Infinitely many radial and non-radial solutions for a fractional Schrödinger equation

Computers and Mathematics with Applications 71 (2016) 737–747

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Infinitely many radial and non-radial solutions for a fractional Schrödinger equation✩ Wen Zhang a , Xianhua Tang a , Jian Zhang b,∗ a

School of Mathematics and Statistics, Central South University, Changsha, 410083 Hunan, PR China

b

School of Mathematics and Statistics, Hunan University of Commerce, Changsha, 410205 Hunan, PR China

article

info

abstract

Article history: Received 10 October 2015 Received in revised form 22 December 2015 Accepted 28 December 2015 Available online 25 January 2016

In this paper, we study the following fractional Schrödinger equation

(−∆)s u + V (|x|)u = f (|x|, u), x ∈ RN , where (−∆)s (s ∈ (0, 1)) denotes the fractional Laplacian. By variational methods, we obtain the existence of a sequence of radial solutions for N ≥ 2, a sequence of non-radial solutions for N = 4 or N ≥ 6, and a non-radial solution for N = 5. © 2016 Elsevier Ltd. All rights reserved.

Keywords: Fractional Schrödinger equations Radial solution Non-radial solution Variational methods

1. Introduction This paper is concerned with the existence and multiplicity of radial and non-radial solutions to the following fractional Schrödinger equation

(−∆)s u + V (|x|)u = f (|x|, u),

x ∈ RN ,

(1.1)

where s ∈ (0, 1), N ≥ 2, V is a potential function and f is a continuous function with some suitable growth conditions. Here (−∆)s is the so-called fractional Laplacian operator of order s ∈ (0, 1), which can be characterized as (−∆)s u = F −1 (|ξ |2s F u), F denotes the usual Fourier transform in RN . In the recent years, the study of fractional calculus and fractional integro-differential equations applied to physics and other areas has grown, see, e.g. [1–3]. Very recently, the fractional Schrödinger equation like Eq. (1.1) was introduced by Laskin (see [4–6]), and comes from an expansion of the Feynman path integral from Brownian-like to Lévy-like quantum mechanical paths. In Laskin’s studies, the Feynman path integral leads to the classical Schrödinger equation and the path integral over Lévy trajectories leads to the fractional Schrödinger equation. The fractional Schrödinger equation appears in many areas such as quantum mechanics, financial market, phase transitions, anomalous diffusions, crystal dislocations, soft thin films, semipermeable membranes, flame propagations, conservation laws, ultra-relativistic limits of quantum mechanics, quasi-geostrophic flows, minimal surfaces, materials science, and water waves. For details see [7–9] for an introduction to its applications.

✩ This work is partially supported by the NNSF (Nos. 11571370, 11471137, 11471278, 11301297, 11261020), and Hunan Provincial Innovation Foundation For Postgraduate (No. CX2014A003). ∗ Corresponding author. E-mail addresses: [email protected] (W. Zhang), [email protected] (X. Tang), [email protected] (J. Zhang).

http://dx.doi.org/10.1016/j.camwa.2015.12.036 0898-1221/© 2016 Elsevier Ltd. All rights reserved.

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W. Zhang et al. / Computers and Mathematics with Applications 71 (2016) 737–747

It is known, but not completely trivial, that (−∆)s reduces to the standard Laplacian −∆ as s → 1. When s = 1, the classical nonlinear Schrödinger equation has been extensively studied in the last thirty years. For example, see [10–24] and the references therein. In a pioneering paper, Strauss [17] first proved the following classical semilinear Schrödinger equation

− 1u + V (|x|)u = f (|x|, u),

x ∈ RN ,

(1.2)

has a sequence of radial solutions {uk } with the energy that goes to infinity as k → ∞ provided N ≥ 2, V ≡ 1 and f (|x|, u) = f (u) is odd and satisfies various growth conditions. This result has been generalized by Berestycki and Lions [10,11] and Struwe [18] under almost necessary growth conditions on f at zero and infinity. The open problem of the existence of non-radial solutions of (1.2) was answered by Bartsch and Willem [12] for the case N = 4 and N ≥ 6. They proved the existence of an unbounded sequence of non-radial solutions in invariant functions space under the classical Ambrosetti–Rabinowitz condition ((AR) for short), i.e., there exists θ > 2 such that 0 < θ F (x, t ) ≤ tf (x, t ),

∀(x, t ) ∈ RN × R,

(1.3)

where F is the primitives of f . Also they constructed the existence of infinitely many radial nodal solutions in [13]. For the case N = 5, a nonradial solution of (1.2) was obtained in [14] under the condition (1.3), which completed the well-known work of Bartsch and Willem [12]. The problem of the existence of non-radial solution of (1.2) keeps open for the case N = 3. On the contrary, as we know, a great attention has been devoted to the fractional and nonlocal operators of elliptic type like (1.1), both for their interesting theoretical structure and in view of concrete applications in many fields. From the mathematicians point of view, one of the main difficulties lies in that the fractional Schrödinger equation involves the fractional Laplacian (−∆)s , which is a nonlocal operator. The greatest achievement in overcoming this difficulty is the extension theorem by Caffarelli and Silvestre in [25]. After some extension, the authors transformed the nonlocal problem into a local problem, and established some existence and nonexistence of Dirichlet problem involving the fractional Laplacian on bounded domain. After the work [25], a great deal of progress has been made to the fractional Laplacian equations. For example, the papers [26–34] studied the existence results or multiplicity results of solutions (bound or ground states), for the regularity result, maximum principle, uniqueness result and other properties, see [35,36], and for other related topics including the superquadratic singular perturbation problem and concentration phenomenon of semiclassical states, see [37,38] and the references therein. Recently, for the investigations about radial solutions, in the spirit of [10], Dipierro et al. [39] proved the existence of a positive and spherically symmetric solution to the equation

(−∆)s u + u = |u|p−1 u,

x ∈ RN ,

(1.4)

for subcritical exponents 1 < p < (N + 2s)/(N − 2s), which generalized the results in [10] from the classical Schrödinger equation to the fractional Schrödinger equation. On the other hand, the approach they used is a constrained minimization in [10]. But this approach cannot expect to work when V is non-constant. When the nonlinearity f satisfies the general hypotheses introduced by Berestycki and Lions [10], Chang and Wang [26] and Secchi [31] also proved the existence of a radially symmetric solution with the help of the Pohozaev identity for (1.1). Motivated by the above facts, in the present paper we will study the fractional Schrödinger equation (1.1) with nonconstant potential V and without (AR) type superlinear condition. Moreover, we use some original arguments in [12–14] to establish the existence of infinitely many radial and non-radial solutions. To state our results, we make the following assumptions: (V ) V ∈ C ([0, +∞)) is bounded from below by a positive constant V0 ; (F1 ) f ∈ C ([0, ∞) × R, R), and there exist constants a1 , a2 > 0 and q ∈ (2, 2∗s ) such that

|f (r , u)| ≤ a1 + a2 |u|q−1 , ∗

∀r ≥ 0, u ∈ R,

2N ; N −2s

where 2s = (F2 ) f (r , u) = o(|u|) as |u| → 0 uniformly in r; F (r ,u) (F3 ) lim|u|→∞ |u|2 = ∞, a.e. in r ∈ [0, +∞), and there exists r0 ≥ 0 such that F (r , u) ≥ 0, (F4 ) F (r , u) =

1 f 2

for any u ∈ R and |u| ≥ r0 ;

(r , u)u − F (r , u) ≥ 0, and there exist c1 > 0 and κ > max{1, N /2s} such that

|F (r , u)|κ ≤ c1 |u|2κ F (r , u),

∀ |u| ≥ r0 ;

(F5 ) f (r , −u) = −f (r , u) for any u ∈ R and r ≥ 0. For convenience, let I(u) denote the energy of the solution u (I will be defined later). The main results of this paper are the following theorems. Theorem 1.1. Assume that N ≥ 2, (V ) and (F1 )–(F5 ) hold. Then the problem (1.1) has a sequence of radial solutions {un } such that I(un ) → ∞ as n → ∞.

W. Zhang et al. / Computers and Mathematics with Applications 71 (2016) 737–747

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Theorem 1.2. Assume that N = 4 or N ≥ 6, (V ) and (F1 )–(F5 ) hold. Then the problem (1.1) has a sequence of non-radial solutions {un } such that I(un ) → ∞ as n → ∞. Theorem 1.3. Assume that (V ) and (F1 )–(F4 ) hold, N = 5 and (F6 ) For all z = (x, y) ∈ R × R4 and for all g ∈ O(R4 ) f (|(x + 1, y)|, u) = f (|x, g (y)|, u) and

V (|(x + 1, y)|) = V (|(x, g (y))|),

where O(R ) is the orthogonal transform group in R4 . 4

Then problem (1.1) has a nontrivial non-radial solution. Remark 1.4. The conditions (F3 ), (F4 ) and (F6 ) were introduced in [40,19,41] ((F3 ) and (F4 )) and [14] ((F6 )) where the author studied the classical Schrödinger equation, respectively. And our conditions are weaker than (AR) condition used in [12–14]. Hence our results can be viewed as extension to the results in [12–14] from the classical Schrödinger equation to the fractional Schrödinger equation. Even if for the case s = 1, our results are also new. 2. Preliminaries In the sequel, s will denote a fixed number, s ∈ (0, 1), we denote by ∥ · ∥p the usual norm of the space Lp (RN ), c , ci or Ci stand for different positive constants. Recall that the fractional Sobolev space is defined by

 H (R ) = s

u ∈ L (R ) :

N

2

N

|u(x) − u(y)| N

|x − y| 2 +s

 ∈ L (R × R ) 2

N

N

and endowed with the standard norm



|u|2 dx +

∥ u∥ H s =



RN

 RN

RN

|u(x) − u(y)|2 dxdy |x − y|N +2s

 21

,

while

 [ u]

Hs



= RN

RN

 21 |u(x) − u(y)|2 dxdy |x − y|N +2s

is the Gagliardo (semi)norm. The space H s (RN ) can also be described by means of the Fourier transform. Indeed, it is defined by H s (RN ) =



u ∈ L2 ( R N ) :

 RN

 (1 + |ξ |2 )s (|F u(ξ )|2 )dξ < ∞ ,

and the norm is defined as

 ∥ u∥ H s = RN

(1 + |ξ |2 )s (|F u(ξ )|2 )dξ

 12

.

Now, we introduce the definition of Schwartz function S (is dense in H s (RN )), that is, the rapidly decreasing C ∞ function on RN . If u ∈ S , the fractional Laplacian (−∆)s acts on u as

(−∆)s u(x) = C (N , s)P .V .

u(x) − u(y)



dy

|x − y|N +2s  u(x) − u(y) = C (N , s) lim dy, N +2s + N ϵ→0 R \B(0,ϵ) |x − y| RN

the symbol P .V . represents the principal value integrals, the constant C (N , s) depends only on the space dimension N and the order s. In [42], the authors show that for u ∈ S ,

(−∆)s u = F −1 (|ξ |2s F u) and that

[u]2H s =

2 C (N , s)

 RN

|ξ |2s |F u|2 dξ .

Moreover, by the Plancherel formula in Fourier analysis, we get

[u]2H s =

2 C (N , s)

  s 2  (−∆) 2  . 2

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Therefore, the norms on H s (RN ) defined below,

|u(x) − u(y)|2 u → |u| dx + dxdy |x − y|N +2s RN RN RN   12 u → (1 + |ξ |2 )s (|F u(ξ )|2 )dξ 

2







|ξ |2s |F u(ξ )|2 dξ

 21

RN



|u|2 dx +

u → RN

 u → RN

RN

 21

   21 s 2  2 |u| dx + (−∆) 2  2

are all equivalent. For our problem (1.1), we define the working space H as follows

 H =

u ∈ H s (RN ) :

 RN

|ξ |2s |F u|2 dξ +



 RN

V (|x|)|u|2 dx < +∞

and it is endowed with the inner product and norm given by

(u, v) =

 RN

|ξ |2s F u(ξ )F v(ξ )dξ +

 RN

V (|x|)uv dx

and



|ξ | |F u(ξ )| dξ + 2s

∥ u∥ = RN

2

 RN

 21

V (|x|)|u| dx 2

.

For the proof of Theorem 1.1, we denote by E the radial symmetric functions space of H, namely, E := Hr = {u ∈ H : u(x) = u(|x|)} . For the proof of Theorem 1.2, following [12], we choose an integer 2 ≤ m ≤ N2 with 2m ̸= N − 1, write elements of RN = Rm × Rm × RN −2m as x = (x1 , x2 , x3 ) with x1 , x2 ∈ Rm and x3 ∈ RN −2m . Consider the action of Gm := O(m) × O(m) × O(N − 2m) on H is defined by gu(x) := u(g −1 x). Let τ ∈ O(N ) be the involution given by τ (x1 , x2 , x3 ) = (x2 , x1 , x3 ). The action of G := {id, τ } on Fix(Gm ) := {u ∈ H : gu = u, ∀g ∈ Gm } is defined by hu(x) :=



u(x), −u(h−1 x),

if h = id, if h = τ .

Set E := Fix(G) = {u ∈ Fix(Gm ) : hu = u, ∀ h ∈ G}. Note that 0 is the only radially symmetric function in E for this case. Moreover, we need the following embedding theorem also due to [43]. Lemma 2.1. E embeds continuously into Lp (RN ) for all p ∈ [2, 2∗s ], and E embeds compactly into Lp (RN ) for all p ∈ (2, 2∗s ). It follows from Lemma 2.1 that there exists constant γp > 0 such that

∥u∥p ≤ γp ∥u∥,

∀ u ∈ E , p ∈ [2, 2∗s ].

(2.1)

Next, on E we define the following functional

I(u) =

1 2

∥ u∥ 2 −

 RN

F (|x|, u)dx,

and we have the following fact.

(2.2)

W. Zhang et al. / Computers and Mathematics with Applications 71 (2016) 737–747

741

Lemma 2.2. Assume that (V ) and (F1 )–(F2 ) hold, then I ∈ C 1 (E , R) and

⟨I′ (u), v⟩ = (u, v) −

 RN

f (|x|, u)v dx.

(2.3)

Furthermore, the critical points of I are solutions of problem (1.1). Proof. For convenience, set

J ( u) =

 RN

F (|x|, u)dx.

By (F1 ) and (F2 ), for any ε > 0, there is Cε > 0 such that

|f (|x|, u)| ≤ ε|u| + Cε |u|q−1 and |F (|x|, u)| ≤

ε 2

|u|2 +

Cε q

|u|q .

(2.4)

For any u, v ∈ E and 0 < |t | < 1, by mean value theorem and (2.4), there exists 0 < θ < 1 such that

|F (|x|, u + t v) − F (|x|, u)| ≤ |t | ≤ ≤ ≤

|f (|x|, u + θ t v)v| ε|u + θ t v| |v| + Cε |u + θ t v|q−1 |v| ε|u| |v| + ε|v|2 + Cε |u + θ t v|q−1 |v| ε|u| |v| + ε|v|2 + 2q−1 Cε (|u|q−1 |v| + |v|q ).

The Hölder inequality implies that

ε|u| |v| + ε|v|2 + 2q−1 Cε (|u|q−1 |v| + |v|q ) ∈ L1 (RN ). Consequently, by the Lebesgue’s Dominated Theorem, we have

⟨J ′ (u), v⟩ =

 RN

f (|x|, u)v dx,

∀ u, v ∈ E .

Next, we show that J ′ : E → E ∗ is weak continuous. Assume that un ⇀ u in E, by Lemma 2.1, we get un → u

in Lp (RN ), for p ∈ (2, 2∗s ).

Note that

∥J ′ (un ) − J ′ (u)∥E ∗ = sup |⟨J ′ (un ) − J ′ (u), v⟩| ∥v∥≤1  ≤ sup |f (|x|, un ) − f (|x|, u)| |v|dx. ∥v∥≤1 RN

By the Hölder inequality and Theorem A.4 in [44], we have

 sup ∥v∥≤1 RN

|f (|x|, un ) − f (|x|, u)| |v|dx → 0 as n → ∞,

thus,

∥J ′ (un ) − J ′ (u)∥E ∗ → 0 as n → ∞. Therefore, I ∈ C 1 (E , R). Moreover, it is standard way to verify that critical points of I are solutions of problem (1.1) (see [44]).  To prove our results, we need the principle of symmetric criticality theorem (see [44, Theorem 1.28]) as follows. Lemma 2.3. Assume that the action of the topological group G on the Hilbert space X is isometric. If Φ ∈ C 1 (X , R) is invariant and if u is a critical point of Φ restricted to Fix(G), then u is a critical point of Φ . It follows from Lemma 2.3 that we know that if u is a critical point of Φ := I|E , then u is a critical point of I. Moreover, we say that Φ ∈ C 1 (E , R) satisfies (C )c -condition if any sequence {un } such that

Φ (un ) → c ,

∥Φ ′ (un )∥(1 + ∥un ∥) → 0

has a convergent subsequence. Lemma 2.4. Assume that (F1 )–(F4 ) hold. Then any (C )c -sequence {un } of Φ is bounded.

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W. Zhang et al. / Computers and Mathematics with Applications 71 (2016) 737–747

Proof. Let {un } ⊂ E be a (C )c -sequence, then

Φ (un ) → c > 0,

⟨Φ ′ (un ), un ⟩ → 0.

(2.5)

To prove the boundedness of {un }, arguing by contradiction, assume that ∥un ∥ → ∞. Let vn = then ∥vn ∥ = 1 and ∗ ∥vn ∥p ≤ γp ∥vn ∥ = γp for 2 ≤ s < 2s . Passing to a subsequence, we may assume that vn ⇀ v in E, vn → v in Lp for 2 < p < 2∗s and vn ⇀ v a.e. in RN . Observe that for n large un , ∥un ∥

1

c0 = c + 1 ≥ Φ (un ) −

2

⟨Φ ′ (un ), un ⟩ =

 RN

F (|x|, un )dx.

(2.6)

Combining (2.2) with (2.5), we get



1

≤ lim sup

2

RN

n→∞

|F (|x|, un )| dx. ∥ un ∥ 2

(2.7)

For 0 ≤ a < b, let

Ωn (a, b) = {x ∈ RN : a ≤ |un | < b} and Cab

 := inf

F (|x|, u) u2



: x ∈ R and u ∈ R with a ≤ |u| < b . N

Since F (|x|, u) > 0 if u ̸= 0, we have Cab > 0 and

F (|x|, un ) ≥ Cab |un |2 for all x ∈ Ωn (a, b). First, we consider the case that v = 0, then vn → 0 in Lp for 2 < p < 2∗s . By virtue of (F1 ), we can find some number r1 > 0 such that r0 > r1 and

|f (|x|, u)| < ε|u|,

for |u| ≤ r1 ,

where r0 is given in (F3 ). Thus F (|x|, un )

 Ωn (0,r1 )

| un | 2

ε



2

|vn | dx ≤

| un | 2 ε |vn |2 dx ≤ ∥vn ∥22 . |un |2 2

2

Ωn (0,r1 )

(2.8)

It follows from (2.6) that

 c0 ≥

Ωn (0,r1 )

 ≥ Ωn (0,r1 )

F (|x|, un )dx +

 Ωn (r1 ,r0 )

F (|x|, un )dx + Crr10

F (|x|, un )dx +

 Ωn (r1 ,r0 )

|un |2 dx +

 Ωn (r0 ,∞)

 Ωn (r0 ,∞)

F (|x|, un )dx

F (|x|, un )dx,

(2.9)

thus, we have

 Ωn (r1 ,r0 )

|un |2 dx ≤

c0 r

Cr10

.

(2.10)

From (2.4), we know that F (|x|, un )

 Ωn (r1 ,r0 )

|un |2

|vn |2 dx ≤

ε



2

Ωn (r1 ,r0 )

 ≤

ε 2

+

Cε q

Cε q

|un |2 +

q −2

|un |q

| un | 2 

r0

Ωn (r1 ,r0 )

|vn |2 dx

|vn |2 dx

(2.11)

and

 Ωn (r1 ,r0 )

|vn |2 dx =



1

∥ un ∥

2

Ωn (r1 ,r0 )

|un |2 dx ≤

1

c0

∥un ∥ Crr10 2

.

(2.12)

Combining (2.11) with (2.12), we get F (|x|, un )

 Ωn (r1 ,r0 )

|un |2

2

|vn | dx ≤



ε 2

+

Cε q

q−2 r0



c0 r Cr10

∥ un ∥ 2

→ 0,

as n → ∞.

(2.13)

W. Zhang et al. / Computers and Mathematics with Applications 71 (2016) 737–747

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κ Set κ ′ = κ− . Since κ > max{1, N /2s}, we get 2κ ′ ∈ (2, 2∗s ). Hence, from (F4 ) and (2.6), we have 1

 Ωn (r0 ,∞)

|F (|x|, un )| |vn |2 dx ≤ |un |2





Ωn (r0 ,∞) 1

≤ c1κ

|F (|x|, un )| |un |2

Ωn (r0 ,∞)

Ωn (r0 ,∞) 1

≤ (c1 c0 ) κ

F (|x|, un )dx



|vn |2κ dx ′

Ωn (r0 ,∞)

|vn |2κ dx ′

dx

 κ1 



→ 0,

 κ1 

κ

Ωn (r0 ,∞)

 1′ κ

 1′ κ ′ |vn |2κ dx

 1′ κ

as n → ∞.

(2.14)

Combining (2.8), (2.13) with (2.14), we get

 RN

 |F (|x|, un )| |F (|x|, un )| 2 |v | dx + |vn |2 dx n 2 | un | |un |2 Ωn (0,r1 ) Ωn (r1 ,r0 )  |F (|x|, un )| + |vn |2 dx | un | 2 Ωn (r0 ,∞) → 0, as n → ∞

|F (|x|, un )| dx = ∥un ∥2



which contradicts (2.7). Next we consider the case that v ̸= 0. Set A := {x ∈ RN : v ̸= 0}, then meas(A) > 0. For x ∈ A, we have |un | → ∞ as n → ∞. Hence, x ∈ Ωn (r0 , ∞) for large n ∈ N, which implies that χΩn (r0 ,∞) (|x|) = 1 for large n, where χΩn denotes the characteristic function on Ω , r0 is given in (F3 ). Since vn → v a.e. in RN , we have χΩn (r0 ,∞) (|x|)vn → v a.e. in A. It follows from (F3 ), (2.5) and Fatou’s Lemma that

Φ (un ) ∥ un ∥ ∥ un ∥ 2    1 F (|x|, un ) 2 = lim − |v | dx n n→∞ 2 | un | 2 RN     F (|x|, un ) F (|x|, un ) 1 2 2 − |v | dx − |v | dx = lim n n n→∞ 2 |un |2 |un |2 Ωn (0,r0 ) Ωn (r0 ,∞)      ε Cε 1 F (|x|, un ) 2 + + r0q−2 |v | dx ≤ lim sup |vn |2 dx − n 2 2 q |un |2 n→∞ RN Ωn (r0 ,∞)    1 ε Cε F (|x|, un ) ≤ + + r0q−2 γ22 − lim inf |vn |2 dx n→∞ 2 2 q |un |2 Ωn (r0 ,∞)    Cε F (|x|, un ) 1 ε = + + r0q−2 γ22 − lim inf [χΩn (r0 ,∞) (|x|)]|vn |2 dx n→∞ 2 2 q |un |2 RN    1 ε C ε q −2 F (|x|, un ) ≤ + + r0 γ22 − lim inf [χΩn (r0 ,∞) (|x|)]|vn |2 dx n →∞ N 2 2 q |un |2 R    1 ε C ε q −2 F (|x|, un ) 2 ≤ + + r0 γ2 − lim inf [χΩn (r0 ,∞) (|x|)]|vn |2 dx n →∞ 2 2 q | un | 2 A

0 = lim

c + o(1)

n→∞

= −∞,

2

= lim

n→∞

as n → ∞,

(2.15)

which is a contradiction. Thus {un } is bounded in E.



Lemma 2.5. Assume that (F1 )–(F4 ) hold. Then Φ satisfies the (C )c -condition. Proof. Let {un } ⊂ E be a (C )c -sequence of Φ . By Lemma 2.4, {un } is bounded. Passing to a subsequence if necessary, we may assume un ⇀ u in E and un → u in Lp for all 2 < p < 2∗s . According to (2.4), we have

 RN

|f (|x|, un ) − f (|x|, u)| |un − u|dx ≤

 N

R ≤

RN

(|f (|x|, un )| + |f (|x|, u)|)|un − u|dx   ε|un | + Cε |un |q−1 + ε|u| + Cε |u|q−1 |un − u|dx

744

W. Zhang et al. / Computers and Mathematics with Applications 71 (2016) 737–747

≤ ε C + Cε → 0,



q

 q−q 1 

|un | dx RN

 1q |un − u|q dx

RN

as n → ∞.

(2.16)

[f (|x|, un ) − f (|x|, u)](un − u)dx.

(2.17)

Observe that

∥un − u∥2 = ⟨Φ ′ (un ) − Φ ′ (u), un − u⟩ +

 RN

It is clear that

⟨Φ ′ (un ) − Φ ′ (u), un − u⟩ → 0,

as n → ∞.

(2.18)

From (2.16)–(2.18), we get

∥ un − u∥ → 0 ,

as n → ∞. 

3. Proof of Theorems 1.1 and 1.2 To prove our results, we need the following Symmetric Mountain Pass Theorem [45, Theorem 9.12]. Lemma 3.1. Let X be an infinite dimensional Banach space, X = Y ⊕ Z , and Y is finite dimensional. If Φ ∈ C 1 (X , R) satisfies (C )c -condition for all c > 0, and

(I1 ) Φ (0) = 0, Φ (−u) = Φ (u) for all u ∈ X ; (I2 ) there exist constants ρ, α > 0 such that Φ |∂ Bρ ∩Z ≥ α ; (I3 ) for any finite dimensional subspace X˜ ⊂ X , there is R = R(X˜ ) > 0 such that Φ (u) ≤ 0 on X˜ \ BR ; then Φ possesses an unbounded sequence of critical values. ∞ Let {ei }be a total orthonormal basis of E and define Xi = Rei , then E = i=1 Xi , let n ∞   Yn = Xi , Zn = Xi , n ∈ Z. i=1

j=n+1

Then E = Yn ⊕ Zn , and Yn is a finite dimensional space. Lemma 3.2. Assume that (F1 )–(F4 ) hold. Then there are constants ρ , α > 0 such that Φ |∂ Bρ ∩Zn ≥ α . Proof. For any u ∈ Zn , from (2.1) and (2.4), we can choose ε small enough such that

Φ ( u) =

≥ ≥

1 2 1 2 1 2

2



∥ u∥ − RN

F (|x|, u)dx

ε Cε ∥u∥2 − ∥u∥22 − ∥u∥qq 2

∥ u∥ 2 −

q

γ22 ε 2

∥ u∥ 2 −

γqq Cε q

∥ u∥ q .

Thus, we can choose constants ρ, α > 0 such that Φ |∂ Bρ ∩Zn ≥ α , the proof is complete.



Lemma 3.3. Assume that (F1 )–(F4 ) hold, for any finite dimensional subspace E˜ ⊂ E, there holds

Φ (u) → −∞,

∥u∥ → ∞,

u ∈ E˜ .

(3.1)

Proof. Arguing indirectly, assume that for some sequence {un } ⊂ E˜ with ∥un ∥ → ∞, there exists M > 0 such that Φ (un ) ≥ −M for all n ∈ N. Let vn = ∥uunn ∥ , then ∥vn ∥ = 1. Passing to a subsequence, we may assume that vn ⇀ v1 in E. Since E˜ is finite dimensional, then vn → v1 ∈ E˜ in E, vn → v1 a.e. on RN , and so ∥v1 ∥ = 1. Hence, we can conclude a contradiction by a similar fashion as (2.15). Then, the desired conclusion is obtained.



Corollary 3.4. Assume that (F1 )–(F4 ) hold, for any finite dimensional subspace E˜ ⊂ E, there exists R = R(E˜ ) > 0 such that

Φ (u) ≤ 0,

∀u ∈ E˜ ,

∥u∥ ≥ R.

(3.2)

Proof of Theorem 1.1. We only need to verify the conditions of Lemma 3.1. Let X = E, Y = Yn and Z = Zn , then E = Y ⊕ Z . Moreover, Y is finite dimensional. From (F5 ), we know Φ is even. Clearly, Φ (0) = 0. Lemma 2.5 implies Φ satisfies (C )c condition. Lemma 3.2 and Corollary 3.4 imply (I2 ) and (I3 ) of Lemma 3.1 are satisfied. Thus, by Lemma 3.1, Φ possesses a

W. Zhang et al. / Computers and Mathematics with Applications 71 (2016) 737–747

745

sequence of radial critical points {un } ⊂ E such that Φ (un ) → ∞ as n → ∞, i.e., the problem (1.1) has a sequence of radial solutions {un } such that Φ (un ) → ∞ as n → ∞.  Proof of Theorem 1.2. By a similar way as Theorem 1.1, we can complete the proof of Theorem 1.2.



4. Proof of Theorem 1.3 In order to prove Theorem 1.3, we need the following Mountain Pass Theorem without compactness [44, Theorem 1.15]. Lemma 4.1. Let X be an Hilbert space, Φ ∈ C 1 (E , R), e ∈ X and r > 0 such that ∥e∥ > r and inf∥u∥=r Φ (u) > Φ (0) ≥ Φ (e). Then there exists a sequence {un } such that Φ (un ) → c and (1 + ∥un ∥)Φ ′ (un ) → 0, where c = inf max Φ (γ (t )) > 0 γ ∈Γ t ∈[0,1]

and Γ = {γ ∈ C ([0, 1], X ) : γ (0) = 0, γ (1) = e}. The following lemmas play a crucial role to obtain some kind of compactness in problem (1.1). Lemma 4.2. Let {Ωj }j∈N be a sequence of open subsets of R such that

¯ j and Ωi ∩ Ωj = ∅, if i ̸= j; (i) R = ∪j∈N Ω (ii) there exists a constant c0 > 0 such that for all j ∈ N, ∥ u∥

≤ c0 ∥u∥H s (Ωj ×R4 ) ,

10

L 5−2s (Ωj ×R4 )

∀u ∈ H s (Ωj × R4 ).

Let {un } be a bounded sequence of H s (R5 ). If



|un |k dx → 0,

sup Ωj ×R4

j∈N



for k ∈ 2,



10 5 − 2s

,

10 then un → 0 in Lq (R5 ), for all 2 < q < 5− . 2s

Proof. The proof can be found in [14]. Here, we give some details for completeness. Let σ = inequality and the Sobolev embedding theorem, we know 2ks

∥u∥σLσ (Ω ×R4 ) ≤ ∥u∥Lk5(Ω ×R4 ) ∥u∥2 j

+ 2, from the Hölder

2ks

10 L 5−2s

j

2ks 5

(Ωj ×R4 )

≤ c02 ∥u∥Lk5(Ω ×R4 ) ∥u∥2H s (Ω ×R4 ) . j

j

Hence,

 σ

∥u∥Lσ (R5 ) ≤

c02

 2s5



k

sup j∈N

Ωj ×R4

|un | dx

∥u∥2H s (R5 ) .

Under the assumption of the lemma, un → 0 in Lσ (R5 ). Since 2 < σ < embedding theorem again, we have un → 0 in L (R ), for all 2 < q < q

5

10 5−2s

10 , 5−2s

by the Hölder inequality and the Sobolev



For the proof of Theorem 1.3, following [14], let G be a subgroup of O(R4 ). It is obvious that R4 is compatible with G if for some r > 0 m(y, r , G) = lim m(y, r , G) = +∞,

(4.1)

|y|→∞

where m(y, r , G) := sup n ∈ N : ∃ g1 , . . . , gn ∈ G such that i ̸= j ⇒ B(gi (y)) ∩ B(gj (y)) = ∅ .





n∈N

Observe that R is compatible with O(R4 ) and with O(R2 ) × O(R2 ) (see [44]). For simplicity of notation, we denote G := O(R2 ) × O(R2 ). Consider the action of G on H given by 4

(gu)(x, y) = u(x, g −1 y),

for all (x, y) ∈ R × R4 , g ∈ O(R4 ).

Let HG := {u ∈ H : gu = u, ∀ g ∈ G} and τ ∈ O(N ) be the involution in R5 = R × R2 × R2 given by τ (x1 , x2 , x3 ) = (x2 , x1 , x3 ). We define an action of the group G1 := {id, τ } on H by hu(x) :=

u(x), −u(h−1 x),



if h = id, if h = τ .

746

W. Zhang et al. / Computers and Mathematics with Applications 71 (2016) 737–747

Let HG1 := {u ∈ H : hu = u, ∀ h ∈ G1 }. Set E := HG ∩ HG1 . It is clear that u = 0 is only radial function in E, which is a Hilbert space with the inner product of H. 10 Lemma 4.3. The embedding HG ↩→ Lp (Ω × R4 ) is compact, where Ω is a bounded subset of R and p ∈ (2, 5− ). 2s

Proof. Suppose un ⇀ 0 in HG and let r be the number given in (4.1). For all n ∈ N m(y, r , G)

m(y,r ,G)

 Ω ×Br (y)

 

|un |2 dx =

Ω ×B(gi (y))

i =1

 ≤ Ω ×R 4

|un |2 dx

|un |2 dx

≤ sup ∥un ∥22 .

(4.2)

n

Let ε > 0, since R4 is compatible with G, together with (4.2), we know that there exists R > 0 such that

 sup |y|>R Ω ×Br (y)

|un |2 dx ≤ ε.

(4.3)

By Rellich’s theorem

 sup |y|≤R Ω ×Br (y)

|un |2 dx → 0,

as n → ∞.

(4.4)

From (4.3) and (4.4), we get

 sup Ω ×Br (y)

y∈R4

|un |2 dx → 0,

as n → ∞.

10 Arguing as in the proof of Lemma 4.2, we can conclude that un → 0 in Lp (Ω × R4 ) for 2 < p < 5− . 2s



Proof of Theorem 1.3. Similar to Lemma 3.2 and Corollary 3.4, it is easy to verify that Φ satisfies the conditions of Lemma 4.1. Then, there exists a (C )c -sequence {un } ⊂ E i.e., {un } satisfies

Φ ( un ) → c ,

(1 + ∥un ∥)Φ ′ (un ) → 0,

where c is the Mountain Pass value given in Lemma 4.1. According to (2.1), we obtain on (1) = ∥Φ ′ (un )∥(1 + ∥un ∥) ≥ ⟨Φ ′ (un ), un ⟩

 = R5

   2 s   2 2 (−∆) un  + V (|x|)|un | dx −

≥ (1 − εγ22 )∥un ∥2 − Cε

 R5

R5

f (|x|, un )un dx

|un |q dx.

(4.5)

Hence, without loss of generality, we can choose θ > 0 such that

 R5

|un |q dx ≥ c1 ∥un ∥2 ≥ θ ,

where c1 =

1 − εγ22 Cε

.

(4.6)

Otherwise, (4.5) implies that un → 0 in E and hence c = 0, which leads to a contradiction. ¯ j . We may claim that there exists δ > 0 such that Let Ωj = (j, j + 1), then R = ∪j∈N Ω

 sup j∈N

Ωj ×R4

|un (x, y)|2 dxdy ≥ 2δ.

10 10 Otherwise, by Lemma 4.2, we have un → 0 in Lq (R5 ) for 2 < q < 5− , which contradicts (4.6). Since 2 < q < 5− , for 2s 2s every n, there exists jn such that

 Ωjn ×R4

|un (x, y)|2 dxdy ≥ δ.

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747

Making change of variable x = x′ + jn , we have

 Ω ×R4

|un (x′ + jn , y)|2 dx′ dy ≥ δ,

where Ω = (0, 1). Let vn (x′ , y) = un (x′ + jn , y), then

 Ω ×R 4

|vn (x′ , y)|2 dx′ dy ≥ δ.

(4.7)

Observe that vn is also a (C )c -sequence of Φ . Hence,

vn ⇀ v in E , vn → v in Lploc (R5 ), where 2 < p <

10 5 − 2s

,

vn → v a.e. on R5 . It is standard to prove that v is a critical point of Φ . Moreover, (4.7) implies v ̸= 0, i.e., the problem (1.1) has a nontrivial nonradial solution u. This completes the proof.  References [1] A. Iomin, G.M. Zaslavsky, Quantum manifestation of Lévy-type flights in a chaotic system, Chem. Phys. 284 (2002) 3–11. [2] R. Metzler, J. Klafter, The random walk’s guide to anomalous diffusion: a fractional dynamics approach, Phys. Rep. 339 (2000) 1–77. [3] R. Metzler, J. Klafter, The restaurant at the random walk: recent developments in the description of anomalous transport by fractional dynamics, J. Phys. A 37 (2004) 161–208. [4] N. Laskin, Fractional quantum mechanics, Phys. Rev. E 62 (2000) 3135. [5] N. Laskin, Fractional quantum mechanics and Levy path integrals, Phys. Lett. A 268 (2000) 298–305. [6] N. Laskin, Fractional Schrödinger equation, Phys. Rev. E 66 (2002) 056108. [7] J. 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