Volume 2008, Article ID 217636,10pages doi:10.1155/2008/217636
Research Article
Existence Result for a Class of
Elliptic Systems with Indefinite Weights in
R
2
Guoqing Zhang1and Sanyang Liu2
1College of Sciences, University of Shanghai for Science and Technology, Shanghai 200093, China 2Department of Applied Mathematics, Xidian University, Xi’an 710071, China
Correspondence should be addressed to Guoqing Zhang,[email protected]
Received 31 October 2007; Accepted 4 March 2008
Recommended by Zhitao Zhang
We obtain the existence of a nontrivial solution for a class of subcritical elliptic systems with indefinite weights inR2. The proofs base on Trudinger-Moser inequality and a generalized linking theorem introduced by Kryszewski and Szulkin.
Copyrightq2008 G. Zhang and S. Liu. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
1. Introduction
In this paper, we study the existence of a nontrivial solution for the following systems of two semilinear coupled Poisson equations
P
−Δuugx, v, x∈R2,
−Δvvfx, u, x∈R2, 1.1
wherefx, tandgx, tare continuous functions onR2×Rand have the maximal growth on
twhich allows to treat problemPvariationally,Δis the Laplace operator.
Recently, there exists an extensive bibliography in the study of elliptic problem in RN 1–6. As dimensionsN ≥ 3, in 1998, de Figueiredo and Yang 5considered the following coupled elliptic systems:
−Δuugx, v, x∈RN,
where f, g are radially symmetric in x and satisfied the following Ambrosetti-Rabinowitz condition:
t
0
fx, sds≥c|t|2δ1,
t
0
gx, sds≥c|t|2δ2, ∀t∈R, 1.3
and for someδ1>0, δ2>0. They obtained the decay, symmetry, and existence of solutions for
problem1.2. In 2004, Li and Yang6proved that problem1.2possesses at least a positive solution when the nonlinearitiesfx, tandgx, tare “asymptotically linear” at infinity and “superlinear” at zero, that is,
1limt→∞fx, t/t l >1, limt→∞gx, t/t m >1,uniformly inx∈RN;
2limt→0fx, t/t limt→0gx, t/t 0,uniformly with respect tox∈RN.
In 2006, Colin and Frigon 1 studied the following systems of coupled Poission equations with critical growth in unbounded domains:
−Δu|v|2∗−2v,
−Δv|u|2∗−2u,
1.4
where 2∗ 2N/N −2is critical Sobolev exponent,u, v ∈ D01,2Ω∗andΩ∗ RN \Ewith
E a∈ZNaω∗ for a domain containing the origin ω∗ ⊂ ω∗ ⊂ B0,1/2. Here, B0,1/2 denotes the open ball centered at the origin of radius 1/2. The existence of a nontrivial solution was obtained by using a generalized linking theorem.
As it is well known in dimensions N ≥ 3, the nonlinearities are required to have polynomial growth at infinity, so that one can define associated functionals in Sobolev spaces. Coming to dimension N 2,much faster growth is allowed for the nonlinearity. In fact, the Trudinger-Moser estimates inN2 replace the Sobolev embedding theorem used inN ≥3.
In dimensionN 2, Adimurth and Yadava7, de Figueiredo et al.8discussed the solvability of problems of the type
−Δufx, u, x∈Ω,
u0, x∈∂Ω, 1.5
where Ω is some bounded domain inR2. Shen et al. 9considered the following nonlinear
elliptic problems with critical potential:
Δu−μ u
|x|logR/|x|2 fx, u, x∈Ω u0, x∈∂Ω,
1.6
and obtained some existence results. In the whole space R2, some authors considered the
following single semilinear elliptic equations:
As the potentialVxand the nonlinearityfx, tare asymptotic to a constant function, Cao
10obtained the existence of a nontrivial solution. As the potentialVxand the nonlinearity
fx, tare asymptotically periodic at infinity, Alves et al.11proved the existence of at least one positive weak solution.
Our aim in this paper is to establish the existence of a nontrivial solution for problem
P in subcritical case. To our knowledge, there are no results in the literature establishing the existence of solutions to these problems in the whole space. However, it contains a basic difficulty. Namely, the energy functional associated with problem P has strong indefinite quadratic part, so there is not any more mountain pass structure but linking one. Therefore, the proofs of our main results cannot rely on classical min-max results. Combining a generalized linking theorem introduced by Kryszewski and Szulkin12and Trudinger-Moser inequality, we prove an existence result for problemP.
The paper is organized as follows. InSection 2, we recall some results and state our main results. InSection 3, main result is proved.
2. Preliminaries and main results
Consider the Hilbert space13
H1R2u∈L2R2,∇u∈L2R2, 2.1
and denote the product spaceZH1R2×H1R2endowed with the inner product:
u, v,φ, ψ
R2∇u∇φuφdx
R2∇v∇ψvψdx, ∀φ, ψ∈Z. 2.2 If we define
Z{u, u∈Z}, Z−{v,−v∈Z}. 2.3
It is easy to check thatZZ⊕Z−,since
u, v 1
2uv, uv 1
2u−v, v−u. 2.4
Let us denote byPresp.,Qthe projection ofZon toZresp.,Z−, we have
1
2Pu, v
2−
Qu, v2 1
2
12uv, uv
2 −1
2
12u−v, v−u
2
1
4 R2
|∇u|2|∇v|22∇u∇vdx
R2
|u|2|v|22uvdx
−
R2
|∇u|2|∇v|2−2∇u∇vdx−
R2
|u|2|v|2−2uvdx
R2∇u∇vuvdx.
Now, we define the functional
Iu, v
R2∇u∇vuvdx−
R2
Fx, u Gx, vdx
Pu, v 2
2 −
Qu, v2
2 −ϕu, v, ∀u, v∈Z,
2.6
where
ϕu, v
R2
Fx, u Gx, vdx. 2.7
Letz0 ∈Z\ {0}and letR > r >0,we define
Mzz−λz0 :z−∈Z−,z ≤R, λ≥0
,
M0
zz−λz0:z− ∈Z−, zRandλ≥0 orz ≤Randλ≥0
,
N z∈Z:zr.
2.8
Here, we assume the following condition:
H1f, g ∈CR2×R, R;
H2limt→0fx, t/t limt→0gx, t/t 0 uniformly with respect tox∈R2; H3there existμ >2 andη >0 such that
0< μFx, t≤tfx, t, 0< μGx, t≤tgx, t, ∀|t| ≥η. 2.9
Lemma 2.1see12,14. Assume (H1), (H2), and (H3), and suppose
1Iz 1/2P z2−Qz2−ϕz,whereϕ∈C1Z, Ris sequentially lower semicontinu-ous, bounded below, and∇ϕis weakly sequentially continuous;
2there existz0∈Z\ {0}, α >0, andR > r >0, such that
inf
N Iz≥α >0, supM0 Iz≤0. 2.10
Then, there existc >0 and a sequencezn⊂Zsuch that
Izn−→c, Izn−→0, as n−→ ∞. 2.11
Moreover,c≥α.
In the whole space R2, do ´O and Souto 15 proved a version of Trudinger-Moser
inequality, that is,
iifu∈H1R2, β >0, we have
R2
expβ|u|2−1dx <∞; 2.12
iiif 0< β <4π and|u|L2R2≤c,then there exists a constantc2c1c, βsuch that
sup
|∇u|L2R2≤1
R2
expβ|u|2−1dx < c2. 2.13
Definition 2.3. We sayfx, t has subcritical growth at∞,if for allβ >0,there exists a positive constantc3such that
fx, t≤c3exp
βt2, ∀x, t∈R2×0,∞. 2.14
3. Proof ofTheorem 2.2
In this section, we will proveTheorem 2.2. under our assumptions and2.14, there existcε >
0, β >0 such that
Fx, t,Gx, t≤ t2
2εcε
expβt2−1, ∀ε >0,∀t∈R. 3.1
Then, we obtain
Fx, u, Gx, v∈L2R2, ∀u, v∈H1R2. 3.2
Therefore, the functionalIu, vis well defined. Furthermore, using standard arguments, we obtain the functionalIu, visC1functional inZand
Iu, vφ, ψ
R2
∇u∇ψuψdx
R2
∇v∇φvφdx
−
R2
fx, uφgx, vψdx, ∀φ, ψ∈Z.
3.3
Consequently, the weak solutions of problemPare exactly the critical points ofIu, vinZ.
Now, we prove that the functionalIu, vsatisfied the geometry ofLemma 2.1.
Lemma 3.1. There existr >0andα >0such thatinfNIu, u≥α >0.
Proof. By2.14and assumptionH2, there existscε >0 such that
Fx, t, Gx, t≤ t
2
2εcεt
and thus onN,we have
Iu, u≥
R2
|∇u|2u2dx−
R2
εu2cεu3
expβu2−1dx
≥
R2
|∇u|2u2dx−ε
R2
u2dx−cε R2
u6dx
1/2
R2
expβu2−12dx
1/2
≥
R2
|∇u|2u2dx−ε
R2
u2dx−cεu3 R2
expβu2−1dx
1/2
.
3.5
So, by the Sobolev embedding theorem and2.12, we can chooser >0 sufficiently small, such that
Iu, u≥α >0, whenever ur. 3.6
Lemma 3.2. There existu0, u0∈Z\ {0}andR > r >0such thatsupM0I≤0.
Proof. 1By assumptionH3, we have onZ−
Iu, u
R2
|∇u|2u2dx−
R2
Fx, u Gx,−udx≤0 3.7
becauseFx, t≥0, Gx, t≥0 for anyx, t∈R2×R.
2AssumptionH3implies that there existc4>0, c5>0 such that
Fx, t, Gx, t≥c4tμ−c5, ∀t∈R. 3.8
Now, we chooseu0, u0∈Z\ {0}such thatu0, u0r, then
I−v, v λu0, u0
λ2
R2
|∇u0|2u20dx−
R2
|∇v|2v2dx
−
R2
Fλu0v
Gλu0−v
dx
≤ −
R2
|∇u|2u2dxcλ2−λμ.
3.9
Becauseμ >2,it follows that forw∈M0
Iw−→ −∞, whenever w −→ ∞, 3.10
Proof ofTheorem 2.2. ByLemma 3.1, there existr >0 andα >0 such that infNIu, u≥α >0.By
Lemma 3.2, there existu0, u0∈Z\{0}andR > r >0 such that supM0I≤0.SinceZZ⊕Z−,
we have
Iu, v
R2
∇u∇vuvdx−
R2
Fx, u Gx, vdx
Pu, v 2
2 −
Qu, v2
2 −ϕu, v, ∀u, v∈Z.
3.11
From2.14,3.1, and assumptionH3,ϕu, v∈ C1, ϕu, v ≥ 0 andϕu, vis sequentially
lower semicontinuous byZ ⊂L2locR2×L2
locR2and Fatou’s lemma;∇ϕis weakly sequentially
continuous. Thus, byLemma 2.1 there exists a sequenceun, vn⊂Zsuch that
Iun, vn−→c≥α, Iun, vn−→0. 3.12
Claim 3.3. There isc < ∞,such that un, vn ≤ cfor anyn.Indeed, from3.12, we obtain
that the sequenceun, vn⊂Zsatisfies
Iun, vn
cδn, I
un, vn
φ, ψ εnun, vn, asn−→ ∞, 3.13
where φ, ψ ∈ {un, vn}, δn → 0, εn → 0 as n → ∞. Taking φ, ψ {un, vn} in 3.13 and
assumptionH3, we have
R2
fx, un
ung
x, vn vn dx ≤2 R2
Fx, un
Gx, vn
dx2c2δnεnun, vn
≤ 2
μ
R2
fx, un
ung
x, vn
vn
dxC2δnεnun, vn,
3.14
whereCdepends only oncandηin assumptionH3. Sinceμ >2,we have1−2/μ>0,and thus
1− 2
μ
R2
fx, un
ung
x, vn
vn
dx≤C2δnεnun, vn, ∀n∈N. 3.15
On the other hand, letφ, ψ vn,0,φ, ψ 0, unin3.13, we obtain
vn2−
εnvn≤
R2f
x, un
vndx, un2−εnun≤
R2g
x, vn
undx. 3.16
that is,
vn≤ R2
fx, un vn
vndxεn, un≤
R2
gx, vn un
Now, we recall the following inequalitysee7, Lemma 2.4: mn≤ ⎧ ⎪ ⎨ ⎪ ⎩
en2
−1mlogm1/2, n≥0, m≥e1/4,
en2−11 2m
2, n≥0, 0≤m≤e1/4. 3.18
Letnvn/vnandmfx, un/c3,wherec3is defined in2.14, we have
c3
R2
fx, un
c3
vn
vndx≤c3 R2 exp vn vn 2 −1 dx c3
{x∈R2,fx,u
n/c3≥e1/4}
fx, un
c3
logf
x, un
c3
1/2
dx
c3
{x∈R2,fx,u
n/c3≤e1/4}
fx, u n c3 2 dx. 3.19
By2.12, we haveR2expvn/vn2−1dx <∞.By2.14, we have
logfx, t
c3 1/2
≤β1/2t. 3.20
Hence, we have
c3
R2
fx, un
c3
vn
vndx≤c6β1/2
R2f
x, un
undx 3.21
for some positive constantc6.So we have vn≤c6β1/2
R2f
x, un
undxεn. 3.22
Using a similar argument, we obtain
un≤c7β1/2
R2
gx, vn
vndxεn 3.23
for some positive constantc7.Combining3.22and3.23, we have un, vn≤c8
1δnεnun, vnεn
3.24
for some positive constantc8,which implies thatun, vn ≤ c. Thus, for a subsequence still
denoted byun, vn,there isu0, v0∈Zsuch that
un, vn
−→u0, v0
weakly in Z, as n−→ ∞,
un, vn
−→u0, v0
in LslocR2×LslocR2for s≥1, as n−→ ∞,
unx, vnx−→u0x, v0x
, almost every, inR2, as n−→ ∞.
Then, there exists hx ∈ H1R2such that |unx| ≤ h,∀x ∈ R2, ∀n ∈ N.From 2.12 and 2.14, we haveR2expβh2x−1dx < c,this implies
R2
fx, un
φdx−→
R2
fx, u0
φdx, as n−→ ∞. 3.26
Similarly, we can obtain
R2
gx, vn
ψdx−→
R2
gx, v0
ψdx, asn−→ ∞. 3.27
From these, we haveIun, vnφ, ψ 0, sou0, v0is weak solution of problemP.
Claim 3.4. u0, v0 is nontrivial. By contradiction, since fx, t has subcritical growth, from 2.14and H ¨older inequality, we have
R2
fx, un
undx≤c
R2
un
expβu2n−1dx
≤c
R2|un| qdx
1/q
R2
expβqu2n−1dx
1/q
,
3.28
where 1/q1/q1.Choosing suitableβandq,we have
R2
expβqu2n−1dx≤c. 3.29
Then, we obtain
R2f
x, un
undx≤c
R2
unq
dx
1/q
. 3.30
Sinceun→0 inLq
R2,asn→ ∞,this will lead to
R2
fx, un
undx−→0, asn−→ ∞. 3.31
Similarly, we have
R2
gx, vn
vndx−→0, as n−→ ∞. 3.32
Using assumptionH3, we obtain
R2
Fx, un
dx−→0,
R2
Gx, vn
dx−→0, asn−→ ∞. 3.33
This together withIun, vnun, vn→0,we have
R2
∇un∇vnunvn
dx−→0, asn−→ ∞. 3.34
Thus, we see that
Iun, vn
−→0, as n−→ ∞. 3.35
which is a contradiction toIun, vn→c≥α >0,asn→ ∞.
Acknowledgment
This work is supported by Innovation Program of Shanghai Municipal Education Commission under Grant no. 08 YZ93.
References
1F. Colin and M. Frigon, “Systems of coupled Poisson equations with critical growth in unbounded domains,”Nonlinear Differential Equations and Applications, vol. 13, no. 3, pp. 369–384, 2006.
2Y. Ding and S. Li, “Existence of entire solutions for some elliptic systems,”Bulletin of the Australian Mathematical Society, vol. 50, no. 3, pp. 501–519, 1994.
3D. G. de Figueiredo, “Nonlinear elliptic systems,”Anais da Academia Brasileira de Ciˆencias, vol. 72, no. 4, pp. 453–469, 2000.
4D. G. de Figueiredo, J. M. do ´O, and B. Ruf, “Critical and subcritical elliptic systems in dimension two,”Indiana University Mathematics Journal, vol. 53, no. 4, pp. 1037–1054, 2004.
5D. G. de Figueiredo and J. Yang, “Decay, symmetry and existence of solutions of semilinear elliptic systems,”Nonlinear Analysis: Theory, Methods&Applications, vol. 33, no. 3, pp. 211–234, 1998.
6G. Li and J. Yang, “Asymptotically linear elliptic systems,” Communications in Partial Differential Equations, vol. 29, no. 5-6, pp. 925–954, 2004.
7Adimurthi and S. L. Yadava, “Multiplicity results for semilinear elliptic equations in a bounded domain ofR2involving critical exponents,”Annali della Scuola Normale Superiore di Pisa, vol. 17, no. 4, pp. 481–504, 1990.
8D. G. de Figueiredo, O. H. Miyagaki, and B. Ruf, “Elliptic equations inR2 with nonlinearities in the critical growth range,”Calculus of Variations and Partial Differential Equations, vol. 3, no. 2, pp. 139–153, 1995.
9Y. Shen, Y. Yao, and Z. Chen, “On a class of nonlinear elliptic problem with critical potential inR2,”
Science in China Series A, vol. 34, pp. 610–624, 2004.
10D. M. Cao, “Nontrivial solution of semilinear elliptic equation with critical exponent in R2,”
Communications in Partial Differential Equations, vol. 17, no. 3-4, pp. 407–435, 1992.
11C. O. Alves, J. M. do ´O, and O. H. Miyagaki, “On nonlinear perturbations of a periodic elliptic problem inR2involving critical growth,”Nonlinear Analysis: Theory, Methods&Applications, vol. 56, no. 5, pp. 781–791, 2004.
12W. Kryszewski and A. Szulkin, “Generalized linking theorem with an application to a semilinear Schr ¨odinger equation,”Advances in Differential Equations, vol. 3, no. 3, pp. 441–472, 1998.
13M. Willem, Minimax Theorems, vol. 24 of Progress in Nonlinear Differential Equations and Their Applications, Birkh¨auser, Boston, Mass, USA, 1996.
14G. Li and A. Szulkin, “An asymptotically periodic Schr ¨odinger equation with indefinite linear part,”
Communications in Contemporary Mathematics, vol. 4, no. 4, pp. 763–776, 2002.
