
The decomposition of the triangular set
Plant probiotic bacteria are a versatile group of bacteria isolated from different environmental sources to improve plant productivity and immunity. The potential of plant probiotic-based formulations is successfully seen as growth enhancement in economically important plants. For instance, endophytic Bacillus species acted as plant growth-promoting bacteria, influenced crops such as cowpea and lady's finger, and increased phytochemicals in crops such as high antioxidant content in tomato fruits. The present review aims to summarize the studies of Bacillus species retaining probiotic properties and compare them with the conventional fertilizers on the market. Plant probiotics aim to take over the world since it is the time to rejuvenate and restore the soil and achieve sustainable development goals for the future. Comprehensive coverage of all the Bacillus species used to maintain plant health, promote plant growth, and fight against pathogens is crucial for establishing sustainable agriculture to face global change. Additionally, it will give the latest insight into this multifunctional agent with a detailed biocontrol mechanism and explore the antagonistic effects of Bacillus species in different crops.
Citation: Shubhra Singh, Douglas J. H. Shyu. Perspective on utilization of Bacillus species as plant probiotics for different crops in adverse conditions[J]. AIMS Microbiology, 2024, 10(1): 220-238. doi: 10.3934/microbiol.2024011
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Plant probiotic bacteria are a versatile group of bacteria isolated from different environmental sources to improve plant productivity and immunity. The potential of plant probiotic-based formulations is successfully seen as growth enhancement in economically important plants. For instance, endophytic Bacillus species acted as plant growth-promoting bacteria, influenced crops such as cowpea and lady's finger, and increased phytochemicals in crops such as high antioxidant content in tomato fruits. The present review aims to summarize the studies of Bacillus species retaining probiotic properties and compare them with the conventional fertilizers on the market. Plant probiotics aim to take over the world since it is the time to rejuvenate and restore the soil and achieve sustainable development goals for the future. Comprehensive coverage of all the Bacillus species used to maintain plant health, promote plant growth, and fight against pathogens is crucial for establishing sustainable agriculture to face global change. Additionally, it will give the latest insight into this multifunctional agent with a detailed biocontrol mechanism and explore the antagonistic effects of Bacillus species in different crops.
We consider systems of semilinear elliptic equations
−Δu(x)+Fu(x,u)=0 |
where
(
When
All the above results are based on the ordered structure of the set of minimal solutions of (PDE) in the case
The study of (PDE) when
M0={u∈E0∣J0(u)=c0:=infE0J0(u)}≠∅. |
Paul H. Rabinowitz studied the case of spatially reversible potentials
J(u)=∑p∈ZJp,0(u):=∑p∈Z(∫Tp,0L(u)dx−c0), | (1) |
(where
Γ(v−,v+)={u∈W1,2(R×Tn−1,Rm)∣‖u−v±‖L2(Tp,0,Rm)→0 as p→±∞}. |
In [30] the existence of minimal double heteroclinics was obtained assuming that the elements of
The proof of these results does not use the ordering property of the solutions and adapts to the study of (PDE) some of the ideas developed to obtain multi-transition solutions for Hamiltonian systems (see e.g. [3], [28] and the references therein). Aim of the present paper is to show how these methods, in particular a refined study of the concentrating properties of the minimal heteroclinic solutions to (PDE), can be used in a symmetric setting to obtain saddle type solutions to (PDE).
Saddle solutions were first studied by Dang, Fife and Peletier in [16]. In that paper the authors considered Allen-Cahn equations
We refer to [14,15,6,7,27] for the study of saddle solutions in higher dimensions and to [1,20,8] for the case of systems of autonomous Allen-Cahn equations. Saddle solutions can be moreover viewed as particular
In [5] the existence of saddle type solutions was studied for non autonomous Allen-Cahn type equations and this work motivated the paper [2] where solutions of saddle type for (PDE) were found in the case
In the present paper we generalize the setting considered in [2] to the case
(
(
F(x1,x2,x3,...,xn,u)=F(x2,x1,x3,...,xn,u) on Rn×Rm. |
By [29] the set
As recalled above, in [2], where
(
J″0(v)h⋅h=∫[0,1]n|∇h|2+Fu,u(x,v(x))h⋅hdx≥α0‖h‖2L2([0,1]n,Rm) |
for every
The assumption (
Γ(v)={u∈W1,2(R×Tn−1,Rm)∣u is odd in x1,limp→+∞‖u−v‖L2([p,p+1]×Tn−1,Rm)=0}. |
In §4, setting
c(v)=infu∈Γ(v)J(u) for v∈M0 |
we show that
Mmin0={v0∈M0∣c(v0)=minv∈M0c(v)}≠∅ |
and that
M(v0)={u∈Γ(v0)∣J(u)=c(v0)} |
is not empty and compact with respect to the
‖u−v0‖W1,2([p,p+1]×Tn−1,Rm)→0 as p→+∞. |
Our main result can now be stated as follows
Theorem 1.1. Assume
Moreover there exists
distW1,2(Rk,Rm)(w,M(v0))→0,ask→+∞, | (2) |
where
Note that by
The proof of Theorem 1.1 uses a variational approach similar to the one already used in previous papers like [5,2]. To adapt this approach to the case
In this section we recall some results obtained by Rabinowitz in [29], on minimal periodic solutions to (PDE). Moreover, following the argument in [2], we study some symmetry properties related to the assumptions (
(
Let us introduce the set
E0=W1,2(Tn,Rm)={u∈W1,2(Rn,Rm)∣u is 1-periodic in all its variables} |
with the norm
‖u‖W1,2([0,1]n,Rm)=(m∑i=1∫[0,1]n(|∇ui|2+|ui|2)dx)12. |
We define the functional
J0(u)=∫[0,1]n12|∇u|2+F(x,u)dx=∫[0,1]nL(u)dx. | (3) |
and consider the minimizing set
M0={u∈E0|J0(u)=c0} where c0=infu∈E0J0(u) |
Then in [29], [30] it is shown
Lemma 2.1. Assume
1.
2. if
3. For every
distW1,2([0,1]n,Rm)(u,M0):=infv∈M0‖u−v‖W1,2([0,1]n,Rm)>ρ, |
then
4. If
5. If
Assumption
Lemma 2.2. Assume
Proof. It is sufficient to observe that if
ⅰ)
ⅱ)
Property (ⅱ) gives the second part of the statement while by (i) and the unique extension property proved in [29], we obtain that the components of
On the other hand, assumption
˜u(x)={u(x),x∈T+,u(x2,x1,x3,…,xn),x∈[0,1]n∖T+. | (4) |
Then, we have
Lemma 2.3. If
Proof. Given
∫T+L(u)dx≤∫[0,1]n∖T+L(u)dx. |
Since
c0=J0(u)=∫T+L(u)dx+∫[0,1]n∖T+L(u)dx≥2∫T+L(u)dx=J0(˜u)≥c0. |
Hence, again by Lemma 2.1-(5),
As an immediate consequence, using Lemma 2.1-(5), we have the following.
Lemma 2.4. There results
minu∈W1,2(T+,Rm)∫T+L(u)dx=c02. | (5) |
Moreover, if
Remark 1. Lemma 2.3 tells us that the elements of
σ0={x∈R×[0,1]n−1|x2−1≤x1≤x2}. |
More precisely we have
Note that by Lemma 2.1-(1) and the assumption (N) we plainly derive that (
where we recall that
Note finally that by
r0:=min{‖u−v‖L2(Tn,Rm)∣u,v∈M0,u≢v}, | (6) |
we have
This section is devoted to introduce the variational framework to study solutions of (PDE) which are heteroclinic between minimal periodic solutions. We follow some arguments in [29], [26], introducing the renormalized functional
Let us define the set
E={u∈W1,2loc(Rn,Rm)∣u is 1-periodic in x2,…xn}. |
For any
J(u)=∑p∈ZJp,0(u), |
where, denoting
Jp,0(u)=∫Tp,0L(u)dx−c0,∀p∈Z. |
Denoting briefly
Jp,0(u)=∫[0,1]nL(u(⋅+p))dx−c0=J0(u(⋅+p))−c0,∀p∈Z. |
Then, by Lemma 2.1, we have
Lemma 3.1. The functional
Proof. Consider a sequence
lim infkJ(uk)≥lim infkℓ∑p=−ℓJp,0(uk)≥ℓ∑p=−ℓJp,0(u)>J(u)−ε, |
thus finishing the proof.
Using the notation introduced above, note that if
First of all, let us consider the functional
minu∈EJp,0(u)+Jp+1,0(u)=0 |
and the set of minima coincide with
distp(u,A)=inf{‖u−v‖W1,2(Tp,0∪Tp+1,0,Rm)∣v∈A}. |
Remark 2. Let us fix some constants that will be used in rest of the paper. By Lemma 2.1-(3), we have that for any
if u∈E satisfies Jp,0(u)+Jp+1,0(u)≤λ(r) for a p∈Z, then distp(u,M0)≤r. | (7) |
It is not restrictive to assume that the function with
On the other hand for every
ρ(λ)=sup{distp(u,M0)∣u∈E with Jp,0(u)+Jp+1,0(u)≤λ,p∈Z} |
we get
Λ(r)=sup{Jp,0(u)∣u∈E and p∈Z are such that distp(u,M0)≤2r} | (8) |
which is non-decreasing and
We say that a set
Lemma 3.2. Given
Proof. Let
‖u−vp‖W1,2(Tp,0∪Tp+1,0,Rm)≤r04 |
from which
Moreover, using the notations introduced above, we have
Lemma 3.3. If
‖u(⋅+p)−u(⋅+(p+1))‖2L2([0,1]n,Rm)≤2(Jp,0(u)+Jp+1,0(u)+2c0). |
Proof. Setting
‖u(⋅+p)−u(⋅+(p+1))‖2L2([0,1]n,Rm)=∫p+1p∫[0,1]n−1|u(x1+1,y)−u(x1,y)|2dydx1 |
and so there exists
∫[0,1]n−1|u(ˉx1+1,y)−u(ˉx1,y)|2dy≥‖u(⋅+p)−u(⋅+(p+1))‖2L2([0,1]n,Rm). |
On the other hand, by Hölder inequality,
2(Jp,0(u)+Jp+1,0(u)+2c0)≥∫p+2p∫[0,1]n−1|∂x1u(x1,y)|2dydx1≥∫[0,1]n−1∫ˉx1+1ˉx1|∂x1u(x1,y)|2dx1dy≥∫[0,1]n−1|u(ˉx1+1,y)−u(ˉx1)|2dy≥‖u(⋅+p)−u(⋅+(p+1))‖2L2([0,1]n,Rm) |
completing the proof.
By the previous lemmas we obtain that the elements in the sublevels of
Lemma 3.4. For every
Proof. Let
‖u(⋅+p)−u(⋅+q)‖L2([0,1]n,Rm)≤l(u)supk∈J(u)‖u(⋅+k)−u(⋅+k+1)‖L2([0,1]n,Rm)+ˉl(u)∑i=1supp,q∈Ii(u)‖u(⋅+p)−u(⋅+q)‖L2([0,1]n,Rm)≤l(u)(2(Λ+2c0))12+ˉl(u)r02. | (9) |
where the first term in (9) follows by the application of Lemma 3.3, since
2(Jk,0(u)+Jk+1,0(u)+2c0)≤2(J(u)+2c0)≤2(Λ+2c0),∀k∈Z, |
while the second one follows by the definition of
Since
The following lemma states the weak compactness of the sublevels of the functional
Lemma 3.5. Given any
Proof. First note that, by Lemma 3.4, there exists
‖u−v‖L2(Tp,0,Rm)=‖u(⋅+p)−v‖L2([0,1]n,Rm)≤‖u(⋅+p)−u(⋅+ℓ)‖L2([0,1]n,Rm)+‖u(⋅+ℓ)−v‖L2([0,1]n,Rm)≤R+ˉR. |
Consider now a sequence as in the statement, setting
‖uk−v‖2L2(QL,Rm)+‖∇uk‖2L2(QL,Rm)≤2L(ˉR+R)2+4Lc0+2Λ. |
Hence,
By Lemma 3.2 we also deduce the following result concerning the asymptotic behaviour of the functions in the sublevels of
Lemma 3.6. If
‖u−v±‖W1,2(Tp,0,Rm)→0asp→±∞. |
Proof. Since
Hence the sequence
By Lemma 3.6, if
Γ(v−,v+)={u∈E∣‖u−v±‖W1,2(Tp,0,Rm)→0asp→±∞} |
where
We note that by Lemma 3.5, every sequence
In particular, given
c(v−,v+)=infu∈Γ(v−,v+)J(u), |
as in [29], we obtain that for any
Finally, we have that
Lemma 3.7. For every
Proof. Assume that there exists
In order to prove the second part of the statement, assume the existence of two sequences
‖v+k−v−k‖L2([0,1]n,Rm)≤‖v−k−uk(⋅+pk)‖L2([0,1]n,Rm)≤+‖uk(⋅+pk)−uk(⋅+qk)‖L2([0,1]n,Rm)≤+‖v+k−uk(⋅+qk)‖L2([0,1]n,Rm)≤ε+R+ε |
since, by periodicity,
We focalize now in the study of heteroclinic solutions which are odd in the first variable, hence we will consider a subset of
Eodd={u∈E∣u is odd with respect to x1}, |
In what follows, when we will consider functions
J+(u)=∑p≥0Jp,0(u). |
For any
Γ(v)={u∈Eodd∣‖u−v‖W1,2(Tp,0,Rm)→0 as p→+∞}⊆Γ(−v,v). |
In this setting we can rewrite Lemma 3.6 as follows.
Lemma 4.1. For every
We are going to look for minimizer of
c(v)=infu∈Γ(v)J(u) and M(v)={u∈Γ(v)∣J(u)=c(v)}. | (10) |
Notice that for any
Lemma 4.2. For any
Moreover, note that, by assumption (
c=minv∈M0c(v) | (11) |
is well defined and the set
Mmin0={v∈M0∣c(v)=c} | (12) |
is nonempty and consists of a finite number of elements. In particular, we have
minv∈M0∖Mmin0c(v)>c. | (13) |
The following lemma provides a concentration property for
Lemma 4.3. For any
Proof. Note that
To prove
˜u(x1,y)={u(x1,y)if x1∈[0,p0], u(x1,y)(p0+1−x1)+v0(x1,y)(x1−p0)if x1∈(p0,p0+1), v0(x1,y)if x1∈[p0+1,+∞) |
Hence,
12c≤12c(v0)≤12J(˜u)=J+(˜u)=J+(u)−+∞∑p=p0Jp,0(u)+Jp0,0(˜u). |
By definition, on
12c≤12J(˜u)≤J+(u)−∑+∞p=p0Jp,0(u)+Λ(r)≤12c−∑+∞p=p0Jp,0(u)+32Λ(r). |
Then
By the previous lemma we get
Lemma 4.4. For any
Proof. Note that the existence of
As a direct consequence of Lemmas 4.3 and 4.4 we obtain the following concentration result.
Lemma 4.5. For any
Proof. The existence of
Finally,
We are now able to prove the existence of a minimum of
Theorem 4.6. Let
Proof. Let
‖uk−v‖W1,2(Tp,0,Rm)≤r1 for every p≥˜ℓ(r1). | (14) |
By Lemma 3.5, since
‖u−v‖W1,2(Tp,0,Rm)≤r1 for every p≥˜ℓ(r1). | (15) |
Therefore, by Lemma 3.6, we conclude that
By Theorem 4.6 we know that for every
Lemma 4.7. Given
∫R×[0,1]n−1∇ˉu⋅∇ψ+Fu(x,ˉu)ψdx=0. |
The proof can be adapted by the one of Lemma 3.3 of [4] or Lemma 5.2 of [6]. Therefore we get that any
Finally, we now study further compactness properties for the functional
Lv:W2,2([0,1]n,Rm)⊂L2([0,1]n,Rm)→L2([0,1]n,Rm), |
Lvh=−Δh+Fu,u(⋅,v(⋅))h |
has spectrum which does not contain
(
J″0(v)h⋅h=∫[0,1]n|∇h(x)|2+Fu,u(x,v(x))|h(x)|2dx≥α0‖h‖2L2([0,1]n,Rm) |
for every
As a consequence of
Lemma 4.8. There exist
ω0‖u−v‖2W1,2(Tp,0,Rm)≤Jp,0(u)≤ω1‖u−v‖2W1,2(Tp,0,Rm). | (16) |
Proof. Notice that, by (
∫[0,1]n|∇h(x)|2+Fu,u(x,v(x))|h(x)|2dx≥α0‖h‖2L2([0,1]n,Rm)≥−α0f0∫[0,1]nFu,u(x,v(x))|h(x)|2dx, |
where
∫[0,1]n11+α0f0|∇h(x)|2dx+∫[0,1]nFu,u(x,v(x))|h(x)|2dx≥0. |
We conclude that
J″0(v)h⋅h=∫[0,1]n|∇h(x)|2+Fu,u(x,v(x))|h(x)|2dx≥α0f01+α0f0‖∇h‖2L2([0,1]n,Rm) |
and so, using
Since by Taylor's formula we have
(17) |
On the other hand, again Taylor's expansion gives us
and we deduce that there exists
(18) |
The lemma follows by periodicity from (17) and (18) recalling that
Remark 3. In connection with Remark 1, arguing as in Remark 3.8 of [2], we can prove that (16) holds true also for the functional
(19) |
Hence, recalling the definition (10), plainly adapting the proof of Lemma 3.10 in [2], we obtain
Lemma 4.9. Let
In this section we prove our main theorem. To this aim, following and adapting the argument in [2], we will first prove the existence of a solution of (PDE) on the unbounded triangle
satisfying Neumann boundary conditions on
Then, by recursive reflections with respect to the hyperplanes
Let us introduce now some notations. We define the squares
and the horizontal strips
The intersection between the strip
where
For every
and the normalized functionals on the bounded strips
for every
Remark 4. Notice that
Then, we can set
We plainly obtain that
Lemma 5.1. We have
We can now introduce on the set
the functional
Notice that
Lemma 5.2. If
We now look for a minimum of the functional
Lemma 5.2, gives that
Proposition 1. We have
Arguing as in [2,4,6] (see e.g. the argument in Lemma 3.3 of [4] or Lemma 5.2 of [6]), we can prove that if
In the next lemma we finally characterise the asymptotic behavior of the solution
Lemma 5.3. Let
Proof. Let
(20) |
We have
Now, for every
A computation gives
Now, consider
and hence, since
and since
(21) |
In particular
(22) |
Let us now consider, for every
Arguing as above
thus giving
We now prove that the sequence
As a consequence, by definition of
(23) |
provided that
(24) |
Finally, recalling (6), since both (23) and (24) holds, we must have
Moreover, we have proved that
Hence we obtain that
(25) |
Finally, for every
Notice that since
Hence, by (25), we conclude
The previous lemma gives the asymptotic estimate in Theorem 1.1 since
We can conclude now the proof of Theorem 1.1 proving the sign property
Finally, for any
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