EXISTENCE OF SOLUTIONS FOR A VARIABLE EXPONENT SYSTEM ...

Electronic Journal of Differential Equations, Vol. 2015 (2015), No. 63, pp. 1–23. ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu ftp ejde.math.txstate.edu

EXISTENCE OF SOLUTIONS FOR A VARIABLE EXPONENT SYSTEM WITHOUT PS CONDITIONS LI YIN, YUAN LIANG, QIHU ZHANG, CHUNSHAN 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 + Fu (x, u, v) − div |∇v|

q(x)−2

∇v) + |v|

q(x)−2

v = λb(x)|v|

δ(x)−2

v + Fv (x, u, v)

in RN , in RN ,

u ∈ W 1,p(·) (RN ), v ∈ W 1,q(·) (RN ), 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.

1. 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 [1, 7, 19, 26]. The following variable exponent flow is an important model in image processing [7]: ut − div |∇u|p(x)−2 ∇u) + λ(u − u0 ) = 0,

in Ω × [0, T ],

on ∂Ω × [0, T ],

u(x, t) = g(x),

u(x, 0) = u0 . The main benefit of this flow is the manner in which it accommodates the local image information. We refer to [14, 18, 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 + Fu (x, u, v)

in RN ,

− div |∇v|q(x)−2 ∇v) + |v|q(x)−2 v = λb(x)|v|δ(x)−2 v + Fv (x, u, v)

in RN ,

u ∈ W 1,p(·) (RN ),

v ∈ W 1,q(·) (RN ),

2000 Mathematics Subject Classification. 35J47. Key words and phrases. Variable exponent system; integral functional; PS condition; variable exponent Sobolev space. c

2015 Texas State University - San Marcos. Submitted September 29, 2014. Published March 13, 2015. 1

(1.1)

2

L. YIN, Y. LIANG, Q. ZHANG, C. ZHAO

EJDE-2015/63

where p, q ∈ C(RN ) are Lipschitz continuous and p(·), q(·) >> 1, the notation h1 (·) >> h2 (·) means ess inf x∈RN (h1 (x) − h2 (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 [15]). 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 [21, Theorem 2.8], if a C 1 (X, R) functional f satisfies the Mountain Pass Geometry, then it has a PS sequence {xn } which satisfies f (xn ) → c which is the mountain pass level and f 0 (xn ) → 0. By [21, Theorem 2.9] it follows that if f also satisfies the PS condition, passing to a subsequence, then xn → x0 in X, and then x0 is a critical point of f , that is f 0 (x0 ) = 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 RN 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 [16, 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 Schr¨ odinger 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 [13] 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 (1.1). Our method is to compare the original equation with its limiting equation at infinity without perturbation. These results also partially generalize the results in [13] and [20]. In this article, we make the following assumptions. (A0) p(·), q(·) are Lipschitz continuous, 1 0 : | | dx + | | dx ≤ 1}. λ λ Ω Ω It is easy to see that the norm k · k0p(·),Ω is equivalent to k · kp(·),Ω on W 1,p(·) (Ω), and k · k0q(·),Ω is equivalent to k · kq(·),Ω on W 1,q(·) (Ω). In the following, we will use k · k0p(·),Ω instead of k · kp(·),Ω on W 1,p(·) (Ω), and use k · k0q(·),Ω instead of k · kq(·),Ω 1,p(·)

on W 1,q(·) (Ω). We denote by W0

(Ω) the closure of C0∞ (Ω) in W 1,p(·) (Ω). 1,p(·)

Proposition 2.6 ([8, 9, 11]). (i) W 1,p(·) (Ω) and W0 (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 1,p(·) (RN ) to Lα(·) (RN ) is continuous; (iii) If Ω is bounded, α ∈ C+ (Ω) and α(·) < p∗ (·) for any x ∈ Ω, then the imbedding from W 1,p(·) (Ω) to Lα(·) (Ω) is compact and continuous. Proposition 2.7 ([12, Lemma 3.1]). Assume that p : RN → R is a uniformly continuous function, if {un } is bounded in W 1,p(·) (RN ) and Z sup |un |ρ(x) dx → 0, n → +∞, y∈RN

B(y,r)

N for some r > 0 and some ρ ∈ L∞ + (R ) satisfying

p(·) ≤ ρ(·) 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 (t1/θ1 u, t1/θ2 v) ∈ N if and only if 0 tg (t) = 0; that is, Z 1 p(x) 1 p(x) [|∇u|p(x) t θ1 dx + |u|p(x) t θ1 ] dx θ1 θ1 RN Z q(x) 1 q(x) 1 + [|∇v|q(x) t θ2 dx + |v|q(x) t θ1 ] dx θ θ N 2 2 ZR Z 1 1 1/θ 1/θ 1/θ Fe1 (t 1 u, t 2 v) t 1 u dx + = Fe2 (t1/θ1 u, t1/θ2 v) t1/θ2 v dx, θ θ 1 2 Ω Ω which can be rearranged as Z p(x) p(x) 1 [|∇u|p(x) t θ1 −1 dx + |u|p(x) t θ1 −1 ] dx RN θ1 Z q(x) q(x) 1 + [|∇v|q(x) t θ2 −1 dx + |v|q(x) t θ1 −1 ] dx RN θ2 Z Z 1 Fe1 (t1/θ1 u, t1/θ2 v) 1 Fe2 (t1/θ1 u, t1/θ2 v) = u dx + v dx. θ1 −1 θ2 −1 Ω θ1 Ω θ2 t θ1 t θ2 It follows from (3.1) and (3.2) that the left hand is strictly decreasing with respect to t, while the right hand is increasing. Thus g 0 (t) = 0 has a unique solution t(u, v) such that ((t(u, v))1/θ1 u, (t(u, v))1/θ2 v) ∈ N . We claim that g(t) is increasing on [0, t(u, v)], and decreasing on [t(u, v), +∞). b 1/θ1 u∗ , t1/θ2 v∗ ). Denote (u∗ , v∗ ) = ((t(u, v))1/θ1 u, (t(u, v))1/θ2 v). Define ρ(t) = J(t We only need to prove that ρ(t) is increasing on [0, 1], and ρ(t) is decreasing on [1, +∞). From (1), it is easy to see that there exists t# > 0 such that b 1/θ1 u∗ , t1/θ2 v∗ ), ρ(t# ) = max J(t t≥0

0

therefore ρ (t# ) = 0. Suppose t > 1. By (A2), we have ρ0 (t) Z =

Z 1 p(x) 1 q(x) p(x) p(x) θ1 −1 t (|∇u∗ | + |u∗ | ) dx + t θ2 −1 (|∇v∗ |q(x) + |v∗ |q(x) ) dx θ θ N N 2 R R Z 1 Z 1 1 1 1 −1 1/θ 1/θ − Fe1 (t 1 u∗ , t 2 v∗ ) t θ1 u∗ dx − Fe2 (t1/θ1 u∗ , t1/θ2 v∗ ) t θ2 −1 v∗ dx θ1 θ2 N RN Z R Z 1 1 < (|∇u∗ |p(x) + |u∗ |p(x) ) dx + (|∇v∗ |q(x) + |v∗ |q(x) ) dx RN θ1 RN θ2 Z Z 1 1 e − F1 (u∗ , v∗ ) u∗ dx − Fe2 (u∗ , v∗ ) v∗ dx θ θ N N 1 2 R R

18

L. YIN, Y. LIANG, Q. ZHANG, C. ZHAO

EJDE-2015/63

1 1 = Jb0 (u∗ , v∗ )( u∗ , v∗ ) = 0. θ1 θ2 Thus ρ(t) is strictly decreasing when t > 1. Suppose t < 1. Similarly, we have 1 1 ρ0 (t) > Jb0 (u∗ , v∗ )( u∗ , v∗ ) = 0. θ1 θ2 Thus ρ(t) is strictly increasing when t < 1. 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 (um , vm ) → (u, v) in X, 1 b 1/θ1 um , t1/θ2 vm ) → J(t b θ1 u, t1/θ2 v). We choose a constant t0 large enough then J(t 1 b θ1 u, t1/θ2 v) < 0, then there exists a M > 0 such that such that J(t 0

0

b 1/θ1 um , t1/θ2 vm ) < 0 J(t 0 0 for any m > M . Therefore t(um , vm ) < t0 when m > M , then {t(um , vm )} has a convergent subsequence {t(umj , vmj )} satisfying t(umj , vmj ) → t∗ . Thus 1/θ1 b b ∗1/θ1 u, t∗1/θ2 v). J((t(u umj , (t(umj , vmj ))1/θ2 vmj ) → J(t mj , vmj ))

From (1) we know that 1

1/θ2 θ b J((t(u vm j ) mj , vmj )) 1 umj , (t(umj , vmj ))

b ≥ J((t(u, v))1/θ1 umj , (t(u, v))1/θ2 vmj ), and hence letting j → ∞, we obtain 1/θ 1 b 1/θ b J(t u, t∗ 2 v) ≥ J((t(u, v))1/θ1 u, (t(u, v))1/θ2 v). ∗

From (1), we have t∗ = t(u, v). Thus t(u, v) is continuous.



b that is Proof of Theorem 3.2. Let {(un , vn )} ⊂ N be a minimizing sequences of J, b n , vn ) = J ∞ > 0. lim J(u

n→+∞

Similar to the proof of Lemma 3.5, we can see {(un , vn )} is bounded in X. Thus there exists a positive constant κ > 1 such that Z (|un |p(x) + |vn |q(x) ) dx ≤ κ, n = 1, 2, . . . . (3.35) RN

We claim that for any fixed δ > 0 andp(·) 0,

n = 1, 2, . . . .

Suppose λ is small enough. From the above inequality, we have 1 1 1/θ J(t∗ 1 un (yn − xn + x), t∗θ2 vn (ηn − ξn + x)) ≥ t∗ C3 > 0, n = 1, 2, . . . . (3.46) 2 1/θ1 Obviously, J(0, 0) = 0 and J(t un (yn − xn + x), t1/θ2 vn (ηn − ξn + x)) → −∞ as t → +∞. Thus there exist tn ∈ (0, +∞) such that 2 1 vn (ηn − ξn + x)) un (yn − xn + x), t1/θ J(t1/θ n n 1

= max J(t θ1 un (yn − xn + x), t1/θ2 vn (ηn − ξn + x)) > 0.

(3.47)

t≥0

It follows from (3.46) and the boundedness of {(un , vn )} that there exist a positive constant  such that tn ≥ , n = 1, 2, . . . . (3.48) Denote 1 b 1/θ1 un (yn − xn + x), t θ2 vn (ηn − ξn + x)). g(t) = J(t Since {(un , vn )} ⊂ N , Lemma 3.7 implies b n (yn − xn + x), vn (ηn − ξn + x)) J(u b 1/θ1 un (yn − xn + x), t1/θ2 vn (ηn − ξn + x)). = max J(t t≥0

Suppose λ is small enough. From (3.3), (3.47), (3.48) and (3.49), we have max J(t1/θ1 un (yn − xn + x), t1/θ2 vn (ηn − ξn + x)) t≥0

(3.49)

EJDE-2015/63

EXISTENCE OF SOLUTIONS

21

1 = J(t1/θ un (yn − xn + x), tn1/θ2 vn (ηn − ξn + x)) n

b 1/θ1 un (yn − xn + x), t1/θ2 vn (ηn − ξn + x)) ≤ J(t n n Z |a(x)| 1/θ1 |b(x)| 1/θ2 + λ[ |tn un (yn − xn + x)|γ(x) + |t vn (ηn − ξn + x)|δ(x) ] dx γ(x) δ(x) n RN Z 1 τ − + |tnθ1 un (yn − xn + x)|αo (x) dx 2αo B(xn ,δ) Z 1 τ − + |tnθ2 vn (ηn − ξn + x)|βo (x) dx 2βo B(ξn ,δ) +

+

max{αo ,βo } 1 b 1/θ1 un , tnθ2 vn ) − τ ε0  ≤ J(t n 2 max{αo+ , βo+ } Z |b(x)| 1/θ2 |a(x)| 1/θ1 vn (ηn − ξn + x)|δ(x) ] dx |tn un (yn − xn + x)|γ(x) + |t + λ[ γ(x) δ(x) n RN +

b n , vn ) − ≤ J(u

+

τ ε0 max{αo ,βo } < J∞ 4 max{αo+ , βo+ }

where ζ ∈ Q(xo , A) .

This completes the proof.



3.3. Proof of Theorem 3.3. According to the [17, Theorems 2.2 and 3.2], u and v are locally bounded. From [10, Theorem 1.2], u and v are locally C 1,α continuous. Similar to the proof of [13, Proposition 2.5], we obtain that u, v ∈ C 1,α (RN ), u(x) → 0, |∇u(x)| → 0, v(x) → 0 and |∇v(x)| → 0 as |x| → ∞. Note 1. Let us consider the existence of solutions for the system − div |∇ui |pi (x)−2 ∇ui ) + |ui |pi (x)−2 ui = λai (x)|ui |γi (x)−2 ui + Fui (x, u1 , . . . , un )

in RN ,

ui ∈ W 1,pi (·) (RN ), i = 1, . . . , n, where u = (u1 , . . . , un ), suppose λ is small enough, then the system has a nontrivial solution if it satisfies the following assumptions: (H0) pi (·) are Lipschitz continuous, 1