Electronic Journal of Differential Equations, Vol. 205 (205), o. 63, pp. 23. ISS: 072-669. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu ftp ejde.math.txstate.edu EXISTECE OF SOLUTIOS FOR A VARIABLE EXPOET SYSTEM WITHOUT PS CODITIOS LI YI, YUA LIAG, QIHU ZHAG, CHUSHA ZHAO Abstract. In this article, we study the existence of solution for the following elliptic system of variable exponents with perturbation terms div u p(x) 2 u) + u p(x) 2 u = λa(x) u γ(x) 2 u + F u(x, u, v) in R, div v q(x) 2 v) + v q(x) 2 v = λb(x) v δ(x) 2 v + F v(x, u, v) in R, u W,p( ) (R ), v W,q( ) (R ), where the corresponding functional does not satisfy PS conditions. We obtain a sufficient condition for the existence of solution and also present a result on asymptotic behavior of solutions at infinity.. Introduction The study of differential equations and variational problems with variable exponent has attracted intense research interests in recent years. Such problems arise from the study of electrorheological fluids, image processing, and the theory of nonlinear elasticity [, 7, 9, 26]. The following variable exponent flow is an important model in image processing [7]: u t div u p(x) 2 u) + λ(u u 0 ) = 0, in Ω [0, T ], u(x, t) = g(x), on Ω [0, T ], u(x, 0) = u 0. The main benefit of this flow is the manner in which it accommodates the local image information. We refer to [4, 8, 24] for the existence of solution of variable exponent problems on bounded domain. In this article, we consider the existence of solutions for the system div u p(x) 2 u) + u p(x) 2 u = λa(x) u γ(x) 2 u + F u (x, u, v) in R, div v q(x) 2 v) + v q(x) 2 v = λb(x) v δ(x) 2 v + F v (x, u, v) in R, u W,p( ) (R ), v W,q( ) (R ), (.) 2000 Mathematics Subject Classification. 35J47. Key words and phrases. Variable exponent system; integral functional; PS condition; variable exponent Sobolev space. c 205 Texas State University - San Marcos. Submitted September 29, 204. Published March 3, 205.
2 L. YI, Y. LIAG, Q. ZHAG, C. ZHAO EJDE-205/63 where p, q C(R ) are Lipschitz continuous and p( ), q( ) >>, the notation h ( ) >> h 2 ( ) means ess inf x R (h (x) h 2 (x)) > 0, p(x) u := div u p(x) 2 u) which is called the p(x)-laplacian. When p( ) p (a constant), p(x)-laplacian becomes the usual p-laplacian. The terms λa(x) u γ(x) 2 u and λb(x) v δ(x) 2 v are the perturbation terms. The p(x) -Laplacian possesses more complicated nonlinearities than the p-laplacian (see [5]). Many methods and results for p-laplacian are invalid for p(x)-laplacian. The PS condition is very important in the study of the existence of solution via variational methods. According to [2, Theorem 2.8], if a C (X, R) functional f satisfies the Mountain Pass Geometry, then it has a PS sequence {x n } which satisfies f(x n ) c which is the mountain pass level and f (x n ) 0. By [2, Theorem 2.9] it follows that if f also satisfies the PS condition, passing to a subsequence, then x n x 0 in X, and then x 0 is a critical point of f, that is f (x 0 ) = 0. In the study of this problems in the bounded domain, since we have the compact embedding from a Sobolev space to a Lebesgue space, so we have the PS condition when we study the case of subcritical growth condition. For the unbounded domain, we cannot get the compact embedding in general, so we do not have the PS condition. It is well known that a main difficulty in the study of elliptic equations in R is the lack of compactness. Many methods have been used to overcome this difficulty. One type of methods is that under some additional conditions we can recover the required compact imbedding theorem, for example, the weighting method [6, 25], and the symmetry method [23]. If equations are periodic, the corresponding energy functionals are invariant under period-translation. We refer to [2] [5] and references cited therein for the applications of this method to the p-laplacian equations, the Schrödinger equations and the biharmonic equations etc. Sometimes we can compare the original equation with its limiting equation at infinity. Especially, we can compare the corresponding critical values of the functionals for these two equations when the existence of the ground state solution for the limiting equation is known. Usually the limiting equations are homogeneous, but in [2]-[5] the limiting equations are periodic. We also refer to [3] for the existence of solution for p(x)-laplacian equations with periodic conditions. In this article we consider the existence and the asymptotic behavior of solutions near infinity for a variable exponent system with perturbations that does not satisfy periodic conditions, which implies the corresponding functional does not satisfy PS conditions on unbounded domain. We will also give a sufficient condition for the existence of solutions for the system (.). Our method is to compare the original equation with its limiting equation at infinity without perturbation. These results also partially generalize the results in [3] and [20]. In this article, we make the following assumptions. (A0) p( ), q( ) are Lipschitz continuous, << p( ), q( ) <<, << γ( ) << p( ) p( ), a( ) L p( ) γ( ) (R ), << δ( ) << q( ), b( ) L q( ) δ( ) (R ), F C (R R 2, R) satisfies F u (x, u, v) C( u p(x) + u α(x) + v q(x)/α0 (x) + v β(x)/α0 (x) ), F v (x, u, v) C( v q(x) + v β(x) + u p(x)/β0 (x) + u α(x)/β0 (x) ), q( )
EJDE-205/63 EXISTECE OF SOLUTIOS 3 where F u = u F, F v = v F, α, β C(R ) satisfy p( ) << α( ) << p ( ), q( ) << β( ) << q ( ), h 0 ( ) denotes the conjugate function of h( ), that is { p p(x)/( p(x)), p(x) <, (x) =, p(x). h(x) + h 0 (x), and (A) There exist constants > p + and 2 > q +, such that F satisfies the following conditions 0 sf s (x, s, t), 0 tf t (x, s, t), (x, s, t) R R R, 0 < F (x, s, t) sf s (x, s, t) + 2 tf t (x, s, t), x R, (s, t) R R\{(0, 0)}. (A2) For (s, t) R 2, the function sf s (x, τ / s, τ /2 t)/τ and the function tf t (x, τ / s, τ /2 t)/τ 2 2 are increasing with respect to τ > 0. (A3) There is a measurable function F (s, t) such that lim F (x, s, t) = F (s, t) x + for bounded s + t uniformly, and F (s, t) + F s (s, t)s + F t (s, t)t C( s p+ + s α + t q+ + t β ), (s, t) R 2, and when x R the following inequalities hold F (x, s, t) F (s, t) ε(r)( s p(x) + s p (x) + t q(x) + t q (x) ), F s (x, s, t) F s (s, t) ε(r)( s p(x) + s p (x) + t q(x)(p (x) )/p (x) + t q (x)(p (x) )/p (x) ), F t (x, s, t) F t (s, t) ε(r)( s p(x)(q (x) )/q (x) + s p (x)(q (x) )/q (x) + t q(x) + t q (x) ), where ε(r) satisfies lim R + ε(r) = 0. This article is organized as follows. In Section 2, we introduce some basic properties of the Lebesgue-Sobolev spaces with variable exponents and p(x)-laplacian. In Section 3, we give the main results and the proofs. 2. otation and preliminary results Throughout this paper, the letters c, c i, C i, i =, 2,..., denote positive constants which may vary from line to line but are independent of the terms which will take part in any limit process. To discuss problem (.), we need some preparations on space W,p( ) (Ω) which we call variable exponent Sobolev space, where Ω R is an open domain. Firstly, we state some basic properties of spaces W,p( ) (Ω) which we will use later (for details, see [9,, 2, 4]). Denote C + (Ω) = {h C(Ω), h(x) for x Ω}, h + Ω = ess sup x Ω h(x), h Ω = ess inf x Ω h(x), for any h L (Ω), h + = ess sup x R h(x), h = ess inf x R h(x), for any h L (R ), S(Ω) = {u : u is a real-valued measurable function on Ω},
4 L. YI, Y. LIAG, Q. ZHAG, C. ZHAO EJDE-205/63 L p( ) (Ω) = {u S(Ω) : Ω u(x) p(x) dx < }. In this section, p( ) and p i ( ) are Lipschitz continuous unless otherwise noted. We introduce the norm on L p( ) (Ω) by u p( ),Ω = inf{λ > 0 : u(x) λ p(x) dx }, and (L p( ) (Ω), p( ),Ω ) becomes a Banach space, we call it variable exponent Lebesgue space. If Ω = R, we will simply denote by p( ) the norm on L p( ) (R ). Proposition 2. ([9]). (i) The space (L p( ) (Ω), p( ),Ω ) is a separable, uniform convex Banach space, and its conjugate space is L p0 ( ) (Ω), where p(x) + p 0 (x). For any u L p( ) (Ω) and v L p0 ( ) (Ω), we have uv dx ( Ω p + Ω (p 0 ) ) u p( ),Ω v p0 ( ),Ω. Ω (ii) If Ω is bounded, p, p 2 C + (Ω), p ( ) p 2 ( ) for any x Ω, then L p2( ) (Ω) L p( ) (Ω), and the imbedding is continuous. Proposition 2.2 ([9]). If f : Ω R R is a Caratheodory function and satisfies f(x, s) h(x) + d s p(x)/p2(x) for any x Ω, s R, where p, p 2 C + (Ω), h L p2( ) (Ω), h(x) 0, d 0, then the emytskii operator from L p( ) (Ω) to L p2( ) (Ω) defined by ( f u)(x) = f(x, u(x)) is continuous and bounded. Proposition 2.3 ([9]). If we denote ρ(u) = u p(x) dx, Ω Ω u L p( ) (Ω), then (i) u p( ),Ω < (= ; > ) ρ(u) < (= ; > ); (ii) u p( ),Ω > = u p p( ),Ω ρ(u) u p+ p( ),Ω ; u p( ),Ω < = u p p( ),Ω ρ(u) u p+ p( ),Ω ; (iii) u p( ),Ω 0 ρ(u) 0; u p( ),Ω ρ(u). Proposition 2.4 ([9]). If u, u n L p( ) (Ω), n =, 2,..., then the following statements are equivalent. () lim n u n u p( ),Ω = 0; (2) lim n ρ(u n u) = 0; (3) u n u in measure in Ω and lim n ρ(u n ) = ρ(u). Denote Y = k i= Lpi( ) (Ω) with the norm k y Y = y i pi( ),Ω, y = (y,..., y k ) Y, i= where p i ( ) C + (Ω), i =,..., m, then Y is a Banach space. With a proof similar to proof in [6], we have:
EJDE-205/63 EXISTECE OF SOLUTIOS 5 Proposition 2.5. Suppose f(x, y) : Ω R k R m is a Caratheodory function; that is, f satisfies (i) For a.e. x Ω, y f(x, y) is a continuous function from R k to R m, (ii) For any y R k, x f(x, y) is measurable. If there exist p ( ),..., p k ( ) C + (Ω), β( ) C(Ω), ρ( ) L β( ) (Ω) and positive constant c > 0 such that k f(x, y) ρ(x) + c y i pi(x)/β(x) for any x Ω, y R k, i= then the emytskii operator from Y to (L β( ) (Ω)) m defined by ( f u)(x) = f(x, u(x)) is continuous and bounded. The space W,p( ) (Ω) is defined by with the norm W,p( ) (Ω) = {u L p( ) (Ω) : u (L p( ) (Ω)) }, u p( ),Ω = u p( ),Ω + u p( ),Ω, Ω u W,p( ) (Ω). If Ω = R, we will denote the norm on W,p( ) (R ) as u p( ). Denote u p( ),Ω = inf{λ > 0 : u Ω λ p(x) dx + u(x) Ω λ p(x) dx }, v q( ),Ω = inf{λ > 0 : v λ q(x) dx + v(x) λ q(x) dx }. It is easy to see that the norm p( ),Ω is equivalent to p( ),Ω on W,p( ) (Ω), and q( ),Ω is equivalent to q( ),Ω on W,q( ) (Ω). In the following, we will use p( ),Ω instead of p( ),Ω on W,p( ) (Ω), and use q( ),Ω instead of q( ),Ω on W,q( ) (Ω). We denote by W,p( ) 0 (Ω) the closure of C 0 (Ω) in W,p( ) (Ω). Proposition 2.6 ([8, 9, ]). (i) W,p( ) (Ω) and W,p( ) 0 (Ω) are separable reflexive Banach spaces; (ii) If p( ) is Lipschitz continuous, α( ) is measurable, and satisfies p( ) α( ) p ( ) for any x Ω, then the imbedding from W,p( ) (R ) to L α( ) (R ) is continuous; (iii) If Ω is bounded, α C + (Ω) and α( ) < p ( ) for any x Ω, then the imbedding from W,p( ) (Ω) to L α( ) (Ω) is compact and continuous. Proposition 2.7 ([2, Lemma 3.]). Assume that p : R R is a uniformly continuous function, if {u n } is bounded in W,p( ) (R ) and u n ρ(x) dx 0, n +, sup y R B(y,r) for some r > 0 and some ρ L + (R ) satisfying p( ) ρ( ) << p ( ), then u n 0 in L α( ) (R ) for any α satisfying p( ) << α( ) << p ( ), where B(y, r) is an open ball with center y and radius r. Ω
6 L. YI, Y. LIAG, Q. ZHAG, C. ZHAO EJDE-205/63 Denote X = W,p( ) (R ), X 2 = W,q( ) (R ), X = X X 2. Let us endow the norm on X as (u, v) = max{ u p( ), v q( ) }. The dual space of X will be denoted by X, then for any Θ X, there exist f (W,p( ) (R )) and g (W,q( ) (R )) such that Θ(u, v) = f(u) + g(v). We denote,,p( ) and,q( ) the norms of X, (W,p( ) (R )) and (W,q( ) (R )), respectively. Obviously X = (W,p( ) (R )) (W,q( ) (R )) and Θ = f,p( ) + g,q( ), Θ X. For every (u, v) and (ϕ, ψ) in X, set Φ (u) = R p(x) u p(x) dx + R p(x) u p(x) dx, Φ 2 (v) = R q(x) v q(x) dx + R q(x) v q(x) dx, Φ(u, v) = Φ (u) + Φ 2 (v), Ψ(u, v) = {λ[ a(x) R γ(x) u γ(x) + b(x) δ(x) v δ(x) ] + F (x, u, v)} dx, Ψ (u, v) = λ a(x) R γ(x) u γ(x) dx + λ b(x) R δ(x) v δ(x) dx. It follows from Proposition 2.5 that Φ C (X, R), then Φ (u, v)(ϕ, ψ) = D Φ(u, v)(ϕ) + D 2 Φ(u, v)(ψ), (ϕ, ψ) X, Ψ (u, v)(ϕ, ψ) = D Ψ(u, v)(ϕ) + D 2 Ψ(u, v)(ψ), (ϕ, ψ) X, where D Φ(u, v)(ϕ) = u p(x) 2 u ϕ dx + R u p(x) 2 uϕ dx = Φ (u)(ϕ), R ϕ X, D 2 Φ(u, v)(ψ) = v p(x) 2 v ψ dx + R v p(x) 2 vψ dx = Φ 2(v)(ψ), R ψ X 2, D Ψ(u, v)(ϕ) = [λa(x) u γ(x) 2 u + R u F (x, u, v)]ϕ dx, ϕ X, D 2 Ψ(u, v)(ψ) = [λb(x) v δ(x) 2 v + R v F (x, u, v)]ψ dx, ψ X 2. The integral functional associated with the problem (.) is J(u, v) = Φ(u, v) Ψ(u, v). Without loss of generality, we may assume that F (x, 0, 0) = 0, then we have F (x, u, v) = 0 [u 2 F (x, tu, tv) + v 3 F (x, tu, tv)]dt, where j denotes the partial derivative of F with respect to its j-th variable. The condition (A0) holds F (x, u, v) c( u p(x) + u α(x) + v q(x) + v β(x) ). (2.)
EJDE-205/63 EXISTECE OF SOLUTIOS 7 From Proposition 2.5 and condition (A0), it is easy to see that J C (X, R) and satisfies where J (u, v)(ϕ, ψ) = D J(u, v)(ϕ) + D 2 J(u, v)(ψ), (ϕ, ψ) X, D J(u, v)(ϕ) = D Φ(u, v)(ϕ) D Ψ(u, v)(ϕ), ϕ X, D 2 J(u, v)(ψ) = D 2 Φ(u, v)(ψ) D 2 Ψ(u, v)(ψ), ψ X 2. Obviously, J (u, v) = D J(u, v),p( ) + D 2 J(u, v),q( ). We say (u, v) X is a critical point of J if J (u, v)(ϕ, ψ) = 0, (ϕ, ψ) X. Proposition 2.8 ([22]). (i) Φ is a convex functional; (ii) Φ is strictly monotone, that is, for any (u, v ), (u 2, v 2 ) X with (u, v ) (u 2, v 2 ), we have (Φ (u, v ) Φ (u 2, v 2 ))(u u 2, v v 2 ) > 0, (iii) Φ is a mapping of type (S + ), that is if (u n, v n ) (u, v) in X and lim sup[φ (u n, v n ) Φ (u, v)](u n u, v n v) 0, n then (u n, v n ) (u, v) in X. (iv) Φ : X X is a bounded homeomorphism. Theorem 2.9. Ψ C (X, R) and Ψ,Ψ are weakly-strongly continuous, that is, (u n, v n ) (u, v) implies Ψ (u n, v n ) Ψ (u, v) and Ψ (u n, v n ) Ψ (u, v). The proof is similar to the proof of [25, Theorem 3.2], we omit it here. 3. Main results and their proofs In this section, we state the main results at first, and using the critical point theory, we prove the existence of solutions for problem (.), and the asymptotic behavior of solutions near infinity. We say that (u, v) X is a weak solution for (.), if u p(x) 2 u ϕ dx + u p(x) 2 u ϕ dx R R = {λa(x) u γ(x) 2 u + F u (x, u, v)}ϕ dx, ϕ X, R v q(x) 2 v ψ dx + v q(x) 2 v ψ dx R R = {λb(x) v δ(x) 2 v + F v (x, u, v)}ψ dx, ψ X 2. R It is easy to see that the critical point of J is a solution for (.). Similar to the proof of [8, Theorem 5], from (A) we have F (x, τ / s, τ /2 t) τf (x, s, t), (x, s, t) R R 2, τ, (3.) F (x, τ / s, τ /2 t) τf (x, s, t), (x, s, t) R R 2, 0 τ. (3.2) In fact, from (A0) and (A) we have
8 L. YI, Y. LIAG, Q. ZHAG, C. ZHAO EJDE-205/63 (A0 ) 0 F (x, s, t) σ s p(x) + C(σ) s α(x) + σ t q(x) + C(σ) t β(x), where σ is a small enough positive constant. Denote Γ = {γ C([0, ], X) : γ(0) = (0, 0), γ() = (u, v )}, c = inf max γ Γ (u,v) γ J(u, v), where (u, v ) X satisfies J(u, v ) < 0. Denote Ĵ(u, v) = R p(x) ( u p(x) + u p(x) ) dx + R q(x) ( v q(x) + v q(x) ) dx R F (u, v) dx, = {(u, v) X : Ĵ (u, v)( u, 2 v) = 0, (u, v) 0}, J = ow our results can be stated as follows. inf Ĵ(u, v). (u,v) Theorem 3.. If F satisfies (A0) (A3), the positive parameter λ is small enough and c < J, then (.) possesses a nontrivial solution. ext, we give an application of Theorem 3., that is, a sufficient condition for c < J. We say h(x) is periodic and its period is A = {a, a 2,..., a } where a i 0, i, if h(x) = h(x + n i a i e i ), x R, where n i are integers and e,..., e is the standard basis of R. Denote Q(x o, A) = {x R : (x x o )e i [0, a i ]}. Theorem 3.2. If p( ), q( ) are periodic and their periods is A, F (x, s, t) satisfies (A0) (A3), and there exist τ, δ > 0 and p( ) << α o ( ) << p ( ), q( ) << β o ( ) << q ( ) such that F (x, s, t) F (s, t), (x, s, t) R R + R +, F (x, s, t) F (s, t) + τs αo(x) + τt βo(x), (x, s, t) B(Q(x o, A), δ) R + R +, (3.3) then (.) possesses at least one nontrivial solution when λ is small enough. ext we give the behavior of solutions near infinity. Theorem 3.3. Suppose (A0) (A3) hold, a, b L (R ). If u is a weak solution for problem (.), then u, v C,α (R ), u(x) 0, u(x) 0, v(x) 0 and v(x) 0 as x.
EJDE-205/63 EXISTECE OF SOLUTIOS 9 3.. Proof of Theorem 3.. For the proof, we need to do some preparations. Lemma 3.4. If F satisfies (A0) (A2), and the parameter λ is small enough, then J satisfies the Mountain Pass Geometry, that is, (i) There exist positive numbers ρ and α such that J(u, v) α for any (u, v) X with (u, v) = ρ; (ii) J(0, 0) = 0, and there exists (u, v) X with (u, v) > ρ such that J(u, v) < 0. Proof. (i) Recall that (A0) (A) imply (A0 ). Then from (A0 ) we have F (x, u, v) ε( u p+ + v q+ ) + C(ε)( u α(x) + v β(x) ). Suppose ε and λ are small enough. We have J(u, v) = R p(x) ( u p(x) + u p(x) ) dx + R q(x) ( v q(x) + v q(x) ) dx λ[ a(x) R γ(x) u γ(x) + b(x) δ(x) v δ(x) ] dx F (x, u, v) dx R Φ(u, v) λ ( u p(x) + v q(x) ) dx λc R ε [ u p + + v q+ ] dx C(ε) ( u α(x) + v β(x) ) dx R R 2 Φ(u, v) C(ε) R ( u α(x) + v β(x) ) dx. Since p( ) << α( ) << p ( ), q( ) << β( ) << q ( ), there exists a positive constant ε 0 such that α( ) p( ) 2ε 0 and β( ) q( ) 2ε 0. Since p is Liptchitz cntinuous, we can divide R into countable disjoint cube Ω n, n =, 2,..., each one has the same side length, such that n=ω n = R and for any n =, 2,..., the following inequalities hold inf α(x) sup p(x) ε 0 and inf β(x) sup q(x) ε 0. x Ω n x Ω n x Ω n x Ω n Denote α Ω n = inf x Ωn α(x), p + Ω n = sup x Ωn p(x). Suppose the positive number ρ < is small enough and (u, v) = ρ, from Propositions 2.3 and 2.6, it follows that C(ε) u α(x) C(ε) u α Ωn α( ),Ω n Ω n 8 + 8p + u p Ωn p( ),Ω n Ω n p(x) ( u p(x) + u p(x) ) dx. Similarly, we have C(ε) v β(x) dx Ω n 8 Ω n q(x) ( v q(x) + v q(x) ) dx. Thus, we have J(u, v) 2 Φ(u, v) C(ε) R ( u α(x) + v β(x) ) dx
0 L. YI, Y. LIAG, Q. ZHAG, C. ZHAO EJDE-205/63 = 2 n= Ω n p(x) ( u p(x) dx + u p(x) ) dx + 2 n= Ω n q(x) ( v q(x) + v q(x) ) dx C(ε) ( u α(x) + v β(x) ) dx Ω n n= Φ(u, v). 8 It means that assertion (i) holds. (ii) Obviously, J(0, 0) = 0. When t, by (3.), we have J(t / u, t /2 v) = R p(x) t p(x) ( u p(x) + u p(x) ) dx + R q(x) t q(x) 2 ( v q(x) + v q(x) ) dx λ[ a(x) R γ(x) t γ(x) u γ(x) + b(x) δ(x) t δ(x) 2 v δ(x) ] dx F (x, t / u, t /2 v) dx R R p(x) t p(x) ( u p(x) + u p(x) ) dx + R q(x) t q(x) 2 ( v q(x) + v q(x) ) dx λ[ a(x) R γ(x) t γ(x) u γ(x) + b(x) δ(x) t δ(x) 2 v δ(x) ] dx tf (x, u, v) dx. R ote that γ(x) << p(x), δ(x) << q(x), > p + and 2 > q +, then for any nontrivial (u, v) X, it is not hard to check J(t / u, t /2 v) We remark that it is easy to see that J > 0. as t +. Lemma 3.5. If F satisfies (A0) (A2), {(u n, v n )} is a PS sequence of J, that is J(u n, v n ) c which is the mountain pass level, and J (u n, v n ) 0, then {(u n, v n )} is bounded. Proof. Since << γ( ) << p( ), a( ) L p( ) γ( ) (R ), << δ( ) << q( ), b( ) q( ) L q( ) δ( ) (R ), we have λ[ a(x) R γ(x) u γ(x) + b(x) δ(x) v δ(x) ] dx [ λa(x) u γ(x) + λb(x) v δ(x) ] dx R [ γ(x) R p(x) (ε ) p(x) γ(x) u p(x) p(x) γ(x) + p(x) λa(x) p(x) γ(x) ] dx p(x) ε + [ δ(x) R q(x) (ε ) q(x) δ(x) v q(x) q(x) δ(x) + q(x) λb(x) q(x) δ(x) ] dx q(x) ε p( ) ε R [ u p(x) + v q(x) ] dx + C(ε ), where ε is a positive small enough constant.
EJDE-205/63 EXISTECE OF SOLUTIOS By (A), we have for large values of n c + J(u n, v n ) = R p(x) ( u n p(x) + u n p(x) ) dx + R q(x) ( v n q(x) + v n q(x) ) dx λ[ a(x) R γ(x) u n γ(x) + b(x) δ(x) v n δ(x) ] dx F (x, u n, v n ) dx R R p(x) { u n p(x) + u n p(x) u n F u (x, u n, v n )} dx + R q(x) { v n q(x) + v n q(x) v n F v (x, u n, v n )} dx 2 λ[ a(x) R γ(x) u n γ(x) + b(x) δ(x) v n δ(x) ] dx, = ( R p(x) )( u n p(x) + u n p(x) ) dx + J (u n, v n )( u n, v n ) 2 + ( R q(x) )( v n q(x) + v n q(x) ) dx l [ u n p(x) + v n q(x) ] dx C 2 2 R l ( u n p(x) + u n p(x) ) dx + l ( v n q(x) + v n q(x) ) dx 2 R 2 R D J(u n, v n ),p( ) u n p( ) 2 D 2 J(u n, v n ),q( ) v n q( ) C, where l = min{( p + ), ( q + 2 )}. Without loss of generality, we assume that v n q( ) u n p( ), n =, 2,.... Therefore for large enough n, we have c + l 2 u n p p( ) ( D J(u n, v n ),p( ) + 2 D 2 J(u n, v n ),q( ) ) u n p( ) C. This is a contradiction. Thus { u n p( ) } and { v n q( ) } are bounded. Lemma 3.6.. Suppose F satisfies (A0) (A3), {(u n, v n )} satisfy J(u n, v n ) c > 0, where c is the mountain pass level, J (u n, v n ) 0, λ is small enough, passing to a subsequence still labeled by n, we have (i) {(u n, v n )} has a nontrivial weak limit (u, v) X or Fu (u n, v n )u n dx + Fv (u n, v n )v n dx δ > 0; R R (ii) If c < J, then {(u n, v n )} has a nontrivial weak limit. Proof. (i) It follows from Lemma 3.5 that {(u n, v n )} is bounded in X. Without loss of generality, we may assume that (u n, v n ) (u, v) in X. If (u, v) = (0, 0), then Proposition 2.6 implies u n 0 v n 0 in L α( ) loc (R ), p( ) α( ) < p ( ), in L β( ) loc (R ), q( ) β( ) < q ( ). (3.4)
2 L. YI, Y. LIAG, Q. ZHAG, C. ZHAO EJDE-205/63 Since (u n, v n ) (0, 0), then Theorem 2.9 implies λ a(x) R γ(x) u n γ(x) dx = o() = λ b(x) R δ(x) v n δ(x) dx. (3.5) Recall that (A0) (A) imply (A0 ). Then from (A0 ), (A3) and (3.4), it follows that (F u (x, u n, v n ) F u (u n, v n ))u n dx R F u (x, u n, v n ) F u (u n, v n ) u n dx x R + C ε(r) x R x R ( u n p(x) + u n α(x) + v n q(x) + v n β(x) ) dx ( u n p(x) + u n α(x) + v n q(x) + v n β(x) ) dx + C ( u n p(x) + u n α(x) + v n q(x) + v n β(x) ) dx, x R which implies F u (x, u n, v n )u n dx = Fu (u n, v n )u n dx + o() R R as n +. (3.6) Similar to the proof of (3.6), we can verify F v (x, u n, v n )v n dx = Fv (u n, v n )v n dx + o() when n +, (3.7) R R F (x, u n, v n ) dx = F (un, v n ) dx + o() R R as n +. (3.8) Since F (x, u, v) 0 and J(u n, v n ) c > 0, we have Φ(u n, v n ) Ψ (u n, v n ) J(u n, v n ) C > 0, for n, (3.9) which together with (3.5)-(3.8) and J (u n, v n ) 0 implies Fu (u n, v n )u n dx + Fv (u n, v n )v n dx δ > 0. (3.0) R R (ii) By (A0), (3.) and (3.2), there exist t n > 0 such that (t / n u n, t /2 n v n ) ; that is, ( tn / u n p(x) + tn / u n p(x)) dx R + ( t /2 n v n q(x) + t /2 v n q(x)) dx 2 R = R Fu (t / n + 2 R Fv (t / n n u n, t 2 n v n )t / u n dx n u n, t 2 n v n )t /2 v n dx. n (3.) Suppose (u, v) is trivial, then (3.0) is valid. oting that {(u n, v n )} is bounded in X. Obviously, there exist positive constants c and c 2 such that c t n c 2. (3.2)
EJDE-205/63 EXISTECE OF SOLUTIOS 3 From (3.5) and (3.2), we have λ a(x) R γ(x) t/ n u n γ(x) dx = o() = R λ b(x) δ(x) t/2 n v n δ(x) dx. (3.3) Since J (u n, v n ) 0 and {(u n, v n )} is bounded in X, it follows from (3.6) and (3.7) that ( u n p(x) + u n p(x) ) dx R (3.4) = F u (x, u n, v n )u n dx + o() = Fu (u n, v n )u n dx + o(), R R ( v n q(x) + v n q(x) ) dx R (3.5) = F v (x, u n, v n )v n dx + o() = Fv (u n, v n )v n dx + o(). R R Obviously, there exist ξ n, η n R such that ( t / n u n p(x) + t / n u n p(x)) p(ξn) dx = t n R ( t /2 n v n q(x) + t /2 n v n q(x)) q(ηn) dx = t 2 n R R ( u n p(x) + u n p(x)) dx, R ( v n q(x) + v n q(x) ) dx, which together with (3.), (3.4) and (3.5) implies p(ξn) t n [ Fu (u n, v n )u n dx + o()] + q(ηn) t 2 n [ Fv (u n, v n )v n dx + o()] R 2 R = Fu (t R / n u n, t 2 n v n )t / n u n dx + Fv (t / n u n, t 2 n v n )tn /2 v n dx. 2 R Thus t p(ξn) n { + q(ηn) t 2 n { 2 = 0. [ F u (t / n R [ F v (t / n R u n, t 2 n v n )t p(ξn) n u n, t 2 n v n )t From (A2), it is easy to see that q(ξn) 2 n u n F u (u n, v n )u n ] dx + o()} v n F v (u n, v n )v n ] dx + o()} 2 F (x, τ / s, τ /2 t)s/ τ and 3 F (x, τ / s, τ /2 t)t/ τ 2 2 (3.6) are increasing about τ when τ > 0; obviously, F (τ / s, τ /2 t)s/ τ and 2 F (τ / s, τ /2 t)t/ τ 2 2 are increasing when τ > 0. By (A) and (3.6), we have () If t n, then 0 p(ξn) t p(ξn) n (t n ) Fu (u n, v n )u n dx R 2 q(ξn) n (t 2 n ) Fv (u n, v n )v n dx o(); + q(ηn) t 2 2 R (3.7)
4 L. YI, Y. LIAG, Q. ZHAG, C. ZHAO EJDE-205/63 (2) If t n <, then 0 p(ξn) p(ξn) t n ( t n ) R + q(ηn) 2 q(ξn) t 2 n ( t 2 n ) 2 R Fu (u n, v n )u n dx From (3.0), (3.2), (3.7) and (3.8), it follows that Fv (u n, v n )v n dx o(). (3.8) lim t n =. (3.9) n Together with (3.5), (3.8) and the definition of (u n, v n ), we have c = J(u n, v n ) + o() = Ĵ(u n, v n ) + o(). (3.20) From the bounded continuity of emytskii operator, we can see Ĵ(u n, v n ) = Ĵ(t/ n u n, t /2 n v n ) + o(). (3.2) ote that (t / n u n, t /2 n v n ). It follows from (3.9), (3.20) and (3.2) that This is a contradiction. c = Ĵ(t/ n u n, t /2 n v n ) + o() J + o() J > c. Proof of Theorem 3.. From Lemmas 3.4 and 3.5, we know that there exist a bounded PS sequence {(u n, v n )} X such that J(u n, v n ) c > 0, J (u n, v n ) 0, where c is the mountain pass level of J. Moreover, from Proposition 2.6 we have then u n u v n v in L α( ) loc (R ), p( ) α( ) << p ( ), in L β( ) loc (R ), q( ) β( ) << q ( ), (3.22) u n u a.e. in R, and v n v a.e. in R. (3.23) Since c < J, Lemma 3.6 implies that (u, v) is nontrivial. It only remains to prove that (u, v) is a solution for (.). Since J (u n, v n ) 0 as n, for any (ϕ, ψ) X, we have ( u n p(x) 2 u n ϕ + u n p(x) 2 u n ϕ) dx R {λa(x) u n γ(x) 2 u n + F u (x, u n, v n )}ϕ dx 0 R ( v n q(x) 2 v n ψ + v n q(x) 2 v n ψ) dx R {λb(x) v n δ(x) 2 v n + F v (x, u n, v n )}ψ dx 0. R Since { u n p( ) } and { v n q( ) } are bounded, for any (ϕ, ψ) X, it is easy to see that the following two groups are uniformly integrable in R, {( u n p(x) + λa(x) u n γ(x) + F u (x, u n, v n ) ) ϕ }, {( v n q(x) + λb(x) v n δ(x) + F v (x, u n, v n ) ) ψ }.
EJDE-205/63 EXISTECE OF SOLUTIOS 5 Combining this, (3.23) and Vitali convergent theorem implies u n p(x) 2 u n ϕ dx {λa(x) u n γ(x) 2 u n + F u (x, u n, v n )}ϕ dx (3.24) R R u p(x) 2 uϕ dx {λa(x) u γ(x) 2 u + F u (x, u, v)}ϕ dx as n, R R v n q(x) 2 v n ψ dx {λb(x) v n δ(x) 2 v n + F v (x, u n, v n )}ψ dx (3.25) R R v q(x) 2 vψ dx {λb(x) v δ(x) 2 v + F v (x, u, v)}ψ dx as n. R R Thus, to prove that (u, v) is a weak solution of (.), we only need to prove that for any (ϕ, ψ) X there holds u n p(x) 2 u n ϕ dx u p(x) 2 u ϕ dx, n +, R R (3.26) v n q(x) 2 v n ψ dx v q(x) 2 v ψ dx, n +. R R Choose φ C 0 (R ) with 0 φ, we have R φ( u n p(x) 2 u n w p(x) 2 w)( u n w) dx 0, w X. (3.27) Since {(u n, v n )} is bounded in X and J (u n, v n ) 0, we have [ u n p(x) 2 u n (φ(u n w)) + u n p(x) 2 u n φ(u n w)] dx R = [λa(x) u n γ(x) 2 u n + F u (x, u n, v n )]φ(u n w) dx + o(), R for all w X. It follows from (3.27) and (3.28) that {[λa(x) u n γ(x) 2 u n + F u (x, u n, v n )] u n p(x) 2 u n }φ(u n w) dx R (u n w) u n p(x) 2 u n φ dx R φ w p(x) 2 w( u n w) dx + o() 0, w X. R ote that (u n, v n ) is bounded in X, we may assume (u n, v n ) (u, v) in X, (3.28) (3.29) u n u in (L p( ) (R )), (3.30) v n v in (L q( ) (R )) u n p(x) 2 u n T in (L p0 ( ) (R )), (3.3) v n q(x) 2 v n S in (L q0 ( ) (R )).
6 L. YI, Y. LIAG, Q. ZHAG, C. ZHAO EJDE-205/63 ote that φ has compact support, letting n +, according to (3.24), (3.29), (3.30) and (3.3), we obtain {[λa(x) u γ(x) 2 u + F u (x, u, v)] u p(x) 2 u}φ(u w) dx R (3.32) (u w)t φ dx φ w p(x) 2 w( u w) dx 0, R R for all w X. On the other hand u n p(x) 2 u n T J (u n, v n ) 0, which implies that (T ϕ + u p(x) 2 uϕ) dx = R [λa(x) u γ(x) 2 u + F u (x, u, v)]ϕ dx. R (3.33) Set ϕ = φ(u w), it follows from (3.32) and (3.33) that R φ(t w p(x) 2 w)( u w) dx 0, w X. (3.34) Set w = u εξ, where ξ X, ε > 0. From (3.34) we have then it is easy to see that R φ(t u p(x) 2 u) ξ dx 0, ξ X, R φ(t u p(x) 2 u) ξ dx = 0, ξ X, Thus (3.26) is valid. Therefore R (T u p(x) 2 u) ξ dx = 0, ξ X. ( u p(x) 2 u ϕ + u p(x) 2 uϕ) dx R = [λa(x) u γ(x) 2 u + F u (x, u, v)]ϕ dx, ϕ X. R Similarly, we have ( v q(x) 2 v ψ + v q(x) 2 vψ) dx R = [λb(x) v δ(x) 2 v + F v (x, u, v)]ψ dx, ψ X 2. R Thus (u, v) is a solution of (.). 3.2. Proof of Theorem 3.2. Motivated by the property of translation invariant for p-laplacian, we get a sufficient condition for c < J. To prove Theorem 3.2, we need the following Lemma. Lemma 3.7. If F satisfies (A0) (A2), then for any (u, v) X\{(0, 0)}, there exists a unique t(u, v) > 0 such that () Ĵ((t(u, v))/ u, (t(u, v)) 2 v) = max s [0,+ ) Ĵ(s u, s /2 v), (2) ((t(u, v)) / u, (t(u, v)) /2 v),
EJDE-205/63 EXISTECE OF SOLUTIOS 7 (3) The operator (u, v) t(u, v) is continuous from X\{(0, 0)} to (0, + ), and the operator (u, v) ((t(u, v)) u, (t(u, v)) /2 v) is a homeomorphism from the unit sphere in X to. Proof. For any (u, v) X\{(0, 0)}, define g(t) = Ĵ(t/ u, t /2 v), t [0, + ). () Similar to the proof of (3.), we have g(t) > 0 as t > 0 is small enough, and g(t) < 0 as t +. Obviously, g is continuous, then g attains it s maximum in (0, + ). (2) From (A2), it is not hard to check that (t / u, t /2 v) if and only if tg (t) = 0; that is, [ u p(x) t p(x) R + = [ v q(x) t q(x) 2 R 2 Ω dx + u p(x) t p(x) ] dx dx + v q(x) t q(x) 2 F (t / u, t /2 v) t / u dx + R Ω ] dx which can be rearranged as [ u p(x) t p(x) dx + u p(x) t p(x) ] dx + = R Ω [ v q(x) t q(x) 2 2 F (t / u, t /2 v) t dx + v q(x) t q(x) ] dx u dx + Ω F 2 (t / u, t /2 v) 2 t /2 v dx, F2 (t / u, t /2 v) v dx. 2 t 2 2 It follows from (3.) and (3.2) that the left hand is strictly decreasing with respect to t, while the right hand is increasing. Thus g (t) = 0 has a unique solution t(u, v) such that ((t(u, v)) / u, (t(u, v)) /2 v). We claim that g(t) is increasing on [0, t(u, v)], and decreasing on [t(u, v), + ). Denote (u, v ) = ((t(u, v)) / u, (t(u, v)) /2 v). Define ρ(t) = Ĵ(t/ u, t /2 v ). We only need to prove that ρ(t) is increasing on [0, ], and ρ(t) is decreasing on [, + ). From (), it is easy to see that there exists t # > 0 such that therefore ρ (t # ) = 0. Suppose t >. By (A2), we have ρ(t # ) = max Ĵ(t / u, t /2 v ), t 0 ρ (t) = t p(x) ( u p(x) + u p(x) ) dx + R R F (t / u, t /2 v ) t u dx R < ( u p(x) + u p(x) ) dx + R F (u, v ) u dx F2 (u, v ) v dx 2 R R t q(x) 2 ( v q(x) + v q(x) ) dx 2 R F2 (t / u, t /2 v ) 2 t 2 v dx R 2 ( v q(x) + v q(x) ) dx
8 L. YI, Y. LIAG, Q. ZHAG, C. ZHAO EJDE-205/63 = Ĵ (u, v )( u, 2 v ) = 0. Thus ρ(t) is strictly decreasing when t >. Suppose t <. Similarly, we have ρ (t) > Ĵ (u, v )( u, 2 v ) = 0. Thus ρ(t) is strictly increasing when t <. Therefore g(t) is increasing on [0, t(u, v)] and decreasing on [t(u, v), + ). (3) We only need to proof that t(, ) is continuously. Let (u m, v m ) (u, v) in X, then Ĵ(t/ u m, t /2 v m ) Ĵ(t u, t /2 v). We choose a constant t 0 large enough such that Ĵ(t 0 u, t/2 0 v) < 0, then there exists a M > 0 such that Ĵ(t / 0 u m, t /2 0 v m ) < 0 for any m > M. Therefore t(u m, v m ) < t 0 when m > M, then {t(u m, v m )} has a convergent subsequence {t(u mj, v mj )} satisfying t(u mj, v mj ) t. Thus Ĵ((t(u mj, v mj )) / u mj, (t(u mj, v mj )) /2 v mj ) Ĵ(t/ u, t /2 v). From () we know that and hence letting j, we obtain Ĵ((t(u mj, v mj )) u mj, (t(u mj, v mj )) /2 v mj ) Ĵ(t / Ĵ((t(u, v))/ u mj, (t(u, v)) /2 v mj ), u, t /2 v) Ĵ((t(u, v))/ u, (t(u, v)) /2 v). From (), we have t = t(u, v). Thus t(u, v) is continuous. Proof of Theorem 3.2. Let {(u n, v n )} be a minimizing sequences of Ĵ, that is lim Ĵ(u n, v n ) = J > 0. n + Similar to the proof of Lemma 3.5, we can see {(u n, v n )} is bounded in X. Thus there exists a positive constant κ > such that R ( u n p(x) + v n q(x) ) dx κ, n =, 2,.... (3.35) We claim that for any fixed δ > 0 andp( ) << α( ) << p ( ), q( ) << β( ) << q ( ), there exist a ε o > 0 such that sup y R u n α(x) dx + sup y R v n β(x) dx 2ε o, n =, 2,.... (3.36) B(y,δ) B(y,δ) Indeed, suppose otherwise. Then it follows from Proposition 2.7 that We claim that u n 0 in L α( ) (R ), p( ) << α( ) << p ( ), (3.37) v n 0 in L β( ) (R ), q( ) << β( ) << q ( ). (3.38) R Fu (u n, v n )u n dx 0, n +. (3.39)
EJDE-205/63 EXISTECE OF SOLUTIOS 9 For any ε > 0, (A0) (A) imply F u (u n, v n )u n ε 2κ ( u n p(x) + v n q(x) ) + C(ε)( u n α(x) + v n β(x) ), and R ε 2κ Fu (u n, v n )u n dx ( u n p(x) + v n q(x) ) dx + C(ε) ( u n α(x) + v n β(x) ) dx. R R (3.40) Combining (3.37) and (3.38), there exist 0 > 0 such that C(ε) ( u n α(x) + v n β(x) ) dx ε R 2, n 0. (3.4) From (3.35), (3.40) and (3.4), we have Thus (3.39) is valid. Similarly, we can get R Fu (u n, v n )u n dx ε, n 0. R Fv (u n, v n )v n dx 0, n +. (3.42) ote that (u n, v n ). It follows from (3.39), (3.42) and (u n, v n ) 0 that Ĵ(u n, v n ) 0. This is a contradiction to lim n + Ĵ(u n, v n ) = J > 0. Thus (3.36) is valid. From (3.36), without loss of generality, we assume that We may assume that sup u n α(x) dx ε o, n =, 2,... y R B(y,δ) B(y n,δ) B(η n,δ) u n α(x) dx 2 sup y R v n β(x) dx 2 sup y R B(y,δ) B(y,δ) u n α(x) dx, v n β(x) dx. From p( ) begin periodic, for any y n,η n, there exist x n, ξ n Q(x o, A) such that p(x) = p(y n x n + x), x R, q(x) = q(η n ξ n + x), x R.
20 L. YI, Y. LIAG, Q. ZHAG, C. ZHAO EJDE-205/63 ow, we consider J(t / u n (y n x n + x), t /2 v n (η n ξ n + x)). Denote J (u, v) = Φ(u, v) F (x, u, v) dx. It follows from (A2) that F (x, t R u, t /2 v)/t is increasing with respect to t. Suppose t (0, ), it follows from (3.2) that J (t / u n (y n x n + x), t /2 v n (η n ξ n + x)) = Φ(t / u n, t /2 v n ) F (x, t / u n (y n x n + x), t /2 v n (η n ξ n + x)) dx R t max{ p +, q+ } 2 Φ(u n, v n ) t F (x, u n (y n x n + x), v n (η n ξ n + x)) dx R = t{t max{ p +, q+ } 2 Φ(u n, v n ) F (x, u n (y n x n + x), v n (η n ξ n + x)) dx}. R (3.43) From (3.36) and the boundedness of {(u n, v n )}, we can see that there exists positive constants C, C 2 such that C (u n, v n ) C 2. (3.44) Since > p + and 2 > q +, there exists a fixed t (0, ) such that, for any n =, 2,..., we have t max{ p +, q+ } 2 Φ(u n, v n ) F (x, u n (y n x n + x), v n (η n ξ n + x)) dx R (3.45) 2 Φ(u n, v n ) C 3 > 0. From (3.43) and (3.45), we obtain J (t / u n (y n x n + x), t /2 v n (η n ξ n + x)) t C 3 > 0, n =, 2,.... Suppose λ is small enough. From the above inequality, we have J(t / u n (y n x n + x), t 2 v n (η n ξ n + x)) 2 t C 3 > 0, n =, 2,.... (3.46) Obviously, J(0, 0) = 0 and J(t / u n (y n x n + x), t /2 v n (η n ξ n + x)) as t +. Thus there exist t n (0, + ) such that n u n (y n x n + x), t /2 n v n (η n ξ n + x)) J(t / = max J(t u n (y n x n + x), t /2 v n (η n ξ n + x)) > 0. t 0 (3.47) It follows from (3.46) and the boundedness of {(u n, v n )} that there exist a positive constant ɛ such that t n ɛ, n =, 2,.... (3.48) Denote g(t) = Ĵ(t/ u n (y n x n + x), t 2 v n (η n ξ n + x)). Since {(u n, v n )}, Lemma 3.7 implies Ĵ(u n (y n x n + x), v n (η n ξ n + x)) = max Ĵ(t / u n (y n x n + x), t /2 v n (η n ξ n + x)). t 0 Suppose λ is small enough. From (3.3), (3.47), (3.48) and (3.49), we have max t 0 J(t/ u n (y n x n + x), t /2 v n (η n ξ n + x)) (3.49)
EJDE-205/63 EXISTECE OF SOLUTIOS 2 = J(t / n u n (y n x n + x), tn /2 v n (η n ξ n + x)) Ĵ(t/ n u n (y n x n + x), tn /2 v n (η n ξ n + x)) + λ[ a(x) R γ(x) t/ n u n (y n x n + x) γ(x) + b(x) τ 2α o + t n u n (y n x n + x) αo(x) dx B(x n,δ) τ t 2 n v n (η n ξ n + x) βo(x) dx 2β + o Ĵ(t/ n B(ξ n,δ) + o,β+ o } δ(x) t/2 n u n, t 2 n v n ) τε 0ɛ max{α 2 max{α o +, β o + } + λ[ a(x) R γ(x) t/ n u n (y n x n + x) γ(x) + b(x) δ(x) t/2 n Ĵ(u n, v n ) τε 0ɛ max{α+ o,β+ o } 4 max{α + o, β + o } < J where ζ Q(x o, A). This completes the proof. v n (η n ξ n + x) δ(x) ] dx v n (η n ξ n + x) δ(x) ] dx 3.3. Proof of Theorem 3.3. According to the [7, Theorems 2.2 and 3.2], u and v are locally bounded. From [0, Theorem.2], u and v are locally C,α continuous. Similar to the proof of [3, Proposition 2.5], we obtain that u, v C,α (R ), u(x) 0, u(x) 0, v(x) 0 and v(x) 0 as x. ote. Let us consider the existence of solutions for the system div u i pi(x) 2 u i ) + u i pi(x) 2 u i = λa i (x) u i γi(x) 2 u i + F ui (x, u,..., u n ) in R, u i W,pi( ) (R ), i =,..., n, where u = (u,..., u n ), suppose λ is small enough, then the system has a nontrivial solution if it satisfies the following assumptions: (H0) p i ( ) are Lipschitz continuous, << p i ( ) <<, << γ i ( ) << p i ( ), a i ( ) L p i ( ) p i ( ) γ i ( ) (R ), F C (R R n, R) and satisfies F ui (x, u,..., u n ) C( u i pi(x) + u i αi(x) + j n, j i [ u j pj(x)/α0 i (x) + u j αj(x)/α0 i (x) ]), where F ui = u i F, α i C(R ), and p i ( ) α i ( ) << p i ( ), where { p p i (x)/( p i (x)), p i (x) <, i (x) =, p(x), (H) F C (R R n ) and satisfies the following conditions 0 s i F si (x, s,..., s n ), (x, s,..., s n ) R R n, i =,..., n, 0 < F (x, s,..., s n ) s i F si (x, s,..., s n ), (x, s,..., s n ) R R n ; i i n
22 L. YI, Y. LIAG, Q. ZHAG, C. ZHAO EJDE-205/63 (H2) For any (s, t) (R R), F si (x, τ / s,..., τ /n s n )/τ i i (i =,..., n) are increasing respect to τ > 0; (H3) There is a measurable function F (s,..., s n ) such that lim F (x, s,..., s n ) = F (s,..., s n ) x + for bounded i n s i uniformly, F (s,..., s n ) + s i Fsi (s,..., s n ) C ( s i p+ i + s i α i ), i n for all (s,..., s n ) R n, and F (x, s,..., s n ) F (s,..., s n ) ε(r) F si (x, s,..., s n ) F si (s,..., s n ) i n i n ( s i pi(x) + s i p i (x) ) when x R, ε(r){ s i pi(x) + s i p i (x) + [ s j pj(x)(p i (x) )/p i (x) + s j p j (x)(p i (x) )/p i (x) ]} when x R, (H4) j n,j i where ε(r) satisfies lim R + ε(r) = 0. F (x, s,..., s n ) F (s,..., s n ), (x, s,..., s n ) R (R + ) n, F (x, s,..., s n ) F (s,..., s n ) + i n τs α# i (x) where p i ( ) << α # i ( ) << p i ( ). i, (x, s,..., s n ) B(Q(x o, A), δ) (R + ) n, Acknowledgments. The authors would like to thank Professor Julio G. Dix for his suggestions, and the anonymous referees for their valuable comments. This research was partly supported by the ational atural Science Foundation of China (3266, 097087), and by the key projects of Science and Technology Research of the Henan Education Department (4A00). References [] E. Acerbi, G. Mingione; Regularity results for a class of functionals with nonstandard growth, Arch. Ration. Mech. Anal., 56 (200), 2 40. [2] C. O. Alves, P. C. Carriä, O. H. Miyagaki; onlinear perturbations of a periodic elliptic problem with critical growth, J. Math. Anal. Appl. 260 (200) 33 46. [3] C. O. Alves, J. Marcos do Ó, O. H. Miyagaki; On perturbations of a class of a periodic m-laplacian equation with critical growth, onlinear Anal. 45 (200) 849 863. [4] C. O. Alves, J. Marcos do Ó, O. H. Miyagaki; ontrivial solutions for a class of semilinear biharmonic problems involving critical exponents, onlinear Anal. 46 (200) 2 33. [5] C. O. Alves, J. Marcos do Ó, O. H. Miyagaki; On nonlinear perturbations of a periodic elliptic problem in R 2 involving critical growth, onlinear Anal. 56 (2004) 78 79. [6] K. C. Chang; Critical point theory and applications, shanghai Scientific and Technology press, Shanghai, 986. [7] Y. Chen, S. Levine, M. Rao; Variable exponent, linear growth functionals in image restoration, SIAM J. Appl. Math., 66 (2006), o.4,383 406. [8] L. Diening, P. Harjulehto, P. Hästö, M. Růžička; Lebesgue and Sobolev spaces with variable exponents, Lecture otes in Mathematics, vol. 207, Springer-Verlag, Berlin, 20.
EJDE-205/63 EXISTECE OF SOLUTIOS 23 [9] X. L. Fan, D. Zhao; On the spaces L p(x) (Ω) and W m,p(x) (Ω), J. Math. Anal. Appl., 263 (200), 424-446. [0] X. L. Fan; Global C,α regularity for variable exponent elliptic equations in divergence form, J. Diff. Equa. 235 (2007) 397-47. [] X. L. Fan, J. S. Shen, D. Zhao; Sobolev embedding theorems for spaces W k,p(x), J. Math. and Appl., 262 (200), 749-760. [2] X. L. Fan, Y. Z. Zhao, D. Zhao; Compact imbedding theorems with symmetry of Strauss- Lions type for the spaces W,p(x), J. Math. Anal. Appl., 255 (200), 333-348. [3] X. L. Fan; p(x)-laplacian equations in R with periodic data and nonperiodic perturbations, J. Math. Anal. Appl., 34 (2008) 03-9. [4] X. L. Fan, Q. H. Zhang; Existence of solutions for p(x)-laplacian Dirichlet problem, onlinear Anal. 52 (2003), 843-852. [5] X. L. Fan, Q. H. Zhang, D. Zhao; Eigenvalues of p(x)-laplacian Dirichlet problem, J. Math. Anal. Appl. 302 (2005), 306-37. [6] X. L. Fan; Solutions for p(x)-laplacian Dirichlet problems with singular coefficients, J. Math. Anal. Appl. 32 (2005) 464 477. [7] X. L. Fan, D. Zhao; The quasi-minimizer of integral functionals with m(x) growth conditions, onlinear Anal. 39, (2000), 807-86. [8] A. El Hamidi; Existence results to elliptic systems with nonstandard growth conditions, J. Math. Anal. Appl. 300 (2004) 30-42. [9] M. Růžička; Electrorheological fluids: modeling and mathematical theory, Lecture otes in Math 748, Springer-Verlag, Berlin, 2000. [20] Y. T. Shen, S. S. Yan; The Variational Methods for Quasilinear Elliptic Equations, South China Univ. of Technology Press, Guang Zhou, 995 (in Chinese). [2] M. Willem; Minimax Theorems, Birkhäuser, Boston, 996. [22] X. C. Xu, Y. K. An; Existence and multiplicity of solutions for elliptic systems with nonstandard growth condition in R, onlinear Analysis, 68 (2008) 956 968. [23] Q. H. Zhang; Existence of radial solutions for p(x)-laplacian equations in R, J. Math. Anal. Appl. 35 (2006), o.2, 506-56. [24] Q. H. Zhang, Z. M. Qiu, R. Dong; Existence and asymptotic behavior of positive solutions for a variable exponent elliptic system without variational structure, onlinear Analysis, 72 (200) 354-363. [25] Q. H. Zhang; Existence of solutions for p(x)-laplacian equations with singular coefficients in R, J. Math. Anal. Appl., 348 (2008), o., 38-50. [26] V. V. Zhikov; Averaging of functionals of the calculus of variations and elasticity theory, Math. USSR. Izv. 29 (987), 33-36. Li Yin College of Information and Management Science, Henan Agricultural University, Zhengzhou, Henan 450002, China E-mail address: mathsr@63.com Yuan Liang Junior College, Zhejiang Wanli University, ingbo, Zhejiang 3500, China E-mail address: ly0432@63.com Qihu Zhang (corresponding author) College of Mathematics and Information Science, Zhengzhou University of Light Industry, Zhengzhou, Henan 450002, China E-mail address: zhangqihu@yahoo.com, zhangqh999@yahoo.com.cn Chunshan Zhao Department of Mathematical Sciences, Georgia Southern University, Statesboro, GA 30460, USA E-mail address: czhao@georgiasouthern.edu