Composites are widely used in different areas of engineering due to their remarkable mechanical properties; however, it has been evidenced that laminated composites exhibit certain vulnerabilities, particularly in interlaminar regions, which can lead to failures. To address this issue, efforts have been made to enhance interlaminar strength, with one notable approach being the incorporation of nano-reinforcements that serve as bridges between the laminate layers. Among these nano-reinforcements, carbon nanotubes (CNTs) have emerged as a highly promising material to mitigate the deficiencies in interlaminar zones. Despite their potential, integrating CNTs into structural laminates presents significant challenges. This research focuses on developing a strategy to effectively incorporate well-dispersed multi-walled carbon nanotubes (MWCNTs) into structural laminate composites to enhance interlaminar toughness. The study explored three different processes for integrating MWCNTs: hand lay-up, vacuum bagging, and liquid resin infusion, each with varying percentages of MWCNT addition. The aim was to determine the most efficient method for achieving uniform dispersion and improved mechanical properties. The results of this investigation demonstrated that well-dispersed MWCNTs significantly enhance the interlaminar and overall mechanical properties of composites. Each method showed varying degrees of success, but the overarching conclusion is clear: MWCNTs, when properly integrated, offer a viable solution to the inherent weaknesses of laminated composites. This advancement holds substantial promise for the future of composite materials, particularly in applications requiring enhanced durability and strength. The findings pave the way for further research and development in optimizing nano-reinforcement techniques, ultimately contributing to the creation of more robust and reliable composite structures.
Citation: Mateo Duarte, Johan A. Oquendo, Sebastián Vallejo, Johnattan Vargas, Yamile Cardona-Maya, Cesar A. Isaza. Fabrication of FRP/CNT hybrid laminate composites and their effect on interlaminar and mechanical properties[J]. AIMS Materials Science, 2024, 11(6): 1125-1144. doi: 10.3934/matersci.2024054
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Composites are widely used in different areas of engineering due to their remarkable mechanical properties; however, it has been evidenced that laminated composites exhibit certain vulnerabilities, particularly in interlaminar regions, which can lead to failures. To address this issue, efforts have been made to enhance interlaminar strength, with one notable approach being the incorporation of nano-reinforcements that serve as bridges between the laminate layers. Among these nano-reinforcements, carbon nanotubes (CNTs) have emerged as a highly promising material to mitigate the deficiencies in interlaminar zones. Despite their potential, integrating CNTs into structural laminates presents significant challenges. This research focuses on developing a strategy to effectively incorporate well-dispersed multi-walled carbon nanotubes (MWCNTs) into structural laminate composites to enhance interlaminar toughness. The study explored three different processes for integrating MWCNTs: hand lay-up, vacuum bagging, and liquid resin infusion, each with varying percentages of MWCNT addition. The aim was to determine the most efficient method for achieving uniform dispersion and improved mechanical properties. The results of this investigation demonstrated that well-dispersed MWCNTs significantly enhance the interlaminar and overall mechanical properties of composites. Each method showed varying degrees of success, but the overarching conclusion is clear: MWCNTs, when properly integrated, offer a viable solution to the inherent weaknesses of laminated composites. This advancement holds substantial promise for the future of composite materials, particularly in applications requiring enhanced durability and strength. The findings pave the way for further research and development in optimizing nano-reinforcement techniques, ultimately contributing to the creation of more robust and reliable composite structures.
In this paper, we study a class of Schrödinger-Poisson system with the following version
{−Δu+u+K(x)ϕu=|u|p−1u+μh(x)u in R3,−Δϕ=K(x)u2 in R3, | (1) |
where
{−Δu+u+ϕu=f(u) in R3,−Δϕ=u2 in R3, | (2) |
which has been derived from finding standing waves of the Schrödinger-Poisson system
{iψt−Δψ+ϕψ=f(ψ) in R3,−Δϕ=|ψ|2 in R3. |
A starting point of studying system (1) is the following fact. For any
ϕu(x)=14π∫R3K(y)|u(y)|2|x−y|dy |
such that
−Δu+u+K(x)ϕuu=|u|p−1u+μh(x)u,u∈H1(R3). | (3) |
Problem (3) can be also looked on as a usual semilinear elliptic equation with an additional nonlocal perturbation
From Lemma 2.1, we know that under the condition (A1), the following eigenvalue problem
−Δu+u=μh(x)u,u∈ H1(R3) |
has a first eigenvalue
F(u):=∫R3K(x)ϕu(x)|u(x)|2dx |
and introduce the energy functional
Iμ(u)=12‖u‖2+14F(u)−∫R3(1p+1|u|p+1+μ2h(x)u2)dx, |
where
⟨I′μ(u),v⟩=∫R3(∇u∇v+uv+K(x)ϕuuv−|u|p−1uv+μh(x)uv)dx. |
It is known that there is a one to one correspondence between solutions of (3) and critical points of
Theorem 1.1. Suppose that the assumptions of
The second result is about
Theorem 1.2. Under the assumptions of
The proofs of Theorem 1.1 and Theorem 1.2 are based on critical point theory. There are several difficulties in the road of getting critical points of
While for nonautonomous version of Schrödinger-Poisson system, only a few results are known in the literature. Jiang et.al.[21] have studied the following Schrödinger-Poisson system with non constant coefficient
{−Δu+(1+λg(x))u+θϕ(x)u=|u|p−2u in R3,−Δϕ=u2in R3,lim|x|→∞ϕ(x)=0, |
in which the authors prove the existence of ground state solution and its asymptotic behavior depending on
{−Δu+u+ϕu=V(x)|u|4u+μP(x)|u|q−2uin R3,−Δϕ=u2in R3,2<q<6,μ>0 |
has been studied by Zhao et al. [31]. Besides some other conditions, Zhao et. al. [31] assume that
{−Δu+u+L(x)ϕu=g(x,u) in R3,−Δϕ=L(x)u2 in R3. | (4) |
Besides some other conditions and the assumption
−Δu=a(x)|u|p−2u+˜μk(x)u, in RN, | (5) |
Costa et.al.[14] have proven the mountain pass geometry for the related functional of (5) when
This paper is organized as follows. In Section 2, we give some preliminaries. Special attentions are focused on several lemmas analyzing the Palais-Smale conditions of the functional
Notations. Throughout this paper,
Sp+1=infu∈H1(R3)∖{0}∫R3(|∇u|2+|u|2)dx(∫R3|u|p+1dx)2p+1. |
For any
In this section, we give some preliminary lemmas, which will be helpful to analyze the (PS) conditions for the functional
Lu(v)=∫R3K(x)u2vdx,v∈D1,2(R3), |
one may deduce from the Hölder and the Sobolev inequalities that
|Lu(v)|≤C‖u‖2L125‖v‖L6≤C‖u‖2L125‖v‖D1,2. | (6) |
Hence, for any
ϕu(x)=14π∫R3K(y)u2(y)|x−y|dy. |
Clearly
‖ϕu‖2D1,2=∫R3|∇ϕu|2dx=∫R3K(x)ϕuu2dx. | (7) |
Using (6) and (7), we obtain that
‖ϕu‖L6≤C‖ϕu‖D1,2≤C‖u‖2L125≤C‖u‖2. | (8) |
Then we deduce that
∫R3K(x)ϕu(x)u2(x)dx≤C‖u‖4. | (9) |
Hence on
F(u)=∫R3K(x)ϕu(x)u2(x)dx | (10) |
and
Iμ(u)=12‖u‖2+14F(u)−∫R3(1p+1|u|p+1+μ2h(x)u2)dx | (11) |
are well defined and
⟨I′μ(u),v⟩=∫R3(∇u∇v+uv+K(x)ϕuuv−|u|p−1uv−μh(x)uv)dx. |
The following Lemma 2.1 is a direct consequence of [28,Lemma 2.13].
Lemma 2.1. Assume that the hypothesis
Using the spectral theory of compact symmetric operators on Hilbert space, the above lemma implies the existence of a sequence of eigenvalues
−Δu+u=μh(x)u,inH1(R3) |
with
μ1=infu∈H1(R3)∖{0}‖u‖2∫R3h(x)u2dx,μn=infu∈S⊥n−1∖{0}‖u‖2∫R3h(x)u2dx, |
where
Next we analyze the
Definition 2.2. For
Lemma 2.3. Let
Proof. For
d+1+o(1)‖un‖=Iμ(un)−14⟨I′μ(un),un⟩=14‖un‖2−μ4∫R3h(x)u2ndx+p−34(p+1)∫R3|un|p+1dx. | (12) |
Note that
∫R3h(x)u2ndx≤(∫R3|un|p+1dx)2p+1(∫R3|h(x)|p+1p−1dx)p−1p+1≤2ϑp+1∫R3|un|p+1dx+p−1p+1ϑ−2p−1∫R3|h(x)|p+1p−1dx. |
Choosing
d+1+o(1)‖un‖≥14‖un‖2−D(p,h)μp+1p−1, | (13) |
where
The following lemma is a variant of Brezis-Lieb lemma. One may find the proof in [20].
Lemma 2.4. [20] If a sequence
limn→∞F(un)=F(u0)+limn→∞F(un−u0). |
Lemma 2.5. There is a
Iμ(u)>−p−12(p+1)Sp+1p−1p+1. |
Proof. Since
Iμ(u)=12(‖u‖2−μ∫R3h(x)u2dx)+14F(u)−1p+1∫R3|u|p+1dx=p−12(p+1)(‖u‖2−μ∫R3h(x)u2dx)+p−34(p+1)F(u). |
Noticing that
Iμ1(u)≥p−34(p+1)F(u)>0. |
Next, we claim: there is a
Iμ(u)>−p−12(p+1)Sp+1p−1p+1. |
Suppose this claim is not true, then there is a sequence
Iμ(n)(uμ(n))≤−p−12(p+1)Sp+1p−1p+1. |
Note that
Iμ(n)(uμ(n))+o(1)‖uμ(n)‖≥Iμ(n)(uμ(n))−14⟨I′μ(n)(uμ(n)),uμ(n)⟩≥14‖uμ(n)‖2−D(p,h)(μ(n))p+1p−1. |
This implies that
‖uμ(n)‖2−μ(n)∫R3h(x)(uμ(n))2dx≥(1−μ(n)μ1)‖uμ(n)‖2→0 |
because
Iμ(n)(uμ(n))=p−12(p+1)(‖uμ(n)‖2−μ(n)∫R3h(x)(uμ(n))2dx)+p−34(p+1)F(uμ(n)), |
we deduce that
lim infn→∞Iμ(n)(uμ(n))≥p−34(p+1)lim infn→∞F(uμ(n))≥0, |
which contradicts to the
Iμ(n)(uμ(n))≤−p−12(p+1)Sp+1p−1p+1. |
This proves the claim and the proof of Lemma 2.5 is complete.
Lemma 2.6. If
Proof. Let
d+o(1)=12‖un‖2−μ2∫R3h(x)u2ndx+14F(un)−1p+1∫R3|un|p+1dx |
and
⟨I′μ(un),un⟩=‖un‖2−μ∫R3h(x)u2ndx+F(un)−∫R3|un|p+1dx. |
Then we can prove that
‖un‖2=‖u0‖2+‖wn‖2+o(1), |
F(un)=F(u0)+F(wn)+o(1) |
and
‖un‖p+1Lp+1=‖u0‖p+1Lp+1+‖wn‖p+1Lp+1+o(1). |
Using Lemma 2.1, we also have that
d+o(1)=Iμ(un)=Iμ(u0)+12‖wn‖2+14F(wn)−1p+1∫R3|wn|p+1dx. | (14) |
Noticing
‖u0‖2−μ∫R3h(x)u20dx+F(u0)=∫R3|u0|p+1dx. | (15) |
Since
o(1)=‖un‖2−μ∫R3h(x)u2ndx+F(un)−∫R3|un|p+1dx. |
Combining this with (15) as well as Lemma 2.1, we obtain that
o(1)=‖wn‖2+F(wn)−∫R3|wn|p+1dx. | (16) |
Recalling the definition of
Suppose that the case (ⅰ) occurs. We may obtain from (16) that
‖wn‖2≥Sp+1(‖wn‖2+F(wn)−o(1))2p+1. |
Hence we get that for
‖wn‖2≥Sp+1p−1p+1+o(1). | (17) |
Therefore using (14), (16) and (17), we deduce that for
d+o(1)=Iμ(un)=Iμ(u0)+12‖wn‖2+14F(wn)−1p+1∫R3|wn|p+1dx.=Iμ(u0)+p−12(p+1)‖wn‖2+p−34(p+1)F(wn)>−p−12(p+1)Sp+1p−1p+1+p−12(p+1)‖wn‖2+p−34(p+1)F(wn)>0, | (18) |
which contradicts to the condition
Next we give a mountain pass geometry for the functional
Lemma 2.7. There exist
Proof. For any
u=te1+v,where∫R3(∇v∇e1+ve1)dx=0. | (19) |
Hence we deduce that
‖u‖2=‖∇(te1+v)‖2L2+‖te1+v‖2L2=t2+‖v‖2, | (20) |
μ2∫R3h(x)v2dx≤‖v‖2,μ1∫R3h(x)e21dx=‖e1‖2=1 | (21) |
and
μ1∫R3h(x)e1vdx=∫R3(∇v∇e1+ve1)dx=0. | (22) |
We first consider the case of
Iμ1(u)=12‖u‖2+14F(u)−μ12∫R3h(x)u2dx−1p+1∫R3|u|p+1dx=12‖te1+v‖2+14F(te1+v)−μ12∫R3h(x)(te1+v)2dx−1p+1∫R3|te1+v|p+1dx≥12(1−μ1μ2)‖v‖2+14F(te1+v)−1p+1∫R3|te1+v|p+1dx≥θ1‖v‖2+14F(te1+v)−C1|t|p+1−C2‖v‖p+1. |
Next we estimate the term
F(te1+v)=14π∫R3×R3K(x)K(y)(te1(y)+v(y))2(te1(x)+v(x))2|x−y|dydx. |
Since
(te1(y)+v(y))2(te1(x)+v(x))2=t4(e1(y))2(e1(x))2+(v(y))2(v(x))2+2t3(e1(y)(e1(x))2v(y)+e1(x)(e1(y))2v(x))+2t(e1(x)v(x)(v(y))2+e1(y)v(y)(v(x))2)+t2((e1(x))2(v(y))2+4e1(y)e1(x)v(y)v(x)+(e1(y))2(v(x))2), |
we know that
|∫R3×R3K(x)K(y)(e1(y)(e1(x))2v(y)+e1(x)(e1(y))2v(x))|x−y|dydx|≤C‖v‖; | (23) |
|∫R3×R3K(x)K(y)(2(e1(x))2(v(y))2+4e1(y)e1(x)v(y)v(x))|x−y|dydx|≤C‖v‖2 | (24) |
and
|∫R3×R3K(x)K(y)(e1(x)v(x)(v(y))2+e1(y)v(y)(v(x))2)|x−y|dydx|≤C‖v‖3. | (25) |
Hence
Iμ1(u)≥θ1‖v‖2+θ2|t|4−C1|t|p+1−C2‖v‖p+1−C3|t|3‖v‖−C4|t|2‖v‖2−C5|t|‖v‖3+14F(v), |
where
t2‖v‖2≤2p+1|t|p+1+p−1p+1‖v‖2(p+1)p−1, |
|t|‖v‖3≤1p+1|t|p+1+pp+1‖v‖3(p+1)p |
and for some
|t|3‖v‖≤1q0‖v‖q0+q0−1q0|t|3q0q0−1. |
Therefore we deduce that
Iμ1(u)≥θ1‖v‖2+θ2|t|4−C3q0‖v‖q0−C3(q0−1)q0|t|3q0q0−1−2C4p+1|t|p+1−(p−1)C4p+1‖v‖2(p+1)p−1−C5p+1|t|p+1−pC5p+1‖v‖3(p+1)p−C|t|p+1−C‖v‖p+1. | (26) |
From
Iμ1(u)≥θ3‖v‖2+θ4|t|4 |
provided that
Iμ1(u)≥θ5‖u‖4for‖u‖2≤˜θ25. | (27) |
Set
Iμ(u)=Iμ1(u)+12(μ1−μ)∫Rh(x)u2dx≥θ5‖u‖4−μ−μ12μ1‖u‖2=‖u‖2(θ5‖u‖2−μ−μ12μ1)≥‖u‖2(12θ5˜θ25−14θ5˜θ25)=14θ5˜θ25‖u‖2 |
for
In this section, our aim is to prove Theorem 1.1. For
Proposition 3.1. Let the assumptions
dμ1=infγ∈Γ1supt∈[0,1]Iμ1(γ(t)) |
with
Γ1={γ∈C([0,1],H1(R3)) : γ(0)=0, Iμ1(γ(1))<0}. |
Then
Before proving Proposition 3.1, we analyze the
Lemma 3.2. If the assumptions
Proof. It suffices to find a path
supt∈[0,1]Iμ1(γ(t))<p−12(p+1)Sp+1p−1p+1. |
Define
‖UR‖2−μ1∫h(x)U2Rdx+T2RF(UR)−Tp−1R∫Up+1Rdx=0. |
If
1T2R(‖UR‖2−μ1∫h(x)U2Rdx)+F(UR)=Tp−3R∫Up+1Rdx→∞, |
which is impossible either. Hence we only need to estimate
Iμ1(tUR)≤g(t)+CF(UR), |
where
g(t)=t22(‖UR‖2−μ1∫R3h(x)U2Rdx)−|t|p+1p+1∫R3Up+1Rdx. |
Noting that under the assumptions
F(UR)≤(∫R3K(x)65U125Rdx)56(∫R3ϕ6URdx)16≤C(∫R3e−65a|x+Rθ|(U(x))125dx)56≤C(∫R3e−65aRe(65a−125(1−ε))|x|dx)56≤Ce−aR | (28) |
since
∫R3h(x)U2Rdx=∫R3h(x+Rθ)U2(x)dx≥C∫R3e−b|x+Rθ|U2(x)dx≥C∫R3e−b|x|−bRU2(x)dx≥Ce−bR∫R3e−b|x|U2(x)dx≥Ce−bR. | (29) |
It is now deduced from (28) and (29) that
supt>0Iμ1(tUR)≤supt>0g(t)+Ce−aR≤p−12(p+1)(‖UR‖2−μ1∫R3h(x)U2Rdx)p+1p−1(‖UR‖−2Lp+1)p+1p−1+Ce−aR≤p−12(p+1)Sp+1p−1p+1−Ce−bR+o(e−bR)+Ce−aR<p−12(p+1)Sp+1p−1p+1 |
for
Lemma 3.3. Under the assumptions
Proof. Let
d+o(1)=12‖un‖2−μ12∫R3h(x)u2ndx+14F(un)−1p+1∫R3|un|p+1dx |
and
⟨I′μ1(un),un⟩=‖un‖2−μ1∫R3h(x)u2ndx+F(un)−∫R3|un|p+1dx. |
Hence we can deduce that
‖un‖2=‖u0‖2+‖wn‖2+o(1),F(un)=F(u0)+F(wn)+o(1) |
and
‖un‖p+1Lp+1=‖u0‖p+1Lp+1+‖wn‖p+1Lp+1+o(1). |
Since
d+o(1)=Iμ1(un)=Iμ1(u0)+12‖wn‖2+14F(wn)−1p+1∫R3|wn|p+1dx. | (30) |
From
‖u0‖2−μ1∫R3h(x)u20dx+F(u0)=∫R3|u0|p+1dx |
and then
Iμ1(u0)≥p−12(p+1)(‖u0‖2−μ1∫R3h(x)u20dx)+p−34(p+1)F(u0)≥0. |
Now using an argument similar to the proof of (16), we obtain that
o(1)=‖wn‖2+F(wn)−∫R3|wn|P+1dx. | (31) |
By the relation
Suppose that the case (I) occurs. Then up to a sbusequence, we may obtain from (31) that
‖wn‖2≥Sp+1(‖wn‖2+F(wn)−o(1))2p+1, |
which implies that for
‖wn‖2≥Sp+1p−1p+1+o(1). |
It is deduced from this and (30) that
Proof of Proposition 3.1. Since
Proof of Theorem 1.1. By Proposition 3.1, the
dμ1≤maxt∈[0,1]Iμ1(γn(t))<dμ1+1n. | (32) |
By Ekeland's variational principle [5], there exists
{dμ1≤maxt∈[0,1]Iμ1(γ∗n(t))≤maxt∈[0,1]Iμ1(γn(t))<dμ1+1n;maxt∈[0,1]‖γn(t)−γ∗n(t)‖<1√n; there existstn∈[0,1]such thatzn=γ∗n(tn) satisfies:Iμ1(zn)=maxt∈[0,1]Iμ1(γ∗n(t)),and‖I′μ1(zn)‖≤1√n. | (33) |
By Lemma 3.2 and Lemma 3.3 we get a convergent subsequence (still denoted by
In this section, we always assume the conditions
inf{Iμ(u) : u∈M},M={u∈H1(R3) : ⟨I′μ(u),u⟩=0} |
to get a ground state solution. But for
N={u∈H1(R3)∖{0}:I′μ(u)=0}. |
And then we consider the following minimization problem
c0,μ=inf{Iμ(u):u∈N}. | (34) |
Lemma 4.1. Let
d0,μ=inf‖u‖<ρIμ(u). |
Then the
Proof. Firstly, we prove that
Iμ(u)=12‖u‖2−μ2∫R3h(x)u2dx+14F(u)−1p+1∫R3|u|p+1dx≥12‖u‖2−μ2μ1‖u‖2−C‖u‖p+1>−∞ |
as
Iμ(te1)=t22‖e1‖2−μt22∫R3h(x)e21dx+t44F(e1)−tp+1p+1∫R3|e1|p+1dx. |
It is now deduced from
Iμ(te1)=t22(1−μμ1)‖e1‖2+t44F(e1)−tp+1p+1∫R3|e1|p+1dx. |
Since
Secondly, let
Iμ(un)→d0,μandI′μ(un)→0. |
Then we can prove that
We emphasize that the above lemma does NOT mean that
Lemma 4.2. For
Proof. By Lemma 4.1, we know that
For any
Iμ(u)=Iμ(u)−14⟨I′μ(u),u⟩≥14‖u‖2−D(p,h)μp+1p−1. |
Therefore the
Now let
Iμ(un)→c0,μandI′μ(un)=0. |
Since
Next, to analyze further the
Lemma 4.3. There exists
Proof. The proof is divided into two steps. In the first place, for
‖u‖2−μ1∫R3h(x)u2dx+F(u)=∫R3|u|p+1dx |
and hence
Iμ1(u)=p−12(p+1)(‖u‖2−μ1∫R3h(x)u2dx)+p−34(p+1)F(u). |
Since
I_{\mu_1}(u) \geq \frac{p-3}{4(p+1)}F(u) > 0. |
In the second place, denoted by
I'_{\mu^{(n)}}(u_{0,\mu^{(n)}}) = 0 |
and we also have that
c_{0, \mu^{(n)}} = I_{\mu^{(n)}}(u_{0,\mu^{(n)}}) < 0. |
Hence we deduce that
\begin{array}{rl} I_{\mu^{(n)}}(u_{0,\mu^{(n)}}) & = \frac{p-1}{2(p+1)} \left(\|u_{0,\mu^{(n)}}\|^2 - \mu^{(n)}\int_{\mathbb{R}^3} h(x)(u_{0,\mu^{(n)}})^2dx\right)\\ & \qquad \quad + \frac{p-3}{4(p+1)}F(u_{0,\mu^{(n)}}). \end{array} |
Using the definition of
\|u_{0,\mu^{(n)}}\|^2 - \mu^{(n)}\int_{\mathbb{R}^3} h(x)(u_{0,\mu^{(n)}})^2dx \geq \left(1-\frac{\mu^{(n)}}{\mu_1}\right)\|u_{0,\mu^{(n)}}\|^2\to 0 |
because
Claim. As
Proof of the Claim. From
\begin{eqnarray} o(1) + I_{\mu^{(n)}}(u_{0,\mu^{(n)}}) & = & I_{\mu^{(n)}}(\tilde{u}_0) + \frac{1}{2}\|\tilde{w}_n\|^2 \\ & & + \frac{1}{4}F(\tilde{w}_n) - \frac{1}{p+1}\int_{\mathbb{R}^3}|\tilde{w}_n|^{p+1}dx, \end{eqnarray} | (35) |
where
Now we distinguish two cases:
Suppose that the case (ⅰ) occurs. We may deduce from a proof similar to Lemma 2.6 that
I_{\mu^{(n)}}(u_{0,\mu^{(n)}}) + o(1) \geq I_{\mu_1}(\tilde{u}_0) + \frac{p-1}{2(p+1)} S_{p+1}^{\frac{p+1}{p-1}}, |
which is a contradiction because
Next we prove that
\liminf\limits_{n\to\infty} I_{\mu^{(n)}}(u_{0,\mu^{(n)}}) \geq \frac{p-3}{4(p+1)} F(\tilde{u}_0) > 0, |
which is also a contradiction since
Hence there is
Remark 4.4. The proof of Lemma 4.3 implies that (1) of Theorem 1.2 holds.
In the following, we are going to prove the existence of another nonnegative bound state solution of (3). To obtain this goal, we have to analyze further the
Lemma 4.5. Under the assumptions of
Proof. Let
d + o(1) = \frac{1}{2}\|u_n\|^2 - \frac{\mu}{2}\int_{\mathbb{R}^3}h(x)u_n^2dx + \frac{1}{4}F(u_n) -\frac{1}{p+1}\int_{\mathbb{R}^3}|u_n|^{p+1}dx |
and
\langle I'_\mu (u_n), u_n\rangle = \|u_n\|^2 - \mu\int_{\mathbb{R}^3}h(x)u_n^2dx + F(u_n) - \int_{\mathbb{R}^3}|u_n|^{p+1}dx. |
Similar to the proof in Lemma 2.3, we can deduce that
\|u_n\|^2 = \|u_0\|^2 + \|w_n\|^2 +o(1), |
F(u_n) = F(u_0) + F(w_n) +o(1) |
and
\|u_n\|^{p+1}_{L^{p+1}} = \|u_0\|^{p+1}_{L^{p+1}} + \|w_n\|^{p+1}_{L^{p+1}} +o(1). |
Using Lemma 2.1, we also have that
\begin{eqnarray} & & d + o(1) = I_\mu(u_n) = I_\mu(u_0) + \frac{1}{2}\| w_n\|^2\\ & & \qquad \quad + \frac{1}{4}F(w_n) -\frac{1}{p+1}\int_{\mathbb{R}^3}|w_n|^{p+1}dx. \end{eqnarray} | (36) |
Since
I_\mu(u_0) \geq c_{0,\mu} |
and
\|u_0\|^2 - \mu\int_{\mathbb{R}^3}h(x)u_0^2dx + \int_{\mathbb{R}^3}\phi_{u_0}u_0^2 = \int_{\mathbb{R}^3}|u_0|^{p+1}dx. |
Note that
o(1) = \|u_n\|^2 - \mu\int_{\mathbb{R}^3}h(x)u_n^2dx + F(u_n) - \int_{\mathbb{R}^3}|u_n|^{p+1}dx |
imply that
\begin{equation} o(1) = \| w_n\|^2 + F(w_n) -\int_{\mathbb{R}^3}|w_n|^{p+1}dx. \end{equation} | (37) |
Using
Suppose (I) occurs. Up to a subsequence, we may obtain from (37) that
\|w_n\|^2 \geq S_{p+1} \left(\|w_n\|^2 + F(w_n) - o(1)\right)^{\frac2{p+1}}. |
Hence we get that for
\begin{equation} \|w_n\|^2 \geq S_{p+1}^{\frac{p+1}{p-1}} + o(1). \end{equation} | (38) |
Therefore using (36) and (38), we deduce that for
\begin{eqnarray} & d & + o(1) = I_\mu(u_n) \\ & = & I_\mu(u_0) + \frac{1}{2}\|w_n\|^2 + \frac{1}{4}F(w_n) -\frac{1}{p+1}\int_{\mathbb{R}^3}|w_n|^{p+1}dx \\ & = & I_\mu(u_0) + \frac{p-1}{2(p+1)}\|w_n\|^2 + \frac{p-3}{4(p+1)}F(w_n) \\ & \geq & c_{0,\mu} + \frac{p-1}{2(p+1)}\|w_n\|^2 + \frac{p-3}{4(p+1)}F(w_n) \\ & > & c_{0,\mu} + \frac{p-1}{2(p+1)} S_{p+1}^{\frac{p+1}{p-1}}, \end{eqnarray} | (39) |
which contradicts to the assumption
Next, for the
d_{2, \mu} = \inf\limits_{\gamma \in \Gamma_2}\sup\limits_{t\in [0,1]}I_\mu(\gamma(t)) |
with
\Gamma_2 = \{ \gamma \in C([0,1], H^1(\mathbb{R}^3))\ : \ \gamma(0) = w_{0, \mu},\ I_\mu(\gamma(1)) < c_{0,\mu} \}. |
Lemma 4.6. Suppose that the conditions
d_{2, \mu} < c_{0,\mu} + \frac{p-1}{2(p+1)} S_{p+1}^{\frac{p+1}{p-1}}. |
Proof. It suffices to find a path starting from
\begin{array}{rl} I_\mu(w_0 + tU_R) & = \frac{1}{2}\left(\left\|w_0 + tU_R\right\|^2 - \mu \int_{\mathbb{R}^3} h(x)|w_0 + t U_R|^2dx\right) \\ & \quad + \frac{1}{4}F(w_0 + tU_R) - \frac1{p+1} \int_{\mathbb{R}^3}|w_0 + t U_R|^{p+1}dx \\ & = I_\mu(w_0) + A_1 + A_2 + A_3 +\frac{t^2}{2}\|U_R\|^2 - \frac{\mu}{2}\int_{\mathbb{R}^3}h(x)U_R^2dx, \end{array} |
where
A_1 = \langle w_0, tU_R\rangle - \mu t\int_{\mathbb{R}^3} h(x)w_0 U_R dx, |
A_2 = \frac{1}{4}\left(F(w_0 + tU_R) - F(w_0)\right) |
and
A_3 = \frac1{p+1} \int_{\mathbb{R}^3}\left(|w_0|^{p+1} - |w_0 + t U_R|^{p+1}\right)dx. |
Since
A_1 = \int_{\mathbb{R}^3}(w_0)^p t U_R dx - \int_{\mathbb{R}^3} K(x)\phi_{w_0}w_0 t U_R dx. |
From an elementary inequality:
(a+b)^q - a^q \geq b^q + q a^{q-1}b,\qquad q > 1,\ \ a > 0, b > 0, |
we deduce that
|A_3| \leq - \frac1{p+1}\int_{\mathbb{R}^3} |t U_R|^{p+1}dx - \int_{\mathbb{R}^3} |w_0|^p tU_Rdx. |
For the estimate of
\begin{array}{rl} |A_2| &\leq t\int_{\mathbb{R}^3}K(x)\phi_{w_0}w_0 U_R dx + \frac{t^2}{2}\int_{\mathbb{R}^3}K(x)\phi_{w_0} (U_R)^2 dx \\ & \qquad \quad + \frac{t^4}{4}\int_{\mathbb{R}^3} K(x)\phi_{U_R}(U_R)^2 dx + t^3\int_{\mathbb{R}^3} K(x)\phi_{U_R}w_0 U_R dx \\ & \qquad \qquad \quad + t^2\int_{\mathbb{R}^3\times\mathbb{R}^3}\frac{K(x)K(y)w_0(x)w_0(y)U_R(x)U_R(y)}{|x-y|}dxdy. \end{array} |
Since
\begin{array}{rl} & \int_{\mathbb{R}^3\times\mathbb{R}^3}\frac{K(x)K(y)w_0(x)w_0(y)U_R(x)U_R(y)}{|x-y|}dxdy \\ & = \int_{\mathbb{R}^3}K(x)\phi_{\sqrt{w_0 U_R}}w_0 U_R dx\\ & \leq \|\phi_{\sqrt{w_0 U_R}}\|_{L^6} \left(\int_{\mathbb{R}^3} K(x)^{\frac65}(w_0U_R)^{\frac65} dx\right)^{\frac56} \\ & \leq C \left(\int_{\mathbb{R}^3} e^{-\frac65 aR} e^{\left(\frac65 a - \frac65(1-\delta)\right)|x|} dx\right)^{\frac56}\\ & \leq C e^{-a R}\quad \hbox{since}\quad 0 < a < 1. \end{array} |
Similarly we can deduce that for
\int_{\mathbb{R}^3}K(x)\phi_{w_0}w_0 U_R dx \leq C e^{-a R}, \ \ \int_{\mathbb{R}^3}K(x)\phi_{w_0} (U_R)^2 dx \leq C e^{-a R}, |
\int_{\mathbb{R}^3} K(x)\phi_{U_R}(U_R)^2 dx \leq C e^{-a R}\ \ \hbox{and}\ \ \int_{\mathbb{R}^3} K(x)\phi_{U_R}w_0 U_R dx \leq C e^{-a R}. |
Since
\begin{array}{rl} & I_\mu(w_0 + tU_R) \leq I_\mu(w_0) +\frac{t^2}{2}\|U_R\|^2dx - \frac{\mu}{2}\int_{\mathbb{R}^3}h(x)U_R^2dx\\ & \qquad \qquad - \frac1{p+1}\int_{\mathbb{R}^3} |t U_R|^{p+1}dx+C e^{-a R}\\ & \leq I_\mu(w_0) + \frac{p-1}{2(p+1)}S_{p+1}^{\frac{p+1}{p-1}} +C e^{-a R} - C e^{-b R} + o(e^{-b R})\\ & < c_{0,\mu} + \frac{p-1}{2(p+1)}S_{p+1}^{\frac{p+1}{p-1}} \end{array} |
for
Proposition 4.7. Under the conditions (A1)-(A4), if
Proof. Since for
Proof of Theorem 1.2. The conclusion (1) of Theorem 1.2 follows from Lemma 4.3 and Remark 4.4. It remains to prove (2) of Theorem 1.2. By Proposition 4.7, the
\begin{equation} d_{2, \mu} \leq \max\limits_{t\in [0, 1]} I_{\mu} (\gamma_n(t)) < d_{2, \mu} + \frac{1}{n}. \end{equation} | (40) |
By Ekeland's variational principle, there exists
\begin{equation} \left\{ \begin{array}{ll} d_{2, \mu} \leq \max\limits_{t\in [0, 1]} I_{\mu} (\gamma_n^*(t))\leq \max\limits_{t\in [0, 1]} I_{\mu} (\gamma_n(t)) < d_{2, \mu} +\frac{1}{n};\\ \max\limits_{t\in [0, 1]}\|\gamma_n(t)-\gamma_n^*(t)\| < \frac{1}{\sqrt{n}}; \\ \hbox{ there exists} \; t_n\in [0, 1]\; \hbox{such that}\; z_n : = \gamma_n^*(t_n) \hbox{ satisfies}: \\ I_{\mu}(z_n) = \max\limits_{t\in [0, 1]} I_{\mu} (\gamma_n^*(t)), \;\hbox{and}\; \|I'_{\mu}(z_n)\|\leq\frac{1}{\sqrt{n}}. \end{array} \right. \end{equation} | (41) |
By Lemma 4.6 we get a convergent subsequence (still denoted by
Next, let
0 < \alpha \leq d_{2, \mu^{(n)}}\leq \max\limits_{s > 0}I_{\mu^{(n)}}(w_{0,\mu^{(n)}} + sU_R) |
and
I_{\mu^{(n)}}(w_{0,\mu^{(n)}} + sU_R) \leq \frac{p-1}{2(p+1)}S_{p+1}^{\frac{p+1}{p-1}} +C e^{-a R} - C e^{-b R} + o(e^{-b R}), |
\begin{equation} \limsup\limits_{n\to\infty} d_{2, \mu^{(n)}} \leq \frac{p-1}{2(p+1)}S_{p+1}^{\frac{p+1}{p-1}}. \end{equation} | (42) |
Next, similar to the proof in Lemma 2.3, we can deduce that
I_{\mu^{(n)}}(u_{2,\mu^{(n)}}) \geq I_{\mu_1}(\tilde{u}_2) + \frac{p-1}{2(p+1)}S_{p+1}^{\frac{p+1}{p-1}}, |
which contradicts to (42). Hence
The author thanks the unknown referee for helpful comments.
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1. | Chao Yang, Sharp condition of global well-posedness for inhomogeneous nonlinear Schrödinger equation, 2021, 14, 1937-1632, 4631, 10.3934/dcdss.2021136 |