Hindawi Publishing Corporation Boundary Value Problems Volume 2008, Article ID 217636,10 pages doi:10.1155/2008/217636

Research Article

Existence Result for a Class of

Elliptic Systems with Indefinite Weights in R2

Guoqing Zhang1 and Sanyang Liu2

1 College of Sciences, University of Shanghai for Science and Technology, Shanghai 200093, China

2 Department of Applied Mathematics, Xidian University, Xi'an 710071, China

Correspondence should be addressed to Guoqing Zhang, zgqw2001@sina.com.cn 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 in R2. The proofs base on Trudinger-Moser inequality and a generalized linking theorem introduced by Kryszewski and Szulkin.

Copyright © 2008 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

~Au + u = g(X/V)/ x e R2' (11) [-Av + v = f (X/U), x e R2/

where f (x, t) and g(x, t) are continuous functions on R2 x R and have the maximal growth on t which allows to treat problem (P) variationally, A is the Laplace operator.

Recently, there exists an extensive bibliography in the study of elliptic problem in RN [1-6]. As dimensions N > 3, in 1998, de Figueiredo and Yang [5] considered the following coupled elliptic systems:

-Am + u = g (x, v), x e RN, -Av + v = f (x, u), x e Rn,

where f, g are radially symmetric in x and satisfied the following Ambrosetti-Rabinowitz condition:

if (x,s)ds > c\t\2+61, f g(x,s)ds > c\t\2+62, Wt e R, (1.3)

and for some S1 > 0, S2 > 0. They obtained the decay, symmetry, and existence of solutions for problem (1.2). In 2004, Li and Yang [6] proved that problem (1.2) possesses at least a positive solution when the nonlinearities f (x, t) and g(x, t) are "asymptotically linear" at infinity and "superlinear" at zero, that is,

(1) limt^(f (x,t)/t) = I > 1, limt^^(g(x,t)/t) = m> 1, uniformly in x e RN;

(2) limt^0(f (x, t)/t) = limt^0(g(x, t)/t) = 0, uniformly with respect to x e RN.

In 2006, Colin and Frigon [1] studied the following systems of coupled Poission equations with critical growth in unbounded domains:

-Au = \v\T 2v, -Av = \m\2*-2m,

where 2* = 2N/(N - 2) is critical Sobolev exponent, u,v e d1,2(Q*) and Q* = RN \ E with E = UaeZNa + w* for a domain containing the origin w* c w* c B(0,1/2). Here, B(0,1/2) 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 in N = 2 replace the Sobolev embedding theorem used in N > 3.

In dimension N = 2, Adimurth and Yadava [7], de Figueiredo et al. [8] discussed the solvability of problems of the type

-Au = f (x,u), x e Q,,

u = 0, x e dQ,

where Q is some bounded domain in R2. Shen et al. [9] considered the following nonlinear elliptic problems with critical potential:

Au - u-- = f (x,u), x e Q

(\x\log (R/\x\))2 (1.6)

u = 0, x e dQ,

and obtained some existence results. In the whole space R2, some authors considered the following single semilinear elliptic equations:

-Au + V(x)u = f (x,u), x e R2.

As the potential V(x) and the nonlinearity f (x,t) are asymptotic to a constant function, Cao [10] obtained the existence of a nontrivial solution. As the potential V(x) and the nonlinearity f (x,t) are asymptotically periodic at infinity, Alves et al. [11] proved 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 Szulkin [12] and Trudinger-Moser inequality, we prove an existence result for problem (P).

The paper is organized as follows. In Section 2, we recall some results and state our main results. In Section 3, main result is proved.

2. Preliminaries and main results

Consider the Hilbert space [13]

H 1(R2) = {u e L2(R2), Vu e L2(R2)}, (2.1)

and denote the product space Z = H 1(R2) x H 1(R2) endowed with the inner product:

((u,v), ($,f)) = (VuVfy + u<p)dx + (VvVf + Vf)dx, V($,f) e Z. (2.2)

Jr2 Jr2

If we define

Z+ = {(u,u) e Z}, Z- = {(v,-v) e Z}. It is easy to check that Z = Z+ e Z-, since

11 (u,v) = ^ (u + v,u + v) + ^ (u - v,v - u).

Let us denote by P (resp., Q) the projection of Z on to Z+ (resp., Z-), we have

1 (HP(u,v)\\2 - \\Q(u,v)\\2) = 1

2 (u + v,u + v)

2 (u - v,v - u)

4(12(|Vu|2 + |Vv|2 + 2VuVv)dx + j 2(|u|2 + |v|2 + 2uv)dx

-| (|Vu|2 + |Vv|2 - 2VuVv)dx -| (| u |2 + |v|2 - 2uv)dx) Jr2 Jr2 /

(VuVv + uv)dx.

Now, we define the functional

I(u,v) = (VuVv + uv)dx - (F(x,u) + G(x,v))dx JR2 JR2

\\P(u,v)\\2 \\Q(u,v)\\2 = 11 V ' J" - "Qy ' J" - q(u,v), V(u,v) e Z,

y(u,v) = (F(x,u) + G(x,v)) dx. (2.7)

Let z0 e Z+ \{0} and let R> r > 0, we define

M = {z = z- + Xz0 : z e Z||zM <R, ^ > 0},

Mo = {z = z~ + Xz0 : z~ e Z~, ||z| = R and 1 > 0or ||z| <Rand 1 > 0}, (2.8)

N = {z e Z+ : ||z|| = r}.

Here, we assume the following condition: (H1) f,g e C(R2 x R, R);

(H2) limf^0(f (x, t)/t) = limf^0(g(x, t)/t) = 0 uniformly with respect to x e R2; (H3) there exist ¡i> 2 and n> 0 such that

0 < ¡F(x, t) < tf (x, t), 0 < ¡¡G(x, t) < tg(x, t), V|t| > n. (2.9)

Lemma 2.1 (see [12,14]). Assume (H1), (H2), and (H3), and suppose

(1) I (z) = (1/2)(||Pz||2-||Qz||2) - y(z), where y e C1(Z,R) is sequentially lower semicontinu-ous, bounded below, and Vy is weakly sequentially continuous;

(2) there exist z0 e Z+ \ {0}, a > 0, and R> r > 0, such that

inf I(z) > a > 0, sup I(z) < 0. (2.10)

Then, there exist c > 0 and a sequence (zn) c Z such that

I (zn) —> c, I'(zn) —> 0, as n —. (2.11)

Moreover, c > a.

Theorem 2.2. Under the assumptions (H1), (H2), and (H3), if f and g has subcritical growth (see definition below), problem (P) possesses a nontrivial weak solution.

In the whole space R2, do Ô and Souto [15] proved a version of Trudinger-Moser inequality, that is,

(i) if u e H1 (R2), ¡> 0, we have

| ( exp (¡|u|2) - 1)dx < +œ; (2.12)

(ii) if 0 < ¡3 < 4n and |u|L2(R2) < c, then there exists a constant c2 = c1 (c,p) such that

sup ( exp (¡|u|2) - 1)dx<c2. (2.13)

|Vu|12(R2)<1 Jr1

Definition 2.3. We say f (x, t) has subcritical growth at +<x>, if for all ¡3 > 0, there exists a positive constant c3 such that

f (x,t) < c3 exp (312), V(x,t) e R2 x [0, +œ). (2.14)

3. Proof of Theorem 2.2

In this section, we will prove Theorem 2.2. under our assumptions and (2.14), there exist c£ > 0,3 > 0 such that

\F(x,t)\,\G(x,t)\< - £ + c£( exp (pt2) - 1), V£> 0, Vt e R. (3.1)

Then, we obtain

F (x, u), G(x, v) e L2 (R2), Vu, v e H1 (R2). (3.2)

Therefore, the functional I(u,v) is well defined. Furthermore, using standard arguments, we obtain the functional I (u,v) is C1 functional in Z and

I'(u,v)(fy,f) = (VuVf + uf)dx + (VvVty + v$)dx

Jr2 Jr2 (3.3)

- (f(x,u)fy + g(x,v)f)dx, V(<p,f) e Z.

Consequently, the weak solutions of problem (P) are exactly the critical points of I(u,v) in Z. Now, we prove that the functional I(u, v) satisfied the geometry of Lemma 2.1.

Lemma 3.1. There exist r > 0 and a> 0 such that infNI(u, u) > a> 0.

Proof. By (2.14) and assumption (H2), there exists c£ > 0 such that

t2 ( ( ) ) F(x, t),G(x, t) < - £ + c£t3(exp (fit2) - 1), Vt e R, (3.4)

and thus on N, we have

I(u,u) > (|Vu|2 + u2)dx - (eu2 + c£u3( exp (ßu2) - 1))dx Jr2 Jr2

>J (|Vu|2 + u2)dx - eJ u2dx - cu6dx^ ^ J (exp (ßu2) - 1))2dx^ >J (|Vu|2 + u2)dx - ej u2dx - ce||u||3^J exp ((ßu2) - 1)dx^ .

So, by the Sobolev embedding theorem and (2.12), we can choose r > 0 sufficiently small, such that

I(u,u) > a> 0, whenever ||u|| = r. (3.6)

Lemma 3.2. There exist (u0,u0) e Z+ \ {0} and R> r > 0 such that supM I < 0. Proof. (1) By assumption (H3), we have on Z-

I(u,u)=\ (|Vu|2 + u2)dx -| (F(x,u) + G(x, -u))dx < 0 (3.7)

Jr2 Jr2

because F(x, t) > 0, G(x, t) > 0 for any (x, t) e R2 x R.

(2) Assumption (H3) implies that there exist c4 > 0, c5 > 0 such that

F(x, t), G(x, t) > c^ - c5, Vt e R. (3.8)

Now, we choose (u0,u0) e Z+ \ {0} such that || (u0, u0) | = r, then

I((-v,v) + X(u0,u0)) = X2 (|Vuo|2 + u0)dx - (|Vv|2 + v2)dx

Jr2 Jr2

- I (F(Xu0 + v) + G(Xu0 - v))dx (3.9)

<-f (|Vu|2 + u2)dx + c(X2 -X^). Jr2

Because ¡> 2, it follows that for w e M0

I(w) —> -to, whenever ||w|| —> to, (3.10)

and so, taking R> r large, we get supM I < 0. □

Proof of Theorem 2.2. By Lemma 3.1, there exist r > 0 and a> 0 such that infNI (u,u) > a> 0. By Lemma 3.2, there exist (u0,u0) e Z+ \{0} and R> r > 0 such that supM I < 0. Since Z = Z+e Z-, we have

I(u,v) = (VuVv + uv)dx - (F(x,u) +G(x,v))dx jr2 jr2

2 2 (3.11)

= HP M\\2 - \\Q(u2V)\\ - ^v), ,uv) e Z.

From (2.14), (3.1), and assumption (H3), y(u,v) e C1, y(u,v) > 0 and y(u,v) is sequentially lower semicontinuous by Z c Ljoc(R2) x Ljoc(R2) and Fatou's lemma; Vy is weakly sequentially continuous. Thus, by Lemma 2.1 there exists a sequence (un,vn) c Z such that

I(un,vn) —> c > a, I'(un,vn) —> 0. (3.12)

Claim 3.3. There is c < such that \\(un,vn)\\ < c for any n. Indeed, from (3.12), we obtain that the sequence (un,vn) c Z satisfies

I(un,vn) = c + 6n, I'(un,vn)($,f) = en\\ (un,vn)\\, as n —> to, (3.13)

where e {un,vn}, 6n ^ 0, en ^ 0 as n ^ to. Taking = {un,vn} in (3.13) and

assumption (H3), we have

(f(x,un)un + g(x,vn)vn)dx

< 2 (F(x,un) + G(x,vn))dx + 2c + 26n + £n\\(un,vn)\\ (3.14)

< - \ ((f(x,un)un + g(x,vn)vn))dx + C + 26n + £n\\(un,vn)\\, ft Jr2

where C depends only on c and n in assumption (H3). Since ¡i> 2, we have (1 - 2/p) > 0, and thus

0 - m)/ ((f (x,un)un + g(x,vn)vn))dx < C + 26„ + £n\\ (un, vn) \\, Vn e N. (3.15) On the other hand, let ($, f) = (vn, 0), ($, f) = (0, un) in (3.13), we obtain

\\vn\\2 - £„\\v„\\ < f(x,un)vndx, \\un\\2 - £n \ un \ < g(x,vn)undx. (3.16)

Jr2 Jr2

that is,

( ) vn ( ) un

vn\ < f(x,un) \ n\ dx + sn, \un\ < g{x,vn) \ n\ dx + sn. (3.17) Jr2 \\vn\ Jr2 \\un\\

Now, we recall the following inequality (see [7, Lemma 2.4]):

(e"2 - 1) + m(log m)1/2, n > 0, m > e1/4,

(e"2 - 1) + 2m2,

>0, 0 < m < e1/4.

Let n = vn/||vn|| and m = f (x,un)/c3, where c3 is defined in (2.14), we have

f f(x,un) vn ,

c3 -71—iïdx < C3

J R2 C3 IIv"M

f x, un

{xeR2,f(x,un)/c3>e1/4} c3

f x, un

{xeR2,f(x,un)/c3<e1/4} \ c3

f x, un

By (2.12), we have ¿2 [exp (vn/||vn||)2 - 1]dx < +œ. By (2.14), we have

f (x,t)

< ß1/2t.

Hence, we have

f f x, un vn

c^ -71—rdx < c6 +

JR2 c3 \\vJ\

ß1/2i f (x,un) R2

u^n/ undx

for some positive constant c6. So we have

\\ vn \ < c6 + ß1/2\ f (x,un) R2

un} undx + £n.

Using a similar argument, we obtain

\\un\\ < c7 + ft/2\ g(x,vn) JR.2

vn) vn dx + £n

for some positive constant c7. Combining (3.22) and (3.23), we have

\\ (un, vn) \\ < c^1 + 6n + £n\\ (un, vn) \\ + £n)

(3.18)

(3.19)

(3.20)

(3.21)

(3.22)

(3.23)

(3.24)

for some positive constant c8, which implies that H(un,vn)H < c. Thus, for a subsequence still denoted by (un,vn), there is (u0,v0) e Z such that

(■un,vn) —> (u0,v0) weakly in Z, as n —> to (■un(x),vn(x)) —> (u0(x),v0(x)), almost every, inR2, as n —> to

(un,vn) —> (uo,vo) in Lfoc(R2) X LSoc(R2) for s > 1, as n —> to, (3.25)

Then, there exists h(x) e H1(R2) such that un(x^ < h, Vx e R2, Vn e N. From (2.12) and (2.14), we have R-(exp(ph2(x)) - 1)dx < c, this implies

f{x,un)fydx —> f(x,u0)fydx, as n —. (3.26)

jr2 jr2

Similarly, we can obtain

g(x,vn)fdx —> g(x,v0)fdx, as n —. (3.27)

jr2 jr2

From these, we have I'(un,vn)(ty,y) = 0, so (u0,v0) is weak solution of problem (P).

Claim 3.4. (u0,v0) is nontrivial. By contradiction, since f (x,t) has subcritical growth, from (2.14) and Holder inequality, we have

(3.28)

f(x,un)undx < c un( exp (fiu^) - 1)d. Jr2 Jr2

< c| |un|q dx^j (exp (fiqu2n) - 1)dx^j ^

where 1/q1 + 1/q = 1. Choosing suitable fi and q, we have

(exp (fiqu2n) - 1)dx < c. (3.29)

Then, we obtain

j" f (x,un)undx < c^J \un\q dx^j . (3.30)

Since un ^ 0 in Lq(R2), as n ^to, this will lead to

f(x,un)undx —> 0, as n —> to. (3.31)

Similarly, we have

g(x,vn)vndx —> 0, as n —> to. (3.32)

Using assumption (H3), we obtain

F(x,un)dx —> 0, G(x,vn)dx —> 0, as n —>to. (3.33)

Jr2 Jr2

(3.34)

This together with I'(un,vn)(un,vn) ^ 0, we have

(Vu„Vv„ + unvn)dx —> 0, as n —>to.

Thus, we see that

I(un,vn) —> 0, as n —> œ. (3.35)

which is a contradiction to I(un, vn) ^ c > a> 0, as n ^œ.

Consequently, we have a nontrivial critical point of the functional I(u,v) and conclude the proof of Theorem 2.2. □

Acknowledgment

This work is supported by Innovation Program of Shanghai Municipal Education Commission

under Grant no. 08 YZ93.

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