Industrial wastewater contains non-biodegradable dyes that are highly toxic to humans and aquatic life. As solution from photocatalytic degradation, TiO2 is one of the effective photocatalysts for wastewater degradation, but it has low adsorption power. To overcome this deficiency, this study synthesized a new photocatalyst by Fe-TiO2/zeolite H-A. The photocatalyst was successfully synthesized by the impregnation method and was systematically characterized by XRD, XRF, SEM, FT-IR and UV-Vis DRS. XRD diffractogram at 2θ = 25.3° showed anatase phase of the photocatalyst. SEM results showed a rough and soft surface with a size of 491.49 nm. FT-IR analysis obtained the zeolite-A characteristic band, vibration of Ti-O-Ti groups and the vibration of the Fe-O group. The bandwidth of the band gap was 3.16 eV. The photocatalytic efficiency of methylene blue degradation reached 89.58% yield with optimum conditions: irradiation time of 50 min, pH 9 and concentration of methylene blue about 20 mg/L. Fe-TiO2/zeolite H-A as a new photocatalyst can be an alternative photocatalyst to purify methylene blue.
Citation: Ririn Cahyanti, Sumari Sumari, Fauziatul Fajaroh, Muhammad Roy Asrori, Yana Fajar Prakasa. Fe-TiO2/zeolite H-A photocatalyst for degradation of waste dye (methylene blue) under UV irradiation[J]. AIMS Materials Science, 2023, 10(1): 40-54. doi: 10.3934/matersci.2023003
[1] | Jiayin Liu . On stability and instability of standing waves for the inhomogeneous fractional Schrodinger equation. AIMS Mathematics, 2020, 5(6): 6298-6312. doi: 10.3934/math.2020405 |
[2] | Meixia Cai, Hui Jian, Min Gong . Global existence, blow-up and stability of standing waves for the Schrödinger-Choquard equation with harmonic potential. AIMS Mathematics, 2024, 9(1): 495-520. doi: 10.3934/math.2024027 |
[3] | Liu Gao, Chunfang Chen, Jianhua Chen, Chuanxi Zhu . Existence of nontrivial solutions for Schrödinger-Kirchhoff type equations involving the fractional p-Laplacian and local nonlinearity. AIMS Mathematics, 2021, 6(2): 1332-1347. doi: 10.3934/math.2021083 |
[4] | M. Hafiz Uddin, M. Ali Akbar, Md. Ashrafuzzaman Khan, Md. Abdul Haque . New exact solitary wave solutions to the space-time fractional differential equations with conformable derivative. AIMS Mathematics, 2019, 4(2): 199-214. doi: 10.3934/math.2019.2.199 |
[5] | Haikun Liu, Yongqiang Fu . On the variable exponential fractional Sobolev space Ws(·),p(·). AIMS Mathematics, 2020, 5(6): 6261-6276. doi: 10.3934/math.2020403 |
[6] | Paul Bracken . Applications of the lichnerowicz Laplacian to stress energy tensors. AIMS Mathematics, 2017, 2(3): 545-556. doi: 10.3934/Math.2017.2.545 |
[7] | Yunmei Zhao, Yinghui He, Huizhang Yang . The two variable (φ/φ, 1/φ)-expansion method for solving the time-fractional partial differential equations. AIMS Mathematics, 2020, 5(5): 4121-4135. doi: 10.3934/math.2020264 |
[8] | Yongbin Wang, Binhua Feng . Instability of standing waves for the inhomogeneous Gross-Pitaevskii equation. AIMS Mathematics, 2020, 5(5): 4596-4612. doi: 10.3934/math.2020295 |
[9] | Chengbo Zhai, Yuanyuan Ma, Hongyu Li . Unique positive solution for a p-Laplacian fractional differential boundary value problem involving Riemann-Stieltjes integral. AIMS Mathematics, 2020, 5(5): 4754-4769. doi: 10.3934/math.2020304 |
[10] | Yang Pu, Hongying Li, Jiafeng Liao . Ground state solutions for the fractional Schrödinger-Poisson system involving doubly critical exponents. AIMS Mathematics, 2022, 7(10): 18311-18322. doi: 10.3934/math.20221008 |
Industrial wastewater contains non-biodegradable dyes that are highly toxic to humans and aquatic life. As solution from photocatalytic degradation, TiO2 is one of the effective photocatalysts for wastewater degradation, but it has low adsorption power. To overcome this deficiency, this study synthesized a new photocatalyst by Fe-TiO2/zeolite H-A. The photocatalyst was successfully synthesized by the impregnation method and was systematically characterized by XRD, XRF, SEM, FT-IR and UV-Vis DRS. XRD diffractogram at 2θ = 25.3° showed anatase phase of the photocatalyst. SEM results showed a rough and soft surface with a size of 491.49 nm. FT-IR analysis obtained the zeolite-A characteristic band, vibration of Ti-O-Ti groups and the vibration of the Fe-O group. The bandwidth of the band gap was 3.16 eV. The photocatalytic efficiency of methylene blue degradation reached 89.58% yield with optimum conditions: irradiation time of 50 min, pH 9 and concentration of methylene blue about 20 mg/L. Fe-TiO2/zeolite H-A as a new photocatalyst can be an alternative photocatalyst to purify methylene blue.
In this paper, we discuss the existence and multiplicity of standing wave solutions for the following perturbed fractional p-Laplacian systems with critical nonlinearity
{εps(−Δ)spu+V(x)|u|p−2u=K(x)|u|p∗s−2u+Fu(x,u,v),x∈RN,εps(−Δ)spv+V(x)|v|p−2v=K(x)|v|p∗s−2v+Fv(x,u,v),x∈RN, | (1.1) |
where ε is a positive parameter, N>ps,s∈(0,1),p∗s=NpN−ps and (−Δ)sp is the fractional p-Laplacian operator, which is defined as
(−Δ)spu(x)=limε→0∫RN∖Bε(x)|u(x)−u(y)|p−2(u(x)−u(y))|x−y|N+psdy,x∈RN, |
where Bε(x)={y∈RN:|x−y|<ε}. The functions V(x),K(x) and F(x,u,v) satisfy the following conditions:
(V0)V∈C(RN,R),minx∈RNV(x)=0andthereisaconstantb>0suchthattheset Vb:={x∈RN:V(x)<b}hasfiniteLebesguemeasure;
(K0)K∈C(RN,R),0<infK≤supK<∞;
(F1)F∈C1(RN×R2,R)andFs(x,s,t),Ft(x,s,t)=o(|s|p−1+|t|p−1) uniformlyinx∈RNas|s|+|t|→0;
(F2)thereexistC0>0andp<κ<p∗ssuchthat |Fs(x,s,t)|,|Ft(x,s,t)|≤C0(1+|s|κ−1+|t|κ−1);
(F3)thereexistl0>0,d>pandμ∈(p,p∗s)suchthatF(x,s,t)≥l0(|s|d+|t|d)and 0<μF(x,s,t)≤Fs(x,s,t)s+Ft(x,s,t)tforall(x,s,t)∈RN×R2;
(F4)Fs(x,−s,t)=−Fs(x,s,t)andFt(x,s,−t)=−Ft(x,s,t)forall(x,s,t)∈RN×R2.
Conditions (V0),(K0), suggested by Ding and Lin [11] in studying perturbed Schrödinger equations with critical nonlinearity, and then was used in [28,32,33].
In recent years, a great deal of attention has been focused on the study of standing wave solutions for perturbed fractional Schrödinger equation
ε2s(−Δ)su+V(x)u=f(u)inRN, | (1.2) |
where s∈(0,1), N>2s and ε>0 is a small parameter. It is well known that the solution of (1.2) is closely related to the existence of solitary wave solutions for the following eqation
iεωt−ε2(−Δ)sω−V(x)ω+f(ω)=0,(x,t)∈RN×R, |
where i is the imaginary unit. (−Δ)s is the fractional Laplacian operator which arises in many areas such as physics, phase transitions, chemical reaction in liquids, finance and so on, see [1,6,18,22,27]. Additionally, Eq (1.2) is a fundamental equation of fractional quantum mechanics. For more details, please see [17,18].
Equation (1.2) was also investigated extensively under various hypotheses on the potential and the nonlinearity. For example, Floer and Weinstein [12] first considered the existence of single-peak solutions for N=1 and f(t)=t3. They obtained a single-peak solution which concentrates around any given nondegenerate critical point of V. Jin, Liu and Zhang [16] constructed a localized bound-state solution concentrating around an isolated component of the positive minimum point of V, when the nonlinear term f(u) is a general critical nonlinearity. More related results can be seen in [5,7,10,13,14,26,43] and references therein. Recently, Zhang and Zhang [46] obtained the multiplicity and concentration of positive solutions for a class of fractional unbalanced double-phase problems by topological and variational methods. Related to (1.2) with s=1, see [31,39] for quasilinear Schrödinger equations.
On the other hand, fractional p-Laplacian operator can be regarded as an extension of fractional Laplacian operator. Many researchers consider the following equation
εps(−Δ)spu+V(x)|u|p−2u=f(x,u). | (1.3) |
When f(x,u)=A(x)|u|p∗s−2u+h(x,u), Li and Yang [21] obtained the existence and multiplicity of weak solutions by variational methods. When f(x,u)=λf(x)|u|q−2u+g(x)|u|r−2u, under suitable assumptions on nonlinearity and weight functions, Lou and Luo [19] established the existence and multiplicity of positive solutions via variational methods. With regard to the p-fractional Schrödinger-Kirchhoff, Song and Shi [29] considered the following equation with electromagnetic fields
{εpsM([u]ps,Aε)(−Δ)sp,Aεu+V(x)|u|p−2u=|u|p∗s−2u+h(x,|u|p)|u|p−2u,x∈RN,u(x)→0,as→∞. | (1.4) |
They obtained the existence and multiplicity solutions for (1.4) by using the fractional version of concentration compactness principle and variational methods, see also [24,25,34,35,38,41] and references therein. Related to (1.3) with s=1, see [15,23].
Recently, from a mathematical point of view, (fractional) elliptic systems have been the focus for many researchers, see [2,8,9,20,30,37,42,44,45]. As far as we know, there are few results concerned with the (fractional) p-Laplacian systems with a small parameter. In this direction, we cite the work of Zhang and Liu [40], who studied the following p-Laplacian elliptic systems
{−εpΔpu+V(x)|u|p−2u=K(x)|u|p∗−2u+Hu(u,v),x∈RN,−εpΔpv+V(x)|v|p−2v=K(x)|v|p∗−2v+Hv(u,v),x∈RN. | (1.5) |
By using variational methods, they proved the existence of nontrivial solutions for (1.5) provided that ε is small enough. In [36], Xiang, Zhang and Wei investigated the following fractional p-Laplacian systems without a small parameter
{(−Δ)spu+a(x)|u|p−2u=Hu(x,u,v),x∈RN,(−Δ)sqv+b(x)|v|p−2v=Hv(x,u,v),x∈RN. | (1.6) |
Under some suitable conditions, they obtained the existence of nontrivial and nonnegative solutions for (1.6) by using the mountain pass theorem.
Motivated by the aforementioned works, it is natural to ask whether system (1.5) has a nontrivial solution when the p-Laplacian operator is replaced by the fractional p-Laplacian operator. As far as we know, there is no related work in this direction so far. In this paper, we give an affirmative answer to this question considering the existence and multiplicity of standing wave solutions for (1.1).
Now, we present our results of this paper.
Theorem 1.1. Assume that (V0), (K0) and (F1)–(F3) hold. Then for any τ>0, there is Γτ>0 such that if ε<Γτ, system (1.1) has at least one solution (uε,vε)→(0,0) in W as ε→0, where W is stated later, satisfying:
μ−pμp[∫∫R2Nεps(|uε(x)−uε(y)|p|x−y|N+ps+|vε(x)−vε(y)|p|x−y|N+ps)dxdy+∫RNV(x)(|uε|p+|vε|p)dx]≤τεN |
and
sN∫RNK(x)(|uε|p∗s+|vε|p∗s)dx+μ−pp∫RNF(x,uε,vε)dx≤τεN. |
Theorem 1.2. Let (V0), (K0) and (F1)–(F4) hold. Then for any m∈N and τ>0 there is Γmτ>0 such that if ε<Γmτ, system (1.1) has at least m pairs of solutions (uε,vε), which also satisfy the above estimates in Theorem 1.1. Moreover, (uε,vε)→(0,0) in W as ε→0.
Remark 1.1. On one hand, our results extend the results in [40], in which the authors considered the existence of solutions for perturbed p-Laplacian system, i.e., system (1.1) with s=1. On the other hand, our results also extend the results in [21] to a class of perturbed fractional p-Laplacian system (1.1).
Remark 1.2. Compared with the results obtained by [12,13,14,15,16], when ε→0, the solutions of Theorems 1.1 and 1.2 are close to trivial solutions.
In this paper, our goal is to prove the existence and multiplicity of standing wave solutions for (1.1) by variational approach. The main difficulty lies on the lack of compactness of the energy functional associated to system (1.1) because of unbounded domain RN and critical nonlinearity. To overcome this difficulty, we adopt some ideas used in [11] to prove that (PS)c condition holds.
The rest of this article is organized as follows. In Section 2, we introduce the working space and restate the system in a equivalent form by replacing ε−ps with λ. In Section 3, we study the behavior of (PS)c sequence. In Section 4, we complete the proof of Theorems 2.1 and 2.2, respectively.
To obtain the existence and multiplicity of standing wave solutions of system (1.1) for small ε, we rewrite (1.1) in a equivalent form. Let λ=ε−ps, then system (1.1) can be expressed as
{(−Δ)spu+λV(x)|u|p−2u=λK(x)|u|p∗s−2u+λFu(x,u,v),x∈RN,(−Δ)spv+λV(x)|v|p−2v=λK(x)|v|p∗s−2v+λFv(x,u,v),x∈RN, | (2.1) |
for λ→+∞.
We introduce the usual fractional Sobolev space
Ws,p(RN):={u∈Lp(RN):[u]s,p<∞} |
equipped with the norm
||u||s,p=(|u|p+[u]ps,p)1p, |
where |⋅|p is the norm in Lp(RN) and
[u]s,p=(∫∫R2N|u(x)−u(y)|p|x−y|N+psdxdy)1p |
is the Gagliardo seminorm of a measurable function u:RN→R. In this paper, we continue to work in the following subspace of Ws,p(RN) which is defined by
Wλ:={u∈Ws,p(RN):∫RNλV(x)|u|pdx<∞,λ>0} |
with the norm
||u||λ=([u]ps,p+∫RNλV(x)|u|pdx)1p. |
Notice that the norm ||⋅||s,p is equivalent to ||⋅||λ for each λ>0. It follows from (V0) that Wλ continuously embeds in Ws,p(RN). For the fractional system (2.1), we shall work in the product space W=Wλ×Wλ with the norm ||(u,v)||p=||u||pλ+||v||pλ for any (u,v)∈W.
We recall that (u,v)∈W is a weak solution of system (2.1) if
∫∫R2N|u(x)−u(y)|p−2(u(x)−u(y))(ϕ(x)−ϕ(y))|x−y|N+psdxdy+λ∫RNV(x)|u|p−2uϕdx+∫∫R2N|v(x)−v(y)|p−2(v(x)−v(y))(ψ(x)−ψ(y))|x−y|N+psdxdy+λ∫RNV(x)|v|p−2vψdx=λ∫RNK(x)(|u|p∗s−2uϕ+|v|p∗s−2vψ)dx+λ∫RN(Fu(x,u,v)ϕ+Fv(x,u,v)ψ)dx |
for all (ϕ,ψ)∈W.
Note that the energy functional associated with (2.1) is defined by
Φλ(u,v)=1p∫∫R2N|u(x)−u(y)|p|x−y|N+psdxdy+1p∫RNλV(x)|u|pdx+1p∫∫R2N|v(x)−v(y)|p|x−y|N+psdxdy+1p∫RNλV(x)|v|pdx−λp∗s∫RNK(x)(|u|p∗s+|v|p∗s)dx−λ∫RNF(x,u,v)dx=1p||(u,v)||p−λp∗s∫RNK(x)(|u|p∗s+|v|p∗s)dx−λ∫RNF(x,u,v)dx. |
Clearly, it is easy to check that Φλ∈C1(W,R) and its critical points are weak solution of system (2.1).
In order to prove Theorem 1.1 and 1.2, we only need to prove the following results.
Theorem 2.1. Assume that (V0), (K0) and (F1)–(F3) hold. Then for any τ>0, there is Λτ>0 such that if λ≥Λτ, system (2.1) has at least one solution (uλ,vλ)→(0,0) in W as λ→∞, satisfying:
μ−pμp[∫∫R2N(|uλ(x)−uλ(y)|p|x−y|N+ps+|vλ(x)−vλ(y)|p|x−y|N+ps)dxdy+∫RNλV(x)(|uλ|p+|vλ|p)dx]≤τλ1−Nps | (2.2) |
and
sN∫RNK(x)(|uλ|p∗s+|vλ|p∗s)dx+μ−pp∫RNF(x,uλ,vλ)dx≤τλ−Nps. | (2.3) |
Theorem 2.2. Assume that (V0), (K0) and (F1)–(F4) hold. Then for any m∈N and τ>0 there is Λmτ>0 such that if λ≥Λmτ, system (2.1) has at least m pairs of solutions (uλ,vλ), which also satisfy the estimates in Theorem 2.1. Moreover, (uλ,vλ)→(0,0) in W as λ→∞.
In this section, we are focused on the compactness of the functional Φλ.
Recall that a sequence {(un,vn)}⊂W is a (PS)c sequence at level c, if Φλ(un,vn)→c and Φ′λ(un,vn)→0. Φλ is said to satisfy the (PS)c condition if any (PS)c sequence contains a convergent subsequence.
Proposition 3.1. Assume that the conditions (V0),(K0) and (F1)–(F3) hold. Then there exists a constant α>0 independent of λ such that, for any (PS)c sequence {(un,vn)}⊂W for Φλ with (un,vn)⇀(u,v), either (un,vn)→(u,v) or c−Φλ(u,v)≥αλ1−Nps.
Corollary 3.1. Under the assumptions of Proposition 3.1, Φλ satisfies the (PS)c condition for all c<αλ1−Nps.
The proof of Proposition 3.1 consists of a series of lemmas which will occupy the rest of this section.
Lemma 3.1. Assume that (V0),(K0) and (F3) are satisfied. Let {(un,vn)}⊂W be a (PS)c sequence for Φλ. Then c≥0 and {(un,vn)} is bounded in W.
Proof. Let {(un,vn)} be a (PS)c sequence for Φλ, we obtain that
Φλ(un,vn)→c,Φ′λ(un,vn)→0,n→∞. |
By (K0) and (F3), we deduce that
c+o(1)||(un,vn)||=Φλ(un,vn)−1μ⟨Φ′λ(un,vn),(un,vn)⟩=(1p−1μ)||(un,vn)||p+λ(1μ−1p∗s)∫RNK(x)(|u|p∗s+|v|p∗s)dx+λ∫RN[1μ(Fu(x,un,vn)un+Fv(x,un,vn)vn)−F(x,un,vn)]dx≥(1p−1μ)||(un,vn)||p, | (3.1) |
which implies that there exists M>0 such that
||(un,vn)||p≤M. |
Thus, {(un,vn)} is bounded in W. Taking the limit in (3.1), we show that c≥0. This completes the proof.
From the above lemma, there exists (u,v)∈W such that (un,vn)⇀(u,v) in W. Furthermore, passing to a subsequence, we have un→u and vn→v in Lγloc(RN) for any γ∈[p,p∗s) and un(x)→u(x) and vn(x)→v(x) a.e. in RN. Clearly, (u,v) is a critical point of Φλ.
Lemma 3.2. Let {(un,vn)} be stated as in Lemma 3.1 and γ∈[p,p∗s). Then there exists a subsequence {(unj,vnj)} such that for any ε>0, there is rε>0 with
limsupj→∞∫Bj∖Br|unj|γdx≤ε,limsupj→∞∫Bj∖Br|vnj|γdx≤ε, |
for all r≥rε, where, Br:={x∈RN:|x|≤r}.
Proof. The proof is similar to the one of Lemma 3.2 of [11]. We omit it here.
Let σ:[0,∞)→[0,1] be a smooth function satisfying σ(t)=1 if t≤1, σ(t)=0 if t≥2. Define ¯uj(x)=σ(2|x|j)u(x), ¯vj(x)=σ(2|x|j)v(x). It is clear that
||u−¯uj||λ→0and||v−¯vj||λ→0asj→∞. | (3.2) |
Lemma 3.3. Let {(unj,vnj)} be stated as in Lemma 3.2, then
limj→∞∫RN[Fu(x,unj,vnj)−Fu(x,unj−¯uj,vnj−¯vj)−Fu(x,¯uj,¯vj)]ϕdx=0 |
and
limj→∞∫RN[Fv(x,unj,vnj)−Fv(x,unj−¯uj,vnj−¯vj)−Fv(x,¯uj,¯vj)]ψdx=0 |
uniformly in (ϕ,ψ)∈W with ||(ϕ,ψ)||≤1.
Proof. By (3.2) and the local compactness of Sobolev embedding, we know that for any r>0,
limj→∞∫Br[Fu(x,unj,vnj)−Fu(x,unj−¯uj,vnj−¯vj)−Fu(x,¯uj,¯vj)]ϕdx=0, | (3.3) |
uniformly for ||ϕ||≤1. For any ε>0, there exists rε>0 such that
limsupj→∞∫Bj∖Br|¯uj|γdx≤∫RN∖Br|u|γdx≤ε, |
for all r≥rε, see [Lemma 3.2, 11]. From (F1) and (F2), we obtain
|Fu(x,u,v)|≤C0(|u|p−1+|v|p−1+|u|κ−1+|v|κ−1). | (3.4) |
Thus, from (3.3), (3.4) and the Hölder inequality, we have
limsupj→∞∫RN[Fu(x,unj,vnj)−Fu(x,unj−¯uj,vnj−¯vj)−Fu(x,¯uj,¯vj)]ϕdx≤limsupj→∞∫Bj∖Br[Fu(x,unj,vnj)−Fu(x,unj−¯uj,vnj−¯vj)−Fu(x,¯uj,¯vj)]ϕdx≤C1limsupj→∞∫Bj∖Br[(|unj|p−1+|¯uj|p−1+|vnj|p−1+|¯vj|p−1)]ϕdx+≤C2limsupj→∞∫Bj∖Br[(|unj|κ−1+|¯uj|κ−1+|vnj|κ−1+|¯vj|κ−1)]ϕdx≤C1limsupj→∞[|unj|p−1Lp(Bj∖Br)+|¯uj|p−1Lp(Bj∖Br)+|vnj|p−1Lp(Bj∖Br)+|¯vj|p−1Lp(Bj∖Br)]|ϕ|p+C2limsupj→∞[|unj|κ−1Lκ(Bj∖Br)+|¯uj|κ−1Lκ(Bj∖Br)+|vnj|κ−1Lκ(Bj∖Br)+|¯vj|κLκ(Bj∖Br)]|ϕ|κ≤C3εp−1p+C4εκ−1κ, |
where C1,C2,C3 and C4 are positive constants. Similarly, we can deduce that the other equality also holds.
Lemma 3.4. Let {(unj,vnj)} be stated as in Lemma 3.2, the following facts hold:
(i)Φλ(unj−¯uj,vnj−¯vj)→c−Φλ(u,v);
(ii)Φ′λ(unj−¯uj,vnj−¯vj)→0inW−1(thedualspaceofW).
Proof. (i) We have
Φλ(unj−¯uj,vnj−¯vj)=Φλ(unj,vnj)−Φλ(¯uj,¯vj)+λp∗s∫RNK(x)(|unj|p∗s−|unj−¯uj|p∗s−|¯uj|p∗s+|vnj|p∗s−|vnj−¯vj|p∗s−|¯vj|p∗s)dx+λ∫RN(F(x,unj,vnj)−F(x,unj−¯uj,vnj−¯vj)−F(x,¯uj,¯vj))dx. |
Using (3.2) and the Brézis-Lieb Lemma [4], it is easy to get
limj→∞∫RNK(x)(|unj|p∗s−|unj−¯uj|p∗s−|¯uj|p∗s+|vnj|p∗s−|vnj−¯vj|p∗s−|¯vj|p∗s)dx=0 |
and
limj→∞∫RN(F(x,unj,vnj)−F(x,unj−¯uj,vnj−¯vj)−F(x,¯uj,¯vj))dx=0. |
Using the fact that Φλ(unj,vnj)→c and Φλ(¯uj,¯vj)→Φλ(u,v) as j→∞, we have
Φλ(unj−¯uj,vnj−¯vj)→c−Φλ(u,v). |
(ii) We observe that for any (ϕ,ψ)∈W satisfying ||(ϕ,ψ)||≤1,
⟨Φ′λ(unj−¯uj,vnj−¯vj),(ϕ,ψ)⟩=⟨Φ′λ(unj,vnj),(ϕ,ψ)⟩−⟨Φ′λ(¯uj,¯vj),(ϕ,ψ)⟩+λ∫RNK(x)[(|unj|p∗s−2unj−|unj−¯uj|p∗s−2(unj−¯uj)−|¯uj|p∗s−2¯uj)ϕ+(|vnj|p∗s−2vnj−|vnj−¯vj|p∗s−2(vnj−¯vj)−|¯vj|p∗s−2¯vj)ψ]dx+λ∫RN[(Fu(x,unj,vnj)−Fu(x,unj−¯uj,vnj−¯vj)−Fu(x,¯uj,¯vj))ϕ+(Fv(x,unj,vnj)−Fv(x,unj−¯uj,vnj−¯vj)−Fv(x,¯uj,¯vj))ψ]dx. |
It follows from a standard argument that
limj→∞∫RNK(x)(|unj|p∗s−2unj−|unj−¯uj|p∗s−2(unj−¯uj)−|¯uj|p∗s−2¯uj)ϕdx=0 |
and
limj→∞∫RNK(x)(|vnj|p∗s−2vnj−|vnj−¯vj|p∗s−2(vnj−¯vj)−|¯vj|p∗s−2¯vj)ψdx=0 |
uniformly in ||(ϕ,ψ)||≤1. By Lemma 3.3, we obtain Φ′λ(unj−¯uj,vnj−¯vj)→0. We complete this proof.
Set u1j=unj−¯uj, v1j=vnj−¯vj, then unj−u=u1j+(¯uj−u), vnj−v=v1j+(¯vj−v). From (3.2), we have (unj,vnj)→(u,v) if and only if (u1j,v1j)→(0,0). By Lemma 3.4, one has along a subsequence that Φλ(u1j,v1j)→c−Φλ(u,v) and Φ′λ(u1j,v1j)→0.
Note that ⟨Φ′λ(u1j,v1j),(u1j,v1j)⟩=0, by computation, we get
∫∫R2N|u1j(x)−u1j(y)|p|x−y|N+psdxdy+∫RNλV(x)|u1j|pdx+∫∫R2N|v1j(x)−v1j(y)|p|x−y|N+psdxdy+∫RNλV(x)|v1j|pdx−λ∫RNK(x)(|u1j|p∗s+|v1j|p∗s)dx−λ∫RNF(x,u1j,v1j)dx=0 | (3.5) |
Hence, by (F3) and (3.5), we have
Φλ(u1j,v1j)−1p⟨Φ′λ(u1j,v1j),(u1j,v1j)⟩=(1p−1p∗s)λ∫RNK(x)(|u1j|p∗s+|v1j|p∗s)dx+λ∫RN[1p(Fu(x,u1j,v1j)u1j+Fu(x,u1j,v1j)v1j)−F(x,u1j,v1j)]dx≥λsKminN∫RN(|u1j|p∗s+|v1j|p∗s)dx, |
where Kmin=infx∈RNK(x)>0. So, it is easy to see that
|u1j|p∗sp∗s+|v1j|p∗sp∗s≤N(c−Φλ(u,v))λsKmin+o(1). | (3.6) |
Denote Vb(x)=max{V(x),b}, where b is the positive constant from assumption of (V0). Since the set Vb has finite measure and (u1j,v1j)→(0,0) in Lploc×Lploc, we obtain
∫RNV(x)(|u1j|p+|v1j|p)dx=∫RNVb(x)(|u1j|p+|v1j|p)dx+o(1). | (3.7) |
By (K0),(F1) and (F2), we can find a constant Cb>0 such that
∫RNK(x)(|u1j|p∗s+|v1j|p∗s)dx+∫RN(Fu(x,u1j,v1j)u1j+Fv(x,u1j,v1j)v1j)dx≤b(|u1j|pp+|v1j|pp)+Cb(|u1j|p∗sp∗s+|v1j|p∗sp∗s). | (3.8) |
Let S is fractional Sobolev constant which is defined by
S|u|pp∗s≤∫∫R2N|u(x)−u(y)|p|x−y|N+psdxdyforallu∈Ws,p(RN). | (3.9) |
Proof of Proposition 3.1. Assume that (unj,vnj)↛(u,v), then liminfj→∞||(u1j,v1j)||>0 and c−Φλ(u,v)>0.
From (3.5), (3.7), (3.8) and (3.9), we deduce
S(|u1j|pp∗s+|v1j|pp∗s)≤∫∫R2N|u1j(x)−u1j(y)|p|x−y|N+psdxdy+∫RNλV(x)|u1j|pdx+∫∫R2N|v1j(x)−v1j(y)|p|x−y|N+psdxdy+∫RNλV(x)|v1j|pdx−∫RNλV(x)(|u1j|p+|v1j|p)dx=λ∫RNK(x)(|u1j|p∗s+|v1j|p∗s)dx+λ∫RN(Fu(x,u1j,v1j)u1j+Fv(x,u1j,v1j)v1j)dx−λ∫RNVb(x)(|u1j|p+|v1j|p)dx≤λCb(|u1j|p∗sp∗s+|v1j|p∗sp∗s)+o(1). |
Thus, by (3.6), we have
S≤λCb(|u1j|p∗sp∗s+|v1j|p∗sp∗s)p∗s−pp∗s+o(1)≤λCb(N(c−Φλ(u,v))λsKmin)sN+o(1), |
or equivalently
αλ1−Nps≤c−Φλ(u,v), |
where α=sKminN(SCb)Nps. The proof is complete.
Lemma 4.1. Suppose that (V0), (K0),(F1),(F2) and (F3) are satisfied, then the functional Φλ satisfies the following mountain pass geometry structure:
(i) there exist positive constants ρ and a such that Φλ(u,v)≥a for ||(u,v)||=ρ;
(ii) for any finite-dimensional subspace Y⊂W,
Φλ(u,v)→−∞,as(u,v)∈W,||(u,v)||→+∞. |
(iii) for any τ>0 there exists Λτ>0 such that each λ≥Λτ, there exists ˜ωλ∈Y with ||˜ωλ||>ρ, Φλ(˜ωλ)≤0 and
maxt≥0Φλ(t˜ωλ)≤τλ1−Nps. |
Proof. (i) From (F1),(F2), we have for any ε>0, there is Cε>0 such that
1p∗s∫RNK(x)(|u|p∗s+|v|p∗s)dx+∫RNF(x,u,v)dx≤ε|(u,v)|pp+Cε|(u,v)|p∗sp∗s. | (4.1) |
Thus, combining with (4.1) and Sobolev inequality, we deduce that
Φλ(u,v)=1p||(u,v)||p−λp∗s∫RNK(x)(|u|p∗s+|v|p∗s)dx−λ∫RNF(x,u,v)dx≥1p||(u,v)||p−λεC5||(u,v)||p−λC6Cε||(u,v)||p∗s, |
where ε is small enough and C5,C6>0, thus (i) is proved because p∗s>p.
(ii) By (F3), we define the functional Ψλ∈C1(W,R) by
Ψλ(u,v)=1p||(u,v)||p−λl0∫RN(|u|d+|v|d)dx. |
Then
Φλ(u,v)≤Ψλ(u,v),forall(u,v)∈W. |
For any finite-dimensional subspace Y⊂W, we only need to prove
Ψλ(u,v)→−∞,as(u,v)∈Y,||(u,v)||→+∞. |
In fact, we have
Ψλ(u,v)=1p||(u,v)||p−λl0|(u,v)|dd. |
Since all norms in a finite dimensional space are equivalent and p<d<p∗s, thus (ii) holds.
(iii) From Corollary 3.1, for λ large and c small enough, Φλ satisfies (PS)c condition. Thus, we will find a special finite dimensional-subspace by which we construct sufficiently small minimax levels for Φλ when λ large enough.
Recall that
inf{∫R2N|φ(x)−φ(y)|p|x−y|N+psdxdy:φ∈C∞0(RN),|φ|d=1}=0,p<d<p∗s, |
see [40] for this proof. For any 0<ε<1, we can take φε∈C∞0(RN) with |φε|d=1, supp φε⊂Brε(0) and [φε]pp,s<ε.
Let
¯ωλ(x):=(ωλ(x),ωλ(x))=(φε(λ1psx),φε(λ1psx)). |
For t≥0, (F3) imply that
Φλ(t¯ωλ)≤2tpp∫∫R2N|ωλ(x)−ωλ(y)|p|x−y|N+psdxdy+2tpp∫RNλV(x)|ωλ|pdx−λ∫RNF(x,tωλ,tωλ)dx≤λ1−Nps{2tpp∫∫R2N|φε(x)−φε(y)|p|x−y|N+psdxdy+2tpp∫RNV(λ−1psx)|φε|pdx−2l0td∫RN|φε|ddx}≤λ1−Nps2l0(d−p)p(∫∫R2N|φε(x)−φε(y)|p|x−y|N+psdxdy+∫RNV(λ−1psx)|φε|pdxl0d)dd−p. |
Indeed, for t>0, define
g(t)=2tpp∫∫R2N|φε(x)−φε(y)|p|x−y|N+psdxdy+2tpp∫RNλV(λ−1psx)|φε|pdx−2l0td∫RN|φε|ddx. |
It is easy to show that t0=(∫∫R2N|φε(x)−φε(y)|p|x−y|N+psdxdy+∫RNV(λ−1psx)|φε|pdxl0d)1d−p is a maximum point of g and
maxt≥0g(t)=g(t0)=2l0(d−p)p(∫∫R2N|φε(x)−φε(y)|p|x−y|N+psdxdy+∫RNV(λ−1psx)|φε|pdxl0d)dd−p. |
Since V(0)=0 and supp φε⊂Brε(0), there exists Λε>0 such that
V(λ−1psx)<ε|φε|pp,∀|x|≤rε,λ>Λε. |
Hence, we have
maxt≥0Φλ(t¯ωλ)≤2l0(d−p)p(1l0d)dd−p(2ε)dd−pλ1−Nps,∀λ>Λε. |
Choose ε>0 such that
2l0(d−p)p(1l0d)dd−p(2ε)dd−p≤τ, |
and taking Λτ=Λε, from (ii), we can take ¯t large enough and define ˜ωλ=¯t¯ωλ, then we have
Φλ(˜ωλ)<0andmax0≤t≤1Φλ(t˜ωλ)≤τλ1−Nps. |
Proof of Theorem 2.1. From Lemma 4.1, for any 0<τ<α, there exists Λτ>0 such that for λ≥Λτ, we have
c=infη∈Γλmaxt∈[0,1]Φλ(η(t))≤τλ1−Nps, |
where Γλ={η∈C([0,1],W):η(0)=0,η(1)=˜ωλ}. Furthermore, in virtue of Corollary 3.1, we obtain that (PS)c condition hold for Φλ at c. Therefore, by the mountain pass theorem, there is (uλ,vλ)∈W such that Φ′λ(uλ,vλ)=0 and Φλ(uλ,vλ)=c.
Finally, we prove that (uλ,vλ) satisfies the estimates in Theorem 2.1.
Since (uλ,vλ) is a critical point of Φλ, there holds for θ∈[p,p∗s]
τλ1−Nps≥Φλ(uλ,vλ)−1θ⟨Φ′λ(uλ,vλ),(uλ,vλ)⟩≥(1p−1θ)||(uλ,vλ)||p+λ(1θ−1p∗s)∫RNK(x)(|uλ|p∗s+|vλ|p∗s)dx+λ(μθ−1)∫RNF(x,uλ,vλ)dx. |
Taking θ=μ, we get the estimate (2.2) and taking θ=p yields the estimate (2.3).
To obtain the multiplicity of critical points, we will adopt the index theory defined by the Krasnoselski genus.
Proof of Theorem 2.2. Denote the set of all symmetric (in the sense that −A=A) and closed subsets of A by ∑. For any A∈∑ let gen (A) be the Krasnoselski genus and
i(A)=mink∈Υgen(k(A)⋂∂Bρ), |
where Υ is the set of all odd homeomorphisms k∈C(W,W) and ρ is the number from Lemma 4.1. Then i is a version of Benci's pseudoindex [3]. (F4) implies that Φλ is even. Set
cλj:=infi(A)≥jsup(u,v)∈AΦλ(u,v),1≤j≤m. |
If cλj is finite and Φλ satisfies (PS)cλj condition, then we know that all cλj are critical values for Φλ.
Step 1. We show that Φλ satisfies (PS)cλj condition at all levels cλj<τλ1−Nps.
To complete the claim, we need to estimate the level cλj in special finite-dimensional subspaces.
Similar to proof in Lemma 4.1, for any m∈N, ε>0 and j=1,2,⋅⋅⋅,m, one can choose m functions φjε∈C∞0(RN) with supp φiε⋂ supp φjε=∅ if i≠j, |φjε|d=1 and [φjε]pp,s<ε.
Let rmε>0 be such that supp φjε⊂Brmε(0). Set
¯ωjλ(x):=(ωjλ(x),ωjλ(x))=(φjε(λ1psx),φjε(λ1psx)) |
and define
Fmλ:=Span{¯ω1λ,¯ω2λ,⋅⋅⋅,¯ωmλ}. |
Then i(Fmλ)=dimFmλ=m. Observe that for each ˜ω=∑mj=1tj¯ωjλ∈Fmλ,
Φλ(˜ω)=m∑j=1Φλ(tj¯ωjλ) |
and for tj>0
Φλ(tj¯ωjλ)≤2tpjp∫∫R2N|ωjλ(x)−ωjλ(y)|p|x−y|N+psdxdy+2tpjp∫RNλV(x)|ωjλ|pdx−λ∫RNF(x,tjωjλ,tjωjλ)dx≤λ1−Nps{2tpjp∫∫R2N|φjε(x)−φjε(y)|p|x−y|N+psdxdy+2tpjp∫RNV(λ−1psx)|φjε|pdx−2l0tdj∫RN|φjε|ddx}. |
Set
ηε:=max{|φjε|pp:j=1,2,⋅⋅⋅,m}. |
Since V(0)=0 and supp φjε⊂Brmε(0), there exists Λmε>0 such that
V(λ−1psx)<εηε,∀|x|≤rmε,λ>Λmε. |
Consequently, there holds
sup˜w∈FmλΦλ(˜w)≤ml0(2ε)dd−pλ1−Nps,∀λ>Λmε. |
Choose ε>0 small that ml0(2ε)dd−p<τ. Thus for any m∈N and τ∈(0,α), there exists Λmτ=Λmε such that λ>Λmτ, we can choose a m-dimensional subspace Fmλ with maxΦλ(Fmλ)≤τλ1−Nps and
cλ1≤cλ2≤⋅⋅⋅≤sup˜w∈FmλΦλ(˜w)≤τλ1−Nps. |
From Corollary 3.1, we know that Φλ satisfies the (PS) condition at all levels cλj. Then all cλj are critical values.
Step 2. We prove that (2.1) has at least m pairs of solutions by the mountain-pass theorem.
By Lemma 4.1, we know that Φλ satisfies the mountain pass geometry structure. From step 1, we note that Φλ also satisfies (PS)cλj condition at all levels cλj<τλ1−Nps. By the usual critical point theory, all cλj are critical levels and Φλ has at least m pairs of nontrivial critical points satisfying
a≤Φλ(u,v)≤τλ1−Nps. |
Thus, (2.1) has at least m pairs of solutions. Finally, as in the proof of Theorem 2.1, we know that these solutions satisfy the estimates (2.2) and (2.3).
In this paper, we have obtained the existence and multiplicity of standing wave solutions for a class of perturbed fractional p-Laplacian systems involving critical exponents by variational methods. In the next work, we will extend the study to the case of perturbed fractional p-Laplacian systems with electromagnetic fields.
The author is grateful to the referees and the editor for their valuable comments and suggestions.
The author declares no conflict of interest.
[1] |
Areerob Y, Cho KY, Oh WC (2017) Microwave assisted synthesis of graphene-Bi8La10O27-Zeolite nanocomposite with efficient photocatalytic activity towards organic dye degradation. J Photoch Photobio A 340: 157–169. https://doi.org/10.1016/j.jphotochem.2017.03.018 doi: 10.1016/j.jphotochem.2017.03.018
![]() |
[2] |
Karuppasamy P, Nisha NRN, Pugazhendhi A, et al. (2021) An investigation of transition metal doped TiO2 photocatalysts for the enhanced photocatalytic decoloration of methylene blue dye under visible light irradiation. J Environ Chem Eng 9: 105254. https://doi.org/10.1016/j.jece.2021.105254 doi: 10.1016/j.jece.2021.105254
![]() |
[3] |
Qaderi J, Mamat CR, Jalil AA (2021) Preparation and characterization of copper, iron, and nickel doped titanium dioxide photocatalysts for decolorization of methylene blue. Sains Malays 50: 135–149. https://doi.org/10.17576/jsm-2021-5001-14 doi: 10.17576/jsm-2021-5001-14
![]() |
[4] |
Javanbakht V, Ghoreishi SM (2017) Application of response surface methodology for optimization of lead removal from an aqueous solution by a novel superparamagnetic nanocomposite. Adsorpt Sci Technol 35: 241–60. https://doi.org/10.1177/0263617416674474 doi: 10.1177/0263617416674474
![]() |
[5] |
Aghajari N, Ghasemi Z, Younesi H, et al. (2019) Synthesis, characterization and photocatalytic application of Ag-doped Fe-ZSM-5@TiO2 nanocomposite for degradation of reactive red 195 (RR 195) in aqueous environment under sunlight irradiation. J Environ Health Sci 17: 219–232. https://doi.org/10.1007/s40201-019-00342-5 doi: 10.1007/s40201-019-00342-5
![]() |
[6] |
Znad H, Abbas K, Hena S, et al. (2018) Synthesis a novel multilamellar mesoporous TiO2/ZSM-5 for photo-catalytic degradation of methyl orange dye in aqueous media. J Environ Chem Eng 6: 218–227. https://doi.org/10.1016/j.jece.2017.11.077 doi: 10.1016/j.jece.2017.11.077
![]() |
[7] |
Shan AY, Ghazi TIM, Rashid SA (2010) Immobilisation of titanium dioxide onto supporting materials in heterogeneous photocatalysis: A review. Appl Catal A-Gen 389: 1–8. https://doi.org/10.1016/j.apcata.2010.08.053 doi: 10.1016/j.apcata.2010.08.053
![]() |
[8] |
Vaez Z, Javanbakht V (2020) Synthesis, characterization and photocatalytic activity of ZSM-5/ZnO nanocomposite modified by Ag nanoparticles for methyl orange degradation. J Photoch Photobiol A 388: 112064. https://doi.org/10.1016/j.jphotochem.2019.112064 doi: 10.1016/j.jphotochem.2019.112064
![]() |
[9] |
Badvi K, Javanbakht V (2021) Enhanced photocatalytic degradation of dye contaminants with TiO2 immobilized on ZSM-5 zeolite modified with nickel nanoparticles. J Clean Prod 280: 124518. https://doi.org/10.1016/j.jclepro.2020.124518 doi: 10.1016/j.jclepro.2020.124518
![]() |
[10] | Massoudinejad M, Sadani M, Gholami Z, et al. (2019) Optimization and modeling of photocatalytic degradation of Direct Blue 71 from contaminated water by TiO2 nanoparticles: Response surface methodology approach (RSM). Iran J Catal 9: 121–132. |
[11] |
Ghasemi Z, Younesi H, Zinatizadeh AA (2016) Preparation, characterization and photocatalytic application of TiO2/Fe-ZSM-5 nanocomposite for the treatment of petroleum refinery wastewater: Optimization of process parameters by response surface methodology. Chemosphere 159: 552–564. https://doi.org/10.1016/j.chemosphere.2016.06.058 doi: 10.1016/j.chemosphere.2016.06.058
![]() |
[12] |
Safajou H, Khojasteh H, Salavati-niasari M, et al. (2017) Enhanced photocatalytic degradation of dyes over graphene/Pd/TiO2 nanocomposites: TiO2 nanowires versus TiO2 nanoparticles. J Colloid Interf Sci 498: 423–432. https://doi.org/10.1016/j.jcis.2017.03.078 doi: 10.1016/j.jcis.2017.03.078
![]() |
[13] |
Bian H, Zhang Z, Xu X, et al. (2020) Photocatalytic activity of Ag/ZnO/AgO/TiO2 composite. Physica E 124: 114236. https://doi.org/10.1016/j.physe.2020.114236 doi: 10.1016/j.physe.2020.114236
![]() |
[14] |
Derakhshan-Nejad A, Rangkooy HA, Cheraghi M, et al. (2020) Removal of ethyl benzene vapor pollutant from the air using TiO2 nanoparticles immobilized on the ZSM-5 zeolite under UVradiation in lab scale. J Environ Health Sci 18: 201–209. https://doi.org/10.1007/s40201-020-00453-4 doi: 10.1007/s40201-020-00453-4
![]() |
[15] |
Pedroza-herrera G, Medina-ramírez IE, Lozano-álvarez JA, et al. (2020) Evaluation of the photocatalytic activity of copper doped TiO2 nanoparticles for the purification and/or disinfection of industrial effluents. Catal Today 341: 37–48. https://doi.org/10.1016/j.cattod.2018.09.017 doi: 10.1016/j.cattod.2018.09.017
![]() |
[16] |
Al-Mamun MR, Kader S, Islam MS, et al. (2019) Photocatalytic activity improvement and application of UV-TiO2 photocatalysis in textile wastewater treatment: A review. J Environ Chem Eng 7: 103248. https://doi.org/10.1016/j.jece.2019.103248 doi: 10.1016/j.jece.2019.103248
![]() |
[17] | Arifiyana D, Murwani IK (2013) Pengaruh doping Logam Fe pada CaF2 terhadap Struktur Ca1–xFexF2. J Sains dan Seni Pomits 2: 54–56. |
[18] |
Khatamian M, Hashemian S, Yavari A, et al. (2012) Preparation of metal ion (Fe3+ and Ni2+) doped TiO2 nanoparticles supported on ZSM-5 zeolite and investigation of its photocatalytic activity. Mater Sci Eng B-Adv 177: 1623–1627. https://doi.org/10.1016/j.mseb.2012.08.015 doi: 10.1016/j.mseb.2012.08.015
![]() |
[19] |
Khairy M, Zakaria W (2014) Effect of metal-doping of TiO2 nanoparticles on their photocatalytic activities toward removal of organic dyes. Egypt J Pet 23: 419–426. https://doi.org/10.1016/j.ejpe.2014.09.010 doi: 10.1016/j.ejpe.2014.09.010
![]() |
[20] |
Hosseini MS, Ebratkhahan M, Shayegan Z, et al. (2020) Investigation of the effective operational parameters of self-cleaning glass surface coating to improve methylene blue removal efficiency; application in solar cells. Sol Energy 207: 398–408. https://doi.org/10.1016/j.solener.2020.06.109 doi: 10.1016/j.solener.2020.06.109
![]() |
[21] |
Noorjahan M, Kumari VD, Subrahmanyam M, et al. (2004) A novel and efficient photocatalyst: TiO2-HZSM-5 combinate thin film. Appl Catal B-Environ 47: 209–213. https://doi.org/10.1016/j.apcatb.2003.08.004 doi: 10.1016/j.apcatb.2003.08.004
![]() |
[22] |
Mahalakshmi M, Priya SV, Arabindoo B, et al. (2009) Photocatalytic degradation of aqueous propoxur solution using TiO2 and Hβ zeolite-supported TiO2. J Hazard Mater 161: 336–343. https://doi.org/10.1016/j.jhazmat.2008.03.098 doi: 10.1016/j.jhazmat.2008.03.098
![]() |
[23] |
Loiola AR, Andrade JCRA, Sasaki JM, et al. (2012) Structural analysis of zeolite NaA synthesized by a cost-effective hydrothermal method using kaolin and its use as water softener. J Colloid Interf Sci 367: 34–39. https://doi.org/10.1016/j.jcis.2010.11.026 doi: 10.1016/j.jcis.2010.11.026
![]() |
[24] | Wuntu AD, Kamu VS, Kumaunang M (2019) Crystallization temperature of NaA zeolite prepared from gel and aluminum hydroxide. Chem Prog 4: 1–4. |
[25] | Afrozi AS, Salam R, R A, et al. (2016) Pengolahan limbah methylen blue secara fotokatalisis TiO2 dengan penambahan Fe dan zeolit. Prosiding Seminar Nasional Teknologi Pengelolaan Limbah, 29–36. Available from: http://repo-nkm.batan.go.id/7417/. |
[26] |
Pratama NA, Artsanti P (2019) Effect of aeration treatment on methylene blue removal using TiO2-zeolite. Proceeding International Conference on Science and Engineering 2: 219–224. https://doi.org/10.14421/icse.v2.89 doi: 10.14421/icse.v2.89
![]() |
[27] | Estiaty LM (2015) Synthesis and characterization of zeolite-TiO2 from modified natural zeolite. Teknol Miner dan Batubara 11: 181–190. |
[28] | Erwanto E, Yulinda Y, Nabela Q (2020) Pengaruh Penambahan Ion Nitrat (NO3–) terhadap Kinetika Fotodegradasi Zat Warna Metilen Biru Menggunakan Zeolit-TiO2. J Inovasi Teknik Kimia 5: 59–67. Available from: http://dx.doi.org/10.31942/inteka.v5i2.3812. |
[29] |
Septian DD, Sugiarti S (2019) Modifikasi Zeolit Alam Ende dengan Garam Logam serta Potensinya Sebagai Katalis Transformasi Glukosa Menjadi 5-Hidroksimetilfurfural (HMF). ALCHEMY J Penelit Kim 15: 203–218. https://doi.org/10.20961/alchemy.15.2.28180.203-218 doi: 10.20961/alchemy.15.2.28180.203-218
![]() |
[30] |
Rigo RT, Prigol C, Antunes Â, et al. (2013) Synthesis of ZK4 zeolite: An LTA-structured zeolite with a Si/Al ratio greater than 1. Mater Lett 102: 87–90. https://doi.org/10.1016/j.matlet.2013.03.120 doi: 10.1016/j.matlet.2013.03.120
![]() |
[31] |
Ginting SB, Sari DP, Iryani DA, et al. (2019) Sintesis ZeolitT Lynde Type-A (LTA) Dari Zeolit Alam Lampung (ZAL) Menggunakan Metode Step Change Temperature Of Hydrotermal Dengan Variasi SiO2/Al2O3 Diaplikasikan Untuk Dehidrasi Etanol. J Chem Process Eng 4: 32–44. https://doi.org/10.33536/jcpe.v4i1.324 doi: 10.33536/jcpe.v4i1.324
![]() |
[32] |
Asrori MR, Santoso A, Sumari S (2022) Initial defect product on immiscible mixture of palm oil: Ethanol by amphiphilic chitosan/Zeolite LTA as optimization of microemulsion fuel. Ind Crops Prod 180: 114727. https://doi.org/10.1016/j.indcrop.2022.114727 doi: 10.1016/j.indcrop.2022.114727
![]() |
[33] | Treacy MMJ, Higgins JB (2007) Collection of Simulated XRD Powder Patterns for Zeolites, 5 Eds., Elsevier. |
[34] |
Sescu AM, Favier L, Lutic D, et al. (2021) TiO2 doped with noble metals as an efficient solution for the photodegradation of hazardous organic water pollutants at ambient conditions. Water (Switzerland) 13: 1–18. https://doi.org/10.3390/w13010019 doi: 10.3390/w13010019
![]() |
[35] |
Sood S, Umar A, Mehta SK, et al. (2015) Highly effective Fe-doped TiO2 nanoparticles photocatalysts for visible-light driven photocatalytic degradation of toxic organic compounds. J Colloid Interf Sci 450: 213–223. https://doi.org/10.1016/j.jcis.2015.03.018 doi: 10.1016/j.jcis.2015.03.018
![]() |
[36] | Cacciato G, Zimbone M, Ruffino F, et al. (2016) TiO2 nanostructures and nanocomposites for sustainable photocatalytic water purification, In: Larramendy ML, Soloneski S, Green Nanotechnology-Overview and Further Prospects, Croatia: Janeza Trdine. https://doi.org/10.5772/62620 |
[37] |
Pratiwi E, Harlia H, Aritonang AB (2020) Sintesis TiO2 terdoping Fe3+ untuk Degradasi Rhodamin B Secara Fotokatalisis dengan Bantuan Sinar Tampak. Positron 10: 57–63. https://doi.org/10.26418/positron.v10i1.37739 doi: 10.26418/positron.v10i1.37739
![]() |
[38] |
Al-Harbi LM, Kosa SA, Abd El Maksod IH, et al. (2015) The photocatalytic activity of TiO2-zeolite composite for degradation of dye using synthetic UV and Jeddah sunlight. J Nanomater 16: 46. https://doi.org/10.1155/2015/565849 doi: 10.1155/2015/565849
![]() |
[39] |
Li H, Zhang W, Liu Y (2020) HZSM-5 zeolite supported boron-doped TiO2 for photocatalytic degradation of ofloxacin. J Mater Res Technol 9: 2557–2567. https://doi.org/10.1016/j.jmrt.2019.12.086 doi: 10.1016/j.jmrt.2019.12.086
![]() |
[40] |
Sahel K, Perol N, Chermette H, et al. (2007) Photocatalytic decolorization of Remazol Black 5 (RB5) and Procion Red MX-5B-Isotherm of adsorption, kinetic of decolorization and mineralization. Appl Catal B-Environ 77: 100–109. https://doi.org/10.1016/j.apcatb.2007.06.016 doi: 10.1016/j.apcatb.2007.06.016
![]() |
[41] |
Khan I, Saeed K, Zekker I, et al. (2022) Review on methylene blue: Its properties, uses, toxicity and photodegradation. Water 14: 242. https://doi.org/10.3390/w14020242 doi: 10.3390/w14020242
![]() |
[42] | Sutanto H, Wibowo S (2015) Semikonduktor Fotokatalis Seng Oksida dan Titania: Sintesis, Deposisi dan Aplikasi. Available from: http://eprints.undip.ac.id/49049/. |
[43] |
Asiltürk M, Sayılkan F, Arpaç E (2009) Effect of Fe3+ ion doping to TiO2 on the photocatalytic degradation of Malachite Green dye under UV and vis-irradiation. J Photoch Photobiol A 203: 64–71. https://doi.org/10.1016/j.jphotochem.2008.12.021 doi: 10.1016/j.jphotochem.2008.12.021
![]() |