Global asymptotic behavior of solutions to a class of Kirchhoff equations
aa r X i v : . [ m a t h . A P ] J un Global asymptotic behavior of solutions to aclass of Kirchhoff equations
Yuzhu Han † School of Mathematics, Jilin University, Changchun 130012, P.R. China
Abstract
In this paper, a parabolic type Kirchhoff equation and its stationary counterpartare considered. For the evolution problem, the precise decay rates of the weak solution and ofthe corresponding energy functional are derived. For the stationary problem, a ground-statesolution is obtained by applying Lagrange multiplier method. Moreover, the asymptotic be-haviors of the general global solutions are also described. These results extend some recentones obtained in [Threshold results for the existence of global and blow-up solutions to Kirch-hoff equations with arbitrary initial energy, Computers and Mathematics with Applications,75(2018), 3283-3297] by Han and Li.
Keywords
Kirchhoff equation; decay rate; ground-state solution; asymptotic behavior.
AMS Mathematics Subject Classification 2010:
In this paper, we are interested in the following initial boundary value problem (IBVP) fora class of nonlocal parabolic equation u t − M ( R Ω |∇ u | d x )∆ u = | u | q − u, ( x, t ) ∈ Ω × (0 , T ) ,u = 0 , ( x, t ) ∈ ∂ Ω × (0 , T ) ,u ( x,
0) = u ( x ) , x ∈ Ω , (1.1)where M ( s ) = a + bs with a, b >
0, Ω ⊂ R n ( n ≥
1) is a bounded smooth domain with ∂ Ω asits boundary, 3 < q < ∗ −
1, where 2 ∗ is the Sobolev conjugate of 2, i.e. 2 ∗ = + ∞ for n = 1 , ∗ = 2 nn − n ≥
3. Moreover, the initial datum u belongs to the energy space H (Ω).When he describes the transversal oscillations of a stretched string by taking into accountthe change in the string length, Kirchhoff [1] first proposed the hyperbolic counterpart of the † Corresponding author.Email addresses: [email protected](Y. Han). ∗ The project is supported by NSFC (11401252), by Science and Technology Development Project of JilinProvince (20160520103JH) and by The Education Department of Jilin Province (JJKH20190018KJ). L (Ω) norm of the gradient of the unknown, which means they are no longerpointwise identities. Therefore, such equations are usually referred to as Kirchhoff equations ornonlocal equations. When the nonlinearity | u | q − u in (1.1) is replaced by some L (Ω) function f ( x ), Chipot et al. [2] studied the local and global existence, uniqueness and global asymptoticbehavior of weak solutions to (1.1). Recently, Han et al. [3, 4] considered the global existenceand finite time blow-up of solutions to IBVP (1.1) by applying the modified potential wellmethod, which was first proposed by Sattinger [5], Payne and Sattinger [6], improved by Liu[7] and Xu et al. [8] and then applied to other evolution problems, for example, in [9, 10].The purpose of this paper is to generalize some results obtained in [3]. As in [3], we willdenote by k u k r the L r (Ω) norm of a Lebesgue function u ∈ L r (Ω) for r ≥ · , · ) theinner product in L (Ω). By H (Ω) we denote the Sobolev space such that both u and |∇ u | belong to L (Ω) for any u ∈ H (Ω). Recalling Poincar´e’s inequality, we can equip H (Ω) withthe equivalent norm k u k H (Ω) = k∇ u k . We denote by H − (Ω) the dual space to H (Ω). For u ∈ H (Ω), set J ( u ) = a k∇ u k + b k∇ u k − q + 1 k u k q +1 q +1 , (1.2) I ( u ) = a k∇ u k + b k∇ u k − k u k q +1 q +1 , (1.3)and define the Nehari manifold N = { u ∈ H (Ω) | I ( u ) = 0 , k∇ u k = 0 } . (1.4)Since q + 1 < ∗ , it can be directly checked that both J ( u ) and I ( u ) are well defined in H (Ω)and J, I ∈ C ( H (Ω) , R ). Moreover, for any u, v ∈ H (Ω), h J ′ ( u ) , v i = a ( ∇ u, ∇ v ) + b k∇ u k ( ∇ u, ∇ v ) − ( | u | q − u, v ) , (1.5)and h I ′ ( u ) , v i = 2 a ( ∇ u, ∇ v ) + 4 b k∇ u k ( ∇ u, ∇ v ) − ( q + 1)( | u | q − u, v ) , (1.6)where h· , ·i denotes the pairing between H − (Ω) and H (Ω). From (1.2) and (1.3) it followsthat J ( u ) = a ( q − q + 1) k∇ u k + b ( q − q + 1) k∇ u k + 1 q + 1 I ( u )= a k∇ u k + q − q + 1) k u k q +1 q +1 + 14 I ( u ) . (1.7)The potential well and its corresponding set are defined, respectively, by W = { u ∈ H (Ω) | I ( u ) > , J ( u ) < d } ∪ { } ,V = { u ∈ H (Ω) | I ( u ) < , J ( u ) < d } , where d = inf = u ∈ H (Ω) sup λ> J ( λu ) = inf u ∈N J ( u ) (1.8)2s the depth of the potential well W . By Lemma 2.1 in [3], d ≥ d := a ( q − q + 1) ( aS q +1 ) q − , (1.9)where S > H (Ω) to L q +1 (Ω), i.e.,1 S = inf = w ∈ H (Ω) k∇ w k k w k q +1 . (1.10)In view of the definition of d , a natural question is that can d be attained? We shall give apositive answer to this question and show that it is closely related to the stationary problem ofIBVP (1.1), i.e., the following boundary value problem (BVP) − M ( R Ω |∇ u | d x )∆ u = | u | q − u, x ∈ Ω ,u = 0 , x ∈ ∂ Ω . (1.11)It is well known that u ∈ H (Ω) is a weak solution to (1.11) if and only if it is a critical pointof J . Let S denote the set of all weak solutions to BVP (1.11). Then we have the followingtheorem. Theorem 1.1.
Let N and d be defined in (1.4) and (1.8) , respectively. Then there is a v ∈ N such that(1) J ( v ) = inf v ∈N J ( v ) = d ; (2) v is a ground-state solution to BVP (1.11) , i.e., v ∈ S \ { } and J ( v ) = inf v ∈S\{ } J ( v ) . Remark 1.1.
Although we stated Theorem 1.1 when < q < ∗ − , it is also valid when q = 3 , provided we assume in addition that < b < S , where S > is given in (1.10) . Infact, noticing that the constant S > in (1.10) is attained, it is easily checked that the Nehari’smanifold N is non-empty and the potential depth d is still positive in this case. The remainingproof is similar to that of Theorem 1.1, and interested readers may check the details. The weak solution to IBVP (1.1) is defined as follows.
Definition 1.1. ( Weak solution ) Let
T > . A function u = u ( x, t ) ∈ L ∞ (0 , T ; H (Ω)) with u t ∈ L (0 , T ; L (Ω)) is called a weak solution to IBVP (1.1) on Ω × [0 , T ) , if u ( x,
0) = u ∈ H (Ω) and satisfies ( u t , φ ) + (cid:16) ( a + b Z Ω |∇ u | d x ) ∇ u, ∇ φ (cid:17) = ( | u | q − u, φ ) , a. e. t ∈ (0 , T ) , (1.12) for any φ ∈ H (Ω) . Moreover, u ( x, t ) satisfies Z t k u τ k d τ + J ( u ( x, t )) = J ( u ) , a. e. t ∈ (0 , T ) . (1.13)3ometimes, we think of the function u ( x, t ) of the space variable x as an element of H (Ω),and briefly denote the element of H (Ω) that arises this way by u ( t ); therefore u ( t ) ∈ H (Ω).When J ( u ) < d , the existence of global solutions to IBVP (1.1), which can be summarized inthe following theorem, was obtained in [3]. Theorem 1.2. [3, Theorem 3.1] Assume a, b > , < q < ∗ − and u ∈ H (Ω) . If J ( u ) < d and I ( u ) > , then IBVP (1.1) admits a global weak solution u ∈ L ∞ (0 , ∞ ; H (Ω)) with u t ∈ L (0 , ∞ ; L (Ω)) and u ( t ) ∈ W for ≤ t < ∞ . Moreover, k u k ≤ k u k e − Ct , where C > is a constant. In addition, the weak solution is unique if it is bounded. In view of Theorem 1.2, we observe that the following three problems are unsolved:(i) Can we give the exact value or can we find some positive bounds for the exponential rate C in Theorem 1.2?(ii) Only the decay rate of k u ( t ) k was derived. Can we also estimate the decay rate of k∇ u ( t ) k and k u ( t ) k q +1 ?(iii) It follows from (1.13) that J ( u ( t )) is decreasing with respect to t . Can we derive thedecay rate of J ( u ( t ))?The second result in this paper is concerned with the above problems, which can be solvedfor the case of J ( u ) < d ≤ d . Theorem 1.3.
Assume a, b > , < q < ∗ − and u ∈ H (Ω) . If J ( u ) < d and I ( u ) > , then IBVP (1.1) admits a global weak solution u ∈ L ∞ (0 , ∞ ; H (Ω)) with u t ∈ L (0 , ∞ ; L (Ω)) and u ( t ) ∈ W for ≤ t < ∞ . Moreover, k u k ≤ k u k e − C t , k∇ u ( t ) k ≤ q + 1) a ( q −
1) [ J ( u ) + k u k ] e − C t , k u ( t ) k q +1 ≤ S ( q + 1) a ( q −
1) [ J ( u ) + k u k ] e − C t ,J ( u ( t )) + k u ( t ) k ≤ [ J ( u ) + k u k ] e − C t , where C = 2 aλ h − (cid:0) J ( u ) d (cid:1) q − i > , C = aλ α ( q − aλ ( q −
1) + 2( q + 1) > , λ > is thefirst eigenvalue of − ∆ in Ω under homogeneous Dirichlet boundary condition and α = 8 h − (cid:0) J ( u ) d (cid:1) q − ih − q + 1 (cid:0) J ( u ) d (cid:1) q − i − > . Remark 1.2.
From (1.7) and I ( u ) > one sees that J ( u ) > . Therefore, all the termsin Theorem 1.3 make sense. In [3], the authors also investigated the global existence and finite time blow-up of solutionsto IBVP (1.1) when J ( u ) > d . To introduce these results, set N + = { u ∈ H (Ω) | I ( u ) > } , N − = { u ∈ H (Ω) | I ( u ) < } ,J s = { u ∈ H (Ω) | J ( u ) < s } , N s = N ∩ J s . s > d , N s = n u ∈ N : a ( q − q + 1) k∇ u k + b ( q − q + 1) k∇ u k < s o = ∅ . (1.14)Define λ s := inf {k u k | u ∈ N s } , Λ s := sup {k u k | u ∈ N s } . (1.15)With the help of the notations given above, Han and Li gave another global existence andfinite time blow-up result for J ( u ) > d . Theorem 1.4. [3, Theorem 5.1] Let < q < ∗ − . Assume that J ( u ) > d , then thefollowing statements hold.(i) If u ∈ N + and k u k ≤ λ J ( u ) , then all the solutions to IBVP (1.1) exist globally andtend to zero in H (Ω) as t → ∞ ;(ii) If u ∈ N − and k u k ≥ Λ J ( u ) , then all the solutions to IBVP (1.1) blow up in finitetime. To make Theorem 1.4 nontrivial, it is necessary to show that K ≤ λ J ( u ) ≤ Λ J ( u ) ≤ K for some positive constants K and K , which was not done in [3]. Moreover, from Theorem1.4 we know that the global solutions converge to 0 in H (Ω) as t → ∞ when the initial datasatisfy some specific conditions. Can we describe the asymptotic behaviors of the general globalsolutions? At the end of this section, we shall answer these two questions. Our results in thesetwo directions can be summarized into the following two theorems. Theorem 1.5.
Let < q < ∗ − . Then for any s > d , λ s and Λ s defined in (1.15) satisfy K ≤ λ s ≤ Λ s ≤ K , where K = (cid:16) aG (cid:17) /γ θ − n ( q − γ , if q ≤ n ; (cid:16) aG (cid:17) /γ h q + 1) sa ( q − i − n ( q − γ , if q > n ,γ = q +1 − n ( q − > , θ = (cid:16) q + 1) dq − (cid:17) q +1 S − , G > is the constant in Gagliardo-Nirenberginequality and K = s q + 1) saλ ( q − . Theorem 1.6.
Assume a, b > , < q < ∗ − and u ∈ H (Ω) . Let u = u ( t ) be a globalsolution to IBVP (1.1) . Then there exists a u ∗ ∈ S and an increasing sequence { t k } ∞ k =1 with t k → ∞ as k → ∞ such that lim k →∞ k u ( t k ) − u ∗ k H (Ω) = lim k →∞ k∇ u ( t k ) − ∇ u ∗ k = 0 . Proofs of the main results.
Proof of Theorem 1.1. (1)
From (1.7) and the definitions of d and N it follows that d = inf v ∈N J ( v ) = inf v ∈N h a ( q − q + 1) k∇ v k + b ( q − q + 1) k∇ v k i . Let { v k } ∞ k =1 ⊂ N be a minimizing sequence of J . Thenlim k →∞ J ( v k ) = lim k →∞ h a ( q − q + 1) k∇ v k k + b ( q − q + 1) k∇ v k k i = d. (2.1)Noticing that q >
3, we obtain from (2.1) that { v k } ∞ k =1 ⊂ N is bounded in H (Ω), which,together with the fact that q + 1 < ∗ , implies that there is a subsequence of { v k } ∞ k =1 ⊂ N ,which we still denote by { v k } ∞ k =1 ⊂ N , and a v ∈ H (Ω) such that v k ⇀ v weakly in H (Ω) as k → ∞ ,v k → v strongly in L q +1 (Ω) as k → ∞ . (2.2)Since { v k } ∞ k =1 ⊂ N , we have a k∇ v k k + b k∇ v k k = k v k k q +1 q +1 . Combining this identity with the weakly lower semi-continuity of k · k and recalling (2.2) weget a k∇ v k + b k∇ v k ≤ lim inf k →∞ [ a k∇ v k k + b k∇ v k k ]= lim k →∞ [ a k∇ v k k + b k∇ v k k ]= lim k →∞ k v k k q +1 q +1 = k v k q +1 q +1 . (2.3)We claim that I ( v ) = 0, or equivalently k∇ v k = lim k →∞ k∇ v k k . If not, then by (2.3) itmust hold that a k∇ v k + b k∇ v k < k v k q +1 q +1 and k∇ v k = 0. By Lemma 2.2 in [3], thereexists a unique λ ∗ > λ ∗ v ∈ N , i.e., I ( λ ∗ v ) = 0. Therefore, aλ ∗ k∇ v k + bλ ∗ k∇ v k = λ ∗ q +1 k v k q +1 q +1 > λ ∗ q +1 ( a k∇ v k + b k∇ v k ) , i.e., a ( λ ∗ q +1 − λ ∗ ) k∇ v k + b ( λ ∗ q +1 − λ ∗ ) k∇ v k < . This implies that λ ∗ < k∇ v k < lim k →∞ k∇ v k k to obtain a ( q − q + 1) k∇ v k + b ( q − q + 1) k∇ v k < a ( q − q + 1) lim k →∞ k∇ v k k + b ( q − q + 1) lim k →∞ k∇ v k k = lim k →∞ h a ( q − q + 1) k∇ v k k + b ( q − q + 1) k∇ v k k i = d.
6n the other hand, it follows from I ( λ ∗ v ) = 0, λ ∗ < d ≤ J ( λ ∗ v ) = a ( q − q + 1) λ ∗ k∇ v k + b ( q − q + 1) λ ∗ k∇ v k < a ( q − q + 1) k∇ v k + b ( q − q + 1) k∇ v k , a contradiction. Therefore, I ( v ) = 0 and k∇ v k = lim k →∞ k∇ v k k . Together with the weakconvergence in (2.2) and the uniform convexity of H (Ω), it implies that v k → v strongly in H (Ω) as k → ∞ . Moreover, J ( v ) = a ( q − q + 1) k∇ v k + b ( q − q + 1) k∇ v k + 1 q + 1 I ( v )= a ( q − q + 1) k∇ v k + b ( q − q + 1) k∇ v k = lim k →∞ h a ( q − q + 1) k∇ v k k + b ( q − q + 1) k∇ v k k i = d, which implies that v = 0. Therefore v ∈ N and J ( v ) = inf v ∈N J ( v ) = d . (2) Since J ( v ) = inf v ∈N J ( v ) = d , by the theory of Lagrange multipliers, there exists aconstant σ ∈ R such that J ′ ( v ) − σI ′ ( v ) = 0 , (2.4)which then ensures σ h I ′ ( v ) , v i = h J ′ ( v ) , v i = I ( v ) = 0 . (2.5)Recalling (1.6) and the fact that I ( v ) = 0 we obtain h I ′ ( v ) , v i = 2 a k∇ v k + 4 b k∇ v k − ( q + 1) k v k q +1 q +1 = 2 a k∇ v k + 4 b k∇ v k − ( q + 1)[ a k∇ v k + b k∇ v k ]= − a ( q − k∇ v k − b ( q − k∇ v k < . Therefore, σ = 0, which in turn implies J ′ ( v ) = 0 by (2.4). So v ∈ S \ { } . Noticing that S \ { } ⊂ N , we have J ( v ) = inf v ∈S\{ } J ( v ) and the proof of Theorem 1.1 is complete. ✷ Proof of Theorem 1.3.
Since d ≤ d , it follows from Theorem 1.2 that IBVP (1.1) admitsa global weak solution u ∈ L ∞ (0 , ∞ ; H (Ω)) with u t ∈ L (0 , ∞ ; L (Ω)) and u ( t ) ∈ W for0 ≤ t < ∞ .We first derive the decay rate of k u ( t ) k . Since u ( t ) ∈ W for 0 ≤ t < ∞ , we have I ( u ( t )) ≥ , t ≥ . (2.6)7y the first equality in (1.7), the energy identity (1.13) and (2.6) we have J ( u ) ≥ J ( u ( t )) = a ( q − q + 1) k∇ u ( t ) k + b ( q − q + 1) k∇ u ( t ) k + 1 q + 1 I ( u ( t )) ≥ a ( q − q + 1) k∇ u ( t ) k , (2.7)which, together with (1.10), implies that k u ( t ) k q +1 ≤ S k∇ u ( t ) k ≤ S h q + 1) a ( q − J ( u ) i . (2.8)With the help of (1.9), (1.10) and (2.8) we arrive at k u ( t ) k q +1 q +1 ≤ S k u ( t ) k q − q +1 k∇ u ( t ) k ≤ S q +1 h q + 1) a ( q − J ( u ) i q − k∇ u ( t ) k = h J ( u ) d i q − a k∇ u ( t ) k . (2.9)Taking φ = u in (1.12) one gets ddt k u ( t ) k = − (cid:16) a k∇ u ( t ) k + b k∇ u ( t ) k − k u ( t ) k q +1 q +1 (cid:17) = − I ( u ( t )) . (2.10)Combining (2.9) with (2.10) we have ddt k u ( t ) k ≤ − h a k∇ u ( t ) k − k u ( t ) k q +1 q +1 i ≤ − a h − (cid:0) J ( u ) d (cid:1) q − i k∇ u ( t ) k ≤ − aλ h − (cid:0) J ( u ) d (cid:1) q − i k u ( t ) k , which implies k u ( t ) k ≤ k u k e − C t , where C = 2 aλ h − (cid:0) J ( u ) d (cid:1) q − i > λ > − ∆ in Ω underhomogeneous Dirichlet boundary condition.Next, we estimate the decay rate of k∇ u ( t ) k , k u ( t ) k q +1 and J ( u ( t )). By the definition of I ( u ) and (2.9) we have I ( u ) ≥ a h − (cid:0) J ( u ) d (cid:1) q − i k∇ u k . (2.11)Define an auxiliary function H ( t ) by H ( t ) = J ( u ( t )) + k u ( t ) k . (2.12)Then from (2.7) it follows that J ( u ( t )) ≤ H ( t ) ≤ J ( u ( t )) + 1 λ k∇ u ( t ) k ≤ h q + 1) aλ ( q − i J ( u ( t )) . (2.13)8urthermore, the second equality in (1.7), (1.13), (2.10) and (2.11) guarantee, for any α > H ′ ( t ) = −k u t ( t ) k − I ( u ( t )) ≤ − I ( u ( t )) − αJ ( u ( t )) + aα k∇ u ( t ) k + α ( q − q + 1) k u ( t ) k q +1 q +1 + α I ( u ( t )) ≤ − αJ ( u ( t )) + βI ( u ( t )) , (2.14)where β = − α α h − (cid:0) J ( u ) d (cid:1) q − i − n q − q + 1 (cid:0) J ( u ) d (cid:1) q − o . Set α = 8 h − (cid:0) J ( u ) d (cid:1) q − ih − q + 1 (cid:0) J ( u ) d (cid:1) q − i − , then β = 0 and from (2.13) and (2.14)we further obtain H ′ ( t ) ≤ − αJ ( u ( t ) ≤ − α h q + 1) aλ ( q − i − H ( t ) := − C H ( t ) . (2.15)where C = aλ α ( q − aλ ( q −
1) + 2( q + 1) >
0. Integrating (2.15) over [0 , t ] to obtain J ( u ( t )) + k u ( t ) k = H ( t ) ≤ H (0) e − C t = [ J ( u ) + k u k ] e − C t . (2.16)By (2.7) and (2.16) we have k∇ u ( t ) k ≤ q + 1) a ( q − J ( u ( t )) ≤ q + 1) a ( q −
1) [ J ( u ( t )) + k u ( t ) k ] ≤ q + 1) a ( q −
1) [ J ( u ) + k u k ] e − C t . (2.17)Finally, it follows from (2.9) and (2.17) that k u ( t ) k q +1 ≤ S k∇ u ( t ) k ≤ S ( q + 1) a ( q −
1) [ J ( u ) + k u k ] e − C t . (2.18)The proof of Theorem 1.3 is complete. ✷ Proof of Theorem 1.5.
We need the following Gagliardo-Nirenberg inequality (see [11])to derive the lower bound of λ s : k u k q +1 q +1 ≤ G k∇ u k n ( q − / k u k γ , ∀ u ∈ H (Ω) , (2.19)where G > n and q and γ = q + 1 − n ( q − > q < ∗ − u ∈ N , by (2.19) we have a k∇ u k ≤ k u k q +1 q +1 ≤ G k∇ u k n ( q − / k u k γ , (2.20)which implies k u k ≥ (cid:16) aG (cid:17) /γ k∇ u k γ − n ( q − γ = (cid:16) aG (cid:17) /γ k∇ u k − n ( q − γ , ∀ u ∈ N . (2.21)9y the definition of d , (1.7), (1.10) and (2.20), it is seen, for any u ∈ N , that d ≤ J ( u ) = a k∇ u k + q − q + 1) k u k q +1 q +1 + 14 I ( u )= a k∇ u k + q − q + 1) k u k q +1 q +1 ≤ h
14 + q − q + 1) i k u k q +1 q +1 ≤ q − q + 1) S q +1 k∇ u k q +12 , which guarantees that k∇ u k ≥ θ := (cid:16) q + 1) dq − (cid:17) q +1 S − , ∀ u ∈ N . (2.22)If 4 − n ( q − ≥
0, then by combining (2.21) and (2.22) we obtain λ s = inf u ∈N s k u k ≥ inf u ∈N k u k ≥ (cid:16) aG (cid:17) /γ θ − n ( q − γ . (2.23)If 4 − n ( q − <
0, then from (2.21) and (1.14) it follows that λ s = inf u ∈N s k u k ≥ (cid:16) aG (cid:17) /γ h sup u ∈N s k∇ u k i − n ( q − q +1) − n ( q − ≥ (cid:16) aG (cid:17) /γ h q + 1) sa ( q − i − n ( q − γ . (2.24)Recalling (1.14) and Poincar´e’s inequality k u k ≤ √ λ k∇ u k for u ∈ H (Ω) we haveΛ s = sup u ∈N s k u k ≤ s q + 1) saλ ( q − . (2.25)The proof of Theorem 1.5 is complete. ✷ Proof of Theorem 1.6.
Let u = u ( t ) be a global solution to IBVP (1.1). Without loss ofgenerality, we may assume that0 ≤ J ( u ( t )) ≤ J ( u ) , t ∈ [0 , ∞ ) . (2.26)In fact, the second inequality follows from (1.13). If J ( u ( t )) < t >
0, then I ( u ( t )) < u = u ( t ) blows up in finite time, which is acontradiction.Since J ( u ( t )) is non-increasing in t and bounded from below, there exists a constant J ≥ t →∞ J ( u ( t )) = J . (2.27)Letting t → ∞ in (1.13) and noticing (2.27) we obtain Z ∞ k u τ k d τ = J ( u ) − J ≤ J ( u ) , { t k } ∞ k =1 with t k → ∞ as k → ∞ such thatlim k →∞ k u t ( t k ) k = 0 . (2.28)Denote u k = u ( t k ). We shall show that { u k } ∞ k =1 is bounded in H (Ω). For any v ∈ H (Ω),it follows from (1.5) that h J ′ ( u k ) , v i = a ( ∇ u k , ∇ v ) + b k∇ u k k ( ∇ u k , ∇ v ) − ( | u k | q − u k , v ) = − ( u t ( t k ) , v ) , which implies k J ′ ( u k ) k H − (Ω) = sup k∇ v k≤ |h J ′ ( u k ) , v i| ≤ sup k∇ v k≤ k u t ( t k ) k k v k ≤ √ λ sup k∇ v k≤ k u t ( t k ) k k∇ v k = 1 √ λ k u t ( t k ) k → , as k → ∞ . (2.29)Therefore, there exists a positive constant κ such that | I ( u k ) | = |h J ′ ( u k ) , u k i| ≤ k J ′ ( u k ) k H − (Ω) k∇ u k k ≤ κ k∇ u k k . Recalling (1.7) again, one gets J ( u ) + κq + 1 k∇ u k k ≥ J ( u k ) − q + 1 I ( u k )= a ( q − q + 1) k∇ u k k + b ( q − q + 1) k∇ u k k ≥ a ( q − q + 1) k∇ u k k , which implies that there exists a constant Θ > k∇ u k k ≤ Θ , k = 1 , , · · · . (2.30)Therefore, there exists a subsequence of { u k } ∞ k =1 , which we still denote by { u k } ∞ k =1 , and a u ∗ ∈ H (Ω) such that u k ⇀ u ∗ weakly in H (Ω) as k → ∞ ,u k → u ∗ strongly in L q +1 (Ω) as k → ∞ . (2.31)For u ∈ H (Ω), set E ( u ) = a k∇ u k + b k∇ u k . Then by Lemma 3.1 in [3], E ′ : H (Ω) → H − (Ω) is a strong monotone operator, which satisfies h E ′ ( u ) − E ′ ( v ) , u − v i ≥ a k∇ u − ∇ v k , ∀ u, v ∈ H (Ω) . (2.32)Since J ( u ) = E ( u ) + | u | q − u , so h J ′ ( u k ) − J ′ ( u ∗ ) , u k − u ∗ i = h E ′ ( u k ) − E ′ ( u ∗ ) , u k − u ∗ i + ( | u k | q − u k − | u ∗ | q − u ∗ , u k − u ∗ ) ≥ a k∇ u k − ∇ u ∗ k + ( | u k | q − u k − | u ∗ | q − u ∗ , u k − u ∗ ) . (2.33)11y (2.29) and (2.30), |h J ′ ( u k ) , u k − u ∗ i| ≤ k J ′ ( u k ) k H − (Ω) → , as k → ∞ . (2.34)By (2.31), |h J ′ ( u ∗ ) , u k − u ∗ i| → , as k → ∞ . (2.35)By H¨older’s inequality, (2.30) and (2.31), | ( | u k | q − u k − | u ∗ | q − u ∗ , u k − u ∗ ) | ≤ ( k u k k qq +1 + k u ∗ k qq +1 ) k u k − u ∗ k q +1 ≤ ( S q k∇ u k k q + k u ∗ k qq +1 ) k u k − u ∗ k q +1 ≤ ( S q Θ q + k u ∗ k qq +1 ) k u k − u ∗ k q +1 → , as k → ∞ . (2.36)Substituting (2.34)-(2.36) into (2.33) we see that k u k − u ∗ k H (Ω) = k∇ u k − ∇ u ∗ k → as k → ∞ . Therefore J ′ ( u ∗ ) = lim k →∞ J ′ ( u k ) , in H − (Ω) , which, together with (2.29), guarantees that J ′ ( u ∗ ) = 0, i.e., u ∗ ∈ S . The proof of Theorem 1.6is complete. ✷ Acknowledgements
The author would like to express his sincere gratitude to Professor Wenjie Gao for his enthusi-astic guidance and constant encouragement.
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