Mathematical Problems in Engineering

STANDING WAVE SOLUTIONS OF SCHRÖDINGER SYSTEMS WITH DISCONTINUOUS NONLINEARITY IN ANISOTROPIC MEDIA

TEODORA-LILIANA DINU

Received 5 July 2005; Revised 14 April 2006; Accepted 5 July 2006

We establish the existence of an entire solution for a class of stationary Schr¨odinger sys- tems with subcritical discontinuous nonlinearities and lower bounded potentials that blow up at infinity. The proof is based on the critical point theory in the sense of Clarke and we apply Chang’s version of the mountain pass lemma for locally Lipschitz func- tionals. Our result generalizes in a nonsmooth framework a result of Rabinowitz (1992) related to entire solutions of the Schr¨odinger equation.

Copyright © 2006 Hindawi Publishing Corporation. All rights reserved.

1. Introduction and the main result

The Schr¨odinger equation plays the role of Newton’s laws and conservation of energy in classical mechanics, that is, it predicts the future behavior of a dynamic system. The linear form of Schr¨odinger’s equation is

Δψ+8π²m ²

E(x)−V(x)ψ=0, (1.1)

whereψ is the Schr¨odinger wave function,mis the mass,denotes Planck’s constant, Eis the energy, and V stands for the potential energy. The structure of the nonlinear Schr¨odinger equation is much more complicated. This equation describes various phe- nomena arising in self-channelling of a high-power ultra-short laser in matter, in the theory of Heisenberg ferromagnets and magnons, in dissipative quantum mechanics, in condensed matter theory, in plasma physics (e.g., the Kurihara superfluid film equation).

We refer to [7,14] for a modern overview, including applications.

Consider the model problem iψ_t= −²

2mΔψ+V(x)ψ−γ|ψ|^p−1ψ inR^N(N≥2), (1.2)

Hindawi Publishing Corporation

International Journal of Mathematics and Mathematical Sciences Volume 2006, Article ID 73619, Pages1–13

DOI10.1155/IJMMS/2006/73619

wherep <2N/(N−2) ifN≥3 andp <+∞ifN=2. In the study of this equation, Oh [12]

supposed that the potentialVis bounded and possesses a nondegenerate critical point at x=0. More precisely, it is assumed thatV belongs to the class (V_a) (for somea) intro- duced by Kato in [10]. Takingγ >0 and>0 sufficiently small and using a Lyapunov- Schmidt-type reduction, Oh [12] proved the existence of a standing wave solution of problem (1.2), that is, a solution of the form

ψ(x,t)=e^−iEt/u(x). (1.3)

Note that substituting the ansatz (1.3) into (1.2) leads to

−²

2Δu+V(x)−Eu= |u|^p−1u. (1.4) The change of variabley=⁻¹x(and replacingybyx) yields

−Δu+ 2V(x)−Eu= |u|^p−1u inR^N, (1.5) whereV(x)=V(x).

In a celebrated paper, Rabinowitz [13] continued the study of standing wave solutions of nonlinear Schr¨odinger equations. After making a standing wave ansatz, Rabinowitz reduces the problem to that of studying the semilinear elliptic equation

−Δu+b(x)u=f(x,u) inR^N, (1.6) under suitable conditions onband assuming that f is smooth, superlinear, and subcriti- cal.

Inspired by Rabinowitz’ paper, we consider the following class of coupled elliptic sys- tems inR^N(N≥3):

−Δu1+a(x)u1=fx,u1,u2

inR^N,

−Δu2+b(x)u2=gx,u1,u2

inR^N. (1.7)

We point out that coupled nonlinear Schr¨odinger systems describe some physical phe- nomena such as the propagation in birefringent optical fibers or Kerr-like photorefractive media in optics. Another motivation to the study of coupled Schr¨odinger systems arises from the Hartree-Fock theory for the double condensate, that is, a binary mixture of Bose-Einstein condensates in two different hyperfine states (cf. [5]). System (1.7) is also important for industrial applications in fiber communications systems [8] and all-optical switching devices [9].

Throughout this paper, we assume thata,b∈L^∞_loc(R^N) and there exista,b >0 such thata(x)≥a,b(x)≥ba.e. inR^N, and esslim_|x|→∞a(x)=esslim_|x|→∞b(x)=+∞. Our aim in this paper is to study the existence of solutions to the above problem in the case when f,gare not continuous functions. Our goal is to show how variational methods can be used to find existence results for stationary nonsmooth Schr¨odinger systems.

Throughout this paper, we assume that f(x,·,·),g(x,·,·)∈L^∞_loc(R²). Denote fx,t1,t2

=lim

δ→0essinffx,s1,s2

;t_i−s_i≤δ;i=1, 2, fx,t1,t2

=lim

δ→0esssupfx,s1,s2

;t_i−s_i≤δ;i=1, 2, gx,t1,t2

=lim

δ→0essinfgx,s1,s2

;ti−si≤δ;i=1, 2, gx,t1,t2

=lim

δ→0esssupgx,s1,s2

;ti−si≤δ;i=1, 2.

(1.8)

Under these conditions, we reformulate problem (1.7) as follows:

−Δu1+a(x)u1∈

fx,u1(x),u2(x),fx,u1(x),u2(x) a.e.x∈R^N,

−Δu2+b(x)u2∈

gx,u1(x),u2(x),gx,u1(x),u2(x) a.e.x∈R^N.

(1.9)

LetH¹=H(R^N,R²) denote the Sobolev space of allU=(u1,u2)∈(L²(R^N))² with weak derivatives∂u1/∂xj,∂u2/∂xj (j=1,. . .,N) also inL²(R^N), endowed with the usual norm

U²_H1=

R^N

|∇U|²+|U|² dx=

R^N

∇u1²+∇u2²+u²₁+u²₂dx. (1.10)

Given the functionsa,b:R^N→Ras above, define the subspace E=

U=

u1,u2

∈H¹;

R^N

∇u1²+∇u2²+a(x)u²₁+b(x)u²₂dx <+∞

. (1.11) Then the spaceEendowed with the norm

U²_E=

R^N

∇u1²+∇u2²+a(x)u²₁+b(x)u²₂dx (1.12)

becomes a Hilbert space.

Sincea(x)≥a >0,b(x)≥b >0, we have the continuous embeddingsH¹L^q(R^N,R²) for all 2≤q≤2^∗=2N/(N−2).

We assume throughout the paper that f,g:R^N×R²→Rare nontrivial measurable functions satisfying the following hypotheses:

f(x,t)≤C|t|+|t|^p

for a.e. (x,t)∈R^N×R², g(x,t)≤C|t|+|t|^p

for a.e. (x,t)∈R^N×R², (1.13) wherep <2^∗;

limδ→0esssupf(x,t)

|t| ; (x,t)∈R^N×(−δ, +δ)²

=0,

limδ→0esssupg(x,t)

|t| ; (x,t)∈R^N×(−δ, +δ)²

=0;

(1.14)

f andgare chosen so that the mappingF:R^N×R²→Rdefined byF(x,t1,t2) :=_t1

0 f(x, τ,t2)dτ+₀^t2g(x, 0,τ)dτsatisfies

Fx,t1,t2

= ^t2

0 g(x,t1,τ)dτ+

t1

0 f(x,τ, 0)dτ, Fx,t1,t2

=0 ifft1=t2=0;

(1.15) there existsμ >2 such that for anyx∈R^N,

0≤μFx,t1,t2

≤

⎧⎪

⎪⎪

⎪⎪

⎪⎪

⎪⎨

⎪⎪

⎪⎪

⎪⎪

⎪⎪

⎩

t1fx,t1,t2

+t2gx,t1,t2

, t1,t2∈[0, +∞), t1fx,t1,t2

+t2gx,t1,t2

, t1∈[0, +∞),t2∈(−∞, 0], t1fx,t1,t2

+t2gx,t1,t2

, t1,t2∈(−∞, 0], t1fx,t1,t2

+t2gx,t1,t2

, t1∈(−∞, 0],t2∈[0, +∞).

(1.16)

Definition 1.1. A functionU=(u1,u2)∈Eis called solution to the problem (1.9) if there exists a functionW=(w1,w2)∈L²(R^N,R²) such that

(i)

fx,u1(x),u2(x)≤w1(x)≤fx,u1(x),u2(x) a.e.xinR^N,

gx,u1(x),u2(x)≤w2(x)≤gx,u1(x),u2(x) a.e.xinR^N; (1.17) (ii)

R^N

∇u1∇v1+∇u2∇v2+a(x)u1v1+b(x)u2v2

dx

= R^N

w1v1+w2v2

dx, ∀ v1,v2

∈E.

(1.18)

Our main result is the following.

Theorem 1.2. Assume that conditions (1.13)–(1.16) are fulfilled. Then problem (1.9) has at least a nontrivial solution inE.

2. Auxiliary results

We first recall some basic notions from the Clarke gradient theory for locally Lipschitz functionals (see [3,4] for more details). LetEbe a real Banach space and assume that I:E→Ris a locally Lipschitz functional. Then the Clarke generalized gradient is defined by

∂I(u)=

ξ∈E^∗;I⁰(u,v)≥ ξ,v,∀v∈E, (2.1) whereI⁰(u,v) stands for the directional derivative ofIatuin the directionv, that is,

I⁰(u,v)=lim sup

w→u λ0

I(w+λv)−I(w)

λ . (2.2)

LetΩbe an arbitrary domain inR^N. Set E_Ω=

U=

u1,u2

∈H¹Ω;R²

; Ω

∇u1²+∇u2²+a(x)u²₁+b(x)u²₂dx <+∞

, (2.3) which is endowed with the norm

U²_EΩ=

Ω

∇u1²+∇u2²+a(x)u²₁+b(x)u²₂dx. (2.4)

ThenE_Ωbecomes a Hilbert space.

Lemma 2.1. The functionalΨΩ:E_Ω→R,ΨΩ(U)=

ΩF(x,U)dxis locally Lipschitz onE_Ω. Proof. We first observe that

F(x,U)=Fx,u1,u2

= ^u1

0 fx,τ,u2 dτ+

u2

0 g(x, 0,τ)dτ

= ^u2

0 gx,u1,τdτ+

u1

0 f(x,τ, 0)dτ

(2.5)

is a Carath´eodory functional which is locally Lipschitz with respect to the second variable.

Indeed, by (1.13),

Fx,t1,t−Fx,s1,t= ^t1

s1

f(x,τ,t)dτ

≤ ^t1

s1

C|τ,t|+|τ,t|^p dτ

≤kt1,s1,tt1−s1.

(2.6)

Similarly,

Fx,t,t2

−Fx,t,s2≤kt2,s2,tt2−s2. (2.7) Therefore,

Fx,t1,t2

−Fx,s1,s2≤Fx,t1,t2

−Fx,s1,t2+Fx,t1,s2

−Fx,s1,s2

≤k(V)t2,s2

− t1,s1,

(2.8) whereV is a neighborhood of (t1,t2), (s1,s2).

Set

χ1(x)=maxu1(x),v1(x), χ2(x)=maxu2(x),v2(x), ∀x∈Ω. (2.9)

It is obvious that if U=(u1,u2), V =(v1,v2) belong toE_Ω, then (χ1,χ2)∈E_Ω. So, by H¨older’s inequality and the continuous embeddingE_Ω⊂L^p(Ω;R²),

ΨΩ(U)−ΨΩ(V)≤Cχ1,χ2

EΩ

U−VEΩ, (2.10)

which concludes the proof.

The following result is a generalization of [11, Lemma 6].

Lemma 2.2. LetΩbe an arbitrary domain inR^Nand let f :Ω×R²→Rbe a Borel function such that f(x,·)∈L^∞_loc(R²). Then f and f are Borel functions.

Proof. Since the requirement is local, we may suppose that f is bounded byMand it is nonnegative. Denote

fm,n x,t1,t2

= ^t1+1/n

t1−1/n t2+1/n t2−1/n

fx,s1,s2^mds1ds2

1/m

. (2.11)

Since f(x,t1,t2)=lim_δ→0esssup{f(x,s1,s2); |t_i−s_i| ≤δ;i=1, 2}, we deduce that for ev- eryε >0, there existsn∈N^∗such that for|ti−si|<1/n(i=1, 2), we have|esssupf(x,s1, s2)−f(x,t1,t2)|< εor, equivalently,

fx,t1,t2

−ε <esssupfx,s1,s2

< fx,t1,t2

+ε. (2.12)

By the second inequality in (2.12), we obtain fx,s1,s2

≤fx,t1,t2

+ε a.e. x∈Ωforti−si<1

n (i=1, 2) (2.13) which yields

f_m,nx,t1,t2

≤

fx,t1,t2

+ε) 4

n² 1/m

. (2.14)

Let

A= s1,s2

∈R²;t_i−s_i< 1

n (i=1, 2); fx,t1,t2

−ε≤fx,s1,s2

. (2.15)

By the first inequality in (2.12) and the definition of the essential supremum, we obtain that|A|>0 and

f_m,n≤

A

fx,s1,s2

m

ds1ds2

1/m

≥

fx,s1,s2

−ε|A|^1/m. (2.16)

Since (2.14) and (2.16) imply that fx,t1,t2

=lim

n→∞ lim

m→∞fm,n

x,t1,t2

, (2.17)

it suffices to prove thatfm,nis Borel. Let ᏹ=

f :Ω×R²−→R;|f| ≤Mand f is a Borel function, ᏺ=

f ∈ᏹ; f_m,nis a Borel function. (2.18) ᏹis the smallest set of functions having the following properties (cf. [1, page 178]):

(i){f ∈C(Ω×R²;R);|f| ≤M} ⊂ᏹ;

(ii) f^(k)∈ᏹand f^(k)−→^k f imply that f ∈ᏹ.

Sinceᏺcontains obviously the continuous functions and (ii) is also true forᏺ, then by the Lebesgue dominated convergence theorem, we obtain thatᏹ=ᏺ. For f, we note that f = −(−f) and the proof ofLemma 2.2is complete.

Let us now assume thatΩ⊂R^Nis a bounded domain. By the continuous embedding L^p+1(Ω;R²)L²(Ω;R²), we may define the locally Lipchitz functionalΨΩ:L^p+1(Ω;R²)→ RbyΨΩ(U)=

ΩF(x,U)dx.

Lemma 2.3. Under the above assumptions and for anyU∈L^p+1(Ω;R²),

∂ΨΩ(U)(x)⊂

fx,U(x),fx,U(x)×

gx,U(x),gx,U(x) a.e.xinΩ, (2.19) in the sense that ifW=(w1,w2)∈∂ΨΩ(U)⊂L^p+1(Ω;R²), then

fx,U(x)≤w1(x)≤fx,U(x) a.e.xinΩ, (2.20) gx,U(x)_≤w2(x)_≤gx,U(x) a.e.xinΩ. (2.21) Proof. By the definition of the Clarke gradient, we have

Ω

w1v1+w2v2

dx≤Ψ⁰Ω(U,V) ∀V= v1,v2

∈L^p+1Ω;R²

. (2.22) ChooseV=(v, 0) such thatv∈L^p+1(Ω),v≥0 a.e. inΩ. Thus, byLemma 2.2,

Ωw1v≤lim sup

(h1,h2)_→U λ0

Ω

_h1(x)+λv(x)

h1(x) fx,τ,h2(x)dτdx λ

≤ Ω

⎛

⎜⎜

⎝lim sup

(h1,h2)_→U λ0

1 λ

h1(x)+λv(x)

h1(x) fx,τ,h2(x)dτ

⎞

⎟⎟

⎠dx

≤ Ωfx,u1(x),u2(x)v(x)dx.

(2.23)

Analogously, we obtain

Ωfx,u1(x),u2(x)v(x)dx≤

Ωw1v dx ∀v≥0 inΩ. (2.24)

Arguing by contradiction, suppose that (2.20) is false. Then there exist ε >0, a set A⊂Ωwith|A|>0, andw1as above such that

w1(x)> fx,U(x)+ε inA. (2.25) Takingv=1Ain (2.23), we obtain

Ωw1v dx=

Aw1dx≤

Afx,U(x)dx, (2.26)

which contradicts (2.25). Proceeding in the same way, we obtain the corresponding result

forgin (2.21).

ByLemma 2.3, Chang [2, Lemma 2.1], and the embeddingE_ΩL^p+1(Ω,R²), we ob- tain also that forΨΩ:E_Ω→R,ΨΩ(U)=

ΩF(x,U)dx, we have

∂ΨΩ(U)(x)⊂

fx,U(x),fx,U(x)×

gx,U(x),gx,U(x) a.e.x∈Ω. (2.27) LetV∈E_Ω. ThenV^#∈E, whereV# :R^N→R²is defined by

V# =

⎧⎨

⎩

V(x) xinΩ,

0 otherwise. (2.28)

ForW∈E^∗, we considerW_Ω∈E^∗_Ω such thatW_Ω,V = W,V# for allV in E_Ω. Set Ψ:E→R,Ψ(U)=

R^NF(x,U).

Lemma 2.4. LetW∈∂Ψ(U), whereU∈E. ThenW_Ω∈∂ΨΩ(U), in the sense thatW_Ω∈

∂ΨΩ(U|Ω).

Proof. By the definition of the Clarke gradient, we deduce thatW,V^# ≤Ψ⁰(U,V^#) for all VinE_Ω,

Ψ⁰(U,V^#)= lim sup

H→U,H∈E λ→0

Ψ(H+λV^#)−Ψ(H) λ

= lim sup

H→U,H∈E λ→0

R^N

F(x,H+λV#)−F(x,H)dx λ

= lim sup

H→U,H∈E λ→0

Ω

F(x,H+λV^#)−F(x,H)dx λ

= lim sup

H→U,H∈EΩ

λ→0

Ω

F(x,H+λV^#)−F(x,H)dx λ

=Ψ⁰_Ω(U,V).

(2.29)

HenceW_Ω,V ≤Ψ⁰_Ω(U,V), which implies thatW_Ω∈∂Ψ⁰_Ω(U).

By Lemmas2.3and2.4, we obtain that for anyW∈∂Ψ(U) (withU∈E),W_Ω sat- isfies (2.20) and (2.21). We also observe that for Ω1,Ω2⊂R^N, we have W_Ω1|Ω1∩Ω2= W_Ω2|Ω1∩Ω2.

LetW0:R^N→R, whereW0(x)=W_Ω(x) ifx∈Ω. ThenW0is well defined and W0(x)∈

fx,U(x),fx,U(x)×

gx,U(x),gx,U(x) a.e.x∈R^N, (2.30) and for allϕ∈C_c^∞(R^N,R²),W,ϕ =

R^NW0ϕ. By density ofC_c^∞(R^N,R²) inE, we deduce thatW,V =

R^NW0V dxfor allV inE. Hence W(x)=W0(x)∈

fx,U(x),fx,U(x)×

g(x,U(x),gx,U(x) a.e.x∈R^N. (2.31) 3. Proof ofTheorem 1.2

Define the energy functionalI:E→Rby I(U)=1

2 R^N

∇u1²+∇u2²+a(x)u²₁+b(x)u²₂dx−

R^NF(x,U)dx

=1

2U²_E−Ψ(U).

(3.1)

The existence of solutions to problem (1.9) will be justified by a nonsmooth variant of the mountain pass theorem (see [2]) applied to the functionalI, even if the PS condition is not fulfilled. More precisely, we check the following geometric hypotheses:

I(0)=0, ∃V∈E, such thatI(V)≤0; (3.2)

∃β,ρ >0 such thatI≥βonU∈E;UE=ρ. (3.3) Verification of (3.2). It is obvious thatI(0)=0. For the second assertion, we need the following lemma.

Lemma 3.1. There exist two positive constantsC1andC2such that

f(x,s, 0)≥C1s^μ−1−C2, for a.e.x∈R^N;s∈[0, +∞). (3.4) Proof. We first observe that (1.16) implies that

0≤μF(x,s, 0)≤

⎧⎪

⎨

⎪⎩

s f(x,s, 0), s∈[0, +∞),

s f(x,s, 0), s∈(−∞, 0], (3.5) which places us in the conditions of [11, Lemma 5].

Verification of (3.2) continued. Choosev∈C_c^∞(R^N)− {0}so thatv≥0 inR^N. We have

R^N|∇v|²+a(x)v²<∞, hencet(v, 0)∈Efor allt∈R. Thus byLemma 3.1, we obtain It(v, 0)=t²

2 R^N|∇v|²+a(x)v²dx−

R^N tv

0 f(x,τ, 0)dτ

≤t²

2 R^N|∇v|²+a(x)v²dx−

R^N tv 0

C1τ^μ−1−C2

dτ

=t²

2 R^N|∇v|²+a(x)v²dx+C2t

R^Nv dx−C1t^μ

R^Nv^μdx <0

(3.6)

fort >0 large enough.

Verification of (3.3). We observe that (1.14), (1.15), and (1.16) imply that for anyε >0, there exists a constantAε>0 such that

f(x,s)≤ε|s|+A_ε|s|^p

g(x,s)≤ε|s|+Aε|s|^p for a.e. (x,s)∈R^N×R². (3.7) By (3.7) and Sobolev’s embedding theorem, we have for anyU∈E,

Ψ(U)=Ψu1,u2

≤ R^N

|u1| 0

fx,τ,u2dτ+

R^N u2

0

g(x, 0,τ)dτ

≤ R^N

ε

2u1,u2²+ Aε

p+ 1u1,u2^p+1 dx + R^N

ε

2u2²+ Aε

p+ 1u2^p+1 dx

≤εU²_L²+ 2Aε

p+ 1U_L^p+1p+1

≤εC3U²_E+C4U_E^p+1,

(3.8)

whereεis arbitrary andC4=C4(ε). Thus I(U)=1

2U²_E−Ψ(U)≥1

2U²_E−εC3U²_E−C4U_E^p+1≥β >0, (3.9) forUE=ρ, withρ,ε, andβsufficiently small positive constants.

Denote

ᏼ=

γ∈C[0, 1],E;γ(0)=0,γ(1)=0,Iγ(1)≤0, c=inf

γ∈ᏼmax

t∈[0,1]Iγ(t). (3.10)

Set

λI(U)= min

ξ∈∂I(U)ξE^∗. (3.11)

Thus, by the nonsmooth version of the mountain pass lemma [2], there exists a sequence {UM} ⊂Esuch that

IUm

−→c, λI

Um

−→0. (3.12)

So, there exists a sequence{Wm} ⊂∂Ψ(Um),Wm=(w¹_m,w²_m), such that

−Δu¹m+a(x)u¹_m−w_m¹,−Δu²m+a(x)u²_m−w_m²_−→0 inE^∗. (3.13)

Note that by (1.16), Ψ(U)≤1

μ u1≥0u1(x)f(x,U)dx+

u1≤0u1(x)f(x,U)dx + u2≥0u1(x)g(x,U)dx+

u2≤0u2(x)g(x,U)dx

.

(3.14)

Therefore, by (2.31), Ψ(U)≤1

μ _R^NU(x)W(x)dx=1 μ _R^N

u1w1+u2w2

dx, (3.15)

for everyU∈EandW∈∂Ψ(U). Hence, if·,·denotes the duality pairing betweenE^∗ andE, we have

IU_m=μ−2 2μ _R^N

∇u¹_m²+∇u_m²+a(x)u_m¹+b(x)u_m²dx

+1 μ

$−Δu¹m+a(x)u¹_m−w¹_m,−Δu²m+b(x)u²_m−w²_m,U_m^%

+1 μ

$Wm,Um%

−ΨUm

≥μ−2 2μ _R^N

∇u¹_m²+∇u²_m²+a(x)u¹_m²+b(x)u²_m²dx

+1 μ

$−Δu¹m+a(x)u¹_m−w¹_m,−Δu²m+b(x)u²_m−w²_m,U_m^%

≥μ−2 2μ Um²

E−o(1)Um

E.

(3.16)

This relation in conjunction with (3.12) implies that the Palais-Smale sequence {Um} is bounded in E. Thus, it converges weakly (up to a subsequence) in Eand strongly in L²_loc(R^N) to some U. Taking into account that W_m∈∂Ψ(U_m) and U_mU in E, we deduce from (3.13) that there exists W∈E^∗ such that WmW in E^∗ (up to a subsequence). Since the mappingU→F(x,U) is compact from EtoL¹, it follows that

W∈∂Ψ(U). Therefore, W(x)∈

fx,U(x),fx,U(x)×

gx,U(x),gx,U(x) a.e.xinR^N, −Δu¹m+a(x)u¹_m−w¹_m,−Δu²m+b(x)u²_m−w²_m

=0⇐⇒

R^N

∇u1∇v1+∇u2∇v2+a(x)u1v1+b(x)u2v2

dx

= R^N

w1v1+w2v2

dx ∀ v1,v2

∈E.

(3.17)

These last two relations show thatUis a solution of the problem (1.9).

It remains to prove thatU≡0. If{Wm}is as in (3.13), then by (1.16), (2.31), (3.12), and for largem,

c

2^≤IUm

−1 2

$−Δu¹m+a(x)u¹_m−w_m¹,−Δu²m+b(x)u²_m−w²_m,Um

%

=1 2

$Wm,Um

%−

R^NFx,Um

dx

≤1

2 u1≥0u1(x)f(x,U)dx+

u1≤0u1(x)f(x,U)dx +

u2≥0u1(x)g(x,U)dx+

u2≤0u2(x)g(x,U)dx

.

(3.18)

Now, taking into account the definition of f,f,g,g, we deduce that f,f,g,gverify (3.2) too. So by (3.18), we obtain

c 2^≤ R^N

εU_m²+A_εu_m^p+1=εU_m²_L2+A_εU_mL^p+1p+1. (3.19)

So,{U_m}does not converge strongly to 0 inL^p+1(R^N;R²). From now on, with the same arguments as in the proof of [6, Theorem 1], we deduce thatU≡0, which concludes our

proof.

References

[1] S. K. Berberian, Measure and Integration, The Macmillan, New York, 1967.

[2] K. C. Chang, Variational methods for nondifferentiable functionals and their applications to partial differential equations, Journal of Mathematical Analysis and Applications 80 (1981), no. 1, 102–

129.

[3] F. H. Clarke, Generalized gradients and applications, Transactions of the American Mathematical Society 205 (1975), 247–262.

[4] , Generalized gradients of Lipschitz functionals, Advances in Mathematics 40 (1981), no. 1, 52–67.

[5] B. D. Esry, C. H. Greene, J. P. Burke Jr., and J. L. Bohn, Hartree-Fock theory for double condensates, Physical Review Letters 78 (1997), no. 19, 3594–3597.

[6] F. Gazzola and V. R˘adulescu, A nonsmooth critical point theory approach to some nonlinear elliptic equations inRⁿ, Differential Integral Equations 13 (2000), no. 1–3, 47–60.