Infinitely many sign-changing solutions for a Schrö dinger equation

Abstract

We study a superlinear Schrö dinger equation in the whole Euclidean space ℝⁿ. By using a suitable sign-changing critical point, we prove that the problem admits infinitely many sign-changing solutions, under weaker conditions.

1 Introduction

In this paper, we consider the following Schrö dinger equation,

$$\left\{\begin{array}{c}-\Delta u+V\left(x\right)u=f\left(x,u\right),x\in {ℝ}^N\\ u\left(x\right)\to 0,\left|x\right|\to \infty .\end{array}\right.$$

(1.1)

In order to overcome the lack of compactness of the problem, we assume that the potential V (x) has a "good" behavior at infinity, in such a way the Schrö dinger operator - Δ + V (x) on L ²(ℝ^N) has a discrete spectrum. More precisely, we suppose $\left(V_1\right)V\in L_{loc}^2\left({ℝ}^N\right)$, V is bounded from below;

(V ₂) There exists r ₀ > 0 such that for any h > 0

$$\text{meas}\left(B_{r_0}\left(y\right)\cap V_h\right)\to 0,\left|y\right|\to +\infty ,$$

where meas(A) denotes the Lebesgue measure of A on ℝ^N, $B_{r_0}\left(y\right)$ is the ball centered at y with radius r ₀ and V ^h = {x ∈ ℝ^N : V (x) < h}.

Of course, V (x) above can satisfy the condition (S ₁) or $\left(\left({\bar{S}}_1\right),\left({\tilde{S}}_1\right)\right)$ in [1], so that the Schrö dinger operator could have the same good properties.

We denote {λ _j } to be the eigenvalues sequence of - Δ+V (x) (see Proposition 2.1 in Section 2). Set $F\left(x,t\right)={\int }_0^{t}f\left(x,s\right)ds,\mathcal{F}\left(x,t\right)=f\left(x,t\right)t-2F\left(x,t\right)$.

We assume the following conditions.

(f ₁) f : ℝ^N × ℝ → ℝ is a Carathé odory function with a subcritical growth,

$$\left|f\left(x,t\right)\right|\le c\left(1+|t{|}^{s-1}\right),t\in ℝ,x\in {ℝ}^N,$$

where s ∈ (2, 2*), f(x, t) ≥ 0 for all (x, t) ∈ ℝ^N × ℝ and f(x, t) = o(|t|) as |t| → 0.

(f ₂) $\underset{\left|t\right|\to +\infty }{lim}\frac{f\left(x,t\right)t}{|t{|}^2}=+\infty$ uniformly for x ∈ ℝ^N.

(f ₃) There exist θ ≥ 1, s ∈ [0, 1] s.t.

$$\theta \mathcal{F}\left(x,t\right)\ge \mathcal{F}\left(x,st\right),\left(x,t\right)\in {ℝ}^N\times ℝ.$$

(1.2)

(f ₄) f (x, t) is odd in t.

Let us point out that, under our assumptions on f(x, t), we can assume without loss of generality that V is strictly positive just replacing V (x) with V (x) + L and f(x, u) with f(x, u) + Lu, L large enough. We shall prove the following result.

Theorem 1.1 Under assumptions (V ₁), (V ₂), (f ₁) - (f ₄), problem (1.1) has infinitely many sign-changing solutions.

Remark 1.1 In [2, 3], they got sign-changing solutions for elliptic problem with Dirichlet boundary value. Those abstract results involved a Banach space of continuous functions in the Hilbert space, where the cone has a nonempty interior. This plays a crucial role. While the abstract theory in this paper only involved a Hilbert space, where the cone has an empty interior.

Remark 1.2 In [4], they showed infinitely many solutions for p-Laplace equation with Dirichlet boundary value, while we get infinitely many sign-changing solutions under similar conditions.

Remark 1.3 Equation 1.1 has been studied in [5], where they obtained the existence for sign-changing solutions in a asymptotically case.

Remark 1.4 In [1, §5.3], they also obtained infinitely many sign-changing solutions for elliptic problem with Dirichlet boundary value, under (AR) condition stronger than (f ₂) and (f ₃) above.

Remark 1.5 In [1, §6.4], Equation 1.1 has been studied the existence for infinitely many sign-changing solutions under conditions stronger than ours above.

2 Preliminaries

We consider the Hilbert space

$$E=\left\{u\in H^1\left({ℝ}^N\right):{\int }_{{ℝ}^N}\left(\mid ▿u{\mid }^2+V\left(x\right)u^2\right)dx<\infty \right\}$$

endowed with the inner product $\left(u,v\right)={\int }_{{ℝ}^N}\left(▿u▿v+V\left(x\right)uv\right)dx$ for u, v ∈ E and norm $\parallel u\parallel ={\left(u,u\right)}^{\frac{1}{2}}$. Clearly it is E ≲ H ¹(ℝ^N). Denote |u| _q to be the norm of u in L ^q(ℝ^N). In order to overcome the lack of compactness of the problem, the following proposition is crucial.

Proposition 2.1 [1, 5] Assume V (x) satisfies condition (V ₁) and (V ₂), or (S ₁) or $\left({\bar{S}}_1\right)$ and $\left({\tilde{S}}_1\right)$ in [1]. Then the imbedding E ≲ L ^q(ℝ^N) is continuous if q ∈ [2, 2*] and compact if q ∈ [2, 2*[. Hence, the eigenvalue problem

$$-\Delta u+V\left(x\right)u=\lambda u,x\in {ℝ}^N$$

possesses a sequence of positive eigenvalue

$$0<{\lambda }_1<{\lambda }_3<\cdots <{\lambda }_k<\cdots \to \infty$$

with finite multiplicity for each λ _k . Moreover, the principle eigenvalue λ ₁ is simple with a positive eigenfunction φ ₁, and the eigenfunctions φ _k corresponding to λ _k , k ≥ 2 are sign changing.

Let us consider the functional J : E → ℝ

$$J\left(u\right)=\frac{1}{2}{∥u∥}^2-{\int }_{{ℝ}^N}F\left(x,u\right)dx.$$

(2.1)

Then J ∈ C ¹(E, ℝ) and J' = id (-Δ + V )^-1 f = id - K _J . The critical point of J is just the weak solution of problem (1.1).

The proof if our main results will be obtained by a suitable applications of an abstract critical point theorem stated in [1]. For completeness, we recall here this theorem.

Let E be Hilbert space with norm ||u||, and Y, M be two subspaces of E with dim Y < ∞, dim Y - co dim M ≥ 1. Let G be C ¹ - functional on E with G'(u) = u - K _G (u) and P denote a closed convex positive cone of E. Denote ±D ₀ by open convex subsets of E, containing the positive cone P in its interior and K = {u ∈ E : G'(u) = 0}, K[a, b] = {u ∈ K : G(u) ∈ [a, b]}. Set D = D ₀ ∪ (-D ₀), S = E \ D. In applications, D contains all positive and negative critical points, and S includes all possible sign-changing critical points. Hence, nontrivial sign-changing solutions can be obtained by different choose of ±D ₀ and S.

Next, we assume that there is another norm || · ||_* of E such that ||u||_* ≤ c _*||u|| for all u ∈ E, where c _* > 0 is a constant. Moreover, we assume that ||u _n - u||_* → 0 whenever u _n ⇀ u weakly in (E, || · ||). Write E = M ₁ ⊕ M.

Let

$$Q^{*}\left(\rho \right)=\left\{u\in M:\frac{{∥u∥}_{*}^p}{{∥u∥}^2}+\frac{∥u∥{∥u∥}_{*}}{∥u∥+D_{*}{∥u∥}_{*}}=\rho \right\}$$

where ρ > 0, D _* > 0, p > 2 are fixed constants. Let Q** = Q*(ρ) ∩ G ^β ⊂ S and $\gamma =\underset{Q**}{inf}G$, where G ^β = {u ∈ E : G(u) ≤ β}, then β ≥ γ.

Let us assume that

(A) K _G (±D ₀) ⊂ ±D ₀;

$\left(A_1^{*}\right)$ Assume that for any a, b > 0, there is a c ₂ = c ₂(a, b) > 0 such that G(u) ≤ a and ||u||_* ≤ b ⇒ ||u|| ≤ c ₂;

$\left(A_2^{*}\right)\underset{u\in Y,∥u∥\to \infty }{lim}=-\infty ,\underset{Y}{sup}G=\beta$.

In the sequel, we shall consider the following Palais-Smale condition, shortly (w* - PS) condition.

Definition 2.1 The functional G is said to satisfy the (w* - PS) condition if any sequence {u _n } such that {G(u _n )} is bounded and G'(u _n ) → 0, we have either {u _n } is bounded and has a convergent subsequence or ∃σ, R, β > 0 s.t. for any u ∈ J ^-1([c - σ, c + σ]) with ||u|| ≥ R, ||J'(u)|| ||u|| ≥ β. If in particular, {G(u _n )} → c, we say that (w*- PS) _c is satisfied.

The following results hold (see [1, Theorem 5.6]).

Theorem 2.1 Assume (A) and $\left(A_1^{*}\right)$ and $\left(A_2^{*}\right)$. If the even functional G satisfies the (w* - PS) _c condition at lever c for each c ∈ [r, β], then

$$K\left[r-\epsilon ,\beta +\epsilon \right]\cap \left(E\backslash P\cup \left(-P\right)\right)\ne \varnothing$$

for all ε > 0 small.

3 Proof of the main theorems

From now on, we will denote by N _k the eigenspace of λ _k . Then dim N _k < ∞. We fix k and let E _k = N ₁ ⊕ ⋯ ⊕ N _k . In order to give the proof of Theorem 1.1, first we state some useful lemmas.

Lemma 3.1 J(u) → -∞, as ||u|| → ∞, for all u ∈ E _k .

Proof. Because dim E _k < ∞, all norms in it are equivalent, then by (f ₂),

$$\frac{J\left(u\right)}{\parallel u|{|}^2}\le \frac{1}{2}-{\int }_{{ℝ}^N}\frac{F\left(x,u\right)}{\parallel u|{|}^2}dx\to -\infty .$$

Consider another norm ||·|| _* := ||·|| _s of E, s ∈ (2, 2*). Then ||u|| _s ≤ C _* ||u|| for all u ∈ E, here C _* > 0 is a constant and by lemma 2.1 ||u _n - u|| _* → 0 whenever u _n ⇀ u weakly in E. Write $E=E_{k-1}\oplus E_{k-1}^{\perp }$. Let

$$Q^{*}\left(\rho \right)=\left\{u\in E_{k-1}^{\perp }:\frac{\parallel u|{|}_s^s}{\parallel u|{|}^2}+\frac{\parallel u\parallel \parallel u|{|}_s}{\parallel u\parallel +D_{*}\parallel u|{|}_s}=\rho \right\}$$

where ρ, D _* are fixed constants.

Lemma 3.2 ||u|| _s ≤ c ₁, ∀u ∈ Q*(ρ), where c ₁ > 0 is a constant.

Proof. If ||u|| _s → ∞, then so does ||u|| → ∞. Hence

$$\frac{\parallel u\parallel \parallel u|{|}_s}{\parallel u\parallel +D_{*}\parallel u|{|}_s}\to \infty ,$$

a contradiction.

By (f ₁), there exist C _F > 0, s ∈ (2, 2*) such that

$$\left|F\left(x,u\right)\right|\le \frac{{\lambda }_1}{4}{u}^2+C_F|u{|}^s,x\in {ℝ}^N,u\in ℝ.$$

(3.1)

Therefore, for any a, b > 0, there is a c ₂ = c ₂(a, b) > 0 such that

$$J\left(u\right)\le a,\parallel u|{|}_s\le b\Rightarrow \parallel u\parallel \le c_2.$$

By lemma 3.1,

$$\underset{u\in Y,\parallel u\parallel \to \infty }{lim}J\left(u\right)=-\infty ,$$

where Y = E _k . Then, conditions $\left(A_1^{*}\right)$ and $\left(A_2^{*}\right)$ are satisfied. We define

$$\underset{Y}{sup}G:=\beta .$$

Let

$$Q^{**}:=Q^{*}\left(\rho \right)\cap J^{\beta }\subset S,\underset{Q^{**}}{inf}J:=\gamma .$$

Set P = {u ∈ E : u(x) ≥ 0 for a.e. x ∈ ℝ^N}. Then, P(-P) is the positive (negative) cone of E and weakly closed. By Lemma 5.4 or Lemma 6.8 [1], there is a δ := δ(β) such that dist(Q**, P) = δ(β) > 0. We define

$$D\left({\mu }_0\right):=\left\{u\in E:dist\left(u,P\right)<{\mu }_0\right\},$$

where μ ₀ us determined by the following lemma.

Lemma 3.3 Under the assumptions (V ₁), (V ₂), and (f ₁), there is a μ ₀ ∈ (0, δ) (may be chosen small enough) such that K _J (±D(μ ₀)) ⊂ ±D(μ ₀). Therefore, (A) is satisfied.

Proof. Please see Lemma 2.9 of [1] for the similar proof.

Let D := -D(μ ₀) ∪ D(μ ₀), S := E \ D. By Lemma 3.3, we may assume Q** ⊂ S.

Lemma 3.4 Let us assume that (V ₁), (V ₂) and (f ₂), (f ₃) hold. Then, the functional J satisfies the (w*-PS) condition.

Proof. As the sequence {u _n } such that {G(u _n )} is bounded and G'(u _n ) → 0, if {u _n } is bounded, then by Proposition 2.1 and the compact imbedding E ≲ L ^q(ℝ^N), q ∈ [2, 2*[, we have {u _n } possesses a convergent subsequence.

Next to prove another case. If not, there exist c ∈ ℝ and {u _n } ⊂ E satisfying, as n → ∞

$$J\left(u_n\right)\to c,\parallel u_n\parallel \to \infty ,\parallel J^{\prime }\left(u_n\right)\parallel \parallel u_n\parallel \to 0$$

(3.2)

then we have

$$\begin{array}{c}\underset{n\to \infty }{lim}{\int }_{{ℝ}^N}\left(\frac{1}{2}f\left(x,u_n\right)u_n-F\left(x,u_n\right)\right)dx\\ =\underset{n\to \infty }{lim}\left(J\left(u_n\right)-\frac{1}{2}<J^{\prime }\left(u_n\right),u_n>\right)=c.\end{array}$$

(3.3)

Denote $v_n=\frac{u_n}{\parallel u_n\parallel }$, then ||v _n || = 1, that is {v _n } is bounded in E. Thus, up to a subsequence, for some v ∈ E, we get

$$\begin{array}{c}{v}_n\rightharpoonup vinE,\\ v_n\to vin{L}^p\left({ℝ}^N\right),for2\le p<2^{*},\\ v_n\left(x\right)\to v\left(x\right)a.e.x\in {ℝ}^N.\end{array}$$

(3.4)

If v ≢ 0, because ||J'(u _n )|| ||u _n || → 0, as the similar proof in Lemma 6.22 of [2] or Lemma 2.2 of [4], we get a contradiction.

If v = 0, by condition (f ₃), as the similar proof in Lemma 6.22 of [2] or Lemma 2.2 of [4], we also have

$${\int }_{{ℝ}^N}\left(\frac{1}{2}f\left(x,u_n\right)u_n-F\left(x,u_n\right)\right)dx\to \infty ,$$

(3.5)

which contradicts (3.3).

This proves that J satisfies the (w*-PS) condition.

Remark 3.1 Our condition (f ₃) here is different from (P ₃) of [1, Theorem 6.14 ], which is used to prove the (w*-PS) condition; furthermore, it is more weaker.

Proof of Theorem 1.1. By Theorem 2.1,

$$K\left[r-\epsilon ,\beta +\epsilon \right]\cap \left(E\backslash P\cup \left(-P\right)\right)\ne \varnothing$$

for all ε > 0 small. That is there exists a u _k ∈ E \ (- P ∪ P) (sign-changing critical point) such that

$$J^{\prime }\left(u_k\right)=0,J\left(u_k\right)\in \left[r-1,\beta +1\right].$$

Next, we estimate the $\gamma =\underset{Q^{**}}{inf}J$. Because of Proposition 2.1, we can adopt the similar method as in [1, p. 67]. Similar to Lemma 2.23 of [1], by choosing the constants D _* and ρ, for all u ∈ Q*(ρ), we may get

$$\left|\right|u\left|\right|\ge {\Lambda }_s^{*}\min \{{\lambda }_k^{(1-\alpha )(s-2)/2},{\lambda }_k^{(1-\alpha )/2}\}\min \{\rho ,{\rho }^{1/(s-2)}\}.$$

By Lemma 2.26 of [1], for any u ∈ Q*(ρ), we have that

$$J\left(u\right)\ge \frac{1}{8}{\left({\Lambda }_s^{*}\right)}^2{T}_1{T}_2,$$

where ${\Lambda }_s^{*}$, T ₁, T ₂ are defined in (2.49)-(2.51) in [1] with p replaced by s ∈ (2, 2*), α ∈ (0, 1) is a constant, and ${\Lambda }_s^{*}$, T ₂ are independent of k. In particular, since λ _k → ∞, we get

$$T_1:=\min \{{\lambda }_k^{(1-\alpha )(s-2)/2},{\lambda }_k^{(1-\alpha )/2}\}\to \infty ,ask\to \infty .$$

Therefore, γ → ∞ as k → ∞; hence the proof of Theorem 1.1 is finished.