Existence of solutions for Kirchhoff-type fractional Dirichlet problem with p -Laplacian
aa r X i v : . [ m a t h . A P ] N ov Existence of solutions for Kirchhoff-type fractional Dirichletproblem with p -Laplacian Danyang Kang , Cuiling Liu , Xingyong Zhang , ∗ Faculty of Science, Kunming University of Science and Technology, Kunming, Yunnan, 650500, P.R. China. School of Mathematics and Statistics, Central South University, Changsha, Hunan, 410083, P.R. China.
Abstract:
In this paper, we investigate the existence of solutions for a class of p -Laplacian fractional orderKirchhoff-type system with Riemann-Liouville fractional derivatives and a parameter λ . By mountainpass theorem, we obtain that system has at least one non-trivial weak solution u λ under some localsuperquadratic conditions for each given large parameter λ . We get a concrete lower bound of the parameter λ , and then obtain two estimates of weak solutions u λ . We also obtain that u λ → λ tends to ∞ . Finally,we present an example as an application of our results. Keywords:
Kirchhoff-type system; Fractional p -Laplacian; Local superquadratic nonlinearity; Mountainpass theorem; Existence
1. Introduction and main results
In this paper, we are concerned with the following system A ( u ( t ))[ t D αT φ p ( D αt u ( t )) + V ( t ) φ p ( u ( t ))] = λ ∇ F ( t, u ( t )) , a.e. t ∈ [0 , T ] ,u (0) = u ( T ) = 0 , (1.1)where A ( u ( t )) = " a + b Z T ( | D αt u ( t ) | p + V ( t ) | u ( t ) | p ) dt p − ,a, b, λ > p > /p < α ≤ p is an integer, u ( t ) = ( u ( t ) , · · · , u N ( t )) τ ∈ R N for a.e. t ∈ [0 , T ], T >
0, and N is a given positive integer, ( · ) τ denote the transpose of a vector, V ( t ) ∈ C ([0 , T ] , R )with min t ∈ [0 ,T ] V ( t ) > D αt and t D αT are the left and right Riemann-Liouville fractional derivatives, respectively, ∗ Corresponding author, E-mail address: [email protected] p ( s ) := | s | p − s , ∇ F ( t, x ) is the gradient of F with respect to x = ( x , · · · , x N ) ∈ R N , that is, ∇ F ( t, x ) =( ∂F∂x , · · · , ∂F∂x N ) τ , and F : [0 , T ] × R N → R satisfies the following condition: (H0) there exists a constant δ > such that F ( t, x ) is continuously differentiable in x ∈ R N with | x | ≤ δ for a.e. t ∈ [0 , T ] , measurable in t for every x ∈ R N with | x | ≤ δ , and there exist a ∈ C ( R + , R + ) and b ∈ L ([0 , T ]; R + ) such that | F ( t, x ) | , |∇ F ( t, x ) | ≤ a ( | x | ) b ( t ) for all x ∈ R N with | x | ≤ δ and a.e. t ∈ [0 , T ] . When α = 1, the operator t D αT ( D αt u ( t )) reduces to the usual second order differential operator − d /dt .Hence, if α = 1, p = 2, N = 1, λ = 1 and V ( t ) = 0 for a.e. t ∈ [0 , T ], system (1.1) becomes the equation withDirichlet boundary condition − (cid:16) a + b R T | u ′ ( t ) | dt (cid:17) u ′′ ( t ) = f ( t, u ( t )) , a.e. t ∈ [0 , T ] ,u (0) = u ( T ) = 0 , (1.2)where f ( t, x ) = ∂F ( t,x ) ∂x and F : [0 , T ] × R → R . It is well known that equation (1.2) is related to the stationaryproblem of a classical model introduced by Kirchhoff [1]. To be precise, in [1], Kirchhoff introduced the model ρ ∂ u∂t = P + Eh L Z L (cid:18) ∂u∂y (cid:19) dy ! ∂ u∂y , (1.3)where 0 ≤ y ≤ L , t ≥ u is the lateral deflection, ρ is the mass density, h is the cross-sectional area, L is thelength, E is the Youngs modulus and P is the initial axial tension. ( Notations : in model (1.3), (1.7) and(1.8) below, t is time variable and y is spatial variable, which are conventional notations in partial differentialequations. One need to distinguish them to t in (1.1), (1.2), (1.4), (1.5) and (1.6) below, where t correspondsto the spatial variable x ). The model (1.3) is used to describe small vibrations of an elastic stretched string.Equation (1.3) has been studied extensively, for instance, [2], [3], [4], [5], [6], [7], [8], [9], [10], [11], [12], [13], [14]and reference therein. For p >
1, the reader can consult [15], [16], [17], [18], [19] and references therein.When α < D αt and t D αT are the left and right Riemann-Liouville fractional derivatives, respectively, whichhas been given some physical interpretations in [20]. Moreover, it is also applied to describe the anomalousdiffusion, L´evy flights and traps in [21] and [22]. Fractional differential equations have been proved to provide anatural framework in the modeling of many real phenomena such as viscoelasticity, neurons, electrochemistry,control, porous media, electromagnetic (the reader can consult [23] in which a collection of references is given).In [23], Jiao and Zhou considered the system t D αT ( D αt u ( t )) = ∇ F ( t, u ( t )) , a.e. t ∈ [0 , T ] ,u (0) = u ( T ) = 0 . (1.4)2hey successfully applied critical point theory to investigate the existence of weak solutions for system (1.4).To be precise, they obtained that system (1.4) has at least one weak solution when F has a quadratic growthor a superquadratic growth by using the least action principle and mountain pass theorem. Subsequently, thistopic related to system (1.4) attracted lots of attention, for example, [24], [25], [26], [27], [28], [29] and referencestherein. It is obvious that system (1.1) is much more complicated than system (1.4) since the appearance ofnonlocal term A ( u ( t )) and p -Laplacian term φ p ( s ). Recently, in [30], the following fractional Kirchhoff equationwith Dirichlet boundary condition was investigated (cid:16) a + b R T | D αt u ( t ) | dt (cid:17) t D αT ( D αt u ( t )) + λV ( t ) u ( t ) = f ( t, u ( t )) , a.e. t ∈ [0 , T ] ,u (0) = u ( T ) = 0 , (1.5)where a, b, λ > f ∈ C ([0 , T ] × R , R ). By using the mountain pass theorem in [42] and the linking theorem in[43], the authors established some existence results of nontrivial solutions for system (1.5) if f satisfies (f1) there exist constants µ > , < τ < and a nonnegative function g ∈ L − τ such that F ( t, x ) − µ f ( t, x ) x ≤ g ( t ) | x | τ , for a.e. t ∈ [0 , T ] , x ∈ R ; (f2) there exists θ > such that lim | x |→∞ inf t ∈ [0 ,T ] F ( t,x ) | x | θ > ;(or (f2) ′ there exists θ > such that lim | x |→∞ inf t ∈ [0 ,T ] F ( t,x ) | x | θ > );(f3) there exists σ > such that lim | x |→ sup t ∈ [0 ,T ] F ( t,x ) | x | σ < ∞ , and some other reasonable conditions.In [31], Chen-Liu investigated the Kirchhoff-type fractional Dirichlet problem with p -Laplacian (cid:16) a + b R T | D αt u ( t ) | p dt (cid:17) p − t D αT φ p ( D αt u ( t )) = f ( t, u ( t )) , t ∈ (0 , T ) ,u (0) = u ( T ) = 0 . (1.6)where a, b, λ > f ∈ C ([0 , T ] × R , R ). By using the Nehari method, they established the existence result ofground state solution for system (1.6) if f satisfies (f4) f ( t, x ) = o ( | x | p − ) as | x | → uniformly for all t ∈ [0 , T ] , and the following well-known Ambrosetti-Rabinowitz (AR for short) condition (AR) there exist two constants µ > p , R > such that < µF ( t, x ) ≤ xf ( t, x ) , for ∀ t ∈ [0 , T ] , x ∈ R with | x | ≥ R, where F ( t, x ) = R x f ( t, s ) ds , and some additional conditions. It is easy to see that all of these conditions (f1),(f2), (f2) ′ and (AR) imply that F ( t, x ) needs to have a growth near the infinity for x , and (f3) and (f4) implythat F ( t, x ) needs to have a growth near 0 for x . 3n this paper, we investigate the existence and concentration of solutions for system (1.1) under localassumptions only near 0 for the nonlinear term F . Our work is mainly motivated by [32] and [12]. In [32],Costa and Wang investigated the multiplicity of both signed and sign-changing solutions for the one-parameterfamily of elliptic problems ∆ u = λf ( u ) in Ω u ( y ) = 0 in ∂ Ω , (1.7)where λ > R N ( N ≥
3) and f ∈ C ( R , R ). They assumedthat the nonlinearity f ( u ) has superlinear growth in a neighborhood of u = 0 and then obtained the number ofsigned and sign-changing solutions which are dependent on the parameter λ . The idea in [32] has been appliedto some different problems, for example, [33] and [35] for quasilinear elliptic problems with p -Laplacian operator,[34] for an elliptic problem with fractional Laplacian operator, [36] for Schr¨odinger equations, [11] for Neumannproblem with nonhomogeneous differential operator and critical growth, and [38] for quasilinear Schr¨odingerequations. Especially, in [12], Li and Su investigated the Kirchhoff-type equations − (cid:2) R R ( |∇ u | + V ( y ) u ) dy (cid:3) [∆ u + V ( y ) u ] = λQ ( y ) f ( u ) , y ∈ R ,u ( y ) → , as | y | → ∞ , (1.8)where λ > V, Q are radial functions and f ∈ C (( − δ , δ ) , R ) for some δ >
0. Via the idea in [32], they alsoestablished the existence result of solutions when f ( u ) has superlinear growth in a neighborhood of u = 0. Itis worthy to note that λ usually needs to be sufficiently large, that is, λ has a lower bound λ ∗ . However, theconcrete values of λ ∗ are not given in these references. Similar to system (1.8), comparing with equation (1.5)and equation (1.6), we add a nonlocal term R T V ( t ) | u ( t ) | p dt in system (1.1) where min t ∈ [0 ,T ] V ( t ) >
0, andmultiply V ( t ) φ p ( u ( t )) by the nonlocal part A ( u ( t )). Moreover, we consider the high-dimensional case, that is, N ≥
1. Since min t ∈ [0 ,T ] V ( t ) >
0, system (1.1) is different from equation (1.2), (1.5), (1.6) and system (1.4).More importantly, we present a concrete value of the lower bound λ ∗ for system (1.1) and then obtain twoestimates of the solutions family { u λ } for all λ > λ ∗ . Next, we make some assumptions for F . (H1) there exist constants q > p , q ∈ ( p , q ) , M > and M > such that M | x | q ≤ F ( t, x ) ≤ M | x | q for all x ∈ R N with | x | ≤ δ and a.e. t ∈ [0 , T ] ;(H2) there exists a constant β > p such that ≤ βF ( t, x ) ≤ ( ∇ F ( t, x ) , x ) for all x ∈ R N with | x | ≤ δ and a.e. t ∈ [0 , T ] . heorem 1.1. Suppose that (H0)-(H2) hold. Then system (1.1) has at least a nontrivial weak solution u λ forall λ > λ ∗ := max { Λ , Λ , Λ } and k u λ k pV ≤ p θa p − ( θ − p ) C ∗ λ − p − q − p ≤ p θa p − ( θ − p ) C ∗ max { Λ , Λ , Λ } − p − q − p , k u λ k ∞ ≤ T α − p Γ( α )( αq − q + 1) q · p θa p − ( θ − p ) C ∗ λ − p − q − p ≤ T α − p Γ( α )( αq − q + 1) q · p θa p − ( θ − p ) C ∗ max { Λ , Λ , Λ } − p − q − p , lim λ →∞ k u λ k V = 0 = lim λ →∞ k u λ k ∞ , where θ = min { β, q } , q = pp − , k u λ k V = Z T | D αt u λ ( t ) | p dt + Z T V ( t ) | u λ ( t ) | p dt ! /p , k u λ k ∞ = max t ∈ [0 ,T ] u λ ( t ) , (1.9)Λ = max V ∞ a p − (Γ( α )( αq − q + 1) q G ) q − p p M T ( α − p )( q − p ) ( δ min { , V ∞ } D ) q − p , bp ( a + bδ p G p max { , V ∞ } ( D p + G p )) p M δ q G q T − q p D q , (1.10)Λ = (cid:20) a + b [max { , V ∞ } ] p δ p G p ( D p + G p ) (cid:21) q ( p − , (1.11)Λ = T pα − h Γ( α )( αq − q + 1) q i p · p θC ∗ a p − ( θ − p ) · p δ p q − pp − , (1.12) V ∞ = max t ∈ [0 ,T ] V ( t ) , V ∞ = min t ∈ [0 ,T ] V ( t ) ,C ∗ = p ( M q ) pq − p − M ( M q ) q q − p ! (cid:18) [max { , V ∞ } ] /p ( D p + G p ) /p T q − p D (cid:19) pq q − p , (1.13) D = (cid:16) T p +1 π p +1 · p − p !! (cid:17) p , if p is odd , (cid:16) T p +1 π p · ( p − p !! (cid:17) p , if p is even ,G = (cid:18) T p +1 − pα [Γ(2 − α )] p ( p + 1 − pα ) (cid:19) /p ,G = T α − p Γ( α )( αq − q + 1) q G, Γ( z ) = Z ∞ t z − e − t dt ( for all z > . We organize this paper as follows. In section 2, we recall some preliminary results including the definitionsof Riemann-Liouville fractional derivatives and working spaces, some conclusions for the working spaces andmountain pass theorem. In section 3, we give the proof of Theorem 1.1. In section 4, we apply Theorem 1.1 toan example and compute the value of lower bound λ ∗ . 5 . Preliminaries In this section, we mainly recall some basic definitions and results.
Definition 2.1. (Left and Right Riemann-Liouville Fractional Integrals [39, 24]) Let f be a function definedon [ a, b ]. The left and right Riemann-Liouville fractional integrals of order γ > f denoted by a D − γt f ( t ) and t D − γb f ( t ) , respectively, are defined by a D − γt f ( t ) = 1Γ( γ ) Z ta ( t − s ) γ − f ( s ) ds, t ∈ [ a, b ] , γ > , t D − γb f ( t ) = 1Γ( γ ) Z bt ( s − t ) γ − f ( s ) ds, t ∈ [ a, b ] , γ > , provided the right-hand sides are pointwise defined on [ a, b ], where Γ > Definition 2.2. (Left and Right Riemann-Liouville Fractional Derivatives[39, 24]) Let f be a function definedon [ a, b ]. The left and right Riemann-Liouville fractional derivatives of order γ > f denoted by a D γt f ( t ) and t D γb f ( t ), respectively, are defined by a D γt f ( t ) = d n dt n a D γ − nt f ( t ) = 1Γ( n − γ ) d n dt n (cid:18)Z ta ( t − s ) n − γ − f ( s ) ds (cid:19) , t D γb f ( t ) = ( − n d n dt n t D γ − nb f ( t ) = ( − n Γ( n − γ ) d n dt n Z bt ( s − t ) n − γ − f ( s ) ds ! , where t ∈ [ a, b ] , n − ≤ γ < n and n ∈ N . Definition 2.3. ([23]) Let 0 < α ≤ < p < ∞ . The fractional derivative space E α,p is defined by theclosure of C ∞ ([0 , T ] , R N ) with the norm k u k = Z T | D αt u ( t ) | p dt + Z T | u ( t ) | p dt ! /p , ∀ u ∈ E α,p . From the definition of E α,p , it is apparent that the fractional derivative space E α,p is the space of functions u : [0 , T ] → R N which is absolutely continuous and has an α -order left and right Riemann-Liouville fractionalderivative D αt u ∈ L p ([0 , T ] , R N ) and u (0) = u ( T ) = 0 and one can define the norm on L p ([0 , T ] , R N ) as k u k L p = Z T | u ( t ) | p dt ! /p .E α,p is uniformly convex by the uniform convexity of L p . Remark 2.1.
It is easy to see that k u k V defined by (1.9) is also a norm on E α,p and k u k V and k u k areequivalent and min { , V ∞ }k u k p ≤ k u k pV ≤ max { , V ∞ }k u k p . (2.14)6 emma 2.1. ([23]) Let 0 < α ≤ < p < ∞ . E α,p is a reflexive and separable Banach space. Lemma 2.2. ([23]) Let 0 < α ≤ < p < ∞ . For all u ∈ E α,p , there has k u k L p ≤ C p k D αt u k L p , where C p = T α Γ( α + 1) > . Moreover, if α > p , then k u k ∞ ≤ T α − p Γ( α )( αq − q + 1) q k D αt u k L p , p + 1 q = 1 . (2.15) Lemma 2.3. ([23]) Let 1 /p < α ≤ < p < ∞ . The imbedding of E α,p in C ([0 , T ] , R N ) is compact.Let X be a Banach space. ϕ ∈ C ( X, R ) and c ∈ R . A sequence { u n } ⊂ X is called (PS) c sequence (namedafter R. Palais and S. Smale) if the sequence { u n } satisfies ϕ ( u n ) → c, ϕ ′ ( u n ) → . Lemma 2.4. (Mountain Pass Theorem [44, 40]) Let X be a Banach space, ϕ ∈ C ( X, R ), w ∈ X and r > k w k > r and b := inf k u k = r ϕ ( u ) > ϕ (0) ≥ ϕ ( w ) . Then there exists a (PS) c sequence with c := inf γ ∈ Γ max t ∈ [0 , ϕ ( γ ( t )) , Γ := { γ ∈ ([0 , , X ]) : γ (0) = 0 , γ (1) = w } . As in [31], for each λ >
0, we can define the functional I λ : E α,p → R as I λ ( u ) = 1 bp a + b Z T ( | D αt u ( t ) | p + V ( t ) | u ( t ) | p ) dt ! p − λ Z T F ( t, u ( t )) dt − a p bp . It is easy to see that the assumption (H0)-(H2) can not ensure that I λ is well defined on E α,p . So we followthe idea in [32] and simply sketch the outline of proof here. We use Lemma 2.4 to complete the proof. Since F satisfies the growth condition only near 0 by (H0)-(H2), in order to use the conditions globally, we modifyand extend F to ¯ F defined in section 3, and the corresponding functional is defined as ¯ I λ . Next we prove that¯ I λ has mountain pass geometry on E α,p . Then Lemma 2.4 implies that ¯ I λ has a (PS) c λ sequence. Then by astandard analysis, a convergent subsequence of the (PS) c λ sequence is obtained to ensure that c λ is the critical7alue of ¯ I λ . Finally, by an estimate about k u λ k ∞ , we obtain that the critical point u λ of ¯ I λ with k u λ k ∞ ≤ δ/ λ > λ ∗ for some concrete λ ∗ .
3. Proofs
Define m ( s ) ∈ C ( R , [0 , sm ′ ( s ) ≤ m ( s ) = , if | s | δ/ , , if | s | > δ. (3.1)Define ¯ F : [0 , T ] × R N → R as ¯ F ( t, x ) = m ( | x | ) F ( t, x ) + (1 − m ( | x | )) M | x | q . We define the variational functional corresponding to ¯ F as¯ I λ ( u ) = 1 bp a + b Z T ( | D αt u ( t ) | p + V ( t ) | u ( t ) | p ) dt ! p − λ Z T ¯ F ( t, u ( t )) dt − a p bp = 1 bp ( a + b k u k pV ) p − λ Z T ¯ F ( t, u ( t )) dt − a p bp (3.2)for all u ∈ E α,p . By (H0) and the definition of ¯ F , it is easy to obtain that ¯ F satisfies (H0) ′ ¯ F ( t, x ) is continuously differentiable in R N for a.e. t ∈ [0 , T ] , measurable in t for every x ∈ R N , andthere exists b ∈ L ([0 , T ]; R + ) such that | ¯ F ( t, x ) | ≤ a b ( t ) + M | x | q , |∇ ¯ F ( t, x ) | ≤ (1 + m ) a b ( t ) + M q | x | q − + m M | x | q for all x ∈ R N and a.e. t ∈ [0 , T ] , a = max s ∈ [0 ,δ ] a ( s ) and m = max s ∈ [ δ ,δ ] | m ′ ( s ) | . Hence, a standard argument shows that ¯ I λ ∈ C ( E α,p , R ) and h ¯ I ′ λ ( u ) , v i = ( a + b k u k pV ) p − Z T [ | D αt u ( t ) | p − ( D αt u ( t ) , D αt v ( t )) dt + V ( t ) | u ( t ) | p − ( u ( t ) , v ( t ))] dt ! − λ Z T ( ∇ ¯ F ( t, u ( t )) , v ( t )) dt for all u, v ∈ E α,p . Hence h ¯ I ′ λ ( u ) , u i = ( a + b k u k pV ) p − k u k pV − λ Z T ( ∇ ¯ F ( t, u ( t )) , u ( t )) dt for all u ∈ E α,p . 8 emma 3.1. Assume that (H1)-(H2) hold. Then(H1) ′ ≤ ¯ F ( t, x ) ≤ M | x | q , for all x ∈ R N ; (H2) ′ < θ ¯ F ( t, x ) ≤ ( ∇ ¯ F ( t, x ) , x ) , for all x ∈ R N / { } , where θ = min { q , β } . Proof. • If | x | ≤ δ , then by (H1), the conclusion (H1) ′ holds;If δ < | x | ≤ δ , by (H1), we have0 ≤ ¯ F ( t, x ) = m ( | x | ) F ( t, x ) + (1 − m ( | x | )) M | x | q ≤ m ( | x | ) M | x | q + (1 − m ( | x | )) M | x | q = M | x | q ;If | x | ≥ δ , then by the definition of m , we have ¯ F ( t, x ) = M | x | q . • For all x ∈ R N / { } , we have ∇ ¯ F ( t, x ) = m ′ ( | x | ) x | x | F ( t, x ) + m ( | x | ) ∇ F ( t, x ) + (1 − m ( | x | )) q M | x | q − x − m ′ ( | x | ) x | x | M | x | q . Then ( ∇ ¯ F ( t, x ) , x ) = | x | m ′ ( | x | )( F ( t, x ) − M | x | q ) + m ( | x | )( ∇ F ( t, x ) , x ) + (1 − m ( | x | )) q M | x | q . and θ ¯ F ( t, x ) − ( ∇ ¯ F ( t, x ) , x ) = m ( | x | )( θF ( t, x ) − ( ∇ F ( t, x ) , x )) + ( θ − q )(1 − m ( | x | )) M | x | q −| x | m ′ ( | x | )( F ( t, x ) − M | x | q ) . Apparently, the conclusion holds for 0 ≤ | x | ≤ δ/ | x | ≥ δ . If δ/ < | x | < δ , by using θ ≤ q , the conclusion(H1), (H2) and the fact sm ′ ( s ) ≤ s ∈ R , we can get the conclusion (H2) ′ . Lemma 3.2. ¯ I λ satisfies the mountain pass geometry for all λ > Λ , where Λ is defined in (1.10). Proof.
Note that q > p > p . By Lemma 3.1 and (2.15) , we have¯ I λ ( u ) = 1 bp ( a + b k u k pV ) p − λ Z T ¯ F ( t, u ( t )) dt − a p bp ≥ a p bp + a p − p k u k pV − λM Z T | u ( t ) | q dt − a p bp ≥ a p − p k u k pV − λM k u k q − p ∞ Z T | u ( t ) | p dt a p − p k u k pV − λM T α − p Γ( α )( αq − q + 1) q ! q − p k u k q − pV k u k pL p ≥ a p − p k u k pV − λ M V ∞ T α − p Γ( α )( αq − q + 1) q ! q − p k u k q V . We choose ν λ = a p − V ∞ p λM Tα − p Γ( α )( αq − q +1) 1 q ! q − p q − p for any given λ >
0. Then we have¯ I λ ( u ) > d λ := a p − p ν pλ − λ M V ∞ T α − p Γ( α )( αq − q + 1) q ! q − p ν q λ > , for all k u k V = ν λ . (3.3)Choose e = (cid:18) Tπ sin πtT , , · · · , (cid:19) ∈ E α,p . (3.4)Then k e k L p = D := (cid:16) T p +1 π p +1 p − p !! (cid:17) p , if p is odd , (cid:16) T p +1 π p ( p − p !! (cid:17) p , if p is even (3.5)and k D αt e k L p ≤ G := T p +1 − pα Γ p (2 − α )( p + 1 − pα ) . (3.6)By (2.15), k e k ∞ ≤ T α − p Γ( α )( αq − q + 1) q k D αt e k L p ≤ G := T α − p Γ( α )( αq − q + 1) q G. (3.7)Note thatΛ = max V ∞ a p − (Γ( α )( αq − q + 1) q G ) q − p p M T ( α − p )( q − p ) ( δ min { , V ∞ } D ) q − p , bp ( a + bδ p G p max { , V ∞ } ( D p + G p )) p M δ q G q T − q p D q . Then k δG e k V ≥ δ min { , V ∞ } G k e k L p ≥ ν λ for all λ > Λ . By (3.7), we have k δG e k ∞ ≤ δ . By the definition of ¯ F and (H1), we have ¯ F ( t, x ) = F ( t, x ) ≥ M | x | q for all | x | ≤ δ/
2, and¯ F ( t, x ) = m ( | x | ) F ( t, x ) + (1 − m ( x )) M | x | q ≥ m ( | x | ) M | x | q + (1 − m ( x )) M | x | q = M | x | q δ < | x | ≤ δ . Hence, by H¨older inequality, we have¯ I λ ( δG e ) = 1 bp ( a + b k δG e k pV ) p − λ Z T ¯ F ( t, δG e ( t )) dt − a p bp ≤ bp ( a + b k δG e k pV ) p − λM Z T | δG e ( t ) | q dt − a p bp ≤ bp ( a + bδ p G p max { , V ∞ }k e k p ) p − λM δ q G q T − q p k e k q L p ≤ bp ( a + bδ p G p max { , V ∞ } ( D p + G p )) p − λM δ q G q T − q p D q < λ > Λ .Let w = δG e and ϕ = ¯ I λ . Then for any given λ > Λ , Lemma 3.2 and Lemma 2.4 imply that ¯ I λ has a(PS) c λ sequence { u n } := { u n,λ } , that is, there exists a sequence { u n } satisfying¯ I λ ( u n ) → c λ , ¯ I ′ λ ( u n ) → , as n → ∞ , (3.8)where c λ := inf γ ∈ Γ max t ∈ [0 , ¯ I λ ( γ ( t )) , (3.9)Γ := { γ ∈ ([0 , , X ]) : γ (0) = 0 , γ (1) = w } . Lemma 3.3.
The (PS) c λ sequence { u n } has a convergent subsequence. Proof.
By virtue of Lemma 3.1, (3.8) and θ = min { q , β } > p , there exists a positive constant M > M + k u n k V ≥ ¯ I λ ( u n ) − θ h ¯ I ′ λ ( u n ) , u n i = ( a + b k u n k pV ) p − (cid:20) bp ( a + b k u n k pV ) − θ k u n k pV (cid:21) − λ Z T (cid:20) ¯ F ( t, u n ) − θ ( ∇ ¯ F ( t, u n ) , u n ) (cid:21) dt − a p bp ≥ ( a + b k u n k pV ) p − (cid:20) bp ( a + b k u n k pV ) − θ k u n k pV (cid:21) − a p bp ≥ a p − (cid:20) abp + (cid:18) p − θ (cid:19) k u n k pV (cid:21) − a p bp = a p − (cid:18) p − θ (cid:19) k u n k pV (3.10)for n large enough, which shows that { u n } is bounded in E α,p by p >
1. By Lemma 2.1, we can assume that,up to a subsequence, for some u λ ∈ E α,p , u n ⇀ u λ in E α,p , (3.11)11 n → u λ in C ([0 , T ] , R N ) . The following argument is similar to [45] with some modification. Since D I ′ λ ( u n ) , u n − u λ E = ( a + b k u k pV ) p − (cid:18) Z T ( | D αt u n | p − D αt u n , D αt ( u n − u λ )) dt + Z T V ( t )( | u n | p − u n , u n − u λ ) dt (cid:19) − λ Z T ( ∇ ¯ F ( t, u n ) , u n − u λ ) dt, (3.12)we have D I ′ λ ( u n ) − I ′ λ ( u λ ) , u n − u λ E = ( a + b k u n k pV ) p − (cid:18) Z T ( | D αt u n | p − D αt u n , D αt ( u n − u λ )) dt + Z T V ( t )( | u n | p − u n , u n − u λ ) dt (cid:19) − λ Z T ( ∇ ¯ F ( t, u n ) , u n − u λ ) dt − (cid:20) ( a + b k u λ k pV ) p − (cid:18) Z T ( | D αt u λ | p − D αt u λ , D αt ( u n − u λ )) dt + Z T V ( t )( | u λ | p − u λ , u n − u λ ) dt (cid:19) − λ Z T ( ∇ ¯ F ( t, u λ ) , u n − u λ ) dt (cid:21) = ( a + b k u n k pV ) p − (cid:18) k u n k pV − Z T ( | D αt u n | p − D αt u n , D αt u λ ) dt − Z T V ( t )( | u n | p − u n , u λ ) dt (cid:19) − ( a + b k u λ k pV ) p − (cid:18) − k u λ k pV + Z T ( | D αt u λ | p − D αt u λ , D αt u n ) dt + Z T V ( t )( | u λ | p − u λ , u n ) dt (cid:19) − λ Z T ( ∇ ¯ F ( t, u n ) − ∇ ¯ F ( t, u λ ) , u n − u λ ) dt ≥ ( a + b k u n k pV ) p − k u n k pV + ( a + b k u λ k pV ) p − k u λ k pV − ( a + b k u n k pV ) p − (cid:20) k D αt u n k p − L p k D αt u λ k L p + (cid:18) Z T V ( t ) | u n | p dt (cid:19) ( p − /p (cid:18) Z T V ( t ) | u λ | p dt (cid:19) /p (cid:21) − ( a + b k u λ k pV ) p − (cid:20) k D αt u λ k p − L p k D αt u n k L p + (cid:18) Z T V ( t ) | u λ | p dt (cid:19) ( p − /p (cid:18) Z T V ( t ) | u n | p dt (cid:19) /p (cid:21) − λ Z T ( ∇ ¯ F ( t, u n ) − ∇ ¯ F ( t, u λ ) , u n − u λ ) dt ≥ ( a + b k u n k pV ) p − k u n k pV + ( a + b k u λ k pV ) p − k u λ k pV − ( a + b k u n k pV ) p − (cid:18) k D αt u n k pL p + Z T V ( t ) | u n | p dt (cid:19) ( p − /p (cid:18) k D αt u λ k pL p + Z T V ( t ) | u λ | p dt (cid:19) /p ( a + b k u λ k pV ) p − (cid:18) k D αt u λ k pL p + Z T V ( t ) | u λ | p dt (cid:19) ( p − /p (cid:18) k D αt u n k pL p + Z T V ( t ) | u n | p dt (cid:19) /p − λ Z T ( ∇ ¯ F ( t, u n ) − ∇ ¯ F ( t, u λ ) , u n − u λ ) dt = ( a + b k u n k pV ) p − k u n k pV + ( a + b k u λ k pV ) p − k u λ k pV − ( a + b k u n k pV ) p − k u n k p − V k u λ k V − ( a + b k u λ k pV ) p − k u n k V k u λ k p − V − λ Z T ( ∇ ¯ F ( t, u n ) − ∇ ¯ F ( t, u λ ) , u n − u λ ) dt = ( a + b k u n k pV ) p − k u n k p − V ( k u n k V − k u λ k V )+( a + b k u λ k pV ) p − k u λ k p − V ( k u λ k V − k u n k V ) − λ Z T ( ∇ ¯ F ( t, u n ) − ∇ ¯ F ( t, u λ ) , u n − u λ ) dt = (cid:18) ( a + b k u n k pV ) p − k u n k p − V − ( a + b k u λ k pV ) p − k u λ k p − V (cid:19) ( k u n k V − k u λ k V ) − λ Z T ( ∇ ¯ F ( t, u n ) − ∇ ¯ F ( t, u λ ) , u n − u λ ) dt. (3.13)Note that λ Z T ( ∇ ¯ F ( t, u n ) − ∇ ¯ F ( t, u λ ) , u n − u λ ) dt ≤ λ Z T |∇ ¯ F ( t, u n ) − ∇ ¯ F ( t, u λ ) || u n − u λ | dt → , (3.14)by u n → u λ in C ([0 , T ] , R N ) and |∇ ¯ F ( t, u n ) − ∇ ¯ F ( t, u λ ) | is bounded in [0 , T ] because of (H0) ′ and the bound-edness of { u n } in E α,p , and (3.8) and (3.11) imply that D I ′ λ ( u n ) − I ′ λ ( u λ ) , u n − u λ E → , as n → ∞ . (3.15)So by (3.13), (3.14) and (3.15), we have k u n k V → k u λ k V , as n → ∞ . By the uniform convexity of E α,p and u n ⇀ u λ , it follows from the Kadec-Klee property (see [41]) and (2.14), u n → u λ in E α,p .By the continuity of ¯ I λ , we obtain that ¯ I λ ( u ) = c λ , where c λ is defined by (3.9). Then (3.3) implies that c λ ≥ d λ >
0. Hence u λ is a nontrivial critical point of I λ in E α,p for any given λ > Λ .Next, we show that u λ precisely is the nontrivial weak solution of system (1.1) for any given λ > λ ∗ . In orderto get this, we need to make an estimate for the critical level c λ . We introduce the functional e J λ : E α,p → R asfollows e J λ ( u ) = 1 bp ( a + b k u k pV ) p − λM Z T | u ( t ) | q dt − a p bp . Lemma 3.4.
For all λ ≥ max { Λ , Λ } , c λ ≤ C ∗ λ − p − q − p , C ∗ is defined by (1.13) which is obviously independent of λ . Proof.
Define f i : [0 , ∞ ) → R , i = 1 ,
2, by f ( s ) = 1 bp ( a + bs p k e k pV ) p − λ q k e k pV s p p − a p bp ,f ( s ) = − λM s q Z T | e | q dt + λ q k e k pV s p p , where e = δG e and e is defined in (3.4). Then f ( s ) + f ( s ) = e J λ ( se ). Let f ′ ( s ) = − λM q k e k q L q s q − + λ q k e k pV s p − = 0 . Thus for each given λ >
0, we have s = λ q k e k pV λM q k e k q L q ! q − p . Thenmax s ≥ f ( s ) = p ( M q ) pq − p − M ( M q ) q q − p ! (cid:18) k e k V k e k L q (cid:19) pq q − p λ − p − q − p . Obviously, f (0) = 0 and f ′ ( s ) = ( a + bs p k e k pV ) p − k e k pV s p − − λ q k e k pV s p − . So if λ > Λ := (cid:20) a + b [max { , V ∞ } ] p δ p G p ( D p + G p ) (cid:21) q ( p − = (cid:18) a + b [max { , V ∞ } ] p δ p G p k e k p (cid:19) q ( p − ≥ ( a + bs p k e k pV ) q ( p − ,f ( s ) is decreasing on s ∈ [0 ,
1] and then f ( s ) < s ∈ [0 , k se k ∞ ≤ k δG e k ∞ ≤ δ (3.16)for all s ∈ [0 , λ > Λ , by (H1) ′ , (3.5), (3.6) and H¨older inequality, we have c λ ≤ max s ∈ [0 , ¯ I λ ( se ) ≤ max s ∈ [0 , e J λ ( se ) ≤ max s ∈ [0 , f ( s ) + max s ≥ f ( s ) ≤ max s ≥ f ( s ) = p ( M q ) pq − p − M ( M q ) q q − p ! (cid:18) k e k V k e k L q (cid:19) pq q − p λ − p − q − p ≤ p ( M q ) pq − p − M ( M q ) q q − p ! [max { , V ∞ } ] /p ( D p + G p ) /p T q − p k e k L p ! pq q − p λ − p − q − p = C ∗ λ − p − q − p . roof of Theorem 1.1. Note that u λ is a critical point of ¯ I λ with critical value c λ . Since h ¯ I ′ ( u λ ) , u λ i = 0,similar to the argument in (3.10) and by Lemma 3.4, we have k u λ k pV ≤ p θa p − ( θ − p ) ¯ I λ ( u λ )= p θa p − ( θ − p ) c λ ≤ p θa p − ( θ − p ) C ∗ λ − p − q − p . (3.17)Since λ > Λ = T pα − h Γ( α )( αq − q + 1) q i p · p θC ∗ a p − ( θ − p ) · p δ p q − pp − , by (3.17), we have k u λ k ∞ ≤ T α − p Γ( α )( αq − q + 1) q k u λ k V ≤ δ/ . (3.18)So for all λ > Λ , | u λ ( t ) | ≤ k u λ k ∞ ≤ δ/ t ∈ [0 , T ] and then ¯ F ( t, u ( t )) = F ( t, u ( t )) for a.e. t ∈ [0 , T ].Furthermore, ¯ I λ ( u λ ) = I λ ( u λ ) = c λ > h ¯ I ′ ( u λ ) , v i = h I ′ ( u λ ) , v i = 0 for all v ∈ E α,p . Thus u λ is preciselythe nontrivial weak solution of system (1.1) when λ > λ ∗ := max { Λ , Λ , Λ } . Note that p > q > p . By(3.17) and (3.18), it is obvious that lim λ →∞ k u λ k V = 0 = lim λ →∞ k u λ k ∞ .
4. Example
Assume that N = 2 , a = b = T = 1, p = 3 and δ = 1. Then q = . Let q = 12 , q = 10 , F ( t, x ) = ( t + 1) | x | for a.e. t ∈ [0 ,
1] and all x ∈ R N with | x | ≤ V ( t ) = 7 t + 1 for all t ∈ [0 , V ∞ = 8 and V ∞ = 1.Choose α = . Consider the system A ( u ( t ))[ t D / φ ( D / t u ( t )) + (7 t + 1) φ ( u ( t ))] = 11 λ ( t + 1) | u | u, a.e. t ∈ [0 , ,u (0) = u (1) = 0 , (4.1)where A ( u ( t )) = (cid:20) Z ( | D / t u ( t ) | + (7 t + 1) | u ( t ) | ) dt (cid:21) . By Theorem 1.1, we can obtain that system (4.1) has at least a nontrivial solution u λ in E , for each λ > . and lim λ →∞ k u λ k V = 0 = lim λ →∞ k u λ k ∞ . 15n fact, we can verify that F ( t, u ) satisfies (H0)-(H2) as follows.i) Note that | F ( t, x ) | = ( t + 1) | x | , |∇ F ( t, x ) | = 11( t + 1) | x | for all | x | ≤ δ . Set a ( | x | ) = | x | + 1 for all x ∈ R N with | x | ≤ b ( t ) = 11( t + 1) for a.e t ∈ [0 , T ]. Thenassumption (H0) is satisfied.ii) Note that q = 12 > q = 10 > p = 9, and | x | ≤ F ( t, x ) = ( t + 1) | x | ≤ | x | , for all | x | ≤ δ and a.e. t ∈ [0 , M = 1 and M = 2. Then assumption (H1) is also satisfied.iii) Let β = 10 > p = 9. Then0 ≤ t + 1) | x | = βF ( t, x ) ≤ t + 1) | x | = ( ∇ F ( t, x ) , x )holds for all x ∈ R and a.e. t ∈ [0 , λ ∗ by the formulas in Theorem 1.1. Note that Γ( ) = √ π, Γ(2 − ) = √ π . We obtain D = (cid:18) T p +1 π p +1 p − p !! (cid:19) p = (cid:18) (cid:19) π − ,G = (cid:18) T p +1 − pα Γ p (2 − α )( p + 1 − pα ) (cid:19) /p = (cid:18) (cid:19) π − ,G = T α − p Γ( α )( αq − q + 1) q G = 5 − · π − . Then by θ = min { β, q } = 10, (1.10), (1.11) and (1.12), we haveΛ = max ( r π / , / · π
625 (1 + 524 π + π / ) , Λ = (cid:18) π − + 32 π / (cid:19) , Λ = (cid:18) · C ∗ π / (cid:19) , and by (1.13), C ∗ = 16 (cid:18) · √ − √ (cid:19)(cid:18) π / (cid:19) / . , Λ and Λ , it is easy to see λ ∗ = Λ ≈ . . Acknowledgement
This project is supported by Yunnan Ten Thousand Talents Plan Young & Elite Talents Project, CandidateTalents Training Fund of Yunnan Province (No: 2016PY027) and National Natural Science Foundation of China(11301235).
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