Development of sustainable materials has become one common goal across the globe to meet the ever-increasing demand for the construction materials. High volume fly ash (HVFA) concrete is one such sustainable construction material which utilizes fly ash in concrete as a partial replacement of cement. Though the existing literature focuses abundantly on high volume fly ash concrete, the present work aimed to explore the intricate hydration process thorough a systematic experimental program. A series of experiments including compressive strength, rapid chloride permeability, UPV, acid resistance, X-ray diffraction, SEM and EDAX were performed to examine the effect of varying proportions of fly ash (0%, 20%, 40%, 60%) on cement replacement. Analysis of results indicated formation of hydration compounds in the form of alite, belite, celite, portlandite and tobermorite (C-S-H gel). Results of mechanical and durability tests showed that, to achieve maximum benefits, cement can be replaced to an optimum value of fly ash of 40%. The authors believe that the formation of hydration compounds tobermorite and celite resulted in attaining enhanced durability and strength in high volume fly ash concrete.
Citation: M. Kanta Rao, Ch. N. Satish Kumar. Influence of fly ash on hydration compounds of high-volume fly ash concrete[J]. AIMS Materials Science, 2021, 8(2): 301-320. doi: 10.3934/matersci.2021020
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Development of sustainable materials has become one common goal across the globe to meet the ever-increasing demand for the construction materials. High volume fly ash (HVFA) concrete is one such sustainable construction material which utilizes fly ash in concrete as a partial replacement of cement. Though the existing literature focuses abundantly on high volume fly ash concrete, the present work aimed to explore the intricate hydration process thorough a systematic experimental program. A series of experiments including compressive strength, rapid chloride permeability, UPV, acid resistance, X-ray diffraction, SEM and EDAX were performed to examine the effect of varying proportions of fly ash (0%, 20%, 40%, 60%) on cement replacement. Analysis of results indicated formation of hydration compounds in the form of alite, belite, celite, portlandite and tobermorite (C-S-H gel). Results of mechanical and durability tests showed that, to achieve maximum benefits, cement can be replaced to an optimum value of fly ash of 40%. The authors believe that the formation of hydration compounds tobermorite and celite resulted in attaining enhanced durability and strength in high volume fly ash concrete.
We study the following Neumann problem of Kirchhoff type equation with critical growth
{−(a+b∫Ω|∇u|2dx)Δu+u=Q(x)|u|4u+λP(x)|u|q−2u,in Ω,∂u∂v=0,on ∂Ω, | (1.1) |
where Ω ⊂ R3 is a bounded domain with a smooth boundary, a,b>0, 1<q<2, λ>0 is a real parameter. We assume that Q(x) and P(x) satisfy the following conditions:
(Q1) Q(x)∈C(ˉΩ) is a sign-changing;
(Q2) there exists xM∈Ω such that QM=Q(xM)>0 and
|Q(x)−QM|=o(|x−xM|)asx→xM; |
(Q3) there exists 0∈∂Ω such that Qm=Q(0)>0 and
|Q(x)−Qm|=o(|x|)asx→0; |
(P1) P(x) is positive continuous on ˉΩ and P(x0)=maxx∈ˉΩP(x);
(P2) there exist σ>0, R>0 and 3−q<β<6−q2 such that P(x)≥σ|x−y|−β for |x−y|≤R, where y is xM∈Ω or 0∈∂Ω.
In recent years, the following Dirichlet problem of Kirchhoff type equation has been studied extensively by many researchers
{−(a+b∫Ω|∇u|2dx)Δu=f(x,u),in Ω,u=0,on ∂Ω, | (1.2) |
which is related to the stationary analogue of the equation
utt−(a+b∫Ω|∇u|2dx)Δu=f(x,u) | (1.3) |
proposed by Kirchhoff in [13] as an extension of the classical D'Alembert's wave equation for free vibrations of elastic strings. Kirchhoff's model takes into account the changes in length of the string produced by transverse vibrations. In (1.2) and (1.3), u denotes the displacement, b is the initial tension and f(x,u) stands for the external force, while a is related to the intrinsic properties of the string (such as Young's modulus). We have to point out that such nonlocal problems appear in other fields like biological systems, such as population density, where u describes a process which depends on the average of itself (see Alves et al. [2]). After the pioneer work of Lions [18], where a functional analysis approach was proposed. The Kirchhoff type Eq (1.2) with critical growth began to call attention of researchers, we can see [1,9,14,17,23,24,28,30] and so on.
Recently, the following Kirchhoff type equation has been well studied by various authors
{−(a+b∫R3|∇u|2dx)Δu+V(x)u=f(x,u),inR3,u>0,u∈H1(R3). | (1.4) |
There has been much research regarding the concentration behavior of the positive solutions of (1.4), we can see [10,11,12,25,33]. Many papers studied the existence of ground state solutions of (1.4), for example [5,8,15,16,21,22,24]. In addition, the authors established the existence of sign-changing solutions of (1.4) in [20,31]. In papers [27,32] proved the existence and multiplicity of nontrivial solutions of (1.4) by using mountain pass theorem.
In particular, Chabrowski in [6] studied the solvability of the Neumann problem
{−Δu=Q(x)|u|2∗−2u+λf(x,u),in Ω,∂u∂v=0,on ∂Ω, |
where Ω ⊂ RN is a smooth bounded domain, 2∗=2NN−2(N≥3) is the critical Sobolev exponent, λ>0 is a parameter. Assume that Q(x)∈C(¯Ω) is a sign-changing function and ∫ΩQ(x)dx<0, under the condition of f(x,u). Using the space decomposition H1(Ω)=span1⊕V, where V={v∈H1(Ω):∫Ωvdx=0}, the author obtained the existence of two distinct solutions by the variational method.
In [14], Lei et al. considered the following Kirchhoff type equation with critical exponent
{−(a+b∫Ω|∇u|2dx)Δu=u5+λuq−1|x|β,in Ω,u=0,on ∂Ω, |
where Ω ⊂ R3 is a smooth bounded domain, a,b>0, 1<q<2, λ>0 is a parameter. They obtained the existence of a positive ground state solution for 0≤β<2 and two positive solutions for 3−q≤β<2 by the Nehari manifold method.
In [34], Zhang obtained the existence and multiplicity of nontrivial solutions of the following equation
{−(a+b∫Ω|∇u|2dx)Δu+u=λ|u|q−2u+f(x,u)+Q(x)u5,in Ω,∂u∂v=0,on ∂Ω, | (1.5) |
where Ω is an open bounded domain in R3, a,b>0, 1<q<2, λ≥0 is a parameter, f(x,u) and Q(x) are positive continuous functions satisfying some additional assumptions. Moreover, f(x,u)∼|u|p−2u with 4<p<6.
Comparing with the above mentioned papers, our results are different and extend the above results to some extent. Specially, motivated by [34], we suppose Q(x) changes sign on Ω and f(x,u)≡0 for (1.5). Since (1.1) is critical growth, which leads to the cause of the lack of compactness of the embedding H1(Ω)↪L6(Ω), we overcome this difficulty by using P.Lions concentration compactness principle [19]. Moreover, note that Q(x) changes sign on Ω, how to estimate the level of the mountain pass is another difficulty.
We define the energy functional corresponding to problem (1.1) by
Iλ(u)=12‖u‖2+b4(∫Ω|∇u|2dx)2−16∫ΩQ(x)|u|6dx−λq∫ΩP(x)|u|qdx. |
A weak solution of problem (1.1) is a function u∈H1(Ω) and for all φ∈H1(Ω) such that
∫Ω(a∇u∇φ+uφ)dx+b∫Ω|∇u|2dx∫Ω∇u∇φdx=∫ΩQ(x)|u|4uφdx+λ∫ΩP(x)|u|q−2uφdx. |
Our main results are the following:
Theorem 1.1. Assume that 1<q<2 and Q(x) changes sign on Ω. Then there exists Λ0>0 such that for every λ∈(0,Λ0), problem (1.1) has at least one nontrivial solution.
Theorem 1.2. Assume that 1<q<2, 3−q<β<6−q2 and Q(x) changes sign on Ω, there exists Λ∗>0 such that for all λ∈(0,Λ∗). Then problem (1.1) has at least two nontrivial solutions.
Throughout this paper, we make use of the following notations:
● The space H1(Ω) is equipped with the norm ‖u‖2H1(Ω)=∫Ω(|∇u|2+u2)dx, the norm in Lp(Ω) is denoted by ‖⋅‖p.
● Define ‖u‖2=∫Ω(a|∇u|2+u2)dx for u∈H1(Ω). Note that ‖⋅‖ is an equivalent norm on H1(Ω) with the standard norm.
● Let D1,2(R3) is the completion of C∞0(R3) with respect to the norm ‖u‖2D1,2(R3)=∫R3|∇u|2dx.
● 0<QM=maxx∈ˉΩQ(x), 0<Qm=maxx∈∂ΩQ(x).
● Ω+={x∈Ω:Q(x)>0} and Ω−={x∈Ω:Q(x)<0}.
● C,C1,C2,… denote various positive constants, which may vary from line to line.
● We denote by Sρ (respectively, Bρ) the sphere (respectively, the closed ball) of center zero and radius ρ, i.e. Sρ={u∈H1(Ω):‖u‖=ρ}, Bρ={u∈H1(Ω):‖u‖≤ρ}.
● Let S be the best constant for Sobolev embedding H1(Ω)↪L6(Ω), namely
S=infu∈H1(Ω)∖{0}∫Ω(a|∇u|2+u2)dx(∫Ω|u|6dx)1/3. |
● Let S0 be the best constant for Sobolev embedding D1,2(R3)↪L6(R3), namely
S0=infu∈D1,2(R3)∖{0}∫R3|∇u|2dx(∫R3|u|6dx)1/3. |
In this section, we firstly show that the functional Iλ(u) has a mountain pass geometry.
Lemma 2.1. There exist constants r,ρ,Λ0>0 such that the functional Iλ satisfies the following conditions for each λ∈(0,Λ0):
(i) Iλ|u∈Sρ≥r>0; infu∈BρIλ(u)<0.
(ii) There exists e∈H1(Ω) with ‖e‖>ρ such that Iλ(e)<0.
Proof. (i) From (P1), by the H¨older inequality and the Sobolev inequality, for all u∈H1(Ω) one has
∫ΩP(x)|u|qdx≤P(x0)∫Ω|u|qdx≤P(x0)|Ω|6−q6S−q2‖u‖q, | (2.1) |
and there exists a constant C>0, we get
|∫ΩQ(x)|u|6dx|≤C∫Ω|u|6dx≤CS−3‖u‖6. | (2.2) |
Hence, combining (2.1) and (2.2), we have the following estimate
Iλ(u)=12‖u‖2+b4(∫Ω|∇u|2dx)2−16∫ΩQ(x)|u|6dx−λq∫ΩP(x)|u|qdx≥12‖u‖2−C6∫Ω|u|6dx−λqP(x0)|Ω|6−q6S−q2‖u‖q≥‖u‖q(12‖u‖2−q−C6S−3‖u‖6−q−λqP(x0)|Ω|6−q6S−q2). |
Set h(t)=12t2−q−C6S−3t6−q for t>0, then there exists a constant ρ=(3(2−q)S3C(6−q))14>0 such that maxt>0h(t)=h(ρ)>0. Letting Λ0=qSq2P(x0)|Ω|6−q6h(ρ), there exists a constant r>0 such that Iλ|u∈Sρ≥r for every λ∈(0,Λ0). Moreover, for all u∈H1(Ω)∖{0}, we have
limt→0+Iλ(tu)tq=−λq∫ΩP(x)|u|qdx<0. |
So we obtain Iλ(tu)<0 for every u≠0 and t small enough. Therefore, for ‖u‖ small enough, one has
m≜infu∈BρIλ(u)<0. |
(ii) Let v∈H1(Ω) be such that supp v⊂Ω+, v≢0 and t>0, we have
Iλ(tv)=t22‖v‖2+bt44(∫Ω|∇v|2dx)2−t66∫ΩQ(x)|v|6dx−λtqq∫ΩP(x)|v|qdx→−∞ |
as t→∞, which implies that Iλ(tv)<0 for t>0 large enough. Therefore, we can find e∈H1(Ω) with ‖e‖>ρ such that Iλ(e)<0. The proof is complete.
Denote
{Θ1=abS304QM+b3S6024Q2M+aS0√b2S40+4aS0QM6QM+b2S40√b2S40+4aS0QM24Q2M,Θ2=abS3016Qm+b3S60384Q2m+aS0√b2S40+16aS0Qm24Qm+b2S40√b2S40+16aS0Qm384Q2m. |
Then we have the following compactness result.
Lemma 2.2. Suppose that 1<q<2. Then the functional Iλ satisfies the (PS)cλ condition for every cλ<c∗= min {Θ1−Dλ22−q,Θ2−Dλ22−q}, where D=2−q3q(6−q4P(x0)S−q2|Ω|6−q6)22−q.
Proof. Let {un}⊂H1(Ω) be a (PS)cλ sequence for
Iλ(un)→cλandI′λ(un)→0asn→∞. | (2.3) |
It follows from (2.1), (2.3) and the H¨older inequality that
cλ+1+o(‖un‖)≥Iλ(un)−16⟨I′λ(un),un⟩≥13‖un‖2+b12(∫Ω|∇un|2dx)2−λ(1q−16)P(x0)S−q2|Ω|6−q6‖un‖q≥13‖un‖2−λ(6−q)6qP(x0)S−q2|Ω|6−q6‖un‖q. |
Therefore {un} is bounded in H1(Ω) for all 1<q<2. Thus, we may assume up to a subsequence, still denoted by {un}, there exists u∈H1(Ω) such that
{un⇀u,weaklyinH1(Ω),un→u,stronglyinLp(Ω)(1≤p<6),un(x)→u(x),a.e.inΩ, | (2.4) |
as n→∞. Next, we prove that un→u strongly in H1(Ω). By using the concentration compactness principle (see [19]), there exist some at most countable index set J, δxj is the Dirac mass at xj⊂ˉΩ and positive numbers {νj}, {μj}, j∈J, such that
|un|6dx⇀dν=|u|6dx+∑j∈Jνjδxj,|∇un|2dx⇀dμ≥|∇u|2dx+∑j∈Jμjδxj. |
Moreover, numbers νj and μj satisfy the following inequalities
S0ν13j≤μjifxj∈Ω,S0223ν13j≤μjifxj∈∂Ω. | (2.5) |
For ε>0, let ϕε,j(x) be a smooth cut-off function centered at xj such that 0≤ϕε,j≤1, |∇ϕε,j|≤2ε, and
ϕε,j(x)={1, in B(xj,ε2)∩ˉΩ,0, in Ω∖B(xj,ε). |
There exists a constant C>0 such that
limε→0limn→∞∫ΩP(x)|un|qϕε,jdx≤P(x0)limε→0limn→∞∫B(xj,ε)|un|qdx=0. |
Since |∇ϕε,j|≤2ε, by using the H¨older inequality and L2(Ω)-convergence of {un}, we have
limε→0limn→∞(a+b∫Ω|∇un|2dx)∫Ω⟨∇un,∇ϕε,j⟩undx≤Climε→0limn→∞(∫Ω|∇un|2dx)12(∫Ω|un|2|∇ϕε,j|2dx)12≤Climε→0(∫B(xj,ε)|u|6dx)16(∫B(xj,ε)|∇ϕε,j|3dx)13≤Climε→0(∫B(xj,ε)|u|6dx)16(∫B(xj,ε)(2ε)3dx)13≤C1limε→0(∫B(xj,ε)|u|6dx)16=0, |
where C1>0, and we also derive that
limε→0limn→∞∫Ω|∇un|2ϕε,jdx≥limε→0∫Ω|∇u|2ϕε,jdx+μj=μj, |
limε→0limn→∞∫ΩQ(x)|un|6ϕε,jdx=limε→0∫ΩQ(x)|u|6ϕε,jdx+Q(xj)νj=Q(xj)νj, |
limε→0limn→∞∫Ωu2nϕε,jdx=limε→0∫Ωu2ϕε,jdx≤limε→0∫B(xj,ε)u2dx=0. |
Noting that unϕε,j is bounded in H1(Ω) uniformly for n, taking the test function φ=unϕε,j in (2.3), from the above information, one has
0=limε→0limn→∞⟨I′λ(un),unϕε,j⟩=limε→0limn→∞{(a+b∫Ω|∇un|2dx)∫Ω⟨∇un,∇(unϕε,j)⟩dx+∫Ωu2nϕε,jdx−∫ΩQ(x)|un|6ϕε,jdx−λ∫ΩP(x)|un|qϕε,jdx}=limε→0limn→∞{(a+b∫Ω|∇un|2dx)∫Ω(|∇un|2ϕε,j+⟨∇un,∇ϕε,j⟩un)dx−∫ΩQ(x)|un|6ϕε,jdx}≥limε→0{(a+b∫Ω|∇u|2dx+bμj)(∫Ω|∇u|2ϕε,jdx+μj)−∫ΩQ(x)|u|6ϕε,jdx−Q(xj)νj}≥(a+bμj)μj−Q(xj)νj, |
so that
Q(xj)νj≥(a+bμj)μj, |
which shows that {un} can only concentrate at points xj where Q(xj)>0. If νj>0, by (2.5) we get
ν13j≥bS20+√b2S40+4aS0QM2QMifxj∈Ω,ν13j≥bS20+√b2S40+16aS0Qm273Qmifxj∈∂Ω. | (2.6) |
From (2.5) and (2.6), we have
μj≥bS30+√b2S60+4aS30QM2QMifxj∈Ω,μj≥bS30+√b2S60+16aS30Qm8Qmifxj∈∂Ω. | (2.7) |
To proceed further we show that (2.7) is impossible. To obtain a contradiction assume that there exists j0∈J such that μj0≥bS30+√b2S60+4aS30QM2QM and xj0∈Ω. By (2.1), (2.3) and (2.4), one has
cλ=limn→∞{Iλ(un)−16⟨I′λ(un),un⟩}=limn→∞{a3∫Ω|∇un|2dx+b12(∫Ω|∇un|2dx)2+13∫Ωu2ndx−λ6−q6q∫ΩP(x)|un|qdx}≥a3(∫Ω|∇u|2dx+∑j∈Jμj)+b12(∫Ω|∇u|2dx+∑j∈Jμj)2+13∫Ωu2dx−λ6−q6qP(x0)S−q2|Ω|6−q6‖u‖q≥a3μj0+b12μ2j0+13‖u‖2−λ6−q6qP(x0)S−q2|Ω|6−q6‖u‖q. |
Set
g(t)=13t2−λ6−q6qP(x0)S−q2|Ω|6−q6tq,t>0, |
then
g′(t)=23t−λ6−q6P(x0)S−q2|Ω|6−q6tq−1=0, |
we can deduce that mint≥0g(t) attains at t0>0 and
t0=(λ6−q4P(x0)S−q2|Ω|6−q6)12−q. |
Consequently, we obtain
cλ≥abS304QM+b3S6024Q2M+aS0√b2S40+4aS0QM6QM+b2S40√b2S40+4aS0QM24Q2M−Dλ22−q=Θ1−Dλ22−q, |
where D=2−q3q(6−q4P(x0)S−q2|Ω|6−q6)22−q. If μj0≥bS30+√b2S60+16aS30Qm8Qm and xj0∈∂Ω, then, by the similar calculation, we also get
cλ≥abS3016Qm+b3S60384Q2m+aS0√b2S40+16aS0Qm24Qm+b2S40√b2S40+16aS0Qm384Q2m−Dλ22−q=Θ2−Dλ22−q. |
Let c∗=min{Θ1−Dλ22−q,Θ2−Dλ22−q}, from the above information, we deduce that cλ≥c∗. It contradicts our assumption, so it indicates that νj=μj=0 for every j∈J, which implies that
∫Ω|un|6dx→∫Ω|u|6dx | (2.8) |
as n→∞. Now, we may assume that ∫Ω|∇un|2dx→A2 and ∫Ω|∇u|2dx≤A2, by (2.3), (2.4) and (2.8), one has
0=limn→∞⟨I′λ(un),un−u⟩=limn→∞[(a+b∫Ω|∇un|2dx)(∫Ω|∇un|2dx−∫Ω∇un∇udx)+∫Ωun(un−u)dx−∫ΩQ(x)|un|5(un−u)dx−λ∫ΩP(x)|un|q−1(un−u)dx]=(a+bA2)(A2−∫Ω|∇u|2dx). |
Then, we obtain that un→u in H1(Ω). The proof is complete.
As well known, the function
Uε,y(x)=(3ε2)14(ε2+|x−y|2)12,foranyε>0, |
satisfies
−ΔUε,y=U5ε,yinR3, |
and
∫R3|∇Uε,y|2dx=∫R3|Uε,y|6dx=S320. |
Let ϕ∈C1(R3) such that ϕ(x)=1 on B(xM,R2), ϕ(x)=0 on R3−B(xM,R) and 0≤ϕ(x)≤1 on R3, we set vε(x)=ϕ(x)Uε,xM(x). We may assume that Q(x)>0 on B(xM,R) for some R>0 such that B(xM,R)⊂Ω. From [4], we have
{‖∇vε‖22=S320+O(ε),‖vε‖66=S320+O(ε3),‖vε‖22=O(ε),‖vε‖2=aS320+O(ε). | (2.9) |
Moreover, by [28], we get
{‖∇vε‖42≤S30+O(ε),‖∇vε‖82≤S60+O(ε),‖∇vε‖122≤S90+O(ε). | (2.10) |
Then we have the following Lemma.
Lemma 2.3. Suppose that 1<q<2, 3−q<β<6−q2, QM>4Qm, (Q1) and (Q2), then supt≥0Iλ(tvε)<Θ1−Dλ22−q.
Proof. By Lemma 2.1, one has Iλ(tvε)→−∞ as t→∞ and Iλ(tvε)<0 as t→0, then there exists tε>0 such that Iλ(tεvε)=supt>0Iλ(tvε)≥r>0. We can assume that there exist positive constants t1,t2>0 and 0<t1<tε<t2<+∞. Let Iλ(tεvε)=β(tεvε)−λψ(tεvε), where
β(tεvε)=t2ε2‖vε‖2+bt4ε4‖∇vε‖42−t6ε6∫ΩQ(x)|vε|6dx, |
and
ψ(tεvε)=tqεq∫ΩP(x)|vε|qdx. |
Now, we set
h(t)=t22‖vε‖2+bt44‖∇vε‖42−t66∫ΩQ(x)|vε|6dx. |
It is clear that limt→0h(t)=0 and limt→∞h(t)=−∞. Therefore there exists T1>0 such that h(T1)=maxt≥0h(t), that is
h′(t)|T1=T1‖vε‖2+bT31‖∇vε‖42−T51∫ΩQ(x)|vε|6dx=0, |
from which we have
‖vε‖2+bT21‖∇vε‖42=T41∫ΩQ(x)|vε|6dx. | (2.11) |
By (2.11) we obtain
T21=b‖∇vε‖42+√b2‖∇vε‖82+4‖vε‖2∫ΩQ(x)|vε|6dx2∫ΩQ(x)|vε|6dx. |
In addition, by (Q2), for all η>0, there exists ρ>0 such that |Q(x)−QM|<η|x−xM| for 0<|x−xM|<ρ, for ε>0 small enough, we have
|∫ΩQ(x)v6εdx−∫ΩQMv6εdx|≤∫Ω|Q(x)−QM|v6εdx<∫B(xM,ρ)η|x−xM|(3ε2)32(ε2+|x−xM|2)3dx+C∫Ω∖B(xM,ρ)(3ε2)32(ε2+|x−xM|2)3dx≤Cηε3∫ρ0r3(ε2+r2)3dr+Cε3∫Rρr2(ε2+r2)3dr≤Cηε∫ρ/ε0t3(1+t2)3dt+C∫R/ερ/εt2(1+t2)3dt≤C1ηε+C2ε3, |
where C1,C2>0 (independent of η, ε). From this we derive that
lim supε→0|∫ΩQ(x)v6εdx−∫ΩQMv6εdx|ε≤C1η. | (2.12) |
Then from the arbitrariness of η>0, by (2.9) and (2.12), one has
∫ΩQ(x)|vε|6dx=QM∫Ω|vε|6dx+o(ε)=QMS320+o(ε). | (2.13) |
Hence, it follows from (2.9), (2.10) and (2.13) that
β(tεvε)≤h(T1)=T21(13‖vε‖2+bT2112‖∇vε‖42)=b‖∇vε‖42‖vε‖24∫ΩQ(x)|vε|6dx+b3‖∇vε‖12224(∫ΩQ(x)|vε|6dx)2+‖vε‖2√b2‖∇vε‖82+4‖vε‖2∫ΩQ(x)|vε|6dx6∫ΩQ(x)|vε|6dx+b2‖∇vε‖82√b2‖∇vε‖82+4‖vε‖2∫ΩQ(x)|vε|6dx24(∫ΩQ(x)|vε|6dx)2≤b(S30+O(ε))(aS320+O(ε))4(QMS320+o(ε))+b3(S90+O(ε))24(QMS320+o(ε))2+(aS320+O(ε))√b2(S60+O(ε))+4(aS320+O(ε))(QMS320+o(ε))6(QMS320+o(ε))+b2(S60+O(ε))√b2(S60+O(ε))+4(aS320+O(ε))(QMS320+o(ε))24(QMS320+o(ε))2≤abS304QM+b3S6024Q2M+aS0√b2S40+4aS0QM6QM+b2S40√b2S40+4aS0QM24Q2M+C3ε=Θ1+C3ε, |
where the constant C_3 > 0 . According to the definition of v_\varepsilon , from [29], for \frac{R}{2} > \varepsilon > 0 , there holds
\begin{align} \psi(t_{\varepsilon}v_{\varepsilon}) &\geq \frac{1}{q}3^{\frac{q}{4}}t_1^q\int_{B(x_M, \frac{R}{2})} \frac{\sigma\varepsilon^\frac{q}{2}}{(\varepsilon^2+|x-x_M|^2)^{\frac{q}{2}}|x-x_M|^\beta}dx \\ &\geq C\varepsilon^\frac{q}{2}\int_0^{R/2}\frac{r^2}{(\varepsilon^2+r^2)^{\frac{q}{2}}r^\beta}dr \\ & = C\varepsilon^{\frac{6-q}{2}-\beta}\int_0^{R/2\varepsilon}\frac{t^2}{(1+t^2)^{\frac{q}{2}}t^\beta}dt \\ &\geq C\varepsilon^{\frac{6-q}{2}-\beta}\int_0^1t^{2-\beta}dt \\ & = C_4 \varepsilon^{\frac{6-q}{2}-\beta}, \end{align} | (2.14) |
where C_4 > 0 (independent of \varepsilon, \lambda ). Consequently, from the above information, we obtain
\begin{equation*} \begin{aligned} I_{\lambda}(t_{\varepsilon}v_{\varepsilon})& = \beta(t_{\varepsilon}v_{\varepsilon})-\lambda\psi(t_{\varepsilon}v_{\varepsilon})\\ &\leq \Theta_1+C_3\varepsilon-C_4\lambda \varepsilon^{\frac{6-q}{2}-\beta}\\ & \lt \Theta_1-D\lambda^{\frac{2}{2-q}}. \end{aligned} \end{equation*} |
Here we have used the fact that \beta > 3-q and let \varepsilon = \lambda^{\frac{2}{2-q}} , 0 < \lambda < \Lambda_1 = \min\{1, (\frac{C_3+D}{C_4})^{\frac{2-q}{6-2q-2\beta}}\} , then
\begin{equation} \begin{aligned} C_3\varepsilon-C_4\lambda \varepsilon^{{\frac{6-q}{2}-\beta}}& = C_3\lambda^{\frac{2}{2-q}}-C_4\lambda^{\frac{8-2q-2\beta}{2-q}}\\ & = \lambda^{\frac{2}{2-q}}(C_3-C_4\lambda^{\frac{6-2q-2\beta}{2-q}})\\ & \lt -D\lambda^{\frac{2}{2-q}}. \end{aligned} \end{equation} | (2.15) |
The proof is complete.
We assume that 0\in\partial\Omega and Q_m = Q(0) . Let \varphi\in C^1(\mathbb{R}^3) such that \varphi(x) = 1 on B(0, \frac{R}{2}) , \varphi(x) = 0 on \mathbb{R}^3-B(0, R) and 0\leq\varphi(x)\leq 1 on \mathbb{R}^3 , we set u_\varepsilon(x) = \varphi(x)U_\varepsilon(x) , the radius R is chosen so that Q(x) > 0 on B(0, R)\cap \Omega . If H(0) denotes the mean curvature of the boundary at 0 , then the following estimates hold (see [6] or [26])
\begin{equation} \begin{cases} \|u_\varepsilon\|_2^2 = O(\varepsilon), \\ \frac{\|\nabla u_{\varepsilon}\|_2^2}{\|u_{\varepsilon}\|_6^2}\leq\frac{S_0}{2^\frac{2}{3}}-A_3H(0)\varepsilon \log\frac{1}{\varepsilon}+O(\varepsilon), \end{cases} \end{equation} | (2.16) |
where A_3 > 0 is a constant. Then we have the following lemma.
Lemma 2.4. Suppose that 1 < q < 2 , 3-q < \beta < \frac{6-q}{2} , Q_M\leq4Q_m , H(0) > 0 , Q is positive somewhere on \partial\Omega , (Q_1) and (Q_3) , then \sup_{t\geq0}I_{\lambda}(tu_\varepsilon) < \Theta_2-D\lambda^{\frac{2}{2-q}}.
Proof. Similar to the proof of Lemma 2.3, we also have by Lemma 2.1, there exists t_{\varepsilon} > 0 such that I_{\lambda}(t_{\varepsilon} u_{\varepsilon}) = \sup_{t > 0}I_{\lambda}(tu_{\varepsilon})\geq r > 0 . We can assume that there exist positive constants t_1, t_2 > 0 such that 0 < t_1 < t_{\varepsilon} < t_2 < +\infty . Let I_{\lambda}(t_{\varepsilon}u_{\varepsilon}) = A(t_{\varepsilon}u_{\varepsilon})-\lambda B(t_{\varepsilon}u_{\varepsilon}) , where
\begin{equation*} A(t_{\varepsilon}u_{\varepsilon}) = \frac{ t_{\varepsilon}^2}{2}\| u_{\varepsilon}\|^2+\frac{b t_{\varepsilon}^4}{4}\|\nabla u_{\varepsilon}\|_2^{4}-\frac{t_{\varepsilon}^6}{6}\int_{\Omega}Q(x)|u_{\varepsilon}|^{6}dx, \end{equation*} |
and
\begin{equation*} B(t_{\varepsilon}u_{\varepsilon}) = \frac{ t_{\varepsilon}^q}{q}\int_{\Omega}P(x)|u_{\varepsilon}|^qdx. \end{equation*} |
Now, we set
\begin{equation*} f(t) = \frac{t^2}{2}\| u_{\varepsilon}\|^2+\frac{b t^4}{4}\|\nabla u_{\varepsilon}\|_2^{4}-\frac{t^6}{6}\int_{\Omega}Q(x)|u_{\varepsilon}|^{6}dx. \end{equation*} |
Therefore, it is easy to see that there exists T_{2} > 0 such that f(T_{2}) = \max_{f\geq0}f(t) , that is
\begin{equation} f'(t)|_{T_{2}} = T_{2}\| u_{\varepsilon}\|^2+bT_{2}^3\|\nabla u_{\varepsilon}\|_2^{4}- T_{2}^5 \int_{\Omega}Q(x)|u_{\varepsilon}|^{6}dx = 0. \end{equation} | (2.17) |
From (2.17) we obtain
\begin{equation*} T_{2}^2 = \frac{b\|\nabla u_{\varepsilon}\|_2^4+\sqrt{b^2\|\nabla u_{\varepsilon}\|_2^8+4\| u_{\varepsilon}\|^2 \int_{\Omega}Q(x)|u_{\varepsilon}|^{6}dx}}{2\int_{\Omega}Q(x)|u_{\varepsilon}|^{6}dx}. \end{equation*} |
By the assumption (Q_3) , we have the expansion formula
\begin{equation} \int_{\Omega}Q(x)|u_\varepsilon|^6dx = Q_m\int_{\Omega}|u_\varepsilon|^6dx+o(\varepsilon). \end{equation} | (2.18) |
Hence, combining (2.16) and (2.18), there exists C_5 > 0 , such that
\begin{align*} A(t_{\varepsilon}u_{\varepsilon})&\leq f(T_{2})\\ & = T_{2}^2\left(\frac{1}{3}\| u_{\varepsilon}\|^2+\frac{b T_{2}^2}{12}\|\nabla u_{\varepsilon}\|_2^4\right)\\ & = \frac{b\|\nabla u_{\varepsilon}\|_2^4\|u_{\varepsilon}\|^2}{4\int_{\Omega}Q(x)|u_\varepsilon|^6dx} +\frac{b^3\|\nabla u_\varepsilon\|_2^{12}}{24(\int_{\Omega}Q(x)|u_{\varepsilon}|^6dx)^2}\\ &\quad+ \frac{\|u_\varepsilon\|^2\sqrt{b^2\|\nabla u_\varepsilon\|_2^8 +4\|u_\varepsilon\|^2\int_{\Omega}Q(x)|u_\varepsilon|^6dx}}{6\int_{\Omega}Q(x)|u_\varepsilon|^6dx}\\ &\quad+ \frac{b^2\|\nabla u_\varepsilon\|_2^8\sqrt{b^2\|\nabla u_\varepsilon\|_2^8 +4\|u_\varepsilon\|^2\int_{\Omega}Q(x)| u_\varepsilon|^6dx}}{24(\int_{\Omega}Q(x)|u_\varepsilon|^6dx)^2}\\ &\leq \frac{ab}{4Q_m}\left(\frac{\|\nabla u_{\varepsilon}\|_2^6}{\int_{\Omega}|u_\varepsilon|^6dx}+O(\varepsilon)\right)+\frac{b^3}{24Q_m^2}\left(\frac{\|\nabla u_\varepsilon\|_2^{12}}{(\int_{\Omega}|u_\varepsilon|^6dx)^2}+O(\varepsilon)\right)\\ &\quad+ \frac{a}{6Q_m}\left(\frac{\|\nabla u_\varepsilon\|_2^2}{(\int_{\Omega}|u_\varepsilon|^6dx)^\frac{1}{3}}\sqrt{\frac{b^2\|\nabla u_\varepsilon\|_2^8}{(\int_{\Omega}|u_\varepsilon|^6dx)^\frac{4}{3}}+\frac{4aQ_m\|\nabla u_\varepsilon\|_2^2}{(\int_{\Omega}|u_\varepsilon|^6dx)^\frac{1}{3}}}+O(\varepsilon)\right)\\ &\quad+ \frac{b^2}{24Q_m^2}\left(\frac{\|\nabla u_\varepsilon\|_2^8}{(\int_{\Omega}|u_\varepsilon|^6dx)^\frac{4}{3}}\sqrt{\frac{b^2\|\nabla u_\varepsilon\|_2^8}{(\int_{\Omega}|u_\varepsilon|^6dx)^\frac{4}{3}}+\frac{4aQ_m\|\nabla u_\varepsilon\|_2^2}{(\int_{\Omega}|u_\varepsilon|^6dx)^\frac{1}{3}}}+O(\varepsilon)\right)\\ &\leq \frac{abS_0^3}{16Q_m}+\frac{b^3S_0^6}{384Q_m^2}+\frac{aS_0\sqrt{b^2S_0^4+16aS_0Q_m}}{24Q_m}\\ &\quad+ \frac{b^2S_0^4\sqrt{b^2S_0^4+16aS_0Q_m}}{384Q_m^2}+C_5\varepsilon\\ & = \Theta_2+C_5\varepsilon. \end{align*} |
Consequently, by (2.14) and (2.15), similarly, there exists \Lambda_2 > 0 such that 0 < \lambda < \Lambda_2 , we get
\begin{equation*} \begin{aligned} I_{\lambda}(t_{\varepsilon}u_{\varepsilon})& = A(t_{\varepsilon}u_{\varepsilon})-\lambda B(t_{\varepsilon}u_{\varepsilon})\\ &\leq \Theta_2+C_5\varepsilon-C_6\lambda \varepsilon^{\frac{6-q}{2}-\beta}\\ & \lt \Theta_2-D\lambda^{\frac{2}{2-q}}. \end{aligned} \end{equation*} |
where C_6 > 0 (independent of \varepsilon, \lambda ). The proof is complete.
Theorem 2.5. Assume that 0 < \lambda < \Lambda_0 ( \Lambda_0 is as in Lemma 2.1) and 1 < q < 2 . Then problem (1.1) has a nontrivial solution u_\lambda with I_\lambda(u_\lambda) < 0 .
Proof. It follows from Lemma 2.1 that
\begin{equation*} m\triangleq\inf\limits_{u\in \overline{B_\rho(0)}}I_\lambda(u) \lt 0. \end{equation*} |
By the Ekeland variational principle [7], there exists a minimizing sequence \{u_n\}\subset {\overline{B_\rho(0)}} such that
\begin{equation*} I_\lambda(u_n)\leq \inf\limits_{u\in{\overline{B_\rho(0)}}}I_\lambda(u)+\frac{1}{n}, \; \; I_\lambda(v)\geq I_\lambda(u_n)-\frac{1}{n}\|v-u_n\|, \; \; v\in{\overline{B_\rho(0)}}. \end{equation*} |
Therefore, there holds I_\lambda(u_n)\rightarrow m and I_\lambda'(u_n)\rightarrow 0 . Since \{u_n\} is a bounded sequence and {\overline{B_\rho(0)}} is a closed convex set, we may assume up to a subsequence, still denoted by \{u_n\} , there exists u_{\lambda}\in{\overline{B_\rho(0)}}\subset H^{1}(\Omega) such that
\begin{align*} \begin{cases} u_n\rightharpoonup u_{\lambda}, \; \; \; \mathrm{weakly\; in}\; H^{1}(\Omega), \\ u_n\rightarrow u_{\lambda}, \; \; \; \mathrm{strongly\; in}\; L^{p}(\Omega), \; 1\leq p \lt 6, \\ u_n(x)\rightarrow u_{\lambda}(x), \; \; \; \mathrm{a.e.\; in}\; \Omega. \end{cases} \end{align*} |
By the lower semi-continuity of the norm with respect to weak convergence, one has
\begin{equation*} \begin{aligned} m&\geq\liminf\limits_{n\rightarrow \infty}\left[I_\lambda(u_n)-\frac{1}{6}\langle I_\lambda'(u_n), u_n\rangle\right]\\ & = \liminf\limits_{n\rightarrow \infty}\bigg[\frac{1}{3}\int_\Omega \left(a|\nabla u_n|^2+u_n^2\right)dx +\frac{b}{12}\left(\int_\Omega|\nabla u_n|^2dx\right)^2\\ &\quad+ \lambda\left(\frac{1}{6} -\frac{1}{q}\right)\int_\Omega P(x)|u_n|^{q}dx\bigg]\\ &\geq \frac{1}{3}\int_\Omega \left(a|\nabla u_\lambda|^2+u_\lambda^2\right)dx +\frac{b}{12}\left(\int_\Omega|\nabla u_\lambda|^2dx\right)^2\\ &\quad+ \lambda\left(\frac{1}{6} -\frac{1}{q}\right)\int_\Omega P(x)|u_\lambda|^{q}dx\\ & = I_\lambda(u_\lambda)-\frac{1}{6}\langle I_\lambda'(u_\lambda), u_\lambda\rangle = I_\lambda(u_\lambda) = m. \end{aligned} \end{equation*} |
Thus I_{\lambda}(u_\lambda) = m < 0 , by m < 0 < c_\lambda and Lemma 2.2, we can see that \nabla u_n\rightarrow \nabla u_\lambda in L^2(\Omega) and u_\lambda\not\equiv0 . Therefore, we obtain that u_\lambda is a weak solution of problem (1.1). Since I_\lambda(|u_\lambda|) = I_\lambda(u_\lambda) , which suggests that u_\lambda\geq0 , then u_\lambda is a nontrivial solution to problem (1.1). That is, the proof of Theorem 1.1 is complete.
Theorem 2.6. Assume that 0 < \lambda < \Lambda_{*} (\Lambda_{*} = \min\{\Lambda_0, \Lambda_1, \Lambda_2\}) , 1 < q < 2 and 3-q < \beta < \frac{6-q}{2} . Then the problem (1.1) has a nontrivial solution u_{1}\in H^1(\Omega) such that I_{\lambda}(u_{1}) > 0 .
Proof. Applying the mountain pass lemma [3] and Lemma 2.2, there exists a sequence \{u_n\}\subset H^1(\Omega) such that
\begin{equation*} I_{\lambda}(u_n)\rightarrow c_\lambda \gt 0\; \; \mathrm{and}\; \; I'_\lambda(u_n)\rightarrow0\; \mathrm{as}\; n\rightarrow \infty, \end{equation*} |
where
c_\lambda = \inf\limits_{\gamma\in\Gamma}\max\limits_{t\in[0, 1]}I_{\lambda}(\gamma(t)), |
and
\Gamma = \left\{\gamma\in C([0, 1], H^{1}(\Omega)): \gamma(0) = 0, \gamma(1) = e\right\}. |
According to Lemma 2.2, we know that \{u_n\}\subset H^1(\Omega) has a convergent subsequence, still denoted by \{u_n\} , such that u_n\rightarrow u_{1} in H^1(\Omega) as n\rightarrow\infty ,
\begin{equation*} I_{\lambda}(u_{1}) = \lim\limits_{n\rightarrow\infty}I_{\lambda}(u_n) = c_\lambda \gt r \gt 0, \end{equation*} |
which implies that u_{1}\not\equiv0 . Therefore, from the continuity of I'_\lambda , we obtain that u_{1} is a nontrivial solution of problem (1.1) with I_{\lambda}(u_{1}) > 0 . Combining the above facts with Theorem 2.5 the proof of Theorem 1.2 is complete.
In this paper, we consider a class of Kirchhoff type equations with Neumann conditions and critical growth. Under suitable assumptions on Q(x) and P(x) , using the variational method and the concentration compactness principle, we proved the existence and multiplicity of nontrivial solutions.
This research was supported by the National Natural Science Foundation of China (Grant Nos. 11661021 and 11861021). Authors are grateful to the referees for their very constructive comments and valuable suggestions.
The authors declare no conflict of interest in this paper.
[1] | Kumar M, Sinha AK, Kujur J (2021) Mechanical and durability studies on high‐volume fly‐ash concrete. Struct Concr 22: E1036-E1049. |
[2] |
Prakash R, Thenmozhi R, Raman SN, et al. (2020) Characterization of eco‐friendly steel fiber‐reinforced concrete containing waste coconut shell as coarse aggregates and fly ash as partial cement replacement. Struct Concr 21: 437-447. doi: 10.1002/suco.201800355
![]() |
[3] |
Rashad AM (2013) A comprehensive overview about the influence of different additives on the properties of alkali-activated slag—A guide for Civil Engineer. Constr Build Mater 47: 29-55. doi: 10.1016/j.conbuildmat.2013.04.011
![]() |
[4] |
Uzbaş B, Aydın AC (2019) Analysis of fly ash concrete with scanning electron microscopy and X-ray diffraction. Adv Sci Technol Res J 13: 100-110. doi: 10.12913/22998624/114178
![]() |
[5] |
Yu J, Li G, Leung CKY (2018) Hydration and physical characteristics of ultrahigh-volume fly ash-cement systems with low water/binder ratio. Constr Build Mate 161: 509-518. doi: 10.1016/j.conbuildmat.2017.11.104
![]() |
[6] | Feng J, Sun J, Yan P (2018) The influence of ground fly ash on cement hydration and mechanical property of mortar. Adv Civ Eng 2018: 4023178. |
[7] |
Saha AK (2018) Effect of class F fly ash on the durability properties of concrete. Sustainable Environ Res 28: 25-31. doi: 10.1016/j.serj.2017.09.001
![]() |
[8] |
Chindaprasirt P, Chotithanorm C, Cao HT, et al. (2007) Influence of fly ash fineness on the chloride penetration of concrete. Constr Build Mater 21: 356-361. doi: 10.1016/j.conbuildmat.2005.08.010
![]() |
[9] | Singh GVPB, Subramaniam KVL (2016) Concrete using siliceous fly ash at very high levels of cement replacement: Influence of lime content and temperature. 2nd RN Raikar Memorial International Conference and Banthia-Basheer International Symposium on Advances in Science and Technology of Concrete, Mumbai, India, 1-11. |
[10] | Liu Z, Xu D, Zhang Y (2017) Experimental investigation and quantitative calculation of the degree of hydration and products in fly ash-cement mixtures. Adv Mater Sci Eng 2017: 2437270. |
[11] | Atiş CD (2003) High-volume fly ash concrete with high strength and low drying shrinkage. J Mater Civil Eng 15: 153-156. |
[12] |
Plowman C, Cabrera JG (1996) The use of fly ash to improve the sulphate resistance of concrete. Waste Manage 16: 145-149. doi: 10.1016/S0956-053X(96)00055-4
![]() |
[13] |
Hemalatha T, Ramaswamy A (2017) A review on fly ash characteristics—Towards promoting high volume utilization in developing sustainable concrete. J Cleaner Prod 147: 546-559. doi: 10.1016/j.jclepro.2017.01.114
![]() |
[14] |
Hemalatha T, Mapa M, George N, et al. (2016) Physico-chemical and mechanical characterization of high volume fly ash incorporated and engineered cement system towards developing greener cement. J Cleaner Prod 125: 268-281. doi: 10.1016/j.jclepro.2016.03.118
![]() |
[15] |
Jung SH, Saraswathy V, Karthick S, et al. (2018) Microstructure characteristics of fly ash concrete with rice husk ash and lime stone powder. Int J Concr Struct Mater 12: 1-9. doi: 10.1186/s40069-018-0237-8
![]() |
[16] | Patil RA, Zodape SP (2011) X-ray diffraction and sem investigation of solidification/stabilization of nickel and chromium using fly ash. E-J Chem 8: S395-S403. |
[17] |
Pyatina T, Sugama T (2016) Acid resistance of calcium aluminate cement-fly ash F blends. Adv Cem Res 28: 433-457. doi: 10.1680/jadcr.15.00139
![]() |
[18] | Wang A, Zhang C, Sun W (2003) Fly ash effects: I. The morphological effect of fly ash. Cement Concrete Res 33: 2023-2029. |
[19] |
Donatello S, Palomo A, Fernández-Jiménez A (2013) Durability of very high volume fly ash cement pastes and mortars in aggressive solutions. Cem Concr Compos 38: 12-20. doi: 10.1016/j.cemconcomp.2013.03.001
![]() |
[20] |
Garcia-Lodeiro I, Fernandez-Jimenez A, Palomo A (2013) Hydration kinetics in hybrid binders: Early reaction stages. Cem Concr Compos 39: 82-92. doi: 10.1016/j.cemconcomp.2013.03.025
![]() |
[21] | Wang A, Zhang C, Sun W (2004) Fly ash effects: Ⅲ. The microaggregate effect of fly ash. Cement Concrete Res 34: 2061-2066. |
[22] |
De Weerdt K, Haha MB, Le Saout G, et al. (2011) Hydration mechanisms of ternary Portland cements containing limestone powder and fly ash. Cement Concrete Res 41: 279-291. doi: 10.1016/j.cemconres.2010.11.014
![]() |
[23] |
Tang SW, Cai XH, He Z, et al. (2016) Hydration process of fly ash blended cement pastes by impedance measurement. Constr Build Mater 113: 939-950. doi: 10.1016/j.conbuildmat.2016.03.141
![]() |
[24] | Kayali O (2004) Effect of high volume fly ash on mechanical properties of fiber reinforced concrete. Mater Struct 37: 318-327. |
[25] |
Siddique R (2004) Performance characteristics of high-volume Class F fly ash concrete. Cement Concrete Res 34: 487-493. doi: 10.1016/j.cemconres.2003.09.002
![]() |
[26] | IS 516-1959 (reaffirmed 2004): Indian Standard Methods of Tests for Strength of Concrete. BIS, 1999. Available form: https://www.iitk.ac.in/ce/test/IS-codes/is.516.1959.pdf. |
[27] | ASTM C1202-12, Standard Test Method for Electrical Indication of Concrete's Ability to Resist Chloride Ion Penetration. ASTM International, 2012. Available form https://www.astm.org/DATABASE.CART/HISTORICAL/C1202-12.htm. |
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