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Solutions for a class of quasilinear Schrödinger equations with critical Sobolev exponents

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In this paper, we study a class of quasilinear Schrödinger equations involving a critical exponent which arises in plasma physics. By using the change of variable and variational approaches combining the concentration-compactness principle, the existence of a positive solution which has a local maximum point and decays exponentially is obtained.
Solutions for a class of quasilinear Schrödinger equations with critical
Sobolev exponents
Zhouxin Li and Yimin Zhang
Citation: J. Math. Phys. 58, 021501 (2017); doi: 10.1063/1.4975009
View online: http://dx.doi.org/10.1063/1.4975009
View Table of Contents: http://aip.scitation.org/toc/jmp/58/2
Published by the American Institute of Physics
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JOURNAL OF MATHEMATICAL PHYSICS 58, 021501 (2017)
Solutions for a class of quasilinear Schrödinger equations
with critical Sobolev exponents
Zhouxin Li1,a) and Yimin Zhang2,a)
1School of Mathematics and Statistics, Central South University, Changsha 410083,
People’s Republic of China
2Department of Mathematics, School of Science, Wuhan University of Technology,
Wuhan 430070, People’s Republic of China
(Received 15 October 2015; accepted 11 January 2017; published online 2 February 2017)
In this paper, we study a class of quasilinear Schrödinger equations involving a critical
exponent which arises in plasma physics. By using the change of variable and varia-
tional approaches combining the concentration-compactness principle, the existence
of a positive solution which has a local maximum point and decays exponentially is
obtained. Published by AIP Publishing. [http://dx.doi.org/10.1063/1.4975009]
I. INTRODUCTION
In this paper, we are interested in the existence of a positive solution for the following quasilin-
ear Schrödinger equation:
ε2u+V(x)ukαε2((|u|2α))|u|2α2u=|u|q2u+|u|2(2α)2u,u>0,xRN,(1.1)
where V(x)is a given potential, ε > 0 is a real parameter, kis a real constant, q2, α > 1/2 is
a constant, 2(2α)=2×2α, and 2=2N
N2is the critical Sobolev exponent. Solutions of problem
(1.1) are related to the standing wave solutions of certain quasilinear Schrödinger equations which
arise in several physical phenomena such as the theory of superfluid film in plasma physics, see
Refs. 7and 12 and the references therein for more backgrounds. For this reason, problem (1.1) is
still called a quasilinear Schrödinger equation.
For α=1, the existence of nontrivial solutions for the quasilinear schrödinger equation was
extensively studied in the past decades. We can see Refs. 4,5, and 13 for the subcritical case
and Refs. 14,16,19, and 20 for the critical case. In Ref. 13, by using a change of variable, they
transform the equation to a semilinear one, then the existence of solutions was obtained via varia-
tional methods under dierent types of potentials V(x). This method is significant and was widely
used in the studies of this kind of problems. For general α, there are few results for this case, as far
as we know, just Refs. 1,2,12, and 15. In Ref. 12, for α > 1
2, 2 <q+1<2(2α), Liu and Wang
considered Equation (1.1) without a critical term, that is,
u+V(x)ukα((|u|2α))|u|2α2u=λ|u|q2u,u>0,xRN,(1.2)
where λ > 0 is a parameter. They obtained the existence of a solution for problem (1.2) by using
the method of Lagrange multiplier. In Ref. 15, the fibering method was employed to obtain the
existence of at least one or sometimes two standing wave solutions for the following quasilinear
Schrödinger equation:
u+V(x)ukα((|u|2α))|u|2α2u=µf(x)|u|q2u,u>0,xRN,
where V(x)=λg(x)and g(x)ia a positive function and µand λare positive numbers. Employing
the change of variable method just as in Ref. 13, the authors of Ref. 1obtained the existence of at
least one positive solution for problem (1.2) by using variational approaches. Moreover, in Ref. 2,
a)Electronic addresses: lzx@math.pku.edu.cn and zhangym802@126.com
0022-2488/2017/58(2)/021501/15/$30.00 58, 021501-1 Published by AIP Publishing.
021501-2 Z. Li and Y. Zhang J. Math. Phys. 58, 021501 (2017)
for V(x)=µ,α > 1
2, they obtained the unique existence of a positive radial solution of problem
(1.2) under some suitable conditions and λ=1. All these results are obtained when the nonlinear
term satisfies subcritical growth. If the right side of Equation (1.2) has a critical term, as far as we
know, there are no results for general α.16
In this paper, our aim is to study the existence of positive solutions of (1.1) with general
α > 1/2 and at 2(2α)growth. Problem of (1.1) at 2(2α)growth has two diculties. First, the
embedding H1(RN)L2(2α)(RN)is not compact, so it is hard to prove the Palais-Smale (PS in
short) condition. Second, even if we can obtain the compactness result of PS sequence, it only holds
at some level of a positive upper bound and it is dicult for us to prove that the functional has such
a minimax level.
We assume that V(x)is locally Hölder continuous and
(V)V>V0>0 such that minxRNV(x)=V0and lim|x|→∞ V(x)=V.
Under assumption (V), we define a space
XBuH1(RN):RN
V(x)u2<
with the norm u2
X=RN|u|2+RNV(x)u2.
For the simplicity of notation, we let m=2α, ¯q=q/m, and kα=1 in this paper. Set
g(t)=|t|q2t+|t|2m2tand G(t)=t
0
g(s)ds.
We formulate problem (1.1) in the variational structure in the space Xas follows:
I(u)=ε2
2RN
(1+m|u|2(m1))|u|2+1
2RN
V(x)u2RN
G(u).
Note that Iis lower semicontinuous on X; we define that uXis a weak solution for (1.1) if
uXL(RN)and it is a critical point of I.
First, for an arbitrary ε > 0, we have
Theorem 1.1. Assume that q (2m,2m)and that condition (V) holds. Moreover, assume that
one of the following conditions holds
(i) 1<m<2and ¯q>4
N2+2
m;
(ii) m 2and ¯q>21.
Then for ε > 0small enough, problem (1.1) has a positive weak solution uεXL(RN)with
lim
ε0uεX=0and uε(x)Cexp(β
ε|xxε|),
where C >0, β > 0are constants, x εRNis a local maximum point of uε.
Next, we consider the case ε=1. We have the following result:
Theorem 1.2. Assume that all conditions in Theorem 1.1 hold and that ε=1, then problem
(1.1) has a positive weak solution u1XL(RN).
This paper is organized as follows. In Section II, we first use a change of variable to reformulate
the problem, then we modify the functional in order to regain the PS condition. In Section III, we
prove that the functional satisfies the PS condition, which is a crucial job of this paper. Finally, in
Section IV, we prove the main theorems, which involves the construction of a mountain pass level at
a certain height.
021501-3 Z. Li and Y. Zhang J. Math. Phys. 58, 021501 (2017)
II. PRELIMINARIES
Since Iis lower semicontinuous on X, we follow the idea in Refs. 5and 13 and make the
change of variables v=f1(u), where fis defined by
f(0)=0,
f(v)=(1+m|f(v)|2(m1))1/2,on [0,+),
f(v)=f(v),on (−∞,0].
The above function f(t)and its derivative satisfy the following properties (see Refs. 1,2, and 13):
Lemma 2.1. For m >1, we have
(1) f is uniquely defined, C2and invertible;
(2) |f(t)| 1for all t R;
(3) |f(t)| |t|for all t R;
(4) f (t)/t1as t 0;
(5) |f(t)| m1/2m|t|1/mfor all t R;
(6) 1
mf(t)t f (t)f(t)for all t >0;
(7) f (t)/m
tm1/2mas t +.
According to Ref. 6(see Corollary 2.1 and Proposition 2.2 in it, note that the embedding in
Corollary 2.1 of Ref. 6is also compact.), we have
Lemma 2.2. The map: v→ f(v)from X into Lr(RN)is continuous for 1r2m and is
compact for 1r<2m.
Using this change of variable, we rewrite the functional I(u)to
J(v)=I(f(v)) =ε2
2RN
|v|2+1
2RN
V(x)f2(v)RN
G(f(v)).
The critical point of Jis the weak solution of the equation
ε2v+V(x)f(v)f(v)=g(f(v)) f(v),xRN.(2.1)
Now we define a suitable modification of the functional Jin order to regain the Palais-Smale
condition. In this time, we make use of the method in Ref. 17.
Let lbe a positive constant such that
l=sup{s>0 : g(t)
tV0
kfor every 0 ts}(2.2)
for some k> θ/(θ2)with θ(2m,q]. We define the functions,
γ(s)=
g(s),s>0,
0,s0¯γ(s)=
γ(s),0sl,
V0
ks,s>l
and
p(x,s)=χR(x)γ(s)+(1χR(x)) ¯γ(s),
P(x,s)=s
0
p(x,t)dt,
where χRdenotes the characteristic function of the set BR(the ball centered at 0 and with radius R
in RN) and R>0 is suciently large such that
min
BR
V(x)<min
BR
V(x).
By definition, the function p(x,s)is measurable in x, of class Cin sand satisfies
(p1) 0 < θP(x,s)p(x,s)sfor every xBRand sR+.
021501-4 Z. Li and Y. Zhang J. Math. Phys. 58, 021501 (2017)
(p2) 0 2P(x,s)p(x,s)s1
kV(x)s2for every xBc
RBRN\BRand sR+.
Now we study the existence of solutions for the deformed equation
ε2v+V(x)f(v)f(v)=p(x,f(v)) f(v),xRN.(2.3)
The corresponding functional of (2.3) is given by
¯
J(v)=ε2
2RN
|v|2+1
2RN
V(x)f2(v)RN
P(x,f(v)).
For vX, since
|(| f(v)|m)|2=m2|f(v)|2(m1)
1+m|f(v)|2(m1)|v|2m|v|2,(2.4)
we infer that |f(v)|mX. By Sobolev inequality, we have
f(v)∥2m=| f(v)|m1/m
2C(| f(v)|m)∥1/m
2Cv1/m
X.(2.5)
It results that f(v)L2m(RN). Using interpolation inequality, we obtain that f(v)Lq(RN). Thus
¯
Jis well defined on X. Let (vn)X, v Xwith vnvin X. Then from Lemma 2.2, we infer that
V(x)f2(vn)V(x)f2(v)in L1(RN)and that f(vn)f(v)in Lq(RN). Thus ¯
Jis continuous on X.¯
J
is Gateaux-dierentiable in Xand the G-derivative is
¯
J(v), ϕ=ε2RNvϕ+RN
V(x)f(v)f(v)ϕRN
p(x,f(v)) f(v)ϕ, ϕX.
Then if vXL(RN)is a critical point of ¯
J, and v(x)aBf1(l),xBc
R, we have u=
f(v)XL(RN)(note that we have |u||v|and |u||v|by the properties of f) is a solution
of (1.1).
III. COMPACTNESS OF PS SEQUENCE
In this section, we show that the functional ¯
Jsatisfies the PS condition which is a crucial job
and its proof is composed of four steps. Let Sdenote the best Sobolev constant, we have
Lemma 3.1. Assume that condition (V) holds and q (2m,2m). Then ¯
J satisfies the PS condi-
tion at level cε<1
N m εNSN/2.
Proof. Let (vn)Ebe a PS sequence of ¯
Jat level cε, that is, (vn)satisfies
¯
J(vn)=ε2
2RN
|vn|2+1
2RN
V(x)f2(vn)RN
P(x,f(vn)) =cε+o(1)(3.1)
and
¯
J(vn), ϕ=ε2RNvnϕ+RN
V(x)f(vn)f(vn)ϕ
RN
p(x,f(vn)) f(vn)ϕ=o(1)∥ϕX,ϕX.(3.2)
We divide the proof into four steps.
Step 1: The sequence RN(|vn|2+V(x)f2(vn)) is bounded. Multiplying (3.1) by θ(θis given
in Section II) and using (p1)-(p2), we get
θε2
2RN
|vn|2+θ
2RN
V(x)f2(vn)
BR
p(x,f(vn)) f(vn)+θ
2kBc
R
V(x)f2(vn)+θcε+o(1).
021501-5 Z. Li and Y. Zhang J. Math. Phys. 58, 021501 (2017)
On the other hand, taking ϕ=f(vn)/f(vn)in (3.2), we get
RN
ε2(1+m(m1)| f(vn)|2(m1)
1+m|f(vn)|2(m1))|vn|2+RN
V(x)f2(vn)
=RN
p(x,f(vn)) f(vn)+o(∥vnX)BR
p(x,f(vn)) f(vn)+o(1)∥vnX.
Combining the above two inequalities, we get
(θ
2m)ε2RN
|vn|2+(θ
2θ
2k1)RN
V(x)f2(vn)
θε2
2RN
|vn|2RN
ε2(1+m(m1)| f(vn)|2(m1)
1+m|f(vn)|2(m1))|vn|2
+θ
2RN
V(x)f2(vn)θ
2kBc
R
V(x)f2(vn)RN
V(x)f2(vn)
θcε+o(1)+o(1)∥vnX.(3.3)
Since θ > 2mand k>θ
θ2, we get the conclusion from (3.3).
Step 2: For every δ > 0, there exists R1R>0 such that
lim sup
nBc
2R1
(|vn|2+V(x)f2(vn)) < δ. (3.4)
We consider a cutofunction ψR1=0 on BR1,ψR1=1 on Bc
2R1, and |ψR1|C/R1on RNfor some
constant C>0. On the one hand, taking ϕ=f(vn)/f(vn), we compute ¯
J(vn), ϕψR1and get
o(1)∥vnX=RN
ε2(1+m(m1)| f(vn)|2(m1)
1+m|f(vn)|2(m1))|vn|2ψR1
+RN
ε2ϕvnψR1+RN
V(x)f2(vn)ψR1
RN
p(x,f(vn)) f(vn)ψR1
RN
ε2|vn|2ψR1+RN
ε2ϕvnψR1
+(11
k)RN
V(x)f2(vn)ψR1.(3.5)
On the other hand, by Hölder inequality,
RN
ϕvnψR1C
R1
vnL2(RN)ϕL2(RN).(3.6)
Note that vnL2(RN)is bounded, and
ϕ2
L2(RN)=RN
f2(vn)(1+m|f(vn)|2(m1))
=RN
f2(vn)+mRN
|f(vn)|2m,(3.7)
by (2.5), ϕL2(RN)is also bounded. Therefore, it follows from (3.5)-(3.7) that
lim sup
nBc
2R1
(|vn|2+V(x)f2(vn)) C
R1
for R1suciently large, which yields (3.4).
Step 3: There exists vXsuch that
lim
n→ ∞ RN
p(x,f(vn)) f(vn)=RN
p(x,f(v)) f(v).(3.8)
021501-6 Z. Li and Y. Zhang J. Math. Phys. 58, 021501 (2017)
First, by step 1, there exists vXsuch that up to a subsequence, vnvweakly in Xand vnv
a.e. in RN. Since we may replace vnby |vn|, we assume vn0 and v0. By (3.4), for any δ > 0,
there exists R1>0 suciently large such that
lim sup
nBc
2R1
(|vn|2+V(x)f2(vn)) kδ.
Therefore, by (p2) we have
lim sup
nBc
2R1
p(x,f(vn)) f(vn)lim sup
nBc
2R1
V(x)
kf2(vn)δ, (3.9)
and by Fatou lemma,
Bc
2R1
p(x,f(v)) f(v)δ. (3.10)
Second, we prove that
B2R1
p(x,f(vn)) f(vn)B2R1
p(x,f(v)) f(v).(3.11)
Then from this, (3.9)-(3.10), and the arbitrariness of δ, we get (3.8). In fact, since (vn)is bounded
in X, we have (f(vn)) is also bounded. Thus there exists a wXsuch that f(vn)⇀ w in X,
f(vn)win Lr(BR1)for 1 r<2, and f(vn)wa.e. in BR1. According to (2.4), (| f(vn)|m)
is also bounded in X. By a normal argument, we have |f(vn)|m|w|min X,|f(vn)|m|w|min
Lr(BR1)for 1 r<2, and |f(vn)|m|w|ma.e. in BR1. Applying Lions’ concentration compact-
ness principle11 to (| f(vn)|m)on ¯
BR1, we obtain that there exist two nonnegative measures µ, ν, a
countable index set K, positive constants {µk},{νk},kK, and a collection of points {xk},kK
in ¯
BR1, such that
(i) ν=|w|2m+
kK
νkδxk;
(ii) µ=|(|w|m)|2+
kK
µkδxk;
(iii) µkSν2/2
k,
where δxkis the Dirac measure at xkand Sis the best Sobolev constant. We claim that νk=0 for
all kK. In fact, let xkbe a singular point of measures µand ν, as in Ref. 9, we define a function
φC
0(RN)by
φ(x)=
1,Bρ(xk),
0,RN\B2ρ(xk),
φ0,|φ|1
ρ,B2ρ(xk)\Bρ(xk),
where Bρ(xk)is a ball centered at xkand with radius ρ > 0. We take ϕ=φf(vn)/f(vn)as test
functions in ¯
J(vn), ϕand get
RN
ε2(1+m(m1)| f(vn)|2(m1)
1+m|f(vn)|2(m1))|vn|2·φ
+RN
ε2vnφ·f(vn)/f(vn)+RN
V(x)f2(vn)φ
RN
p(x,f(vn)) f(vn)φ=o(1)∥vnφX.(3.12)
Then Lions’ concentration compactness principle implies that
BR1
||f(vn)|m|2φBR1
φdµ, BR1
|f(vn)|2mφBR1
φdν. (3.13)
021501-7 Z. Li and Y. Zhang J. Math. Phys. 58, 021501 (2017)
Since xkis singular point of ν, by the continuity of f, we have
f(vn(x))|(B2ρ\{xk}) → ∞
as ρ0. Thus
1+m(m1)| f(vn)|2(m1)
1+m|f(vn)|2(m1)=mo(ρ)
on B2ρfor ρsuciently small. Then by (2.4) we get from (3.12) that
BR1
ε2φdµBR1
φdν
=lim
nBR1
ε2||f(vn)|m|2φBR1
|f(vn)|2mφ
lim
nBR1
mε2|vn|2φBR1
|f(vn)|2mφ
lim
nBR1
ε2(1+m(m1)| f(vn)|2(m1)
1+m|f(vn)|2(m1))|vn|2φ
+o(ρ)BR1
ε2|vn|2φBR1
|f(vn)|2mφ
lim
nBR1
ε2vnφ·f(vn)/f(vn)+λBR1
|f(vn)|qφ
+o(ρ)BR1
ε2|vn|2φ+o(1)∥vnφX.(3.14)
We prove that the last inequality in (3.14) tends to zero as ρ0. By Hölder inequality, we have
lim
n→ ∞ BR1vnφ·f(vn)/f(vn)
lim sup
n(BR1
|vn|2)1/2
·(BR1
|[ f(vn)/f(vn)] · ∇φ|2)1/2
.(3.15)
Since |f(vn)/f(vn)|2=f2(vn)+m|f(vn)|2m, using Hölder inequality we have
lim
n→ ∞ BR1
|[ f(vn)/f(vn)] · ∇φ|2
Cρw2
L2(B2ρ(xj)) +w2
L2m(B2ρ(xj))0
as ρ0. Thus we obtain that the right hand side of (3.15) tends to 0. On the other hand, since
q(2m,2m), by Lemma 2.2, we can prove that g(x,h(vn))h(vn)φg(x, w )in L1(BR1)and
BR1g(x, w)w φ 0 as ρ0. All these facts imply that the last inequality in (3.14) tends to zero as
ρ0. Thus νkε2µk. This means that either νk=0 or νkεNSN/2by virtue of Lions’ concen-
tration compactness principle. We claim that the latter is impossible. Indeed, if νkεNSN/2holds
for some kK, then
cε=lim
n¯
J(vn)1
2m¯
J(vn),f(vn)/f(vn)⟩
lim
n(1
2m1
2m)RN
|f(vn)|2m(1
2m1
2m)RN
dν
(1
2m1
2m)RN
|w|2m+(1
2m1
2m)SN/2εN1
N m εNSN/2,
which is a contradiction. Thus νk=0 for all kK, and it implies that f(vn)L2m(BR1)
wL2m(BR1). By the uniform convexity of L2m(BR1), we have f(vn)wstrongly in L2m(BR1).
021501-8 Z. Li and Y. Zhang J. Math. Phys. 58, 021501 (2017)
Finally, since p(x,f(vn)) f(vn)is sub-(2m)growth on B2R1\BR1, we conclude that (3.11) holds. This
proves (3.8).
Step 4: (vn)is compact in X. Since we have (3.8), the proof of the compactness is trivial. This
completes the proof of the lemma.
IV. PROOF OF MAIN RESULTS
Before we prove Theorem 1.1, we will show first some properties about the change of variable
f.
Lemma 4.1. Let f 1(v)=|f(v)|m/v , v ,0and f1(0)=0, then f 1is continuous, odd, nonde-
creasing and
lim
v0f1(v)=0and lim
|v|+|f1(v)| =m.(4.1)
Proof. In fact, by (6) of Lemma 2.1,
f
1(v)=v2(m|f(v)|m2f(v)f(v)v|f(v)|m)0,
so f1is nondecreasing. By (4) of Lemma 2.1, f1(v)0 as v0. Finally, according to L’Hospital’s
principle,
lim
v+f1(v)=lim
v+
|f(v)|m
v=lim
v+m|f(v)|m2f(v)f(v)=m.
This shows that (4.1) holds.
Lemma 4.2. There exists d0>0such that
lim
v+(mvfm(v)) d0.
Proof. Assume that v > 0. Since by (6) of Lemma 2.1, f(v)m f (v)v, we have
mvfm(v)mvm f m1(v)f(v)v
=1+m f 2(m1)(v)m f m1(v)
1+m f 2(m1)(v)
mv
=mv
1+m f 2(m1)(v)+m f m1(v)1+m f 2(m1)(v)
mv
2(1+m f 2(m1)(v)) fm(v)
4m f 2(m1)(v)
=1
4m f m2(v)Bd(m, v ).(4.2)
In the last inequality, we have used the fact that mvfm(v)and that m f 2(m1)(v)>1 for v > 0
suciently large.
If 1 <m<2, then d(m, v )+as v+. If m=2, then d(m, v )=1/8. If m>2, we
claim that mvfm(v)0 as v+is impossible. In fact, assume on the contrary, then using
L’Hospital’s principle, we get
0<lim
v+
mvfm(v)
f2m(v)=lim
v+
mm f m1(v)f(v)
(2m)f1m(v)f(v)
=lim
v+
m
(2m)f1m(v)m1+m f 2(m1)(v)+m f m1(v)
=1
2(2m)<0.
021501-9 Z. Li and Y. Zhang J. Math. Phys. 58, 021501 (2017)
This is a contradiction. Thus for all m>1, there exists d0>0 such that there holds
lim
v+(mvfm(v)) d0.
This completes the proof.
Lemma 4.3. We have
(i) If 1<m<2, then
lim
v+
mvfm(v)
f2m(v)=1
2(2m).
(ii) If m 2, then
lim
v+
mvfm(v)
log f(v)
1
2,m=2,
0,m>2.
Proof. First, we prove part (i). According to (4.2) in Lemma 4.2, we have mvfm(v)+
as v+. Thus by L’Hospital’s principle, we get
lim
v+
mvfm(v)
f2m(v)=lim
v+
mm f m1(v)f(v)
(2m)f1m(v)f(v)=1
2(2m).
Next, we prove part (ii). If there exists a constant C>0 such that mvfm(v)C, then
the conclusion holds. Otherwise, assume that mvfm(v)+as v+. Then again by
L’Hospital’s principle, we have
lim
v+
mvfm(v)
log f(v)=lim
v+
mm f m1(v)f(v)
f(v)/f(v)=
1
2,m=2,
0,m>2.
This completes the proof.
To prove Theorem 1.1, it is crucial to prove that ¯
Jhas the mountain pass level cε<1
N m εNSN/2.
Let us consider the following family of functions in Ref. 3:
v
ω(x)=[N(N2)ω2](N2)/4
[ω2+|x|2](N2)/2,
which solves the equation u=u21in RNand satisfies v
ω2
L2=v
ω2
L2=SN/2. Let ωbe
such that 2ω < Rand let ηω(x)[0,1]be a positive smooth cutofunction with ηω(x)=1 in Bω,
ηω(x)=0 in BR\B2ω. Let vω=ηωv
ω. For all ω > 0, there exists tω>0 such that ¯
J(tωvω)<0 for
all t>tω. Define the class of paths
Γ={γC([0,1],X):γ(0)=0, γ(1)=tωvω},
and the minimax level
cε=inf
γΓmax
t[0,1]
¯
J(γ(t)).
Let tωbe such that
¯
J(tωvω)=max
t0
¯
J(tvω).
Note that the sequence (vω)is uniformly bounded in X, then if ¯
J(tωvω)0 as tω0, we are done;
on the other hand, if tω+, then ¯
J(tωvω)→ −∞, which is impossible, so it remains to consider
the case where the sequence (tω)is upper and lower bounded by two positive constants. According
to Ref. 3, we have, as ω0,
vω2
L2=SN/2+O(ωN2),vω2
L2=SN/2+O(ωN).
021501-10 Z. Li and Y. Zhang J. Math. Phys. 58, 021501 (2017)
Let a(0,ε(N2)/2
2m),b(2ε(N2)/2
m,+)be such that tω[a,b],ω(00), where ω0>0 small
enough. By computing d
dt ¯
J(tvω)=0, we obtain tω=ε(N2)/2
m+o(1). Let
H(v)=1
2V(x)f2(v)+λ
q|f(v)|q1
2m|mv|2+1
2m|f(v)|2m,
then by (4.1) and (4) of Lemma 2.1, for m>1, we have
lim
|v|+H(v)/|v|2=0 and lim
v0H(v)/v2=1
2V(x).
Thus H(v)is sub-(2)growth.
The following proposition is important to the computation of a mountain pass level cε<
1
N m εnSN/2.
Proposition 4.4. Under the assumptions of Theorem 1.1, there exists a function τ=τ(ω)such
that limω0τ(ω)= +and for ωsmall enough,
RN
H(tωvω)τ(ω)·ωN2.
Proof. We divide the proof into three steps.
Step 1: We prove that
1
ωN2Bω
H(tωvω)τ1(ω)(4.3)
with limω0τ1(ω)= +.
By the definition of vω, for xBω, there exist constants c2c1>0 such that for ωsmall
enough, we have
c1ω(N2)/2vω(x)c2ω(N2)/2
and
c1ω(N2)/2fm(vω(x)) c2ω(N2)/2.(4.4)
On the one hand, by (7) of Lemma 2.1, (4.4) and the continuity of V(x)in ¯
Bω, there exists C1>0
such that
Bω
V(x)f2(tωvω)C1ωN2
mN2
2=C1ω(2
21
m)(N2).(4.5)
Similarly, there exists C2>0 such that
Bω
fq(tωvω)C2ωNq
mN2
2=C2ω(2
2¯q
2)(N2),(4.6)
where ¯q=q/m. On the other hand, since the function ζ(t)=t2is convex, using (5) of Lemma 2.1
and Hölder inequality, we have
1
2mBω(mtωvω)2(fm(tωvω))2
1
mBω
(mtωvω)21[mtωvωfm(tωvω)]
1
m(Bω
(mtωvω)2)(21)/2(Bω
[mtωvωfm(tωvω)]2)1/2.(4.7)
Case 1: 1 <m<2. From (4.7) and (i) of Lemma 4.3, we obtain that there exists C3>0 such
that
1
2mBω(mtωvω)2(fm(tωvω))2C3ω[N(2
m1)N2
22]1
2=C3ω(11
m)(N2).(4.8)
021501-11 Z. Li and Y. Zhang J. Math. Phys. 58, 021501 (2017)
Combining (4.5), (4.6), and (4.8), we have
1
ωN2Bω
H(tωvω)≥ −C1ω(2
21
m1)(N2)+C2ω(2
2¯q
21)(N2)C3ω1
m(N2)Bτ1(ω).
It is obvious that 2
21
m1>1
m. Using the condition (i) in Theorem 1.1, we have 2
2¯q
21<
1
m. It results that τ1(ω)+as ω0.
Case 2:m2. Note that for any δ(0,m), limv+log f(v)/fδ(v)=0,we have log f(v)
fδ(v)for v > 0 large enough. Thus for ω > 0 small enough, from (4.7) and (ii) of Lemma 4.3, we
get
1
2mBω(mtωvω)2(fm(tωvω))2C
3ω[Nδ
mN2
22]1
2=C
3ω1
2(1δ
m)(N2).(4.9)
Combining (4.5), (4.6), and (4.9), we have
1
ωN2Bω
H(tωvω)≥ −C1ω(2
21
m1)(N2)+C2ω(2
2¯q
21)(N2)C
3ω1
2(1+δ
m)(N2)Bτ1(ω).
Since m2, we have 2
21
m1>1
2(1+δ
m). From condition (ii) in Theorem 1.1, there exists a
δ=δ(N,¯q)>0 (depends on Nand ¯q) small enough such that 2
2¯q
21<1
2(1+δ
m). It results
that τ1(ω)+as ω0.
Cases 1 and 2 show that (4.3) holds.
Step 2: We prove that there exists C4>0 such that
1
ωN2B2ω\Bω
H(tωvω)≥ −C4ω(2
21
m1)(N2)Bτ2(ω).(4.10)
Note that for xB2ω\Bω, we have
vω(x)v
ω(x)c2ω(N2)/2.(4.11)
Since ηωis a positive smooth cutofunction, without the loss of generality, we may assume that ηω
is such that
B2ω\Bω
|vω|2B2ω\Bω
V(x)f2(vω).
By (4.1) and (4.11), we have fm(vω(x)) c2ω(N2)/2for xB2ω\Bω. Thus
1
ωN2B2ω\Bω
H(tωvω)
≥ − 1
2ωN2B2ω\Bω
V(x)f2(tωvω)1
2mωN2B2ω\Bω
|mtωvω|2
≥ − C5
ωN2B2ω\Bω
V(x)f2(tωvω)
≥ −C4ωN2
mN2
2(N2)=C4ω(2
21
m1)(N2),
where C4>0, C5>0 are constants. This shows that (4.10) holds.
Step 3: To conclude, let τ(ω)=τ1(ω)+τ2(ω), we have τ(ω)+as ω0. This implies the
conclusion of the proposition.
Proof of Theorem 1.1. By (4) and (7) of Lemma 2.1, it is easy to verify that ¯
Jhas the moun-
tain pass geometry. Lemma 3.1 shows that ¯
Jsatisfies the PS condition. We prove that ¯
Jhas the
mountain pass level cε<1
N m εNSN/2. Let
F(t)=ε2
2(tvω)∥2
L21
2mmtvω2
L2.
Then we have
F(t)F(t0)=1
N m εNSN/2+O(ωN2),t0,(4.12)
021501-12 Z. Li and Y. Zhang J. Math. Phys. 58, 021501 (2017)
where t0=ε(N2)/2
m. By (4.12) and Proposition 4.4, we have
¯
J(tωvω)=F(tωvω)RN
H(tωvω)
1
N m εNSN/2+O(ωN2)τ(ω)ωN2
<1
N m εNSN/2.
This shows that ¯
J(v)has a nontrivial critical point vεX, which is a weak solution of (2.3).
We prove that vεis also a weak solution of (2.1). First, we can argue as the proof of Proposition
2.1 in Ref. 17 to obtain that
lim
ε0max
xBR
vε(x)=0.
Thus there exists ε0>0 such that for all ε(0, ε0), we have vε(x)aBf1(l),|x|=R, where l
is given in (2.2). Secondly, we prove that
vε(x)a,ε(0, ε0)and xRN\BR.(4.13)
Taking
ϕ=
(vεa)+,xRN\BR,
0,xBR
as a test function in ¯
J(vε), ϕ=0, we get
ε2RN\BR
|(vεa)+|2+ε2RN\BR(V(x)p(x,f(vε))
f(vε))f(vε)f(vε)(vεa)+=0.(4.14)
By (p2), we have
V(x)p(x,f(vε))
f(vε)>0,xRN\BR.
Therefore, all terms in (4.14) must be equal to zero. This implies vεain RN\BR. This proves
(4.13). Thus vεis a solution of problem (2.1).
To complete the proof, we deduce as the proof for Theorem 4.1 in Ref. 10 to obtain that
vε|BRL(BR). Thus uε=f(vε)XL(RN)is a nontrivial weak solution of (1.1). Finally, by
Proposition 4.5 in the following, we have limε0uεX=0 and uε(x)Ceβ
ε|xxε|.This completes
the proof.
We prove the norm estimate and the exponential decay.
Proposition 4.5. Let vεXL(RN)be a solution of (2.1) and let uε=f(vε), then we have
lim
ε0uεX=0and uε(x)Ceβ
ε|xxε|,
where C >0,β > 0are constants.
Proof. First, let x0BRbe such that V(x0)=V0. Define J0:XRby
J0(v)=1
2RN
|v|2+1
2RN
V0f2(v)RN
G(f(v)).
Let
c0=inf
γΓ0
sup
t[0,1]
J0(γ(t)),
Γ0={vC([0,1],X):γ(0)=0,J0(γ(1)) <0}.
021501-13 Z. Li and Y. Zhang J. Math. Phys. 58, 021501 (2017)
Similar to the proof for estimate (2.4) in Ref. 17 (or Lemma 3.1 in Ref. 18), we can show that
cεεNc0+o(εN)by using the change of coordinates y=(xx0). Arguing as for (3.3), and by
virtue of this energy estimate, we obtain
vεXθcε
min{( θ
2m)ε2,(θ
2θ
2k1)} 2θc0
θ2mεN2+o(εN2)
for ε > 0 suciently small. Let uε=f(vε), then uε,0. Note that |uε||vε|and |uε||vε|, we
get limε0uεX=0.
Second, similar to the proof for Theorem 4.1 in Ref. 10, we conclude that vεL(RN)
and by Ref. 8, we have vεC1(BR). Now let xεdenote the maximum point of vεin BRand
let
σBsup{s>0 : g(t)<V0tfor every t[0,s]}.
Then vε(xε)f1(σ)for ε > 0 small. In fact, assume that vε(xε)<f1(σ)for some ε > 0 su-
ciently small. According to the definition of l(see (2.2)) and σ, we have vε(x)f1(l)<f1(σ)
(note that k>1 in (2.2)), xRN\BR. Thus
V(x)g(f(vε))
f(vε)>0,xRN.
Since vε=f1(uε)is a critical point of Jε, we choose ϕ=f(vε)/f(vε)as a test function in
J
ε(vε), ϕ=0 and get
0=ε2RN(1+m(m1)| f(vn)|2(m1)
1+m|f(vn)|2(m1))|vε|2
+RN
V(x)f2(vε)RN
g(f(vε)) f(vε)
=ε2RN(1+m(m1)| f(vn)|2(m1)
1+m|f(vn)|2(m1))|vε|2
+RN(V(x)g(f(vε))
f(vε))f2(vε).
It turns out that all terms in the above equality must be equal to zero, which means that vε0, a
contradiction.
Now let wε(x)=vε(xε+εx), then wεsolves the equation
wε+V(xε+εx)f(wε)f(wε)=g(f(wε)) f(wε),xRN.
Note that limt0+f(t)f(t)
t=1 by the properties of fand that wε(x)0 as |x|+; we have,
there exists R0>0 such that for all |x|R0,
f(wε(x)) f(wε(x)) 3
4wε(x)(4.15)
and
g(f(wε(x))) f(wε(x)) V0
2wε(x).(4.16)
Let ϕ(x)=Meβ|x|with β2<V0
4and MeβR0wε(x)for all |x|=R0. It is easy to verify that for
x,0,
ϕβ2ϕ. (4.17)
Now define ψε=ϕwε. Using (4.15)-(4.17), we have
ψε+V0
4ψε0,in |x|R0,
ψε0,in |x|=R0,
lim|x|ψε=0.
021501-14 Z. Li and Y. Zhang J. Math. Phys. 58, 021501 (2017)
By the maximum principle, we have ψε0 for all |x|R0. Thus, we obtain that for all |x|
R0,
wε(x)ϕ(x)Meβ|x|.
Using the change of variable, we have that for all |x|R0,
vε(x)=wε(ε1(xxε)) Meβ
ε|xxε|.
Then by the regularity of vεon BRand note that f(t)tfor all t0, we have
uε(x)Ceβ
ε|xxε|
for some C>0. This completes the proof.
Proof of Theorem 1.2. We consider the following equation:
u+V(x)ukα((|u|2α))|u|2α2u=|u|q2u+|u|2(2α)2u,u>0,xRN.(4.18)
Let y=εxwith ε(0, ε0),ε0is given by Theorem 1.1, then we can transform (4.18) into
ε2u+¯
V(y)ukαε2((|u|2α))|u|2α2u=|u|q2u+|u|2(2α)2u,u>0, y RN.(4.19)
Here ¯
V(y)=V(y
ε)still has the properties given in assumption (V). Thus according to Theorem 1.1,
(4.19) has a positive weak solution uε(y)in XL(RN), which implies that (4.18) has a positive
weak solution u1(x)=uε(εx).
ACKNOWLEDGMENTS
The authors would like to express their sincere gratitude to the anonymous referee for his/her
valuable suggestions and comments. The first author was supported by the Natural Science
Foundation of China (No. 11201488) and the Hunan Provincial Natural Science Foundation of
China (No. 14JJ4002). The second author was supported by the Natural Science Foundation of
China (No. 11471330) and the Fundamental Research Funds for the Central Universities(WUT:
2017 IVA 075).
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After the study made in the locally compact case for variational problems with some translation invariance, we investigate here the variational problems (with constraints) for example in RN where the invariance of RN by the group of dilatations creates some possible loss of compactness. This is for example the case for all the problems associated with the determination of extremal functions in functional inequalities (like for example the Sobolev inequalities). We show here how the concentration-compactness principle has to be modified in order to be able to treat this class of problems and we present applications to Functional Analysis, Mathematical Physics, Differential Geometry and Harmonic Analysis.
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We are concerned with a quasilinear elliptic equation of the form -Δu + a(x)u - Δ(|u|α)|u|α-2u = h(u) in ℝN, where α > 1 and N ≥ 1. By using variational approaches, we prove the existence of at least one positive solution of the above equation under suitable conditions on a(x) and h. In particular, we are interested in the situation that a(x) is invariant under the finite group action G.
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For a class of quasilinear Schrödinger equations with critical exponent we establish the existence of both one-sign and nodal ground states of soliton type solutions by the Nehari method. The method is to analyze the behavior of solutions for subcritical problems from our earlier work (Liu et al. Commun Partial Differ Equ 29:879–901, 2004) and to pass limit as the exponent approaches to the critical exponent.
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We are concerned with the uniqueness result of positive solutions for a class of quasilinear elliptic equation arising from plasma physics. We convert a quasilinear elliptic equation into a semilinear one and show the unique existence of positive radial solution for original equation under the suitable conditions on the power of nonlinearity and quasilinearity. We also investigate the non-degeneracy of a positive radial solution for a converted semilinear elliptic equation.
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Nonlinear time evolution of the condensate wave function in superfluid films is studied on the basis of a Schrödinger equation, which incorporates van der Waals potential due to substrate in its fully nonlinear form, and a surface tension term. In the weak nonlinearity limit, our equation reduces to the ordinary (cubic) nonlinear Schrödinger equation for which exact soliton solutions are known. It is demonstrated by numerical analysis that even under strong nonlinearity, where our equation is far different from cubic Schrödinger equation, there exist quite stable composite ``quasi-solitons''. These quasi-solitons are bound states of localized excitations of amplitude and phase of the condensate (superfluid thickness and superfluid velocity, in more physical terms). Thus the present work shows the persistence of the solitonic behavior of superfluid films in the fully nonlinear situation.