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Review

Perspective on utilization of Bacillus species as plant probiotics for different crops in adverse conditions

  • 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

    (F1) FC2(Rn×Rm;R) is 1-periodic sin all its variable, n,m1.

    When n=1 and m1, (PDE) are particular cases of the dynamical systems considered in the Aubry-Mather Theory ([9,23,24]). When n>1 and m=1 equations like (PDE) were studied by Moser in [25] (indeed in a much more general setting), and then by Bangert [13] and Rabinowitz and Stredulinsky [31], extending some of the results of the Aubry-Mather Theory for partial differential equations. These studies show the presence of a very rich structure of the set of minimal (or locally minimal) entire solutions of (PDE). In particular, when m=1 the set M0 of minimal periodic solutions of (PDE) is a non empty ordered set and if M0 is not a continuum then there exists another ordered family M1 of minimal entire solutions which are heteroclinic in one space variable to a couple of (extremal) periodic solutions u<v (a gap pair in M0). If M1 is not a continuum the argument can be iterated to find more complex ordered classes of minimal heteroclinic type solutions and the process continues if the corresponding set of minimal heteroclinics contains gaps. Variational gluing arguments were then employed by Rabinowitz and Stredulinsky to construct various kinds of homoclinic, heteroclinic or more generally multitransition solutions as local minima of renormalized functionals associated to (PDE), see [31]. Other extensions of Moser's results, including changing slope or higher Morse index solutions, have been developed by Bessi [10,11], Bolotin and Rabinowitz [12], de la Llave and Valdinoci [17,33]. Recently, in a symmetric setting and correspondingly to the presence of a gap pair in M0 symmetric with respect to the origin, entire solutions of saddle type were found by Autuori, Alessio and Montecchiari in [2].

    All the above results are based on the ordered structure of the set of minimal solutions of (PDE) in the case m=1 and a key tool in their proofs is the Maximum Principle, which is no longer available when m>1.

    The study of (PDE) when n,m>1 was initiated by Rabinowitz in [29,30]. Denoting L(u)=12|u|2+F(x,u) and Tn=Rn/Zn, periodic solutions to (PDE) were found as minima of the functional J0(u)=TnL(u)dx on E0=W1,2(Tn,Rm) showing that

    M0={uE0J0(u)=c0:=infE0J0(u)}.

    Paul H. Rabinowitz studied the case of spatially reversible potentials F assuming (¯F2) F is even in xi for 1in and proved in [29] that if M0 is constituted by isolated points then for each vM0 there is a v+M0{v} and a solution uC2(R×Tn1,Rm) of (PDE) that is heteroclinic in x1 from v to v+. These solutions were found by variational methods minimizing the renormalized functional

    J(u)=pZJp,0(u):=pZ(Tp,0L(u)dxc0), (1)

    (where Tp,0=[p,p+1]×[0,1]n1) on the space

    Γ(v,v+)={uW1,2(R×Tn1,Rm)uv±L2(Tp,0,Rm)0 as p±}.

    In [30] the existence of minimal double heteroclinics was obtained assuming that the elements of M0 are not degenerate critical points of J0 and that the set M1(v,v+) of the minima of J on Γ(v,v+) is constituted by isolated points. This research line was continued by Montecchiari and Rabinowitz in [26] where, via variational methods, multitransition solutions of (PDE) were found by glueing different integer phase shifts of minimal heteroclinic connections.

    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 Δu+W(u)=0 on R2 with W an even double well potential. They proved the existence of a (unique) saddle solution vC2(R2) of that equation, i.e., a bounded entire solution having the same sign and symmetry of the product function x1x2 and being asymptotic to the minima of the potential W along any directions not parallel to the coordinate axes. The saddle solution can be seen as a phase transition with cross interface.

    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 k-end solutions (see [4,18,22,19]).

    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 m=1.

    In the present paper we generalize the setting considered in [2] to the case m>1. Indeed we consider to have potentials F satisfying (F1) and the symmetry properties

    (F2) F is even in all its variables;

    (F3) F has flip symmetry with respect to the first two variables, i.e.,

    F(x1,x2,x3,...,xn,u)=F(x2,x1,x3,...,xn,u) on Rn×Rm.

    By [29] the set M0 of minimal periodic solution of (PDE) is not empty. The symmetry of F implies that any vM0 has components whose sign is constant on Rn and if vM0 then (ν1v1,,νmvm)M0 for every (ν1,,νm){±1}m (see Lemma 2.2 below). In this sense we can say that M0 is symmetric with respect to the constant function v00.

    As recalled above, in [2], where m=1, a saddle solution was found when M0 has a gap pair symmetric with respect to the origin. In the case m>1 we generalize this gap condition asking that 0M0 and, following [30], we look for saddle solutions of (PDE) when any vM0 is not degenerate for J0. We then assume

    (N) 0M0 and there exists α0>0 such that

    J0(v)hh=[0,1]n|h|2+Fu,u(x,v(x))hhdxα0h2L2([0,1]n,Rm)

    for every hW1,2([0,1]n,Rm) and every vM0.

    The assumption (N) and the symmetries of F allow us to find heteroclinic connections between elements of M0 which are odd in the variable x1. More precisely for vM0 these solutions are searched as minima of the functional J (see (1)) on the space

    Γ(v)={uW1,2(R×Tn1,Rm)u is odd in x1,limp+uvL2([p,p+1]×Tn1,Rm)=0}.

    In §4, setting

    c(v)=infuΓ(v)J(u) for vM0

    we show that

    Mmin0={v0M0c(v0)=minvM0c(v)}

    and that Mmin0 is such that if v0Mmin0 then

    M(v0)={uΓ(v0)J(u)=c(v0)}

    is not empty and compact with respect to the W1,2(R×Tn1,Rn) metric. The elements uM(v0) are classical solutions to (PDE), odd in x1, even and 1-periodic in x2,...,xn and satisfy the asymptotic condition

    uv0W1,2([p,p+1]×Tn1,Rm)0 as p+.

    Our main result can now be stated as follows

    Theorem 1.1. Assume (F1), (F2), (F3) and (N). Then, there exists a classical solution w of (PDE) such that every component wi (for i=1,m) satisfies

    (i) wi0 for x1x2>0;

    (ii) wi is odd in x1 and x2, 1-periodic in x3,...,xn;

    (iii) wi(x1,x2,x3,...,xn)=wi(x2,x1,x3,...,xn) in Rn.

    Moreover there exists v0Mmin0 such that the solution w satisfies the asymptotic condition

    distW1,2(Rk,Rm)(w,M(v0))0,ask+, (2)

    where Rk=[k,k]×[k,k+1]×[0,1]n2.

    Note that by (i) and (ii) any component of w has the same sign as the product function x1x2. Moreover by (2), since w is asymptotic as x2+ to the compact set M(v0) of odd heteroclinic type solutions, the symmetry of w implies that w is asymptotic to v0 or v0 along any direction not parallel to the planes x1=0, x2=0. In this sense w is a saddle solution, representing a multiple transition between the pure phases v0 and v0 with cross interface.

    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 m>1 and so to avoid the use of the Maximum Principle we need a refined analysis of the concentrating properties of the minimizing sequences. For that a series of preliminaries results is given in §2, §3, §4 while the proof of Theorem 1.1 is developed in §5.

    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 (F2) and (F3). Here and in the following we will work under the not restrictive assumption

    (F4) F0 on Rn×Rm.

    Let us introduce the set

    E0=W1,2(Tn,Rm)={uW1,2(Rn,Rm)u is 1-periodic in all its variables}

    with the norm

    uW1,2([0,1]n,Rm)=(mi=1[0,1]n(|ui|2+|ui|2)dx)12.

    We define the functional J0:E0R as

    J0(u)=[0,1]n12|u|2+F(x,u)dx=[0,1]nL(u)dx. (3)

    and consider the minimizing set

    M0={uE0|J0(u)=c0} where c0=infuE0J0(u)

    Then in [29], [30] it is shown

    Lemma 2.1. Assume (F1), then M0. Moreover, setting [u]=[0,1]nudx, we have that

    1. ˆM0={uM0[u][0,1]m} is a compact set in E0;

    2. if (uk)kE0, with [uk][0,1]m, is a minimizing sequence for J0, then there exists uˆM0 such that uku in E0 up to subsequences;

    3. For every ρ>0 there exists β(ρ)>0 such that if uE0 is such that

    distW1,2([0,1]n,Rm)(u,M0):=infvM0uvW1,2([0,1]n,Rm)>ρ,

    then J0(u)c0β(ρ);

    4. If (F2) holds, then any uM0 minimizes also I(u)=[0,12]nL(u)dx on W1,2([0,12]n,Rm). As a consequence, every uM0 is symmetric in xi about xi=0 and xi=12 for every index i and u is even in xi for every index i;

    5. If (F2) holds, there results c0=infuW1,2([0,1]n,Rm)J0(u). Furthermore, if uW1,2([0,1]n,Rm) verifies J0(u)=c0, then for every i=1,2,...,n, u is symmetric in xi about xi=12 and hence uM0.

    Assumption (F2), in particular the even parity of F with respect to the components of u, provides that the elements in M0 have components with definite sign, thanks to the unique extension property (see [29], Proposition 3).

    Lemma 2.2. Assume (F1), (F2) and 0M0. If u=(u1,,um)M0 then, for every i=1,...,m, one has either ui0, or ui0 on [0,1]n and u does not vanish on open sets. Moreover, (ν1u1,,νmum)M0 for every (ν1,,νm){±1}m.

    Proof. It is sufficient to observe that if u=(u1,,um)M0 then, since F is even with respect to the components of u, we have

    ⅰ) ˉu=(|u1|,,|um|)M0 and

    ⅱ) (ν1u1,,νmum)M0 for every (ν1,,νm){±1}m.

    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 u do not change sign. If u vanishes on an open set, the unique continuation property gives u0, giving a contradiction and concluding the proof.

    On the other hand, assumption (F3) gives more structure on the set M0: its elements have a flip symmetry property. Indeed, setting T+={x[0,1]n|x1x2}, for every uW1,2(T+,Rm), let us define ˜uW1,2([0,1]n,Rm) as

    ˜u(x)={u(x),xT+,u(x2,x1,x3,,xn),x[0,1]nT+. (4)

    Then, we have

    Lemma 2.3. If uM0 then, u˜u in [0,1]n.

    Proof. Given uM0, without loss of generality, we assume

    T+L(u)dx[0,1]nT+L(u)dx.

    Since ˜uW1,2([0,1]n,Rm) by Lemma 2.1-(5) we have J0(˜u)c0. By the previous inequality we get

    c0=J0(u)=T+L(u)dx+[0,1]nT+L(u)dx2T+L(u)dx=J0(˜u)c0.

    Hence, again by Lemma 2.1-(5), ˜uM0. By the unique extension property of the solutions of (PDE) (cf. [29], Proposition 3), we have ˜uu in [0,1]n.

    As an immediate consequence, using Lemma 2.1-(5), we have the following.

    Lemma 2.4. There results

    minuW1,2(T+,Rm)T+L(u)dx=c02. (5)

    Moreover, if uW1,2(T+,Rm) verifies T+L(u)dx=c02, then ˜uM0.

    Remark 1. Lemma 2.3 tells us that the elements of M0 are symmetric with respect to the diagonal iperplane {xRnx1=x2} and by Lemma 2.4 they can be found by minimizing T+L(v)dx on W1,2(T+,Rm). Analogously, setting T=[0,1]nT+, we can find the elements of M0 by minimizing TL(v)dx on W1,2(T,Rm) or, by periodicity, by minimizing TL(v)dx on W1,2(T,Rm) whenever T=p+T± with pZn. For future references it is important to note that this property implies in particular that uM0 if and only if u is a minimizer of the functional σ0L(v)dx on W1,2(σ0,Rm) where

    σ0={xR×[0,1]n1|x21x1x2}.

    More precisely we have c0=infvW1,2(σ0,Rm)σ0L(v)dx and uM0 if and only if σ0L(u)dx=c0. From Lemma 2.1-(3) we recover an analogous property in W1,2(σ0,Rm): for any r>0 there exists β(r)>0 such that if uW1,2(σ0,Rm) verifies σ0L(u)dxc0+β(r), then distW1,2(σ0,Rm)(u,M0)r.

    Note that by Lemma 2.1-(1) and the assumption (N) we plainly derive that (N0) ˆM0 is a finite set and 0ˆM0,

    where we recall that ˆM0={uM0[u][0,1]m} and note that M0=ˆM0+Zm.

    Note finally that by (N0), setting

    r0:=min{uvL2(Tn,Rm)u,vM0,uv}, (6)

    we have r0>0.

    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 J and studying some of its basic properties.

    Let us define the set

    E={uW1,2loc(Rn,Rm)u is 1-periodic in x2,xn}.

    For any uE we consider the functional

    J(u)=pZJp,0(u),

    where, denoting Tp,0=[p,p+1]×[0,1]n1,

    Jp,0(u)=Tp,0L(u)dxc0,pZ.

    Denoting briefly u(+p) the shifting of the function u with respect to the first coordinate (that is, u(+p)=u(+pe1) where e1=(1,0,...,0)), note that by periodicity we have

    Jp,0(u)=[0,1]nL(u(+p))dxc0=J0(u(+p))c0,pZ.

    Then, by Lemma 2.1, we have Jp,0(u)0 for any uE and pZ, from which J is non-negative on E.

    Lemma 3.1. The functional J:ER is weakly lower semicontinuous.

    Proof. Consider a sequence (uk)k such that uku weakly in E. Then, for every N, by the weak lower semicontinuity of J0, and hence of Jp,0, we have p=Jp,0(u)lim infkp=Jp,0(uk). If J(u)=+, then we obtain easily lim infkJ(uk)=+. So, let us assume J(u)<+, then for any ε>0 we have that there exists N such that |p|>Jp,0(u)<ε. We get

    lim infkJ(uk)lim infkp=Jp,0(uk)p=Jp,0(u)>J(u)ε,

    thus finishing the proof.

    Using the notation introduced above, note that if uE is such that J(u)<+, then Jp,0(u)0 as |p|+, that is, the sequence (u(+p))pZ is such that J0(u(+p))c0 as p±. Hence, by Lemma 2.1-(3), there exist u±M0 such that, up to a subsequence, u(+p)u± as p± in E0. Using this remark and the local compactness of M0 given by (N0), we are going to prove some concentration properties of the minimizing sequence of the functional J.

    First of all, let us consider the functional Jp,0+Jp+1,0 for a certain fixed integer p. Notice that, by Lemma 2.1-(5),

    minuEJp,0(u)+Jp+1,0(u)=0

    and the set of minima coincide with M0. We introduce the following distance

    distp(u,A)=inf{uvW1,2(Tp,0Tp+1,0,Rm)vA}.

    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 r>0 there exists λ(r)>0 such that

    if uE satisfies Jp,0(u)+Jp+1,0(u)λ(r) for a pZ, then distp(u,M0)r. (7)

    It is not restrictive to assume that the function with rλ(r) is non-decreasing.

    On the other hand for every λ>0 if we set

    ρ(λ)=sup{distp(u,M0)uE with Jp,0(u)+Jp+1,0(u)λ,pZ}

    we get ρ(λ)>0 and that λρ(λ) is non-decreasing. Moreover, for every ε>0, since if Jp,0(u)+Jp+1,0(u)λ(ε) for a certain pZ, then distp(u,M0)ε, we obtain ρ(λ)ρ(λ(ε))ε for every λ(0,λ(ε)], so that limλ0+ρ(λ)=0 holds. Hence, recalling the definition of r0 in (6), we can fix λ0>0 satisfying ρ(λ0)r04. Finally, we can define

    Λ(r)=sup{Jp,0(u)uE and pZ are such that distp(u,M0)2r} (8)

    which is non-decreasing and limr0Λ(r)=0. Then we fix r1(0,r04) such that Λ(r)λ08 for every r(0,r1].

    We say that a set IZ is a set of consecutive integers if it is of the form {Z|p<p+k} or {Z|pk<p} for a pZ and kN{+}. If uE is such that Jp,0 is small enough for some consecutive integers pI, then, using (N0), we can prove that, in the corresponding sets Tp,0, u is ``near'' to an element of M0, the same for all pI. Indeed we have

    Lemma 3.2. Given λ(0,λ02], uE and a set of consecutive integers I, if Jp,0(u)λ for any pI, then there exists vM0 such that uvW1,2(Tp,0,Rm)ρ(2λ)r04, for every pI.

    Proof. Let pI be such that p+1I. Then Jp,0(u)+Jp+1,0(u)2λλ0 and, by Remark 2 and the definition of λ0, distp(u,M0)ρ(2λ)ρ(λ0)r04. Then, by (N0) and the choice of r0 in (6), we can find vpM0 such that

    uvpW1,2(Tp,0Tp+1,0,Rm)r04

    from which uvpW1,2(Tk,0,Rm)r04 for k=p,p+1. If p+2I, repeating the argument with the couple of indices p+1 and p+2 we find vp+1M0 such that uvp+1W1,2(Tk,0,Rm)r04 for k=p+1,p+2. By the choice of r0 in (6), we conclude that vp+1=vp and the lemma follows.

    Moreover, using the notations introduced above, we have

    Lemma 3.3. If uW1,2(Tp,0Tp+1,0,Rm) then

    u(+p)u(+(p+1))2L2([0,1]n,Rm)2(Jp,0(u)+Jp+1,0(u)+2c0).

    Proof. Setting y=(x2,,xn), we have

    u(+p)u(+(p+1))2L2([0,1]n,Rm)=p+1p[0,1]n1|u(x1+1,y)u(x1,y)|2dydx1

    and so there exists ˉx1(p,p+1) such that

    [0,1]n1|u(ˉx1+1,y)u(ˉx1,y)|2dyu(+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]n1|x1u(x1,y)|2dydx1[0,1]n1ˉx1+1ˉx1|x1u(x1,y)|2dx1dy[0,1]n1|u(ˉx1+1,y)u(ˉx1)|2dyu(+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 J satisfy the following boundeness property.

    Lemma 3.4. For every Λ>0 there exists R>0 such that for every uE satisfying J(u)Λ one has u(+p)u(+q)L2([0,1]n,Rm)R for any p,qZ.

    Proof. Let uE be such that J(u)Λ. We define J(u)={kZJk,0(u)λ02} and note that the number l(u) of elements of J(u) is at most [2Λλ0]+1, where [] denotes the integer part. Then, the set ZJ(u) is constituted by ˉl(u) sets of consecutive elements of Z, Ii(u), with ˉl(u)l(u)+1. By the triangular inequality, for any p,qZ, we obtain

    u(+p)u(+q)L2([0,1]n,Rm)l(u)supkJ(u)u(+k)u(+k+1)L2([0,1]n,Rm)+ˉl(u)i=1supp,qIi(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),kZ,

    while the second one follows by the definition of Ii(u) and Lemma 3.2.

    Since ˉl(u)l(u)+1 and l(u)[2cλ0]+1, the lemma follows by choosing R=([2Λλ0]+1)(2(Λ+2c0))12+([2Λλ0]+2)r02.

    The following lemma states the weak compactness of the sublevels of the functional J.

    Lemma 3.5. Given any Λ>0, let (uk)kE be a sequence such that J(uk)Λ for every kN and let (pk)k be a sequence of integers. Assume that there exist ˉR<+ and vM0 such that ukvW1,2(Tpk,0,Rm)ˉR for all kN. Then, there exists uE with J(u)Λ such that, up to a subsequence, uku weakly in E.

    Proof. First note that, by Lemma 3.4, there exists R>0 such that if uE and J(u)Λ then u(+p)u(+q)L2([0,1]n,Rm)R for any p,qZ. If uvW1,2(T,0,Rm)ˉR for some Z and vM0, by triangular inequality for any pZ we obtain

    uvL2(Tp,0,Rm)=u(+p)vL2([0,1]n,Rm)u(+p)u(+)L2([0,1]n,Rm)+u(+)vL2([0,1]n,Rm)R+ˉR.

    Consider now a sequence as in the statement, setting QL=[L,L]×[0,1]n1 for LN, we get

    ukv2L2(QL,Rm)+uk2L2(QL,Rm)2L(ˉR+R)2+4Lc0+2Λ.

    Hence, (ukv)k is bounded in W1,2(QL,Rm) for any LN and, by a diagonal argument and the weak lower semicontinuity of J, the statement follows.

    By Lemma 3.2 we also deduce the following result concerning the asymptotic behaviour of the functions in the sublevels of J.

    Lemma 3.6. If J(u)<+, there exist v±M0 such that

    uv±W1,2(Tp,0,Rm)0asp±.

    Proof. Since J(u)<+, we have Jp,0(u)0 as |p|+ and there exists ˉp such that Jp,0(u)λ02 for any |p|ˉp. Thus, by Lemma 3.2, there exists v±M0 such that uv+W1,2(Tp,0,Rm)r04 for pˉp and uvW1,2(Tp,0,Rm)r04 for pˉp.

    Hence the sequence (u(+p))pN is such that u(+p)v+W1,2([0,1]n,Rm)r04 for every pˉp and J0(u(+p))c0=Jp,0(u)0 as p+. Then, by Lemma 2.1, uv+W1,2(Tp,0,Rm)=u(+p)v+W1,2([0,1]n,Rm)0 as p+. Analogously we obtain that uvW1,2(Tp,0,Rm)0 as p

    By Lemma 3.6, if uE satisfies J(u)<+ we can view it as an heteroclinic or homoclinic connection between two periodic solutions v and v+ belonging to M0. Hence, we can consider elements of E belonging to the classes

    Γ(v,v+)={uEuv±W1,2(Tp,0,Rm)0asp±}

    where v±M0.

    We note that by Lemma 3.5, every sequence (uk)kNΓ(v,v+) with J(uk)Λ for all kN, admits a subsequence which converges weakly to some uE. Indeed, since ukv+W1,2(Tp,0,Rm)0 as p+ for every kN, fixed ˉR>0 there exists pkN such that ukv+W1,2(Tpk,0,Rm)ˉR and since J(uk)Λ, by Lemma 3.5, there exists uE such that, up to a subsequence, uku weakly as k+.

    In particular, given v±M0 and setting

    c(v,v+)=infuΓ(v,v+)J(u),

    as in [29], we obtain that for any vM0 there exist v+M0{v} and uΓ(v,v+) such that c(v,v+)=J(u). Moreover, it can be proved that any uΓ(v,v+) such that c(v,v+)=J(u) is a classical solution of (PDE) (see Theorem 3.3 in [29]).

    Finally, we have that infvv+c(v,v+)>0 as a consequence of the following lemma.

    Lemma 3.7. For every v±M0 with vv+, we have c(v,v+)λ02. Moreover, c(v,v+)+ as v+vW1,2([0,1]n,Rm)+.

    Proof. Assume that there exists uΓ(v,v+) satisfying J(u)<λ02. Then Jp,0(u)<λ02 for every pZ, so that by Lemma 3.2 there exists vM0 such that uvW1,2(Tp,0,Rm)r04 for every pZ. Since uΓ(v,v+) we know that uvW1,2(Tp,0,Rm)0 as p and uv+W1,2(Tp,0,Rm)0 as p+, so that by (6) we would have v=v=v+ giving a contradiction.

    In order to prove the second part of the statement, assume the existence of two sequences (vk)k and (v+k)k in M0 such that (c(vk,v+k))k is bounded while v+kvkW1,2(Tn,Rm)+ as k+. Since (c(vk,v+k))k is bounded, we can find Λ>0 and a sequence (uk)k, with ukΓ(vk,v+k), such that J(uk)Λ, for every index k. Hence, by Lemma 3.4, there exists R>0 such that uk(+p)uk(+q)L2([0,1]n,Rm)R for every kN and p,qZ. Moreover, for every ε>0 and kN, since ukΓ(vk,v+k), there exist pk,qkZ such that ukvkW1,2(Tpk,0,Rm)<ε and ukv+kW1,2(Tqk,0,Rm)<ε for every kN. In particular we get

    v+kvkL2([0,1]n,Rm)vkuk(+pk)L2([0,1]n,Rm)+uk(+pk)uk(+qk)L2([0,1]n,Rm)+v+kuk(+qk)L2([0,1]n,Rm)ε+R+ε

    since, by periodicity, v±kuk(+p)L2([0,1]n,Rm)=v±kukW1,2(L2,Rm) for any kN, pZ. Finally, since v2L2([0,1]n,Rm)2c0 for every vM0, we recover v+vW1,2([0,1]n,Rm)22c0+2ε+R in contradiction with v+kvkW1,2(Tn,Rm)+.

    We focalize now in the study of heteroclinic solutions which are odd in the first variable, hence we will consider a subset of Γ(v,v), vM0, so let us introduce the set

    Eodd={uEu is odd with respect to x1},

    In what follows, when we will consider functions uEodd we often present their properties for x10, avoiding to write the corresponding ones for x1<0. In particular, for every uEodd we have J(u)=2J+(u), where

    J+(u)=p0Jp,0(u).

    For any vM0 let

    Γ(v)={uEodduvW1,2(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 uEodd for which J(u)<+ there exists vM0 such that uvW1,2(Tp,0,Rm)0 as p+, that is uΓ(v).

    We are going to look for minimizer of J in the set Γ(v). So, for every vM0 we set

    c(v)=infuΓ(v)J(u) and M(v)={uΓ(v)J(u)=c(v)}. (10)

    Notice that for any vM0 we have c(v,v)c(v)<+ holds and, by Lemma 3.7 since by (N0), 0M0, we have the following.

    Lemma 4.2. For any vM0, c(v)λ02, and c(v)+ as vW1,2([0,1]n,Rm)+.

    Moreover, note that, by assumption (N0), the intersection between M0 and a bounded set consists of a finite number of elements. Hence, from the previous lemma, the minimum

    c=minvM0c(v) (11)

    is well defined and the set

    Mmin0={vM0c(v)=c} (12)

    is nonempty and consists of a finite number of elements. In particular, we have

    minvM0Mmin0c(v)>c. (13)

    The following lemma provides a concentration property for uEodd such that J(u) is close to the value c: the elements of the sequence (u(+p))pZ remain far from M0 only for a finite number of indexes p. Moreover, (u(+p))pZ approaches an element v0M0 only once. Indeed, recalling the notation introduced in Remark 2, we have

    Lemma 4.3. For any r(0,r1] there exists (r)N, δ(r)(0,r04) with δ(r)0 as r0+ with the following property: if uEodd is such that J(u)c+Λ(r) then

    (i) if distW1,2(Tp,0,Rm)(u,M0)r for every p in a set I of consecutive integers, then Card(I)(r),

    (ii) if uv0W1,2(Tp0,0,Rm)r for some index p00 and v0M0, then uv0W1,2(Tp,0,Rm)δ(r) for every pp0, and +p=p0Jp,0(u)2Λ(r).

    Proof. Note that (i) plainly follows from Lemma 2.1-(3), setting (r)=[c+Λ(r)β(r)]+1, where [] denotes the integer part.

    To prove (ii), we consider ˜uEodd defined for x10 as

    ˜u(x1,y)={u(x1,y)if x1[0,p0]u(x1,y)(p0+1x1)+v0(x1,y)(x1p0)if x1(p0,p0+1)v0(x1,y)if x1[p0+1,+)

    Hence, ˜uΓ(v0) and since ˜uu in [p0,p0]×Rn1, while ˜u=v0 in [p0+1,+)×Rn1, we obtain

    12c12c(v0)12J(˜u)=J+(˜u)=J+(u)+p=p0Jp,0(u)+Jp0,0(˜u).

    By definition, on Tp0,0 we have ˜u(x1,y)v0(x1,y)=(p0+1x1)(u(x1,y)v0(x1,y)) and so ˜uv0W1,2(Tp0,0,Rm)2uv0W1,2(Tp0,0,Rm)2r. Since ˜u=v0 in [p0+1,p0+2]×Rn1, we have distp(˜u,M0)=˜uv0W1,2(Tp0,0,Rm)2r, so that, by Remark 2, we obtain Jp0,0(˜u)Λ(r)λ08 and therefore

    12c12J(˜u)J+(u)+p=p0Jp,0(u)+Λ(r)12c+p=p0Jp,0(u)+32Λ(r).

    Then +p=p0Jp,0(u)32Λ(r) and in particular Jp,0(u)32Λ(r)λ02 for any pp0. Hence, by Lemma 3.2, uv0W1,2(Tp,0,Rm)ρ(3Λ(r))<r0 for any pp0. Hence (ii) follows setting δ(r)=ρ(3Λ(r)). Indeed, by Remark 2, we have limr0+δ(r)=0 and, since Λ(r)λ08 for all r(0,r1], we get δ(r)ρ(λ0)r04 for every r(0,r1).

    By the previous lemma we get

    Lemma 4.4. For any r(0,r1], if uEodd satisfies J(u)c+Λ(r), then there exists v0M0 such that uΓ(v0) and

    (i) if uv0W1,2(Tp0,0,Rm)r for a certain index p00, then we have uv0W1,2(Tp,0,Rm)δ(r) for every pp0, and +p=p0Jp,0(u)2Λ(r).

    (ii) if wM0{v0}, then uwW1,2(Tp,0,Rm)>r1 for every pZ,p0.

    Proof. Note that the existence of v0 such that uΓ(v0) is ensured by Lemma 4.1 and (i) plainly follows from Lemma 4.3-(ii). To prove (ii) we argue by contradiction assuming that there exist ˉp0Z, ˉp00 and wM0{v0} such that uwW1,2(Tˉp0,0,Rm)r1. Again, by Lemma 4.3-(ii) we get uwW1,2(Tp,0,Rm)δ(r1)r04 for every pˉp0 which is in contradiction with uΓ(v0), recalling the definition of r0 in (6).

    As a direct consequence of Lemmas 4.3 and 4.4 we obtain the following concentration result.

    Lemma 4.5. For any ρ(0,r1] there exists ˜Λ(ρ), with ˜Λ(ρ)0 as ρ0+, and ˜(ρ)N such that if uEodd satisfies J(u)c+˜Λ(ρ), then there exists v0M0 such that uΓ(v0) and

    (i) uv0W1,2(Tp,0,Rm)ρfor everyp˜(ρ);

    (ii) +p=˜(ρ)Jp,0(u)2˜Λ(ρ);

    (iii) uwW1,2(Tp,0,Rm)r1 for every pZ,p0 and wM0{v0}.

    Proof. The existence of v0 such that uΓ(v0) is again ensured by Lemma 4.1. By Lemma 4.3, given any ρ(0,r1], there exists r(0,ρ) such that δ(r)ρ. Then, if uΓ(v0) is such that J(u)c+Λ(r), by Lemma 4.3-(i), there exists p0[0,(r)+1] such that distW1,2(Tp0,0,Rm)(u,M0)<r and hence a vM0 such that uvW1,2(Tp0,0,Rm)<r. Therefore, by Lemma 4.3-(ii), we obtain uvW1,2(Tp,0,Rm)<δ(r) for all pp0 and since δ(r)<ρ<r1<r04, we can conclude that vv0 and hence that uv0W1,2(Tp,0,Rm)ρ for every pp0. Moreover, again by Lemma 4.3-(ii), we have +p=p0Jp,0(u)2Λ(r). Hence (i) and (ii) follows setting ˜(ρ)=(r)+1 and ˜Λ(ρ)=Λ(r).

    Finally, uwW1,2(Tp,0,Rm)r1 for every pZ,p0, and wM0{v0} follows directly by Lemma 4.4 -(ii).

    We are now able to prove the existence of a minimum of J in the set Γ(v) for every vMmin0, i.e., that M(v) for all vMmin0.

    Theorem 4.6. Let vMmin0, then there exists uΓ(v) such that J(u)=c(v)=c.

    Proof. Let (uk)kΓ(v) be such that J(uk)c(v). Without loss of generality we can assume that J(uk)c+˜Λ(r1) for any kN. By Lemma 4.5, we obtain that for any kN,

    ukvW1,2(Tp,0,Rm)r1 for every p˜(r1). (14)

    By Lemma 3.5, since Eodd is weakly closed, there exists uEodd such that, along a subsequence, uku weakly in Eodd. Finally, by (14) and the weakly lower semicontinuity of the distance we obtain

    uvW1,2(Tp,0,Rm)r1 for every p˜(r1). (15)

    Therefore, by Lemma 3.6, we conclude that uvW1,2(Tp,0,Rm)0 as p+, so that uΓ(v). Finally, by semicontinuity, J(u)=c(v).

    By Theorem 4.6 we know that for every v0Mmin0, M(v0) is nonempty. One can prove that M(v0) consists of weak solutions of (PDE).

    Lemma 4.7. Given ˉuM(v0), with v0Mmin0, then for any ψC0(R×Tn1,Rm) we have

    R×[0,1]n1ˉuψ+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 uM(v0) is a classical C2(Rn,Rm) solution of (PDE) which is 1-periodic in the variables xi, i2.

    Finally, we now study further compactness properties for the functional J that will be useful in the next section. They will be obtained as consequences of the nondegeneracy property of the elements of M0 asked in (N). In particular assumption (N) asks that, for every vM0, the linearized operator about v

    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 0. This is the assumption made in [30] and it is indeed equivalent to require as in (N) that

    (N1) there exists α0>0 such that

    J0(v)hh=[0,1]n|h(x)|2+Fu,u(x,v(x))|h(x)|2dxα0h2L2([0,1]n,Rm)

    for every hW1,2([0,1]n,Rm) and every vM0.

    As a consequence of (N1) we obtain the following (see also Lemma 3.6 in [2]).

    Lemma 4.8. There exist r2(0,r1) and ω1>ω0>0 such that if uW1,2(Tp,0,Rm), pZ, verifies uvW1,2(Tp,0,Rm)r2 for some vM0 then

    ω0uv2W1,2(Tp,0,Rm)Jp,0(u)ω1uv2W1,2(Tp,0,Rm). (16)

    Proof. Notice that, by (N1), if hW1,2([0,1]n,Rm) and vM0 then

    [0,1]n|h(x)|2+Fu,u(x,v(x))|h(x)|2dxα0h2L2([0,1]n,Rm)α0f0[0,1]nFu,u(x,v(x))|h(x)|2dx,

    where f0=1/Fuu, and so

    [0,1]n11+α0f0|h(x)|2dx+[0,1]nFu,u(x,v(x))|h(x)|2dx0.

    We conclude that

    J0(v)hh=[0,1]n|h(x)|2+Fu,u(x,v(x))|h(x)|2dxα0f01+α0f0h2L2([0,1]n,Rm)

    and so, using and setting , we obtain

    Since by Taylor's formula we have for all and , we obtain that there exists such that if verifies for some , then

    (17)

    On the other hand, again Taylor's expansion gives us

    and we deduce that there exists such that, taking smaller if necessary, if verifies , , then

    (18)

    The lemma follows by periodicity from (17) and (18) recalling that and that for all .

    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 on , that is, if for some then

    (19)

    Hence, recalling the definition (10), plainly adapting the proof of Lemma 3.10 in [2], we obtain

    Lemma 4.9. Let and be such that . Then there exists such that, up to a subsequence, as .

    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 , which is odd in the first variable , asymptotic as to a certain heterocline where

    Then, by recursive reflections with respect to the hyperplanes , we will recover a solution of (PDE) on the whole .

    Let us introduce now some notations. We define the squares

    and the horizontal strips

    The intersection between the strip and the triangle consists of a bounded strip

    where .

    Figure 1. 

    The decomposition of the triangular set

    .

    For every we define the sets of functions

    and the normalized functionals on the bounded strips as

    for every , where .

    Remark 4. Notice that for every , . Indeed, we can view the restriction as a traslation of a function in and the restriction on can be treated similarly using Lemma 2.4, the symmetry of and Remark 1. Moreover, we note that the functional is lower semicontinuous with respect to the weak topology for every .

    Then, we can set

    We plainly obtain that and that the sequence is increasing. Moreover, , evaluating on a function with . Moreover, the non degeneracy assumption () permits us to obtain as in [2] (see Lemma 4.2) the following stronger result.

    Lemma 5.1. We have , in particular as .

    We can now introduce on the set

    the functional

    Notice that for every . Indeed, the restriction and so for any . Moreover, is lower semicontinuous in the weak topology of . By Lemma 5.1 we readily obtain that is finite for at least one .

    Lemma 5.2. If for some , then .

    We now look for a minimum of the functional on , thus we set

    Lemma 5.2, gives that and we can prove the existence of the minimum applying the direct method of the Calculus of Variations (see e.g. the proof of Proposition 4.4 in [2]).

    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 then it is a weak solution of (PDE) on with Neumann boundary condition on . Then we can conclude that every is indeed a classical solution of (PDE). Finally, using (), we can recursively reflect with respect to the hyperplanes , obtaining an entire solution of (PDE) (see e.g. [2]). By construction, it is odd both in and , symmetric with respect to the hyperplanes and it is 1–periodic in . Hence, it satisfies hypotheses - of Theorem 1.1.

    In the next lemma we finally characterise the asymptotic behavior of the solution .

    Lemma 5.3. Let be the function obtained by recursive reflection of a given . Then there exists such that

    Proof. Let be as in the statement, we start proving that there exists such that

    (20)

    We have . Hence, as so that, by Lemma 5.1, as . Therefore, we can find a sequence , with such that as , and in particular . By Lemma 2.1-(3), we get as thus giving the existence of such that

    Now, for every , we define in the horizontal strip the following interpolation between and :

    A computation gives so that

    Now, consider defined on . We have , therefore

    and hence, since and , we obtain as . As a consequence, since , by (13), we can conclude that . Moreover we have

    and since , and , we obtain

    (21)

    In particular , so that , by the symmetry of with respect to . Summing up, using (21), we get , so we can apply Lemma 3.2 and conclude that

    (22)

    Let us now consider, for every , a different interpolation in the horizontal strip between and the periodic solution previously introduced:

    Arguing as above , so that, defining in we find {and hence, } . Since and we obtain, reasoning as above,

    thus giving .

    We now prove that the sequence is indeed a (definitively) constant sequence, i.e. for every sufficiently large. Being , we can assume and since and , we can apply Lemma 4.5 obtaining that

    ;

    ;

    As a consequence, by definition of and recalling that when we obtain and

    (23)

    provided that where . Consider now the vertical rectangle . We have for any in the set of consecutive integers , so that we can argue as in Lemma 3.2 and conclude that there exists such that

    (24)

    Finally, recalling (6), since both (23) and (24) holds, we must have for every . In particular, (22) gives the claim in (20).

    Moreover, we have proved that with and since , we can apply Lemma 4.9 to get that there exists for which, up to a subsequence,

    Hence we obtain that

    (25)

    Finally, for every we have

    Notice that since and using (20), we have

    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 . By Lemma 2.2, for any periodic solution we can define belonging to too, being easily verified. Now, by Theorem 4.6, there exists a heteroclinic solution . We can define the function , such that when , and verify that being .

    Finally, for any we can find as in Lemma 5.3. Similarly as above, we can define such that when . Then, we can verify that verifies Lemma 5.3 with the choice . By reflecting with rispect to the hyperplanes , we obtain the saddle-type solution satisfying in Theorem 1.1, thus completing the proof.


    Acknowledgments



    We would like to thank Mr. Liyu Chiang for his assistance.

    Conflict of interest



    The authors declare no conflict of interest.

    Author Contributions



    S.S.: Conceptualization, Investigation, Data Curation, Validation, Writing – Original Draft Preparation. D.J.H.S: Conceptualization, Supervision, Project Administration, Suggestion, and Writing – Review and Editing.

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