Abstract
In the present paper, a class of Schrödinger equations is investigated, which can be stated as
If the external potential V is radial and coercive, then we give the local Ambrosetti-Rabinowitz super-linear condition on the nonlinearity term f ∈ C(ℝ, ℝ) which assures the problem has not only infinitely many radial sign-changing solutions, but also infinitely many non-radial sign-changing solutions.
1 Introduction
In this paper, a class of Schrödinger equations is studied. It can be stated as
where V is the external potential and the nonlinearity term f : ℝ → ℝ is continuous. For the appliance to the nonlinear optics, the nonlinear Schrödinger equation is receiving increasingly amount of attention from the field of mathematics. Indeed, some nonlinear Schrödinger equations arose from some simplified models. Bergé [10], C. Sulem and P.-L. Sulem [33] studied them from the modelization aspects. On the other hand, quantum field theory, especially the Hartree-Fock theory, is also origin fields of Schrödinger equations. Many researchers, Avron, Herbst and Simon [1], Bialinycki-Birula and Mycielski [11], Gogny and Lions [16], Kato [18], and C. Sulem and P.-L. Sulem [33] has been focusing on it since decades ago. What's more, the nonlinear Schrodinger equation is also a good model dispersion equation, because it is usually technically simpler than other dispersion equations (such as the wave or KdV). In the mind of mathematicians, this kind of problem has been concerned for a long time. Of course, there are rich results about it (e.g. [8, 9, 15, 19, 21]). What we care about is whether there are multiple solutions to this problem. Let's review some of the work in this area.
For the case of autonomy, that is, V > 0 is a constant, to handle the existence of infinitely many solutions to problem (1.1), researchers in [14, 15, 26, 30]) considered it with some special nonlinearities up, p ∈ (1, 2* − 1), where
Notice that the authors in [21] made the standard Ambrosetti-Rabinowitz super-linear condition (there is μ > 2 satisfying sf(s) ≥ μF(s) > 0 for all s ∈ ℝ \ {0}, where
For the case where V is continuous and 1-periodic in x1, · · · , xN, it has been considered by Szulkin and Weth [35], and Tang et al. [37]. In [12, 31] the authors showed that problem (1.1) have infinitely many solutions where V is an asymptotically autonomous potential, namely, lim|x|→∞ V(x) = V∞ > 0. For the case where V is radial potential, i.e., V(x) = V(|x|), provided that r2V(r) has a local maximum point, or local minimum point r0 > 0 with V(r0) > 0, Chen, Wei and Yan [13] then proved that problem (1.1) with critical growth has infinitely many non-radial solutions. If V(y) = (y′, y″), (y′, y″) ∈ ℝ2 × ℝN−2 has a stable critical point, then problem (1.1) with critical growth has infinitely many solutions by using the standard reduction method to locate the concentrating points of the solutions (see [28]).
Hong [17] required that V(x) = λa(x) + 1 and f satisfies the standard Ambrosetti-Rabinowitz super-linear condition, thanks for the principle of symmetric criticality and the fountain theorem, the authors proved that the existence of a sequence of solutions to equation (1.1) and proved the existence of an unbounded sequence of solutions, which is non-radial and sign-changing. As we know, there also exists many similar researches concerning the existence of the nontrivial solutions to problem (1.1). We refer the interested reader to [15, 23, 29, 34, 36, 38] and references therein.
However, it can be seen from the above literature that few articles start from two aspects, that is, to prove the existence of two families of solutions. There is a natural guess: it exists infinitely many radial and non-radial sign-changing solutions for problem (1.1). Fortunately, we will endorse this guess in this work. We introduce our work space given by
endowed with the norm
This is a Hilbert space, and its inner product is denoted by 〈·, ·〉. Here V satisfies
(V1) V ∈ C(ℝN), infx∈ℝN V(x) > 0 and V(x) = V(|x|),
(V2) there exists a constant r > 0 such that, for any M > 0, meas {x ∈ ℝN : |x − y| ≤ r, V(x) ≤ M} → 0 as |y| → +1.
We investigate Schrödinger equations as follows:
where N ≥ 1 and f satisfies
(f1) f ∈ C(ℝ, ℝ) is odd and
(f2)
(f3) f(s) ≥ 0 for any s ≥ 0 and
(f4) there exist L > 0, μ > 2 such that
Our result reads as
Theorem 1.1
Let (V1), (V2) and (f1)–(f4) hold. If N = 2k, k ∈ ℤ+ then problem (1.2) has not only infinitely many radial sign-changing solutions, but also infinitely many non-radial sign-changing solutions.
Remark 1.2
To our knowledge, our result seems to be the first attempt to illustrate that problem (1.2) has two families of infinitely many sign-changing solutions. One can easily see that the local (AR) condition (f4) is weaker than the (AR) condition. For example,
Clearly f does not satisfy the (AR) condition, but satisfies (f4). Therefore, to a certain extent, this work can also be regarded as an extension and supplement to previous research results.
The current work consists of four parts. Let's end the first section here. The second part gives some preliminary results. In Section 3, we introduce a constraint problem. The proof of Theorem 1.1 will be given in Section 4.
2 Preliminaries
Here we give notations which will be used in the rest of this paper.
H1(ℝN) is the usual Sobolev space endowed with the usual norm
Lp(ℝN) is the usual Lebesgue space endowed with the norm
meas Ω denotes the Lebesgue measure of the set Ω.
u± := max{±u, 0}.
C, Ci(i = 1, 2, · · · ) denote positive constants and are possibly different from line to line.
The embedding E ↪ Ls (ℝN) (2 ≤ s < 2*) is compact (see [6, 7]). Now, we have to emphasize that the existence of solutions for problem (1.2) can be proved by using variational methods. As for problem (1.2), the energy functional ℐ : E → ℝ can be constructed as
Thus, based on the conditions of (f1)–(f3), by a standard argument we can affirm that ℐ is well defined and of class C1, and its derivative in E is
Formally, all critical points of ℐ are solutions of problem (1.2).
Lemma 2.1
Assume that (V1), (V2) and (f1)–(f4) hold, then ℐ satisfies the Palais-Smale condition.
Proof
Suppose that {vn} is a Palais-Smale (Shortly: (PS) ) sequence of ℐ for some c ∈ ℝ, that is,
We claim that the sequence {vn} is bounded in E, otherwise there should be a subsequence, still denoted vn, such that ‖vn‖ → ∞ as n → ∞. Let
If meas(Ω) is not equal to 0, then limn→∞ |vn(x)| = +∞ almost everywhere in Ω. With applying (f2) and Fatou's lemma, we calculate directly
Then we obtain that
This contradiction indicates that meas(Ω) = 0, so that u(x) = 0 a.e. x ∈ ℝN and un → 0 in Ls (ℝN) (2 ≤ s < 2*). By (f1), we have
Combining with (f3), we deduce
Hence,
This is a contradiction with μ > 2. So {vn} is bounded in E and then there exists v ∈ E satisfying vn → v in Lq (ℝN) (2 ≤ q < 2*), vn(x) → v(x) a.e. x ∈ ℝN. Generalized control convergence theorem gives us that (f1) and (f2) imply that
Therefore vn → v in E, and hence ℐ satisfies the Palais-Smale condition.
Lemma 2.2
Assume that (V1), (V2) and (f1)–(f4) hold. Then
(i) inf‖u‖=ρ ℐ(u) > 0 holds for some positive constant ρ,
(ii) Let Em be a subspace of E with dim Em < +∞, then there is Rm = R (Em) > 0 satisfying
where BR := {u ∈ E : ‖u‖ ≤ R} and
Proof
(i). From (f1)–(f3), for any fixed δ > 0 we can find Cδ > 0 such that
where p is given in (f2). Then we have
Noting that p > 2, taking δ small enough we conclude that there is 0 < ρ < 1 satisfying
(ii). It follows from (f1)–(f3) that there exists C1, C2 > 0 such that
So we deduce
For μ > 2 and because of the equivalence of norms in finite dimensional space, we see that ℐ(u) = −∞ as ‖u‖ → ∞, u ∈ Em. Therefore the result follows.
3 Radial sign-changing solutions
Set
Clearly, Er is a nonempty complete subspace of E. Now, let problem (1.2) constrain on Er, namely,
Notice that ℐ|Er is the corresponding energy function of problem (3.1). For simplicity, we denote ℐ|Er by ℐr. Next we introduce an auxiliary operator B, which plays an important role in constructing the descending flow for ℐr. Let the operator B be defined as follows: for a fixed u ∈ Er, v = B(u) ∈ Er is the unique solution of
Clearly, the following three expressions are equivalent:
u is a solution of problem (3.1),
u is a critical point of ℐr,
u is a fixed point of B.
In the following lemmas, we introduce some properties of B, which can be similarly proved in [21, 24, 34]. For better viewing, we provide the proof here.
Lemma 3.1
Assume that (V1) and (f1)–(f4) hold. Then
(i) the operator B is well defined as well as continuous,
(ii)
(iii)
(iv) for c < d and a > 0, there exists b > 0 such that ‖u − B(u)‖ ≥ a if u ∈ Er, ℐr(u) ∈ [c, d] and
Proof
(i) Let u ∈ Er, and define for any v ∈ Er,
so that ℰ ∈ C1(Er, ℝ) and ℰ is weakly lower semicontinuous. By (f1)–(f2) and the Sobolev embedding we then obtain
which implies
and this means that ℰ(v) → +∞ as ‖v‖ → ∞, where C = C(u) is a constant depending on u. This shows that ℰ is coercive. Moreover, we know that ℰ is bounded below and it keeps the boundness of sets as a mapping. We now estimate
so that ℰ is also strictly convex. Hence, from [2, Theorems 1.5.6 and 1.5.8], ℰ admits a unique minimizer v = B(u) ∈ Er, which is the unique solution to problem (3.2). Clearly, B also maps bounded sets into bounded sets by (f1)–(f3).
Next, we give the proof of the continuity of B. Assume that {un} ⊂ Er with un → u in Er. Denote v = B(u) and vn = B (un). Since 〈ℰ′(vn), vn〉 = 0 we have
Combining with (f1)–(f2), we can obtain the boundness of {vn} in Er. Assume that vn ⇀ v* in Er, after extracting a subsequence, then for any w ∈ Er
which implies that v* = B(u). So v = v* by the uniqueness.
We show that ‖vn − v‖ → 0 as n → ∞. From 〈ℰ′(v) − ℰ′(vn), v − vn〉 = 0, we can write
So, by the general Lebesgue control convergence theorem, vn → v in Er. Hence, B is continuous.
(ii) By using the fact that B(u) is a solution of problem (3.2), one obtains
for all u ∈ Er.
(iii) Notice that for all
Then we have
(iv) Contrarily, suppose that it exists {un} ⊂ Er with
such that
and hence from (iii) we deduce that
Since the Hilbert space L2 (ℝN) is a separable and Er can be embedded continuously in it, we can conclude that Er possesses a countable orthogonal basis {li}. For any m ∈ ℕ, we denote
In order to prove our results, as in many references such as [21, 23, 34], we introduce the positive and negative cones, which is used to determine the position of the solution. Exactly, set
For ε > 0, let
Similarly, set
Lemma 3.2
Assume that (V1) and (f1)–(f4) hold. Then there exists ε0 > 0 such that for any ε ∈ (0, ε0),
Proof
Let u ∈ Er and v = B(u). Conditions (f1) and (f2) imply that for any ∊ > 0 there exists a constant C∊ > 0 depending on ∊ to satisfy
Based on dist (v, Y−) ≤ ‖v+‖, combined with (3.3) and Sobolev's inequality, it is true that
Let's make ∊ and ε small enough, then
and thus one can find ε0 > 0 such that
As a result, it gives us that
Lemma 3.2 illustrates that every sign-changing solutions of problem (3.1) should be contained in
Lemma 3.3
Assume that (V1) and (f1)–(f4) hold. Then there exists a operator
(i)
(ii)
(iii)
(iv) if f is an odd function, then A is odd.
For a fixed u ∈ Er, we introduce the following initial value problem
where 𝒜(u) = u − A(u). Because A is a locally Lipschitz continuous operator, problem (3.5) has a unique solution by the existence and uniqueness theory of ODE. Denote by τ(t, u) with maximal interval of existence [0, T(u)]. It follows from Lemma 3.3 that ℐr(τ(t, u)) is strictly decreasing in [0, T(u)]. According to [5, Lemma 3.2], through Lemma 3.3 we have the fact that for any ε ∈ (0, ε0] and
For a fixed c ∈ ℝ, set
Lemma 3.4
Assume that (V1), (V2) and (f1)–(f4) hold, and 𝒩 is an open symmetric neighborhood of Kc\W. Then one can find ε0 > 0 such that for any 0 < ε < ε′ < ε0, there is σ ∈ C([0, 1] × Er, Er) satisfying:
(a) σ(t, u) = u if t = 0 or
(b)
(c) if Kc\W = ∅, then
(d) σ(t, − u) = −σ(t, u) for all (t, u) ∈ [0, 1] × Er,
(e)
Proof
Set
Thanks to Lemma 3.1, one can always find b > 0 satisfying
Taking generality into account, we can still assume
Decreasing ε0 if necessary, by Lemma 2.1 there is δ′ > 0 such that
Thus, χ(u) = 0 for any u ∈ ℬδ′ (K). From Lemma 3.3, we know that χ(·)𝒜(·) is locally Lipschitz continuous on Er, where 𝒜(u) = u − A(u). We study the following initial value problem
which deduces that η(t, u) is well-defined and continuous on ℝ+ × Er.
Let
This says that for any t ∈ [0, 1], σ(t, u) ∉ ℬ2δ (Kc). Lemma 3.3 tells us that
so that ℐr(σ(t, u)) is not increased for t ≥ 0, and thus
This shows that for any t ∈ [0, 1] we have
Therefore, from Lemma 3.3, we have
which is a contradiction suggesting that (b) holds.
If Kc \ W = ∅, we can see that ℬ3δ (Kc) ⊂ W. According to the proof of (b), we can derive (c). Finally, (e) is the result of Lemma 3.3 (see [5, 21, 22]). This proves the lemma.
Proposition 3.5
Assume that (V1), (V2) and (f1)–(f4) hold. Then problem (3.1) possesses infinitely many sign-changing solutions.
Proof
Define
where k ≥ 2, γ(P) denotes the Krasnoselskir's genus of P (see [32]) and
Define
We claim that 0 < a ≤ ck < +1. In fact, Gm ≠ ∅ since id ∈ Gm, m ∈ ℕ. Thus, ck < +∞. Now, we seek to prove 0 < a ≤ ck. From [29, Proposition 9.18], Ψk satisfies:
(1) Ψk ≠ ∅ and Ψk+1 ⊂ Ψk for all k ≥ 2,
(2) Let ϕ ∈ C(Er, Er) be an odd function and ϕ(u) = u, u ∈ ∂BRm ∩ Hm, then ϕ( ) ∈ Ψk if Ψ ∈ Ψk for every k ≥ 2,
(3) if Ψ ∈ Ψk, X = −X is an open set and γ(X) ≤ s < k with k − s ≥ 2, then Ψ\X ∈ Ψk−s.
Define the attracting domain of 0 in Er as
where τ is defined in (3.5). Since 0 is a local minimum of ℐr, combining with the continuous dependence of ODE on initial data, we have seen that 𝒪 is open. In fact,
We choose Ψ ∈ Ψk such that
Define
So that 𝒟 is a bounded open symmetric set with 0 ∈ 𝒟 and
According to [32, Proposition 5.4], one can see that
Since
so that
As a result,
Thus, ck ≥ α > 0.
Next, we will show that
Then
Combining with the definition of W, it follows that σ(1, W) \ W = ∅ from (e) of Lemma 3.4. Then, by (3.8), one has see that
which is contrary to the definition of ck, since σ(1, Ψ) ∈ Ψk. So
We also note that ck+1 ≥ ck for any k ≥ 2 because of the property (1) of Ψk. Finally, we’re arguing to prove that ℐr has infinitely many sign-changing critical points by proving that ck → ∞ as k → ∞. By using reduction to absurdity, we assume that ck → c < ∞ as k → ∞. As a result of Lemma 2.2, one can see that Kc is compact and nonempty. In addition, we claim that
Let {uk} be a sequence of sign-changing solutions of problem (3.1) such that ℐr (uk) → c as k → ∞. Since Lemma 2.2, up to a subsequence, there exists
which yields that ‖(uk)±‖ ≥ δ0 > 0, where δ0 is a constant independent of k. Consequently, u is still a sign-changing solution and then
Assume that
Because ck → c as k → ∞, one may choose k large enough to get
In fact, ck+n ≥ ck for any k ≥ 2. The definition of ck+n brings us that there exists Ψ ∈ Ψk+n, i.e.,
where ψ ∈ Gm, m ≥ k + n,
Therefore,
Thus
Notice that
where the fact σ(1, W))\W = ∅ also plays a role. Hence,
4 Proof of Theorem 1.1
Set
Clearly, Eo is a nonempty complete subspace of E. Also because Eo ↪ L2 (ℝ2) and L2 (ℝ2) is separable, Eo admits a countable orthogonal basis
Similarly, let problem (1.2) constrain on Eo, namely,
Proposition 4.1
Assume that (V1), (V2) and (f1)–(f4) hold. Then problem (4.1) has infinitely many solutions in Eo.
Proof
For k ≥ 2, set
and
Define
where ℐo stands for ℐ|Eo. Certainly,
Lemma 2.1 gives us that ℐo satisfies the (PS) condition. Therefore, un,k → uk in Eo, uk is one nontrivial critical point of ℐo and
As in [29, Proposition 9.33], one can prove that
Proof of Theorem 1.1
Recall Z2 = {−1, 1}. Define the action of a topological group Z2 on a normed space E as
Hence, the action is a continuous map such that, for any g, h ∈ Z2, u ∈ E, u ↦ hu is linear and (hg)u = h(gu). Thus, ‖hu‖ = ‖u‖ for any h ∈ Z2, which implies that the action is isometric. The space of invariant points is defined by
If N = 2k, k ∈ ℤ+, then ℐ(−1 ○ u) = ℐ(u), so that ℐ is invariant. In fact, Fix(Z2) = Eo, thus, by principle of symmetric criticality [27, 38], we conclude that if u is a critical point of ℐ restricted to Eo then u is also a critical point of ℐ. Due to the properties of Eo one can see that u is non-radial sign-changing for any u ∈ Eo \ {0}. So problem (1.2) admits infinitely many non-radial sign-changing solutions from proposition 4.1.
Let ℋ be a subgroup of O(N) ([20, 38]) and the action of E on E is defined by
Notice that the subspace of invariant functions is
We also note that ℐ is invariant since (V1) and (f1). Then, also by principle of symmetric criticality, we conclude that if u is a critical point of ℐ restricted to Er then u is also a critical point of ℐ. So problem (1.2) has infinitely many radial sign-changing solutions from proposition 3.5. This completes the proof of Theorem 1.1.
Acknowledgments
The author states no conflict of interest, and would like to express their heartfelt thanks to the anonymous referees for his/her valuable suggestions and comments.
This work was supported by the special (special post) scientific research fund of natural science of Guizhou University (No. (2021)43) and National Natural Science Foundation of China (No.11971393).
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