In this paper, we consider the following Schrödinger-Poisson system
{−Δu+V(x)u+ϕu=|u|p−2u+λK(x)|u|q−2u inR3,−Δϕ=u2 inR3.
Under the weakly coercive assumption on V and an appropriate condition on K, we investigate the cases when the nonlinearities are of concave-convex type, that is, 1<q<2 and 4<p<6. By constructing a nonempty closed subset of the sign-changing Nehari manifold, we establish the existence of least energy sign-changing solutions provided that λ∈(−∞,λ∗), where λ∗>0 is a constant.
Citation: Chen Yang, Chun-Lei Tang. Sign-changing solutions for the Schrödinger-Poisson system with concave-convex nonlinearities in R3[J]. Communications in Analysis and Mechanics, 2023, 15(4): 638-657. doi: 10.3934/cam.2023032
[1] | Xiao Qing Huang, Jia Feng Liao . Existence and asymptotic behavior for ground state sign-changing solutions of fractional Schrödinger-Poisson system with steep potential well. Communications in Analysis and Mechanics, 2024, 16(2): 307-333. doi: 10.3934/cam.2024015 |
[2] | Yan-Fei Yang, Chun-Lei Tang . Positive and sign-changing solutions for Kirchhoff equations with indefinite potential. Communications in Analysis and Mechanics, 2025, 17(1): 159-187. doi: 10.3934/cam.2025008 |
[3] | Fangyuan Dong . Multiple positive solutions for the logarithmic Schrödinger equation with a Coulomb potential. Communications in Analysis and Mechanics, 2024, 16(3): 487-508. doi: 10.3934/cam.2024023 |
[4] | Maomao Wu, Haidong Liu . Multiple solutions for quasi-linear elliptic equations with Berestycki-Lions type nonlinearity. Communications in Analysis and Mechanics, 2024, 16(2): 334-344. doi: 10.3934/cam.2024016 |
[5] | Yonghang Chang, Menglan Liao . Nonexistence of asymptotically free solutions for nonlinear Schrödinger system. Communications in Analysis and Mechanics, 2024, 16(2): 293-306. doi: 10.3934/cam.2024014 |
[6] | Shengbing Deng, Qiaoran Wu . Existence of normalized solutions for the Schrödinger equation. Communications in Analysis and Mechanics, 2023, 15(3): 575-585. doi: 10.3934/cam.2023028 |
[7] | Floyd L. Williams . From a magnetoacoustic system to a J-T black hole: A little trip down memory lane. Communications in Analysis and Mechanics, 2023, 15(3): 342-361. doi: 10.3934/cam.2023017 |
[8] | Rui Sun, Weihua Deng . A generalized time fractional Schrödinger equation with signed potential. Communications in Analysis and Mechanics, 2024, 16(2): 262-277. doi: 10.3934/cam.2024012 |
[9] | Xueqi Sun, Yongqiang Fu, Sihua Liang . Normalized solutions for pseudo-relativistic Schrödinger equations. Communications in Analysis and Mechanics, 2024, 16(1): 217-236. doi: 10.3934/cam.2024010 |
[10] | Zhi-Jie Wang, Hong-Rui Sun . Normalized solutions for Kirchhoff equations with Choquard nonlinearity: mass Super-Critical Case. Communications in Analysis and Mechanics, 2025, 17(2): 317-340. doi: 10.3934/cam.2025013 |
In this paper, we consider the following Schrödinger-Poisson system
{−Δu+V(x)u+ϕu=|u|p−2u+λK(x)|u|q−2u inR3,−Δϕ=u2 inR3.
Under the weakly coercive assumption on V and an appropriate condition on K, we investigate the cases when the nonlinearities are of concave-convex type, that is, 1<q<2 and 4<p<6. By constructing a nonempty closed subset of the sign-changing Nehari manifold, we establish the existence of least energy sign-changing solutions provided that λ∈(−∞,λ∗), where λ∗>0 is a constant.
In the past decades, the following Schrödinger-Poisson system
{−Δu+V(x)u+ϕu=f(x,u)inR3,−Δϕ=u2inR3 | (1.1) |
has been studied extensively by many authors, where V:R3→R and f∈C(R3×R,R). This system can be used to describe the interaction of a charged particle with the electrostatic field in quantum mechanics. In this context, the unknown u and ϕ represent the wave functions related to the particle and electric potentials, respectively. Moreover, the local nonlinearity f(x,u) models the interaction among particles. We refer the reader to [6,20] for more details on its physical background.
It is worth noting that system (1.1) is a nonlocal problem due to the appearance of the term ϕu, where ϕ=ϕu is presented in (1.4) below. This fact states that problem (1.1) is no longer a pointwise identity and brings some essential difficulties. For example, the term ∫R3ϕuu2dx in the corresponding energy functional is homogeneous of degree four, then, compared with the local Schrödinger equation, it seems difficult to obtain the boundedness and compactness for any Palais-Smale sequence. In light of the previous observations, the existence of solutions for problem (1.1) have been widely studied and some open problems have been proposed [3,11,15,16,19,25,28,30,35].
In what follows, we are particularly interested in the existence of sign-changing solutions (also known as nodal solutions) for problem (1.1). From this perspective, Wang and Zhou [29] were concerned with the existence and energy property of sign-changing solutions for problem (1.1) with f(x,u)=|u|p−2u. By introducing appropriate compactness conditions on V, they used methods different from [5] to prove that the so-called sign-changing Nehari manifold is nonempty provided that 4<p<6. Then, combining some analytical techniques and the Brouwer degree theory, the existence of least energy sign-changing solutions was established. After that, the authors in [21] investigated sign-changing solutions of problem (1.1) when f∈C1(R,R) satisfied super-cubic and subcritical growth at infinity, superlinear growth at origin, and a well-known Nehari-type monotonicity condition. In particular, they established the energy doubling [31]. Moreover, the authors in [10,38] obtained the similar existence results if the nonlinearity f satisfied asymptotically cubic and three-linear growth, respectively. On the other hand, when f satisfies three-sublinear growth, the existence and multiplicity of sign-changing solutions can be obtained by invariant sets of descending flow [13,18]. For more interesting results, such as the Sobolev critical exponent or bounded domains, we refer to [1,24,27,34,36,37] and the references therein.
According to the previous statements, we observe that the nonlinearities always satisfy superlinear growth or convexity (i.e. f(x,u)=|u|p−2u, 2<p<6) provided that the sign-changing solution of Schrödinger-Poisson systems in the whole space R3 is considered. Once the nonlinearity is not constrained by the above forms, the methods mentioned previously cannot be directly used. Therefore, in present paper, we focus on a special type of nonlinearities; that is, the concave-convex type, such as f(x,u)=|u|p−2u+|u|q−2u with 4<p<6 and 1<q<2. The concave-convex nonlinearities were introduced in [2], where the authors proved the existence of infinitely many solutions with negative energy for local elliptic problems in bounded domains. After this work, a great attention has been paid to the existence of solutions to elliptic problems with concave-convex nonlinearities. For example, see [7,8,17,32] for local Schrodinger equations, and [9,14,22,23,26,33] for Schrodinger-Poisson systems.
Note that only [7,8,17,33] involve the sign-changing solutions. More precisely, Bobkov [7] considered the following Schrödinger equation
{−Δu=λ|u|q−2u+|u|γ−2uin Ω,u=0in∂Ω, |
where Ω⊂RN is a bounded connected domain with a smooth boundary, N≥1, 1<q<2<γ<2∗ and 2∗ is the well-known Sobolev critical exponent. They proved the existence of a sign-changing solution on the nonlocal interval λ∈(−∞,λ∗0), where λ∗0 is determined by the variational principle of nonlinear spectral analysis through the fibering method. Moreover, the author in [8] obtained similar existence results and some interesting properties for the nodal solutions of the elliptic equation
{−Δu=λk(x)|u|q−2u+h(x)|u|γ−2uinΩ,u=0in∂Ω, |
where 1<q<2<γ<2∗, λ∈R and the weight functions k, h∈L∞(Ω) satisfy the conditions essinfx∈Ωk(x)>0 and essinfx∈Ωh(x)>0. Note that the methods in [7,8] cannot be applied to the nonlocal elliptic problem (1.1). To this end, based on the setting of bounded domains, Yang and Ou [33] studied the following Schrödinger-Poisson system
{−Δu+ϕu=λ|u|p−2u+|u|q−2uinΩ,−Δϕ=u2inΩ,u=0in∂Ω, | (1.2) |
where Ω is a bounded domain with smooth boundary ∂Ω in R3 and 1<p<2, 4<q<6, λ is a constant. By constrained variational method and quantitative deformation lemma, they obtained that the problem (1.2) has a nodal solution uλ with positive energy when λ<λ∗, λ∗ is a constant. Here, we point out that if the bounded domain is involved, the embedding H10(Ω)↪Lp(Ω) is compact for 1≤p<2∗, which not only avoids the verification of compactness but also ensures the boundedness of the concave term. However, once the whole space is considered, these points cannot be directly determined. Therefore, motivated by the works described above, in this paper we focus on the following Schrödinger-Poisson system in the whole space R3 with concave-convex nonlinearities
{−Δu+V(x)u+ϕu=|u|p−2u+λK(x)|u|q−2uinR3,−Δϕ=u2inR3, | (1.3) |
where 1<q<2, 4<p<6, λ>0 and V, K satisfy the assumptions:
(V) V∈C(R3,R) satisfies infx∈R3V(x)≥a>0 for each A>0, meas{x∈R3:V(x)≤A}<∞, where a is a constant and meas denotes the Lebesgue measure in R3;
(K) K is positive and K∈L66−q(R3).
Here the condition (V) is similar to [17]. This condition also ensures the compactness of embedding H↪Lp(R3), 2≤p<2∗, where H is the Hilbert space
H={u∈H1(R3):∫R3V(x)u2dx<+∞} |
endowed with the norm
‖u‖=(∫R3(|∇u|2+V(x)u2)dx)12[4,39, Lemma 3.4]. |
Meanwhile, we point out that the authors in [17] considered the local Schrödinger type equation in RN
−Δu+V(x)u=λ|u|q−2u+μu+ν|u|p−2u, |
where 1<q<2<p<2∗, N≥2 and λ,μ,ν are parameters. The above equation involves a combination of concave and convex terms. They obtained infinitely many nodal solutions by using the method of invariant sets. However, it seems that this method cannot be applied to problem (1.3). In order to overcome the previous difficulties, we introduce the condition (K), which guarantees a weak continuity result (see Lemma 2.2 below). Moreover, conditions (V) and (K) allow us to construct a suitable nonempty closed subset of sign-changing Nehari manifold similar to [33], and then a least energy sign-changing solution can be obtained.
Before proceeding, we discuss the basic framework for dealing with our problem. The usual norm in the Lebesgue space Lr(R3) is denoted by |u|r=(∫R3|u|rdx)1r, r∈[1,+∞). It is well known that, by the Lax-Milgram theorem, when u∈H, there exists a unique ϕu∈D1,2(R3) such that −Δϕu=u2, where
ϕu(x)=14π∫R3u2(y)|x−y|dy. | (1.4) |
Substituting (1.4) into (1.3), we can rewrite system (1.3) as the following equivalent form
−Δu+V(x)u+ϕuu=|u|p−2u+λK(x)|u|q−2u inR3. | (1.5) |
Therefore, the energy functional associated with system (1.3) is defined by
Iλ(u)=12∫R3(|∇u|2+V(x)u2)dx+14∫R3ϕuu2dx−1p∫R3|u|pdx−λq∫R3K(x)|u|qdx,∀u∈H. |
The functional Iλ(u) is well-defined for every u∈H and belongs to C1(H,R). Furthermore, for any v∈H,
⟨I′λ(u),v⟩=∫R3(∇u⋅∇v+V(x)uv)dx+∫R3ϕuuvdx−∫R3|u|p−2uvdx−λ∫R3K(x)|u|q−2uvdx. |
As is well known, the solution of problem (1.5) is the critical point of the functional Iλ(u). Moreover, if u∈H is a solution of problem (1.5) and u±≠0, then u is a sign-changing solution of system (1.3), where
u+(x)=max{u(x),0}andu−(x)=min{u(x),0}. |
Naturally, we introduce the Nehari manifold of Iλ as
Nλ={u∈H∖{0}:⟨I′λ(u),u⟩=0}, |
which is related to the behavior of the map φu:r→Iλ(ru) (r>0) (see [12] for the introduction of this map). For u∈H, we have
φu(r)=12r2||u||2+14r4∫R3ϕuu2dx−1prp|u|pp−λqrq∫R3K(x)|u|qdx. |
It is well known that, for any u∈H∖{0}, φ′u(r)=0 if and only if ru∈Nλ, which also implies that φ′u(1)=0 if and only if u∈Nλ. This manifold is always used to find the positive ground state solution. In order to obtain sign-changing solutions of problem (1.3), it is necessary to consider the sign-changing Nehari manifold
Mλ={u∈H:u±≠0,⟨I′λ(u),u±⟩=0}. |
Hoeever, this manifold cannot be directly applied due to appearance of concave term λk(x)|u|q−2u. As we will see, inspired by [33], we can construct the set M∗λ⊂Mλ and prove this set is a nonempty closed set, where M∗λ is defined by (2.5) below. We then show that the minimization problem mλ:=infu∈M∗λIλ(u) is attained by some uλ∈M∗λ with positive energy. Finally, the classical deformation lemma [32, Lemma 2.3] states that the uλ is a weak solution of problem (1.3). Up to now, the main results can be stated as follows.
Theorem 1.1. Assume that (V) and (K) hold. Then there exists a constant λ∗>0 (determined in (2.13)) such that for any λ∈(−∞,λ∗), problem (1.3) possesses a least energy sign-changing solution uλ with positive energy.
Remark 1.2. As mentioned previously, our Theorem 1.1 extends the result of [33] to the whole space R3. Moreover, for nonlinearities that do not involve concave terms, such as f(x,u)=|u|p−2u (4<p<6), with the aid of classic techniques in [29], one can obtain that the sign-changing Nehari manifold is nonempty. However, in our paper, 1<q<2 and 4<p<6 mean that λK(x)|u|q−2u is concave and |u|p−2u is convex, which is different from [29]. At this point, it is difficult to directly prove that the set Mλ is nonempty. To this end, we carefully analyze the behavior of fu(s,t)=Iλ(su++tu−) and then introduce the set M∗λ. In particular, by determining some important lower bound estimates, we verify that M∗λ≠∅. On the other hand, we point out that the range of parameter λ can be negative, which is similar to previous results.
The remainder of this paper is organized as follows. In section 2, we present some preliminary lemmas that are crucial for proving our main results. Section 3 is devoted to proving Theorem 1.1.
In this section, we present some preliminary lemmas that are crucial for proving our main results. First, we recall some well-known properties of ϕu that are a collection of results in [11,19].
Lemma 2.1. For any u∈H, we get
(i) there exists C>0 such that ∫R3ϕuu2dx≤C‖u‖4;
(ii) ϕu≥0, for any u∈H;
(iii) ϕtu=t2ϕu, for any t>0 and u∈H;
(iv) if un⇀u in H, then ϕun⇀ϕu in D1,2(R3) and
limn→∞∫R3ϕunu2ndx=∫R3ϕuu2dx. |
Next, we verify a weak continuity of the concave term. The proof is similar to [32, Lemma 2.13], but we state the proof here for the readers convenience.
Lemma 2.2. Assume that 1<q<2 and (K) hold, then the functional
G:H→R:u↦∫R3K(x)|u|qdx |
is weakly continuous.
Proof. Undoubtedly, it is sufficient to prove that if un⇀u in H, then ∫R3K(x)|un|qdx→∫R3K(x)|u|qdx as n→∞. In fact, if un⇀u in H, going if necessary to a subsequence, we can assume that un→u a.e. on R3. Since un⇀u in H, we get that {un} is bounded in L6(R3) and {uqn} is bounded in L6q(R3). Therefore, uqn⇀uq in L6q(R3). Combining with (K) and the definition of weak convergence, we obtain
∫R3K(x)|un|qdx→∫R3K(x)|u|qdxas n→∞. |
Now, for any u∈H with u±≠0, we introduce the map fu:[0,+∞)×[0,+∞)→R defined by fu(s,t)=Iλ(su++tu−), i.e.,
fu(s,t)=12s2||u+||2+14s4∫R3ϕu+(u+)2dx+12s2t2∫R3ϕu+(u−)2dx−1psp|u+|pp−λqsq∫R3K(x)|u+|qdx+12t2||u−||2+14t4∫R3ϕu−(u−)2dx−1ptp|u−|pp−λqtq∫R3K(x)|u−|qdx. | (2.1) |
Here we have used the fact that
∫R3ϕu+|u−|2dx=∫R3ϕu−|u+|2dx. |
Moreover, we have that
∇fu(s,t)=(⟨I′λ(su++tu−),u+⟩,⟨I′λ(su++tu−),u−⟩)=(1s⟨I′λ(su++tu−),su+⟩,1t⟨I′λ(su++tu−),tu−⟩), |
which implies that for any u∈H with u±≠0, su++tu−∈Mλ if and only if the pair (s,t) is a critical point of fu.
Lemma 2.3. Assume that 1<q<2, 4<p<6 and the assumption (K) hold, then there exists a constant λ1>0 such that for any u∈H with u±≠0, there holds that
(i) if λ∈(0,λ1), then for any fixed t≥0, fu(s,t) has exactly two critical points, 0<s1(t)<s2(t); s1(t) is the minimum point and s2(t) is the maximum point; moreover, if λ≤0, then for any fixed t≥0, fu(s,t) has exactly one critical point, s3(t)>0, and it is the maximum point;
(ii) if λ∈(0,λ1), then for any fixed s≥0, fu(s,t) has exactly two critical points, 0<t1(s)<t2(s); t1(s) is the minimum point and t2(s) is the maximum point; moreover, if λ≤0, then for any fixed s≥0, fu(s,t) has exactly one critical point, t3(s)>0, and it is the maximum point.
Proof. We define fμ(s,t):[0,+∞)×[0,+∞)→R by
fμ(s,t)=12||su++tu−||2+14μ∫R3ϕsu++tu−(su++tu−)2dx−1p|su++tu−|pp−λq∫R3K(x)|su++tu−|qdx, |
where μ is a nonnegative parameter.
(i) For any fixed t≥0, a direct calculation gives
∂fμ∂s(s,t)=s||u+||2+μs3∫R3ϕu+(u+)2dx+μst2∫R3ϕu+(u−)2dx−sp−1|u+|pp−λsq−1∫R3K(x)|u+|qdx=sq−1(s2−q||u+||2+μs4−q∫R3ϕu+(u+)2dx+μs2−qt2∫R3ϕu+(u−)2dx−sp−q|u+|pp−λ∫R3K(x)|u+|qdx). |
Then, if s>0, ∂fμ∂s(s,t)=0 is equivalent to
βμ(s)=s2−q||u+||2+μs4−q∫R3ϕu+(u+)2dx+μs2−qt2∫R3ϕu+(u−)2dx−sp−q|u+|pp−λ∫R3K(x)|u+|qdx=0. |
For βμ(s), we can obtain that
β′μ(s)=s1−q((2−q)||u+||2+μ(4−q)s2∫R3ϕu+(u+)2dx+μ(2−q)t2∫R3ϕu+(u−)2dx−(p−q)sp−2|u+|pp). |
Clearly, for any fixed t≥0, 1<q<2 and 4<p<6, we can infer that βμ has exactly one critical point sμ>0, where sμ is related to t for any μ≥0. Moreover, βμ is strictly increasing in (0,sμ) and strictly decreasing in (sμ,+∞).
Noting that if μ=0, we have
β0(s)=s2−q||u+||2−sp−q|u+|pp−λ∫R3K(x)|u+|qdx |
and
β′0(s)=s1−q((2−q)||u+||2−(p−q)sp−2|u+|pp). |
Hence,
s0=((2−q)||u+||2(p−q)|u+|pp)1p−2 |
and
β0(s0)=p−2p−q(2−qp−q)2−qp−2||u+||2(p−q)p−2|u+|p(2−q)p−2p−λ∫R3K(x)|u+|qdx. |
Let
αμ(s)=s2−q||u+||2+μs4−q∫R3ϕu+(u+)2dx+μs2−qt2∫R3ϕu+(u−)2dx−sp−q|u+|pp |
and
λ+1=infu∈H,u±≠0α0(s0)∫R3K(x)|u+|qdx, |
then it follows from Sobolev embedding that
λ+1≥p−2p−q(2−qp−q)2−qp−21Sq6Sp(2−q)p−2p|K|66−q>0. | (2.2) |
Therefore, if λ∈(0,λ+1), we can deduce βμ(s0)>β0(s0)>0 for any u∈H with u±≠0. For any fixed t≥0, if λ∈(0,λ+1), there exist unique s1(t) and s2(t) with 0<s1(t)<s2(t), such that βμ(s)=0 and β′μ(s1(t))>0, β′μ(s2(t))<0. On the other hand, when λ≤0, there is a unique s3(t)>0 such that βμ(s)=0 and β′μ(s3(t))<0.
Finally, considering that
∂2fμ∂s2(s,t)=||u+||2+3μs2∫R3ϕu+(u+)2dx+μt2∫R3ϕu+(u−)2dx−(p−1)sp−2|u+|pp−λ(q−1)sq−2∫R3K(x)|u+|qdx=sq−2(s2−q||u+||2+3μs4−q∫R3ϕu+(u+)2dx+μs2−qt2∫R3ϕu+(u−)2dx−(p−1)sp−q|u+|pp−λ(q−1)∫R3K(x)|u+|qdx), |
we can find that
∂2fμ∂s2(s,t)=(q−1)sq−2βμ(s)+sq−1β′μ(s). |
Note that ∂fμ∂s(s,t)=0 is equivalent to βμ(s)=0, then we have
∂2fμ∂s2(s,t)=sq−1β′μ(s) if ∂fμ∂s(s,t)=0. |
Thus, the facts that β′μ(s1(t))>0,β′μ(s2(t))<0 and β′μ(s3(t))<0 signify that
∂2fμ∂s2(s1(t),t)>0,∂2fμ∂s2(s2(t),t)<0and∂2fμ∂s2(s3(t),t)<0. |
In particular, when μ=1, the results still hold.
(ii) For any fixed s≥0, let
λ−1=infu∈H,u±≠0α0(s0)∫R3K(x)|u−|qdx=infu∈H,u±≠0p−2p−q(2−qp−q)2−qp−2||u−||2(p−q)p−2|u−|p(2−q)p−2p∫R3K(x)|u−|qdx. | (2.3) |
Analogously with the proof (i), we can derive the conclusion.
At last, it is easy to see that λ+1=λ−1. Indeed, u±≠0 indicates that (−u)±≠0, which yields that λ+1=λ−1. Let λ1=λ+1=λ−1, from (i) and (ii), then the proof is completed.
Let
λ2=infu∈H∖{0}p−2p−q(2−qp−q)2−qp−2||u||2(p−q)p−2|u|p(2−q)p−2p∫R3K(x)|u|qdx. |
Undoubtedly,
λ1≥λ2≥p−2p−q(2−qp−q)2−qp−21Sq6Sp(2−q)p−2p|K|66−q>0. | (2.4) |
Here, the notation Sp represents the embedding constant of H↪Lp(R3), which has a value depending on p∈[2,6]. According to Lemma 2.3, the following corollary is a direct result.
Corollary 2.4. Assume that 1<q<2, 4<p<6, the assumption (K) and 0<λ<λ2 hold, then for any u∈H∖{0}, φu(r) has exactly two critical points, 0<r1(u)<r2(u) and φ′′u(r1(u))>0, φ′′u(r2(u))<0. On the other hand, when λ≤0, φu(r) has exactly one critical point, r3(u)>0, and φ′′u(r3(u))<0.
Lemma 2.5. Assume that 1<q<2, 4<p<6, the assumption (K) and λ<λ1 hold, then for any u∈Mλ, (∂2fu/∂s2)(1,1)≠0 and (∂2fu/∂t2)(1,1)≠0. Moreover, Iλ(u)→+∞ as ||u||→+∞, i.e., the functional Iλ is coercive and bounded from below on Nλ.
Proof. If 0<λ<λ1, from Lemma 2.3, it follows that fu(s,1) has exactly two critical points s1(1), s2(1) and ∂2fu∂s2(s1(1),1)>0, ∂2fu∂s2(s2(1),1)<0. Since u∈Mλ, we have ∂fu∂s(1,1)=0, which means that s1(1)=1 or s2(1)=1. Hence, ∂2fu∂s2(1,1)≠0. Analogously, we can conclude that ∂2fu∂t2(1,1)≠0.
If λ≤0, it follows from Lemma 2.3 that fu(s,1) has exactly one critical point s3(1) and ∂2fu∂s2(s3(1),1)<0. Combining with u∈Mλ, we have ∂fu∂s(1,1)=0, which shows that s3(1)=1. Therefore, we get ∂2fu∂s2(1,1)≠0. Similarly, we can deduce that t3(1)=1, ∂2fu∂t2(1,t3(1))<0 and the claim is clearly true.
Note that u∈Mλ⊂Nλ, then the Sobolev embedding indicates that
Iλ(u)=Iλ(u)−14⟨I′λ(u),u⟩=14||u||2+(14−1p)∫R3|u|pdx+λ(14−1q)∫R3K(x)|u|qdx≥14||u||2+λ(14−1q)|K|66−q|u|q6≥14||u||2+λ1(14−1q)|K|66−qSq6||u||q. |
Hence, combining 1<q<2 and 4<p<6, we derive Iλ(u)→+∞ as ||u||→+∞. That is, the functional Iλ(u) is coercive and bounded from below on Nλ. The proof is completed.
Similarly, we obtain the following result.
Corollary 2.6. Assume that 1<q<2, 4<p<6, the assumption (K) and λ<λ2 hold, then for any u∈Nλ, φ′′u(1)≠0.
In what follows, we construct the following sets
M−λ={u∈Mλ:∂2fu∂s2(1,1)<0,∂2fu∂t2(1,1)<0} |
and
M∗λ={u∈Mλ:∂2fu∂s2(1,1)<0,∂2fu∂t2(1,1)<0,φ′′u(1)<0}. | (2.5) |
Unquestionably, M∗λ⊂M−λ. According to the properties of fu mentioned above, we can verify that the set M∗λ is nonempty and M∗λ=M−λ (see Lemma 2.9). To this end, we first get the following fact.
Lemma 2.7. Assume that 1<q<2, 4<p<6 and the assumption (K) hold, there exists σ>0, which is independent of u and λ, such that
||u±||>σ>0 |
for any u∈M−λ.
Proof. For any u∈M−λ, from ∂2fu∂s2(1,1)<0, ∂2fu∂t2(1,1)<0 and Sobolev embedding, it follows that
(2−q)||u±||2<(2−q)||u±||2+(4−q)∫R3ϕu±(u±)2dx+(2−q)∫R3ϕu+(u−)2dx<(p−q)|u±|pp≤(p−q)Spp||u±||p, |
which implies
||u±||>(2−q(p−q)Spp)1p−2:=σ>0. | (2.6) |
Hence, the proof is finished.
In order to prove that M∗λ≠∅, let us define
λ3=infu∈M−λ{(p−2)||u+||2+(p−4)∫R3ϕu+(u+)2dx(p−q)∫R3K(x)|u+|qdx,(p−2)||u−||2+(p−4)∫R3ϕu−(u−)2dx(p−q)∫R3K(x)|u−|qdx}. |
From Sobolev embedding and Lemma 2.7, it follows that
(p−2)||u±||2+(p−4)∫R3ϕu±(u±)2dx(p−q)∫R3K(x)|u±|qdx≥(p−2)||u±||2−q(p−q)Sq6|K|66−q>(p−2)σ2−q(p−q)Sq6|K|66−q>0, |
where σ is given by (2.6). Therefore, we have
λ3≥(p−2)σ2−q(p−q)Sq6|K|66−q>0. | (2.7) |
Furthermore, we compute that
∂2fu∂s2(s,t)=||u+||2+3s2∫R3ϕu+(u+)2dx+t2∫R3ϕu+(u−)2dx−(p−1)sp−2|u+|pp−λ(q−1)sq−2∫R3K(x)|u+|qdx, |
∂2fu∂t2(s,t)=||u−||2+3t2∫R3ϕu−(u−)2dx+s2∫R3ϕu+(u−)2dx−(p−1)tp−2|u−|pp−λ(q−1)tq−2∫R3K(x)|u−|qdx |
and
∂2fu∂s∂t(s,t)=2st∫R3ϕu+(u−)2dx,∂2fu∂t∂s(s,t)=2st∫R3ϕu+(u−)2dx. |
For any u∈Mλ, we obtain
∂2fu∂s2(1,1)=||u+||2+3∫R3ϕu+(u+)2dx+∫R3ϕu+(u−)2dx−(p−1)|u+|pp−λ(q−1)∫R3K(x)|u+|qdx=(2−q)||u+||2+(4−q)∫R3ϕu+(u+)2dx+(2−q)∫R3ϕu+(u−)2dx−(p−q)|u+|pp=(2−p)||u+||2+(4−p)∫R3ϕu+(u+)2dx+(2−p)∫R3ϕu+(u−)2dx−λ(q−p)∫R3K(x)|u+|qdx |
and
∂2fu∂t2(1,1)=||u−||2+3∫R3ϕu−(u−)2dx+∫R3ϕu+(u−)2dx−(p−1)|u−|pp−λ(q−1)∫R3K(x)|u−|qdx=(2−q)||u−||2+(4−q)∫R3ϕu−(u−)2dx+(2−q)∫R3ϕu+(u−)2dx−(p−q)|u−|pp=(2−p)||u−||2+(4−p)∫R3ϕu−(u−)2dx+(2−p)∫R3ϕu+(u−)2dx−λ(q−p)∫R3K(x)|u−|qdx. |
Lemma 2.8. Assume that 1<q<2, 4<p<6, the assumption (K) and λ<λ1 hold, then for any u∈H with u±≠0, there exists a unique pair (su,tu)∈R+×R+ such that suu++tuu−∈M−λ. Moreover, if λ<λ3, then Iλ(suu++tuu−)=maxs,t>0Iλ(su++tu−).
Proof. First, we only prove the case of 0<λ<λ1 since the proof of λ≤0 is very similar. Let u∈H with u±≠0, from the proof of Lemma 2.3, then we have that ∂fu∂s(s,t) satisfies the conditions:
(i) ∂fu∂t(s,t2(s))=0 for all s≥0;
(ii) ∂fu∂t(s,t) is continuous and has continuous partial derivatives in [0,+∞)×[0,+∞);
(iii) ∂2fu∂t2(s,t2(s))<0 for all s≥0.
Hence, we can obtain that if 0<λ<λ1, ∂fu∂t(s,t)=0 determines an implicit function t2(s) with continuous derivative on [0,+∞) by using the implicit function theorem. Analogously, if 0<λ<λ1, ∂fu∂s(s,t)=0 determines an implicit function s2(t) with continuous derivative on [0,+∞).
On the other hand, for every s≥0, from ∂fu∂t(s,t2(s))=0 and ∂fu∂t(s,t)≤0 for sufficiently large t>0, we can show that
t2(s)<s for large enough s. | (2.8) |
Otherwise, if t2(s)≥s, where s is large enough, it follows from the definition of ∂fu∂t(s,t) that ∂fu∂t(s,t2(s))<0, which contradicts ∂fu∂t(s,t2(s))=0. Similarly, we get
s2(t)<t for sufficiently large t. | (2.9) |
Therefore, by (2.8), (2.9), t2(0)>0,s2(0)>0, the continuity of t2(s) and s2(t), we conclude that the curves of t2(s) and s2(t) must intersect at some point (su,tu)∈R+×R+. That is, ∂fu∂t(su,tu)=∂fu∂s(su,tu)=0. Additionally, noting that
t′2(s)=−(∂2fu∂t∂s/∂2fu∂t2)(s,t2(s))>0 |
for any s>0, we obtain that the function t2(s) is strictly increasing in (0,+∞). Similarly, the function s2(t) is strictly increasing in (0,+∞). Consequently, there is a unique pair (su,tu)∈R+×R+ such that
∂fu∂s(su,tu)=∂fu∂t(su,tu)=0 |
and
∂2fu∂s2(su,tu)<0,∂2fu∂t2(su,tu)<0; |
that is, suu++tuu−∈M−λ.
Next, we prove that (su,tu) is the unique maximum point of fu(s,t) on [0,+∞)×[0,+∞). In fact, if u∈M−λ, we only show that (su,tu)=(1,1) is the pair of numbers such that Iλ(suu++tuu−)=maxs,t>0Iλ(su++tu−). Define
H(u)=(∂2fu∂s2∂2fu∂t2−∂2fu∂t∂s∂2fu∂s∂t)|(1,1). |
If we verify that H(u)>0, then (1,1) is a local maximum point of fu(s,t). Combining uniqueness of (su,tu), we have (1,1) as a global maximum point of fu(s,t). Let u∈M−λ, then
H(u)=(∂2fu∂s2∂2fu∂t2−∂2fu∂t∂s∂2fu∂s∂t)|(1,1)=((2−p)||u+||2+(4−p)∫R3ϕu+(u+)2dx+(2−p)∫R3ϕu+(u−)2dx−λ(q−p)∫R3K(x)|u+|qdx)×((2−p)||u−||2+(4−p)∫R3ϕu−(u−)2dx+(2−p)∫R3ϕu+(u−)2dx−λ(q−p)∫R3K(x)|u−|qdx)−4(∫R3ϕu+(u−)2dx)2. |
From ∂2fu∂s2(1,1)<0, ∂2fu∂t2(1,1)<0, if λ<λ3, we derive
−∂2fu∂s2(1,1)−2∫R3ϕu+(u−)2dx=(p−2)||u+||2+(p−4)∫R3ϕu+(u+)2dx+(p−4)∫R3ϕu+(u−)2dx−λ(p−q)∫R3K(x)|u+|qdx>(p−2)||u+||2+(p−4)∫R3ϕu+(u+)2dx−λ3(p−q)∫R3K(x)|u+|qdx>0 |
and
−∂2fu∂t2(1,1)−2∫R3ϕu+(u−)2dx>(p−2)||u−||2+(p−4)∫R3ϕu−(u−)2dx−λ3(p−q)∫R3K(x)|u−|qdx>0, |
which show that H(u)>0.
If u∉M−λ, then there exists a unique pair (s′u,t′u) of positive numbers such that s′uu++t′uu−∈M−λ. Let v=s′uu++t′uu−, i.e., v∈M−λ. Repeat the above steps and we will get H(v)>0. Hence, the proof is completed.
Lemma 2.9. If 1<q<2, 4<p<6, the assumption (K) and λ<min{λ1,λ2,λ3} hold, then M∗λ≠∅. Moreover, we get M∗λ=M−λ.
Proof. By the definitions of M−λ and M∗λ, M∗λ⊂M−λ is obvious. Hence, we only need to prove that if λ<min{λ1,λ2,λ3}, then M−λ⊂M∗λ. That is, for any u∈M−λ, φu reaches its maximum at point r=1. It follows from λ<λ1 and Lemma 2.8 that M−λ≠∅, and from Lemma 2.8, for any u∈M−λ, we obtain H(u)>0 when λ<λ3. Combining fu(r,r)=φu(r), it follows that r=1 is a maximum of φu. Therefore, M−λ⊂M∗λ. This completes the proof of Lemma 2.9.
Corollary 2.10. If 1<q<2, 4<p<6, the assumption (K) and λ<min{λ1,λ2,λ3} hold, for u∈H and u±≠0, then there exists a unique pair (su,tu)∈R+×R+ such that suu++tuu−∈M∗λ and Iλ(suu++tuu−)=maxs,t>0Iλ(su++tu−).
Lemma 2.11. If 1<q<2, 4<p<6, the assumption (K) and λ<λ4 hold, for all u∈H∖{0}, then there exists ru>0 such that φu(ru)>0, where λ4>0.
Proof. Fixed u∈H∖{0}, let
Eu(r)=12r2||u||2−1prp|u|pp |
for any r≥0, then we have
φu(r)=12r2||u||2+14r4∫R3ϕuu2dx−1prp|u|pp−λqrq∫R3K(x)|u|qdx≥12r2||u||2−1prp|u|pp−λqrq∫R3K(x)|u|qdx=Eu(r)−λqrq∫R3K(x)|u|qdx. | (2.10) |
Considering Eu(r), we obtain that there is a unique r1(u)=(||u||2|u|pp)1p−2>0 such that Eu(r) achieves its maximum at r1(u) and the maximum value is Eu(r1(u))=p−22p(||u|||u|p)2pp−2. Moreover, from Sobolev embedding and (2.10), it is clear to calculate that
φu(r1(u))≥Eu(r1(u))−λq(r1(u))q∫R3K(x)|u|qdx≥Eu(r1(u))−λq(r1(u))q|K|66−qSq6||u||q=Eu(r1(u))−λqSq6|K|66−q(2pp−2)q2(Eu(r1(u)))q2=(Eu(r1(u)))q2((Eu(r1(u)))2−q2−λqSq6|K|66−q(2pp−2)q2). | (2.11) |
Consequently, by taking
λ4=(p−2)q2p|K|66−qSq6infu∈H∖{0}(||u|||u|p)p(2−q)p−2≥(p−2)q2p|K|66−qSq6Sp(2−q)p−2p>0, | (2.12) |
we conclude that if λ<λ4, it holds
λqSq6|K|66−q(2pp−2)q2<λ4qSq6|K|66−q(2pp−2)q2≤1qSq6|K|66−q(2pp−2)q2(p−2)q2pSq6|K|66−q(||u|||u|p)p(2−q)p−2=(Eu(r1(u)))2−q2 |
for any u∈H∖{0}. This together with (2.11) yields that φu(r1(u))>0 for any λ<λ4.
Let
λ∗=min{λ1,λ2,λ3,λ4}, | (2.13) |
then it follows from (2.2), (2.4), (2.7) and (2.12) that λ∗>0. Now, we consider the properties of the set M∗λ.
Lemma 2.12. If 1<q<2, 4<p<6, λ<λ∗ and the assumptions (V) and (K) hold, then M∗λ is a closed set.
Proof. Letting {un}⊂M∗λ satisfy un→u0 as n→∞ in H, we now prove that u0∈M∗λ. From {un}⊂M∗λ, we obtain
⟨I′λ(u0),u±0⟩=limn→∞⟨I′λ(un),u±n⟩=0, | (2.14) |
∂2fu0∂s2(1,1)=limn→∞∂2fun∂s2(1,1)≤0, | (2.15) |
∂2fu0∂t2(1,1)=limn→∞∂2fun∂t2(1,1)≤0, | (2.16) |
φ′′u0(1)=limn→∞φ′′un(1)≤0. | (2.17) |
From Lemma 2.7, it follows that ||u±n||>σ>0 for any un∈M−λ and hence ||u±0||=limn→∞||u±n||>σ>0, which indicates u±0≠0. Using this and (2.14), we obtain u0∈Mλ and r=1 is a critical point of φu0. Consequently, by (2.15)-(2.17), Lemma 2.5 and Corollary 2.6, we derive that
∂2fu0∂s2(1,1)<0,∂2fu0∂t2(1,1)<0,φ′′u0(1)<0. |
Hence, u0∈M∗λ and M∗λ is a closed set.
Lemma 2.13. If 1<q<2, 4<p<6, λ<λ∗ and assumptions (V) and (K) hold, then the infimum mλ:=infu∈M∗λIλ(u) can be achieved by some uλ∈M∗λ and mλ>0.
Proof. According to Lemma 2.5, mλ>−∞ when λ<λ∗. Let {un}⊂M∗λ be a minimizing sequence for the functional Iλ, namely Iλ(un)→mλ as n→∞. Since the functional Iλ is coercive on M∗λ, then {un} is bounded in H. Going if necessary to a subsequence, we may assume that
un⇀uλinH,un→uλinLp(R3). |
Now, we first claim that u±λ≠0. In fact, from Lemma 2.2, Lemma 2.7 and the convergence of {un} in Lp(R3), for any λ<λ∗, we conclude that
|u±λ|pp+λ∫R3K(x)|u±λ|qdx=limn→∞(|u±n|pp+λ∫R3K(x)|u±n|qdx)=limn→∞(||u±n||2+∫R3ϕu±n(u±n)2dx+∫R3ϕu+n(u−n)2dx)≥limn→∞||u±n||2>σ2>0. |
This means that u±λ≠0 for any λ<λ∗.
Next, we proof that un→uλ in H. Arguing by contradiction, suppose that
||u+λ||<limn→∞inf||u+n||or||u−λ||<limn→∞inf||u−n||. |
From Corollary 2.10, there exists a unique pair (suλ,tuλ) such that ˜uλ=suλu+λ+tuλu−λ∈M∗λ and Iλ(u+n+u−n)=maxs,t>0Iλ(suλu+n+tuλu−n). Consequently,
mλ≤Iλ(˜uλ)<limn→∞infIλ(suλu+n+tuλu−n)≤limn→∞infIλ(u+n+u−n)=mλ. |
That is, we get a contradiction. Therefore, un→uλ in H and mλ is achieved by uλ. Combining the fact that M∗λ is closed, so uλ∈M∗λ.
Finally, it follows from uλ∈M∗λ that φuλ(r) reached its global maximum at r=1. By this and Lemma 2.11, we can deduce that φuλ(1)>0, i.e., mλ>0. This finishes the proof of Lemma 2.13.
The main aim of this section is to prove our results. Thanks to Lemma 2.13, it suffices to check that the minimizer uλ for mλ is a sign-changing of problem (1.3).
Proof of Theorem 1.1. Since uλ∈M∗λ, according to Corollary 2.10, we obtain that
Iλ(su+λ+tu−λ)<Iλ(u+λ+u−λ)=mλ,for(s,t)∈(R+×R+)∖(1,1). |
Moreover, we get Iλ(uλ)>0, φ′′uλ(1,1)<0, (∂2fuλ/∂s2)(1,1)<0 and (∂2fuλ/∂t2)(1,1)<0.
Let D=(1−δ,1+δ)×(1−δ,1+δ) and h:D→H by h(s,t)=su+λ+tu−λ for any (s,t)∈D. Then there is a constant 0<δ<1 such that
0<m:=max∂DIλ(h(s,t))<mλ,max(s,t)∈D∂2fh(s,t)∂s2(1,1)<0, | (3.1) |
max(s,t)∈D∂2fh(s,t)∂t2(1,1)<0,max(s,t)∈Dφ′′h(s,t)(1)<0. | (3.2) |
By the quantitative deformation lemma, we prove that I′λ(uλ)=0. Suppose by contradiction that I′λ(uλ)≠0, then there exist λ1>0 and ξ>0 such that
||I′λ(v)||≥λ1forallv∈H,||v−uλ||≤3ξ. |
Let ε=min{mλ−m3,λ1ξ8} and sξ={u∈H:||u−uλ||≤ξ}, then the deformation lemma (see[32], Lemma 2.3) shows that there is a deformation η∈C([0,1]×H,H) such that
(i) η(d,u)=u if u∉I−λ([mλ−2ε,mλ+2ε])∩s2ξ,d∈[0,1];
(ii) Iλ(η(d,u))≤Iλ(u) for all u∈H, d∈[0,1];
(iii) Iλ(η(d,u))<mλ, ∀u∈Imλλ∩sξ, ∀d∈(0,1].
First, we need to prove that
max(s,t)∈DIλ(η(d,h(s,t)))<mλforalld∈(0,1]. | (3.3) |
In fact, for any d∈(0,1], it follows from Corollary 2.10 and (ii) that
max{(s,t)∈D:h(s,t)∉sξ}Iλ(η(d,h(s,t)))≤max{(s,t)∈D:h(s,t)∉sξ}Iλ(h(s,t))<mλ. |
Moreover, Corollary 2.10 and (iii) imply that
max{(s,t)∈D:h(s,t)∈sξ}Iλ(η(d,h(s,t)))<mλforalld∈(0,1]. |
Hence, (3.3) holds. From the continuity of η and (3.1)-(3.2), there exists a constant d0∈(0,1] such that
max(s,t)∈D∂2fη(d0,h(s,t))∂s2(1,1)<0,max(s,t)∈D∂2fη(d0,h(s,t))∂t2(1,1)<0,max(s,t)∈Dφ′′η(d0,h(s,t))(1)<0. | (3.4) |
In the following, we prove that η(d0,h(D))∩M∗λ≠∅, which contradicts the definition of mλ. In fact, let g(s,t)=η(d0,h(s,t)) and
ψ1(s,t)=(⟨I′λ(h(s,t)),u+λ⟩,⟨I′λ(h(s,t)),u−λ⟩), |
ψ2(s,t)=(1s⟨I′λ(g(s,t)),g+(s,t)⟩,1t⟨I′λ(g(s,t)),g−(s,t)⟩). |
Then Corollary 2.10 and the degree theory yield deg(ψ1,D,0)=1. On the other hand, we know ε<mλ−m3, m<mλ−2ε. Hence, from (i) we have η(d,h(s,t))=h(s,t) for d∈(0,1], (s,t)∈∂D, and it follows that
ψ1(s,t)=ψ2(s,t)forany(s,t)∈∂D. |
Combining the homotopy invariance property of the degree, we get deg(ψ2,D,0) = deg(ψ1,D,0)=1. That is, there exists (s0,t0)∈D such that ψ2(s0,t0)=0. Therefore, using (3.4) and ψ2(s0,t0)=0, we have η(d0,h(s0,t0))∈M∗λ, i.e., η(d0,h(D))∩M∗λ≠∅. From this, uλ is a critical point of Iλ, i.e., I′λ(uλ)=0.
The authors declare they have not used Artificial Intelligence (AI) tools in the creation of this article.
This research is supported by National Natural Science Foundation of China [No.11971393].
The authors declare there is no conflict of interest.
[1] | C. O. Alves, M. A. S. Souto, Existence of least energy nodal solution for a Schrödinger-Poisson system in bounded domains, Z. Angew. Math. Phys., 65 (2014), 1153–1166. https://doi.org/10.1007/s00033-013-0376-3 |
[2] | A. Ambrosetti, H. Brezis, G. Cerami, Combined effects of concave and convex nonlinearities in some elliptic problems, J. Funct. Anal., 122 (1994), 519–543. https://doi.org/10.1006/jfan.1994.1078 |
[3] | A. Azzollini, A. Pomponio, Ground state solutions for the nonlinear Schrödinger-Maxwell equations, J. Math. Anal. Appl., 345 (2008), 90–108. https://doi.org/10.1016/j.jmaa.2008.03.057 |
[4] | T. Bartsch, A. Pankov, Z. Wang, Nonlinear Schrödinger equations with steep potential well, Commun. Contemp. Math., 3 (2001), 549–569. https://doi.org/10.1142/S0219199701000494 |
[5] |
T. Bartsch, T. Weth, Three nodal solutions of singularly perturbed elliptic equations on domains without topology, Ann. Inst. H. Poincaré Anal. Non Linéaire, 22 (2005), 259–281. https://doi.org/10.1016/j.anihpc.2004.07.005 doi: 10.1016/j.anihpc.2004.07.005
![]() |
[6] | V. Benci, D. Fortunato, An eigenvalue problem for the Schrödinger-Maxwell equations, Topol. Methods Nonlinear Anal., 11 (1998), 283–293. http://dx.doi.org/10.12775/TMNA.1998.019 |
[7] | V. E. Bobkov, On existence of nodal solution to elliptic equations with convex-concave nonlinearities, Ufa Math. J., 5 (2013), 18–30. http://dx.doi.org/10.13108/2013-5-2-18 |
[8] | V. E. Bobkov, On the existence of a continuous branch of nodal solutions of elliptic equations with convex-concave nonlinearities, Differ. Equ., 50 (2014), 765–776. https://doi.org/10.1134/S0012266114060056 |
[9] | S. Chen, L. Li, Infinitely many large energy solutions for the Schrödinger-Poisson system with concave and convex nonlinearities, Appl. Math. Lett., 112 (2021), 106789. https://doi.org/10.1016/j.aml.2020.106789 |
[10] | S. Chen, X. Tang, Ground state sign-changing solutions for asymptotically cubic or super-cubic Schrödinger-Poisson systems without compact condition, Comput. Math. Appl., 74 (2017), 446–458. https://doi.org/10.1016/j.camwa.2017.04.031 |
[11] |
T. D'Aprile, D. Mugnai, Solitary waves for nonlinear Klein-Gordon-Maxwell and Schrödinger-Maxwell equations, Proc. Roy. Soc. Edinburgh Sect. A, 134 (2004), 893–906. https://doi.org/10.1017/S030821050000353X doi: 10.1017/S030821050000353X
![]() |
[12] |
P. Drábek, S. I. Pohožaev, Positive solutions for the p-Laplacian: application of the fibering method, Proc. Roy. Soc. Edinburgh Sect. A, 127 (1997), 703–726. https://doi.org/10.1017/S0308210500023787 doi: 10.1017/S0308210500023787
![]() |
[13] | L. Gu, H. Jin, J. Zhang, Sign-changing solutions for nonlinear Schrödinger-Poisson systems with subquadratic or quadratic growth at infinity, Nonlinear Anal., 198 (2020), 111897. https://doi.org/10.1016/j.na.2020.111897 |
[14] | C. Lei, H. Suo, Positive solutions for a Schrödinger-Poisson system involving concave-convex nonlinearities, Comput. Math. Appl., 74 (2017), 1516–1524. https://doi.org/10.1016/j.camwa.2017.06.029 |
[15] | W. Li, V. D. Rădulescu, B. Zhang, Infinitely many solutions for fractional Kirchhoff-Schrödinger-Poisson systems, J. Math. Phys., 60 (2019), 011506. https://doi.org/10.1063/1.5019677 |
[16] | Z. Liu, L. Tao, D. Zhang, S. Liang, Y. Song, Critical nonlocal Schrödinger-Poisson system on the Heisenberg group, Adv. Nonlinear Anal., 11 (2022), 482–502. https://doi.org/10.1515/anona-2021-0203 |
[17] | Z. Liu, Z. Wang, Schrödinger equations with concave and convex nonlinearities, Z. Angew. Math. Phys., 56 (2005), 609–629. http://dx.doi.org/10.1007/s00033-005-3115-6 |
[18] | Z. Liu, Z. Wang, J. Zhang, Infinitely many sign-changing solutions for the nonlinear Schrödinger-Poisson system, Ann. Mat. Pura Appl., 195 (2016), 775–794. https://doi.org/10.1007/s10231-015-0489-8 |
[19] | D. Ruiz, The Schrödinger-Poisson equation under the effect of a nonlinear local term, J. Funct. Anal., 237 (2006), 655–674. https://doi.org/10.1016/j.jfa.2006.04.005 |
[20] | Ó. Sánchez, J. Soler, Long-time dynamics of the Schröginger-Poisson-Salter system, J. Stat. Phys., 144 (2004), 179–204. https://doi.org/10.1023/B: JOSS.0000003109.97208.53 |
[21] | W. Shuai, Q. Wang, Existence and asymptotic behavior of sign-changing solutions for the nonlinear Schrödinger-Poisson system in R3, Z. Angew. Math. Phys., 66 (2015), 3267–3282. https://doi.org/10.1007/s00033-015-0571-5 |
[22] | M. Shao, A. Mao, Multiplicity of solutions to Schrödinger-Poisson system with concave-convex nonlinearities, Appl. Math. Lett., 83 (2018), 212–218. https://doi.org/10.1016/j.aml.2018.04.005 |
[23] | M. Shao, A. Mao, Schrödinger-Poisson system with concave-convex nonlinearities, J. Math. Phys., 60 (2019), 061504. https://doi.org/10.1063/1.5087490 |
[24] | K. Sofiane, Least energy sign-changing solutions for a class of Schrödinger-Poisson system on bounded domains, J. Math. Phys., 62 (2021), 031509. https://doi.org/10.1063/5.0040741 |
[25] |
J. Sun, T. Wu, Z. Feng, Multiplicity of positive solutions for a nonlinear Schrödinger-Poisson system, J. Differential Equations, 260 (2016), 586–627. https://doi.org/10.1016/j.jde.2015.09.002 doi: 10.1016/j.jde.2015.09.002
![]() |
[26] | J. Sun, T. Wu, On Schrödinger-Poisson systems involving concave-convex nonlinearities via a novel constraint approach, Commun. Contemp. Math., 23 (2021), 2050048. https://doi.org/10.1142/S0219199720500480 |
[27] | D. Wang, H. Zhang, W. Guan, Existence of least-energy sign-changing solutions for Schrödinger-Poisson system with critical growth, J. Math. Anal. Appl., 479 (2019), 2284–2301. https://doi.org/10.1016/j.jmaa.2019.07.052 |
[28] | X. Wang, F. Chen, F. Liao, Existence and nonexistence of nontrivial solutions for the Schrödinger-Poisson system with zero mass potential, Adv. Nonlinear Anal., 12, 20220319 (2023). https://doi.org/10.1515/anona-2022-0319 |
[29] |
Z. Wang, H. Zhou, Sign-changing solutions for the nonlinear Schrödinger-Poisson system in R3, Calc. Var. Partial Differential Equations, 52 (2015), 927–943. https://doi.org/10.1007/s00526-014-0738-5 doi: 10.1007/s00526-014-0738-5
![]() |
[30] | L. Wen, S. Chen, V. D. Rădulescu, Axially symmetric solutions of the Schrödinger-Poisson system with zero mass potential in R2, Appl. Math. Lett., 104, (2020), 106244. https://doi.org/10.1016/j.aml.2020.106244 |
[31] |
T. Weth, Energy bounds for entire nodal solutions of autonomous superlinear equations, Calc. Var. Partial Differential Equations, 27 (2006), 421–437. https://doi.org/10.1007/s00526-006-0015-3 doi: 10.1007/s00526-006-0015-3
![]() |
[32] | M. Willem, Minimax Theorems, Birkhäuser Boston, 1996. https://doi.org/10.1007/978-1-4612-4146-1 |
[33] | Z. Yang, Z. Ou, Nodal solutions for Schrödinger-Poisson systems with concave-convex nonlinearities, J. Math. Anal. Appl., 499 (2021), 125006. https://doi.org/10.1016/j.jmaa.2021.125006 |
[34] | S. Yu, Z. Zhang, Sufficient and necessary conditions for ground state sign-changing solutions to the Schrödinger-Poisson system with cubic nonlinearity on bounded domains, Appl. Math. Lett., 123 (2022), 107570. https://doi.org/10.1016/j.aml.2021.107570 |
[35] | J. Zhang, R. Niu, X. Han, Positive solutions for a nonhomogeneous Schrödinger-Poisson system, Adv. Nonlinear Anal., 11 (2022), 1201–1222. https://doi.org/10.1515/anona-2022-0238 |
[36] | Z. Zhang, Y. Wang, Ground state and sign-changing solutions for critical Schrödinger-Poisson system with lower order perturbation, Qual. Theory Dyn. Syst., 22, 76 (2023). https://doi.org/10.1007/s12346-023-00764-5 |
[37] | X. Zhong, C. Tang, Ground state sign-changing solutions for a Schrödinger-Poisson system with a critical nonlinearity in R3, Nonlinear Anal. Real World Appl., 39 (2018), 166–184. https://doi.org/10.1016/j.nonrwa.2017.06.014 |
[38] | X. Zhong, C. Tang, Ground state sign-changing solutions for a Schrödinger-Poisson system with a 3-linear growth nonlinearity, J. Math. Anal. Appl., 455 (2017), 1956–1974. https://doi.org/10.1016/j.jmaa.2017.04.010 |
[39] | W. Zou, M. Schechter, Critical point theory and its applications, Springer, New York, 2006. https://doi.org/10.1007/0-387-32968-4 |
1. | Xueqi Sun, Yongqiang Fu, Sihua Liang, Normalized solutions for pseudo-relativistic Schrödinger equations, 2024, 16, 2836-3310, 217, 10.3934/cam.2024010 |