We investigate a class of fractional Schrödinger-Poisson system via variational methods. By using symmetric mountain pass theorem, we prove the existence of multiple solutions. Moreover, by using dual fountain theorem, we prove the above system has a sequence of negative energy solutions, and the corresponding energy values tend to 0. These results extend some known results in previous papers.

National Natural Science Foundation of China11701346115712091. Introduction

We consider the following system via variational methods:SP-Δαw+Vxw+ϕw=gx,w+λhxwq-2w,x∈R3,-Δβϕ=w2,x∈R3,where λ>0, 1<q<2, α,β∈(0,1], 2β+4α>3. (-Δ)α and (-Δ)β represent the Laplace operator of the fractional order. If α=β=1, then the system SP degenerates into the standard Schrödinger-Poisson system, which describes the interaction between the same charged particles when the magnetic effect can be ignored [1]. In recent years, the existence, multiplicity, and centralization of solutions for the Schrödinger-Poisson system have been deeply studied via variational methods, and a great number of works have been obtained, see, for example, [2–8]. On the other hand, (-Δ)α is a class of nonlocal pseudo-differential operators. Since nonlocal differential equations can better and more fully describe the physical experimental phenomena than classical local differential operators, the study of nonlinear fractional Laplace equation has become one of the most popular research fields in nonlinear analysis.

In the literature [9], Wei considered the following system:(1)-Δαw+Vxw+ϕw=gx,w,x∈R3,-Δαϕ=γαw2,x∈R3.By using the critical point theory, the author obtained infinitely many solutions when α∈0,1. In the literature [10], Teng studied a system of the form(2)-Δαw+Vxw+ϕw=θwq-1w+w2α∗-2w,x∈R3,-Δβϕ=w2,x∈R3,where q∈(1,3+2α/3-2α), α,β∈(0,1), 2β+2α>3. In [11], Zhang, Marcos, and Squassina used perturbation approach to obtain the existence of solutions for the following system when the nonlinear term is subcritical or critical(3)-Δαw+λϕw=gw,x∈R3,-Δβϕ=λw2,x∈R3,where λ>0, α,β∈[0,1]. In [12], Duarte and Souto investigated the following system via variational methods(4)-Δαw+Vxw+ϕw=gw,x∈R3,-Δβϕ=w2,x∈R3,where α∈(3/4,1), β∈(0,1), V:R3→R is a periodic potential. A positive solution and a ground state solution were got in [12]. In [13], Li studied a system of the following form:(5)-Δαw+Vxw+ϕw=gx,w,x∈R3,-Δβϕ=w2,x∈R3,where α,β∈(0,1], 2β+4α>3. Combining the perturbation method with mountain pass theorem, the existence of nontrivial solutions was obtained in [13]. In [14], Yu, Zhao, and Zhao studied the following fractional Schrödinger–Poisson system with critical growth via variational methods(6)ε2α-Δαw+Vxw+ϕw=w2α∗-2w+gw,x∈R3,ε2α-Δαϕ=w2,x∈R3,where α∈(3/4,1),2α∗=6/3-2α, the potential V is continuous with positive infimum, g is continuous and subcritical at infinity. Under some Monotone hypothesis on g, the existence of positive ground state solution is got in [14]. For small ε>0, a multiple result is also got in [14].

Inspired by [9–16], in this paper, we prove the existence of multiple solutions for system SP by symmetric mountain pass theorem. Moreover, we prove the system SP has a sequence of negative energy solutions by dual fountain theorem. The assumptions on V and nonlinearity g in this paper are given below:

(V) V∈C(R3,R), V0≔infx∈R3V(x)>0 and lim|x|→+∞V(x)=+∞;

(H1) g∈C(R3×R,R), and there exists C1>0 such that |g(x,w)|≤C1(|w|+|w|p-1), where p∈(4,2α∗), 2α∗=6/3-2α;

(H2) there exist κ>4 and r>0 such that 0<κG(x,w)≔κ∫0wg(x,t)dt≤g(x,w)w, for |w|≥r. Moreover, infx∈R3,|w|=rG(x,w)>0;

(H3) g(x,-w)=-g(x,w), x∈R3, w∈R;

(H4) h:R3→R+, and h∈L2/2-q(R3).

2. Preliminaries

For 1≤ν<∞, Lν(R3) denotes the usual Lebesgue space with norm ‖w‖ν=(∫R3|w|νdx)1/ν. Fix α∈(0,1), fractional Sobolev space Hα(R3) denoted as(7)HαR3≔w∈L2R3:∫R31+l2αFwl2dl<∞,equipped with the norm(8)wHα=∫R3Fwl2+l2αFwl2dl1/2,where Fw denotes the Fourier transform of function w. Let g∈C0∞(R3); the fractional Laplacian operator (-Δ)α:C0∞(R3)→C0∞R3′ is defined by(9)-Δαg=F-1l2αFg,l∈R3.According to Plancherel theorem [17], one has ‖Fw‖2=‖w‖2, ‖|l|αFw‖2=‖(-Δ)α/2w‖2. By (8), we define the equivalent norm(10)wHα=∫R3-Δα/2wx2+wx2dx1/2.Dα,2(R3) is denoted as(11)Dα,2R3=w∈L2α∗R3:lαFwl∈L2R3.In particular, Dα,2(R3) is the completion of C0∞(R3), with respect to the norm(12)wDα,2=∫R3-Δα/2w2dx1/2=∫R3l2αFwl2dl1/2.Lq(R3,h), q∈(1,2) represents a weighted Lebesgue space, that is,(13)LqR3,h=w:R3→Rismeasurableand∫R3hxwqdx<+∞,equipped with the norm(14)wLqR3,h=∫R3hxwqdx1/q.

For convenience, we use C to represent any positive constants which may change from line to line. According to [18], the embedding Hα(R3)↪Lν(R3) is continuous for all ν∈[2,2α∗], i.e., there exists Mν>0 satisfying(15)wν≤MνwHα,w∈HαR3.So, the embedding Hα(R3)↪Lν(R3) is continuous when 2β+4α>3. Fix w∈Hα(R3); we define the nonlinear operator Lw:Dβ,2(R3)→R by(16)Lwv=∫R3w2vdx.Hence(17)Lwv≤∫R3wx12/3+2βdx3+2β/6∫R3vx2β∗dx1/2β∗≤C∫R3wx12/3+2βdx3+2β/6vDβ,2≤Cw2vDβ,2.By Lax-Milgram theorem, we can find a unique ϕwβ∈Dβ,2(R3) such that(18)∫R3-Δβ/2ϕwβ-Δβ/2vdx=∫R3w2vdx,∀v∈Dβ,2R3,and ϕwβ is expressed as(19)ϕwβx=cβ∫R3w2yx-y3-2βdy,wherecβ=Γ3/2-2βπ3/222βΓβ.According to (19), ϕwβ≥0 for all x∈R3. Since β∈(0,1], 2β+4α>3, we can also get 12/3+2β∈(2,2α∗). Together with (17) and (18),(20)ϕwβDβ,2=∫R3-Δβ/2ϕwβ2dx≤C∫R3wx12/3+2βdx3+2β/6≤Cw2.From Hölder’s inequality and (19),(21)∫R3ϕwβw2dx≤C∫R3wx12/3+2αdx3+2α/6ϕwβDβ,2≤Cw2ϕwβDβ,2≤Cw4.Evidently,(22)∫R3ϕwβw2dx≤Cw4.Substituting (18) into system SP, system SP is equivalent toS-Δαw+Vxw+ϕwβw=gx,w+λhxwq-2w.

For the equation S, we define the work space E as(23)E≔w∈HαR3:∫R3Vxw2dx<∞,with(24)w=∫R3-Δα/2w2+Vxw2dx1/2.

Lemma 1 (see [<xref ref-type="bibr" rid="B19">19</xref>]).

Assume condition (V) holds, α∈(0,1), ν∈[2,2α∗); then the embedding E↪Lν(R3) is compact.

From (V) and (H1)-(H4), I:E→R(25)Iw=12∫R3-Δα/2w2+Vxw2dx+14∫R3ϕwβw2dx-∫R3Gx,wdx-λq∫R3hxwqdx.is well defined. Moreover, by Lemma 1, I∈C1(E,R) with(26)I′w,v=∫R3-Δα/2w-Δα/2v+Vxwvdx+∫R3ϕwβwvdx-∫R3gx,wvdx-λ∫R3hxwq-2wvdx,v∈E

Proposition 2.

(i) If equation S has a solution w∈E, then system SP has a solution (w,ϕ)∈E×Dβ,2(R3).

(ii) If for every v∈E, the following equation(27)∫R3-Δα/2w-Δα/2v+Vxwvdx+∫R3ϕwβwvdx-∫R3gx,wvdx-λ∫R3hxwq-2wvdx=0holds, then w∈E is a solution of S.

Set {ei}i=1∞ as a set of normalized orthogonal basis of E, Xi=Rei. Yk=⊕i=1kXi,Zk=⊕i=k+1∞Xi¯,k∈N. Obviously, E=Yk⊕Zk.

Definition 3 (see [<xref ref-type="bibr" rid="B20">20</xref>, <xref ref-type="bibr" rid="B21">21</xref>]).

Set I∈C1(E,R), c∈R. If any sequence {wn}⊂E satisfying(28)Iwn→c,I′wn→0,n→∞,has a convergent subsequence, then I satisfies the (PS)c condition. Any sequence satisfying (28) is called the (PS)c sequence.

Definition 4 (see [<xref ref-type="bibr" rid="B21">21</xref>, <xref ref-type="bibr" rid="B22">22</xref>]).

Set I∈C1(E,R), c∈R. If every sequence {wnj}⊂E, satisfying(29)wnj∈Ynj,Iwnj→c,IYnj′→0,n→∞,has a convergent subsequence, then I satisfies the (PS)c∗ condition.

Proposition 5 (see [<xref ref-type="bibr" rid="B23">23</xref>]).

Let E=Y⊕Z be a Banach space with dimY<∞. Assume I∈C1(E,R) is an even functional and satisfies the (PS)c condition and

(A1) there exist ω, μ>0 satisfying I∂Bω∩Z=infw∈Z,‖w‖=ωI(w)≥μ;

(A2) for every linear subspace U⊂E with dimU<∞, there exists a constant L=L(U) such that maxw∈U,‖w‖≥LI(w)<0,

then I has a list of unbounded critical points.

Proposition 6 (see [<xref ref-type="bibr" rid="B22">22</xref>]).

Assume that I∈C1(E,R) is an even functional, k0∈N. If for any k>k0, there exist rk>γk>0 such that

bk=infIw:w∈Zk,w=rk≥0;

ak=max{I(w):w∈Yk,‖w‖=γk}<0;

ck=inf{I(w):w∈Zk,‖w‖≤rk}→0, k→∞;

for every c∈[ck0,0], I satisfies the (PS)c∗ condition,

then I has a sequence of negative critical points that converge to 0.

3. Main ResultsLemma 7.

If hypotheses (V) and (H1)-(H4) hold, then for any c∈R, I satisfies the (PS)c condition.

Proof.

First, we prove the (PS)c sequence {wn} of I is bounded. According to (H4), it is easy to get that(30)∫R3hxwnqdx≤∫R3hx2/2-qdx2-q/2∫R3wn2dxq/2≤h2/2-qw2q≤M2qh2/2-qwnq.By condition (H2), there exists r>0 such that(31)gx,ww≥κGx,w,w≥r.Moreover, for any given C0∈(0,1/16V0), we can choose a constant δ>0 such that(32)1κgx,ww-Gx,w≤C0w2,forw≤δ.From condition (H1), when δ≤|u|≤r, one has(33)1κgx,ww-Gx,w≤C1δ+rp-2w2.So, for any |w|≤r,(34)1κgx,ww-Gx,w≤C0w2+C1δ+rp-2w2.For lim|x|→+∞V(x)=+∞, there exists L>r>0 such that(35)116Vx>C1δ+rp-2,x≥L.Now (34) implies(36)14∫R3Vxwn2dx+∫wnx≤r1κgx,wnwn-Gx,wndx≥14∫R3Vxwn2dx-∫wnx≤rC0wn2+C1δ+rp-2wn2dx=18∫R3Vxwn2dx-∫wnx≤rC0wn2dx+18∫R3Vxwn2dx-∫wnx≤rC1δ+rp-2wn2dx≥∫wnx≤r116V0-C0wn2dx+18∫R3Vxwn2dx-C1δ+rp-2r2·measx∈R3∣x≤L≥18∫R3Vxwn2dx-C1δ+rp-2r2·measx∈R3∣x≤L.Since {wn} is the (PS)c sequence, when n is large enough,(37)c+wn≥Iwn-1κI′wn,wn=12-1κ∫R3∇wn2+Vxwn2dx+14-1κ∫R3ϕwn2wn2dx+∫R31κgx,wnwn-Gx,wndx+1κ-1qλ∫R3hxwnqdx≥14∫R3∇wn2+Vxwn2dx+∫R31κgx,wnwn-Gx,wndx-1q-1κλM2qh2/2-qwnq≥14∫R3∇wn2dx+14∫R3Vxwn2dx+∫wnx≤r1κgx,wnwn-Gx,wndx-1q-1κλM2qh2/2-qwnq.Thus, according to (36), when n is large enough,(38)c+wn≥14∫R3∇wn2dx+14∫R3Vxwn2dx+∫wnx≤r1κgx,wnwn-Gx,wndx-1q-1κλM2qh2/2-qwnq≥18wn2-C1δ+rp-2r2·measx∈R3∣x≤L-1q-1κλM2qh2/2-qwnq.Therefore, {wn} is a bounded sequence in E. By Lemma 1, there exists w^∈E such that(39)wn⇀w^inE,wn→w^inLνR3,wnx→w^xa.e. onR3.

Next, we define the linear operator Bφ:E→R as(40)Bφv=-Δα/2φ-Δα/2v.From Hölder’s inequality, we obtain(41)Bφv≤φv,v∈E.Now by Lemma 1 and (22),(42)∫R3ϕwnβwnwn-w^dx≤ϕwnβ2β∗wn12/3+2βwn-w^12/3+2β≤CϕwnβDβ,2wn12/3+2βwn-w^12/3+2β≤Cwn12/3+2β3wn-w^12/3+2β≤Cwn3wn-w^12/3+2β.Similarly, we can also prove(43)∫R3ϕw^βw^wn-w^dx≤Cw^3wn-w^12/3+2β.Since wn→w^ in Lν(R3)(ν∈[2,2α∗)), limn→∞∫R3(ϕwnβwn-ϕw^βw^)(wn-w^)dx=0.

At last, combining Hölder’s inequality with (H1) and (H4), we can easily get(44)limn→∞∫R3gx,wn-gx,w^wn-w^dx=0,limn→∞∫R3hxwnq-2wn-hxw^q-2w^wn-w^dx=0.Thus(45)o1=I′wn-I′w^,wn-w^=Bwnwn-w^-Bwwn-w^+∫R3Vxwnwn-w^-Vxw^wn-w^dx+∫R3gx,wn-gx,w^wn-w^dx+λ∫R3hxwnq-2wn-w^q-2wwn-w^dx+∫R3ϕwnβwn-ϕw^βw^wn-w^dx=Bwnwn-w^-Bwwn-w^+∫R3Vxwnwn-w^-Vxw^wn-w^dx+o1,that is,(46)wn-w^2=Bwnwn-w^-Bwwn-w^+∫R3Vxwn-w^2dx→0.

Lemma 8.

If hypotheses (V) and (H1)-(H4) hold, then I satisfies (PS)c∗ condition for all c∈R.

Proof.

By Definition 4, we just prove the following fact: if for any c∈R, {wnj}⊂E, and wnj∈Ynj,I(wnj)→c,IYnj′→0, as nj→∞, then {wnj} has a convergence subsequence. The proof method is similar to Lemma 7.

Lemma 9 (see [<xref ref-type="bibr" rid="B24">24</xref>]).

For 2≤ν<2α∗, k∈N, set(47)βνk≔supwν:w∈Zk,w=1,and then βν(k)→0, k→∞.

Theorem 10.

If hypotheses (V) and (H1)-(H4) hold, then we can find λ0>0, such that system (SP) has multiple solutions for every λ<λ0. Moreover, the corresponding energy values tend to infinity.

Proof.

According to Lemma 7, I satisfies (PS)c condition. We only need to prove that I satisfies (A1) and (A2). By virtue of (H1),(48)Gx,w≤C12w2+C1pwp,x,w∈R3×R.From Lemma 9, we can get(49)Iw=12∫R3-Δα/2w2+Vxw2dx+14∫R3ϕwβw2dx-∫R3Gx,wdx-λq∫R3hxwqdx≥12w2-C12∫R3w2dx-C1p∫R3wpdx-M2qλh2/2-qwq≥12w2-C12β22kw2-C1pMppwp-M2qλh2/2-qwq≥w212-C12β22k-C1Mppwp-2-M2qλh2/2-qwq-2.Take a sufficiently large k such that β22(k)<1/2C1. Combining the above inequality, we obtain(50)Iw≥w214-C1Mppwp-2-M2qλh2/2-qwq-2.Set(51)ηt=14-C1Mpptp-2-M2qλh2/2-qtq-2,t>0.Since 1<q<2<p, there exists(52)ωλ≔λ2-qM2qh2/2-qC1Mppp-21/p-q>0,such that maxt∈R+η(t)=η(ωλ). Therefore, foreveryλ<λ0≔(2-q/4C1Mpp(p-q))p-q/p-2·C1(p-2)Mpp/M2q(2-q)‖h‖2/2-q,(53)Iw≥ωλ2ηωλ≔μ>0,withw=ωλ.

On the other hand, by conditions (H1) and (H2), there exist positive constants C2,C3 such that(54)Gx,w≥C2wκ-C3w2,x,w∈R3×R.Since all the norms are equivalent in every finite linear subspace U⊂E, then for w∈U(55)Iw=12∫R3-Δα/2w2+Vxw2dx+14∫R3ϕwβw2dx-∫R3Gx,wdx-λq∫R3hxwqdx≤12w2+Cw4-C2wκκ-C3w22-λqwLqR3,hq.For q<2<4<κ, I(w)→-∞ as w→∞. Then there exists L=L(U)>0 such that maxw∈U,‖w‖≥LI(w)<0. Thus, according to Proposition 5, the system (SP) has a list of solutions {(wn,ϕn)}⊂E×Dβ,2(R3), and the corresponding energy values tend to infinity.

Theorem 11.

If hypotheses (V) and (H1)-(H4) hold, then the system (SP) has a sequence of negative energy solutions for all λ>0, and the energy values tend to 0.

Proof.

By Lemma 8, for all c∈R, I satisfies the (PS)c∗ condition. It now remains to show that (C1)-(C3) are satisfied. According to Lemma 9, for every ν∈[2,2α∗), βν(k)→0, as k→∞. Thus there exists k1>0 such that β2(k)≤1/2C1 for k>k1. For 4<p<2α∗, there exists L∈(0,1) such that(56)18w2≥C1pMppwp,withw≤L.

Hence, for w∈Zk with ‖w‖≤L, it follows that(57)Iw=12∫R3-Δα/2w2+Vxw2dx+14∫R3ϕwβw2dx-∫R3Gx,wdx-λq∫R3hxwqdx≥12w2-C12∫R3w2dx-C1p∫R3wpdx-λq∫R3hxwqdx≥12w2-C12β22ku2-C1pMppwp-λq∫R3hxwqdx≥14w2-C1pMppwp-λqh2/2-qβ2qkwq≥18w2-λqh2/2-qβ2qkwq.For every k>k1, let rk=(8/qλ‖h‖2/2-qβ2q(k))1/(2-q). By Lemma 9, rk→0, as k→∞. Thus, there exists k0>k1, such that for every k≥k0, I(w)≥0, for w∈Zk with ‖w‖=rk.

Secondly, since for every fixed k∈N, the norms are equivalent in Yk, when k is sufficiently large, there exists a small enough γk such that 0<γk<rk and I(w)<0 for w∈Yk with ‖w‖=γk.

Finally, according to (C3), when k≥k0, for u∈Zk, with ‖w‖≤rk, one has(58)Iw≥-λqh2/2-qβ2qkwq≥-λqh2/2-qβ2qkrkq.Since β2(k)→0, rk→0, as k→∞, therefore (C3) holds. By Proposition 6, I has a list of solutions {(wn,ϕn)}⊂E×Dβ,2(R3) such that(59)12∫R3-Δα/2wn2+Vxwn2dx+14∫R3ϕnβwn2dx-∫R3Gx,wndx-λq∫R3hxwnqdx→0.

Data Availability

No data were used to support this study.

Conflicts of Interest

The authors declare that they have no competing interests.

Authors’ Contributions

All authors read and approved the final manuscript.

Acknowledgments

This work is supported by the National Natural Science Foundation of China (Grant No. 11701346, 11571209).

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