The main aim of this paper is to find sign-changing solutions of (1.1) whenf is asymptotically linear

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ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu

SIGN-CHANGING SOLUTIONS FOR ASYMPTOTICALLY LINEAR SCHR ¨ODINGER EQUATION IN BOUNDED DOMAINS

SITONG CHEN, YINBIN LI, XIANHUA TANG

Abstract. In this article we study the Schr¨odinger equation

−∆u=f(x, u), x∈Ω, u∈H₀¹(Ω),

where Ω is a bounded domain inR^N andf(x, u) is asymptotically linear at infinity with respect to u. Inspired by the works of Salvatore [14] on sign- changing solutions, in whichf(x, u) is asymptotically linear at zero with respect tou, we prove, via the constraint variational method and the quantitative deformation lemma, that the equation possesses one sign-changing solution with exactly two nodal domains.

1. Introduction and statement of main results In this article, we consider the Schr¨odinger equation

−∆u=f(x, u), x∈Ω,

u∈H₀¹(Ω), (1.1)

where Ω is a bounded domain in R^N andf : Ω×R→Ris continuous. The main aim of this paper is to find sign-changing solutions of (1.1) whenf is asymptotically linear. Precisely, we assume thatf satisfies the following assumptions:

(A1) f ∈C(Ω×R), F(x, t) :=Rt

0f(x, s)ds≥0 and f(x, t) =o(|t|) as|t| →0, uniformly inx∈Ω;

(A2) f(x, t) =V_∞(x)t+f1(x, t),V_∞∈C(Ω), andf1(x, t) =o(|t|) as |t| →+∞, uniformly inx∈Ω;

(A3) t7→f(x, t)/|t|is strictly increasing on (−∞,0)∪(0,∞) for everyx∈Ω;

(A4) Fe(x, t) := ¹₂f(x, t)t−F(x, t)→+∞ast→+∞uniformly inx∈Ω.

The nonlinear Schr¨odinger equation is of interest in many branches of physics.

As we know, the solutions of problems like (1.1) are related to the existence of standing wave solutions for nonlinear Schr¨odinger equation like

i~∂Ψ

∂t =−~²4Ψ +V(x)Ψ−f(x,Ψ) for allx∈Ω, (1.2) where Ω is a domain inR^N, ~>0 and Ψ is the amplitude of the wave. Equation (1.2) is one of the main objects of quantum physics, for it appears in problems

2010Mathematics Subject Classification. 35J10, 35J20.

Key words and phrases. Schr¨odinger equation; sign-changing solutions; asymptotically linear.

c

2016 Texas State University.

Submitted June 22, 2016. Published December 14, 2016.

1

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involving nonlinear optics, plasma physics and condensed matter physics, see An- derson and Bonnedal [1], Chen [6], Chiao et al [8], Gatz and Herrmann [9], Karlsson [10], Sodha et al [17], Stuart [19] and the references therein.

In recent years, problems like (1.1) have been widely studied under variant assumptions on f, and the existence of positive solutions, ground state solutions, multiple solutions and semiclassical states were obtained in many papers, see for example [3, 7, 12, 13, 20, 21, 22, 25, 26] and the references therein. When f is superlinear at infinity in u, the existence of sign-changing solutions of (1.1) was established by Bartsch, Liu and Weth in [2]. For more discussions on the existence of sign-changing solutions of (1.1), in this case, we refer the readers to [4, 5, 11, 27]

and the references therein. Whenf is asymptotically linear at zero inu, that is,f satisfies the condition:

µ1<lim inf

t→0

f(x, t)

t ≤lim sup

t→0

f(x, t)

t < µk uniformly forx∈Ω, (1.3) where{µ_j}is the sequence of eigenvalues of the Schr¨odinger operator−∆+V(x) and V is a linear potential, Salvatore [14] proved the existence of sign-changing solutions.

Note that conditions (A1) and (1.3) are quite different and were considered in different situations. To the best of our knowledge, there are no works concerning the least energy sign-changing solutions for Problem (1.1) with asymptotically linear case at infinity, and it is an interesting problem.

Let H¹(Ω) be the usual Sobolev space with the standard scalar product and norm

(u, v) = Z

Ω

(∇u∇v+uv)dx, kuk²= Z

Ω

|∇u|²+u² dx.

Define the energy functional Φ :H₀¹(Ω)→Rby Φ(u) = 1

2 Z

Ω

|∇u|²dx− Z

Ω

F(x, u)dx. (1.4)

Conditions (A1) and (A2) imply that Φ is a well-defined of classC¹functional, and that

hΦ⁰(u), ϕi= Z

Ω

∇u∇ϕdx− Z

Ω

f(x, u)ϕdx, ∀u, ϕ∈H₀¹(Ω). (1.5) Clearly, critical points of Φ are the weak solutions of (1.1). Furthermore, if u∈H₀¹(Ω) is a solution of (1.1) andu^± 6= 0, thenuis a sign-changing solution of (1.1), where

u⁺(x) := max{u(x),0}, u⁻(x) := min{u(x),0}.

Using (1.4) and (1.5), it is obvious that

Φ(u) = Φ(u⁺) + Φ(u⁻),

hΦ⁰(u), u⁺i=hΦ⁰(u⁺), u⁺i, hΦ⁰(u), u⁻i=hΦ⁰(u⁻), u⁻i.

To obtain a sign-changing solution of (1.1), we first seek a minimizer of the energy functional Φ under the constraint

M={u∈H₀¹(Ω) :u^± 6= 0, hΦ⁰(u), u⁺i=hΦ⁰(u), u⁻i= 0}, then show that the minimizer is a sign-changing solution of (1.1).

To state our results, we make the following assumption:

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(A5) infx∈ΩV∞(x)> µ:= infu∈Πmax{k∇u⁺k²₂, k∇u⁻k²₂}, where Π :={u∈H₀¹(Ω) :u^±6= 0,

Z

Ω

|u^±|²dx= 1}.

Letλ1be the first eigenvalue of −∆, then for anyu∈H₀¹(Ω) andu6= 0, λ1≤ k∇uk²₂

kuk²₂

= k∇u⁺k²₂+k∇u⁻k²₂ ku⁺k²₂+ku⁻k²₂

≤ 2

ku⁺k²₂+ku⁻k²₂max{k∇u⁺k²₂,k∇u⁻k²₂}.

By the definition ofµand (A5), one hasλ₁≤µ <+∞.

Remark 1.1. Using (A1) and (A2), it is obvious that for any ε >0, there exists C_ε>0 such that

|f(x, t)| ≤ε|t|+Cε|t|^p−1 and |F(x, t)| ≤ε|t|²+Cε|t|^p (1.6) for all (x, t)∈Ω×R, where 2< p <2^∗= _N−2^2N . Furthermore, (A1) and (A3) imply

1

2f(x, t)t > F(x, t)>0, ∀t6= 0, x∈Ω. (1.7) It follows from (A1)–(A3) and (A5) that

f1(x, t)

|t| → −V∞(x)<0 as|t| →0, t7→ f1(x, t)

|t| is negative, strictly increasing on (−∞,0)∪(0,∞), which, together withf1(x, t) =o(|t|) as|t| → ∞uniform inx, yields

tf1(x, t)<0, ∀t6= 0. (1.8) Theorem 1.2. Assume (A1)–(A5) are satisfied. Then (1.1) has a sign-changing solution u ∈ M such that Φ(u) = inf_MΦ > 0, which has precisely two nodal domains.

Now, we give an example to illustrate the feasibility of assumptions (A1)–(A5).

Let

F(x, t) =V_∞(x)

2 t² 1− 1 1 +|t|^α

,∀x∈Ω, t∈R,

whereα∈(0,2),V_∞∈C(Ω), infΩV_∞> µ. By elementary computations, it is easy to check thatf satisfies (A1)–(A5).

The main tools this article are the minimization argument and the quantitative deformation lemma. We must point out that the difficulty in proving Theorem 1.2 is to show thatM 6=∅and the minimizer is a critical point of Φ.

This article organized as follows. In Section 2, we prove several preliminary lemmas. The proof of Theorem 1.2 will be given in the last section.

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2. Preliminaries

In this section, we prove that the minimizer of the energy functional Φ under the constraint Mis a critical point. To this end, we showM 6=∅ with the aid of an important behavior of strictly increasing functions.

Lemma 2.1([20, Lemma 2.3]). Suppose thath(x, t)is strictly increasing in t∈R andh(x,0) = 0for anyx∈R^N. Then

1−θ²

2 h(x, τ)τ|τ|>

Z τ

θτ

h(x, s)|s|ds, ∀θ∈[0,1)∪(1,∞), τ ∈R\{0}.

Lemma 2.2. Suppose that (A1)–(A3) are satisfied. Then for any u=u⁺+u⁻∈ H₀¹(Ω) with u^±6= 0,s,t≥0 and(s−1)²+ (t−1)²6= 0,

Φ(u)>Φ(su⁺+tu⁻) +1−s²

2 hΦ⁰(u), u⁺i+1−t²

2 hΦ⁰(u), u⁻i. (2.1) Proof. For anyx∈Ω, from (A3) and Lemma 2.1 it follows that

1−θ²

2 f(x, τ)τ >

Z τ

θτ

f(x, ξ)dξ, ∀θ∈[0,1)∪(1,∞), τ ∈R\{0}. (2.2) By (1.4), (1.5) and (2.2), for anyu=u⁺+u⁻∈H₀¹(Ω) withu^±6= 0,s,t≥0 and (s−1)²+ (t−1)²6= 0, we have

Φ(u)−Φ(su⁺+tu⁻)

= 1 2 Z

Ω

|∇u|²dx− Z

Ω

F(x, u)dx+ Z

Ω

F(x, su⁺+tu⁻)dx−1 2

Z

Ω

|∇(su⁺+tu⁻)|²dx

= 1−s²

2 hΦ⁰(u), u⁺i+1−t²

2 hΦ⁰(u), u⁻i+ Z

Ω

h1−t²

2 f(x, u⁻)u⁻

− Z u⁻

tu⁻

f(x, ξ)dξi dx+

Z

Ω

h1−s²

2 f(x, u⁺)u⁺− Z u⁺

su⁺

f(x, ξ)dξi dx

> 1−s²

2 hΦ⁰(u), u⁺i+1−t²

2 hΦ⁰(u), u⁻i.

This shows that (2.1) holds.

From Lemma 2.2, we have the following two corollaries.

Corollary 2.3. Suppose that(A1)–(A3)are satisfied. Then for anyu=u⁺+u⁻∈ M,

Φ(u)≥Φ(su⁺+tu⁻), ∀s, t≥0.

Corollary 2.4. Suppose that(A1)–(A3)are satisfied. Then for anyu=u⁺+u⁻∈ M,

Φ(u⁺+u⁻) = max

s,t≥0Φ(su⁺+tu⁻).

Define the set E0=

u∈H₀¹(Ω) :k∇u^±k²₂− Z

Ω

V_∞(x)|u^±|²dx <0 . (2.3) Lemma 2.5. Suppose that (A1)–(A3), (A5)are satisfied. Then E₀6=∅ andM ⊂ E₀.

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Proof. In view of (A5), the definition ofµimplies that there existsv∈Π such that max{k∇v⁺k²₂,k∇v⁻k²₂} ≤µ+infΩV_∞−µ

2 = infΩV_∞+µ

2 .

It follows that k∇v^±k²₂−

Z

Ω

V_∞(x)|v^±|²dx≤max{k∇v⁺k²₂,k∇v⁻k²₂} −inf

Ω V_∞

≤ µ−inf_ΩV_∞ 2 <0.

Hence, we have v ∈ E0. This shows that E0 6=∅ because of (A5). Moreover, by (1.5) and (1.8), we can easily derive that for anyu∈ M,

k∇u^±k²₂− Z

Ω

V_∞(x)|u^±|²dx= Z

Ω

f₁(x, u^±)u^±dx <0.

This shows thatM ⊂E₀.

Lemma 2.6. Suppose that (A1)–(A3), (A5) are satisfied. If u ∈ E0, then there exists a unique pair (su, tu)of positive numbers such that suu⁺+tuu⁻∈ M.

Proof. Let

g₁(s) =s² Z

Ω

|∇u⁺|²dx− Z

Ω

f(x, su⁺)su⁺dx, (2.4) g₂(t) =t²

Z

Ω

|∇u⁻|²dx− Z

Ω

f(x, tu⁻)tu⁻dx. (2.5) Clearly, g₁(0) = g₂(0) = 0. Using (A1), (A2), (1.6) and (2.3), we conclude that g1(s)>0 for s >0 small, and

g₁(s) =s² Z

Ω

|∇u⁺|²dx− Z

Ω

f(x, su⁺)su⁺dx

=s² Z

Ω

[|∇u⁺|²−V_∞(x)|u⁺|²]dx− Z

Ω

f1(x, su⁺)

su⁺ (su⁺)²dx <0 for slarge. From the continuity of g1(·), there is a su >0 such that g1(su) = 0.

Using (A3), it is easy to verify thatsuis unique. Then it follows from (1.5) and (2.4) that hΦ⁰(suu⁺), u⁺i= 0. Similarly, there is a unique tu >0 such that g2(tu) = 0,

and sohΦ⁰(tuu⁻), u⁻i= 0.

Lemma 2.7. Suppose that (A1)–(A3), (A5)satisfied. Then

u∈Minf Φ(u) =m= inf

u∈E0

s,t≥0maxΦ(su⁺+tu⁻).

Combining Corollary 2.4, Lemmas 2.5 and 2.6, we obtain the proof of the above lemma.

Lemma 2.8. Suppose that (A1)–(A5)are satisfied. Thenm >0 is achieved.

Proof. Let {un} ⊂ M be such that Φ(un) → m. Next, we prove that {un} is bounded in H₀¹(Ω). Arguing by contradiction, suppose that kunk → ∞. Let v_n = u_n/kunk, then kvnk = 1. By Sobolev imbedding theorem, passing to a subsequence, we may assume that there existsv∈H₀¹(Ω) such thatv_n* v weakly

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in H₀¹(Ω),vn →v strongly inL^s(Ω), 2≤s <2^∗. Ifv= 0, then vn →0 in L^s(Ω), 2≤s <2^∗. FixR >[2(1 +m)]^1/2, by (1.6), one has

lim sup

n→∞

Z

Ω

F(x, Rvn)dx≤R²ε lim

n→∞kvnk²₂+R^pCε lim

n→∞kvnk^p_p= 0. (2.6) Lettn=R/kunk. Then by (2.6) and Corollary 2.3, one has

m= Φ(u_n) +o(1)≥Φ(t_nu_n) +o(1)

=t²_n

2 kunk²− Z

Ω

F(x, tnun)dx+o(1)

=R² 2 −

Z

Ω

F(x, Rv_n)dx+o(1)

=R²

2 +o(1)> m+ 1 +o(1), which is a contradiction. Thusv6= 0.

Forx∈Ω0 :={y ∈Ω : v(y)6= 0}, we have limn→∞|un(x)| =∞. Thus, from (1.4), (1.5), (A3), (A4) and Fatou’s lemma it follows that

m+ 1≥ lim

n→∞

Φ(un)−1

2hΦ⁰(un), uni

≥lim inf

n→∞

Z

Ω₀

Fe(x, un)dx= +∞.

This contradiction shows that{kunk}is bounded. Hence, passing to a subsequence, there existsue∈H₀¹(Ω) such thatu^±_n *ue^±weakly inH₀¹(Ω),u^±_n →ue^± strongly in L^s(Ω), 2≤s <2^∗. Sinceun∈ M, we havehΦ⁰(un), u^±_ni= 0. In view of (1.6) and Sobolev embedding theorem, there existsC1>0 such that

ku^±_nk²= Z

Ω

f(x, u^±_n)u^±_ndx≤1

2ku^±_nk²+C1ku^±_nk²ku^±_nk^p−2_p , which implies

Z

Ω

|u^±_n|^pdx≥( 1 2C1

)^p−2^p .

By the compactness of the embeddingH₀¹(Ω),→L^s(Ω) for 2≤s <2^∗, we obtain Z

Ω

|ue^±|^pdx≥( 1 2C₁)^p−2^p . Thus,ue^±6= 0. Moreover, (A1), (A2) and [24, A.2] imply

n→∞lim Z

Ω

f(x, u^±_n)u^±_ndx= Z

Ω

f(x,eu^±)ue^±dx, (2.7)

n→∞lim Z

Ω

F(x, u^±_n)dx= Z

Ω

F(x,ue^±)dx, (2.8)

n→∞lim Z

Ω

f₁(x, u^±_n)u^±_ndx= Z

Ω

f₁(x,eu^±)ue^±dx. (2.9) From (1.8), (2.9) and the weak semicontinuity of norm, we have

k∇ue^±k²₂− Z

Ω

V_∞(x)|ue^±|²dx≤lim inf

n→∞

k∇u^±_nk²₂− Z

Ω

V_∞(x)|u^±_n|²dx

= lim inf

n→∞

Z

Ω

f1(x, u^±_n)u^±_ndx

= Z

Ω

f1(x,ue^±)eu^±dx <0,

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which shows that eu ∈ E0. By Lemma 2.6, there exist s0 > 0 and t0 > 0 such that s0ue⁺+t0ue⁻ ∈ Mand Φ(s0eu⁺+t0eu⁻)≥ m. By (1.5), (2.7) and the weak semicontinuity of norm, we have

hΦ⁰(u),e ue^±i=kue^±k²− Z

Ω

f(x,ue^±)eu^±dx

≤lim inf

n→∞

ku^±_nk²− Z

Ω

f(x, u^±_n)u^±_ndx = 0.

(2.10)

From (1.4), (1.5), (2.1), (2.7), (2.10), Fatou’s Lemma and Lemma 2.7 it follows that m= lim

n→∞

Φ(u_n)−1

2hΦ⁰(u_n), u_ni

= lim

n→∞

Z

Ω

1

2f(x, u_n)u_n−F(x, u_n) dx

= lim

n→∞

Z

Ω

1

2f(x,eu)eu−F(x,u)e

dx= Φ(u)e −1

2hΦ⁰(eu),eui

≥Φ(s0eu⁺+t0eu⁻) +1−s²₀

2 hΦ⁰(u),e ue⁺i+1−t²₀

2 hΦ⁰(eu),eu⁻i −1

2hΦ⁰(u),e uie

≥m−s²₀

2hΦ⁰(eu),eu⁺i −t²₀

2hΦ⁰(eu),eu⁻i.

This implies thatue∈ Mand Φ(eu) =m.

Lemma 2.9. Suppose that(A1)–(A5)are satisfied. If uˆ∈ MandΦ(ˆu) =m, then ˆ

uis a critical point of Φ.

Proof. Assume that û= û⁺+ û⁻∈ M, Φ(û) =mand Φ⁰(û)6= 0. Then there exist δ >0 andλ >0 such that

kΦ⁰(u)k ≥λ, for allku−uk ≤ˆ 3δ andu∈H₀¹(Ω).

LetD= (1/2,3/2)×(1/2,3/2). It follows from Lemma 2.2 that χ:= max

(s,t)∈∂DΦ(sû⁺+tû⁻)< m. (2.11) Forε:= min{(m−χ)/3, λδ/8},S:=B(û, δ), [24, Lemma 2.3] yields a deforma- tionη∈C([0,1]×H₀¹(Ω)) such that

(i) η(1, u) =uif Φ(u)< m−2εor Φ(u)> m+ 2ε;

(ii) η(1,Φ^m+ε∩B(ˆu, δ))⊂Φ^m−ε; (iii) Φ(η(1, u))≤Φ(u) for allu∈H₀¹(Ω).

We claim that

max

(s,t)∈D

Φ(η(1, sˆu⁺+tˆu⁻))< m. (2.12) Indeed, by Lemma 2.2 and (iii), we have

Φ(η(1, sû⁺+tuˆ⁻))≤Φ(sû⁺+tû⁻)<Φ(û) =m, (2.13) for alls, t≥0,|s−1|²+|t−1|²≥δ²/kûk².

On the other hand, by Corollary 2.4, we have Φ(sû⁺+tû⁻) ≤Φ(û) = m for s, t≥0, then it follows from (ii) that

Φ(η(1, sû⁺+tû⁻))≤m−ε, ∀s, t≥0, |s−1|²+|t−1|²< δ²/kukˆ ². (2.14) Both (2.13) and (2.14) imply that (2.12) holds. Define h(s, t) = sû⁺+tuˆ⁻. We now prove thatη(1, h(D))∩ M 6=∅, contradicting to the definition ofm. We adopt the idea from [16]. Letβ(s, t) :=η(1, h(s, t)) and

Ψ₀(s, t) := Φ⁰(h(s, t))ˆu⁺,Φ⁰(h(s, t))ˆu⁻ ,

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Ψ₁(s, t) := 1

sΦ⁰(β(s, t))(β(s, t))⁺,1

tΦ⁰(β(s, t))(β(s, t))⁻ .

By Lemma 2.6 and degree theory, we can derive that deg(Ψ₀, D,0) = 1. From (2.11) and (i) it follows thatβ=hon∂D. Consequently, deg(Ψ₁, D,0) = deg(Ψ₀, D,0) = 1, and so, Ψ₁(s₀, t₀) = 0 for some (s₀, t₀)∈D, that isη(1, h(s₀, t₀)) =β(s₀, t₀)∈ M, which contradicts (2.12). From this, we conclude that ˆu is a critical point of

Φ.

3. Sign-changing solutions

Proof of Theorem 1.2. In view of Lemmas 2.8 and 2.9, there exists au∈ Msuch that Φ(u) =mand Φ⁰(u) = 0. Now, we show thatuhas exactly two nodal domains.

Setu=u₁+u₂+u₃ andhΦ⁰(u), u_ii= 0 (i= 1,2,3), where u1≥0, u2≤0, Ω1∩Ω2=∅, u3|Ω1∪Ω2 = 0,

Ω1:={x∈Ω :u1(x)>0}, Ω2:={x∈Ω :u2(x)<0}, (3.1) and Ω1, Ω2 are connected open subsets of Ω.

Letv=u1+u2, thenv⁺=u1, v⁻ =u2,v^±6= 0 andhΦ⁰(v), v^±i= 0. By (1.4), (1.5), (1.7), (2.1), (3.1) and Lemma 2.7, we have

m= Φ(u) = Φ(u)−1

2hΦ⁰(u), ui

= Φ(v) + Φ(u₃)−1

2[hΦ⁰(v), vi+hΦ⁰(u₃), u₃i]

≥ sup

s,t≥0

Φ(sv⁺+tv⁻) +1−s²

2 hΦ⁰(v), v⁺i+1−t²

2 hΦ⁰(v), v⁻i + Φ(u3)−1

2[hΦ⁰(v), vi+hΦ⁰(u3), u3]

= sup

s,t≥0

Φ(sv⁺+tv⁻) + Z

Ω

1

2f(x, u₃)u₃−F(x, u₃)

dx≥m,

which shows thatu₃= 0. Therefore,uhas exactly two nodal domains.

Acknowledgements. This work is partially supported by the National Natural Science Foundation of China (No: 11571370). The authors thank the anonymous referees for their valuable suggestions and comments.

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Sitong Chen

School of Mathematics and Statistics, Central South University, Changsha, 410083 Hunan, China

E-mail address:[email protected]

Yinbin Li

Xianhua Tang