In this paper, we prove the existence of positive solutions with prescribed L2-norm to the following Choquard equation:
−Δu−λu=(Iα∗F(u))f(u), x∈R3,
where λ∈R,α∈(0,3) and Iα:R3→R is the Riesz potential. Under the weaker conditions, by using a minimax procedure and some new analytical techniques, we show that for any c>0, the above equation possesses at least a couple of weak solution (ˉuc,ˉλc)∈Sc×R− such that ‖ˉuc‖22=c.
Citation: Shuai Yuan, Sitong Chen, Xianhua Tang. Normalized solutions for Choquard equations with general nonlinearities[J]. Electronic Research Archive, 2020, 28(1): 291-309. doi: 10.3934/era.2020017
[1] | Shuai Yuan, Sitong Chen, Xianhua Tang . Normalized solutions for Choquard equations with general nonlinearities. Electronic Research Archive, 2020, 28(1): 291-309. doi: 10.3934/era.2020017 |
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In this paper, we prove the existence of positive solutions with prescribed L2-norm to the following Choquard equation:
−Δu−λu=(Iα∗F(u))f(u), x∈R3,
where λ∈R,α∈(0,3) and Iα:R3→R is the Riesz potential. Under the weaker conditions, by using a minimax procedure and some new analytical techniques, we show that for any c>0, the above equation possesses at least a couple of weak solution (ˉuc,ˉλc)∈Sc×R− such that ‖ˉuc‖22=c.
This paper is dedicated to deal with the existence of normalized solutions to the generalized Choquard equation as follows:
−Δu−λu=(Iα∗F(u))f(u), x∈R3, | (1.1) |
where
For example, when
−Δu+u=(Iα∗|u|p)|u|p−2u, x∈RN, | (1.2) |
where
Nowadays, since physicist are more and more interested in the normalized solutions, like [1,2,3,13,14,30], mathematical researchers are committed to investigate the solutions with prescribed
IN(u)=12∫RN|∇u|2dx−12∫RN(Iα∗F(u))F(u)dx | (1.3) |
on the constraint
S′c={u∈H1(RN):‖u‖22=c}, | (1.4) |
where
σ(c):=infu∈S′cIN(u), | (1.5) |
then there exists
In [12], Jeanjean proved the existence of normalized solutions of the following Schrödinger equation
−Δu−f(u)=λu in RN, | (1.6) |
where
(f1)
(f2)
αG(s)≤g(s)s≤βG(s), |
where
(f3) let
ˉG′(s)s>2N+4NˉG(s). |
Jeanjean deduced the existence of normalized solutions by dealing with the minimization problem
infu∈H1(RN),‖u‖2=c∫RN[12|∇u|2−F(u)]dx, |
and the author verified the existence of the mountain pass structure on the constraint defined by
˜I(u,t)=e2t2∫RN|∇u|2dx−1eNt∫RNG(eNt2u)dx. |
By applying the new functional
Bellazzini, Jeanjean and Luo [4] verified the existence of standing waves with prescribed
−Δu+(|x|−1∗|u|2)u−|u|q−2u=λu, x∈R3, | (1.7) |
where
ˆI(u)=12∫R3|∇u|2dx+14∫R3∫R3|u(x)|2|u(y)|2|x−y|dxdy−1q∫R3|u|qdx, | (1.8) |
is no more bounded from below on the constraint:
Sc={u∈H1(R3):‖u‖22=c}. | (1.9) |
To overcome this difficulty, they first investigated the mountain-pass structure of
M′c={u∈S′c:ˆJ(u):=ddtˆI(ut)|t=1=0}, | (1.10) |
that is
ˆm(c)=infu∈M′cˆI(u). | (1.11) |
As far as we know, there seems to be only one paper [16] dealing with the Choquard equation in the sense of prescribed
(F1')
lim|s|→0f(s)|s|r−2s=0 and lim|s|→+∞F(s)|s|r=+∞; |
(F2')
(F3') there exists
(F4')
(F5') let
ˉF′(s)s>N+α+2NˉF(s); |
(F6') there exists
F(ts)≤θ2|t|N+α+2NF(s). |
In fact, the nonlinearity term in paper [16] needs the assumption
(F1)
lim|s|→0f(s)|s|r−2s=0, and lim|s|→+∞F(s)|s|r=+∞; |
(F2)
(F3)
(F4) there exists
(F5)
In this paper, we define
I(u)=12∫R3|∇u|2dx−12∫R3(Iα∗F(u))F(u)dx | (1.12) |
and
Mc={u∈Sc:J(u):=ddtI(ut)|t=1=0}, |
where the definition of
Theorem 1.1. Assume that (F1)-(F5) hold. Then for any constant
I(ˉvc)=infv∈McI(v)=infv∈Scmaxt>0I(vt)>0. |
Notice that, we proved the existence of normalized solution of problem (1.1) under the assumptions (F1)-(F5). Compared to [16], case (F5) plays an important role to overcome the difficulty caused by the absence of condition (F5
Now, we give our main idea for the proof of Theorem 1.1.
By (F1) and (F2), there exists some
|F(s)|≤C(|s|r+|s|3+α). | (1.13) |
By Hardy-Littlewood-Sobolev inequality:
∫R3(Iα∗f)gdx≤C‖f‖p‖g‖q, |
we see that
Inspired by [16], (F4) implies that
f(s)s−9+2α6F(s)≥0 for any s∈R. | (1.14) |
Then for any
F(s)≥0 for any s∈R. | (1.15) |
Then by (1.14), we see that
As in [4,12],
Definition 1.2. For given
γ(c)=infg∈Γcmaxτ∈[0,1]I(g(τ))>maxg∈Γcmax{I(g(0)),I(g(1))}, | (1.16) |
where
Let us recall this, to obtain this conclusion, the authors in [4] constructed some sequence of paths
˜I(v,t)=I(β(v,t))=e2t2‖∇v‖22−12e(3+α)t∫R3(Iα∗F(e3t2v))F(e3t2v)dx, |
and also we shall know the fact that
Since we have obtained the boundness of
In [4] the fact
Throughout the paper we use the following notations:
(u,v)=∫R3(∇u⋅∇v+uv)dx, ‖u‖=(u,u)1/2, ∀ u,v∈H1(R3); |
To prove Theorem 1.1, recalling the Gagliardo-Nirenberg inequality, that is, let
‖u‖Lp≤Cp‖∇u‖βL2‖u‖1−βL2, |
where
In the following lemma, we show that
Lemma 2.1. Assume that
Ak1={u∈Sc:‖∇u‖22≤k1,I(u)>0} | (2.1) |
and
Ak2={u∈Sc:‖∇u‖22>k2,I(u)<0}. | (2.2) |
Moreover,
Proof. Given any
Bk={u∈Sc:‖∇u‖22≤k}. | (2.3) |
We shall check that there exist
I(u)>0, ∀ u∈Bk2 and supu∈Bk1I(u)<infu∈∂Bk2I(u). | (2.4) |
We have known that
|∫R3(Iα∗F(u))F(u)dx|≤C(∫R3|F(u)|63+αdx)3+α3≤C(∫R3|u|6r3+αdx+∫R3|u|2∗dx)3+α3≤C[‖u‖2r6r3+α+(∫R3|u|2∗dx)3+α3]≤C(‖∇u‖3r−(3+α)2+‖∇u‖6+2α2). | (2.5) |
Hence, we have that
I(u)≥12‖∇u‖22−C2‖∇u‖3r−(3+α)2−C2‖∇u‖6+2α2. | (2.6) |
Since
infu∈∂Bk2I(u)≥ρ>0 and I(u)>0 for u∈Bk2. | (2.7) |
On the other hand, use (2.5) again, we have
|I(u)|≤12‖∇u‖22+12|∫R3(Iα∗F(u))F(u)dx|≤12‖∇u‖22+C2‖∇u‖3r−(3+α)2+C2‖∇u‖6+2α2. | (2.8) |
which implies
supu∈Bk|I(u)|→0 as k→0. | (2.9) |
Combining (2.7) with (2.9), there exists
supu∈Bk1I(u)<ρ≤infu∈∂Bk2I(u). |
and (2.4) follows.
Let
ut(x)=t3/2u(tx), ∀ t>0,u∈H1(R3). | (2.10) |
Then
I(ut)=t22‖∇u‖22−12t3+α∫R3(Iα∗F(t3/2u))F(t3/2u)dx. | (2.11) |
Using (F1), (2.10) and Fatou's Lemma, which is inspired by [16], we can see that
lim inft→∞∫R3(Iα∗F(t3/2u)|t3/2u||u|r)F(t3/2u)|t3/2u||u|r≥∫R3lim inft→∞[(F(t3/2u)|t3/2u||u|r)(x)F(t3/2u)|t3/2u||u|r]→+∞. | (2.12) |
Hence, we have
I(ut)t3r−3−α=‖∇u‖222t3r−(3+α+2)−12∫R3(Iα∗F(t3/2u)|t3/2u||u|r)F(t3/2u)|t3/2u||u|r→−∞ as t→+∞. | (2.13) |
So
‖∇ut1‖22=t21‖∇u‖22≤k1, ‖∇ut2‖22=t22‖∇u‖22>k2 and I(ut2)<0. | (2.14) |
Set
‖∇u1‖22≤k1, ‖∇u2‖22>k2. |
This fact indicates that
We next claim that
Γc:={g∈C([0,1],Sc):‖∇g(0)‖22≤k1,I(g(1))<0}, |
if
maxt∈[0,1]I(g(t))≥I(g(τ0))≥infu∈∂Bk2I(u)>supu∈Bk1I(u), ∀ g∈Γc, |
which, together with the arbitrariness of
γ(c)=infg∈Γcmaxt∈[0,1]I(g(t))>maxg∈Γcmax{I(g(0)),I(g(1))}. | (2.15) |
Indeed, to obtain the desired conclusion, it suffices to check that
g0(τ)=u(1−τ)t1+τt2, ∀ τ∈[0,1]. |
It follows from (2.14) that
Next, inspired by [6,12], we will show the existence of a (PS) sequence for the functional
J(u)=‖∇u‖22−32∫R3(Iα∗F(u))[f(u)u−3+α3F(u)]dx, ∀ u∈H1(R3). | (2.16) |
To achieve this, we define a continuous map
β(v,t)(x)=e3t2v(etx) for v∈H1(RN), t∈R, and x∈R3, | (2.17) |
where
˜I(v,t)=I(β(v,t))=e2t2‖∇v‖22−12e(3+α)t∫R3(Iα∗F(e3t2v))F(e3t2v)dx. | (2.18) |
It is easy to see that
⟨˜I′(v,t),(w,s)⟩=e2t∫R3∇v⋅∇wdx−12e(3+α)t∫R3(Iα∗F(e3t2v))f(e3t2v)e3t2wdx +e2ts‖∇v‖22+(3+α)s2e(3+α)t∫R3(Iα∗F(e3t2v))F(e3t2v)dx −3s2e(3+α)t∫R3(Iα∗F(e3t2v))f(e3t2v)e3t2vdx. | (2.19) |
Set
˜γ(c):=inf˜g∈˜Γcmaxτ∈[0,1]˜I(˜g(τ)), | (2.20) |
where
˜Γc={˜g∈C([0,1],Sc×R):˜g(0)∈Ak1×{0},˜g(1)∈Ak2×{0}}. |
and the sets
˜γ(c)=γ(c)>maxg∈Γcmax{I(g(0)),I(g(1))}=max˜g∈˜Γcmax{˜I(˜g(0)),˜I(˜g(1))}. | (2.21) |
Following by [29], we recall that for any
Tu={v∈H1(R3):∫R3uvdx=0}. | (2.22) |
The norm of the
‖I|′Sc(u)‖=supv∈Tu,‖v‖=1⟨I′(u),v⟩. | (2.23) |
And the tangent space at
˜Tu,t={(v,s)∈H:∫R3uvdx=0}. | (2.24) |
The norm of the derivative of the
‖˜I|′Sc×R(u,t)‖=sup(v,s)∈˜Tu,t,‖(v,s)‖H=1⟨˜I|′Sc×R(u,t),(v,s)⟩. | (2.25) |
Learning from [12,Proposition 2.2], we have the following proposition.
Proposition 1. Assume that
maxτ∈[0,1]˜I(˜gn(τ))≤˜γ(c)+1n. | (2.26) |
Then there exists a sequence
(ⅰ)
(ⅰ)
(ⅰ)
|⟨˜I′(un,tn),(v,s)⟩|≤2√n‖(v,s)‖H, ∀ (v,s)∈˜Tun,tn. |
Applying proposition 1 to
Lemma 2.2. Assume that
I(vn)→γ(c)>0, I|′Sc(vn)→0 and J(vn)→0. | (2.27) |
Proof. Given
maxτ∈[0,1]I(gn(τ))≤γ(c)+1n. | (2.28) |
In order to obtain the desired sequence, we first apply proposition 1 to
˜gn(τ)=(gn(τ),0), ∀ τ∈[0,1]. |
It is easy to know that
maxτ∈[0,1]˜I(˜gn(τ))≤˜γ(c)+1n. | (2.29) |
From the preceding proposition 1, there exists a sequence
(ⅰ)
(ⅰ)
(ⅰ)
Set
I(vn)→γ(c). | (2.30) |
Accoring to (2.19) and (ⅱ), we derive
⟨I′(vn),w⟩=⟨˜I′(un,tn),(β(w,−tn),0)⟩≤2√n‖(β(w,−tn),0)‖H, ∀ w∈Tvn. | (2.31) |
To prove
∫R3vnwdx=∫R3e3tn2un(etnx)w(x)dx=0, |
we can see that
∫R3un(x)β(w,−tn)dx=∫R3un(x)e−3tn2w(e−tnx)dx=∫R3e3tn2un(etnx)w(x)dx=0, |
follows
(β(w,−tn),0)∈˜Γun,tn. | (2.32) |
Then by (ⅱ), we have
|tn|≤minτ∈[0,1]‖(un,tn)−˜gn(τ)‖H≤1 for large n∈N, |
which leads to
‖(β(w,−tn),0)‖2H=‖β(w,−tn)‖2=e−2tn‖∇w‖22+‖w‖22≤ e2‖w‖2 for large n∈N. | (2.33) |
This shows that
|⟨˜I′(un,tn),(0,1)⟩|=J(β(un,tn))=J(vn)=o(1). | (2.34) |
Hence,
In connection with the additional minimax characterization of
Lemma 2.3. Assume that
h(t):=(3+α)t32N∫R3(Iα∗F(u))F(u)dx+12(t3+α)∫R3(Iα∗F(t32u))F(t32u)dx −t32∫R3(Iα∗F(u))f(u)udx≥0, ∀ t>0. | (2.35) |
Proof. For any
ddth(t)=(3+α)t122∫R3(Iα∗F(u))F(u)dx−3+α2t3+α+1∫R3(Iα∗F(t32u))F(t32u)dx +32t3+α+1∫R3(Iα∗F(t32u))f(t32u)(t32u)dx−3t122∫R3(Iα∗F(u))f(u)udx | (2.36) |
Now we only need to study
q(t,τ1,τ2)=F(t32τ1)[32t3+α+1f(t32τ2)t32τ2−3+α2t3+α+1F(t32τ2)] −F(τ1)[3t2f(τ2)τ2−(3+α)t122F(τ2)] |
=32|τ2|9+2α3t12F(t32τ1)[f(t32τ2)t32τ2−3+α3F(t32τ2)|t32τ2|9+2α3] −32|τ2|9+2α3t12F(τ1)[f(τ2)τ2−3+α3F(τ2)|τ2|9+2α3]{≥0, t≥1,≤0, 0<t<1, |
By (F5) and (1.15), we can easily get the above conclusion, which implies that
By the preceding scaling (2.10), we have
I(ut)=t22‖∇u‖22−12t3+α∫R3(Iα∗F(t3/2u))F(t3/2u)dx. | (2.37) |
It can be easily checked that
h1(t):=4t32−3t2−1, t≥0. | (2.38) |
After basic calculations, we can see
h1(1)=0, h1(t)>0, ∀ t∈[0,1)∪(1,+∞). | (2.39) |
Inspired by [7,25], we obtain the following key inequality.
Lemma 2.4. Assume that
I(u)≥I(ut)+2(1−t32)3J(u)+h1(t)6‖∇u‖22, ∀ u∈H1(R3), t>0 | (2.40) |
and
I(u)≥23J(u)−16‖∇u‖22, ∀ u∈H1(R3). | (2.41) |
Proof. By (1.12), (2.16), (2.35), (2.37), (2.38) and (2.39), we have
I(u)−I(ut)=1−t22∫R3|∇u|2dx−12∫R3(Iα∗F(u))F(u)dx+12t3+α∫R3(Iα∗F(t32u))F(t32u)dx=2(1−t32)3J(u)+(1−t32)∫R3(Iα∗F(u))f(u)udx−3+2(3+α)(1−t32)6∫R3(Iα∗F(u))F(u)dx+4t32−3t2−16‖∇u‖22+12t3+α∫R3(Iα∗F(t32u))F(t32u)dx |
=2(1−t32)3J(u)+h1(t)6‖∇u‖22+h(t)+∫R3(Iα∗F(u))[f(u)u−9+2α6F(u)]≥2(1−t32)3J(u)+h1(t)6‖∇u‖22, ∀ u∈H1(R3), t>0. | (2.42) |
This shows that (2.40) holds. Letting
Following the Lemma 2.4 naturally, we obtain the following corollary.
Corollary 1. Assume that
I(u)=maxt>0I(ut), ∀ u∈Mc. | (2.43) |
Lemma 2.5. Assume that
Proof. Let
ζ′(t)=0⇔ t‖∇u‖22=32t3+α+1∫R3Iα∗F(t3/2u))[f(t3/2u)t3/2u−N+α3F(t3/2u)]dx ⇔ 1tJ(ut)=0 ⇔ ut∈Mc. | (2.44) |
Note that (F1) leads to
|F(s)|≤|s|r for ∀ |s|≤δ, | (2.45) |
From (2.37) and (2.45), we infer that
I(ut)≥t22‖∇u‖22−12t3r−3−α∫R3(Iα∗F(u))F(u)dx, | (2.46) |
which, together with
In order to finish this proof, it is suffices to show that
I(ut1)>I(ut2)+2[t321−t322]3t321J(ut1)=I(ut2)>I(ut1)+2[t322−t321]3t322J(ut2)=I(ut1). | (2.47) |
This contradiction shows us that
Combining the Corollary 1 and Lemma 2.5, we can easily obtain the following facts.
Lemma 2.6. Assume that
infu∈McI(u)=m(c)=infu∈Scmaxt>0I(ut). |
Lemma 2.7. Assume that
Proof. To achieve this purpose, it is sufficient to verify whether in the condition that for any
m(c2)≤m(c1)+ε | (2.48) |
By the definition of
η(x)={1, |x|≤1,∈[0,1], 1≤|x|<2,0, |x|≥2. |
For any small
uδ(x)=η(δx)⋅u(x). | (2.49) |
It is easy to obtain that
I(uδ)→I(u)≤m(c1)+ε4, J(uδ)→J(u)=0. | (2.50) |
From Lemma 2.5, for any
0=limδ→0I(utδδ)t2δ=12‖∇u‖22+limδ→0∫R3|Iα∗F(t3/2δu)t5+α2δ|F(t3/2δ)ut5+α2δdx=−∞, |
which is a contradiction. So we may assume that up to a subsequence,
I(utδδ)≤I(uδ)−2(1−t32δ)3J(uδ)+h1(tδ)6‖∇utδδ‖22, |
which, together with (2.50), implies that there exists
I(utδ0δ0)≤I(uδ0)+ε8≤I(u)+ε4≤m(c1)+ε2. | (2.51) |
Let
v0=c2−‖uδ0‖22‖v‖22v, |
for which we have
dist{suppuδ0,suppvλ0}≥2Rδ0λ−Rδ0=2δ0(2λ−1)>0, | (2.52) |
following which we can easily obtain
|wλ(x)|2=|uδ0(x)+vλ0(x)|2=|uδ0(x)|2+|vλ0(x)|2, | (2.53) |
‖wλ‖22=‖uδ0+vλ0‖22=‖uδ0‖2+‖vλ0‖22=‖uδ0‖2+‖v0‖22, | (2.54) |
‖∇wλ‖22=‖∇uδ0+∇vλ0‖22=‖∇uδ0‖2+‖∇vλ0‖22=‖∇uδ0‖2+λ2‖∇v0‖22, | (2.55) |
∫R3F(wλ)dx=∫R3F(uδ0+vλ0)dx=∫R3F(uδ0)dx+∫R3F(vλ0)dx=∫R3F(uδ0)dx+λ−3∫R3F(λ32v0)dx | (2.56) |
Then (2.55), (2.56) and (F2) imply that as
‖∇wλ‖22→‖∇uδ0‖2, ∫R3F(wλ)dx→∫R3F(uδ0)dx | (2.57) |
and by (2.57), we have
∫R3∫R3F(wλ(x))F(wλ(y))|x−y|3−αdxdy→∫R3∫R3F(uδ0(x)F(uδ0(y))|x−y|3−αdxdy, | (2.58) |
which lead to
I(wλ)→I(uδ0) and J(wλ)→J(uδ0). | (2.59) |
By (2.54), we have
∫R3F(wλtλ)dx→∫R3F(uδ0ˆt)dx. | (2.60) |
and
∫R3∫R3F(wtλλ(x))F(wtλλ(y))|x−y|3−αdxdy→∫R3∫R3F(uˆtδ0(x)F(uˆtδ0(y))|x−y|3−αdxdy | (2.61) |
Deduced by (2.60) and (2.61), there exists
m(c2)≤I(wλtλ)≤I(uδ0ˆt)+ε2≤maxt>0I(uδ0t)+ε2=I(uδ0tδ0)+ε2≤m(c1)+ε. | (2.62) |
The proof is completed.
Inspired by the above works, we have established the additional minimax characterization of
Lemma 2.8. Assume that
Proof. By (2.14), for any
ˉg(τ)=u(1−τ)t1+τt2, ∀ τ∈[0,1], |
we have
γ(c)≤maxτ∈[0,1]I(ˉg(τ))=I(u), |
and so
On the other hand, by (2.41), we have
J(u)≤32I(u)+14‖∇u‖22, ∀ u∈Sc. |
which implies
J(g(1))≤32I(g(1))<0, ∀ g∈Γc. |
Moreover, it is easy to verify that there exists
maxτ∈[0,1]I(g(τ))≥infu∈McI(u)=m(c), ∀ g∈Γc, |
and so
Let
Lemma 2.9. Let
(1)
(1)
Lemma 2.10. Let
(1)
(1)
(1)
Proof. (1) since
(2) Since
I′(vn)−⟨I′(vn),vn⟩vn→0 in H−1(R3). |
It means that for any
⟨I′(vn)−⟨I′(vn),vn⟩vn,ω⟩=∫R3∇vn∇ω−∫R3(Iα∗F(vn))f(vn)vnω−λn∫R3vnw→0, |
where
λn=‖∇vn‖22−∫R3(Iα∗F(vn))f(vn)vn‖vn‖22, |
Then
I′(vn)−λnvn→0 in H−1(R3) | (2.63) |
and
(3) follows immediately (1), (2) and (2.63).
Lemma 2.11. Assume that
∃ C>0 such that for every s∈R,|sf(s)|≤C(|s|N+αN+|s|N+αN−2). |
If
12‖∇u‖22−32λ‖u‖22−3+α2∫R3(Iα∗F(u))F(u)=0. | (2.64) |
Next Lemma can also be found in [16], for the sake of completeness and convenience for reading, we show it here again.
Lemma 2.12. Assume that
Proof. Since
12‖∇ˉvc‖22=32ˉλc‖ˉvc‖22+3+α2∫R3(Iα∗F(ˉvc))F(ˉvc)=32‖∇ˉvc‖22−32∫R3(Iα∗F(ˉvc))f(ˉvc)ˉvc+3+α2∫R3(Iα∗F(ˉvc))F(ˉvc), | (2.65) |
where we use that
ˉλc=‖∇ˉvc‖22−∫R3(Iα∗F(ˉvc))f(ˉvc)ˉvc‖ˉvc‖22. | (2.66) |
Then, we have
‖∇ˉvc‖22+3+α2∫R3(Iα∗F(ˉvc))F(ˉvc)−32∫R3(Iα∗F(ˉvc))f(ˉvc)ˉvc=0. | (2.67) |
i.e.
By (2.66),
ˉλcc=‖∇ˉvc‖22−∫R3(Iα∗F(ˉvc))f(ˉvc)ˉvc=12∫R3(Iα∗F(ˉvc))f(ˉvc)ˉvc−3+α2∫R3(Iα∗F(ˉvc))F(ˉvc)≤0. | (2.68) |
hence
In view of Lemmas 2.8 and 2.11, for each
I(vn)→m(c)>0, I|′Sc(vn)→0 and J(vn)→0. | (3.1) |
By (2.41) and (3.1), we have
m(c)+o(1)=I(vn)−23J(vn)≥−16‖∇vn‖22, | (3.2) |
which, combining with
I(ˉvn)→m(c), J(ˉvn)=o(1), ∫B1(0)|ˉvn|2dx>δ. | (3.3) |
Therefore, there exists
{ˉvn⇀ˉv,in H1(R3);ˉvn→ˉv,in Lsloc(R3), ∀ s∈[1,6);ˉvn→ˉv,a.e. on R3. | (3.4) |
Let
‖ˉv‖22:=ˉc≤c, ‖wn‖22:=ˉcn≤c for large n∈N | (3.5) |
and
I(ˉvn)=I(ˉv)+I(wn)+o(1) and J(ˉvn)=J(ˉv)+J(wn)+o(1). | (3.6) |
Let
Ψ(u):=I(u)−23J(u)=−16‖∇u‖22+∫R3[f(u)u−9+2α6F(u)]dx, ∀ u∈H1(R3). | (3.7) |
Then
Ψ(wn)=m(c)−Ψ(ˉv)+o(1), J(wn)=−J(ˉv)+o(1). | (3.8) |
If there exists a subsequence
which contradicts
which implies
(3.9) |
Since
(3.10) |
Combining (3.9) with (3.10), we have
And by condition (F1) and the strong maximum principle, we conclude that
The authors express their sincere gratitude to the anonymous referee for all insightful comments and valuable suggestions.
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