Multiple solutions of a Schr¨ odinger type semilinear equation
Xiaochun Liu, Jianfu Yang
Abstract. Two nontrivial solutions are obtained for nonhomogeneous semilinear Schr¨o- dinger equations.
Keywords: Schr¨odinger equation, multiple solutions Classification: 35Q55, 35J20, 35J65
1. Introduction
The main purpose of this work is to investigate the existence of multiple solu- tions of the semilinear Schr¨odinger equation
(1.1) −△u+q(x)u=λu+g(x, u) +f in RN, wheref ∈L2(RN),N ≥3.
Throughout this paper we assume that (A1) q∈L∞(RN) is periodic;
(A2) λis in the spectral gap of the operator (−△+q).
It is well known that the spectrumσ(T) of Schr¨odinger operatorT =−△+q is purely continuous. We denote byE the Sobolev spaceH1(RN). For λ∈G, a spectral gap ofT, we may decomposeE corresponding to the spectral gapGinto E=E+LE−such that the quadratic form
Q(u) = Z
RN
(| ▽u|2+qu2−λu2)dx
associated withT−λI,λ∈G, is positive and negative onE+andE−respectively.
BothE+andE−are infinite dimensional, so the operator−△+q−λis strongly indefinite. There are many existence results for the case f ≡0 and we refer to the papers [BJ], [CY], [PP] and references therein. Such a problem is usually resolved by the Linking theorem ([R]), it only yields one solution in general. The nonhomogeneous termf plays a role that the associated functional of (1.1) is no longer even, so the multiple solutions of (1.1) cannot be obtained in a direct way.
There are obtained in [CZ] and [J] some multiplicity results forq= 0 and λ <0.
In this case, the operatorT−λIis positive definite. Our problem is different and more involved. We assume further that
(G1) g(x, t) isC1−function andg′t(x, t)≥0 onRN ×R,
(G2) there existsK∈L1(RN)∩LN−22N (RN) such that|g(x, t)| ≤K(x)(1 +|t|p), wherep∈(1,N+ 2
N−2),N≥3,
(G3) g(x, t) =o(|t|) ast→0 uniformly inx∈RN, (G4) there is a constantβ >2 such that
0< βG(x, t)≤tg(x, t) for allt6= 0 andx∈RN, whereG(x, t) =
Z t
0
g(x, s)ds.
Therefore, the limitsg±= lim
t→±∞
g(x, t)
t = +∞uniformly forx∈Ω⊂⊂RN. It reminds one of a type of Ambrosetti-Prodi problem in bounded domains [AP], [F]
and [FY]. These Ambrosetti-Prodi type of problems can be viewed as a question of characterizing the range of a perturbation of a linear operator by some nonlinear operator.
In this paper, we obtain two solutions for problem (1.1). The solutions of problem (1.1) will be found as critical points of the functional
(1.2) J(u) =1 2
Z
RN
(|∇u|2+qu2−λu2)dx− Z
RN
G(x, u)dx− Z
RN
f u dx.
First we reduce the problem by the Lyapunov-Schmidt reduction to a problem in E+, and then using variational method, we obtain the following result.
Theorem A. Assume(A1)–(A2)and(G1)–(G4). If kfkL2(RN)is small, problem (1.1) possesses at least two solutions.
Section 2 is dealt with Lyapunov-Schmidt reduction, existence result is proved in Section 3.
2. Lyapunov-Schmidt reduction
LetE=E+LE−and the quadratic formQbe defined as in Section 1. It is known thatQis positive onE+and negative onE−. We can define a new scalar product (·,·)E onE with the corresponding normk · kE such that
Q(u) =−kuk2E for u∈E− and Q(u) =kuk+E for u∈E+.
The norm k · kE is equivalent to the original norm on E, see [PP] for details.
LetP+ :E →E+ and P− :E →E− be orthogonal projections of E onto E+ and E− respectively. With the aid of these projections, we can write Qin the
formQ(u) =kP+uk2E− kP−uk2E. One may verify that the functional J defined in (1.2) is well defined andC1 onE. To eliminate the effect of indefinite property, we consider the functional
(2.1) Iv(w) =J(v+w) =1
2(kvk2E−kwk2E)−
Z
RN
G(x, v+w)dx−
Z
RN
f(v+w)dx defined onE−for fixedv∈E+. By (A2), (G4) and H¨older’s inequality, we have (2.2) Iv(w)≤ 1
2(kvk2E− kwk2E) +εkwk2E+Cεkfk2L2 +kfkL2kvkE.
Chooseε >0 sufficiently small in (2.2), then for any fixedv∈E+,Iv(w)→ −∞
askwkE → ∞. It implies thatIv(w) is bounded above onE−. Set
(2.3) M = sup
w∈E−
Iv(w).
Lemma 2.1. LetK(x)be as in(G2). If un n
⇀ uweakly inE, then a subsequence of{un}, still denoted by{un}, satisfies
n→∞lim Z
RN
K(x) un−u
p+1dx= 0.
The conclusion follows by the fact thatK decays uniformly in “average” sense at infinity. For a proof we refer to [L].
Lemma 2.2. M is attained by somew0∈E−. Furthermore,w0 satisfies (2.4) −△w0+qw0 =λw0+g(x, v+w0) +f in (E−)∗.
Proof: We follow some ideas from [BJS]. By Ekeland’s variational principle [E], we may find a maximizing sequence{wn} ⊂E−of problem (2.3) such that (2.5) 1
2(kvk2E− kwnk2E)− Z
RN
G(x, v+wn)dx− Z
RN
f(v+wn)dx=M+o(1), (2.6)
Z
RN
(▽wn▽ϕ+qwnϕ−λwnϕ)dx− Z
RN
g(x, v+wn)ϕ dx− Z
RN
f ϕ dx
=o(1)kϕkE, ∀ϕ∈E−. Takingϕ=−wnin (2.6), we obtain (2.7) kwnk2E +
Z
RN
g(x, v+wn)wndx+ Z
RN
f wndx=o(1)kwnkE.
Therefore kwnk2E+
Z
RN
g(x, v+wn)(v+wn)dx
≤ Z
RN
g(x, v+wn)v dx+CkfkL2kwnkE+o(1)kwnkE. By (G1)–(G4), we have
|g(x, t)|2 ≤Ctg(x, t) if |t| ≤1 and x∈RN,
|g(x, t)|p+1p ≤Ctg(x, t) if |t| ≥1 and x∈RN for some constantC >0. It follows
(2.8)
| Z
RN
g(x, v+wn)v dx|
≤C(
Z
{|v+wn|≤1}|g(x, v+wn)|2dx)12kvkL2 +C(
Z
{|v+wn|≥1}|g(x, v+wn)|
p+1 p dx)
p
p+1kvkLp+1
≤C(
Z
RN
(v+wn)g(x, v+wn)dx)12kvkL2 +C(
Z
RN
(v+wn)g(x, v+wn)dx)
p
p+1kvkLp+1
≤ε Z
RN
(v+wn)g(x, v+wn)dx+Cε(kvk2E+kvkp+1E ).
As a result, we obtain
kwnkE ≤C
by choosingε >0 sufficiently small. Therefore we may assume thatwn n
⇀ w0 in E and wn n
−→w0 in Lrloc(RN) for 2 ≤r <2∗ := 2N
N−2 and we have w0 ∈E− satisfying (2.4). Hence
(2.9) Z
RN
[▽(wn−w0)▽ϕ+q(wn−w0)ϕ−λ(wn−w0)ϕ]dx
= Z
RN
[g(x, v+wn)−g(x, v+w0)]ϕ dx+o(1)kϕkE, ∀ϕ∈E−. Letϕ=−(wn−w0) in (2.9). Then
kwn−w0k2E+ Z
RN
[g(x, v+wn)(wn−w0)−g(x, v+w0)(wn−w0)]dx
=o(1)kwn−w0kE.
By (G2), H¨older’s inequality and Lemma 2.1 we obtain Z
RN
g(x, v+wn)(wn−w0)dx−→n 0, (2.10)
Z
RN
g(x, v+w0)(wn−w0)dx−→n 0.
(2.11)
Actually, by (G2)
(2.12)
| Z
RN
g(x, v+wn)(wn−w0)dx|
≤C Z
RN
K(x)(|v+wn|+|v+wn|p)|wn−w0|dx
≤C Z
RN
K(x)(|wn−w0|2+|wn−w0|p+1)dx
since{wn} is bounded inE. (2.12) and Lemma 2.1 imply (2.10). (2.11) can be obtained in the same way. Consequently,
wn−→n w0 strongly in E.
The assertion follows.
Lemma 2.3. There exists h∈C1(E+, E−)such that
J(v+w)< J(v+h(v)), ∀w∈ E− and w6=h(v).
Moreover,h(v)satisfies(2.4).
Proof: Following arguments in [BJS], we let
k(v, w) =−△w+qw−λw−P−(g(x, v+w) +f), wherev is fixed,w∈E−. By Lemma 2.2 we have
k(v, w0) = 0.
For allz∈E−,z6= 0, we deduce by (G1) that hDwk(v, w0)z, zi=
Z
RN
(| ▽z|2+qz2−λz2)dx− Z
RN
gt′(x, v+w0)z2dx
≤ −kzk2E<0.
Hence Dwk(v, w0) is bounded in E∗, we conclude that its inverse exists and is bounded. The Implicit Function Theorem yields that there existsh∈C1(E+, E−)
such thatw0=h(v).
3. Existence results
In this section we prove Theorem A. The first solution is obtained as a local minimum of a functional in a small ball, the second one is found by the Mountain Pass Theorem ([AR]). Let
F(v) =J(v+h(v)), ∀ v∈ E+. ThenF ∈C1(E+,R). By (2.4) we know that
− Z
RN
f h(0)dx= Z
RN
h(0)g(x, h(0))dx+kh(0)k2E. Using (G4) we obtain
| Z
RN
f h(0)dx| ≥ kh(0)k2E.
IfkP−fkL2(RN)small, the inequality implieskh(0)kE small. Consequently,F(0) is small provided thatkP−fkL2(RN) is small.
Lemma 3.1. If kP−fkL2(RN)is small, there existα, r >0 such that (3.1) F(v)≥α > F(0), ∀ v∈ E+, kvkE =r.
Proof: By (G2), (G3), Lemma 2.3 and H¨older’s inequality, we have (3.2) F(v)≥J(v)≥(1
2−ε)kvk2E−Cε(kvkp+1E +kfk2L2).
On the other hand,
(3.3) F(0)≤CkfkL2kh(0)kE.
Thus, from (3.2) and (3.3) we obtain (3.1) forkvkE andkfkL2 small.
Lemma 3.2. For anyv∈E+,kF′(v)kE∗=kJ′(v+h(v))kE∗.
Proof: See the proof of Lemma 2.2 in [BJS].
A sequence{vn}is said to be the Palais-Smale sequence forF ((P S)-sequence for short) if |F(vn)| ≤ C uniformly in n and F′(vn) −→n 0 in (E+)∗. We say that F satisfies the Palais-Smale condition ((P S) condition for short) if every (P S)-sequence ofF is relatively compact inE+.
Lemma 3.3. F satisfies(P S)condition.
Proof: Letvn⊂E+ be a (P S)-sequence ofF. We may assume that F(vn)−→n c, F′(vn)−→n 0.
By Lemma 3.2 we have
(3.4) J(vn+h(vn))−→n c, J′(vn+h(vn))−→n 0.
Letun=vn+h(vn). Then J(un)−1
2hJ′(un), uni
=1 2
Z
RN
g(x, un)undx− Z
RN
G(x, un)dx+1 2
Z
RN
f undx
≤c+o(1)kunkE+o(1).
By (G4)
(3.5) 1
2− 1 β
Z
RN
g(x, un)undx≤c+o(1)kunkE+o(1).
Sinceh(vn) satisfies (2.4), Q(h(vn)) =
Z
RN
g(x, un)h(vn)dx+ Z
RN
f h(vn)dx.
Hence as (2.9) we deduce (3.6) kh(vn)k2E ≤
Z
RN
|g(x, un)|
p+1 p dxp+1p
kh(vn)kLp+1 +C
Z
RN
|g(x, un)|2dx12
kh(vn)kL2+CkfkL2kh(vn)kE. (3.5) and (3.6) implykh(vn)kE is uniformly bounded inn. In the same way, we infer from
hJ′(un), vni=o(1)kvnkE that
(3.7) kvnk2E≤C+C Z
RN
g(x, un)undx+o(1)kvnkE. SokvnkE is also uniformly bounded. Consequently,
kunkE≤C.
We may assume
vn⇀ vn 0, wn⇀ wn 0 in E andv0∈E+,w0∈E− and
un n
⇀ u0=v0+w0 in E, un n
−→u0 in Lrloc(RN), 2≤r <2∗. We remark thatu0 is a weak solution of problem (1.1). Therefore
Z
RN
[▽(un−u0)▽ϕ+q(un−u0)ϕ−λ(un−u0)ϕ]dx
− Z
RN
[g(x, un)−g(x, u0)]ϕ dx=o(1)kϕkE, ∀ϕ∈E.
Letϕ=vn−v0, then kvn−v0k2E−
Z
RN
g(x, un)(vn−v0)dx−
Z
RN
g(x, u0)(vn−v0)dx=o(1)kvn−v0kE. By H¨older’s inequality and Lemma 2.1 again, we infer that
kvn−v0kE −→n 0.
The proof is completed.
Let
m= inf
v∈Br
F(v),
whereBr={v∈E+ | kvkE < r} andris determined in Lemma 3.1.
Proposition 3.4. If kfkL2 is small,m is attained by somev1∈E+, andv1+ h(v1)is a solution of (1.1).
Proof: Again by the Ekeland’s variational principle, we have a minimizing se- quence{vn} satisfying
F(vn)−→n m, F′(vn)−→n 0 and kvnkE ≤r.
From Lemma 3.3 we know that there exists a subsequence of {vn} convergent strongly in E. Denote by v1 the limit function, then kv1kE ≤ r. Lemma 3.1 implieskv1k < r, so v1 is a critical point ofF. By Lemma 3.2, v1+h(v1) is a
solution of (1.1).
Next, we use the Mountain Pass Theorem to obtain the second solution.
Lemma 3.5. There exists v∈E+,v /∈Br(0) such thatF(v)<0.
Proof: By assumptions (G1) and (G4), there exists a function l(x)>0, ∀x∈ RN such that
G(x, t)≥l(x)|t|β
provided that|t| ≥σfor someσ >0. Choosingv∈E+and kvkE = 1, we claim that
(3.8) F(tv)<0
fort >0 large.
Let {tn} be a sequence of positive numbers, tn n
−→ ∞. Denote un = tnv+ h(tnv), andwn= un
kunkE. We may assume thatwn⇀ wn =w++w−inE, where w±∈E±.
We distinguish two cases:
(i) kh(tnv)kE
tn →+∞;
(ii) kh(tnv)kE
tn →k≥0, wherek is a constant.
In the first case, by (G4) and H¨older’s inequality, we deduce F(tnv) =J(tnv+h(tnv))
≤ 1 2
t2nkvk2E− kh(tnv)k2E
+CkfkL2ktnv+h(tnv)kE
≤ t2n 2
kvk2E− 1
t2nkh(tnv)k2E+C
tnkfkL2kvkE+ C
t2nkfkL2kh(tnv)kE (3.9)
≤ t2n 2
kvk2E− 1
t2n(1−ε)kh(tnv)k2E+Cεkfk2L2+CkfkL2kvkE .
Choosingε >0 sufficiently small, we obtain F(tnv)→ −∞
asn→ ∞.
In the second case, ifkh(tnv)kE/tn→k >0, then we may assumeh(tnv)/tn n
⇀ h1, it follows that w = v+h1
(1 +k2)12 6≡0. In fact, were it not the case, we would havev=−h1, it would yield
0 =Q(v, h1) =Q(v,−v) =−kvk2E a contradiction to the choice ofv. By Lemma 2.1
n→∞lim Z
RN
l(x)|wn|βdx= Z
RN
l(x)|w|βdx.
The limit is positive.
For n large we have kunkE ≥ tn > 1. Let ωn = {x ∈ RN : |tnv(x) + h(tnv(x))| ≥σ}. We estimate by (G2)
Z
RN/ωn
G(x, tnv+h(tnv))dx≤C
and Z
RN/ωn
l(x)
tnv+h(tnv)
βdx≤C, whereC >0 is independent ofn. Hence we deduce
(3.10)
Z
RN
G(x, tnv+h(tnv))dx
= Z
ωn
G(x, tnv+h(tnv))dx+ Z
RN/ωn
G(x, tnv+h(tnv))dx
≥ Z
ωn
l(x)|tnv+h(tnv)|β dx−C
≥ kunkβE Z
RN
l(x)
tnv+h(tnv) kunkE
βdx−C1
≥tβn Z
RN
l(x)|w|βdx+o(1)
−C1. It concludes by (3.10) that
F(tnv)≤t2n 2
kvk2E− 1
t2n(1−ε)kh(tnv)k2E+Cεkfk2L2+CkfkL2kvkE (3.11)
−tβn Z
RN
l(x)|w|βdx+o(1)
−C≤0 fornlarge.
If kh(tnv)kE/tn → 0, then kunkE/tn → 1. By Sobolev embedding, we have h(tnv)/tn→0 a.e. in RN. It results
Z
RN
l(x)
tnv+h(tnv) kunkE
βdx→
Z
RN
l(x)|v|βdx >0.
Then we may argue as before. The conclusion follows.
Proof of Theorem A: By Lemma 3.5, there existse∈E+,e /∈Br such that F(e)<0. Let
Γ ={γ∈C([0,1], E+)|γ(0) =v1, γ(1) =e},
wherev1 is the minimum point ofmobtained in Proposition 3.4. Define c= inf
γ∈Γmax
v∈γF(v).
Lemma 3.3 and the Mountain Pass Theorem implyc is a critical value ofF, and by Lemma 3.2, corresponding critical point v2 gives second solutionv2+h(v2)
of (1.1).
Acknowledgments. The authors would like to thank the referee for his useful comments. The work was supported by Science Program of Nanchang University, NSFJ, NSF and 21 Century Science Program of Jiangxi Province, P.R. of China.
References
[AP] Ambrosetti A., Prodi G.,On the inversion of some differentiable mappings with singu- larities between Banach spaces, Ann. Mat. Pura Appl.93(1972), 231–246.
[AR] Ambrosetti A., Rabinowitz P., Dual variational methods in critical point theory and application, J. Funct. Anal.14(1973), 231–246.
[BJ] Buffoni B., Jeanjean L.,Minimax characterization of solutions for a semilinear elliptic equation with lack of compactness, Ann. Inst. H. Poincar´e Anal. Nonlineaire10(1993), no. 4, 377–404.
[BJS] Buffoni B., Jeanjean L., Stuart C.A.,Existence of a non-trivial solution to a strongly indefinite semilinear equation, Proc. Amer. Math. Soc.119(1993), no. 1, 175–186.
[CY] Chabrowski J., Yang Jianfu,Existence theorems for the Schr¨odinger equation involving a critical Sobolev exponent, Z. Angew. Math. Phys.49(1998), 276–293.
[CZ] Cao D.M., Zhou H.S.,Multiple positive solutions of nonhomogeneous semilinear elliptic equations onRN, Proc. Royal Soc. Edinburgh126A(1996), 443–463.
[E] Ekeland I.,On the variational principle, J. Math. Anal. Appl.47(1974), 324–353.
[F] de Figueiredo D.G.,On superlinear elliptic problems with nonlinearities interacting only with higher eigenvalues, Rocky Mount. J. Math.18(1988), no. 2, 287–303.
[FY] de Figueiredo D.G., Yang Jianfu,Critical superlinear Ambrosetti-Prodi Problem, Topol.
Methods Nonlinear Anal.14(1999), 59–80.
[J] Jeanjean L.,Two positive solutions for a class nonhomogeneous elliptic equations, Dif- ferential Integral Equations10(1997), 609–624.
[L] Liu Xiaochun,Existence theorem of concave and convex effects for nonlinear Schr¨odinger equations, J. Nanchang Univ. Nat. Sci.22(1998), no. 1, 31–38.
[PP] Pankov A.A., Pfl¨uger K.,On a semilinear Schr¨odinger equation with periodic potential, Nonlinear Anal. TMA33(1998), 593–609.
[R] Rabinowitz P.,Minimax methods in critical point theory with applications to differential equations, AMS Conf. Ser. Math. 65 (1986).
Department of Mathematics, Wuhan University, Wuhan 430072, China
Department of Mathematics, Nanchang University, Nanchang 330047, China and
IMECC-UNICAMP, Caixa Postal 6065, 13083-970 Campinas S.P., Brazil (Received August 31, 1999,revised March 10, 2000)
