Here Ω is a bounded smooth domain in N, N3. The constant c is nonnegative and throughout we assume the following:

  • (A1)

    b(x) and h(x) (0) are nonnegative Cα(Ω¯) functions.

  • (A2)

    There exists a smooth region Ω0 such that Ω¯0Ω and b(x)0 for xΩ¯0, and b(x)>0 on ΩΩ¯0.

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 [4], which we recall below.

Theorem 2.1 (Saddle-node bifurcation at a turning point [4]).

Let X and Y be Banach spaces and assume that (λ0,u0)R×X. Let F be a continuously differentiable mapping of an open neighborhood V of (λ0,u0) into Y. Suppose that the following hold:

  • (i)

    dimN(Fu(λ0,u0))=codimR(Fu(λ0,u0))=1, N(Fu(λ0,u0))=span{w0},

  • (ii)

    Fλ(λ0,u0)R(Fu(λ0,u0)).

If Z is a complement of span{w0} in X, then the solutions of F(λ,u)=F(λ0,u0) near (λ0,u0) form a curve (λ(s),u(s))=(λ0+τ(s),u0+sw0+z(s)), where s(τ(s),z(s))R×Z is a continuously differentiable function near s=0 and τ(0)=τ(0)=0, z(0)=z(0)=0. Moreover, if F is k times continuously differentiable, then so are τ(s) and z(s).

Furthermore, by [19, Theorem 2.4], we have

τ′′(0)=-l,Fuu(λ0,u0)[w0,w0]l,Fλ(λ0,u0),

where lY* satisfies N(l)=R(Fu(λ0,u0)).

The following two results provide additional useful information for the pure logistic equation set in a bounded domain Ω with N3:

-Δu=au-b(x)u2,xΩ.(2.1)

Lemma 2.2 ([8]).

Suppose that u>0 and v>0 are, respectively, C2(Ω) sub and super solutions of (2.1), that is,

Δv+av-b(x)v20Δu+au-b(x)u2in Ω.

Furthermore, assume that

lim supxΩ(u-v)0.

Then uv in Ω.

Next we recall a result of Du and Huang (see [8]) on the existence of boundary blow-up solutions for

-Δu=au-b(x)u2,in ΩΩ¯0,u=0on Ω,u=+on Ω0.(2.2)

Theorem 2.3 ([8, Theorem 2.4]).

For any a(-,+), (2.2) has a minimal positive solution U¯a and a maximal positive solution U¯a, in the sense that any positive solution u of (2.2) satisfies U¯a(x)u(x)U¯a(x).

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.

Theorem 2.4.

Suppose that λ1(Ω)<a<λ1(Ω0) and (A1) and (A2) hold. Then there exists c¯a>0 such that the following hold:

  • (i)

    If 0c<c¯a , then equation ( 1.1 ) has a maximal solution u¯a,c , and if c>c¯a , then ( 1.1 ) has no solution.

  • (ii)

    The curve cu¯a,c is decreasing with respect to the parameter c for c[0,c¯a) and u¯a,c is stable, that is, μ1(u¯a,c)>0 . Furthermore, u¯a,c is the unique stable solution of ( 1.1 ).

  • (iii)

    For c=c¯a , there exists a degenerate solution u¯a , i.e., μ1(u¯a)=0 , and equation ( 1.1 ) has another unstable solution for c near and to the left of c¯a.

Proof.

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 (c1,u1), then either u10 or the comparison lemma above applied on the set {x:u1(x)>0} implies that u1+ua, where ua is the unique positive solution of equation (1.2). Multiplying equation (1.1) with φ1, the first eigenfunction of -Δ in Ω with Dirichlet boundary condition, and then integrating over Ω, we obtain

c1(a-λ1(Ω))Ωuaφ1Ωh(x)φ1,

implying that c¯a is well defined. Next for p>n let X={uW2,p(Ω):u=0 on Ω} and Y=Lp(Ω). We define F:×XY by F(c,u)=Δu+au-b(x)u2-ch(x). Note that if F(c1,u1)=0, that is, if u1 is a solution of (1.1) with harvesting rate c=c1, and if μ1(u1)>0, then by applying the implicit function theorem, we can continue the curve of solutions forward (i.e., for cc1) with respect to c. Moreover, denoting the curve of solutions by (c,u)=(c,u(c)), it will be decreasing as v:=uc(c1) solves the equation

-Δv+(-a+2b(x)u1)v=-h(x).

Hence, μ1(u(c))=λ1(-a+2b(x)u(c)) is decreasing with respect to c as well. Therefore, the curve of solutions starting at (0,ua), can be continued to the right until a point (c0,u0) with μ1(u0)=0. Next applying the saddle-node bifurcation (Theorem 2.1), one easily sees that the curve turns back at the degenerate solution u0, therefore generating a second unstable solution in a left neighborhood of c0. 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 (c1,u1), then (1.1) will have a maximal solution (c1,u¯1) with μ1(u¯1)0. Indeed, u1,ua is a pair of sub-super solutions of (2.1), therefore the comparison lemma above applied on the set {x:u1(x)>0} implies that u1+ua. Next, considering u1,ua as an ordered pair of sub-super solutions of (1.1) for c=c1, the standard iteration process starting at the super solution ua will provide the maximal solution u¯1=u¯1(c1) of (1.1) for c=c1, which, by construction, will have μ1(u¯1)0 (see [17]). Also, as u¯1(c1) is a sub-solution of (1.1) with c=c2, if c1>c2, it is clear that u¯(c) is decreasing in c and therefore left continuous. In addition, for 0c<c0, since u(c)u¯(c), we have μ1(u¯(c))μ1(u(c))>0. This implies that the implicit function theorem applies at every point (c,u¯(c)), from which one easily concludes that u¯(c) is right continuous as well. Hence, the curve of maximal solutions cu¯(c) is continuous and decreasing for 0c<c0. Finally, as ua is the unique positive (and therefore maximal) solution of equation (1.1) for c=0, we have u(0)=u¯(0)=ua, which together with the fact that both curves (c,u(c)) and (c,u¯(c)) are continuous, decreasing and constitute of stable solutions yield u(c)=u¯(c) for 0c<c0.

Next we will show that equation (1.1) does not have a solution for c>c0. Otherwise, if u1 is a solution of (1.1) for some c1>c0, then as was shown above, (1.1) has a maximal solution u¯1 at c1 with μ1(u¯1)0. Now depending on whether μ1(u¯1)>0 or μ1(u¯1)=0, we can continue the curve of solutions from (c1,u¯1) backwards (i.e., for c<c1) in an increasing fashion, initially using either the implicit function theorem or the saddle-node bifurcation theorem, and then (as μ1(u(c)) becomes positive) through the implicit function theorem all the way to c=c0. Thus, equation (1.1) will have a solution u at c=c0 with μ1(u)>0. Since uu¯(c0)=u(c0), we get μ1(u(c0))>0, which is a contradiction. Finally, a similar argument yields uniqueness of stable solutions. The proof of the theorem is now complete. ∎

Remark 2.5.

It is obvious that c^ac¯a. In the remarks on page 3614 of [15], Oruganti, Shi and Shivaji claim, in passing, that c¯a=c^a. Although the validity of this claim for arbitrary λ1(Ω)<a<λ1(Ω0) is not clear to us, we are able to show that if h0 in Ω¯Ω0, then c^a=c¯a for a sufficiently close to λ1(Ω0) (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.

Theorem 2.6.

Assume aλ1(Ω0). Then any solution u (not necessary positive) of (1.1) is unstable, that is, μ1(u)<0 .

Proof.

Let u0 be a solution of equation (1.1) with c=c0>0. Let φ>0 denote the first eigenfunction of the linearization of (1.1) at u0, that is,

-Δφ=aφ-2b(x)u0φ+μ1(u0)φ.

Multiplying the above equation by φ1*, the first eigenfunction of -Δ in Ω0 with Dirichlet boundary condition, and integrating over Ω0, we obtain

Ω0φνφ1*+(λ1(Ω0)-a)Ω0φφ1*=μ1(u0)Ω0φφ1*,

which readily yields μ1(u0)<0. ∎

Remark 2.7.

Recalling the classical result (see [17]) 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 μ1(u)0, 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 aλ1(Ω0), u=0 is the unique nonnegative solution of (1.1) with c=0, a simple application of implicit function theorem yields the following result.

Proposition 2.8.

Assume (A1) and (A2), and suppose that a>λ1(Ω0) and aλi(Ω) for all i>1. Then equation (1.1) has a positive solution for 0<c<σ (with some σ>0) if and only if the linear equation

-Δv=av-h(x)in Ω,v=0on Ω(2.3)

has a positive solution.

Remark 2.9.

In regards to the existence of a positive solution to (2.3), we may write h(x)=h1φ1+h~(x), where φ1 is the first eigenfunction of -Δ in Ω with Dirichlet boundary condition and Ωh~(x)φ1=0. Then clearly v, the unique solution of equation (2.3), is given by

v=h1a-λ1(Ω)φ1+(Δ+a)-1(h~(x))>0.

Therefore, there exists h1*>0, depending on a and h~, such that if h1>h1*, then v>0.

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 aλ1(Ω0) and c large. Our first result provides a necessary condition for existence of positive solutions for c large.

Theorem 3.1.

Assume that aλ1(Ω) and h0 in ΩΩ¯0. Then there exists 0<c¯a<+ such that equation (1.1) does not have a nonnegative solution for c>c¯a.

Proof.

Let u be a nonnegative solution of (1.1). Using the comparison Lemma 2.2, we have uU¯a in ΩΩ¯0, where U¯a is the minimal boundary blow-up solution given in Theorem 2.3. Let Ωn={xΩ:d(x,Ω0)>1n}. Clearly, h0 in Ωn for all n large. Fix such an n and let φn,1>0 be the first eigenfunction of -Δ in Ωn with Dirichlet boundary condition. Multiplying equation (1.1) with φn,1 and then integrating over Ωn, we obtain

Ωn-Δuφn,1=aΩnuφn,1-Ωnb(x)u2φn,1-cΩnh(x)φn,1,

which implies

cΩnh(x)φn,1(a-λ1(Ωn))Ωnuφn,1-Ωnuνφn,1,

where ν is the outer normal vector of Ωn. Therefore, we have

cΩnh(x)φn,1(a-λ1(Ωn))ΩnU¯aφn,1-ΩnU¯aνφn,1

if a>λ1(Ωn), and

cΩnh(x)φn,1-ΩnU¯aνφn,1

if aλ1(Ωn). Therefore, c has a bound independent of u. ∎

In the rest of this section we will assume that h0 in Ω¯Ω0. In order to prove the existence of positive solutions of (1.1) for aλ1(Ω0) and c large, we first show that (1.1) has an unstable solution for large c as a approaches λ1(Ω0) from bellow. Using this and the fact that unstable solutions can not blow up as aλ1(Ω0), we prove the existence of a positive solution for a=λ1(Ω0) 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 λ1(Ω0). We start this procedure by first considering the behavior of c^a as aλ1(Ω0).

Theorem 3.2.

Assume that h0 in ΩΩ¯0. Then c^a+ as aλ1(Ω0).

Proof.

Fix c>0 arbitrary large. We will show that (1.1) has a positive solution for a sufficiently close λ1(Ω0) by constructing an ordered pair of sub and super solutions. Clearly, ua is a super solution of (1.1). To construct a sub-solution, we consider

u¯={wn(x),xΩΩ¯0,n+mφ1*(x),xΩ¯0,

where m and n are large positive constants (given below), φ1*(x) denotes the first eigenfunction of -Δ in Ω0 with Dirichlet boundary condition, and wn is the positive solution of the equation

-Δw=aw-b(x)w2in ΩΩ¯0,w=0on Ω,w=nonΩ0,

whose existence is shown in [8, Lemma 2.3]. We claim that u¯ is a weak sub-solution of (1.1). First note that u¯ is continuous. Next let 0ϕCc(Ω). We have

Ωu¯ϕ=ΩΩ¯0wnϕ+mΩ0φ1*ϕ=ΩΩ¯0(-Δwn)ϕ+mΩ0(-Δφ1*)ϕ+Ω0(mνφ1*-νwn)ϕ=aΩ0Ω¯0(wn-b(x)wn2)ϕ+λ1(Ω0)mΩ0φ1*ϕ+Ω0(mνφ1*-νwn)ϕ.

Therefore,

Ωu¯ϕ-Ω(au¯+b(x)u¯2-ch(x))ϕΩ0(m(λ1(Ω0)-a)φ1*+ch(x)-an)ϕ+Ω0(mνφ1*-νwn)ϕ.

Now we assume that a[λ1(Ω0)-ϵ0,λ1(Ω0)) for some ϵ0 small. First we take n large enough such that (λ1(Ω0)-ϵ0)n-ch1. Next, using the fact that maxΩ0νφ1*<0, we choose m sufficiently large so that mνφ1*-νwn<0 on Ω0. Finally, we choose a close enough to λ1(Ω0) such that ua>n+mφ1* in Ω¯0 and m(λ1(Ω0)-a)φ1*<1. Then u¯ is a weak sub-solution and, in addition, u¯ua (note that h0 on ΩΩ¯0). So, equation (1.1) has a positive solution between u¯ and ua. This completes the proof. ∎

Before stating our next result, we recall that by the anti-maximum principle (see [3]), there exists δh>0 such that for λ1(Ω0)<a<λ1(Ω0)+δh, the following equation has a unique positive solution ψa:

-Δψ=aψ-h(x),in Ω0,ψ=0on Ω0.(3.1)

Lemma 3.3.

There exists 0<δ<δh such that if un is a sequence of nonnegative solutions of equation (1.1), with a=an, c=cn such that λ1(Ω)<an<λ1(Ω0)+δ, cn0 and un+. Then un+ uniformly on Ω¯0.

We note that this lemma is in fact valid for arbitrary nonnegative h(x).

Proof.

Up to a subsequence, ana* with λ1(Ω0)a*λ1(Ω0)+δ. We divide the proof into several steps. Step 1: un+ uniformly in every compact subsets of Ω0. Note that cnun. Thus, up to a subsequence, cn/unα<+. Let vn=un/un. From equation (1.1) we have

-Δvn=anvn-b(x)unvn2-cnunh(x).(3.2)

Since -Δvnanvn and vn=1, there exists v0H01(Ω) such that vnv0 weakly in H01(Ω) and strongly in Lp(Ω) for p>1. Multiplying (3.2) by vn/un and then integrating over Ω, we get

1unΩ|vn|2=anunΩvn2-Ωb(x)vn3-cnunΩh(x)vn.

Thus, Ωb(x)v03=0, and so v00 in Ω¯Ω0.

Next, multiplying (3.2) by ϕCc(Ω0) and integrating over Ω0, we have

Ω0vnϕ=anΩ0vnϕn-cnunΩ0h(x)ϕ.

Passing n to infinity, we obtain

Ω0v0ϕ=a*Ω0v0ϕ-αΩ0h(x)ϕ.

Therefore, v0 is a nonnegative weak solution of the equation

-Δv0=a*v0-αh(x),in Ω0,v0=0on Ω0.

If a*=λ1(Ω0), then, thanks to the Fredholm alternative, we have α=0, since h(x)0 and h(x)0. Therefore, v0=0 or v0=φ1*>0 in Ω0 (recall that φ1* is the first eigenfunction of -Δ with Dirichlet boundary condition in Ω0).

We claim that v00. In fact, if v0=0, then, by equation (3.2), -Δvnanvn, so that

0vn(an+1)(-Δ+1)-1vn0

uniformly in Ω, as vn0 in Lp(Ω) for p>1. This contradicts vn=1.

Next if λ1(Ω0)<a*λ1(Ω0)+δ, then assuming that α=0, we have v0=0, as a*λi(Ω0). But this will again yield a contradiction as above. Thus, α>0 and v0=αψa*>0, the unique positive solution of equation (3.1).

Now standard elliptic estimates (see [11]) imply that un+ uniformly in every compact subset of Ω0, completing the proof of step 1.

Next we let un(zn)=minxΩ¯0un(x) and show that un(zn). This will be done in the next two steps. First note that since by assumption Ω0 is C2,γ, it satisfies a uniform interior ball assumption, i.e., there exists R>0 such that for every xΩ0, there exists a ball Bx=BR(y) with radius R and center y such that BxΩ¯0 and BxΩ0={x}. Let K be a compact subset of Ω0 such that BR/2(y)K for all xΩ0.

Step 2: Let un(zn)=minxΩ¯0un(x). If {un(zn)} is bounded, then znΩ0 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 (zn), we have znΩ0. Then step 1 implies znΩ0. Hence, there exists xnΩ0 such that znBxnBR/2(yn). Next we consider the auxiliary functions

ηn(x)=e-σ|x-yn|2-e-σR2,

and show that there exists a sequence βn such that

un(x)un(zn)+βnηn(x)in BxnBR/2(yn).(3.3)

To start with, we fix a suitably chosen large σ>0, so that ηn satisfy the following properties on Bxn:

ηn(x)=0,xBxn,ηn(x)>0,xBxnBR/2(yn),(3.4)ηn(x)=e-σR2/4-e-σR2<e-σR2/4,xBR2(yn),νnηn(xn)=2σR2e-σR2>0,where νn=yn-xn|yn-xn|,(3.5)

and

Δηn+anηn-e-σR2h(x)=(4σ2|x-yn|2-2σN+an)e-σ|x-yn|2-(an+h(x))e-σR2>(4σ2|x-yn|2-2σN-h(x))e-σR2>0,xBxnBR/2(yn).

Note that the choice of σ depends only on R and the function h(x). Let αn=minKun(x). The rest of the proof proceeds by considering the two cases a*=λ1(Ω0) and a*>λ1(Ω0) separately.

If a*=λ1(Ω0), since un/unφ1*>0 in Ω0 and cn/un0, then αn/cn+. Hence, we can choose βn so that cneσR2βn12αneσR2/4. Therefore, for xBxnBR/2(yn), we have

(-Δ-an)(un(x)-(un(zn)+βnηn(x)))(βne-σR2-cn)h(x)0

and, in addition,

un(x)un(zn)+βnηn(x),x(BxnBR/2(yn)).

Since anλ1(Ω0)<λ1(Bxn), the maximum principle finally yields (3.3). Now taking x=zn in (3.3), we have βnηn(zn)0, which, since znBxnBR/2(yn), contradicts (3.4).

In the second case, that is, a*>λ1(Ω0), by step 1 we have 0<α=limn+cn/un and vnv0=αψa*. Thus,

limn+αncn=limn+αnuncnunαminxKψa*(x)αminxKψa*(x).

From (3.1) and Remark 2.9, we have

ψa*=h1a*-λ1(Ω0)φ1*+(Δ+a*)-1(h~(x)).

Hence, by further decreasing δ, we can guarantee that

minxKψa*(x)=minxK(h1a*-λ1(Ω0)φ1*+(Δ+a*)-1(h~(x)))>e3σR2/4.

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 un(zn)=minxΩ¯0un(x). Then un(zn) . To argue by contradiction, we assume that un(zn) is bounded, i.e., un(zn)M for some M>0 independent of n. Then, by step 2, we have znΩ0. We follow the argument in step 2 where now zn=xn, and conclude

un(x)un(xn)+βnηn(x)in BxnBR/2(yn)(3.6)

for some sequence βn. Lemma 2.3 in [8] implies that the equation

-Δwn=anwn-b(x)wn2,in ΩΩ¯0,wn=0on Ω,wn=un(xn)on Ω0,

has a unique positive solution and by the comparison lemma, un(x)wn(x) in Ω¯Ω0. Similarly, w, the unique positive solution of

-Δw=a*w-b(x)w2,in ΩΩ¯0,w=0on Ω,w=Mon Ω0,

satisfies wn(x)w(x) in Ω¯Ω0. Thus, wnL(ΩΩ¯0) is bounded, and therefore standard elliptic estimates imply that {wn} is bounded in C1(Ω¯Ω0), and so, in particular, |wn(xn)| remains uniformly bounded. Since un(xn)=wn(xn) and un(x)wn(x) in Ω¯Ω0, we have

νn(xn)νnwn(xn)M0(3.7)

for some M0>0 independent of n. On the other hand, using (3.6) and taking into account (3.5), we obtain

νnun(xn)βnνnηn(xn)+,

contradicting (3.7). This completes the proof of step 3, and therefore the proof of the lemma. ∎

At this point for λ1(Ω)<a<λ1(Ω0), we define

c¯a=sup{c0:equation (1.1) has a nonnegative solution on P},

where P is the boundary of the cone

P={uC01(Ω¯):u(x)>0 for xΩ and uν(x)<0 for xΩ}

and C01(Ω¯) denotes the subspace of functions in C1(Ω¯) which are zero on the boundary of Ω. Note that u=0 is a nonnegative solution of (1.1) with c=0, so c¯a is well defined and c¯ac^a. In the following lemma we show that c¯a does not go to infinity as aλ1(Ω0).

Lemma 3.4.

lim supaλ1(Ω0)c¯a is bounded.

Proof.

Assume on the contrary that there exists a sequence an such that anλ1(Ω0) and c¯an+. Let unP be a sequence of nonnegative solutions of equation (1.1) with a=an and c¯an-ϵcnc¯an for some ϵ>0. Since cnun, by Lemma 3.3, we have un+ uniformly in Ω¯0. Thus, for large n, we have un>0 on Ω0. Now clearly we can choose Mn>0 large so that

(-Δ+Mn)un=(an+Mn)un-b(x)un20in ΩΩ¯0.

Hence, by the maximum principle and the Hopf boundary lemma un>0 in Ω and u/ν<0 on Ω, contradicting unP. ∎

Our next result shows that for c¯a<c<c^a equation (1.1) has an unstable positive solution.

Lemma 3.5.

There exists δ>0 such that if a(λ1(Ω0)-δ,λ1(Ω0)) and c¯a<c<c^a, then (1.1) has an unstable positive solution.

Proof.

Since c^a+ as a approaches λ1(Ω0), Lemma 3.4 implies the existence of δ>0 such that c¯a<c^a if a(λ1(Ω0)-δ,λ1(Ω0)). Next, given such an a, we fix c¯a<c0<c^a and M>c^a. For the positive constants σ, Kσ and c[c0,M], we define

Tσ={uC01(Ω¯):σφ1uua in Ω and uaνuνσφ1ν on Ω},

where φ1 is the first eigenfunction of -Δ in Ω with Dirichlet boundary condition and Aσ,c:TσC01(Ω¯) is such that

Aσ,c=(-Δ+Kσ)-1((a+Kσ)u-b(x)u2-ch(x)).

Firstly, taking into account the definition of c¯a, a simple limiting argument implies that for σ sufficiently small, equation (1.1) has no solution on Tσ for c[c0,M] (where Tσ denotes the relative boundary of Tσ in P). Fixing such a σ, we then take Kσ sufficiently large so that Aσ,c maps Tσ,c into P. Now for c[c0,M], we have

deg(I-Aσ,c,Tσ,0)=deg(I-Aσ,M,Tσ,0)=0.(3.8)

On the other hand, by Theorem 1.3, for any c<c^a, (1.1) has a unique positive stable solution u¯a,c. Moreover, using the fixed point index calculation of Dancer in [5], it is easily seen that

ind(Aσ,c,u¯a,c)=1.(3.9)

Thus, (3.8) and (3.9) imply that for c[c0,c^a), equation (1.1) has another solution u¯a,c in Tσ which, by the uniqueness of stable solutions, satisfies μ1(u¯a,c)0. We claim that in fact μ1(u¯a,c)<0. Indeed, if μ1(u¯a,c)=0, then applying the saddle-node bifurcation theorem, there exist ϵ>0 and an open neighborhood OC(Ω¯) of u¯a,c, such that the solution set of (1.1) in I=(c-ϵ,c+ϵ)×O 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 c=c^a. This completes the proof. ∎

Theorem 3.6.

Assume that a=λ1(Ω0). There exists c0>lim supaλ1(Ω0)c¯a such that for all c>c0, equation (1.1) has a positive solution.

Proof.

First fix c>lim supaλ1(Ω0)c¯a. Lemma 3.5 implies the existence of a sequence of unstable positive solutions un of equation (1.1) with a=an<λ1(Ω0) and anλ1(Ω0). We claim that un is bounded. Otherwise, un+, and therefore by Lemma 3.3, un+ uniformly in Ω¯0. Hence, there exist n1 and n2 such that un2>un1 in Ω¯0 and an2>an1. Now an application of the comparison lemma in ΩΩ¯0 easily implies un2un1 in all of Ω (note that un2 is a super solution and un1 a solution of the logistic equation -Δu=an1u-b(x)u2 on ΩΩ¯0). Also, as an1<an2, we have that un1 is a sub-solution of (1.1) with a=an2. Therefore, equation (1.1) for a=an2 has a minimal solution u0 between un1 and un2 with μ1(u0)0 (see [17, Theorem 4.1]). Hence, μ1(un2)μ1(u0)0, contradicting the fact that un2 is unstable.

Therefore, un is bounded and unuc* weakly in H01(Ω) and strongly in Lp(Ω) for some uc*H01(Ω). Obviously, uc* is a nonnegative solution of (1.1) with a=λ1(Ω0).

Therefore, for all c>lim supaλ1(Ω0)c¯a, equation (1.1) has a nonnegative solution uc*. Since cuc*, by Lemma 3.3, we get uc*+ uniformly in Ω¯0 as c+. Therefore, there exists c0 such that uc*>0 in Ω¯0 for c>c0, and a further application of the maximum principle in ΩΩ¯0 (see the proof of Lemma 3.4) yields uc*>0 in Ω. ∎

Lemma 3.7.

Let λ1(Ω0)aλ1(Ω0)+δ with δ as in Lemma 3.3. There exists c1=c1(a) such that every positive solution of (1.1) with cc1 is nondegenerate.

Proof.

Fix an a. Assume on the contrary that un is a sequence of solutions of (1.1) corresponding to cn+ and μin(un)=0 for some in. From Lemma 3.3, we have un+ uniformly in Ω¯0. Thus, for a subsequence un1<un2<un3< in Ω¯0 and therefore (by an application of the comparison lemma, as in the proof of Theorem 3.6) in all of Ω. Since μi(u)=λi(-a+2b(x)u), we have μi(unj)>0 for all j>1 and iin1. On the other hand, since μinj(unj)=0, we have inj<in1. Therefore, there exists a fixed kin1 such that μk(unj)=0 for all j. This contradicts the fact that the sequence unj is strictly increasing, and therefore so is μk(unj). ∎

Lemma 3.8.

Let λ1(Ω0)aλ1(Ω0)+δ with δ as in Lemma 3.3. There exists c2=c2(a) such that for cc2, equation (1.1) has a at most one positive solution.

Proof.

Fix an a. Assume on the contrary that un and u¯n are two sequences of positive solutions of equation (1.1) corresponding to cn+. If we subtract the equations for un and u¯n, then we obtain that λin(-a+b(x)(un+u¯n))=0 for some in. By Lemma 3.3, we have un+u¯n+ uniformly in Ω¯0. Thus, there exists a subsequence unj+u¯nj such that un1+u¯n1<un2+u¯n2<un3+u¯n3< in Ω¯0, and therefore by the comparison lemma in all of Ω. Therefore, λi(-a+b(x)(unj+u¯nj))>0 for all j>1 and iin1. On the other hand, since λinj(-a+b(x)(unj+u¯nj))=0, we have inj<in1. 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 λ1(Ω0)aλ1(Ω0)+δ and c large. In what follows, we let

c*=max{c0,c1(λ1(Ω0)),c2(λ1(Ω0))}.

Note that the above results imply that equation (1.1) for a=λ1(Ω0) and cc* has a unique positive solution which, in addition, is nondegenerate.

Lemma 3.9.

Let δ>0 be defined as in Lemma 3.3 and let dc* be a given constant. There exists K>0, depending on δ and d, such that if u is a nonnegative solution of (1.1) with λ1(Ω0)aλ1(Ω0)+δ and c*cd, then u<K.

Proof.

Assume on the contrary that there exists a sequence un of nonnegative solutions of (1.1) with a=an and c*c=cnd, where λ1(Ω0)anλ1(Ω0)+δ, and un+. Now, by Lemma 3.3, we have un+ in Ω¯0. Denoting the unique positive solution of (1.1) for a=λ1(Ω0) and c=d by u*, we have for n0 large, un0>u* in Ω¯0, and then by the comparison lemma in all of Ω (note that un0 is a super solution and u* a solution of the logistic equation -Δu=λ1(Ω0)u-b(x)u2 on ΩΩ¯0). Hence, u*,un0 is an ordered pair of sub-super solution of (1.1) with a=λ1(Ω0) and c=d, and therefore (1.1) has a solution u0 (achieved with the iteration process starting at un0) between uc* and un0 with μ1(u0)0. This contradicts Theorem 2.6. ∎

Lemma 3.10.

There exists c3>0 such that (1.1) has no solution on P, provided that λ1(Ω0)aλ1(Ω0)+δ with δ as in Lemma 3.3 and cc3.

Proof.

Assume on the contrary that unP is a sequence of solutions of (1.1) with a=an and c=cn with λ1(Ω0)aλ1(Ω0)+δ and cn+. By Lemma 3.3, un+ uniformly in Ω¯0, and therefore, by the maximum principle (as in the proof of Lemma 3.4), un>0 in Ω for large n, contradicting unP. ∎

We are now ready to state the main result of this paper.

Theorem 3.11.

Let λ1(Ω0)aλ1(Ω0)+δ with δ as in Lemma 3.3. Then, for cmax{c*,c3} equation (1.1) has an unstable positive solution. Furthermore, for each a, there exists c(a), such that for cc(a), the positive solution is unique and nondegenerate.

Proof.

The only statement that requires a proof is the existence of a positive solution for cmax{c*,c3}. The proof uses similar arguments as in the proof of Theorem 3.6, so we shall brief them here. Fixing a cmax{c*,c3}, we define

Te,K={uC01(Ω¯):eφ1uK in Ω and ν(u-eφ1)0 on Ω},

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 Te,K (the relative boundary of Te,K in P) for λ1(Ω0)aλ1(Ω0)+δ. Next we define Aa:C01(Ω¯)C01(Ω¯) by

Aa(u)=(-Δ+L)-1((a+L)u-b(x)u2-ch(x)).

By taking L>0 sufficiently large, we may assume that (a+L)u-b(x)u2-ch(x) is increasing in [eφ1(x),K] for all xΩ and that Aa,c maps Te,K into P. Indeed, if uTe,K and u0=Aa(u), then we have

u0=(-Δ+L)-1((a+L)u-b(x)u2-ch(x))(-Δ+L)-1((a+L)eφ1-b(x)e2φ12-ch(x))0,

as (a+L)eφ1-b(x)e2φ12-ch(x)0 for L>0 large. Hence, for e small and L large, deg(I-Aa,Te,K,0) is admissible for λ1(Ω0)aλ1(Ω0)+δ. Now, by Lemmas 3.7 and 3.8, we have

deg(I-Aλ1(Ω0),Te,K,0)0.

Hence,

deg(I-Aa,Te,K,0)=deg(Aλ1(Ω0),Te,K,0)0

for all a[λ1(Ω0),λ1(Ω0)+δ]. 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 λ1(Ω0). In particular, for any a>λ1(Ω0), it is easily seen (through a familiar limiting argument) that the existence of a positive solution to the equation

-Δu=au-h(x)in Ω0,u=0on Ω0,

is a necessary condition for existence of a positive solutions to (1.1) as c. 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.

References

  • [1]

    S. Alama and G. Tarantello, On the solvability of a semilinear elliptic equation via an associated eigenvalue problem, Math. Z. 221 (1996), no. 3, 467–493.  CrossrefGoogle Scholar

  • [2]

    A. Ambrosetti and J. L. Gámez, Branches of positive solutions for some semilinear Schrödinger equations, Math. Z. 224 (1997), no. 3, 347–362.  CrossrefGoogle Scholar

  • [3]

    P. Clément and L. A. Peletier, An anti-maximum principle for second-order elliptic operators, J. Differential Equations 34 (1979), no. 2, 218–229.  CrossrefGoogle Scholar

  • [4]

    M. G. Crandall and P. H. Rabinowitz, Bifurcation, perturbation of simple eigenvalues and linearized stability, Arch. Ration. Mech. Anal. 52 (1973), 161–180.  CrossrefGoogle Scholar

  • [5]

    E. N. Dancer, On the indices of fixed points of mappings in cones and applications, J. Math. Anal. Appl. 91 (1983), no. 1, 131–151.  CrossrefGoogle Scholar

  • [6]

    E. N. Dancer, Some remarks on classical problems and fine properties of Sobolev spaces, Differential Integral Equations 9 (1996), no. 3, 437–446.  Google Scholar

  • [7]

    M. A. del Pino, Positive solutions of a semilinear elliptic equation on a compact manifold, Nonlinear Anal. 22 (1994), no. 11, 1423–1430.  CrossrefGoogle Scholar

  • [8]

    Y. Du and Q. Huang, Blow-up solutions for a class of semilinear elliptic and parabolic equations, SIAM J. Math. Anal. 31 (1999), no. 1, 1–18.  CrossrefGoogle Scholar

  • [9]

    Y. Du and J. Shi, Allee effect and bistability in a spatially heterogeneous predator-prey model, Trans. Amer. Math. Soc. 359 (2007), no. 9, 4557–4593.  CrossrefGoogle Scholar

  • [10]

    J. M. Fraile, P. Koch-Medina, J. López-Gómez and S. Merino, Elliptic eigenvalue problems and unbounded continua of positive solutions of a semilinear elliptic equation, J. Differential Equations 127 (1996), no. 1, 295–319.  CrossrefGoogle Scholar

  • [11]

    D. Gilbarg and N. S. Trudinger, Elliptic partial differential equations of second order, Classics Math., Springer, Berlin, 2001.  Google Scholar

  • [12]

    P. Girão and H. Tehrani, Positive solutions to logistic type equations with harvesting, J. Differential Equations 247 (2009), no. 2, 574–595.  CrossrefWeb of ScienceGoogle Scholar

  • [13]

    P. M. Girão, Bifurcation curves of a logistic equation when the linear growth rate crosses a second eigenvalue, Nonlinear Anal. 74 (2011), no. 1, 94–113.  CrossrefWeb of ScienceGoogle Scholar

  • [14]

    P. M. Girão and M. Pérez-Llanos, Bifurcation curves of a diffusive logistic equation with harvesting orthogonal to the first eigenfunction, J. Math. Anal. Appl. 403 (2013), no. 2, 376–390.  CrossrefWeb of ScienceGoogle Scholar

  • [15]

    S. Oruganti, J. Shi and R. Shivaji, Diffusive logistic equation with constant yield harvesting. I: Steady states, Trans. Amer. Math. Soc. 354 (2002), no. 9, 3601–3619.  CrossrefGoogle Scholar

  • [16]

    T. Ouyang, On the positive solutions of semilinear equations Δu+λu-hup=0 on the compact manifolds, Trans. Amer. Math. Soc. 331 (1992), no. 2, 503–527.  Google Scholar

  • [17]

    D. H. Sattinger, Monotone methods in nonlinear elliptic and parabolic boundary value problems, Indiana Univ. Math. J. 21 (1971/72), 979–1000.  Google Scholar

  • [18]

    S. Shabani, Diffusive Holling type-II predator-prey system with harvesting of prey, J. Math. Anal. Appl. 410 (2014), no. 1, 469–482.  Web of ScienceCrossrefGoogle Scholar

  • [19]

    J. Shi, Persistence and bifurcation of degenerate solutions, J. Funct. Anal. 169 (1999), no. 2, 494–531.  CrossrefGoogle Scholar

About the article

Received: 2016-09-22

Accepted: 2017-03-01

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.

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