Citation: Jie Xie, Xingyong Zhang, Cuiling Liu, Danyang Kang. Existence and multiplicity of solutions for a class of damped-like fractional differential system[J]. AIMS Mathematics, 2020, 5(5): 4268-4284. doi: 10.3934/math.2020272
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In this paper, we are concerned with the existence and multiplicity of weak solutions for the damped-like fractional differential system
{−ddt(p(t)(12 0D−ξt(u′(t))+12 tD−ξT(u′(t))))+r(t)(12 0D−ξt(u′(t))+12 tD−ξT(u′(t)))+q(t)u(t)=λ∇F(t,u(t)), a.e. t∈[0,T],u(0)=u(T)=0, | (1.1) |
where 0D−ξt and tD−ξT are the left and right Riemann-Liouville fractional integrals of order 0≤ξ<1, respectively, p,r,q∈C([0,T],R), L(t):=∫t0(r(s)/p(s))ds, 0<m≤e−L(t)p(t)≤M and q(t)−p(t)≥0 for a.e. t∈[0,T], u(t)=(u1(t),u2(t)⋯,un(t))T, (⋅)T denotes the transpose of a vector, n≥1 is a given positive integer, λ>0 is a parameter, ∇F(t,x) is the gradient of F with respect to x=(x1,⋯,xn)∈Rn, that is, ∇F(t,x)=(∂F∂x1,⋯,∂F∂xn)T, and there exists a constant δ∈(0,1) such that F:[0,T]ׯBδ0→R (where ¯Bδ0 is a closed ball in RN with center at 0 and radius δ) satisfies the following condition
(F0) F(t,x) is continuously differentiable in ¯Bδ0 for a.e. t∈[0,T], measurable in t for every x∈¯Bδ0, and there are a∈C(¯Bδ0,R+) and b∈L1([0,T];R+) such that
|F(t,x)|≤a(|x|)b(t) |
and
|∇F(t,x)|≤a(|x|)b(t) |
for all x∈¯Bδ0 and a.e. t∈[0,T].
In recent years, critical point theory has been extensively applied to investigate the existence and multiplicity of fractional differential equations. An successful application to ordinary fractional differential equations with Riemann-Liouville fractional integrals was first given by [1], in which they considered the system
{−ddt((12 0D−ξt(u′(t))+12 tD−ξT(u′(t))))=∇F(t,u(t)), a.e. t∈[0,T],u(0)=u(T)=0. | (1.2) |
They established the variational structure and then obtained some existence results for system (1.2). Subsequently, this topic attracted lots of attention and a series of existence and multiplicity results are established (for example, see [2,3,4,5,6,7,8,9,10,11,12] and reference therein). It is obvious that system (1.1) is more complicated than system (1.2) because of the appearance of damped-like term
r(t)(12 0D−ξt(u′(t))+12 tD−ξT(u′(t))). |
In [13], the variational functional for system (1.1) with λ=1 and N=1 has been established, and in [14], they investigated system (1.1) with λ=1, N=1 and an additional perturbation term. By mountain pass theorem and symmetric mountain pass theorem in [15] and a local minimum theorem in [16], they obtained some existence and multiplicity results when F satisfies superquadratic growth at infinity and some other reasonable conditions at origin.
In this paper, motivated by the idea in [17,18], being different from those in [13,14], we consider the case that F has subquadratic growth only near the origin and no any growth condition at infinity. Our main tools are Ekeland's variational principle in [19], a variant of Clark's theorem in [17] and a cut-off technique in [18]. We obtain that system (1.1) has a ground state weak solution uλ if λ is in some given interval and then some estimates of uλ are given, and when F(t,x) is also even about x near the origin for a.e. t∈[0,T], for each given λ>0, system (1.1) has infinitely many weak solutions {uλn} with ‖uλn‖→0 as n→∞. Next, we make some assumptions and state our main results.
(f0) There exist constants M1>0 and 0<p1<2 such that
F(t,x)≥M1|x|p1 | (1.3) |
for all x∈¯Bδ0 and a.e. t∈[0,T].
(f1) There exist constants M2>0 and 0<p2<p1<2 such that
F(t,x)≤M2|x|p2 | (1.4) |
for all x∈¯Bδ0 and a.e. t∈[0,T].
(f0)′ There exist constants M1>0 and 0<p1<1 such that (1.3) holds.
(f1)′ There exist constants M2>0 and 0<p2<p1<1 such that (1.4) holds.
(f2) There exists a constant η∈(0,2) such that
(∇F(t,x),x)≤ηF(t,x) |
for all x∈¯Bδ0 and a.e. t∈[0,T].
(f3) F(t,x)=F(t,−x) for all x∈¯Bδ0 and a.e. t∈[0,T].
Theorem 1.1. Suppose that (F0), (f0), (f1) and (f2) hold. If
0<λ≤min{|cos(πα)|2C,(1B)2−p2(δ2)2−p2|cos(πα)|2C}, |
then system (1.1) has a ground state weak solution uλ satisfying
‖uλ‖2−p2≤min{1,(1B)2−p2(δ2)2−p2},‖uλ‖2−p2∞≤{B2−p2,(δ2)2−p2}. |
where
B=T2α−12√mΓ(α)(2α−1)12,C=max{p,η}max{M1,M2}Tmaxt∈[0,T]e−L(t)max{Bp1,Bp2}. |
If (f0) and (f1) are replaced by the stronger conditions (f0)′ and (f1)′, then (f2) is not necessary in Theorem 1.1. So we have the following result.
Theorem 1.2. Suppose that (F0), (f0)′ and (f1)′ hold. If
0<λ≤min{|cos(πα)|3C∗,(1B)2−p1(δ2)2−p1|cos(πα)|3C∗}, |
then system (1.1) has a ground state weak solution uλ satisfying
‖uλ‖2−p1≤min{1,(1B)2−p1(δ2)2−p1},‖uλ‖2−p1∞≤{B2−p1,(δ2)2−p1}, |
where C∗=maxt∈[0,T]e−L(t)max{(1+ρ0)a0B∫T0b(t)dt,M1p1TBp1,ρ0M1TBp1+1}, a0=maxs∈[0,δ]a(s) and ρ0=maxs∈[δ2,δ]|ρ′(s)| and ρ(s)∈C1(R,[0,1]) is any given even cut-off function satisfying
ρ(s)={1,if |s|≤δ/2,0,if |s|>δ. | (1.5) |
Theorem 1.3. Suppose that (F0), (f0), (f1) and (f3) hold. Then for each λ>0, system (1.1) has a sequence of weak solutions {uλn} satisfying {uλn}→0, as n→∞.
Remark 1.1. Theorem 1.1-Theorem 1.3 still hold even if r(t)≡0 for all t∈[0,T], that is, the damped-like term disappears, which are different from those in [2,3,4,5,6,7,8,9,10,11,12] because all those assumptions with respect to x in our theorems are made only near origin without any assumption near infinity.
The paper is organized as follows. In section 2, we give some preliminary facts. In section 3, we prove Theorem 1.1–Theorem 1.3.
In this section, we introduce some definitions and lemmas in fractional calculus theory. We refer the readers to [1,9,20,21,22]. We also recall Ekeland's variational principle in [19] and the variant of Clark's theorem in [17].
Definition 2.1. (Left and Right Riemann-Liouville Fractional Integrals [22]) Let f be a function defined on [a,b]. The left and right Riemann-Liouville fractional integrals of order γ for function f denoted by aD−γtf(t) and tD−γbf(t), respectively, are defined by
aD−γtf(t)=1Γ(γ)∫ta(t−s)γ−1f(s)ds,t∈[a,b],γ>0,tD−γbf(t)=1Γ(γ)∫bt(s−t)γ−1f(s)ds,t∈[a,b],γ>0. |
provided the right-hand sides are pointwise defined on [a,b], where Γ>0 is the Gamma function.
Definition 2.2. ([22]) For n∈N, if γ=n, Definition 2.1 coincides with nth integrals of the form
aD−ntf(t)=1(n−1)!∫ta(t−s)n−1f(s)ds,t∈[a,b],n∈N,tD−nbf(t)=1(n−1)!∫bt(t−s)n−1f(s)ds,t∈[a,b],n∈N. |
Definition 2.3. (Left and Right Riemann-Liouville Fractional Derivatives [22]) Let f be a function defined on [a,b]. The left and right Riemann-Liouville fractional derivatives of order γ for function f denoted by aDγtf(t) and tDγbf(t), respectively, are defined by
aDγtf(t)=dndtnaDγ−ntf(t)=1Γ(n−γ)dndtn(∫ta(t−s)n−γ−1f(s)ds),tDγbf(t)=(−1)ndndtntDγ−nbf(t)=(−1)nΓ(n−γ)dndtn(∫bt(s−t)n−γ−1f(s)ds). |
where t∈[a,b],n−1≤γ<n and n∈N. In particular, if 0≤γ<1, then
aDγtf(t)=ddtaDγ−1tf(t)=1Γ(1−γ)ddt(∫ta(t−s)−γf(s)ds),t∈[a,b],tDγbf(t)=−ddttDγ−1bf(t)=−1Γ(1−γ)ddt(∫bt(s−t)−γf(s)ds),t∈[a,b]. |
Remark 2.1. ([9,13]) The left and right Caputo fractional derivatives are defined by the above-mentioned Riemann-Liuville fractional derivative. In particular, they are defined for function belonging to the space of absolutely continuous functions, which we denote by AC([a,b],RN). ACk([a,b],RN)(k=0,1,...) are the space of the function f such that f∈Ck([a,b],RN). In particular, AC([a,b],RN)=AC1([a,b],RN).
Definition 2.4. (Left and Right Caputo Fractional Derivatives [22]) Let γ≥0 and n∈N.
(ⅰ) If γ∈(n−1,n) and f∈ACn([a,b],RN), then the left and right Caputo fractional derivatives of order γ for function f denoted by caDγtf(t) and ctDγbf(t), respectively, exist almost everywhere on [a,b]. caDγtf(t) and ctDγbf(t) are represented by
caDγtf(t)=aDγ−ntfn(t)=1Γ(n−γ)(∫ta(t−s)n−γ−1f(n)(s)ds),ctDγbf(t)=(−1)n tDγ−nbfn(t)=(−1)nΓ(n−γ)(∫bt(s−t)n−γ−1f(n)(s)ds), |
respectively, where t∈[a,b]. In particular, if 0<γ<1, then
caDγtf(t)=aDγ−1tf′(t)=1Γ(1−γ)(∫ta(t−s)−γf′(s)ds),t∈[a,b],ctDγbf(t)=−tDγ−1bf′(t)=−1Γ(1−γ)(∫bt(s−t)−γf′(s)ds),t∈[a,b]. |
(ⅱ) If γ=n−1 and f∈ACn([a,b],RN), then caDγtf(t) and ctDγbf(t) are represented by
caDn−1tf(t)=f(n−1)(t),t∈[a,b],ctDn−1bf(t)=(−1)n−1f(n−1)(t),t∈[a,b]. |
In particular, caD0tf(t)= ctD0bf(t)=f(t), t∈[a,b].
Lemma 2.1. ([22]) The left and right Riemann-Liouville fractional integral operators have the property of a semigroup, i.e.
aD−γ1t(aD−γ2tf(t))=aD−γ1−γ2tf(t),tD−γ1b(tD−γ2bf(t))=tD−γ1−γ2bf(t),∀γ1,γ2>0, |
in any point t∈[a,b] for continuous function f and for almost every point in [a,b] if the function f∈L1([a,b],RN).
For 1≤r<∞, define
‖u‖Lr=(∫T0|u(t)|rdt)1r | (2.1) |
and
‖u‖∞=maxt∈[0,T]|u(t)|. | (2.2) |
Definition 2.5. ([1]) Let 0<α≤1 and 1<p<∞. The fractional derivative space Eα,p0 is defined by closure of C∞0([0,T],RN) with respect to the norm
‖u‖α,p=(∫T0|u(t)|pdt+∫T0|c0Dαtu(t)|pdt)1p. | (2.3) |
Remark 2.2. ([9]) Eα,p0 is the space of functions u∈Lp([0,T],RN) having an α-order Caputo fractional derivative c0Dαtu(t)∈Lp([0,T],RN) and u(0)=u(T)=0.
Lemma 2.2. ([1]) Let 0<α≤1 and 1<p<∞. Eα,p0 is a reflexive and separable Banach space.
Lemma 2.3. ([1]) Assume that 1<p<∞ and α>1p. Then Eα,p0 compactly embedding in C([0,T],RN).
Lemma 2.4. ([1]) Let 0<α≤1 and 1<p<∞. For all u∈Eα,p0, we have
‖u‖Lp≤TαΓ(α+1)‖c0Dαtu‖Lp. | (2.4) |
Moreover, if α>1p and 1p+1q=1, then
‖u‖∞≤Tα−1pΓ(α)((α−1)q+1)1q‖c0Dαtu‖Lp. | (2.5) |
Definition 2.6. ([13]) Assume that X is a Banach space. An operator A:X→X∗ is of type (S)+ if, for any sequence {un} in X, un⇀u and lim supn→+∞⟨A(un),un−u⟩≤0 imply un→u.
Let φ:X→R. A sequence {un}⊂X is called (PS) sequence if the sequence {un} satisfies
φ(un) is bounded, φ′(un)→0. |
Furthermore, if every (PS) sequence {un} has a convergent subsequence in X, then one call that φ satisfies (PS) condition.
Lemma 2.5. ([19]) Assume that X is a Banach space and φ:X→R is Gˆateaux differentiable, lower semi-continuous and bounded from below. Then there exists a sequence {xn} such that
φ(xn)→infXφ,‖φ′(xn)‖∗→0. |
Lemma 2.6. ([17]) Let X be a Banach space, φ∈C1(X,R). Assume φ satisfies the (PS) condition, is even and bounded below, and φ(0)=0. If for any k∈N, there exist a k-dimensional subspace Xk of X and ρk>0 such that supXk∩Spkφ<0, where Sρ={u∈X|‖u‖=ρ}, then at least one of the following conclusions holds.
(ⅰ) There exist a sequence of critical points {uk} satisfying φ(uk)<0 for all k and ‖uk‖→0 as k→∞.
(ⅱ) There exists a constant r>0 such that for any 0<a<r there exists a critical point u such that ‖u‖=a and φ(u)=0.
Remark 2.3. ([17]) Lemma 2.6 implies that there exist a sequence of critical points uk≠0 such that φ(uk)≤0, φ(uk)→0 and ‖uk‖→0 as k→∞.
Now, we establish the variational functional defined on the space Eα,20 with 12<α≤1. We follow the same argument as in [13] where the one-dimensional case N=1 and λ=1 for system (1.1) was investigated. For reader's convenience, we also present the details here. Note that L(t):=∫t0(r(s)/p(s))ds,0<m≤e−L(t)p(t)≤M and q(t)−p(t)≥0 for a.e. t∈[0,T]. Then system (1.1) is equivalent to the system
{−ddt(e−L(t)p(t)(12 0D−ξt(u′(t))+12 tD−ξT(u′(t))))+e−L(t)q(t)u(t)=λe−L(t)∇F(t,u), a.e. t∈[0,T],u(0)=u(T)=0. | (2.6) |
By Lemma 2.1, for every u∈AC([0,T],RN), it is easy to see that system (2.6) is equivalent to the system
{−ddt[e−L(t)p(t)(12 0D−ξ2t(0D−ξ2tu′(t))+12 tD−ξ2T(tD−ξ2Tu′(t)))]+e−L(t)q(t)u(t)=λe−L(t)∇F(t,u), a.e. t∈[0,T],u(0)=u(T)=0, | (2.7) |
where ξ∈[0,1).
By Definition 2.4, we obtain that u∈AC([0,T],RN) is a solution of problem (2.7) if and only if u is a solution of the following system
{−ddt(e−L(t)p(t)(12 0Dα−1t(c0Dαtu(t))−12 tDα−1T(ctDαTu(t))))+e−L(t)q(t)u(t)=λe−L(t)∇F(t,u),u(0)=u(T)=0, | (2.8) |
for a.e. t∈[0,T], where α=1−ξ2∈(12,1]. Hence, the solutions of system (2.8) correspond to the solutions of system (1.1) if u∈AC([0,T],RN).
In this paper, we investigate system (2.8) in the Hilbert space Eα,20 with the corresponding norm
‖u‖=(∫T0e−L(t)p(t)(|c0Dαtu(t)|2+|u(t)|2)dt)12. |
It is easy to see that ‖u‖ is equivalent to ‖u‖α,2 and
m∫T0|c0Dαtu(t)|2dt≤∫T0e−L(t)p(t)|c0Dαtu(t)|2dt≤M∫T0|c0Dαtu(t)|2dt. |
So
‖u‖L2≤Tα√mΓ(α+1)(∫T0e−L(t)p(t)|c0Dαtu(t)|2dt)12, |
and
‖u‖∞≤B‖u‖, | (2.9) |
where
B=T2α−12√mΓ(α)(2α−1)12>0. |
(see [13]).
Lemma 2.7. ([13]) If 12<α≤1, then for every u∈Eα,20, we have
|cos(πα)|‖u‖2≤−∫T0e−L(t)p(t)(c0Dαtu(t),ctDαTu(t))dt+∫T0e−L(t)p(t)|u(t)|2dt≤max{Mm|cos(πα)|,1}‖u‖2. | (2.10) |
We follow the idea in [17] and [18]. We first modify and extend F to an appropriate ˜F defined by
˜F(t,x)=ρ(|x|)F(t,x)+(1−ρ(|x|))M1|x|p1, for all x∈RN, |
where ρ is defined by (1.5).
Lemma 3.1. Let (F0), (f0), (f1) (or (f0)′, (f1)′), (f2) and (f3) be satisfied. Then
(˜F0) ˜F(t,x) is continuously differentiable in x∈RN for a.e. t∈[0,T], measurable in t for every x∈RN, and there exists b∈L1([0,T];R+) such that
|˜F(t,x)|≤a0b(t)+M1|x|p1,|∇˜F(t,x)|≤(1+ρ0)a0b(t)+M1p1|x|p1−1+ρ0M1|x|p1 |
for all x∈RN and a.e. t∈[0,T];
(˜f0) ˜F(t,x)≥M1|x|p1 for all x∈RN and a.e. t∈[0,T];
(˜f1) ˜F(t,x)≤max{M1,M2}(|x|p1+|x|p2) for all x∈RN and a.e. t∈[0,T];
(˜f2) (∇˜F(t,x),x)≤θ˜F(t,x) for all x∈RN and a.e. t∈[0,T], where θ=max{p1,η};
(˜f3) ˜F(t,x)=˜F(t,−x) for all x∈RN and a.e. t∈[0,T].
Proof. We only prove (˜f0), (˜f1) and (˜f2). (˜F0) can be proved by a similar argument by (F0). By the definition of ˜F(t,x), (f0) and (f1) (or (f0)′ and (f1)′), we have
M1|x|p1≤˜F(t,x)=F(t,x)≤M2|x|p2, if |x|≤δ/2, |
˜F(t,x)=M1|x|p1, if |x|>δ, |
˜F(t,x)≤F(t,x)+M1|x|p1≤M1|x|p1+M2|x|p2, if δ/2<|x|≤δ |
and
˜F(t,x)≥ρ(|x|)M1|x|p1+(1−ρ(|x|))M1|x|p1=M1|x|p1, if δ/2<|x|≤δ. |
Hence, (˜f1) holds. Note that
θ˜F(t,x)−(∇˜F(t,x),x)=ρ(|x|)(θF(t,x)−(∇F(t,x),x))+(θ−p1)(1−ρ(|x|))M1|x|p1−|x|ρ′(|x|)(F(t,x)−M1|x|p1). |
It is obvious that the conclusion holds for 0≤|x|≤δ/2 and |x|>δ. If δ/2<|x|≤δ, by using θ≥p1, (f2), (˜f1) and the fact sρ′(s)≤0 for all s∈R, we can get the conclusion (˜f2). Finally, since ρ(|x|) is even for all x∈RN, by (f3) and the definition of ˜F(t,x), it is easy to get (˜f3).
Remark 3.1. From the proof of Lemma 3.1, it is easy to see that (F0), (f0) (or (f0)′) and (f1) (or (f1)′) independently imply (˜F0), (˜f0) and (˜f1), respectively.
Consider the modified system
{−ddt(e−L(t)p(t)(12 0Dα−1t(c0Dαtu(t))−12 tDα−1T(ctDαTu(t))))+e−L(t)q(t)u(t)=λe−L(t)∇˜F(t,u),u(0)=u(T)=0, | (3.1) |
for a.e. t∈[0,T], where α=1−ξ2∈(12,1].
If the equality
∫T0e−L(t)[−12p(t)((c0Dαtu(t),ctDαTv(t))+(ctDαTu(t),c0Dαtv(t)))+p(t)(u(t),v(t))+(q(t)−p(t))(u(t),v(t))−λ(∇˜F(t,u(t)),v(t))]dt=0 |
holds for every v∈Eα,20, then we call u∈Eα,20 is a weak solution of system (3.1).
Define the functional ˜J:Eα,20→R by
˜J(u)=∫T0e−L(t)[12p(t)(−(c0Dαtu(t),ctDαTu(t))+|u(t)|2)+12(q(t)−p(t))|u(t)|2−λ˜F(t,u(t))]dt, for all u∈Eα,20. |
Then (˜F0) and Theorem 6.1 in [9] imply that ˜J∈C1(Eα,20,R), and for every u,v∈Eα,20, we have
⟨˜J′(u),v⟩=∫T0e−L(t)[−12p(t)((c0Dαtu(t),ctDαTv(t))+(ctDαTu(t),c0Dαtv(t)))+p(t)(u(t),v(t))+(q(t)−p(t))(u(t),v(t))−λ(∇˜F(t,u(t)),v(t))]dt. |
Hence, a critical point of ˜J(u) corresponds to a weak solution of problem (3.1).
Let
⟨Au,v⟩:=∫T0e−L(t)[−12p(t)((c0Dαtu(t),ctDαTv(t))+(ctDαTu(t),c0Dαtv(t)))+p(t)(u(t),v(t))+(q(t)−p(t))(u(t),v(t))]dt. |
Lemma 3.2. ([13])
γ1‖u‖2≤⟨Au,u⟩≤γ2‖u‖2,for all u∈Eα,20, | (3.2) |
where γ1=|cos(πα)| and γ2=(max{Mm|cosπα|,1}+maxt∈[0,T](q(t)−p(t))).
Lemma 3.3. Assume that (F0), (f0) and (f1) (or (f0)′ and (f1)′) hold. Then for each λ>0, ˜J is bounded from below on Eα,20 and satisfies (PS) condition.
Proof. By (˜f1), (2.9) and (3.2), we have
˜J(u)=12⟨Au,u⟩−λ∫T0e−L(t)˜F(t,u(t))dt≥γ12‖u‖2−λmax{M1,M2}∫T0e−L(t)(|u(t)|p1+|u(t)|p2)dt≥γ12‖u‖2−λmax{M1,M2}Tmaxt∈[0,T]e−L(t)(‖u‖p1∞+‖u‖p2∞)≥γ12‖u‖2−λmax{M1,M2}Tmaxt∈[0,T]e−L(t)[Bp1‖u‖p1+Bp2‖u‖p2]. |
It follows from 0<p2<p1<2 that
˜J(u)→+∞, as ‖u‖→∞. |
Hence, ˜J is coercive and then is bounded from below. Now we prove that ˜J satisfies the (PS) condition. Assume that {un} is a (PS) sequence of ˜J, that is,
˜J(un) is bounded, ˜J′(un)→0. | (3.3) |
Then by the coercivity of ˜J and (3.3), there exists C0>0 such that ‖un‖≤C0 and then by Lemma 2.3, there exists a subsequence (denoted again by {un}) such that
un⇀u, weakly in Eα,20, | (3.4) |
un→u, a.e. in C([0,T],R). | (3.5) |
Therefore, the boundness of {un} and (3.3) imply that
|⟨˜J′(un),un−u⟩|≤‖˜J′(un)‖(Eα,20)∗‖un−u‖,≤‖˜J′(un)‖(Eα,20)∗(‖un‖+‖u‖)→0, | (3.6) |
where (Eα,20)∗ is the dual space of Eα,20, and (˜F0), (2.9) together with (3.5) imply that
|λ∫T0(∇˜F(t,un(t)),un(t)−u(t))dt|≤λ∫T0|∇˜F(t,un(t))||(un(t)−u(t))|dt≤λ‖un−u‖∞∫T0[(1+ρ0)a0b(t)+M1p1|un(t)|p1−1+ρ0M1|un(t)|p1]dt≤λ‖un−u‖∞[(1+ρ0)a0∫T0b(t)dt+M1p1TBp1−1Cp1−10+M1Tρ0Bp1Cp10]→0. | (3.7) |
Note that
⟨˜J′(un),un−u⟩=⟨Aun,un−u⟩−λ∫T0(∇˜F(t,un(t)),un(t)−u(t))dt. |
Then (3.6) and (3.7) imply that limn→∞⟨Aun,un−u⟩=0. Moreover, by (3.4), we have
limn→∞⟨Au,un−u⟩=0. |
Therefore
limn→∞⟨Aun−Au,un−u⟩=0. |
Since A is of type (S)+ (see [13]), by Definition 2.6, we obtain un→u in Eα,20.
Define a Nehari manifold by
Nλ={u∈Eα,20/{0}|⟨˜J′λ(u),u⟩=0}. |
Lemma 3.4. Assume that (F0) and (f0) (or (f0)′) hold. For each λ>0, ˜Jλ has a nontrivial least energy (ground state) weak solution uλ, that is, uλ∈Nλ and ˜Jλ(uλ)=infNλ˜Jλ. Moreover, the least energy can be estimated as follows
˜Jλ(uλ)≤Gλ:=(p1/γ2)p12−p1[λM1mint∈[0,T]e−L(t)∫T0|w0(t)|p1dt]22−p1(p1−2)2. |
where w0=w‖w‖, and w=(TπsinπtT,0,⋯,0)∈Eα,20.
Proof. By Lemma 3.3 and ˜J∈C1(Eα,20,R), for each λ>0, Lemma 2.5 implies that there exists some uλ∈Eα,20 such that
˜J(uλ)=infv∈Eα,20˜J(v)and ˜J′(uλ)=0. | (3.8) |
By (3.2) and (˜f0), we have
˜Jλ(sw0)=12⟨A(sw0),sw0⟩−λ∫T0e−L(t)˜F(t,sw0(t))dt≤γ22s2‖w0‖2−λ∫T0e−L(t)M1|sw0(t)|p1dt≤γ22s2−λM1mint∈[0,T]e−L(t)sp1∫T0|w0(t)|p1dt. | (3.9) |
for all s∈[0,∞). Define g:[0,+∞)→R by
g(s)=γ22s2−λM1mint∈[0,T]e−L(t)sp1∫T0|w0(t)|p1dt. |
Then g(s) achieves its minimum at
s0,λ=(p1λM1mint∈[0,T]e−L(t)∫T0|w0(t)|p1dtγ2)12−p1 |
and
g(s0,λ)=(p1/γ2)p12−p1[λM1mint∈[0,T]e−L(t)∫T0|w0(t)|p1dt]22−p1(p1−2)2. |
Note that p1<2. So g(s0,λ)<0. Hence, (3.9) implies that
˜Jλ(uλ)=infv∈Eα,20˜Jλ(v)≤˜Jλ(s0,λw0)≤g(s0,λ)<0=˜Jλ(0) |
and then uλ≠0 which together with (3.8) implies that uλ∈Nλ and ˜Jλ(uλ)=infNλ˜Jλ.
Lemma 3.5. Assume that (F0), (f1) and (f2) hold. If 0<λ≤|cos(πα)|2C, then the following estimates hold
‖uλ‖2−p2≤2λC|cos(πα)|,‖uλ‖2−p2∞≤2λCB2−p2|cos(πα)|. |
Proof. It follows from Lemma 3.1, (2.9) and ⟨˜J′(uλ),uλ⟩=0 that
∫T0e−L(t)[−p(t)(c0Dαtuλ(t),ctDαTuλ(t))+p(t)(uλ(t),uλ(t))+(q(t)−p(t))(uλ(t),uλ(t))]dt=λ∫T0e−L(t)(∇˜F(t,uλ(t)),uλ(t))dt≤λθ∫T0e−L(t)˜F(t,uλ(t))dt≤λθmax{M1,M2}maxt∈[0,T]e−L(t)∫T0(|uλ(t)|p1+|uλ(t)|p2)dt≤λθmax{M1,M2}Tmaxt∈[0,T]e−L(t)(‖uλ‖p1∞+‖uλ‖p2∞)≤λθmax{M1,M2}Tmaxt∈[0,T]e−L(t)[Bp1‖uλ‖p1+Bp2‖uλ‖p2]≤λC(‖uλ‖p1+‖uλ‖p2). | (3.10) |
We claim that ‖uλ‖≤1 uniformly for all 0<λ≤|cos(πα)|2C. Otherwise, we have a sequence of {λn≤|cos(πα)|2C} such that ‖uλn‖>1. Thus ‖uλn‖p2<‖uλn‖p1 since p2<p1<2. By (2.10) and (3.10), we obtain
∫T0e−L(t)[−p(t)(c0Dαtuλn(t),ctDαTuλn(t))+p(t)(uλn(t),uλn(t))+(q(t)−p(t))(uλn(t),uλn(t))]dt≥|cos(πα)|‖uλn‖2+∫T0e−L(t)(q(t)−p(t))|uλn(t)|2dt. | (3.11) |
By (3.10) and (3.11), we obtain
|cos(πα)|‖uλn‖2+∫T0e−L(t)(q(t)−p(t))|uλn(t)|2dt≤λnC(‖uλn‖p1+‖uλn‖p2). |
Since q(t)−p(t)>0,
|cos(πα)|‖uλn‖2≤λnC(‖uλn‖p1+‖uλn‖p2)≤2λnC‖uλn‖p1. |
Then
‖uλn‖2−p1≤2λnC|cos(πα)|≤1, |
which contradicts with the assumption ‖uλn‖>1. Now, from (3.10) we can get
|cos(πα)|‖uλ‖2≤λC(‖uλ‖p1+‖uλ‖p2)≤2λC‖uλ‖p2. |
So
‖uλ‖2−p2≤2λC|cos(πα)|. |
By (2.9), we can obtain
‖uλ‖∞≤B‖uλ‖≤B(2λC|cos(πα)|)12−p2. |
Observe that, in the proof of Lemma 3.5, (˜f2) is used only in (3.10). If we directly use (˜F0) to rescale (∇˜F(t,uλ(t)),uλ(t)) in (3.10). Then the assumption (f2) is not necessary but we have to pay the price that p∈(0,1). To be precise, we have the following lemma.
Lemma 3.6. Assume that (F0) and (f0)′ hold. If 0<λ≤|cos(πα)|3C∗, then the following estimates hold
‖uλ‖2−p1≤3λC∗|cos(πα)|,‖uλ‖2−p1∞≤3λC∗B2−p1|cos(πα)|. |
Proof. It follows from (F0), Lemma 3.1, Remark 3.1, (2.9) and ⟨˜J′(uλ),uλ⟩=0 that
∫T0e−L(t)[−p(t)(c0Dαtuλ(t),ctDαTuλ(t))+p(t)(uλ(t),uλ(t))+(q(t)−p(t))(uλ(t),uλ(t))]dt=λ∫T0e−L(t)(∇˜F(t,uλ(t)),uλ(t))dt≤λmaxt∈[0,T]e−L(t)∫T0|∇˜F(t,uλ(t))||uλ(t)|dt≤λmaxt∈[0,T]e−L(t)∫T0[(1+ρ0)a0b(t)|uλ(t)|+M1p1|uλ(t)|p1+ρ0M1|uλ(t)|p1+1]dt≤λmaxt∈[0,T]e−L(t)[(1+ρ0)a0‖uλ‖∞∫T0b(t)dt+M1p1‖uλ‖p1∞+ρ0M1T‖uλ‖p1+1∞]≤λmaxt∈[0,T]e−L(t)[(1+ρ0)a0B‖uλ‖∫T0b(t)dt+M1p1TBp1‖uλ‖p1+ρ0M1TBp1+1‖uλ‖p1+1]≤λC∗(‖uλ‖+‖uλ‖p1+‖uλ‖p1+1). | (3.12) |
We claim that ‖uλ‖≤1 uniformly for all 0<λ≤|cos(πα)|3C∗. Otherwise, we have a sequence of {λn≤|cos(πα)|3C∗} such that ‖uλn‖>1. Thus ‖uλn‖p1<‖uλn‖<‖uλn‖p1+1 since p1<1. By (2.10) and (3.12), we obtain
∫T0e−L(t)[−p(t)(c0Dαtuλn(t),ctDαTuλn(t))+p(t)(uλn(t),uλn(t))+(q(t)−p(t))(uλn(t),uλn(t))]dt≥|cos(πα)|‖uλn‖2+∫T0e−L(t)(q(t)−p(t))|uλn(t)|2dt. | (3.13) |
By (3.12) and (3.13), we obtain
|cos(πα)|‖uλn‖2+∫T0e−L(t)(q(t)−p(t))|uλn|2dt≤λnC∗(‖uλ‖+‖uλ‖p1+‖uλ‖p1+1). |
Since q(t)−p(t)>0,
|cos(πα)|‖uλn‖2≤λnC∗(‖uλ‖+‖uλ‖p1+‖uλ‖p1+1)≤3λnC∗‖uλn‖p1+1. |
Then
‖uλn‖1−p1≤3λnC∗|cos(πα)|≤1, |
which contradicts with the assumption ‖uλn‖>1. Now, we can get from (3.12) that
|cos(πα)|‖uλ‖2≤λC∗(‖uλ‖+‖uλ‖p1+‖uλ‖p1+1)≤3λC∗‖uλ‖p1. |
So
‖uλ‖2−p1≤3λC∗|cos(πα)|. |
By (2.9), we can obtain
‖uλ‖∞≤B‖uλ‖≤B(3λC∗|cos(πα)|)12−p1. |
Proof of Theorem 1.1. Since 0<λ≤min{|cos(πα)|2C,(1B)2−p2(δ2)2−p2|cos(πα)|2C}, Lemma 3.5 implies that
‖uλ‖∞≤δ2. |
Therefore, for all 0<λ≤min{|cos(πα)|2C,(1B)2−p2(δ2)2−p2|cos(πα)|2C}, we have ˜F(t,u(t))=F(t,u(t)) and then uλ is a nontrivial weak solution of the original problem (1.1). Moreover, Lemma 3.5 implies that limλ→0‖uλ‖=0 as λ→0 and
‖uλ‖2−p2≤min{1,(1B)2−p2(δ2)2−p2},‖uλ‖2−p2∞≤B2−p2{1,(1B)2−p2(δ2)2−p2}. |
Proof of Theorem 1.2. Note that 0<λ≤min{|cos(πα)|3C∗,(1B)2−p1(δ2)2−p1|cos(πα)|3C∗}. Similar to the proof of Theorem 1.1, by Lemma 3.6, it is easy to complete the proof.
Proof of Theorem 1.3. By Lemma 3.1 and Lemma 3.3, we obtain that ˜J satisfies (PS) condition and is even and bounded from below, and ˜J(0)=0. Next, we prove that for any k∈N, there exists a subspace k-dimensional subspace Xk⊂Eα,20 and ρk>0 such that
supu∈Xk∩Sρk˜Jλ(u)<0. |
In fact, for any k∈N, assume that Xk is any subspace with dimension k in Eα,20. Then by (2.10) and Lemma 3.1, there exist constants C1,C2>0 such that
˜J(u)≤max{Mm|cos(πα)|,1}‖u‖2+12∫T0e−L(t)(q(t)−p(t))|u(t)|2dt−λ∫T0e−L(t)˜F(t,u(t))dt≤max{Mm|cos(πα)|,1}‖u‖2+C12‖u‖2∞−λC2∫T0˜F(t,u(t))dt≤max{Mm|cos(πα)|,1}‖u‖2+C1B22‖u‖2−λC2M1∫T0|u(t)|p1dt≤[max{Mm|cos(πα)|,1}+C1B22]‖u‖2−λC2M1‖u‖p1Lp1. |
Since all norms on Xk are equivalent and p1<2, for each fixed λ>0, we can choose ρk>0 small enough such that
supu∈Xk∩Sρk˜Jλ(u)<0. |
Thus, by Lemma 2.6 and Remark 2.3. ˜Jλ has a sequence of nonzero critical points {uλn}⊂Eα,20 converging to 0 and ˜Jλ(uλn)≤0. Hence, for each fixed λ>0, (3.1) has a sequence of weak solutions {uλn}⊂Eα,20 with ‖uλn‖→0, as n→∞. Furthermore, there exists n0 large enough such that ‖uλn‖≤δ2B for all n≥n0 and then (2.9) implies that ‖uλn‖∞≤δ2 for all n≥n0. Thus, ˜F(t,u(t))=F(t,u(t)) and then {uλn}∞n0 is a sequence of weak solutions of the original problem (1.1) for each fixed λ>0.
When the nonlinear term F(t,x) is local subquadratic only near the origin with respect to x, system (1.1) with λ in some given interval has a ground state weak solution uλ. If the nonlinear term F(t,x) is also locally even near the origin with respect to x, system (1.1) with λ>0 has infinitely many weak solutions {uλn}.
This project is supported by Yunnan Ten Thousand Talents Plan Young & Elite Talents Project and Candidate Talents Training Fund of Yunnan Province (No: 2017HB016).
The authors declare that they have no conflicts of interest.
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