INFINITELY MANY SOLUTIONS FOR

Electronic Journal of Differential Equations, Vol. 2017 (2017), No. 97, pp. 1–13. ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu

INFINITELY MANY SOLUTIONS FOR FRACTIONAL ¨ SCHRODINGER-POISSON SYSTEMS WITH SIGN-CHANGING POTENTIAL JIANHUA CHEN, XIANHUA TANG, HUXIAO LUO Communicated by Marco Squassina

Abstract. In this article, we prove the existence of multiple solutions for following fractional Schr¨ odinger-Poisson system with sign-changing potential (−∆)s u + V (x)u + λφu = f (x, u), t

2

(−∆) φ = u ,

x ∈ R3 ,

3

x∈R ,

(−∆)α

where denotes the fractional Laplacian of order α ∈ (0, 1), and the potential V is allowed to be sign-changing. Under certain assumptions on f , we obtain infinitely many solutions for this system.

1. Introduction and preliminaries This article concerns the fractional Schrödinger-Poisson system (−∆)s u + V (x)u + λφu = f (x, u), (−∆)t φ = u2 ,

x ∈ R3 ,

x ∈ R3 ,

(1.1)

where (−∆)α denotes the fractional Laplacian operator, λ is a positive parameter and V is allowed to be sign-changing. In (1.1), the first equation is a nonlinear fractional Schr¨ odinger equation in which the potential φ satisfies a nonlinear fractional Poisson equation. For this reason, system (1.1) is called a fractional SchrödingerPoisson system, also known as the fractional Schrödinger-Maxwell system, which is not only a physically relevant generalization of the classical NLS but also an important model in the study of fractional quantum mechanics. For more details about the physical background, we refer the reader to [14, 15] and the references therein. If λ = 1, then system (1.1) reduces to the fractional Schrödinger-Poisson system (−∆)s u + V (x)u + φu = f (x, u), (−∆)t φ = u2 ,

x ∈ R3 ,

x ∈ R3 ,

(1.2)

2010 Mathematics Subject Classification. 35J60, 35J20. Key words and phrases. Fractional Schr¨ odinger-Poisson systems; sign-changing potential; symmetric mountain pass theorem; infinitely many solutions. c

2017 Texas State University. Submitted June 28, 2016. Published April 5, 2017. 1

2

J. CHEN, X. TANG, H. LUO

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which has been studied by Zhang [29] by using the fountain theorem. The author proved the existence of multiple solutions under the condition (A4) and (A5) below. Meanwhile, the author proved that (A4) and (A5) are more weaker than (A8). Let s = 1, t = 1 and λ = 1, then system (1.1) can be simplified to the classical fractional Schr¨ odinger-Poisson system −∆u + V (x)u + φu = f (x, u), −∆φ = u2 ,

x ∈ R3 ,

x ∈ R3 ,

(1.3)

which has been considered to prove the existence of infinitely many solutions for (1.3) via the fountain theorem. For more details, see the references [11, 13, 17, 20, 24] and the references therein, for more results about applying the critical point theory to second-order elliptic equations, we refer the reader to [2, 3, 4, 5, 26, 27, 28, 32] and the references therein. However, it is well known that the fractional Schrödinger-Poisson system was first introduced by Giammetta [10] and the diffusion is fractional only in the Poisson equation. Afterwards, in [23], the authors proved the existence of radial ground state solutions of (1.1) when V (x) ≡ 0 and nonlinearity f (x, u) is of subcritical or critical growth. Recently, in [29], the author proved infinitely many solutions via fountain theorem in (1.1) when λ = 1 and V (x) is positive. However, to the best of our knowledge, for the sign-changing potential case, there are not many results for problem (1.1). In 2013, Tang [21] gave some more weaker conditions and studied the existence of infinitely many solutions for Schrödinger equation via the symmetric mountain pass theorem with sign-changing potential. Using Tang’s conditions, some authors studied the existence of infinitely many solutions for different equations with signchanging potential. See, e.g., [6, 7, 9, 24, 25, 31] and the references quoted in them. These results generalized and extended some known results. In [29], the author proved the existence of multiple solutions for the fractional Schr¨ odinger-Possion equation with the following super-quadratic conditions: (A1) inf x∈R3 V (x) ≥ V0 > 0, where V0 is a constant. Moreover, for every M > 0, meas({x ∈ R3 : V (x) ≤ M }) < ∞, where meas(·) denote the Lebesgue measure in R3 . (A2) There exists a1 > 0 and q ∈ (2, 2∗s ) such that |f (x, u)| ≤ a1 (1 + |u|p−1 ), ∀(x, u) ∈ R3 × R, 6 is the critical exponent in fractional Sobolev inequalities. where 2∗s = 3−2s Moreover, f (x, u) = o(u) as u → 0. (x,u) (A3) lim|u|→∞ F|u| = ∞, uniformly for x ∈ R3 . 4 (A4) there exists a constant θ ≥ 1 such that

θF(x, u) ≥ F(x, τ u),

∀(x, u) ∈ R3 × R, ∀τ ∈ [0, 1],

where F(x, u) := 41 uf (x, u) − F (x, u). (A5) there exists r1 > 0 such that 4F (x, u) ≤ uf (x, u),

∀(x, u) ∈ R3 × R, |u| ≥ r1 .

(A6) f (x, −u) = −f (x, u), ∀ (x, u) ∈ R3 × R. Under the conditions (A1)–(A4), (A6) and (A1)–(A3), (A5) and (A6), respectively, the author obtained multiple solutions for (1.1) in [29, 30]. However, in [12],

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INFINITELY MANY SOLUTIONS

3

Jeanjean gave the following condition and application to Landesman-Lazer type problems in RN . (A7) f (x,u) is increasing in u > 0 and decreasing in u < 0. u3 In [8], the following Ambrosetti and Rabinowitz condition was assumed to prove the existence of high energy solutions. (A8) (Also known as (AR) condition) There exist µ > 4 and r1 > 0 such that 0 < µF (x, u) ≤ uf (x, u), ∀x ∈ R3 , |u| > r1 , Ru where F (x, u) = 0 f (x, η)dη. Inspired by the above results, we consider problem (1.1) with sign-changing potential and without the (AR) type superlinear condition, and establish the existence of infinitely many solutions by the symmetric mountain pass theorem in [21]. To state our results, we use the following conditions on V : (A9) V ∈ C(R3 , R) and inf x∈R3 V (x) > −∞; (A10) there exists a constant d0 > 0 such that lim meas {x ∈ R3 : |x − y| ≤ d0 , V (x) ≤ M } = 0, ∀M > 0, |y|→∞

where meas denotes the Lebesgue measure on R3 . Condition (A10) was first introduced by Bartsch and Wang [4]. From (A9), we give the following equivalent equations for problem (1.1): Remark 1.1. By (A9), we known that V (x) is bounded from below. Hence there exists V0 > 0 such that inf x∈R3 Ve (x) > 0 for all x ∈ R3 , where Ve (x) := V (x) + V0 . Let fe(x, u) := f (x, u) + V0 u. Then problem (1.1) is equivalent to the problem (−∆)s u + Ve (x)u + λφu = fe(x, u), x ∈ R3 , (−∆)t φ = u2 ,

x ∈ R3 .

To achieve our results, we need to make the following assumptions on F and f . (A11) f ∈ C(R3 , R), and there exist c1 > 0, c2 > 0 and q ∈ (4, 2∗s ) such that |f (x, u)| ≤ c1 |u|3 + c2 |u|q−1 , where 2∗s =

(A12)

6 3−2s (x,u)| lim|u|→∞ |F|u| 4

∀(x, u) ∈ R3 × R,

is the critical exponent in fractional Sobolev inequalities. = ∞, a.e. x ∈ R3 and there exists r0 ≥ 0 such that

F (x, u) ≥ 0,

∀(x, u) ∈ R3 × R, |u| ≥ r0 ;

(A13) there exists θ0 > 0 such that 4F (x, u) − uf (x, u) ≤ θ0 u2 ,

∀(x, u) ∈ R3 × R.

Next, we illustrate that F and f satisfying (A11)-(A13) and (A6) is not equivalent to Fe and fe satisfying (A11)–(A13) and (A6). Remark 1.2. First, we prove that fe satisfying (A11) is not equivalent to f satisfying (A11). In fact, if fe satisfies (A11), then we have |f (x, u)| ≤ |fe(x, u) − V0 u| ≤ c1 |u|3 + c2 |u|q−1 + V0 |u|. Thus f does not satisfy (A11). Now, if f satisfy (A11), similar to the discussion of f , we can obtain |fe(x, u)| ≤ c1 |u|3 + c2 |u|q−1 + V0 |u|.

4


EJDE-2017/97

Thus fe does not satisfy (A11). (x,u)| (x,u)| Second, if lim|u|→∞ |F|u| = ∞, a.e. x ∈ R3 then lim|u|→∞ |F|u| = ∞, a.e. 4 4 3 x ∈ R . the converse also holds. Moreover, if F (x, u) ≥ 0 for any (x, u) ∈ R3 × R, |u| ≥ r0 , then Fe(x, u) ≥ 0 for any (x, u) ∈ R3 × R, |u| ≥ r0 . Conversely, it does not hold. Finally, fe satisfying (A13) is not equivalent to f satisfying (A13). In fact, if 4F (x, u) − uf (x, u) ≤ θ0 u2 , then using fe(x, u) := f (x, u) + V0 u, we have 1 4 Fe(x, u) − V0 u2 − u fe(x, u) + V0 u = 4Fe(x, u) − ufe(x, u) − V0 u2 ≤ θ0 u2 2 which implies 4Fe(x, u) − ufe(x, u) ≤ (θ0 + V0 )u2 . This shows that Fe and fe satisfy (A13). On the contrary, if 4Fe(x, u) − ufe(x, u) ≤ θ0 u2 , then similar to the proof the above inequalities, we obtain e

4F (x, u) − uf (x, u) ≤ (θ0 − V0 )u2 . But we do not know whether θ0 > V0 or not. Thus F and f do not satisfy (A13). As for (A6), it is easy to check that f satisfying (A6) is equivalent to fe satisfying (A6). Now, we are ready to state the main results of this paper. Note that the space E is defined in (2.2). Theorem 1.3. Suppose that (A6), (A9)–(A13) are satisfied. Then when s ∈ (3/4, 1), t ∈ (0, 1) satisfying 4s + 2t ≥ 3, problem (1.1) has infinitely many nontrivial solutions. {(uk , φtuk )} in E × Dt,2 (R3 ) satisfying J(uk ) → +∞ as k → ∞, where the functional J is defined in (2.8). In [21], the author used the following conditions to prove the existence of infinitely many solutions for Schr¨ odinger equation. (A15) there exist µ > 4 and % > 0 such that µF (x, u) ≤ uf (x, u) + %u2 ,

∀(x, u) ∈ R3 × R;

(A16) there exist µ > 4 and r1 > 0 such that µF (x, u) ≤ uf (x, u),

∀(x, u) ∈ R3 × R, |u| ≥ r0 ;

It is easy to check that (A15) imply (A13). Thus, we have the following corollary. Corollary 1.4. Suppose that (A6), (A9)–(A12), (A15) are satisfied. Then when s ∈ (3/4, 1), t ∈ (0, 1) satisfy 4s + 2t ≥ 3, problem (1.1) has infinitely many nontrivial solutions. {(uk , φtuk )} in E×Dt,2 (R3 ) satisfying J(uk ) → +∞ as k → ∞, where the functional J is defined in (2.8). It is easy to check that (A11) and (A15) imply (A16). Thus, we have the following corollary. Corollary 1.5. Suppose that (A6), (A9)–(A12), (A16) are satisfied. Then when s ∈ (3/4, 1), t ∈ (0, 1) satisfy 4s + 2t ≥ 3, problem (1.1) has infinitely many nontrivial solutions. {(uk , φtuk )} in E×Dt,2 (R3 ) satisfying J(uk ) → +∞ as k → ∞, where the functional J is defined in (2.8).

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Remark 1.6. In our results, F (x, u) is allowed to be sign-changing. Thus (A12) is much weaker than (A3). In addition, it is obvious that (A15) is somewhat weaker than (A16). Moreover, (A16) implies (A5) and that (A15) implies (A13). Hence, our condition (A13) is somewhat weaker than (A5), (A15), (A16). Remark 1.7. If s ∈ (3/4, 1) then we can infer that 2∗s > 4. Hence (A11) is feasible. 2. Variational framework and main results In this section, we need assumptions (A17) and (A18) instead of (A9) and (A10). (A17) Ve ∈ C(R3 , R) and inf x∈R3 Ve (x) > 0; (A18) there exists a constant d0 > 0 such that lim meas {x ∈ R3 : |x − y| ≤ d0 , Ve (x) ≤ M } = 0, ∀M > 0, |y|→∞

where meas denotes the Lebesgue measure on R3 . We define the Gagliardo seminorm by 1/p Z Z |u(x) − u(y)|p dxdy , [u]α,p = N +αp R3 R3 |x − y| where u : R3 → R is a measurable function. On the one hand, we define fractional Sobolev space by W α,p (R3 ) = {u ∈ Lp (R3 ) : u is measurable and [u]α,p < ∞} endowed with the norm 1/p kukα,p = [u]pα,p + kukpp ,

(2.1)

where kukp =

Z

1/p |u(x)|p dx .

R3

If p = 2, the space W α,2 (R3 ) is an equivalent definition of the fractional Sobolev spaces is based on the Fourier analysis; that is, Z α 3 α,2 3 2 3 H (R ) := W (R ) = u ∈ L (R ) : (1 + |ξ|2α )|e u|2 dξ < ∞ , R3

endowed with the norm kukH α =

Z

|ξ|2α |e u|2 dξ +

R3

Z

|u|2 dξ

1/2

,

R3

where u e denotes the usual Fourier transform of u. Furthermore, we know that k · kH α is equivalent to the norm Z Z 1/2 kukH α = |(−∆)α/2 u|2 dx + u2 dx . R3

R3

On the other hand, in view of the potential Ve (x), we consider the subspace Z E = u ∈ H α (R3 ) : Ve (x)u2 dx < ∞ . (2.2) R3

Thus, E is a Hilbert space with the inner product Z Z 2α (u, v)EV = |ξ| u e(ξ)e v (ξ) + u e(ξ)e v (ξ) dξ + R3

R3

Ve (x)u(x)v(x)dx

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EJDE-2017/97

and the norm kukEV =

Z

Z

|ξ|2α |e u(ξ)|2 + |e u(ξ)|2 dξ +

R3

1/2 Ve (x)u2 (x)dx .

R3

Moreover, k · kEV is equivalent to the norm Z Z kuk := kukE = |(−∆)α/2 u|2 dx + R3

1/2 Ve (x)u2 dx ,

R3

where the corresponding inner product is Z (u, v)E = (−∆)α/2 u(−∆)α/2 v + Ve (x)uv dx. R3

The homogeneous Sobolev space Dα,2 (R3 ) is defined by ∗ Dα,2 (R3 ) = u ∈ L2α (R3 ) : |ξ|α u e(ξ) ∈ L2 (R3 ) , which is the completion of C0∞ (R3 ) under the norm 1/2 Z Z = |(−∆)α/2 u|2 dx kukDα,2 =

|ξ|2α |e u(ξ)|2 dξ

1/2

,

R3

R3

endowed with the inner product Z

(−∆)α/2 u(−∆)α/2 v dx.

(u, v)Dα,2 = R3 ∗

Then Dα,2 (R3 ) ,→ L2α (R3 ); that is, there exists a constant C0 > 0 such that kuk2∗α ≤ C0 kukDα,2 .

(2.3)

Next, we give the following lemmas which discuss the continuous and compact embedding for E ,→ Lp (R3 ) for all p ∈ [2, 2∗α ]. In the rest of this article, we use the norm k · k in E. Motivated by [33, Lemma 3.4], we can prove the following lemma. Here we omit its proof. Lemma 2.1. Space E is continuously embedded in Lp (R3 ) for 2 ≤ p ≤ 2∗α := and compactly embedded in Lp (R3 ) for all s ∈ [2, 2∗α ).

6 3−2α

By Lemma 2.1, we can conclude that there exists a constant γp > 0 such that kukp ≤ γp kuk, p

3

where kukp denotes the usual norm in L (R ) for all 2 ≤ p ≤

(2.4) 2∗α .

Lemma 2.2 ([18, Theorem 6.5]). For any α ∈ (0, 1), Dα,2 (R3 ) is continuously ∗ embedded int L2α (R3 ), that is, there exists Sα > 0 such that Z Z 2/2∗α 2∗ α |u| dx ≤ Sα |(−∆)α/2 u|2 dx ∀u ∈ Dα,2 (R3 ). R3

R3

Next, let α = s ∈ (0, 1). Using Hölder’s inequality, for every u ∈ E and s, t ∈ (0, 1), we have Z Z 3+2t Z 1/2∗t ∗ 12 6 u2 vdx ≤ |u| 3+2t dx |v|2t dx R3 R3 R3 (2.5) 1/2 2 12 S t,2 ≤ γ 3+2t kuk kvk , D t where we use the embedding 12

E ,→ L 3+2t (R3 )

when 2t + 4s ≥ 3.

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By the Lax-Milgram theorem, there exists a unique φtu ∈ Dt,2 (R3 ) such that Z Z Z t t t/2 t t/2 v(−∆) φu dx = (−∆) φu (−∆) vdx = u2 vdx, v ∈ Dt,2 (R3 ). (2.6) R3

R3

R3

Hence, φtu satisfies the Poisson equation (−∆)t φtu = u2 ,

x ∈ R3 .

Moreover, φtu has the integral expression Z u2 (y) t φu (x) = ct dy, 3−2t R3 |x − y|

x ∈ R3 ,

which is called t-Riesz potential, where ct = π −3/2 2−2t

Γ( 32 − 2t) . Γ(t)

Thus φtu (x) ≥ 0 for all x ∈ R3 , from (2.1) and (2.6), we have 1/2

kφtu kDt,2 ≤ St

kuk2

12

L 3+2t

≤ C1 kuk2

when 2t + 4s ≥ 3.

(2.7)

e1 > 0, C e2 > 0 such Therefore, by H¨ older’s inequality and Lemma 2.1, there exist C that Z 3+2t Z 1/2∗t Z 12 6 t 2 t 2∗ t φu u dx ≤ |φu | dx |u| 3+2t dx R3

R3

≤

e1 kφtu kDt,2 kuk2 C

R3

e2 kuk4 . ≤C

Next, we define the energy functional J on E by Z Z 1 λ Fe(x, u)dx, J(u) = kuk2 + φtu u2 dx − 2 4 R3 R3

∀u ∈ E.

(2.8)

By [19], the energy functional J : E → R is well defined and of class C 1 (E, R). Moreover, the derivative of J is Z hJ 0 (u), vi = (−∆)s/2 u(−∆)s/2 v + Ve (x)uv + λφtu uv − fe(x, u)v dx, (2.9) R3

for all u, v ∈ E. Obviously, it can be proved that if u is a critical point of J, then the pair (u, φtu ) is a solution of system (1.1). A sequence {un } ⊂ E is said to be a (C)c -sequence if J(u) → c and kJ 0 (u)k(1 + kun k) → 0. J is said to satisfy the (C)c -condition if any (C)c -sequence has a convergent subsequence. To prove our results, we state the following symmetric mountain pass theorem, see [1, Lemma 2.4] and [19, Lemma 912] Lemma 2.3. Let X be an infinite dimensional Banach space, X = Y ⊕ Z, where Y is finite dimensional. If J ∈ C 1 (X, R) satisfies the (C)c condition for all c > 0, and (1) J(0) = 0, J(−u) = J(u) for all u ∈ X; (2) there exist constants ρ, α > 0 such that J|∂Bρ ∩Z ≥ α; e ⊂ X, there is R = R(X) e > 0 such (3) for any finite dimensional subspace X e that J(u) ≤ 0 on X\BR ; then J possesses an unbounded sequence of critical values.

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Lemma 2.4. Assume that a sequence {un } ⊂ E, un * u in E as n → ∞ and {kun k} is a bounded sequence. Then, as n → ∞, Z (φtun un − φtu u)(un − u)dx → 0. (2.10) R3

Proof. Take a sequence {un } ⊂ E such that un * u in E as n → ∞ and {kun k} is a bounded sequence. By Lemma 2.1, we have un → u in Lp (R3 ) where 2 ≤ p < 2∗s , and un → u a.e. on R3 . Hence supn∈N kun k < ∞ and kuk is finite. Since s ∈ 6 (3/4, 1), then we know that E ,→ L 2s . Hence by (2.3) and (2.7), we have Z (φtu un − φtu u)(un − u)dx n

R3

Z

1/2 Z 1/2 (φtun un − φtu u)2 dx (un − u)2 dx R3 R3 i1/2 √ hZ kun − uk2 (|φtun un |2 + |φtu u|2 ) ≤ 2 ≤

R3

1/2 ≤ C3 kφtun k22∗s kun k26 + kφtu k22∗s kuk26 kun − uk2 2s 2s 1/2 ≤ C3 kun k4 + kuk4 kun − uk2 → 0, as n → ∞.

This completes the proof.

The next lemmas are needed for our proofs. Lemma 2.5 ([21]). Assume that p1 , p2 > 1, r, q ≥ 1 and Ω ⊆ RN . Let g(x, t) be a Carathéodory function on Ω × R and satisfies |g(x, t) ≤ a1 |t|

p1 −1 r

+ a2 |t| p1

p2 −1 r

,

∀(x, t) ∈ Ω × R,

(2.11)

p2

where a1 , a2 ≥ 0. t If un → u in L (Ω) ∩ L (Ω), and un → u a.e. x ∈ Ω, then for any v ∈ Lp1 q (Ω) ∩ Lp2 q (Ω), Z |g(x, un ) − g(x, u)|r |v|q dx = 0. (2.12) lim n→∞

Ω

Lemma 2.6. Suppose that (A9)–(A13) are satisfied. Then any {un } ⊂ E satisfying J(un ) → c > 0,

kJ 0 (un )k(1 + kun k) → 0

(2.13)

is bounded in E. Proof. To prove the boundedness of {un } we argue by contradiction. Assume that kun k → ∞ and let vn = un /kun k. Then kvn k = 1 and kvn kp ≤ γp kvn k = γp for 2 ≤ p < 2∗s . By (A13) and (2.13), for n large enough, we have 1 c + 1 ≥ J(un ) − hJ 0 (un ), un i 4Z 1 1 2 = kun k + fe(x, un )un − Fe(x, un ) dx 4 4 3 ZR h i 1 1 1 (2.14) = kun k2 + f (x, un ) − F (x, un ) − V0 u2 dx 4 4 R3 4 1 1 ≥ kun k2 − (θ0 + V0 ) kun k22 4 4 1 1 2 ≥ kun k − (θ0 + V0 ) kvn k22 kun k2 , 4 4

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which implies c+1 1 1 ≥ − (θ0 + V0 ) kvn k22 . 2 kun k 4 4

(2.15)

For 0 < a < b, let Ωn (a, b) = {x ∈ R3 : a ≤ |un | < b}. (2.16) Passing to a subsequence, we may assume that vn * v in E, then by Lemma 2.1, vn → v in Lp (R3 ), 2 ≤ p < 2∗s , and vn → v a.e. on R3 . Hence, if kun k → ∞ in (2.15), then 4 + on (1), kvn k22 ≥ θ0 + V0 which shows that vn * v 6= 0. We only need to consider the case v 6= 0. Set A := {x ∈ R3 : v(x) 6= 0}. Thus meas(A) > 0. For a.e. x ∈ A, we have limn→∞ |un (x)| = ∞. Hence A ⊂ Ωn (r0 , ∞) for large n ∈ N. which implies that χΩn (r0 ,∞) = 1 for large n, where χΩn denotes the characteristic function on Ωn and Ωn (0, r0 ) is the same as in (2.16). By (A13), then we have 1 (2.17) |Fe(x, un )| ≤ c1 u4n + c2 |un |q + V0 u2n , 2 which implies that there exists a constant C3 > 0 such that |Fe(x, un )| ≤ C3 u2n , for any |un | ≤ r0 . It follows from (2.8), (2.16), (2.17) and Fatou’s Lemma that c + o(1) J(un ) = lim 4 n→∞ kun k n→∞ kun k4 Z Z h 1 i 1 t 2 e(x, un )dx + λ 1 F = lim − φ u dx n→∞ 2kun k2 ku k4 R3 4 kun k4 R3 un n Z n h 1 Fe(x, un )dx = lim − n→∞ kun k4 Ωn (0,r0 ) Z Z i Fe(x, un ) 4 λ 1 − v dx + φtun u2n dx n 4 4 un 4 kun k R3 Ωn (r0 ,∞) Z h 1 e2 Fe(x, un ) 4 i λC 2 ku k vn dx + ≤ lim sup − n 2 4 4 kun k un 4 n→∞ Ωn (r0 ,∞) h kv k2 Z i e2 Fe(x, un ) 4 λC n 2 ≤ lim sup − vn dx + 2 4 kun k un 4 n→∞ Ωn (r0 ,∞) Z h γ2 i e2 Fe(x, un ) 4 λC 2 = lim sup − vn dx + 2 4 kun k un 4 n→∞ Ωn (r0 ,∞) Z e e F (x, un ) 4 λC2 ≤ − lim inf vn dx + 4 n→∞ u 4 Ωn (r0 ,∞) n Z e2 |Fe(x, un )| λC 4 = − lim inf [χ (x)]v dx + Ω (r ,∞) n n 0 n→∞ u4n 4 R3 Z e2 Fe(x, un ) λC = −∞, (2.18) ≤− lim inf [χΩn (r0 ,∞) (x)]vn4 dx + 4 n→∞ un 4 R3

0 = lim

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

Lemma 2.7. Suppose that (A9)–(A13) are satisfied. Then each {un } ⊂ E satisfying (2.13) has a convergent subsequence in E.

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Proof. By Lemma 2.6, {un } is bounded in E. If necessary going to a subsequence, we can assume that un * u in E. From Lemma 2.1, we have un → u in Lp (Ω) for all 2 ≤ p < 2∗s . Hence, by Lemma 2.5, one has Z (fe(x, un ) − fe(x, u))(un − u)dx → 0, as n → ∞. (2.19) R3

Observe that 2

0

Z

0

kun − uk = hJ (un ) − J (u), un − ui + (φtun − φtu )(un − u)dx R3 Z + (fe(x, un ) − fe(x, u))(un − u)dx.

(2.20)

R3

It is clear that hJ 0 (un ) − J 0 (u), un − ui → 0,

as n → ∞.

From (2.19), (2.20) and (2.21), we have kun − uk → 0, as n → ∞.

(2.21)

e ⊂ E, it holds Lemma 2.8. Suppose that (A9)–(A13) are satisfied. Then for each E J(u) → −∞,

kuk → ∞,

e u ∈ E.

(2.22)

e with kun k → Proof. Arguing indirectly, assume that for some sequence {un } ⊂ E ∞, there is M > 0 such that J(un ) ≥ −M for all n ∈ N. Set vn = kuunn k , then e is kvn k = 1. Passing to a subsequence, we may assume that vn * v in E. Since E N e finite dimensional, then vn → v ∈ E in E, vn → v a.e. on R , and so kvk = 1. Hence, we can conclude a contradiction by a similar methods as (2.18). e ⊂ E, there Corollary 2.9. Suppose that (A9)–(A13) are satisfied. Then for any E e exists R = R(E) > 0 such that J(un ) ≤ 0,

kuk ≥ R,

e ∀u ∈ E.

Let {ej } is a total orthonormal basis of E and define Xj = Rej , Yk = ⊕kj=1 Xj ,

Zk = ⊕∞ j=k+1 Xj ,

∀k ∈ Z.

(2.23)

Lemma 2.10. Suppose that (A9) and (A10) are satisfied. Then for 2 ≤ p < 2∗s , we have βk (s) :=

sup

kukp → 0,

k → ∞.

u∈Zk ,kuk=1

Proof. It is clear that 0 < βk+1 ≤ βk , so that βk → β ≥ 0(k → ∞). For every k ∈ N, there exists uk ∈ Zk such that |uk |2 > β2k and kuk k = 1. For any v ∈ E, writing v = Σ∞ j=1 cj ej , we have, by the Cauchy-Schwartz inequality, ∞ |(uk , v)| = |(uk , Σ∞ j=1 cj ej )| = |(uk , Σj=k+1 cj ej )| ∞ 2 1/2 ≤ kuk kkΣ∞ →0 j=k+1 cj ej k = (Σj=k+1 cj )

as k → ∞, which implies that uk * 0. By Lemma 2.1, the compact embedding of E ,→ Lp (R3 ) (2 ≤ p < 2∗s ) implies that uk → 0 in Lp (R3 ). Hence, letting k → ∞, we obtain β = 0, which completes the proof.

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By Lemma 2.10, we can choose an integer m ≥ 1 such that r 1 1 q 2 2 2 kuk2 ≤ kuk , kuk4 ≤ kuk2 , kukqq ≤ kukq , ∀u ∈ Zm , 2V0 c1 4c2

11

(2.24)

where q ∈ (4, 2∗s ). Lemma 2.11. Suppose that (A9)–(A11) are satisfied. Then there exist constants ρ, α > 0 such that J ∂B ∩Z ≥ α. ρ

m

Proof. From (2.17) and (2.24), for u ∈ Zm , choosing ρ := kuk = 21 , we obtain Z Z λ 1 2 t 2 φ u dx − Fe(x, u)dx J(u) = kuk + 2 4 R3 u R3 Z 1 2 Fe(x, u)dx ≥ kuk − 2 R3 1 c1 c2 1 ≥ kuk2 − kuk44 − kukqq − V0 kuk22 2 4 q 2 1 ≥ kuk2 − kuk4 − kukq 4 1 3 1 = − := α > 0, 4 16 2q since q ∈ (4, 2∗s ). This completes the proof.

Proof of Theorem 1.3. Let X = E, Y = Ym and Z = Zm . By Lemmas 2.6, 2.7, 2.11 and Corollary 2.9, all conditions of Lemma 2.3 are satisfied. Thus problem (1.1) possesses infinitely many nontrivial solutions. Acknowledgements. The authors want to than the editor and the anonymous referees for their valuable comments and suggestions. This work was supported by the National Natural Science Foundation of China (Grants No. 11461043, 11571370 and 11601525), and by the Provincial Natural Science Foundation of Jiangxi, China (20161BAB201009) and by the Science and Technology Project of Educational Commission of Jiangxi Province, China (150013), and by the Hunan Provincial Innovation Foundation For Postgraduate (Grant No. CX2016B037). References [1] T. Bartolo, V. Benci, D. Fortunato; Abstract critical point theorems and applications to some nonlinear problems with strong resonance at infinity, Nonlinear Anal., 7 (1983), 981-1012. [2] T. Bartsch; Infinitely many solutions of a symmetric Dirichlet problem. Nonlinear Anal., 20 (1993), 1205-1216. [3] T. Bartsch, M. Willem; On an elliptic equation with concave and convex nonlinearities. Proc. Am. Math. Soc., 123 (1995), 3555-3561. [4] T. Bartsch, Z. Q. Wang; Existence and multiplicity results for some superlinear elliptic problems on RN , Comm. Partial Differ. Equ., 20 (1995), 1725-1741. [5] T. Bartsch, Z. Q. Wang, M. Willem; The Dirichlet problem for superlinear elliptic equations, In: Chipot, M., Quittner, P.: (eds.) Handbook of Differential Equations-Stationary Partial Differential Equations, vol. 2, pp. 1-5. [6] J. Chen, B. Cheng, X. H. Tang; New existence of multiple solutions for nonhomogeneous Schr¨ odinger-Kirchhoff problems involving the fractional p-Laplacian with sign-changing potential, Rev. Real Acad. Cienc. Exactas F., (2016) DOI: 10.1007/s13398-016-0372-5.

12


EJDE-2017/97

[7] J. Chen, X. H. Tang, Z. Gao; Existence of multiple solutions for modified Schr¨ odingerKirchhoff-Poisson type systems via perturbation method with sign-changing potential, Comput. Math. Appl., 73 (2017), 505-519. [8] S. Chen, C. Tang; High energy solutions for the Schr¨ odinger-Maxwell equations, Nonlinear Anal. 71 (2009) 4927-4934. [9] B. Cheng, X. H. Tang; High energy solutions of modified quasilinear fourth-order elliptic equations with sign-changing potential, Comput. Math. Appl., 73 (2017), 27-36. [10] A. R. Giammetta; Fractional Schr¨ odinger-Poisson-Slater system in one dimension, arXiv:1405.2796v1. [11] E. Hebey, J. Wei; Schr¨ odinger-Poisson systems in the 3-sphere, Calc. Var. Partial Differ. Equ., 47 (2013), 25-54. [12] L. Jeanjean; On the existence of bounded Palais-Smale sequences and application to a landesman-Lazer type problem on RN , Proc. Roy. Soc. Edinburgh Sect. A, 129 (1999), 787809. [13] H. Kikuchi; On the existence of solution for elliptic system related to the Maxwell-Schr¨ odinger equations, Nonlinear Anal., 27 (2007), 1445-1456. [14] N. Laskin; Fractional quantum mechanics and L´ evy path integrals, Phys. Lett. A., 268 (2000), 298-305. [15] N. Laskin; Fractional Schr¨ odinger equation, Phys. Rev. E., 66 056108 (2002). [16] S. B. Liu, S. J. Li; Infinitely many solutions for a superlinear elliptic equation, Acta Math. Sinica (Chin. Ser.), 46 (4) (2003), 625-630 (in Chinese). [17] D. Mugnai; The Schr¨ odinger-Poisson system with positive potential, Comm. Partial Differ. Equ., 36 (2011), 1099-1117. [18] E. D. Nezza, G. Palatucci, E. Valdinoci; Hitchhiker’s guide to the fractional Sobolev spaces, Bull. Sci. Math., 136 (2012), 512-573. [19] P. H. Rabinowitz; Minimax methods in critical point theory with applications to differential equations, in: CBMS Reg. Conf. Ser. in Math., vol. 65, Amer. Math. Soc., Providence, RI, 1986. [20] J. Seok; On nonlinear Schr¨ odinger-Poisson equations with general potentials, J. Math. Anal. Appl., 401 (2013), 672-681. [21] X. H. Tang; Infinitely many solutions for semilinear Schr¨ odinger equations with signchanging potential and nonlinearity, J. Math. Anal. Appl., 401 (2013) 407-415. [22] L. Zhao, H. Liu, F. Zhao; Existence and concentration of solutions for the Schr¨ odingerPoisson equations with steep well potential, J. Differential Equations, 255 (2013) 1-23. ´ M. Squassina; Fractional Schr¨ [23] J. Zhang, J. M. do O, odinger-Possion equations with a general subcritical or critical nonlinearity, Adv. Nonlinear Stud., 16 (2016), 15-30. [24] J. Zhang, X. H. Tang, W. Zhang; Infinitely many solutions of quasilinear Schr¨ odinger equation with sign-changing potential, J. Math. Anal. Appl., 420 (2014), 1762-1775. [25] J. Zhang, X. H. Tang, W. Zhang; Existence of multiple solutions of Kirchhoff type equation with sign-changing potential, Appl. Math. Comput., 242 (2014), 491-499. [26] J. Zhang, X. H. Tang, W. Zhang; Existence and multiplicity of stationary solutions for a class of Maxwell-Dirac system, Nonlinear Anal., 127 (2015), 298-311. [27] J. Zhang, X. H. Tang, W. Zhang; Ground states for diffusion system with periodic and asymptotically periodic nonlinearity, Comput. Math. Appl., 71 (2016), 633-641. [28] J. Zhang, W. Zhang, X. L. Xie; Existence and concentration of semiclassical solutions for Hamiltonian elliptic system, Commu. Pure Appl. Anal., 15 (2016), 599-622. [29] J. G. Zhang; Existence and multiplicity results for the Fractional Schr¨ odinger-Poisson systems, arXiv:1507.01205v1. [30] Q. Zhang, Q. Wang; Multiple solutions for a class of sublinear Schr¨ odinger equations, J. Math. Anal. Appl., 389 (2012), 511-518. [31] W. Zhang, X. H. Tang, J. Zhang; Infinitely many solutions for fourth-order elliptic equations with sign-changing potential, Taiwanese J. Math., 18 (2014), 645-659. [32] W. Zhang, X. H. Tang, J. Zhang; Infinitely many radial and non-radial solutions for a fractional Schr¨ odinger equation, Comput. Math. Appl., 71 (2016), 737-747. [33] W. M. Zou, M. Schechter; Critical Point Theory and Its Applications, Springer, New York, 2006.

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Jianhua Chen School of Mathematics and Statistics, Central South University, Changsha, Hunan 410083, China E-mail address: [email protected] Xianhua Tang (corresponding author) School of Mathematics and Statistics, Central South University, Changsha, Hunan 410083, China E-mail address: [email protected] Huxiao Luo School of Mathematics and Statistics, Central South University, Changsha, Hunan 410083, China E-mail address: [email protected]