- Email: [email protected]
Soliton solutions for fractional Schrödinger equations
Accepted Manuscript Soliton solutions for fractional Schr¨odinger equations Quanqing Li, Xian Wu PII: DOI: Reference:
S0893-9659(15)00294-3 http://dx.doi.org/10.1016/j.aml.2015.10.006 AML 4876
To appear in:
Applied Mathematics Letters
Received date: 5 August 2015 Revised date: 12 October 2015 Accepted date: 13 October 2015 Please cite this article as: Q. Li, X. Wu, Soliton solutions for fractional Schr¨odinger equations, Appl. Math. Lett. (2015), http://dx.doi.org/10.1016/j.aml.2015.10.006 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
*Manuscript Click here to view linked References
Soliton solutions for fractional Schr¨odinger equations ∗ Quanqing Li Department of Mathematics and Statistics, Yunnan University, Kunming, Yunnan 650091, P.R.China
Xian Wu† Department of Mathematics, Yunnan Normal University, Kunming, Yunnan 650092, P.R.China
email: [email protected]
ABSTRACT. In this paper, we study the existence of a soliton type solution for the fractional Schr¨odinger equation (−∆)s u + V (x)u + [(−∆)s u2 ]u = λ|u|p−1 u, u > 0, x ∈ RN , where 0 < s < 1, (−∆)s denotes the fractional Laplacian of order s, N > 2s, 2∗s =
2N N −2s ,
1 < p < 2∗s − 1. We prove that the equation has a solution by
using a constrained minimization argument.
1991 Mathematics Subject Classifications: 35J20, 35J70, 35P05, 35P30,34B15, 58E05, 47H04.
Key Words: ∗
fractional Schr¨odinger equation; soliton solution; minimization argument.
This work is supported in part by the National Natural Science Foundation of China (11261070) and
the Scientific New Talent Award for Excellent Doctoral Student of Yunnan Province. † Corresponding author: [email protected]
1
1. Introduction and preliminaries Consider the following fractional Schr¨odinger equation (−∆)s u + V (x)u + [(−∆)s u2 ]u = λ|u|p−1 u, u > 0, x ∈ RN , where N > 2s with s ∈ (0, 1), 2∗s =
2N N −2s ,
(1.1)
1 < p < 2∗s − 1.
Solutions of (1.1) are related to the existence of solitary wave solutions for the following equation iψt = (−∆)s ψ + [V (x) + E]ψ − η(|ψ|2 )ψ + κ[(−∆)s ρ(|ψ|2 )]ρ0 (|ψ|2 )ψ,
(1.2)
where ψ : RN × R → C, κ is a real constant and η, ρ are real functions. Here we are
interested in the existence of solutions for the equation (1.2) with ρ(s) = s. If we take
ψ(x, t) = e−iEt u(x) and κ = 1 in (1.2), where E ∈ R and u > 0 is a real function, then
(1.2) turns into (1.1). For κ = 0, in [1], under appropriate assumptions, Autuori and Pucci
obtained the existence of nontrivial non-negative entire solution of the equation (−∆)s u + a(x)u = λw(x)|u|q−2 u − h(x)|u|r−2 u, in RN , via a modification of the mountain pass theorem of Ambrosetti and Rabinowitz, which N established in [2], where the coefficient a is supposed to be in L∞ loc (R ) and to satisfy
a(x) ≥ κν(x) for some constant κ ∈ (0, 1] and for almost all x ∈ RN , where ν(x) :=
max{a(x), (1+|x|)−2s }. In [3], Fiscella and Valdinoci obtained the existence of non-negative weak solutions of the equation −M (R 2N |u(x) − u(y)|2 K(x − y)dxdy)LK u = λf (x, u) + |u|2∗s −2 u, x ∈ Ω, R u = 0, x ∈ RN \ Ω,
by using the concentration-compactness principle, where λ > 0, Ω ⊂ RN is an open bounded
set and LK is an integro-differential operator with kernel K defined as follows: LK u(x) =
1 2
Z
RN
[u(x + y) + u(x − y) − 2u(x)]K(y)dy
2
for all x ∈ RN . Especially, in [6], by applying Ljusternik-Schnirelemann category theory, Shang and Zhang studied the existence and multiplicity of solutions for the critical fractional Schr¨odinger equation ∗
ε2s (−∆)s u + V (x)u = |u|2s −2 u + λf (u), x ∈ RN , where the potential V : RN → R is a continuous function satisfying 0 < inf V (x) := V0 < x∈RN
lim inf V (x) := V∞ < +∞. It is worth noticing that the case κ = 0 has been extensively |x|→∞
studied in recent years. However, to our best knowledge, there is no any result for the case κ = 1. In this paper, we assume that the following condition holds. (V ) lim V (x) = +∞. |x|→∞
For any 0 < s < 1, the fractional Sobolev space H s (RN ) is defined by H s (RN ) = {u ∈ L2 (RN ) :
|u(x) − u(y)| |x − y|
N +2s 2
∈ L2 (RN × RN )},
endowed with the natural norm kukH s (RN ) = (
Z
RN
u dx + 2
where the term [u]H s (RN ) := (
Z
R2N
Z
R2N
1 |u(x) − u(y)|2 dxdy) 2 , N +2s |x − y|
1 |u(x) − u(y)|2 dxdy) 2 |x − y|N +2s
is the so-called Gagliardo semi-norm of u. Let S be the Schwartz space of rapidly decaying C ∞ functions in RN , for any u ∈ S and s ∈ (0, 1), (−∆)s is defined as Z Z u(x) − u(y) u(x) − u(y) s dy = CN,s lim dy, (−∆) u(x) = CN,s P.V. N +2s N +2s + ε→0 C Bε (x) |x − y| RN |x − y|
(1.3)
where C Bε (x) = RN \ Bε (x). The symbol P.V. stands for the Cauchy principal value and CN,s is a dimensional constant that depends on N and s, precisely given by Z 1 − cosζ1 CN,s = ( dζ)−1 . N +2s |ζ| N R
Indeed, the fractional Laplacian (−∆)s can be viewed as a pseudo-differential operator of symbol |ξ|2s , as stated in the following, refer to Lemma 1.1 in [5]. 3
Let s ∈ (0, 1) and (−∆)s : S → L2 (RN ) be the fractional Laplacian operator defined
by (1.3). Then, for any u ∈ S ,
(−∆)s u(x) = F −1 (|ξ|2s (F u))(x), ∀ξ ∈ RN . N
Here, F u := u ˆ := (2π)− 2
R
RN
u(x)e−ix·ξ dx denotes the Fourier transform of u. Then we
can see that an alternative definition of the fractional Sobolev space H s (RN ) via the Fourier transform as follows: H s (RN ) = {u ∈ L2 (RN ) :
Z
RN
(1 + |ξ|2s )|ˆ u(ξ)|2 dξ < +∞}.
Propositions 3.4 and 3.6 in [5] imply that Z s −1 −1 2CN,s |ξ|2s |ˆ u(ξ)|2 dξ = 2CN,s k(−∆) 2 uk2L2 (RN ) = [u]2H s (RN ) . RN
Consequently, Z Z −1 −1 2s ¯ |ξ| u ˆ(ξ)ϕ(ξ)dξ ˆ = 2CN,s 2CN,s
(−∆) u·ϕdx = s
RN
RN
Z
R2N
[u(x) − u(y)][ϕ(x) − ϕ(y)] dxdy |x − y|N +2s
for all ϕ ∈ H s (RN ). Set H = {u ∈ H s (RN ) : u2 ∈ H s (RN )}. By a similar computation, we have Z −1 c2 (ξ)|2 dξ = 2C −1 k(−∆) 2s u2 k2 2 N = [u2 ]2 s N . 2CN,s |ξ|2s |u N,s L (R ) H (R ) RN
Consequently, Z −1 2CN,s
RN
2s c 2
|ξ| u (ξ)uϕ(ξ)dξ c
−1 =2CN,s
=
Z
Z
[(−∆)s u2 ]uϕdx
RN [u2 (x)
R2N
− u2 (y)][u(x)ϕ(x) − u(y)ϕ(y)] dxdy |x − y|N +2s
for all ϕ ∈ H. Clearly, we can define the energy functional on H associated with (1.1) by Z Z Z Z 1 1 1 λ 2s 2 2 2s c 2 2 J(u) = |ξ| |ˆ u(ξ)| dξ + V (x)u dx + |ξ| |u (ξ)| dξ − |u|p+1 dx 2 RN 2 RN 4 RN p + 1 RN Z Z Z 1 1 1 2s 2 2 |ξ| |ˆ u(ξ)| dξ + V (x)u dx + |ξ|2s |ˆ u(ξ) ∗ u ˆ(ξ)|2 dξ = 2 RN 2 RN 4 RN Z λ |u|p+1 dx, − p + 1 RN 4
R c2 (ξ) = u where u ˆ(ξ) ∗ u ˆ(ξ). Here (f ∗ g)(x) = RN f (y)g(x − y)dy. Moreover, Z Z Z Z 0 s s 2 hJ (u), ϕi = (−∆) uϕdx + V (x)uϕdx + [(−∆) u ]uϕdx − λ |u|p−1 uϕdx N N N N R R R R Z Z Z 2s ¯ˆ(ξ) ∗ ϕ(ξ)]dξ ¯ˆ ¯ |ξ|2s [ˆ u(ξ) ∗ u ˆ(ξ)] · [u = |ξ| u ˆ(ξ)ϕ(ξ)dξ ˆ + V (x)uϕdx + RN RN RN Z −λ |u|p−1 uϕdx N R Z Z 1 [u(x) − u(y)][ϕ(x) − ϕ(y)] = CN,s dxdy + V (x)uϕdx 2 |x − y|N +2s R2N RN Z Z 1 [u2 (x) − u2 (y)][u(x)ϕ(x) − u(y)ϕ(y)] + CN,s dxdy − λ |u|p−1 uϕdx N +2s 2 |x − y| 2N N R R for all u, ϕ ∈ H. Hence u ∈ H is a weak solution of (1.1) if hJ 0 (u), ϕi = 0 for all ϕ ∈ H.
It seems quite clear that H is the working space for studying the problem (1.1). However, H is not a linear space. Consequently, the classical critical point theory and usual min-max techniques can not be directly applied to the energy functional J, and hence we encounter the difficulties caused by the lack of an appropriate working space. Inspired by [4], we use a constrained minimization argument to give a solution of the equation (1.1). Our main result is the following: Theorem 1.1 Let 1 < p < 2∗s − 1. Assume (V ) holds. Then there exist λn → +∞ such that the equation (1.1) with λ = λn has a solution. Set X := {u ∈ H (R ) : s
with the norm kuk2X For any a > 0, we define
=
Z
RN
N
Z
RN
V (x)u2 dx < +∞}
|ξ| |ˆ u(ξ)| dξ + 2s
2
Z
RN
V (x)u2 dx.
ma := inf E(u), u∈Ma
where Ma := {u ∈ H : kukp+1 = a} and E(u) =
Z
RN
|ξ| |ˆ u(ξ)| dξ + 2s
2
Z
RN
V (x)u dx + 2
5
Z
RN
|ξ|2s |ˆ u(ξ) ∗ u ˆ(ξ)|2 dξ.
2. Proof of Theorem 1.1 Proof. We divide the proof into two steps. Step 1: We claim that for each a > 0, ma is achieved at some ua ∈ Ma , which is a weak
solution of equation (1.1) with λ = λa satisfying λa =
ma . ap+1
Indeed, we fix a > 0. Take {un } ⊂ Ma be a minimizing sequence for ma . We may
assume un ≥ 0 for all n. Then by the proof of Lemma 3.4 in [7], we know that the embedding X ,→ Ls (RN ) is compact for 2 ≤ q < 2∗s . Hence, passing to a subsequence, we
have un * ua in X, un → ua in Lq (RN ) for 2 ≤ q < 2∗s and un (x) → ua (x) ≥ 0 for almost
all x ∈ RN . Note that Z Z |ξ|2s |ˆ un (ξ) ∗ u ˆn (ξ)|2 dξ =
c2 (ξ)|2 dξ |ξ|2s |u n Z 1 |u2n (x) − u2n (y)|2 dxdy. = CN,s 2 |x − y|N +2s R2N
RN
RN
By Fatou Lemma we have Z Z |u2a (x) − u2a (y)|2 |u2n (x) − u2n (y)|2 dxdy ≤ lim inf dxdy. n→∞ |x − y|N +2s |x − y|N +2s R2N R2N Consequently,
Z
RN
|ξ| |ˆ ua (ξ) ∗ u ˆa (ξ)| dξ ≤ lim inf 2s
2
n→∞
Z
RN
|ξ|2s |ˆ un (ξ) ∗ u ˆn (ξ)|2 dξ.
Hence by weakly lower semi-continuity of the norm, one has Z Z Z E(ua ) ≤ lim inf [ |ξ|2s |ˆ un (ξ)|2 dξ + V (x)u2n dx] + lim inf n→∞
RN
n→∞
RN
RN
|ξ|2s |ˆ un (ξ) ∗ u ˆn (ξ)|2 dξ
≤ lim inf E(un ). n→∞
Since 2 < p + 1 < 2∗s , a = kun kp+1 → kua kp+1 , and hence kua kp+1 = a, so ma ≤ E(ua ) ≤ lim inf E(un ) = ma , i.e., E(ua ) = ma . Hence ma is achieved at some ua ∈ Ma . Clearly, we n→∞
can conclude that ua is a weak solution of
(−∆)s u + V (x)u + [(−∆)s u2 ]u = λa |u|p−1 u.
(2.1)
Multiplying the equation (2.1) by ua and integrating over RN we have Z Z Z 2s 2 2 p+1 |ξ| |ˆ ua (ξ)| dξ + V (x)ua dx + |ξ|2s |ˆ ua (ξ) ∗ u ˆa (ξ)|2 dξ = λa kua kp+1 . p+1 = λa · a RN
RN
RN
6
Consequently, λa · ap+1 = E(ua ) = ma , i.e., λa =
ma . ap+1
Step 2: We prove that λa → +∞ as a → 0. Indeed, suppose the conclusion is false, then there exists an → 0 such that λn := λan ≤
C1 . Set un := uan . By kun kp+1 = an → 0 as n → ∞, one has Z p+1 kun k2X + |ξ|2s |ˆ un (ξ) ∗ u ˆn (ξ)|2 dξ = λn kun kp+1 → 0. p+1 ≤ C1 an RN
Since 1 < p < 2∗s − 1, there exists a constant θ ∈ (0, 1) such that Z p+1 kun kp+1 = |un |θ(p+1) |un |(1−θ)(p+1) dx
1 p+1
=
θ 2
+ 1−θ 2∗ . Therefore, s
RN
θ(p+1)
≤kun k2
(1−θ)(p+1)
kun k2∗s
+ (1 − θ)kun kp+1 ≤θkun kp+1 2∗s 2 p+1 ≤C[kun kp+1 X + kun kX ] Z p+1 p+1 ≤C2 {kun kX + [ |ξ|2s |ˆ un (ξ) ∗ u ˆn (ξ)|2 dξ] 2 } N Z R p+1 ≤C2 [kun k2X + |ξ|2s |ˆ un (ξ) ∗ u ˆn (ξ)|2 dξ] 2 . RN
Consequently,
kun k2X +
Z
RN
|ξ|2s |ˆ un (ξ) ∗ u ˆn (ξ)|2 dξ
=λn kun kp+1 p+1
≤C1 C2 [kun k2X which implies that kun k2X +
R
RN
+
Z
RN
|ξ|2s |ˆ un (ξ) ∗ u ˆn (ξ)|2 dξ]
p+1 2
,
|ξ|2s |ˆ un (ξ) ∗ u ˆn (ξ)|2 dξ ≥ C3 > 0 since
p+1 2
> 1, a contra-
diction. By Steps 1 and 2 we complete the proof of Theorem 1.1. ¤
References [1] G. Autuori, P. Pucci, Elliptic problems involving the fractional Laplacian in RN , J. Diff. Eqns. 255(2013), 2340-2362. [2] G. Autuori, P. Pucci, Existence of entire solutions for a class of quasilinear elliptic equations, NoDEA Nonlinear Differential Equations Appl. 20(2013), 977-1009. 7
[3] A. Fiscella, E. Valdinoci, A critical Kirchhoff type problem involving a nonlocal operator, Nonlinear Anal. 94(2014), 156-170. [4] J. Liu, Z. Q. Wang, Soliton solutions for quasilinear Schr¨odinger equations: I, Proc. Amer. Math. Soc. 131(2003), 441-448. [5] E. Di Nezza, G. Palatucci, E. Valdinoci, Hitchhiker, s guide to the fractional Sobolev spaces, Bull. Sci. Math. 136(2012), 521-573. [6] X. Shang, J. Zhang, Ground states for fractional Schr¨odinger equations with critical growth, Nonlinearity, 27(2014), 187-207. [7] W. M. Zou, M. Schechter, Critical Point Theory and its Applications, Springer, New York, (2006).
8
S0893-9659(15)00294-3 http://dx.doi.org/10.1016/j.aml.2015.10.006 AML 4876
To appear in:
Applied Mathematics Letters
Received date: 5 August 2015 Revised date: 12 October 2015 Accepted date: 13 October 2015 Please cite this article as: Q. Li, X. Wu, Soliton solutions for fractional Schr¨odinger equations, Appl. Math. Lett. (2015), http://dx.doi.org/10.1016/j.aml.2015.10.006 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
*Manuscript Click here to view linked References
Soliton solutions for fractional Schr¨odinger equations ∗ Quanqing Li Department of Mathematics and Statistics, Yunnan University, Kunming, Yunnan 650091, P.R.China
Xian Wu† Department of Mathematics, Yunnan Normal University, Kunming, Yunnan 650092, P.R.China
email: [email protected]
ABSTRACT. In this paper, we study the existence of a soliton type solution for the fractional Schr¨odinger equation (−∆)s u + V (x)u + [(−∆)s u2 ]u = λ|u|p−1 u, u > 0, x ∈ RN , where 0 < s < 1, (−∆)s denotes the fractional Laplacian of order s, N > 2s, 2∗s =
2N N −2s ,
1 < p < 2∗s − 1. We prove that the equation has a solution by
using a constrained minimization argument.
1991 Mathematics Subject Classifications: 35J20, 35J70, 35P05, 35P30,34B15, 58E05, 47H04.
Key Words: ∗
fractional Schr¨odinger equation; soliton solution; minimization argument.
This work is supported in part by the National Natural Science Foundation of China (11261070) and
the Scientific New Talent Award for Excellent Doctoral Student of Yunnan Province. † Corresponding author: [email protected]
1
1. Introduction and preliminaries Consider the following fractional Schr¨odinger equation (−∆)s u + V (x)u + [(−∆)s u2 ]u = λ|u|p−1 u, u > 0, x ∈ RN , where N > 2s with s ∈ (0, 1), 2∗s =
2N N −2s ,
(1.1)
1 < p < 2∗s − 1.
Solutions of (1.1) are related to the existence of solitary wave solutions for the following equation iψt = (−∆)s ψ + [V (x) + E]ψ − η(|ψ|2 )ψ + κ[(−∆)s ρ(|ψ|2 )]ρ0 (|ψ|2 )ψ,
(1.2)
where ψ : RN × R → C, κ is a real constant and η, ρ are real functions. Here we are
interested in the existence of solutions for the equation (1.2) with ρ(s) = s. If we take
ψ(x, t) = e−iEt u(x) and κ = 1 in (1.2), where E ∈ R and u > 0 is a real function, then
(1.2) turns into (1.1). For κ = 0, in [1], under appropriate assumptions, Autuori and Pucci
obtained the existence of nontrivial non-negative entire solution of the equation (−∆)s u + a(x)u = λw(x)|u|q−2 u − h(x)|u|r−2 u, in RN , via a modification of the mountain pass theorem of Ambrosetti and Rabinowitz, which N established in [2], where the coefficient a is supposed to be in L∞ loc (R ) and to satisfy
a(x) ≥ κν(x) for some constant κ ∈ (0, 1] and for almost all x ∈ RN , where ν(x) :=
max{a(x), (1+|x|)−2s }. In [3], Fiscella and Valdinoci obtained the existence of non-negative weak solutions of the equation −M (R 2N |u(x) − u(y)|2 K(x − y)dxdy)LK u = λf (x, u) + |u|2∗s −2 u, x ∈ Ω, R u = 0, x ∈ RN \ Ω,
by using the concentration-compactness principle, where λ > 0, Ω ⊂ RN is an open bounded
set and LK is an integro-differential operator with kernel K defined as follows: LK u(x) =
1 2
Z
RN
[u(x + y) + u(x − y) − 2u(x)]K(y)dy
2
for all x ∈ RN . Especially, in [6], by applying Ljusternik-Schnirelemann category theory, Shang and Zhang studied the existence and multiplicity of solutions for the critical fractional Schr¨odinger equation ∗
ε2s (−∆)s u + V (x)u = |u|2s −2 u + λf (u), x ∈ RN , where the potential V : RN → R is a continuous function satisfying 0 < inf V (x) := V0 < x∈RN
lim inf V (x) := V∞ < +∞. It is worth noticing that the case κ = 0 has been extensively |x|→∞
studied in recent years. However, to our best knowledge, there is no any result for the case κ = 1. In this paper, we assume that the following condition holds. (V ) lim V (x) = +∞. |x|→∞
For any 0 < s < 1, the fractional Sobolev space H s (RN ) is defined by H s (RN ) = {u ∈ L2 (RN ) :
|u(x) − u(y)| |x − y|
N +2s 2
∈ L2 (RN × RN )},
endowed with the natural norm kukH s (RN ) = (
Z
RN
u dx + 2
where the term [u]H s (RN ) := (
Z
R2N
Z
R2N
1 |u(x) − u(y)|2 dxdy) 2 , N +2s |x − y|
1 |u(x) − u(y)|2 dxdy) 2 |x − y|N +2s
is the so-called Gagliardo semi-norm of u. Let S be the Schwartz space of rapidly decaying C ∞ functions in RN , for any u ∈ S and s ∈ (0, 1), (−∆)s is defined as Z Z u(x) − u(y) u(x) − u(y) s dy = CN,s lim dy, (−∆) u(x) = CN,s P.V. N +2s N +2s + ε→0 C Bε (x) |x − y| RN |x − y|
(1.3)
where C Bε (x) = RN \ Bε (x). The symbol P.V. stands for the Cauchy principal value and CN,s is a dimensional constant that depends on N and s, precisely given by Z 1 − cosζ1 CN,s = ( dζ)−1 . N +2s |ζ| N R
Indeed, the fractional Laplacian (−∆)s can be viewed as a pseudo-differential operator of symbol |ξ|2s , as stated in the following, refer to Lemma 1.1 in [5]. 3
Let s ∈ (0, 1) and (−∆)s : S → L2 (RN ) be the fractional Laplacian operator defined
by (1.3). Then, for any u ∈ S ,
(−∆)s u(x) = F −1 (|ξ|2s (F u))(x), ∀ξ ∈ RN . N
Here, F u := u ˆ := (2π)− 2
R
RN
u(x)e−ix·ξ dx denotes the Fourier transform of u. Then we
can see that an alternative definition of the fractional Sobolev space H s (RN ) via the Fourier transform as follows: H s (RN ) = {u ∈ L2 (RN ) :
Z
RN
(1 + |ξ|2s )|ˆ u(ξ)|2 dξ < +∞}.
Propositions 3.4 and 3.6 in [5] imply that Z s −1 −1 2CN,s |ξ|2s |ˆ u(ξ)|2 dξ = 2CN,s k(−∆) 2 uk2L2 (RN ) = [u]2H s (RN ) . RN
Consequently, Z Z −1 −1 2s ¯ |ξ| u ˆ(ξ)ϕ(ξ)dξ ˆ = 2CN,s 2CN,s
(−∆) u·ϕdx = s
RN
RN
Z
R2N
[u(x) − u(y)][ϕ(x) − ϕ(y)] dxdy |x − y|N +2s
for all ϕ ∈ H s (RN ). Set H = {u ∈ H s (RN ) : u2 ∈ H s (RN )}. By a similar computation, we have Z −1 c2 (ξ)|2 dξ = 2C −1 k(−∆) 2s u2 k2 2 N = [u2 ]2 s N . 2CN,s |ξ|2s |u N,s L (R ) H (R ) RN
Consequently, Z −1 2CN,s
RN
2s c 2
|ξ| u (ξ)uϕ(ξ)dξ c
−1 =2CN,s
=
Z
Z
[(−∆)s u2 ]uϕdx
RN [u2 (x)
R2N
− u2 (y)][u(x)ϕ(x) − u(y)ϕ(y)] dxdy |x − y|N +2s
for all ϕ ∈ H. Clearly, we can define the energy functional on H associated with (1.1) by Z Z Z Z 1 1 1 λ 2s 2 2 2s c 2 2 J(u) = |ξ| |ˆ u(ξ)| dξ + V (x)u dx + |ξ| |u (ξ)| dξ − |u|p+1 dx 2 RN 2 RN 4 RN p + 1 RN Z Z Z 1 1 1 2s 2 2 |ξ| |ˆ u(ξ)| dξ + V (x)u dx + |ξ|2s |ˆ u(ξ) ∗ u ˆ(ξ)|2 dξ = 2 RN 2 RN 4 RN Z λ |u|p+1 dx, − p + 1 RN 4
R c2 (ξ) = u where u ˆ(ξ) ∗ u ˆ(ξ). Here (f ∗ g)(x) = RN f (y)g(x − y)dy. Moreover, Z Z Z Z 0 s s 2 hJ (u), ϕi = (−∆) uϕdx + V (x)uϕdx + [(−∆) u ]uϕdx − λ |u|p−1 uϕdx N N N N R R R R Z Z Z 2s ¯ˆ(ξ) ∗ ϕ(ξ)]dξ ¯ˆ ¯ |ξ|2s [ˆ u(ξ) ∗ u ˆ(ξ)] · [u = |ξ| u ˆ(ξ)ϕ(ξ)dξ ˆ + V (x)uϕdx + RN RN RN Z −λ |u|p−1 uϕdx N R Z Z 1 [u(x) − u(y)][ϕ(x) − ϕ(y)] = CN,s dxdy + V (x)uϕdx 2 |x − y|N +2s R2N RN Z Z 1 [u2 (x) − u2 (y)][u(x)ϕ(x) − u(y)ϕ(y)] + CN,s dxdy − λ |u|p−1 uϕdx N +2s 2 |x − y| 2N N R R for all u, ϕ ∈ H. Hence u ∈ H is a weak solution of (1.1) if hJ 0 (u), ϕi = 0 for all ϕ ∈ H.
It seems quite clear that H is the working space for studying the problem (1.1). However, H is not a linear space. Consequently, the classical critical point theory and usual min-max techniques can not be directly applied to the energy functional J, and hence we encounter the difficulties caused by the lack of an appropriate working space. Inspired by [4], we use a constrained minimization argument to give a solution of the equation (1.1). Our main result is the following: Theorem 1.1 Let 1 < p < 2∗s − 1. Assume (V ) holds. Then there exist λn → +∞ such that the equation (1.1) with λ = λn has a solution. Set X := {u ∈ H (R ) : s
with the norm kuk2X For any a > 0, we define
=
Z
RN
N
Z
RN
V (x)u2 dx < +∞}
|ξ| |ˆ u(ξ)| dξ + 2s
2
Z
RN
V (x)u2 dx.
ma := inf E(u), u∈Ma
where Ma := {u ∈ H : kukp+1 = a} and E(u) =
Z
RN
|ξ| |ˆ u(ξ)| dξ + 2s
2
Z
RN
V (x)u dx + 2
5
Z
RN
|ξ|2s |ˆ u(ξ) ∗ u ˆ(ξ)|2 dξ.
2. Proof of Theorem 1.1 Proof. We divide the proof into two steps. Step 1: We claim that for each a > 0, ma is achieved at some ua ∈ Ma , which is a weak
solution of equation (1.1) with λ = λa satisfying λa =
ma . ap+1
Indeed, we fix a > 0. Take {un } ⊂ Ma be a minimizing sequence for ma . We may
assume un ≥ 0 for all n. Then by the proof of Lemma 3.4 in [7], we know that the embedding X ,→ Ls (RN ) is compact for 2 ≤ q < 2∗s . Hence, passing to a subsequence, we
have un * ua in X, un → ua in Lq (RN ) for 2 ≤ q < 2∗s and un (x) → ua (x) ≥ 0 for almost
all x ∈ RN . Note that Z Z |ξ|2s |ˆ un (ξ) ∗ u ˆn (ξ)|2 dξ =
c2 (ξ)|2 dξ |ξ|2s |u n Z 1 |u2n (x) − u2n (y)|2 dxdy. = CN,s 2 |x − y|N +2s R2N
RN
RN
By Fatou Lemma we have Z Z |u2a (x) − u2a (y)|2 |u2n (x) − u2n (y)|2 dxdy ≤ lim inf dxdy. n→∞ |x − y|N +2s |x − y|N +2s R2N R2N Consequently,
Z
RN
|ξ| |ˆ ua (ξ) ∗ u ˆa (ξ)| dξ ≤ lim inf 2s
2
n→∞
Z
RN
|ξ|2s |ˆ un (ξ) ∗ u ˆn (ξ)|2 dξ.
Hence by weakly lower semi-continuity of the norm, one has Z Z Z E(ua ) ≤ lim inf [ |ξ|2s |ˆ un (ξ)|2 dξ + V (x)u2n dx] + lim inf n→∞
RN
n→∞
RN
RN
|ξ|2s |ˆ un (ξ) ∗ u ˆn (ξ)|2 dξ
≤ lim inf E(un ). n→∞
Since 2 < p + 1 < 2∗s , a = kun kp+1 → kua kp+1 , and hence kua kp+1 = a, so ma ≤ E(ua ) ≤ lim inf E(un ) = ma , i.e., E(ua ) = ma . Hence ma is achieved at some ua ∈ Ma . Clearly, we n→∞
can conclude that ua is a weak solution of
(−∆)s u + V (x)u + [(−∆)s u2 ]u = λa |u|p−1 u.
(2.1)
Multiplying the equation (2.1) by ua and integrating over RN we have Z Z Z 2s 2 2 p+1 |ξ| |ˆ ua (ξ)| dξ + V (x)ua dx + |ξ|2s |ˆ ua (ξ) ∗ u ˆa (ξ)|2 dξ = λa kua kp+1 . p+1 = λa · a RN
RN
RN
6
Consequently, λa · ap+1 = E(ua ) = ma , i.e., λa =
ma . ap+1
Step 2: We prove that λa → +∞ as a → 0. Indeed, suppose the conclusion is false, then there exists an → 0 such that λn := λan ≤
C1 . Set un := uan . By kun kp+1 = an → 0 as n → ∞, one has Z p+1 kun k2X + |ξ|2s |ˆ un (ξ) ∗ u ˆn (ξ)|2 dξ = λn kun kp+1 → 0. p+1 ≤ C1 an RN
Since 1 < p < 2∗s − 1, there exists a constant θ ∈ (0, 1) such that Z p+1 kun kp+1 = |un |θ(p+1) |un |(1−θ)(p+1) dx
1 p+1
=
θ 2
+ 1−θ 2∗ . Therefore, s
RN
θ(p+1)
≤kun k2
(1−θ)(p+1)
kun k2∗s
+ (1 − θ)kun kp+1 ≤θkun kp+1 2∗s 2 p+1 ≤C[kun kp+1 X + kun kX ] Z p+1 p+1 ≤C2 {kun kX + [ |ξ|2s |ˆ un (ξ) ∗ u ˆn (ξ)|2 dξ] 2 } N Z R p+1 ≤C2 [kun k2X + |ξ|2s |ˆ un (ξ) ∗ u ˆn (ξ)|2 dξ] 2 . RN
Consequently,
kun k2X +
Z
RN
|ξ|2s |ˆ un (ξ) ∗ u ˆn (ξ)|2 dξ
=λn kun kp+1 p+1
≤C1 C2 [kun k2X which implies that kun k2X +
R
RN
+
Z
RN
|ξ|2s |ˆ un (ξ) ∗ u ˆn (ξ)|2 dξ]
p+1 2
,
|ξ|2s |ˆ un (ξ) ∗ u ˆn (ξ)|2 dξ ≥ C3 > 0 since
p+1 2
> 1, a contra-
diction. By Steps 1 and 2 we complete the proof of Theorem 1.1. ¤
References [1] G. Autuori, P. Pucci, Elliptic problems involving the fractional Laplacian in RN , J. Diff. Eqns. 255(2013), 2340-2362. [2] G. Autuori, P. Pucci, Existence of entire solutions for a class of quasilinear elliptic equations, NoDEA Nonlinear Differential Equations Appl. 20(2013), 977-1009. 7
[3] A. Fiscella, E. Valdinoci, A critical Kirchhoff type problem involving a nonlocal operator, Nonlinear Anal. 94(2014), 156-170. [4] J. Liu, Z. Q. Wang, Soliton solutions for quasilinear Schr¨odinger equations: I, Proc. Amer. Math. Soc. 131(2003), 441-448. [5] E. Di Nezza, G. Palatucci, E. Valdinoci, Hitchhiker, s guide to the fractional Sobolev spaces, Bull. Sci. Math. 136(2012), 521-573. [6] X. Shang, J. Zhang, Ground states for fractional Schr¨odinger equations with critical growth, Nonlinearity, 27(2014), 187-207. [7] W. M. Zou, M. Schechter, Critical Point Theory and its Applications, Springer, New York, (2006).
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