
Citation: Mohammed Maikudi Usman, Arezoo Dadrasnia, Kang Tzin Lim, Ahmad Fahim Mahmud, Salmah Ismail. Application of biosurfactants in environmental biotechnology; remediation of oil and heavy metal[J]. AIMS Bioengineering, 2016, 3(3): 289-304. doi: 10.3934/bioeng.2016.3.289
[1] | Marco G. Ghimenti, Anna Maria Micheletti . Compactness and blow up results for doubly perturbed Yamabe problems on manifolds with non umbilic boundary. Electronic Research Archive, 2022, 30(4): 1209-1235. doi: 10.3934/era.2022064 |
[2] | Qingming Hao, Wei Chen, Zhigang Pan, Chao Zhu, Yanhua Wang . Steady-state bifurcation and regularity of nonlinear Burgers equation with mean value constraint. Electronic Research Archive, 2025, 33(5): 2972-2988. doi: 10.3934/era.2025130 |
[3] | Yutong Chen, Jiabao Su . Resonant problems for non-local elliptic operators with unbounded nonlinearites. Electronic Research Archive, 2023, 31(9): 5716-5731. doi: 10.3934/era.2023290 |
[4] | Hongyu Li, Liangyu Wang, Yujun Cui . Positive solutions for a system of fractional q-difference equations with generalized p-Laplacian operators. Electronic Research Archive, 2024, 32(2): 1044-1066. doi: 10.3934/era.2024051 |
[5] | Xu Liu, Jun Zhou . Initial-boundary value problem for a fourth-order plate equation with Hardy-Hénon potential and polynomial nonlinearity. Electronic Research Archive, 2020, 28(2): 599-625. doi: 10.3934/era.2020032 |
[6] | Yuhua Long, Dan Li . Multiple nontrivial periodic solutions to a second-order partial difference equation. Electronic Research Archive, 2023, 31(3): 1596-1612. doi: 10.3934/era.2023082 |
[7] | Yijun Chen, Yaning Xie . A kernel-free boundary integral method for reaction-diffusion equations. Electronic Research Archive, 2025, 33(2): 556-581. doi: 10.3934/era.2025026 |
[8] | Qianqian Zhu, Yaojun Ye, Shuting Chang . Blow-up upper and lower bounds for solutions of a class of higher order nonlinear pseudo-parabolic equations. Electronic Research Archive, 2024, 32(2): 945-961. doi: 10.3934/era.2024046 |
[9] | Haixia Li, Xin Liu . Positive solutions for a Caputo-type fractional differential equation with a Riemann-Stieltjes integral boundary condition. Electronic Research Archive, 2025, 33(6): 4027-4044. doi: 10.3934/era.2025179 |
[10] | Qingcong Song, Xinan Hao . Positive solutions for fractional iterative functional differential equation with a convection term. Electronic Research Archive, 2023, 31(4): 1863-1875. doi: 10.3934/era.2023096 |
Let R,Z, and N be the sets of all real numbers, integers, and natural numbers, respectively. Here and below, for a,b∈N with a<b, we use the notation Z(a,b):={a,a+1,⋯,b}.
In recent years, difference equations as mathematical models to describe a variety of practical problems in the fields of economy, biology, disease prevention and control, environmental protection, etc., have attracted great attention and there have been a lot of outstanding works in theory and actual application; see, for example, [1,2].
In the paper, we consider the following discrete Dirichlet boundary value problem involving the mean curvature operator
{−△ϕ(△u(k−1))+q(k)ϕ(u(k))=λf(k,u(k)),k∈Z(1,T),u(0)=u(T+1)=0, | (1.1) |
where ϕ(s)=s√1+κs2,κ>0 is a constant [3], T is a given positive integer, △ is the forward difference operator defined by △u(k)=u(k+1)−u(k),△2u(k)=△(△u(k)),q(k)≥0 for all k∈Z(1,T),λ is a real positive parameter, and f(k,⋅)∈C(R,R) for each k∈Z(1,T).
When q(k)=0 for all k∈Z(1,T), the problem (1.1) is the discrete analogy of one-dimensional prescribed mean curvature equations with Dirichlet boundary conditions
{−(ϕ(u′))′=λf(t,u(t)),t∈(0,1),u(0)=u(1)=0. |
The existence of (positive) solutions of the above problem has been studied by using the variational method, the standard techniques from bifurcation theory, fixed point index, and Kresnoselskii's fixed point theorem (see [4,5,6] and the references therein).
Note that when κ=0, the problem (1.1) degenerates into the classical second-order difference equation boundary value problem
{−△2u(k−1)+q(k)u(k)=λf(k,u(k)),k∈Z(1,T),u(0)=u(T+1)=0. | (1.2) |
The existence of (positive) solutions to the problem (1.2) has been well known with various qualitative assumptions of nonlinearity f ([7,8,9]).
When κ=1, the problem (1.1) is the ordinary discrete mean curvature problem
{−△(u(k−1)√1+u2(k−1))+q(k)u(k)√1+u2(k)=λf(k,u(k)),k∈Z(1,T),u(0)=u(T+1)=0, | (1.3) |
which has aroused the interest of many scholars recently [10,11,12]. Their approaches are variational. The existence of (positive) solutions to the problem (1.3) depends on the behavior at zero or infinity of the potential F(k,u)=∫u0f(k,s)ds. In [10,11], the oscillating behavior of F at +∞ played an important role in obtaining an infinite number of positive solutions to the problem (1.3). Chen and Zhou [12] obtained at least three positive solutions of the problem (1.3) with q(k)=0 for all k∈Z(1,T), where the primitive F on the nonlinear datum satisfies that lim sups→+∞F(k,s)|s|<α (α is a positive constant). Meanwhile, the case that the potential F satisfies lim infs→∞F(k,s)|s|>β (β is a positive constant) has been also discussed. They obtained at least two nontrivial solutions based on a two-critical-point theorem (Theorem 2.1) established in [13]. It is an important tool in obtaining at least two positive solutions of the Laplacian or the algebraic boundary value problems (see, for instance, [15,16]). However, it is relatively less used to look for positive solutions to the mean curvature boundary value problems.
Inspired by this, we will apply the two-critical-point theorem to establish the existence of at least two positive solutions for the problem (1.1) in this paper. As a special case of our main theorem, the existence of two positive solutions for the problem (1.1) with κ=1 and all q(k)=0 is obtained in Remark 4. The result improves Theorem 2 [12], where the author establishes the existence of only two nontrivial solutions without providing sign information for them under stronger hypotheses on the potential F. In Theorem 2 [12], the bilateral limit assumption on F(k,s)|s| at ∞ ensures that the energy functional of that problem is anticoercive, which consequently guarantees the establishment of the Palais-Smale condition—a pivotal requirement for applying critical point theorems. In our paper, the bilateral limit on the potential F is weakened to a unilateral limit at +∞, and then the energy functional of the problem (1.1) loses its anticoercivity. However, with the help of some inequality techniques, it is proven that the energy functional still satisfies the Palais-Smale condition. Moreover, in our main result (Theorem 3.1), the existence of at least two positive solutions is established without any asymptotic condition of the potential F at 0 and with no requiring that f(k,0)>0 for any k∈Z(1,T). In fact, the algebraic conditions in Theorem 3.1 are more general than the conditions that the potential F is subquadratic at 0 and superlinear at +∞ (see Corollary 3.1).
In 2003, Guo and Yu [17] first used the variational method to study the periodic solutions of second-order difference equations. Since then, for nonlinear difference systems, many scholars have used the variational method to study the existence of various solutions, such as periodic solutions, subharmonic solutions, and homoclinic solutions [18,19,20]. For general references on difference equations and their applications, we refer the reader to monographs [21,22] and the references therein.
This paper is organized as follows: In section 2, some definitions and results are collected. Some estimations of equivalence of the norm are provided. Moreover, Lemma 2.1 is presented to guarantee us to obtain positive solutions rather than nontrivial solutions to the problem (1.1). Section 3 is devoted to our main result. Lemma 3.1 is given to ensure the Palais-Smale condition of the functional on this basis. Some consequences of our main result are presented together with three examples.
Consider the T dimensional Banach space
S={u:Z(0,T+1)→Rsuchthatu(0)=u(T+1)=0} |
endowed with the norm ‖⋅‖ as
‖u‖:=(T∑k=0|△u(k)|2)12,∀u∈S. |
Moreover, the space S can also be equipped with the following equivalent norms:
‖u‖∞:=maxk∈Z(1,T)|u(k)|,∀u∈S, |
‖u‖1:=T∑k=1|u(k)|,∀u∈S, |
‖u‖2:=(T∑k=1|u(k)|2)12,∀u∈S, |
respectively. It is easy to know that for all u∈S,
‖u‖1≥ ‖u‖2. | (2.1) |
By Theorem 12.6.1 in [21], we have for all u∈S,
√λ1‖u‖2≤‖u‖≤√λT‖u‖2, | (2.2) |
where λj=4sin2jπ/2(T+1) for all j∈Z(1,T). By virtue of Lemma 2.2 in [23], we get for all u∈S,
‖u‖∞≤√T+12‖u‖. | (2.3) |
Note that v(0)=v(T+1)=0 for all v∈S, we have
T+1∑k=1ϕ(△u(k−1))△v(k−1)=−T∑k=1△ϕ(△u(k−1))v(k) | (2.4) |
for all u,v∈S.
Let
s+=max{s,0}ands−=max{−s,0} |
for all s∈R.
Remark 2.1. In fact, the nonnegative solution of the following problem
{−△ϕ(△u(k−1))+q(k)ϕ(u(k))=λf(k,u+(k)),k∈Z(1,T),u(0)=u(T+1)=0 | (2.5) |
is the nonnegative solution of the problem (1.1). To obtain positive solutions to the problem (1.1), we only need to search for positive solutions to the problem (2.5) now.
Consider two functionals Φ and Ψ defined respectively on S by
Φ(u)=1κT+1∑k=1(√1+κ|△u(k−1)|2−1)+1κT∑k=1q(k)(√1+κ|u(k)|2−1) | (2.6) |
and
Ψ(u)=T∑k=1F+(k,u(k)), | (2.7) |
F+(k,s)=∫s0f(k,t+)dt for all (k,s)∈Z(1,T)×R. Obviously,
F+(k,s)={∫s0f(k,t)dt, if s>0,f(k,0)s, if s≤0, |
for all k∈Z(1,T), and Φ,Ψ∈C1(S,R), which means that Φ and Ψ are two continuously Gâteaux-differentiable functionals defined on S. By (2.4), we have that for any u,v∈S,
Φ′(u)(v)=T+1∑k=1ϕ(△u(k−1))△v(k−1)+T∑k=1q(k)ϕ(u(k))v(k)=T∑k=1[−△ϕ(△u(k−1))+q(k)ϕ(u(k))]v(k) |
and
Ψ′(u)(v)=T∑k=1f(k,u+(k))v(k). |
We define Iλ:S→R by
Iλ(u)=Φ(u)−λΨ(u). |
Clearly, Iλ∈C1(S,R) and for any u,v∈S,
I′λ(u)(v)=T∑k=1[−△ϕ(△u(k−1))+q(k)ϕ(u(k))−λf(k,u+(k))]v(k). |
Remark 2.2. For any u,v∈S, I′λ(u)(v)=0 if and only if
−△ϕ(△u(k−1))+q(k)ϕ(u(k))−λf(k,u+(k))=0 |
for all u∈S and k∈Z(1,T). In other words, a critical point of Iλ on S corresponds to a solution to the problem (2.5).
For the problem (1.1), we are interested in the existence of positive solutions rather than nontrivial ones. To the end, we present an assumption on f(k,0) as follows.
(H1) f(k,0)≥0,∀k∈Z(1,T).
Lemma 2.1. If (H1) holds, then any nonzero critical point of the functional Iλ on S is a positive solution to the problem (1.1).
Proof. In fact, taking Remarks 2.1 and 2.2 into account, it is sufficient to verify that any nontrivial solution u of the problem (2.5) is positive, that is, u(k)>0 for all k∈Z(1,T).
Notice that ϕ is a strictly monotonically increasing and odd function on R. By standard computation, we can conclude that
−ϕ(△u(k−1))△u−(k−1)≥ϕ(△u−(k−1))△u−(k−1)=|△u−(k−1)|2√1+κ|△u−(k−1)|2 | (2.8) |
for all k∈Z(1,T+1). By a direct computation, we get
−q(k)ϕ(u(k))u−(k)=q(k)ϕ(u−(k))u−(k)=q(k)|u−(k)|2√1+κ|u−(k)|2 | (2.9) |
for all k∈Z(1,T). For any nontrivial solution u of the problem (2.5), combining Eqs (2.4), (2.8) and (2.9) with assumption (H1), we obtain that
0=T∑k=1[−△ϕ(△u(k−1))+q(k)ϕ(u(k))−λf(k,u+(k))](−u−(k))=−T+1∑k=1ϕ(△u(k−1))△u−(k−1)−T∑k=1[q(k)ϕ(u(k))−λf(k,u+(k))]u−(k)≥T+1∑k=1ϕ(△u−(k−1))△u−(k−1)+T∑k=1[q(k)ϕ(u−(k))+λf(k,0)]u−(k)≥T+1∑k=1|△u−(k−1)|2√1+κ|△u−(k−1)|2+T∑k=1q(k)|u−(k)|2√1+κ|u−(k)|2≥0. |
Hence △u−(0)=△u−(1)=⋯=△u−(T)=0. Recalling that u(0)=u(T+1)=0 for any u∈S, we have u−(k)=0 for all k∈Z(1,T). Thus u is nonnegative.
Next, we further prove that u is positive. Otherwise, there exists some k0∈Z(1,T) such that u(k0)=0, then
−△ϕ(△u(k0−1))=λf(k0,0)≥0. |
Clearly, ϕ(△u(k0))≤ϕ(△u(k0−1)). Because ϕ is a strictly monotonically increasing homomorphism, we get △u(k0)≤△u(k0−1). Hence, u(k0+1)+u(k0−1)≤0. As a result, u(k0±1)=0. Then iterating the process, we have that u(k)=0 for all k∈Z(1,T). In short, u is zero somewhere in Z(1,T). Then it is zero identically. This contradicts the nontriviality of u. The proof is ended.
Let (X,‖⋅‖) be a real Banach space and φ∈C1(X,R). φ is said to satisfy the Palais-Smale condition ((PS) condition), if any sequence {un}⊂X for which {φ(un)} is bounded and φ′(un)→0 as n→∞ possesses a convergent subsequence in X.
The following two-critical-point theorem is given by Bonanno and D'Aguì in 2016 [13].
Theorem 2.1. [13] Let X be a real Banach space, and let Φ,Ψ:X→R be two functionals of class C1 such that infXΦ=Φ(0)=Ψ(0)=0. Assume that there are r∈R and ˜u∈X, with 0<Φ(˜u)<r, such that
supu∈Φ−1((−∞,r])Ψ(u)r<Ψ(˜u)Φ(˜u) | (2.10) |
for each
λ∈Λ=(Φ(˜u)Ψ(˜u),rsupu∈Φ−1((−∞,r])Ψ(u)), |
the functional Iλ=Φ−λΨ satisfies the (PS) condition, and it is unbounded from below. Then for each λ∈Λ, the functional Iλ admits at least two nonzero critical points uλ,1,uλ,2 such that Iλ(uλ,1)<0<Iλ(uλ,2).
Let
L∞(k):=lim infs→+∞F(k,s)sandL∞:=mink∈Z(1,T)L∞(k), |
where F(k,s)=∫s0f(k,t)dt for all (k,s)∈Z(1,T)×R. Here and below, when L∞=0, we think 1L∞=∞. Before we come to a conclusion, let's give a lemma on the (PS) condition.
Lemma 3.1. If L∞>0 and (H1) hold, then Iλ satisfies the (PS) condition, and it is unbounded from below for all λ∈(√(T+1)λT+Q√κL∞,+∞), where Q:=(∑Tk=1|q(k)|2)12.
Proof. Because S is finite dimensional, it is sufficient to show that any (PS) sequence of Iλ is bounded on S. Let {un}⊂S be a sequence such that {Iλ(un)} is bounded and I′λ(un)→0 as n→∞.
First, we claim that {u−n} is bounded. According to (2.8) and (2.9), we have for all n∈N and k∈Z(1,T+1),
−ϕ(△un(k−1))△u−n(k−1)≥ϕ(△u−n(k−1))△u−n(k−1)=|△u−n(k−1)|2√1+κ|△u−n(k−1)|2 | (3.1) |
and
−q(k)ϕ(un(k))u−n(k)=q(k)ϕ(u−n(k))u−n(k)=q(k)|u−n(k)|2√1+κ|u−n(k)|2. | (3.2) |
By (3.1) and (3.2), we can estimate the derivative of Φ at un in the direction of −u−n that
−Φ′(un)(u−n)=T∑k=1[−△ϕ(△un(k−1))+q(k)ϕ(un(k))](−u−n(k))=−T+1∑k=1ϕ(△un(k−1))△u−n(k−1)−T∑k=1q(k)ϕ(un(k))u−n(k)≥T+1∑k=1|△u−n(k−1)|2√1+κ|△u−n(k−1)|2+T∑k=1q(k)|u−n(k)|2√1+κ|u−n(k)|2=1κT+1∑k=1κ|△u−n(k−1)|2√1+κ|△u−n(k−1)|2+1κT∑k=1q(k)κ|u−n(k)|2√1+κ|u−n(k)|2=1κT+1∑k=1κ|△u−n(k−1)|2+1−1√1+κ|△u−n(k−1)|2+1κT∑k=1q(k)κ|u−n(k)|2+1−1√1+κ|u−n(k)|2≥1κT+1∑k=1(√1+κ|△u−n(k−1)|2−1)+1κT∑k=1q(k)(√1+κ|u−n(k)|2−1). |
To put it simply,
−Φ′(un)(u−n)≥Φ(u−n),∀n∈N. | (3.3) |
And, from the condition (H1), it follows that
Ψ′(un)(u−n)=T∑k=1f(k,u+n(k))u−n(k)=T∑k=1f(k,0))u−n(k)≥0,∀n∈N. | (3.4) |
Combining (3.3) and (3.4), we have
Φ(u−n)≤−Φ′(un)(u−n)≤−Φ′(un)(u−n)+λΨ′(un)(u−n)=−I′λ(un)(u−n) | (3.5) |
for all λ>0 and n∈N. Moreover, by the definition of the functional Φ, we obtain, for all u∈S,
Φ2(u)≥[1κT+1∑k=1(√1+κ|△u(k−1)|2−1)]2≥1κ2T+1∑k=1(√1+κ|△u(k−1)|2−1)2=1κ2[T+1∑k=1(1+κ|△u(k−1)|2)+T+1∑k=11−2T+1∑k=1√1+κ|△u(k−1)|2]=1κT+1∑k=1|△u(k−1)|2−2κ2T+1∑k=1(√1+κ|△u(k−1)|2−1)≥1κ‖u‖2−2κΦ(u), |
which implies that
‖u‖2≤κΦ2(u)+2Φ(u) | (3.6) |
for all u∈S. Thus, it is clear that for all u∈S,
−1κ+√1κ‖u‖2+1κ2≤Φ(u). |
So combining (3.5) and the above inequality, we have
0≤−1κ+√1κ‖u−n‖2+1κ2≤−I′λ(un)(u−n),∀n∈N. |
Also, from limn→∞I′λ(un)=0, it follows that
limn→∞I′λ(un)(u−n)‖u−n‖=limn→∞I′λ(un)(u−n‖u−n‖)=0. |
As a result, we obtain
limn→∞−1κ+√1κ‖u−n‖2+1κ2‖u−n‖=0. |
By standard computation, the above equation is transformed into
limn→∞[√κ+1‖u−n‖2+1‖u−n‖]=+∞, |
which means that limn→∞‖u−n‖=0. Hence our claim is proved. So there is M>0 such that 0≤u−n(k)≤M for all k∈Z(1,T) and n∈N.
Next, we prove that {un} is bounded. If {un} is not bounded, we may assume, going if necessary to a subsequence, that ‖un‖→∞(n→∞). Taking L∞>0 into account, we fix λ>√(T+1)λT+Q√κL∞ and fix l=l(λ) such that for all k∈Z(1,T),
√(T+1)λT+Qλ√κ<l<L∞≤L∞(k)=lim infs→+∞F(k,s)s, |
then for all k∈Z(1,T), there is a δk>0 such that
F+(k,s)=F(k,s)>ls=l|s|,∀s>δk. |
Meanwhile, for all k∈Z(1,T) and s∈[−M,δk],
F+(k,s)≥mins∈[−M,δk]F+(k,s)≥l|s|−lmax{δk,M}+mins∈[−M,δk]F+(k,s)≥l|s|−max{lmax{δk,M}−mins∈[−M,δk]F+(k,s),0}=l|s|−η(k). |
Hence, for all k∈Z(1,T),
F+(k,s)≥l|s|−η(k),∀s≥−M. |
On account of (2.1), we obtain that for all n∈N,
Ψ(un)=T∑k=1F+(k,un(k))≥lT∑k=1|un(k)|−T∑k=1η(k)=l‖un‖1−η≥l‖un‖2−η, |
where η=∑Tk=1η(k). Using the Cauchy-Schwarz inequality, we get that for all n∈N,
Φ(un)=1κT+1∑k=1(√1+κ|△un(k−1)|2−1)+1κT∑k=1q(k)(√1+κ|un(k)|2−1)≤1κT+1∑k=1(1+√κ|△un(k−1)|2−1)+1κT∑k=1q(k)(1+√κ|un(k)|2−1)≤1√κT+1∑k=1|△un(k−1)|+1√κT∑k=1q(k)|un(k)|≤1√κ√T+1‖un‖+1√κQ‖un‖2. |
Therefore, from the previous two inequalities and (2.2), it follows that for all n∈N,
Iλ(un)=Φ(un)−λΨ(un)≤1√κ√T+1‖un‖+1√κQ‖un‖2−λl‖un‖2+λη≤1√κ√(T+1)λT‖un‖2+1√κQ‖un‖2−λl‖un‖2+λη≤(√(T+1)λT+Q√κ−λl)1√λ1‖un‖+λη. |
When ‖un‖→+∞(n→∞), by √(T+1)λT+Q√κ−λl<0, we have limn→∞Iλ(un)=−∞. This leads to a contradiction. Hence {un} is bound and Iλ satisfies the (PS) condition.
Finally, we prove that Iλ is unbounded from below. Let {un} be such that un=u+n for any n∈N and ‖u+n‖→∞(n→∞). Arguing as before, we obtain that
Iλ(un)≤(√(T+1)λT+Q√κ−λl)1√λ1‖un‖+λη,∀n∈N. |
So limn→∞Iλ(un)=−∞. Our conclusion follows.
Below, our main result is presented.
Theorem 3.1. Suppose that (H1) holds and there exist two positive constants c and d with d<c such that
(H2)∑Tk=1maxs∈[0,c]F(k,s)−1κ+√4c2κ(T+1)+1κ2<L∞√κ√(T+1)λT+Q |
and
(H3)∑Tk=1maxs∈[0,c]F(k,s)−1κ+√4c2κ(T+1)+1κ2<κ∑Tk=1F(k,d)(2+q)(√1+κd2−1). |
Then for each
λ∈Λ+:=(max{√(T+1)λT+QL∞√κ,(2+q)(√1+κd2−1)κ∑Tk=1F(k,d)},−1κ+√4c2κ(T+1)+1κ2∑Tk=1maxs∈[0,c]F(k,s)), |
the problem (1.1) possesses at least two positive solutions, where q:=∑Tk=1q(k).
Proof. By Lemma 2.1, it is enough to prove that Iλ has at least two nonzero critical points. We apply Theorem 2.1 by putting X=S,Iλ=Φ−λΨ where Φ,Ψ are the functions introduced in (2.6) and (2.7).
Clearly, infSΦ=Φ(0)=Ψ(0)=0. Also notice that F+(k,s)=F(k,s) for all (k,s)∈Z(1,T)×[0,+∞). From the conditions (H1),(H2) and (H3), it follows that L∞>0 and Λ+ is non-degenerate. Then Lemma 3.1 ensures that the function Iλ satisfies the (PS) condition and it is unbounded from below for λ∈Λ+.
For fixed λ∈Λ+, there exists c>0 such that λ≤−1κ+√4c2κ(T+1)+1κ2∑Tk=1maxs∈[0,c]F(k,s). Put r:=−1κ+√4c2κ(T+1)+1κ2. We claim that
{u∈S:Φ(u)≤r}⊂{u∈S:‖u‖∞≤c}. |
If Φ(u)≤r, by (3.6), we get that ‖u‖2≤κΦ2(u)+2Φ(u)≤κr2+2r=4c2T+1. Thus, from the inequality (2.3), it is clear that
‖u‖∞≤√T+12‖u‖≤√T+12√4c2T+1=c. |
The assertion is verified. Therefore, we obtain
supu∈Φ−1([0,r])Ψ(u)r≤∑Tk=1maxs∈[0,c]F(k,s)−1κ+√4c2κ(T+1)+1κ2. | (3.7) |
Now, we look for ˜u∈S in Theorem 2.1. Define ˜u∈RT+2 as ˜u(k)=d(0<d<c) for all k∈Z(1,T) and ˜u(0)=˜u(T+1)=0. Clearly, ˜u∈S. It is easy to see that
Φ(˜u)=1κ(2+q)(√1+κd2−1), | (3.8) |
and hence we get
Ψ(˜u)Φ(˜u)=κ∑Tk=1F(k,d)(2+q)(√1+κd2−1). | (3.9) |
Therefore, from (3.7), (3.9), and (H3), it follows that
supu∈Φ−1([0,r])Ψ(u)r≤Ψ(˜u)Φ(˜u) |
and Λ+⊂Λ. Moreover, since 0<d<c, and again by virtue of the condition (H3), we obtain that
1κ(2+q)(√1+κd2−1)<−1κ+√4c2κ(T+1)+1κ2=r. | (3.10) |
On account of (3.8) and (3.10), we get that 0<Φ(˜u)<r. Thus, (2.10) holds. Hence Theorem 2.1 ensures that Iλ admits two nonzero critical points. This completes the proof.
Remark 3.1. Indeed, condition (H1) implicitly covers both f(k,0)>0 and f(k,0)=0 cases. Theorem 3.1 holds true under either sub-case of condition (H1).
Remark 3.2. If all f(k,⋅) are nonnegative on [0,c](c>0), then maxs∈[0,c]F(k,s)=F(k,c) for all k∈Z(1,T). According to Theorem 3.1, it is enough to assume that there is a positive constant d with d<c such that
∑Tk=1F(k,c)−1κ+√4c2κ(T+1)+1κ2<min{L∞√κ√(T+1)λT+Q,κ∑Tk=1F(k,d)(2+q)(√1+κd2−1)}. |
Remark 3.3. When κ=1 and q(k)=0 for all k∈Z(1,T) (Q=0 and q=0), the conditions in Remark 3.2 are more general than those of Theorem 2 in [12], where f(k,s) is required to be positive for all (k,s)∈Z(1,T)×[−c,c] rather than nonnegative, and the potential F is assumed to possess asymptotic behavior not only at +∞ but also at −∞. Here, the two solutions we obtain are positive, while the solutions in Theorem 2 of [12] are only nontrivial. Obviously, our main result improves Theorem 2 in [12].
Now, we present two particular cases of Theorem 3.1.
Corollary 3.1. Assume that (H1) holds,
lim sups→0+F(k,s)s2=+∞ | (3.11) |
and
lims→+∞F(k,s)s=+∞ | (3.12) |
for all k∈Z(1,T). Then for each λ∈(0,λ∗), where
λ∗=supc>0−1κ+√4c2κ(T+1)+1κ2∑Tk=1maxs∈[0,c]F(k,s), |
the problem (1.1) admits at least two positive solutions.
Proof. Let λ∈(0,λ∗) and c>0 such that
λ<−1κ+√4c2κ(T+1)+1κ2∑Tk=1maxs∈[0,c]F(k,s). |
Taking (3.11) into account, we get that lim sups→0+F(k,s)√1+κs2−1=+∞. As a result, there is d>0 with d<c such that κ2+q∑Tk=1F(k,d)√1+κd2−1>1λ. Note that L∞=+∞, so Theorem 3.1 ensures the conclusion.
Corollary 3.2. Assume that f(k,⋅)=f for all k∈Z(1,T) and f is a continuous function such that
lims→0+f(s)s=+∞ | (3.13) |
and
lims→+∞f(s)=+∞. | (3.14) |
Then, for each
λ∈(0,supc>0−1κ+√4c2κ(T+1)+1κ2Tmaxs∈[0,c]∫s0f(t)dt), |
the problem (1.1) admits at least two positive solutions.
Proof. In fact, Corollary 3.2 is a consequence of Corollary 3.1. It is easy to see that (3.13) implies f(0)≥0, and (3.11) and (3.12) can be derived from (3.13) and (3.14), respectively.
Example 3.1. For each λ∈(0,−1κ+√4κ(T+1)+1κ2(3e−1)T), the problem
{−△ϕ(△u(k−1))+q(k)ϕ(u(k))=λ(e3√u(k)−1),k∈Z(1,T),u(0)=u(T+1)=0 |
admits at least two positive solutions. Here, the nonlinearity f(t)=e3√t−1 satisfies the conditions of Corollary 3.2 and
supc>0−1κ+√4c2κ(T+1)+1κ2T∫c0f(t)dt≥−1κ+√4κ(T+1)+1κ2(3e−1)T. |
The image of f(t) is shown in Figure 1.
Example 3.2. For each λ∈(0,−1κ+√4κ(T+1)+1κ2T(T+1)), the problem
{−△ϕ(△u(k−1))+q(k)ϕ(u(k))=λk√u(k)(cosu(k)+2),k∈Z(1,T),u(0)=u(T+1)=0 |
admits at least two positive solutions. It is easy to see that f(k,t)=k√t(cost+2) satisfies the assumptions of Corollary 3.1 and
λ∗≥−1κ+√4κ(T+1)+1κ2∑Tk=1∫10f(k,t)dt=−1κ+√4κ(T+1)+1κ2T(T+1)2∫10√t(cost+2)dt≥−1κ+√4κ(T+1)+1κ2T(T+1). |
Figure 2 displays the functional plot of f(k,t) for k∈(0,100).
Remark 3.4. We observe that Theorem 2 in [12] cannot be applied in the previous two examples with κ=1 and q(k)=0 for all k∈Z(1,T), since f(k,0)>0 is assumed there.
Example 3.3. Consider the boundary value problem (1.1) with κ=1,q(k)=0 and
f(k,t)=f(t)={12(4−3t)cost4,t≤83π,2π−1,t>83π |
for all k∈Z(1,T). Clearly, if t∈(0,43), then f(t)>0; and if t∈(43,2π), then f(t)<0. Such a relationship is visually apparent in Figure 3. A straightforward calculation yields
F(k,s)=F(s)={2(4sins4−3ssins4−12coss4+12),s≤83π,(2π−1)s−4(2π−1)(23π+√3),t>83π |
for all k∈Z(1,T).
Obviously, L∞(k)=lim infs→+∞F(k,s)s=2π−1 for all k∈Z(1,T), then
L∞=mink∈Z(1,T)L∞(k)=2π−1≈5.382. |
Letting T=3,c=5, and d=1, we obtain
√(T+1)λT=4sinT2(T+1)π=4sin38π≈3.695, |
T∑k=1maxs∈[0,c]F(k,s)=3∑k=1∫430f(k,t)dt=72(1−cos13)≈3.963, |
and
T∑k=1F(k,d)=3∑k=1∫10f(k,t)dt=6[sin14+12(1−cos14)]≈3.724. |
It follows that
√(T+1)λTL∞≈3.6955.382≈0.687, |
2(√1+d2−1)∑Tk=1F(k,d)≈0.8283.724≈0.222, |
and
−1+√4c2T+1+1∑Tk=1maxs∈[0,c]F(k,s)≈4.0993.963≈1.034. |
From f(k,0)=f(0)=2>0 and the preceding relations, all conditions of Theorem 3.1 hold. Thus for each λ∈(√(T+1)λTL∞,−1+√4c2T+1+1∑Tk=1maxs∈[0,c]F(k,s))≈(0.687,1.034), the boundary value problem (1.1) has at least two positive solutions.
Remark 3.5. The potential function F in the above example lacks superlinearity at +∞, failing to satisfy the conditions of Corollaries 3.1 and 3.2. Nevertheless, Theorem 3.1 still guarantees the existence of two positive solutions for the problem (1.1) in Example 3.3.
In this paper, we consider a generalized difference mean curvature problem, which includes the conventional one and the classical second-order difference equation boundary value problem. The main novelties of this research are as follows:
(1) The existence of two positive solutions rather than nontrivial solutions for our problem is established based on a two-zero critical points theorem and some inequality techniques.
(2) Under the assumption of the unilateral limit of F(k,s)|s| at +∞ on the potential F(k,s)=∫s0f(k,t)dt instead of the bilateral limit at ∞, it is proved that without anticoercivity the energy functional associated with our problem still satisfies the Palais-Smale condition that plays a key role in the critical point theorem.
(3) Our principal result can be applied to cases with nonlinear terms where f(k,0)>0 but also to cases where the nonlinear terms satisfy f(k,0)=0.
(4) It is also worth mentioning that the algebraic conditions in our main result are more general than the subquadraticity at 0 and the superlinearity at +∞ for the potential F.
In fact, due to the previous (1)–(3), our main result extends Theorem 2 in [12].
The authors declare they have not used Artificial Intelligence (AI) tools in the creation of this article.
This work is supported by the National Nature Science Foundation of China (No.10901071, No.11501054, and No.12301186).
The authors declare there is no conflicts of interest.
[1] |
Megharaj M, Ramakrishnan B, Venkateswarlu K, et al. (2011) Bioremediation approaches for organic pollutants: A critical perspective. Environ Int 37: 1362–1375. doi: 10.1016/j.envint.2011.06.003
![]() |
[2] |
Bezza FA, Nkhalambayausi Chirwa EM (2016) Biosurfactant-enhanced bioremediation of aged polycyclic aromatic hydrocarbons (PAHs) in creosote contaminated soil. Chemosphere 144: 635–644. doi: 10.1016/j.chemosphere.2015.08.027
![]() |
[3] |
de la Cueva SC, Rodríguez CH, Cruz NOS, et al. (2016) Changes in Bacterial Populations During Bioremediation of Soil Contaminated with Petroleum Hydrocarbons. Water Air Soil Poll 227: 1–12. doi: 10.1007/s11270-015-2689-7
![]() |
[4] | Dadrasnia A, Shahsavari N, Salmah I (2015) The top 101 cited articles in environmental clean-up: Oil spill remediation. Global NEST J 17: 692–700. |
[5] | Chirwa E, Smit H (2010) Simultaneous Cr (VI) reduction and phenol degradation in a trickle bed bioreactor: shock loading response. Chem Eng Trans 20: 55–60. |
[6] |
Dixit R, Malaviya D, Pandiyan K, et al. (2015) Bioremediation of heavy metals from soil and aquatic environment: an overview of principles and criteria of fundamental processes. Sustainability 7: 2189–2212. doi: 10.3390/su7022189
![]() |
[7] | Bustamante M, Durán N, Diez M (2012) Biosurfactants are useful tools for the bioremediation of contaminated soil: a review. J Soil Sci Plant Nut 12: 667–687. |
[8] |
Abdel-Moghny T, Mohamed RS, El-Sayed E, et al. (2012) Removing of hydrocarbon contaminated soil via air flushing enhanced by surfactant. Appl Petrochem Res 2: 51–59. doi: 10.1007/s13203-012-0008-4
![]() |
[9] |
Dadrasnia A, Ismail S (2015) Biosurfactant Production by Bacillus salmalaya for Lubricating Oil Solubilization and Biodegradation. Int J Environ Res Pub Heal 12: 9848. doi: 10.3390/ijerph120809848
![]() |
[10] | Santos DK, Brandão YB, Rufino RD, et al. (2014) Optimization of cultural conditions for biosurfactant production from Candida lipolytica. Biocat Agr Biotechnol 3: 48–57. |
[11] |
Dyke MIV, Couture P, Brauer M, et al. (1993) Pseudomonas aeruginosa UG2 rhamnolipid biosurfactants: structural characterization and their use in removing hydrophobic compounds from soil. Can J Microbiol 39: 1071–1078. doi: 10.1139/m93-162
![]() |
[12] |
Lima ÁS, Alegre RM (2009) Evaluation of emulsifier stability of biosurfactant produced by Saccharomyces lipolytica CCT-0913. Braz Arch Biol Tech 52: 285–290. doi: 10.1590/S1516-89132009000200004
![]() |
[13] | Nwachi AC, Onochie CC, Iroha IR, et al. (2016) Extraction of Biosurfactants Produced from Bacteria Isolated from Waste-Oil Contaminated Soil in Abakaliki Metropolis, Ebonyi State. J Biotechnol Res 2: 24–30. |
[14] | Walter V, Syldatk C, Hausmann R (2013) Screening Concepts for the Isolation of Biosurfactant Producing Microorganisms. In: Madame Curie Bioscience Database [Internet]. Austin (TX): Landes Bioscience; 2000–2013. |
[15] |
Cortés-Sánchez AdJ, Hernández-Sánchez H, Jaramillo-Flores ME (2013) Biological activity of glycolipids produced by microorganisms: New trends and possible therapeutic alternatives. Microbiol Res 168: 22–32. doi: 10.1016/j.micres.2012.07.002
![]() |
[16] |
Vijayakumar S, Saravanan V (2015) Biosurfactants-Types, Sources and Applications. Res J Microbiol 10: 181. doi: 10.3923/jm.2015.181.192
![]() |
[17] | Neu TR (1996) Significance of bacterial surface-active compounds in interaction of bacteria with interfaces. Microbiol Rev 60: 151–166. |
[18] |
Rosenberg E, Ron E (1999) High-and low-molecular-mass microbial surfactants. Appl Microbiol Biotechnol 52: 154–162. doi: 10.1007/s002530051502
![]() |
[19] | Perfumo A, Smyth TJP, Marchant R, et al. (2010) Production and roles of biosurfactants and bioemulsifiers in accessing hydrophobic substrates. In: Kenneth N. Timmis (ed.), Handbook of Hydrocarbon and Lipid Microbiology, Springer. UK. 2, 1501–1512. |
[20] | Mulligan CN, Gibbs BF (1990) Recovery of biosurfactants by ultrafiltration. J Chem Technol Biotechnol 47: 23–29. |
[21] |
Whang L-M, Liu P-WG, Ma C-C, et al. (2008) Application of biosurfactants, rhamnolipid, and surfactin, for enhanced biodegradation of diesel-contaminated water and soil. J Hazard Mater 151: 155–163. doi: 10.1016/j.jhazmat.2007.05.063
![]() |
[22] |
Souza EC, Vessoni-Penna TC, de Souza Oliveira RP (2014) Biosurfactant-enhanced hydrocarbon bioremediation: An overview. Int Biodeter Biodeg 89: 88–94. doi: 10.1016/j.ibiod.2014.01.007
![]() |
[23] | Uzoigwe C, Burgess JG, Ennis CJ, et al. (2015) Bioemulsifiers are not biosurfactants and require different screening approaches. Front Microbiol 6: 245. |
[24] | Dadrasnia A, Usman MM, Wei KSC, et al. (2016) Native soil bacterial isolate in Malaysia exhibit promising supplements on degrading organic pollutants. Process Saf Environ 100: 264–271. |
[25] |
Rahman PK, Gakpe E (2008) Production, characterisation and applications of biosurfactants-Review. Biotech 7: 360–370 doi: 10.3923/biotech.2008.360.370
![]() |
[26] | Karanth N, Deo P, Veenanadig N (1999) Microbial production of biosurfactants and their importance. Curr Sci 77: 116–126. |
[27] | Sáenz-Marta CI, de Lourdes Ballinas-Casarrubias M, Rivera-Chavira BE, et al. (2015) Biosurfactants as Useful Tools in Bioremediation. Advances in Bioremediation of Wastewater and Polluted Soil: INTECH. DOI: 10.5772/60751 |
[28] | Muthusamy K, Gopalakrishnan S, Ravi TK, et al. (2008) Biosurfactants: Properties, commercial production and application. Curr Sci (00113891) 94. |
[29] |
Gautam K, Tyagi V (2006) Microbial surfactants: a review. J Oleo Sci 55: 155–166. doi: 10.5650/jos.55.155
![]() |
[30] | Rosenberg E, Ron EZ (2013) Biosurfactants. The Prokaryotes: Springer. pp. 281–294. |
[31] |
Silva RdCFS, Almeida DG, Rufino RD, et al. (2014) Applications of Biosurfactants in the Petroleum Industry and the Remediation of Oil Spills. Int J Mol Sci 15: 12523–12542. doi: 10.3390/ijms150712523
![]() |
[32] | Kosaric N (2008) Biosurfactants. Biotechnology Set: Wiley-VCH Verlag GmbH. pp. 659–717. |
[33] | Cooper DG, Paddock DA (1983) Torulopsis petrophilum and Surface Activity. Appl Environ Microbiol 46: 1426–1429. |
[34] |
Thavasi R, Subramanyam Nambaru VRM, Jayalakshmi S, et al. (2011) Biosurfactant Production by Pseudomonas aeruginosa from Renewable Resources. Indian J Microbiol 51: 30–36. doi: 10.1007/s12088-011-0076-7
![]() |
[35] |
Duvnjak Z, Cooper DG, Kosaric N (1982) Production of surfactant by Arthrobacter paraffineus ATCC 19558. Biotechnol Bioeng 24: 165–175. doi: 10.1002/bit.260240114
![]() |
[36] |
Müller MM, Hörmann B, Kugel M, et al. (2011) Evaluation of rhamnolipid production capacity of Pseudomonas aeruginosa PAO1 in comparison to the rhamnolipid over-producer strains DSM 7108 and DSM 2874. Appl Microbiol Biotechnol 89: 585–592. doi: 10.1007/s00253-010-2901-z
![]() |
[37] |
Hewald S, Josephs K, Bölker M (2005) Genetic analysis of biosurfactant production in Ustilago maydis. Appl Environ Microbiol 71: 3033–3040. doi: 10.1128/AEM.71.6.3033-3040.2005
![]() |
[38] | Cooper DG, Paddock DA (1984) Production of a Biosurfactant from Torulopsis bombicola. Appl Environ Microbiol 47: 173–176. |
[39] | Matvyeyeva OL, Vasylchenko, Aliiev§Ñ OR (2014) Microbial Biosurfactants Role in Oil Products Biodegradation. Int J Environ Bioremediat Biodegradat 2: 69–74. |
[40] |
Pacwa-Płociniczak M, Płaza GA, Piotrowska-Seget Z, et al. (2011) Environmental Applications of Biosurfactants: Recent Advances. Intl J Mol Sci 12: 633–654. doi: 10.3390/ijms12010633
![]() |
[41] |
Cameotra SS, Singh P (2009) Synthesis of rhamnolipid biosurfactant and mode of hexadecane uptake by Pseudomonas species. Microb Cell Fact 8: 1–7. doi: 10.1186/1475-2859-8-1
![]() |
[42] | Berg G, Seech AG, Lee H, et al. (1990) Identification and characterization of a soil bacterium with extracellular emulsifying activity. J Environ Sci Heal A 25: 753–764. |
[43] | Sen R (2010) Biosurfactants: Springer New York. 331 p. |
[44] |
Franzetti A, Gandolfi I, Bestetti G, et al. (2010) Production and applications of trehalose lipid biosurfactants. Eur J Lipid Sci Tech 112: 617–627. doi: 10.1002/ejlt.200900162
![]() |
[45] | Chen Y-G, Wang Y-X, Zhang Y-Q, et al. (2009) Nocardiopsis litoralis sp. nov., a halophilic marine actinomycete isolated from a sea anemone. Int J Syst Evol Microbiol 59: 2708–2713. |
[46] |
Bennur T, Kumar AR, Zinjarde S, et al. (2015) Nocardiopsis species: Incidence, ecological roles and adaptations. Microbiol Res 174: 33–47. doi: 10.1016/j.micres.2015.03.010
![]() |
[47] | Jennema GE, McInerney MJ, Knapp R, M., , et al. (1983) A halotolerant, biosurfactants-producing Bacillus species potentially useful for enhanced oil recovery. Dev Ind Microbiol 24: 485–492. |
[48] | Cirigliano MC, Carman GM (1985) Purification and Characterization of Liposan, a Bioemulsifier from Candida lipolytica. Appl Environ Microbiol 50: 846–850. |
[49] | Cameron DR, Cooper DG, Neufeld RJ (1988) The mannoprotein of Saccharomyces cerevisiae is an effective bioemulsifier. Appl Environ Microbiol 54: 1420–1425. |
[50] |
Appanna VD, Finn H, St Pierre M (1995) Exocellular phosphatidylethanolamine production and multiple-metal tolerance in Pseudomonas fluorescens. FEMS Microbiol Lett 131: 53–56. doi: 10.1111/j.1574-6968.1995.tb07753.x
![]() |
[51] |
Silva RdCFS, Almeida DG, Rufino RD, et al. (2014) Applications of Biosurfactants in the Petroleum Industry and the Remediation of Oil Spills. Intl J Mol Sci 15: 12523–12542. doi: 10.3390/ijms150712523
![]() |
[52] | Reis RS, Pacheco GJ, Pereira AG, et al. (2013) Biosurfactants: Production and Applications, Biodegradation - Life of Science, Dr. Rolando Chamy (Ed.), InTech, DOI: 10.5772/56144. |
[53] | Banat IM, Satpute SK, Cameotra SS, et al. (2014) Cost effective technologies and renewable substrates for biosurfactants’ production. Front Microbiol 5: 697. |
[54] |
Banat IM (1995) Biosurfactants production and possible uses in microbial enhanced oil recovery and oil pollution remediation: A review. Bioresource Technol 51: 1–12. doi: 10.1016/0960-8524(94)00101-6
![]() |
[55] |
Banat IM (1995) Characterization of biosurfactants and their use in pollution removal – State of the Art. (Review). Acta Biotechnologica 15: 251–267. doi: 10.1002/abio.370150302
![]() |
[56] | McInerney MJ, Han SO, Maudgalya S, et al. (2003) Development of More Effective Biosurfactants for Enhanced Oil Recovery. U.S. Department of Energy, Tulsa, Oklahoma. DOE/BC/15113e2. Available online at: http://www.netl.doe.gov/KMD/cds/disk44/I-Microbial/BC15113_2.pdf (accessed 28.06.2016.). |
[57] | Bachmann RT, Johnson AC, Edyvean RGJ (2014) Biotechnology in the petroleum industry: An overview. Int Biodeter Biodeg 86, Part C: 225–237. |
[58] |
Mulligan CN (2005) Environmental applications for biosurfactants. Environmental Pollution 133: 183–198. doi: 10.1016/j.envpol.2004.06.009
![]() |
[59] |
Cameotra S, Makkar R (1998) Synthesis of biosurfactants in extreme conditions. Appl Microbiol Biotechnol 50: 520–529. doi: 10.1007/s002530051329
![]() |
[60] | Rodrigues L, Banat IM, Teixeira J, et al. (2006) Biosurfactants: potential applications in medicine. J Antimicrob Chemoth 57: 609–618. |
[61] | Poremba K, Gunkel W, Lang S, et al. (1991) Marine biosurfactants, III. Toxicity testing with marine microorganisms and comparison with synthetic surfactants. Zeitschrift für Naturforschung C 46: 210–216. |
[62] |
Lima TMS, Procópio LC, Brandão FD, et al. (2011) Evaluation of bacterial surfactant toxicity towards petroleum degrading microorganisms. Bioresource Technol 102: 2957–2964. doi: 10.1016/j.biortech.2010.09.109
![]() |
[63] | Gudiña EJ, Rodrigues AI, Teixeira JA, et al. (2015) New microbial surface-active compounds: the ultimate alternative to chemical surfactants? SINAFERM 2015 XX Simpósio Nacional de Bioprocessos. Fortaleza, Brazil, Sep. 1-4, 1-5, 2015. |
[64] | Korcan SE, Ciğerci İH, Konuk M (2013) White-Rot Fungi in Bioremediation. Fungi as Bioremediators: Springer. pp. 371–390. |
[65] |
Barr DP, Aust SD (1994) Mechanisms white rot fungi use to degrade pollutants. Environ Sci Technol 28: 78A–87A. doi: 10.1021/es00051a724
![]() |
[66] | Parmar B, Mervana P, Vyas B (2015) Degradation of textile dyes by white rot basidiomycetes. Lifesci Leaflet 59: 62–75. |
[67] | Cookson J (1995) Bioremediation engineering: design and application: McGraw Hill, New York. |
[68] |
Peng X, Yuan X-z, Liu H, et al. (2015) Degradation of Polycyclic Aromatic Hydrocarbons (PAHs) by Laccase in Rhamnolipid Reversed Micellar System. Appl Biochem Biotechnol 176: 45–55. doi: 10.1007/s12010-015-1508-3
![]() |
[69] |
Liu Z-F, Zeng G-M, Zhong H, et al. (2012) Effect of dirhamnolipid on the removal of phenol catalyzed by laccase in aqueous solution. World J Microbiol Biot 28: 175–181. doi: 10.1007/s11274-011-0806-3
![]() |
[70] |
Shin K-H, Kim K-W, Ahn Y (2006) Use of biosurfactant to remediate phenanthrene-contaminated soil by the combined solubilization–biodegradation process. J Hazard Mater 137: 1831–1837. doi: 10.1016/j.jhazmat.2006.05.025
![]() |
[71] |
Santos D, Rufino R, Luna J, et al. (2016) Biosurfactants: Multifunctional Biomolecules of the 21st Century. Intl J Mol Sci 17: 401. doi: 10.3390/ijms17030401
![]() |
[72] | Wuana RA, Okieimen FE (2011) Heavy Metals in Contaminated Soils: A Review of Sources, Chemistry, Risks and Best Available Strategies for Remediation. ISRN Ecology 2011: 20. |
[73] |
Miller RM (1995) Biosurfactant-facilitated remediation of metal-contaminated soils. Environ Health Persp 103: 59–62. doi: 10.1289/ehp.95103s459
![]() |
[74] |
Babich H, Stotzky G (1983) Temperature, pH, salinity, hardness, and particulates mediate nickel toxicity to eubacteria, an actinomycete, and yeasts in lake, simulated estuarine, and sea waters. Aquat Toxicol 3: 195–208. doi: 10.1016/0166-445X(83)90040-1
![]() |
[75] |
Babich H, Stotzky G (1982) Nickel toxicity to microbes: Influence of pH and implications for acid rain. Environ Res 29: 335–350. doi: 10.1016/0013-9351(82)90035-4
![]() |
[76] |
Sandrin TR, Maier RM (2002) Effect of pH on cadmium toxicity, speciation, and accumulation during naphthalene biodegradation. Environ Toxicol Chem 21: 2075–2079. doi: 10.1002/etc.5620211010
![]() |
[77] |
Olaniran AO, Balgobind A, Pillay B (2013) Bioavailability of Heavy Metals in Soil: Impact on Microbial Biodegradation of Organic Compounds and Possible Improvement Strategies. Intl J Mol Sci 14: 10197–10228. doi: 10.3390/ijms140510197
![]() |
[78] |
Batista S, Mounteer A, Amorim F, et al. (2006) Isolation and characterization of biosurfactant/bioemulsifier-producing bacteria from petroleum contaminated sites. Bioresource Technol 97: 868–875. doi: 10.1016/j.biortech.2005.04.020
![]() |
[79] |
Liu B, Liu J, Ju M, et al. (2016) Purification and characterization of biosurfactant produced by Bacillus licheniformis Y-1 and its application in remediation of petroleum contaminated soil. Mar Pollut Bull 107: 46–51. doi: 10.1016/j.marpolbul.2016.04.025
![]() |
[80] |
Yan P, Lu M, Yang Q, et al. (2012) Oil recovery from refinery oily sludge using a rhamnolipid biosurfactant-producing Pseudomonas. Bioresource Technol 116: 24–28. doi: 10.1016/j.biortech.2012.04.024
![]() |
[81] |
Liu W, Wang X, Wu L, et al. (2012) Isolation, identification and characterization of Bacillus amyloliquefaciens BZ-6, a bacterial isolate for enhancing oil recovery from oily sludge. Chemosphere 87: 1105–1110. doi: 10.1016/j.chemosphere.2012.01.059
![]() |
[82] |
Zhou W, Wang X, Chen C, et al. (2013) Enhanced soil washing of phenanthrene by a plant-derived natural biosurfactant, Sapindus saponin. Colloid Surface A 425: 122–128. doi: 10.1016/j.colsurfa.2013.02.055
![]() |
[83] |
Lau EV, Gan S, Ng HK, et al. (2014) Extraction agents for the removal of polycyclic aromatic hydrocarbons (PAHs) from soil in soil washing technologies. Environ Poll 184: 640–649. doi: 10.1016/j.envpol.2013.09.010
![]() |
[84] | Deziel E, Paquette G, Villemur R, et al. (1996) Biosurfactant production by a soil pseudomonas strain growing on polycyclic aromatic hydrocarbons. Appl Environ Microbiol 62: 1908–1912. |
[85] |
Mao X, Jiang R, Xiao W, et al. (2015) Use of surfactants for the remediation of contaminated soils: A review. J Hazard Mater 285: 419–435. doi: 10.1016/j.jhazmat.2014.12.009
![]() |
[86] |
Ye M, Sun M, Wan J, et al. (2015) Evaluation of enhanced soil washing process with tea saponin in a peanut oil–water solvent system for the extraction of PBDEs/PCBs/PAHs and heavy metals from an electronic waste site followed by vetiver grass phytoremediation. J Chem Technol Biot 90: 2027–2035. doi: 10.1002/jctb.4512
![]() |