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Research article

Sign-changing solutions for the Schrödinger-Poisson system with concave-convex nonlinearities in R3

  • Received: 14 September 2023 Revised: 10 October 2023 Accepted: 12 October 2023 Published: 19 October 2023
  • 35A15, 35J20, 35J50

  • In this paper, we consider the following Schrödinger-Poisson system

    {Δu+V(x)u+ϕu=|u|p2u+λK(x)|u|q2u    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

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  • In this paper, we consider the following Schrödinger-Poisson system

    {Δu+V(x)u+ϕu=|u|p2u+λK(x)|u|q2u    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:R3R and fC(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|p2u. 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 fC1(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|p2u, 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|p2u+|u|q2u 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|q2u+|u|γ2uin Ω,u=0inΩ,

    where ΩRN is a bounded connected domain with a smooth boundary, N1, 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(,λ0), where λ0 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|q2u+h(x)|u|γ2uinΩ,u=0inΩ,

    where 1<q<2<γ<2, λR and the weight functions k, hL(Ω) satisfy the conditions essinfxΩk(x)>0 and essinfxΩh(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|p2u+|u|q2uinΩ,Δϕ=u2inΩ,u=0inΩ, (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 1p<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|p2u+λK(x)|u|q2uinR3,Δϕ=u2inR3, (1.3)

    where 1<q<2, 4<p<6, λ>0 and V, K satisfy the assumptions:

    (V) VC(R3,R) satisfies infxR3V(x)a>0 for each A>0, meas{xR3:V(x)A}<, where a is a constant and meas denotes the Lebesgue measure in R3;

    (K) K is positive and KL66q(R3).

    Here the condition (V) is similar to [17]. This condition also ensures the compactness of embedding HLp(R3), 2p<2, where H is the Hilbert space

    H={uH1(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|q2u+μu+ν|u|p2u,

    where 1<q<2<p<2, N2 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 uH, there exists a unique ϕuD1,2(R3) such that Δϕu=u2, where

    ϕu(x)=14πR3u2(y)|xy|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|p2u+λK(x)|u|q2u   inR3. (1.5)

    Therefore, the energy functional associated with system (1.3) is defined by

    Iλ(u)=12R3(|u|2+V(x)u2)dx+14R3ϕuu2dx1pR3|u|pdxλqR3K(x)|u|qdx,uH.

    The functional Iλ(u) is well-defined for every uH and belongs to C1(H,R). Furthermore, for any vH,

    Iλ(u),v=R3(uv+V(x)uv)dx+R3ϕuuvdxR3|u|p2uvdxλR3K(x)|u|q2uvdx.

    As is well known, the solution of problem (1.5) is the critical point of the functional Iλ(u). Moreover, if uH 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={uH{0}:Iλ(u),u=0},

    which is related to the behavior of the map φu:rIλ(ru) (r>0) (see [12] for the introduction of this map). For uH, we have

    φu(r)=12r2||u||2+14r4R3ϕuu2dx1prp|u|ppλqrqR3K(x)|u|qdx.

    It is well known that, for any uH{0}, φu(r)=0 if and only if ruNλ, which also implies that φu(1)=0 if and only if uNλ. 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={uH:u±0,Iλ(u),u±=0}.

    Hoeever, this manifold cannot be directly applied due to appearance of concave term λk(x)|u|q2u. As we will see, inspired by [33], we can construct the set MλMλ and prove this set is a nonempty closed set, where Mλ is defined by (2.5) below. We then show that the minimization problem mλ:=infuMλIλ(u) is attained by some uλMλ 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|p2u (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|q2u is concave and |u|p2u 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 Mλ. In particular, by determining some important lower bound estimates, we verify that Mλ. 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 uH, we get

    (i) there exists C>0 such that R3ϕuu2dxCu4;

    (ii) ϕu0, for any uH;

    (iii) ϕtu=t2ϕu, for any t>0 and uH;

    (iv) if unu in H, then ϕunϕu in D1,2(R3) and

    limnR3ϕunu2ndx=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:HR:uR3K(x)|u|qdx

    is weakly continuous.

    Proof. Undoubtedly, it is sufficient to prove that if unu in H, then R3K(x)|un|qdxR3K(x)|u|qdx as n. In fact, if unu in H, going if necessary to a subsequence, we can assume that unu a.e. on R3. Since unu in H, we get that {un} is bounded in L6(R3) and {uqn} is bounded in L6q(R3). Therefore, uqnuq in L6q(R3). Combining with (K) and the definition of weak convergence, we obtain

    R3K(x)|un|qdxR3K(x)|u|qdxas  n.

    Now, for any uH 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+14s4R3ϕu+(u+)2dx+12s2t2R3ϕu+(u)2dx1psp|u+|ppλqsqR3K(x)|u+|qdx+12t2||u||2+14t4R3ϕu(u)2dx1ptp|u|ppλqtqR3K(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)=(Iλ(su++tu),u+,Iλ(su++tu),u)=(1sIλ(su++tu),su+,1tIλ(su++tu),tu),

    which implies that for any uH with u±0, su++tuMλ 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 uH with u±0, there holds that

    (i) if λ(0,λ1), then for any fixed t0, 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 λ0, then for any fixed t0, 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 s0, 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 λ0, then for any fixed s0, 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+14μR3ϕsu++tu(su++tu)2dx1p|su++tu|ppλqR3K(x)|su++tu|qdx,

    where μ is a nonnegative parameter.

    (i) For any fixed t0, a direct calculation gives

    fμs(s,t)=s||u+||2+μs3R3ϕu+(u+)2dx+μst2R3ϕu+(u)2dxsp1|u+|ppλsq1R3K(x)|u+|qdx=sq1(s2q||u+||2+μs4qR3ϕu+(u+)2dx+μs2qt2R3ϕu+(u)2dxspq|u+|ppλR3K(x)|u+|qdx).

    Then, if s>0, fμs(s,t)=0 is equivalent to

    βμ(s)=s2q||u+||2+μs4qR3ϕu+(u+)2dx+μs2qt2R3ϕu+(u)2dxspq|u+|ppλR3K(x)|u+|qdx=0.

    For βμ(s), we can obtain that

    βμ(s)=s1q((2q)||u+||2+μ(4q)s2R3ϕu+(u+)2dx+μ(2q)t2R3ϕu+(u)2dx(pq)sp2|u+|pp).

    Clearly, for any fixed t0, 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 μ0. Moreover, βμ is strictly increasing in (0,sμ) and strictly decreasing in (sμ,+).

    Noting that if μ=0, we have

    β0(s)=s2q||u+||2spq|u+|ppλR3K(x)|u+|qdx

    and

    β0(s)=s1q((2q)||u+||2(pq)sp2|u+|pp).

    Hence,

    s0=((2q)||u+||2(pq)|u+|pp)1p2

    and

    β0(s0)=p2pq(2qpq)2qp2||u+||2(pq)p2|u+|p(2q)p2pλR3K(x)|u+|qdx.

    Let

    αμ(s)=s2q||u+||2+μs4qR3ϕu+(u+)2dx+μs2qt2R3ϕu+(u)2dxspq|u+|pp

    and

    λ+1=infuH,u±0α0(s0)R3K(x)|u+|qdx,

    then it follows from Sobolev embedding that

    λ+1p2pq(2qpq)2qp21Sq6Sp(2q)p2p|K|66q>0. (2.2)

    Therefore, if λ(0,λ+1), we can deduce βμ(s0)>β0(s0)>0 for any uH with u±0. For any fixed t0, 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 βμ(s1(t))>0, βμ(s2(t))<0. On the other hand, when λ0, there is a unique s3(t)>0 such that βμ(s)=0 and βμ(s3(t))<0.

    Finally, considering that

    2fμs2(s,t)=||u+||2+3μs2R3ϕu+(u+)2dx+μt2R3ϕu+(u)2dx(p1)sp2|u+|ppλ(q1)sq2R3K(x)|u+|qdx=sq2(s2q||u+||2+3μs4qR3ϕu+(u+)2dx+μs2qt2R3ϕu+(u)2dx(p1)spq|u+|ppλ(q1)R3K(x)|u+|qdx),

    we can find that

    2fμs2(s,t)=(q1)sq2βμ(s)+sq1βμ(s).

    Note that fμs(s,t)=0 is equivalent to βμ(s)=0, then we have

    2fμs2(s,t)=sq1βμ(s)  if  fμs(s,t)=0.

    Thus, the facts that βμ(s1(t))>0,βμ(s2(t))<0 and βμ(s3(t))<0 signify that

    2fμs2(s1(t),t)>0,2fμs2(s2(t),t)<0and2fμs2(s3(t),t)<0.

    In particular, when μ=1, the results still hold.

    (ii) For any fixed s0, let

    λ1=infuH,u±0α0(s0)R3K(x)|u|qdx=infuH,u±0p2pq(2qpq)2qp2||u||2(pq)p2|u|p(2q)p2pR3K(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=λ1. Indeed, u±0 indicates that (u)±0, which yields that λ+1=λ1. Let λ1=λ+1=λ1, from (i) and (ii), then the proof is completed.

    Let

    λ2=infuH{0}p2pq(2qpq)2qp2||u||2(pq)p2|u|p(2q)p2pR3K(x)|u|qdx.

    Undoubtedly,

    λ1λ2p2pq(2qpq)2qp21Sq6Sp(2q)p2p|K|66q>0. (2.4)

    Here, the notation Sp represents the embedding constant of HLp(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 uH{0}, φu(r) has exactly two critical points, 0<r1(u)<r2(u) and φu(r1(u))>0, φu(r2(u))<0. On the other hand, when λ0, φu(r) has exactly one critical point, r3(u)>0, and φu(r3(u))<0.

    Lemma 2.5. Assume that 1<q<2, 4<p<6, the assumption (K) and λ<λ1 hold, then for any uMλ, (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 2fus2(s1(1),1)>0, 2fus2(s2(1),1)<0. Since uMλ, we have fus(1,1)=0, which means that s1(1)=1 or s2(1)=1. Hence, 2fus2(1,1)0. Analogously, we can conclude that 2fut2(1,1)0.

    If λ0, it follows from Lemma 2.3 that fu(s,1) has exactly one critical point s3(1) and 2fus2(s3(1),1)<0. Combining with uMλ, we have fus(1,1)=0, which shows that s3(1)=1. Therefore, we get 2fus2(1,1)0. Similarly, we can deduce that t3(1)=1, 2fut2(1,t3(1))<0 and the claim is clearly true.

    Note that uMλNλ, then the Sobolev embedding indicates that

    Iλ(u)=Iλ(u)14Iλ(u),u=14||u||2+(141p)R3|u|pdx+λ(141q)R3K(x)|u|qdx14||u||2+λ(141q)|K|66q|u|q614||u||2+λ1(141q)|K|66qSq6||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 uNλ, φu(1)0.

    In what follows, we construct the following sets

    Mλ={uMλ:2fus2(1,1)<0,2fut2(1,1)<0}

    and

    Mλ={uMλ:2fus2(1,1)<0,2fut2(1,1)<0,φu(1)<0}. (2.5)

    Unquestionably, MλMλ. According to the properties of fu mentioned above, we can verify that the set Mλ is nonempty and Mλ=Mλ (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 uMλ.

    Proof. For any uMλ, from 2fus2(1,1)<0, 2fut2(1,1)<0 and Sobolev embedding, it follows that

    (2q)||u±||2<(2q)||u±||2+(4q)R3ϕu±(u±)2dx+(2q)R3ϕu+(u)2dx<(pq)|u±|pp(pq)Spp||u±||p,

    which implies

    ||u±||>(2q(pq)Spp)1p2:=σ>0. (2.6)

    Hence, the proof is finished.

    In order to prove that Mλ, let us define

    λ3=infuMλ{(p2)||u+||2+(p4)R3ϕu+(u+)2dx(pq)R3K(x)|u+|qdx,(p2)||u||2+(p4)R3ϕu(u)2dx(pq)R3K(x)|u|qdx}.

    From Sobolev embedding and Lemma 2.7, it follows that

    (p2)||u±||2+(p4)R3ϕu±(u±)2dx(pq)R3K(x)|u±|qdx(p2)||u±||2q(pq)Sq6|K|66q>(p2)σ2q(pq)Sq6|K|66q>0,

    where σ is given by (2.6). Therefore, we have

    λ3(p2)σ2q(pq)Sq6|K|66q>0. (2.7)

    Furthermore, we compute that

    2fus2(s,t)=||u+||2+3s2R3ϕu+(u+)2dx+t2R3ϕu+(u)2dx(p1)sp2|u+|ppλ(q1)sq2R3K(x)|u+|qdx,
    2fut2(s,t)=||u||2+3t2R3ϕu(u)2dx+s2R3ϕu+(u)2dx(p1)tp2|u|ppλ(q1)tq2R3K(x)|u|qdx

    and

    2fust(s,t)=2stR3ϕu+(u)2dx,2futs(s,t)=2stR3ϕu+(u)2dx.

    For any uMλ, we obtain

    2fus2(1,1)=||u+||2+3R3ϕu+(u+)2dx+R3ϕu+(u)2dx(p1)|u+|ppλ(q1)R3K(x)|u+|qdx=(2q)||u+||2+(4q)R3ϕu+(u+)2dx+(2q)R3ϕu+(u)2dx(pq)|u+|pp=(2p)||u+||2+(4p)R3ϕu+(u+)2dx+(2p)R3ϕu+(u)2dxλ(qp)R3K(x)|u+|qdx

    and

    2fut2(1,1)=||u||2+3R3ϕu(u)2dx+R3ϕu+(u)2dx(p1)|u|ppλ(q1)R3K(x)|u|qdx=(2q)||u||2+(4q)R3ϕu(u)2dx+(2q)R3ϕu+(u)2dx(pq)|u|pp=(2p)||u||2+(4p)R3ϕu(u)2dx+(2p)R3ϕu+(u)2dxλ(qp)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 uH with u±0, there exists a unique pair (su,tu)R+×R+ such that suu++tuuMλ. 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 λ0 is very similar. Let uH with u±0, from the proof of Lemma 2.3, then we have that fus(s,t) satisfies the conditions:

    (i) fut(s,t2(s))=0 for all s0;

    (ii) fut(s,t) is continuous and has continuous partial derivatives in [0,+)×[0,+);

    (iii) 2fut2(s,t2(s))<0 for all s0.

    Hence, we can obtain that if 0<λ<λ1, fut(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, fus(s,t)=0 determines an implicit function s2(t) with continuous derivative on [0,+).

    On the other hand, for every s0, from fut(s,t2(s))=0 and fut(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 fut(s,t) that fut(s,t2(s))<0, which contradicts fut(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, fut(su,tu)=fus(su,tu)=0. Additionally, noting that

    t2(s)=(2futs/2fut2)(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

    fus(su,tu)=fut(su,tu)=0

    and

    2fus2(su,tu)<0,2fut2(su,tu)<0;

    that is, suu++tuuMλ.

    Next, we prove that (su,tu) is the unique maximum point of fu(s,t) on [0,+)×[0,+). In fact, if uMλ, 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)=(2fus22fut22futs2fust)|(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 uMλ, then

    H(u)=(2fus22fut22futs2fust)|(1,1)=((2p)||u+||2+(4p)R3ϕu+(u+)2dx+(2p)R3ϕu+(u)2dxλ(qp)R3K(x)|u+|qdx)×((2p)||u||2+(4p)R3ϕu(u)2dx+(2p)R3ϕu+(u)2dxλ(qp)R3K(x)|u|qdx)4(R3ϕu+(u)2dx)2.

    From 2fus2(1,1)<0, 2fut2(1,1)<0, if λ<λ3, we derive

    2fus2(1,1)2R3ϕu+(u)2dx=(p2)||u+||2+(p4)R3ϕu+(u+)2dx+(p4)R3ϕu+(u)2dxλ(pq)R3K(x)|u+|qdx>(p2)||u+||2+(p4)R3ϕu+(u+)2dxλ3(pq)R3K(x)|u+|qdx>0

    and

    2fut2(1,1)2R3ϕu+(u)2dx>(p2)||u||2+(p4)R3ϕu(u)2dxλ3(pq)R3K(x)|u|qdx>0,

    which show that H(u)>0.

    If uMλ, then there exists a unique pair (su,tu) of positive numbers such that suu++tuuMλ. Let v=suu++tuu, i.e., vMλ. 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 Mλ. Moreover, we get Mλ=Mλ.

    Proof. By the definitions of Mλ and Mλ, MλMλ is obvious. Hence, we only need to prove that if λ<min{λ1,λ2,λ3}, then MλMλ. That is, for any uMλ, φu reaches its maximum at point r=1. It follows from λ<λ1 and Lemma 2.8 that Mλ, and from Lemma 2.8, for any uMλ, 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, MλMλ. 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 uH and u±0, then there exists a unique pair (su,tu)R+×R+ such that suu++tuuMλ 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 uH{0}, then there exists ru>0 such that φu(ru)>0, where λ4>0.

    Proof. Fixed uH{0}, let

    Eu(r)=12r2||u||21prp|u|pp

    for any r0, then we have

    φu(r)=12r2||u||2+14r4R3ϕuu2dx1prp|u|ppλqrqR3K(x)|u|qdx12r2||u||21prp|u|ppλqrqR3K(x)|u|qdx=Eu(r)λqrqR3K(x)|u|qdx. (2.10)

    Considering Eu(r), we obtain that there is a unique r1(u)=(||u||2|u|pp)1p2>0 such that Eu(r) achieves its maximum at r1(u) and the maximum value is Eu(r1(u))=p22p(||u|||u|p)2pp2. Moreover, from Sobolev embedding and (2.10), it is clear to calculate that

    φu(r1(u))Eu(r1(u))λq(r1(u))qR3K(x)|u|qdxEu(r1(u))λq(r1(u))q|K|66qSq6||u||q=Eu(r1(u))λqSq6|K|66q(2pp2)q2(Eu(r1(u)))q2=(Eu(r1(u)))q2((Eu(r1(u)))2q2λqSq6|K|66q(2pp2)q2). (2.11)

    Consequently, by taking

    λ4=(p2)q2p|K|66qSq6infuH{0}(||u|||u|p)p(2q)p2(p2)q2p|K|66qSq6Sp(2q)p2p>0, (2.12)

    we conclude that if λ<λ4, it holds

    λqSq6|K|66q(2pp2)q2<λ4qSq6|K|66q(2pp2)q21qSq6|K|66q(2pp2)q2(p2)q2pSq6|K|66q(||u|||u|p)p(2q)p2=(Eu(r1(u)))2q2

    for any uH{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 Mλ.

    Lemma 2.12. If 1<q<2, 4<p<6, λ<λ and the assumptions (V) and (K) hold, then Mλ is a closed set.

    Proof. Letting {un}Mλ satisfy unu0 as n in H, we now prove that u0Mλ. From {un}Mλ, we obtain

    Iλ(u0),u±0=limnIλ(un),u±n=0, (2.14)
    2fu0s2(1,1)=limn2funs2(1,1)0, (2.15)
    2fu0t2(1,1)=limn2funt2(1,1)0, (2.16)
    φu0(1)=limnφun(1)0. (2.17)

    From Lemma 2.7, it follows that ||u±n||>σ>0 for any unMλ and hence ||u±0||=limn||u±n||>σ>0, which indicates u±00. Using this and (2.14), we obtain u0Mλ 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

    2fu0s2(1,1)<0,2fu0t2(1,1)<0,φu0(1)<0.

    Hence, u0Mλ and Mλ 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:=infuMλIλ(u) can be achieved by some uλMλ and mλ>0.

    Proof. According to Lemma 2.5, mλ> when λ<λ. Let {un}Mλ be a minimizing sequence for the functional Iλ, namely Iλ(un)mλ as n. Since the functional Iλ is coercive on Mλ, then {un} is bounded in H. Going if necessary to a subsequence, we may assume that

    unuλinH,unuλinLp(R3).

    Now, we first claim that u±λ0. 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+λR3K(x)|u±λ|qdx=limn(|u±n|pp+λR3K(x)|u±n|qdx)=limn(||u±n||2+R3ϕu±n(u±n)2dx+R3ϕu+n(un)2dx)limn||u±n||2>σ2>0.

    This means that u±λ0 for any λ<λ.

    Next, we proof that unuλ in H. Arguing by contradiction, suppose that

    ||u+λ||<limninf||u+n||or||uλ||<limninf||un||.

    From Corollary 2.10, there exists a unique pair (suλ,tuλ) such that ˜uλ=suλu+λ+tuλuλMλ and Iλ(u+n+un)=maxs,t>0Iλ(suλu+n+tuλun). Consequently,

    mλIλ(˜uλ)<limninfIλ(suλu+n+tuλun)limninfIλ(u+n+un)=mλ.

    That is, we get a contradiction. Therefore, unuλ in H and mλ is achieved by uλ. Combining the fact that Mλ is closed, so uλMλ.

    Finally, it follows from uλMλ 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 uλMλ, according to Corollary 2.10, we obtain that

    Iλ(su+λ+tuλ)<Iλ(u+λ+uλ)=mλ,for(s,t)(R+×R+)(1,1).

    Moreover, we get Iλ(uλ)>0, φuλ(1,1)<0, (2fuλ/s2)(1,1)<0 and (2fuλ/t2)(1,1)<0.

    Let D=(1δ,1+δ)×(1δ,1+δ) and h:DH by h(s,t)=su+λ+tuλ for any (s,t)D. Then there is a constant 0<δ<1 such that

    0<m:=maxDIλ(h(s,t))<mλ,max(s,t)D2fh(s,t)s2(1,1)<0, (3.1)
    max(s,t)D2fh(s,t)t2(1,1)<0,max(s,t)Dφh(s,t)(1)<0. (3.2)

    By the quantitative deformation lemma, we prove that Iλ(uλ)=0. Suppose by contradiction that Iλ(uλ)0, then there exist λ1>0 and ξ>0 such that

    ||Iλ(v)||λ1forallvH,||vuλ||3ξ.

    Let ε=min{mλm3,λ1ξ8} and sξ={uH:||uuλ||ξ}, then the deformation lemma (see[32], Lemma 2.3) shows that there is a deformation ηC([0,1]×H,H) such that

    (i) η(d,u)=u if uIλ([mλ2ε,mλ+2ε])s2ξ,d[0,1];

    (ii) Iλ(η(d,u))Iλ(u) for all uH, d[0,1];

    (iii) Iλ(η(d,u))<mλ, uImλλsξ, 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)D2fη(d0,h(s,t))s2(1,1)<0,max(s,t)D2fη(d0,h(s,t))t2(1,1)<0,max(s,t)Dφη(d0,h(s,t))(1)<0. (3.4)

    In the following, we prove that η(d0,h(D))Mλ, which contradicts the definition of mλ. In fact, let g(s,t)=η(d0,h(s,t)) and

    ψ1(s,t)=(Iλ(h(s,t)),u+λ,Iλ(h(s,t)),uλ),
    ψ2(s,t)=(1sIλ(g(s,t)),g+(s,t),1tIλ(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<mλm3, m<mλ2ε. 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))Mλ, i.e., η(d0,h(D))Mλ. From this, uλ is a critical point of Iλ, i.e., Iλ(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.



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