Existence and a priori estimates of solutions for quasilinear singular elliptic systems with variable exponents
aa r X i v : . [ m a t h . A P ] J u l EXISTENCE AND A PRIORI ESTIMATES OF SOLUTIONS FORQUASILINEAR SINGULAR ELLIPTIC SYSTEMS WITHVARIABLE EXPONENTS
ABDELKRIM MOUSSAOUI AND JEAN V´ELIN
Abstract.
This article sets forth results on the existence, a priori estimatesand boundedness of positive solutions of a singular quasilinear systems ofelliptic equations involving variable exponents. The approach is based onSchauder’s fixed point Theorem. A Moser iteration procedure is also obtainedfor singular cooperative systems involving variable exponents establishing apriori estimates and boundedness of solutions. Introduction
In the present paper we focus on the system of quasilinear elliptic equations( P ) − ∆ p ( x ) u = f ( u, v ) in Ω − ∆ q ( x ) v = g ( u, v ) in Ω u, v > u, v = 0 on ∂ Ω , on a bounded domain Ω in R N ( N ≥
2) with Lipschitz boundary ∂ Ω, which exhibitsa singularity at zero. Here ∆ p ( x ) (resp. ∆ q ( x ) ) stands for the p ( x )-Laplacian (resp. q ( x )-Laplacian) differential operator on W ,p ( x )0 (Ω) (resp. W ,q ( x )0 (Ω)) with p, q :Ω → [1 , ∞ ) , (1.1) 1 < p − ≤ p + < N and 1 < q − ≤ q + < N, which satisfy the log-H¨older continuous condition, i.e., there is constants C , C > | p ( x ) − p ( y ) | ≤ C − ln | x − y | and | q ( x ) − q ( y ) | ≤ C − ln | x − y | , for every x i , y i ∈ Ω with | x i − y i | < / i = 1 , p ∗ and q ∗ the Sobolev critical exponents p ∗ ( x ) = Np ( x ) N − p ( x ) and q ∗ ( x ) = Nq ( x ) N − q ( x ) and we set s − = inf x ∈ Ω s ( x ) and s + = sup x ∈ Ω s ( x ) . A solution ( u, v ) ∈ W ,p ( x )0 (Ω) × W ,q ( x )0 (Ω) of problem ( P ) is understood in theweak sense, that is, it satisfies(1.3) (cid:26) R Ω |∇ u | p ( x ) − ∇ u ∇ ϕ dx = R Ω f ( u, v ) ϕ dx R Ω |∇ v | q ( x ) − ∇ v ∇ ψ dx = R Ω g ( u, v ) ψ dx , Mathematics Subject Classification.
Key words and phrases. p ( x )-Laplacian, variable exponents, fixed point, singular system, reg-ularity, boundedness. for all ( ϕ, ψ ) ∈ W ,p ( x )0 (Ω) × W ,q ( x )0 (Ω).Nonlinear boundary value problems involving p ( x )-Laplacian operator are math-ematically challenging and important for applications. Their study is stimulatedby their applications in physical phenomena related to electrorheological fluids andimage restorations, see for instance [1, 2, 6, 21]. When p ( x ) ≡ p and q ( x ) ≡ q areconstant functions, ∆ p ( x ) and ∆ q ( x ) coincide with the well-known p -Laplacian and q -Laplacian operators. However, it is worth pointing out that p ( x )-Laplacian operatorpossesses more complicated nonlinearity than p -Laplacian since it is inhomogeneousand in general, it has no first eigenvalue, that is, the infimum of the eigenvalues of p ( x )-Laplacian equals 0 (see, e.g., [14, 20]). This point constitute a serious technicaldifficulty in the study of problem ( P ), for which topological methods are difficultto apply. Another serious difficulty encountered in studying system ( P ) is that thenonlinearities f ( u, v ) and g ( u, v ) can exhibit singularities when the variables u and v approach zero. Specifically, we assume that f, g : (0 , + ∞ ) × (0 , + ∞ ) → (0 , + ∞ ) , are continuous functions satisfying the conditions: (H. f ): f ( s , s ) ≤ m (1 + s α ( x )1 )(1 + s β ( x )2 ) for all s , s > , with a constant m > α , β : Ω −→ R ∗ . (H. g ): g ( s , s ) ≤ m (1 + s α ( x )1 )(1 + s β ( x )2 ) for all s , s > , with a constant m > α , β : Ω −→ R ∗ .We explicitly observe that under assumptions (H. f ) and (H. g ) and dependingon the sign of the variable exponents α i ( · ) and β i ( · ) , i = 1 ,
2, system ( P ) presentstwo types of complementary structures:(1.4) α − , β − > α +2 , β +1 < H( f, g ) : σ := min { inf s ,s > f ( s , s ) , inf s ,s > g ( s , s ) } > . This assumption is useful in the subsequent estimates keeping the values of f ( s , s ) and g ( s , s ) above zero. In the case of competitive system ( P ), in additionof (1.5), we assume H( f, g ) : For all constant
M > s → f ( s ,s ) s p −− = + ∞ for all s ∈ (0 , M )and lim s → g ( s ,s ) s q −− = + ∞ for all s ∈ (0 , M ) . UASILINEAR SINGULAR ELLIPTIC SYSTEMS WITH VARIABLE EXPONENTS 3
This type of problem is rare in the literature. Actually, according to our knowl-edge, the only class of singular problems incorporated in statement ( P ) patterns thesystem for f ( u, v ) = u α ( x ) v β ( x ) and g ( u, v ) = u α ( x ) v β ( x ) was studied recently byAlves & Moussaoui [3]. The authors obtained the existence of solutions throughnew theorems involving sub and supersolutions for singular systems with variableexponents by dealing with cooperative and competitive structures. However, whenthe exponent variable functions p ( · ) , q ( · ) , α i ( · ) and β i ( · ), i = 1 ,
2, are reduced tobe constants, problem ( P ) have been thoroughly investigated, we refer to [19] forsystem ( P ) with cooperative structure, while we quote [17, 18] for the study ofcompetitive structure in ( P ). Furthermore, in the constant exponent context, thesingular problem ( P ) arise in several physical situations such as fluid mechanics,pseudoplastics flow, chemical heterogeneous catalysts, non- Newtonian fluids, bio-logical pattern formation, for more details about this subject, we cite the papers ofFulks & Maybe [12], Callegari & Nashman [7, 8] and the references therein.Our goal is to establish the existence and regularity of (positive) solutions forproblem ( P ) by processing the cases (1.4) and (1.5) related to the structure of ( P ).Our main results are stated as follows. Theorem 1.1.
Let assumptions (H. f ), (H. g ), H( f, g ) and (1.4) hold with (1.6) β ( x ) ≤ q ∗ ( x ) p ∗ ( x ) ( p ∗ ( x ) − , α ( x ) ≤ p ∗ ( x ) q ∗ ( x ) ( q ∗ ( x ) − and (1.7) (cid:26) − N < α − ≤ α +1 < − N < β − ≤ β +2 < . Then, problem ( P ) possesses at least one (positive) solution in C (Ω) × C (Ω) satisfying (1.8) u ( x ) , v ( x ) ≥ c d ( x ) , where d ( x ) := d ( x, ∂ Ω) and c is a positive constant. Theorem 1.2.
Under assumptions (H. f ), (H. g ), H( f, g ) and (1.5) with (1.9) max {− N , − α − } < β − ≤ β +1 < < α − ≤ α +1 < p − − and (1.10) max {− N , − β − } < α − ≤ α +2 < < β − ≤ β +2 < q − − , problem ( P ) possesses at least one (positive) solution ( u, v ) in C (Ω) × C (Ω) sat-isfying (1.8). The main technical difficulty consists in the presence of p ( x )-Laplacian and q ( x )-Laplacian operators in the principle parts of equations in ( P ) on the one hand and,on the other the presence of singular terms through variable exponents that canoccur under hypotheses (H. f ) and (H. g ). Under cooperative structure (1.4), byadapting Moser iterations procedure to problem ( P ), together with an adequatetruncation, we prove a priori estimates for an arbitrary solution of ( P ). In par-ticular, it provides that all solution ( u, v ) of ( P ) are bounded in L ∞ (Ω) × L ∞ (Ω).Taking advantage of this boundedness and applying Schauder’s fixed point Theoremwe obtain the existence of a solution of problem ( P ). To the best of our knowledge, ABDELKRIM MOUSSAOUI AND JEAN V´ELIN it is for the first time when Moser iterations method is applied for problems withvariable exponents.For system ( P ) subjected to competitive structure (1.5), we develop some com-parison arguments which provide a priori estimates on solutions of ( P ). In turn,these estimates enable us to obtain our main result by applying the Schauder’sfixed point theorem. It is worth noting that besides our method is different fromthat used by Alves & Moussaoui [3], our assumptions, precisely H( f, g ) , (1.9) and(1.10), are not satisfied by hypotheses considered there.We indicate simple examples showing the applicability of Theorems 1.1 and 1.2.Related to system ( P ) under assumptions above, we can handle singular cooperativesystems of the form − ∆ p ( x ) u = ( u α ( x ) + 1)( v β ( x ) + 1) in Ω − ∆ q ( x ) v = ( u α ( x ) + 1)( v β ( x ) + 1) in Ω u, v > u, v = 0 on ∂ Ω , and singular competitive systems of type − ∆ p ( x ) u = v α ( x ) + v β ( x ) in Ω − ∆ q ( x ) v = u α ( x ) + u β ( x ) in Ω u, v > u, v = 0 on ∂ Ω , with variable exponents α , α , β , β as in hypotheses (1.6), (1.7) and (1.9), (1.10),respectively.The rest of this article is organized as follows. Section 2 deals with a prioriestimates and regularity of solutions of cooperative system ( P ), whereas Section 3presents comparison properties of competitive system ( P ). Sections 4 and 5 containthe proof of Theorems 1.1 and 1.2.2. A priori estimates and regularity
Let L p ( x ) (Ω) be the generalized Lebesgue space that consists of all measurablereal-valued functions u satisfying ρ p ( x ) ( u ) = R Ω | u ( x ) | p ( x ) dx < + ∞ , endowed with the Luxemburg norm k u k p ( x ) = inf { τ > ρ p ( x ) ( uτ ) ≤ } . The variable exponent Sobolev space W ,p ( · )0 (Ω) is defined by W ,p ( x )0 (Ω) = { u ∈ L p ( x ) (Ω) : |∇ u | ∈ L p ( x ) (Ω) } . The norm k u k ,p ( x ) = k∇ u k p ( x ) makes W ,p ( x )0 (Ω) a Banach space. On the basis of(1.2), the following embedding(2.1) W ,p ( x )0 (Ω) ֒ → L r ( x ) (Ω)is continuous with 1 < r ( x ) ≤ p ∗ ( x ) (see [9, Corollary 5.3]). UASILINEAR SINGULAR ELLIPTIC SYSTEMS WITH VARIABLE EXPONENTS 5
Lemma 2.1. ( i ) For any u ∈ L p ( x ) (Ω) we have k u k p − p ( x ) ≤ ρ p ( x ) ( u ) ≤ k u k p + p ( x ) if k u k p ( x ) > , k u k p + p ( x ) ≤ ρ p ( x ) ( u ) ≤ k u k p − p ( x ) if k u k p ( x ) ≤ . ( ii ) For u ∈ L p ( x ) (Ω) \{ } we have (2.2) k u k p ( x ) = a if and only if ρ p ( x ) ( ua ) = 1 . The next result provides a priori estimates for an arbitrary solution of ( P ) sub-jected to cooperative structure. Theorem 2.2.
Assume that (1.4) and the growth conditions (H. f ) and (H. g ) holdwith (2.3) ( α +1 < < β ( x ) ≤ q ∗ ( x ) p ∗ ( x ) ( p ∗ ( x ) − β +2 < < α ( x ) ≤ p ∗ ( x ) q ∗ ( x ) ( q ∗ ( x ) − in Ω .Then there exist positive constants C = C ( m , β , N, Ω , p, q ) and C ′ = C ′ ( m , α , N, Ω , p, q ) such that every solution ( u, v ) ∈ W ,p ( x )0 (Ω) × W ,q ( x )0 (Ω) of ( P ) satisfies the esti-mate (2.4) k u k ∞ ≤ C max(1 , k u k p ∗ ( x ) ) p + /p − (1 + max(1 , k v k β +1 q ∗ ( x ) )) p − ) ∗− p − , (2.5) k v k ∞ ≤ C ′ max(1 , k v k q ∗ ( x ) ) q + /q − (1 + max(1 , k u k α +2 p ∗ ( x ) )) q − ) ∗− q − . In particular, problem ( P ) has only bounded solutions.Proof. Let φ : R −→ [0 ,
1] be a C cut-off function such that φ ( s ) = (cid:26) s ≤ , s ≥ φ ′ ( s ) ≥ , . Given δ > , we define φ δ ( t ) = φ ( t − δ ) for all t ∈ R . It follows that(2.6) φ δ ◦ z ∈ W ,p ( x )0 (Ω) and ∇ ( φ δ ◦ z ) = ( φ ′ δ ◦ z ) ∇ z, for z ∈ W ,p ( x )0 (Ω).Let ( u, v ) ∈ W ,p ( x )0 (Ω) × W ,q ( x )0 (Ω) be a weak solution of ( P ). Acting in thefirst equation in (1.3) with the test function ϕ = ( φ δ ◦ u ) ϕ with ϕ ∈ W ,p ( x )0 (Ω)and ϕ ≥ R Ω |∇ u | p ( x ) − ∇ u ∇ (( φ δ ◦ u ) ϕ ) dx = R Ω f ( u, v )( φ δ ◦ u ) ϕ dx. Hence, by (3.18), we get R Ω |∇ u | p ( x ) ( φ ′ δ ◦ u ) ϕ dx + R Ω |∇ u | p ( x ) − ∇ u ∇ ϕ ( φ δ ◦ u ) dx = R Ω f ( u, v )( φ δ ◦ u ) ϕ dx. Since φ ′ δ ◦ u ≥ , it follows that R Ω |∇ u | p ( x ) − ∇ u ∇ ϕ ( φ δ ◦ u ) dx ≤ R Ω f ( u, v )( φ δ ◦ u ) ϕ dx. Letting δ → R { u> } |∇ u | p ( x ) − ∇ u ∇ ϕ dx ≤ R { u> } f ( u, v ) ϕ dx, for all ϕ ∈ W ,p ( x )0 (Ω) with ϕ ≥ P ), we get(2.8) R { v> } |∇ v | q ( x ) − ∇ v ∇ ψ dx ≤ R { v> } g ( u, v ) ϕ dx, ABDELKRIM MOUSSAOUI AND JEAN V´ELIN for all ψ ∈ W ,q ( x )0 (Ω) with ψ ≥ M >
0, define u M ( x ) = min { u ( x ) , M } , v M ( x ) = min { v ( x ) , M } . Observe that h ( s ) = s k − p + +1 is a C function, h (0) = 0 and there is a constant L > | h ′ ( s ) | ≤ L for all 0 ≤ s ≤ M . By proceeding analogously to the proofof [5, Proposition XI.5, page 155], it follows that u k − p + +1 M ∈ W ,p ( x )0 (Ω) ∩ L ∞ (Ω).Similarly we get v ¯ k − q + +1 M ∈ W ,q ( x )0 (Ω) ∩ L ∞ (Ω).Inserting ( ϕ, ψ ) = ( u k − p + +1 M , v ¯ k − q + +1 M ) in (2.7) and (2.8), where(2.9) (cid:26) ( k ( x ) + 1) p ( x ) = p ∗ ( x ) (cid:0) ¯ k ( x ) + 1 (cid:1) q ( x ) = q ∗ ( x ) , one has(2.10) R { u> } |∇ u | p ( x ) − ∇ u ∇ ( u k − p + +1 M ) dx ≤ R { u> } f ( u, v ) u k − p + +1 M dx and(2.11) R { v> } |∇ v | q ( x ) − ∇ v ∇ ( v ¯ k − q + +1 M ) dx ≤ R { v> } g ( u, v ) v ¯ k − q + +1 M dx, Step 1. Estimation of the left-hand side in (2.10) and (2.11)
In what follows denote by ( s − + := max { s, } for s ≥ |∇ u M | p ( x ) u M k − p ( x ) = k − +1) p ( x ) |∇ ( u M ) k − +1 ) | p ( x ) ≥ k − +1) p + |∇ ( u M ) k − +1 ) | p ( x ) . Then(2.13) R { u> } |∇ u | p ( x ) − ∇ u ∇ ( u k − p + +1 M ) dx = R { u> } |∇ u | p ( x ) − ∇ u ∇ ( u M ) k − p + +1 dx = ( k − p + + 1) R { u M > } |∇ u M | p ( x ) u k − p + M dx ≥ ( k − p + + 1) R { u M > } |∇ u M | p ( x ) u k − p ( x ) M dx ≥ k − p + +1( k − +1) p + R { u M > } |∇ ( u k − +1 M ) | p ( x ) dx. On the other hand, using (2.2) and through the mean value theorem, there exists x ∈ Ω such that(2.14) 1 = R Ω (cid:12)(cid:12)(cid:12)(cid:12) ( u M − + k ( u M − + k ( k − p ∗ ( x ) (cid:12)(cid:12)(cid:12)(cid:12) ( k − +1) p ∗ ( x ) dx = R Ω (cid:12)(cid:12)(cid:12)(cid:12) (( u M − + ) k − k (( u M − + ) k − k p ∗ ( x ) (cid:12)(cid:12)(cid:12)(cid:12) p ∗ ( x ) × k (( u M − + ) k − k p ∗ ( x ) k ( u M − + k k − k − p ∗ ( x ) p ∗ ( x ) dx = k (( u M − + ) k − k p ∗ ( x ) k ( u M − + k k − k − p ∗ ( x ) p ∗ ( x ) , which implies(2.15) k (( u M − + ) k − +1 k p ∗ ( x ) = k ( u M − + k k − +1( k − +1) p ∗ ( x ) . UASILINEAR SINGULAR ELLIPTIC SYSTEMS WITH VARIABLE EXPONENTS 7
Furthermore, from (2.2) one has R Ω | ∇ (( u M − + ) k − k (( u M − + ) k − k ,p ( x ) | p ( x ) dx = 1 . Using the mean value theorem, there exists x M ∈ Ω such that(2.16) R Ω |∇ (( u M − + ) k − +1 | p ( x ) dx = k (( u M − + ) k − +1 k p ( x M )1 ,p ( x ) . Then, (3.4), (2.15), (3.20) and through the Sobolev embedding (2.1), one gets(2.17) k − p + +1( k − +1) p + R { u M > } |∇ ( u k − +1 M ) | p ( x ) dx = k − p + +1( k − +1) p + R Ω |∇ (( u M − + ) k − +1 | p ( x ) dx = k − p + +1( k − +1) p + k (( u M − + ) k − +1 k p ( x M )1 ,p ( x ) ≥ ˆ C k − p + +1( k − +1) p + k (( u M − + ) k − +1 k p ( x M ) p ∗ ( x ) = ˆ C k − p + +1( k − +1) p + k ( u M − + k ( k − +1) p ( x M )( k − +1) p ∗ ( x ) ≥ C k − p + +1( k − +1) p + k ( u M − + k ( k − +1) p ± ( k − +1) p ∗ ( x ) , where C = C ( p, N, Ω) is a positive constant and(2.18) p ± = ( p + if k ( u M − + k ( k − +1) p ∗ ( x ) > p − if k ( u M − + k ( k − +1) p ∗ ( x ) ≤ . Similarly, following the same argument as above leads to(2.19) ¯ k − q + +1(¯ k − +1) q + R { v M > } |∇ ( v ¯ k − +1 M ) | q ( x ) dx ≥ C k − q + +1(¯ k − +1) q + k ( v M − + k (¯ k − +1) q ± (¯ k − +1) q ∗ ( x ) , with positive constants C = C ( q, N, Ω) and(2.20) q ± = ( q + if k ( v M − + k (¯ k − +1) q ∗ ( x ) > q − if k ( v M − + k (¯ k − +1) q ∗ ( x ) ≤ . Step 2. Estimation of the right-hand side in (2.10) and (2.11).
Using (2.7), (H. f ), (2.9), (2.3), (2.1) together with H¨older’s inequality and [4,Proposition 2.3], we get(2.21) R { u> } f ( u, v ) u k − p + +1 M dx ≤ R { u> } f ( u, v ) u k − p + +1 dx ≤ m R { u> } (1 + v β ( x ) ) u k − p + +1 dx = 2 m R Ω (( u − + ) k − p + +1 dx + 2 m R Ω v β ( x ) (( u − + ) k − p + +1 dx ≤ ˆ C (cid:18) k ( u − + k k − p + +1 p ∗ ( x ) + k ( u − + k k − p + +1 p ∗ ( x ) (cid:13)(cid:13) v β ( x ) (cid:13)(cid:13) p ∗ ( x ) p ∗ ( x ) − (cid:19) ≤ ˆ C ′ (cid:18) k ( u − + k k − p + +1 p ∗ ( x ) + k ( u − + k k − p + +1 p ∗ ( x ) k v k β ± β x ) p ∗ ( x ) p ∗ ( x ) − (cid:19) ≤ C k ( u − + k k − p + +1 p ∗ ( x ) (1 + k v k β ± q ∗ ( x ) ) , with a positive constant C = C ( m , β , N, Ω , p, q ) and(2.22) β ± = ( β +1 if k v k q ∗ ( x ) > β − if k v k q ∗ ( x ) ≤ . Similarly, by (2.8), (H. g ), (2.9), (2.3), (2.1), combined with H¨older’s inequality and[4, Proposition 2.3], one has(2.23) R { v> } g ( u, v ) v ¯ k − q + +1 M dx ≤ C ′ (1 + k u k α i p ∗ ( x ) ) k v k ¯ k − q + +1 q ∗ ( x ) , ABDELKRIM MOUSSAOUI AND JEAN V´ELIN where the positive constant C ′ = C ′ ( m , α , N, Ω , p, q ) and(2.24) α i = ( α +2 if k u k p ∗ ( x ) > α − if k u k p ∗ ( x ) ≤ . Step 3. Moser iteration procedure and passage to the limit.
We note that if k ( u − + k p ∗ ( x ) , k ( v − + k q ∗ ( x ) >
1, then there hold(2.25) k ( u − + k k − p + +1 p ∗ ( x ) ≤ k ( u − + k ( k − +1) p + p ∗ ( x ) and k ( v − + k ¯ k − q + +1 q ∗ ( x ) ≤ k ( v − + k (¯ k − +1) q + q ∗ ( x ) because p + , q + >
1. Then, it follows from (2.17) - (2.25) that(2.26) k ( u M − + k ( k − +1) p ∗ ( x ) ≤ C k − k − +1( k − p + +1) p + ! p +( k − p ± k ( u − + k p + /p − p ∗ ( x ) (cid:16) k v k β ± q ∗ ( x ) (cid:17) p − ) ∗ and(2.27) k ( v M − + k (¯ k − +1) q ∗ ( x ) ≤ C k − ¯ k − +1(¯ k − q + +1) q + ! q +(¯ k − q ± k ( v − + k q + /q − q ∗ ( x ) (cid:16) k u k α ± p ∗ ( x ) (cid:17) q − ) ∗ with a constant C = C ( m , α , β , N, Ω , p, q ) > { k n } n ≥ and { k n } n ≥ by defining(2.28) k n ( x ) + 1 = ( k n − ( x ) + 1) p ∗ ( x ) p ( x ) = (cid:16) p ∗ ( x ) p ( x ) (cid:17) n ,k n ( x ) + 1 = ( k n − ( x ) + 1) q ∗ ( x ) q ( x ) = (cid:16) q ∗ ( x ) q ( x ) (cid:17) n , for all n ≥ n that k ( u − + k ( k − n +1) p ∗ ( x ) ≤ k ( v − + k ( k − n +1) q ∗ ( x ) ≤ , then letting n → ∞ we get k ( u k ∞ ≤ k v k ∞ ≤
1, and we are done. If not, itsuffices to consider the case k ( u − + k ( k − n +1) p ( x ) > k ( v − + k ( k − n +1) q ( x ) > n because otherwise the proof reduces to special case of Moser iterationprocedure for an elliptic equation. In this case, we argue as for obtaining (2.26) and(2.27). Namely, proceeding by induction through (2.28) and then letting M → ∞ we arrive at(2.29) k ( u − + k ( k − n +1) p ∗ ( x ) ≤ C k − n +1 (cid:18) k − n +1( k − n p + +1) p + (cid:19) p +( k − n +1) p ± k ( u − + k p + /p − ( k − n − +1) p ∗ ( x ) (1 + k v k β ± q ∗ ( x ) ) k − n − p − ) ∗ UASILINEAR SINGULAR ELLIPTIC SYSTEMS WITH VARIABLE EXPONENTS 9 and(2.30) k ( v − + k ( k − n +1) q ∗ ( x ) ≤ C k − n +1 k − n +1( k − n q + +1) q + ! q +( k − n +1) q ± k ( v − + k q + /q − ( k − n +1) q ∗ ( x ) (1 + k u k α ± p ∗ ( x ) ) k − n − q − ) ∗ , with positive constants C = C ( N, Ω , m , p, β ) and C = C ( N, Ω , m , q, α ). Itturns out from (2.29) that k ( u − + k ( k − n +1) p ∗ ( x ) ≤ C n P i =1 1 k − i +1 n Q i =1 k − i +1 ( k − i p + +1 ) p + ! √ k − i +1 √ k − i +1 p + /p ± k ( u − + k p + /p − p ∗ ( x ) (1 + k v k β ± q ∗ ( x ) ) p − ) ∗ n − P i =1 1 k − i +1 ! . Furthermore, since lim z →∞ (cid:18) z +1( zp + +1) p + (cid:19) √ z +1 = 1, there is a positive constant C for which one has(2.31) k ( u − + k ( k n +1) p ∗ ≤ C n P i =1 1 k − i +1 C p + p ± n P i =1 1 √ k − i +1 k ( u − + k p + /p − p ∗ ( x ) (1 + k v k β ± q ∗ ( x ) ) p ∗ ) − n − P i =1 1 k − i +1 ! . Similarly, we obtain(2.32) k ( v − + k ( k − n +1) q ∗ ( x ) ≤ C n P i =1 1 k − i +1 C q + q ± n P i =1 1 √ k − i +1 k ( v − + k q + /q − q ∗ ( x ) (1 + k u k α ± p ∗ ( x ) ) q − ) ∗ n − P i =1 1¯ k − i +1 ! . Moreover, (2.28) guarantees the convergence of the series in (2.31) and (2.32), forinstance 1 + n − P i =1 1 k − i +1 = n − P i =0 (cid:16) p − ( p − ) ∗ (cid:17) i −→ ( p − ) ∗ ( p − ) ∗ − p − . Letting n → ∞ in (2.31) and (2.32) we derive the estimates (4.7) and (2.5). Thiscompletes the proof. (cid:3) Next result is consequence of Theorem 2.2.
Proposition 2.1.
Under the assumptions of Theorem 1.1, every solutions ( u, v ) of ( P ) is bounded in C ,γ (Ω) × C ,γ (Ω) and there is a constant R > such that k u k C ,γ (Ω) , k v k C ,γ (Ω) < R. Moreover, it holds (2.33) u ( x ) , v ( x ) ≥ c d ( x ) , with some constant c > .Proof. We first show (2.33). Recalling the constant σ > f, g ) , let z and z the only positive solutions of(2.34) (cid:26) − ∆ p ( x ) z = σ in Ω z = 0 on ∂ Ω and (cid:26) − ∆ q ( x ) z = σ in Ω z = 0 on ∂ Ω , which are known to satisfy(2.35) z ( x ) ≥ c d ( x ) and z ( x ) ≥ c ′ d ( x ) in Ω , for certain positive constants c and c ′ (see, e.g., [3]). Then, from ( P ), (2.34) andH( f, g ) , it follows that (cid:26) − ∆ p ( x ) u ≥ − ∆ p ( x ) z in Ω u = z on ∂ Ω and (cid:26) − ∆ q ( x ) v ≥ − ∆ q ( x ) z in Ω v = z on ∂ Ω . Therefore, the weak comparison principle leads to (2.33).By virtue of (H. f ), (H. g ), (2.33), (1.4), (1.5) and (1.7), on account of Theorem2.2, one has(2.36) f ( u, v ) ≤ C d ( x ) α − and f ( u, v ) ≤ C ′ d ( x ) β − in Ω , for some positive constants C and C ′ . Then, the C ,α -boundedness of u and v follows from [3, Lemma 2]. The proof is completed. (cid:3) Comparison properties
In this section, we assume that (1.9) and (1.10) hold. For a fixed δ > u and v in C ,γ (Ω) , for certain γ ∈ (0 , − ∆ p ( x ) u = λ (cid:26) \ Ω δ u − α ( x ) in Ω δ , u > , u = 0 on ∂ Ω(3.2) − ∆ q ( x ) v = λ (cid:26) \ Ω δ v − β ( x ) in Ω δ , v > , v = 0 on ∂ Ω . where λ > δ = { x ∈ Ω : d ( x, ∂ Ω) < δ } . Combining the results in [3, Lemmas 1 and 3] and [11], it is readily seen that for λ > u and v verify(3.3) min { δ, d ( x ) } ≤ u ( x ) ≤ c λ p −− in Ω , and(3.4) min { δ, d ( x ) } ≤ v ( x ) ≤ c λ q −− in Ω , for some positive constant c , c independent of λ and for δ > c , c ′ > u ( x ) ≤ c d ( x ) θ and v ( x ) ≤ c ′ d ( x ) θ in Ω δ , for some constants θ , θ ∈ (0 , , which assumed to satisfy the estimates(3.6) θ ≥ − β − α − and θ ≥ − α − β − . Notice that θ and θ exist since − β − < α − and − α − < β − (see (H. f ) and (H. g )).Now, let consider the functions u and v defined by(3.7) − ∆ p ( x ) u = λ − (cid:26) \ Ω δ − δ , u = 0 on ∂ Ω UASILINEAR SINGULAR ELLIPTIC SYSTEMS WITH VARIABLE EXPONENTS 11 and(3.8) − ∆ q ( x ) v = λ − (cid:26) \ Ω δ − δ , v = 0 on ∂ Ω . where Ω δ is given by(3.9) Ω δ = { x ∈ Ω : d ( x, ∂ Ω) < δ } , with a fixed δ > c d ( x ) ≤ u ( x ) ≤ c λ − p + − and c ′ d ( x ) ≤ v ( x ) ≤ c ′ λ − q + − in Ω , where c , c , c ′ and c ′ are positive constants. Obviously, from (3.1), (3.2), (3.7)and (3.8), we have ( u, v ) ≤ ( u, v ) in Ω for λ > Proposition 3.1.
Assume that (H. f ), (H. g ) and H( f, g ) hold. Then, for λ > large enough, we have (3.11) − ∆ p ( x ) u ≤ f ( u, v ) , − ∆ q ( x ) v ≤ g ( u, v ) in Ω , (3.12) − ∆ p ( x ) u ≥ f ( u, v ) , − ∆ q ( x ) v ≥ g ( u, v ) in Ω . Proof.
For all λ > − λ − u − ( p − − ≤ < − λ − v − ( q − − ≤ < δ . By (3.10), it follows that(3.14) λ − u − ( p − − ≤ λ − ( c d ( x )) − ( p − − ≤ λ − ( c δ ) − ( p − − ≤ \ Ω δ , and(3.15) λ − v − ( q − − ≤ λ − ( c ′ d ( x )) − ( q − − ≤ λ − ( c ′ δ ) − ( q − − ≤ \ Ω δ , provided that λ is sufficiently large. Another hand, by H( f, g ) there exist constants ρ, ¯ ρ > f ( s , s ) ≥ s p − − , for all 0 < s < ρ, for all 0 < s < λ p −− , and(3.17) g ( s , s ) ≥ s q − − , for all 0 < s ≤ λ q −− , for all 0 < s < ¯ ρ. Then, for λ > { c λ − p −− , c ′ λ − q −− } < min { ρ, ¯ ρ } , combining (3.13) - (3.17) together, we infer that (3.11) holds true.Next, we show (3.12). By (H. f ), (H. g ), (3.10), (2.13) and (3.4), it follows that(3.18) f ( u, v ) ≤ M (1 + u α ( x ) )(1 + v β ( x ) ) ≤ M (1 + c α ( x )4 λ α +1 p −− )(1 + ( c ′ d ( x )) β ( x ) ) ≤ λ in Ω \ Ω δ and(3.19) g ( u, v ) ≤ M (1 + u α ( x ) )(1 + v β ( x ) ) ≤ M (cid:0) c d ( x )) α ( x ) (cid:1) (1 + ( c ′ ) β ( x ) λ β +2 q −− ) ≤ λ in Ω \ Ω δ . provided that λ > δ . From (H. f ), (H. g ), (3.10), (2.13), (3.4) and (1.3), we get(3.20) u α ( x ) f ( u, v ) ≤ M ( u α ( x ) + u α ( x ) )(1 + v β ( x ) ) ≤ M (cid:0) ( c d ( x ) θ ) α ( x ) + ( c d ( x ) θ ) α ( x ) (cid:1) (cid:0) c ′ d ( x )) β ( x ) (cid:1) ≤ M max { ( c ) α ( x ) , ( c ) α ( x ) } ( d ( x ) θ α ( x ) + d ( x ) θ α ( x ) ) (cid:0) c ′ d ( x )) β ( x ) (cid:1) ≤ C ( d ( x ) θ α − + d ( x ) θ α − ) (cid:16) d ( x ) β − (cid:17) ≤ λ in Ω δ and similarly(3.21) v β ( x ) g ( u, v ) ≤ M (1 + u α ( x ) )( v β ( x ) + v β ( x ) ) ≤ M (cid:0) c d ( x )) α ( x ) (cid:1) (( c ′ d ( x ) θ ) β ( x ) + ( c ′ d ( x ) θ ) β ( x ) ) ≤ λ in Ω δ , provided that λ > (cid:3) Proof of Theorem 1.1
For every z , z ∈ C (Ω) , let us state the auxiliary problem( P z ) − ∆ p ( x ) u = ˜ f ( z , z ) in Ω , − ∆ q ( x ) v = ˜ g ( z , z ) in Ω ,u, v = 0 on ∂ Ω , where(4.1) ˜ f ( z , z ) = f (˜ z , ˜ z ) and ˜ g ( z , z ) = g (˜ z , ˜ z ) , with(4.2) ˜ z i = min { max { z i , c d ( x ) } , R } for i = 1 , . On account of (4.2) it follows that c d ( x ) ≤ ˜ z i ≤ R for i = 1 , P z ). In addition,it shows that solutions ( u, v ) of problem ( P z ) cannot occur outside the rectangle[ c d ( x ) , L R ] × [ c d ( x ) , L R ] , with a constant L R > Proposition 4.1.
Assume (H. f ), (H. g ) and (1.4) hold. Then all solutions ( u, v ) of ( P z ) belong to C ,γ (Ω) × C ,γ (Ω) for some γ ∈ (0 , and there is a positiveconstante L R , depending on R , such that (4.3) k u k C ,γ (Ω) , k v k C ,γ (Ω) < L R . Moreover, it holds (4.4) u ( x ) , v ( x ) ≥ c d ( x ) in Ω . Proof.
First, we prove the boundedness for solutions of ( P z ) in L ∞ (Ω) × L ∞ (Ω).To this end, we adapt the argument which proves [3, Lemma 2]. For each k ∈ N ,set U k,R = { x ∈ Ω : u ( x ) > kR } and V k,R = { x ∈ Ω : v ( x ) > kR } , where the constant R > u, v ∈ L (Ω), we have(4.5) | U k,R | , | V k,R | → k → + ∞ . UASILINEAR SINGULAR ELLIPTIC SYSTEMS WITH VARIABLE EXPONENTS 13
Using ( u − kR ) + and ( v − kR ) + as a test function in ( P z ), we get(4.6) ( R U k,R |∇ u | p dx = R U k,R f (˜ z , ˜ z )( u − kR ) + dx R V k,R |∇ v | q dx = R V k,R g (˜ z , ˜ z )( v − kR ) + dx. By (H. f ) and (4.2) observe that R Ω | f (˜ z , ˜ z ) | N dx ≤ C R Ω (1 + ˜ z Nα ( x )1 )(1 + ˜ z Nβ ( x )2 ) dx ≤ C (1 + R Nβ + ) R Ω (1 + ( c d ( x )) Nα ( x ) ) dx ≤ ˆ C R Ω (1 + d ( x ) Nα − ) dx. Since
N α − > − Z Ω d ( x ) Nα − dx < ∞ . Then, it follows that f (˜ z , ˜ z ) ∈ L N (Ω) and therefore(4.7) k f (˜ z , ˜ z ) k L N ( U k,R ) → k → + ∞ . Similarly, we obtain(4.8) k g (˜ z , ˜ z ) k L N ( V k,R ) → k → + ∞ . Now, proceeding analogously to the proof of [3, Lemma 2] provides a constant k ≥ | u ( x ) | , | v ( x ) | ≤ k R a.e in Ω . Consider now functions w and w defined by(4.9) (cid:26) − ∆ w = ˜ f ( z , z ) in Ω w = 0 on ∂ Ω and (cid:26) − ∆ w = ˜ g ( z , z ) in Ω w = 0 on ∂ Ω . On account of (4.1), (H. f ), (H. g ), (4.2), (1.4) and (1.7), one has(4.10) ˜ f ( z , z ) ≤ C d ( x ) α − and ˜ g ( z , z ) ≤ C ′ d ( x ) β − in Ω , for some positive constants C and C ′ . On the basis of (1.7) and Thanks to [16,Lemma in page 726], the right-hand side of problems in (4.9) belongs to H − (Ω).Consequently, the Minty-Browder theorem (see [5, Theorem V.15]) implies the ex-istence and uniqueness of w and w in (4.9). Moreover, bearing in mind (1.7) and(4.10), the regularity theory found in [15, Lemma 3.1] implies that w and w arebounded in C ,γ (Ω), for certain γ ∈ (0 , P z ) yields − div ( |∇ u | p ( x ) − ∇ u − ∇ w ) = 0 and − div ( |∇ v | q ( x ) − ∇ v − ∇ w ) = 0 , and the C ,α -boundedness of u and v follows from [10, Theorem 1.2]. Summariz-ing, we have obtained that solutions ( u, v ) of ( P z ) belong to C ,γ (Ω) × C ,γ (Ω),for certain γ ∈ (0 , L R > (cid:3) Next we prove the existence result for cooperative system ( P ). Proof of Theorem 1.1.
Denote by B (0 , L R ) = { ( u, v ) ∈ C (Ω) × C (Ω) : k u k C (Ω) + k v k C (Ω) < L R } and O = { ( u, v ) ∈ B (0 , L R ) : u ( x ) , v ( x ) ≥ c d ( x ) in Ω } . Let us introduce the operator P : O → C (Ω) × C (Ω) by P ( z , z ) = ( u, v ), where( u, v ) is the solution of problem ( P z ). Bearing in mind (4.10) and (1.7), the Minty-Browder theorem together with [3, Lemma 2] garantee that problem ( P z ) has aunique solution ( u, v ) in C ,γ (Ω) × C ,γ (Ω), for certain γ ∈ (0 , P is well defined. Moreover, analysis similar to that in the proof ofTheorem 3 in [3] imply that P is continuous and compact operator. On the otherhand, according to Proposition 4.1, it follows that O is invariant by P , that is, P ( O ) ⊂ O . Therefore we are in a position to apply Schauder’s fixed point Theoremto the set O and the map P : O → O . This ensures the existence of ( u, v ) ∈ O satisfying P ( u, v ) = ( u, v ), that is, ( u, v ) ∈ C (Ω) × C (Ω) is a solution of problem − ∆ p ( x ) u = ˜ f ( u, v ) in Ω , − ∆ q ( x ) v = ˜ g ( u, v ) in Ω ,u, v = 0 on ∂ Ω . Finally, thank’s to proposition 2.1, it turns out that ( u, v ) ∈ C (Ω) × C (Ω) is a(positive) solution of problem ( P ). (cid:3) Proof of Theorem 1.2
The proof is based on Schauder’s fixed point Theorem. Using the functions ( u, v )and ( u, v ) given in (3.1), (3.2), (3.7) and (3.8) let introduce the set K = (cid:8) ( y , y ) ∈ C (Ω) × C (Ω) : u ≤ y ≤ u and v ≤ y ≤ v in Ω (cid:9) , which is closed, bounded and convex in C (Ω) × C (Ω). Then we define the operator T : K → C (Ω) × C (Ω) by T ( y , y ) = ( u, v ), where ( u, v ) is required to satisfy( P y ) − ∆ p ( x ) u = f ( y , y ) in Ω − ∆ q ( x ) v = g ( y , y ) in Ω u, v = 0 on ∂ Ω . For ( y , y ) ∈ K , we derive from (H. f ), (H .g ), (2.13), (3.4), (3.5), and (3.10) theestimates(5.1) f ( y , y ) ≤ m (1 + u α ( x ) )(1 + v β ( x ) ) ≤ C d ( x ) β ( x ) in Ωand(5.2) g ( y , y ) ≤ m (1 + u α ( x ) )(1 + v β ( x ) ) ≤ C d ( x ) α ( x ) in Ω , with positive constants C , C . We point out that estimates (5.1) and (5.2) com-bined with (1.9) and (1.10) enable us to deduce that f ( y , y ) ∈ W − ,p ′ ( x ) (Ω) and g ( y , y ) ∈ W − ,q ′ ( x ) (Ω). Then the unique solvability of ( u, v ) in ( P y ) is readilyderived from Minty-Browder theorem (see, e.g., [5]). Hence, the operator T is welldefined.Using the regularity theory up to the boundary (see [3, Lemma 2]), it followsthat ( u, v ) ∈ C ,β (Ω) × C ,β (Ω), with some β ∈ (0 , M > k u k C ,β (Ω) , k v k C ,β (Ω) ≤ M, whenever ( u, v ) = T ( y , y ) with UASILINEAR SINGULAR ELLIPTIC SYSTEMS WITH VARIABLE EXPONENTS 15 ( y , y ) ∈ K . Then, analysis similar to that in the proof of [3, Theorem 3] implythat T is continuous and compact operator.The next step in the proof is to show that T ( K ) ⊂ K . Let ( y , y ) ∈ K and denote( u, v ) = T ( y , y ). Using the definitions of K and T , on the basis of Proposition3.1, (H .f ) and (H .g ), it follows that − ∆ p ( x ) u ( x ) = f ( y ( x ) , y ( x )) ≤ f ( u ( x ) , v ( x )) ≤ − ∆ p ( x ) u ( x ) in Ω , and similarly − ∆ q ( x ) v ( x ) = g ( y ( x ) , y ( x )) ≤ g ( u ( x ) , v ( x )) ≤ − ∆ q ( x ) v ( x ) in Ω . Proceeding in the same way, via Proposition 3.1 and hypotheses (H .f ), (H .g ), leadsto − ∆ p ( x ) u ( x ) = f ( y ( x ) , y ( x )) ≥ f ( u, v ) ≥ − ∆ p ( x ) u ( x ) in Ω , and similarly − ∆ q ( x ) v ( x ) = g ( y ( x ) , y ( x )) ≥ g ( u, v ) ≥ − ∆ q ( x ) v ( x ) in Ω . Then from the strict monotonicity of the operators − ∆ p ( x ) and − ∆ q ( x ) we get that( u, v ) ∈ K , which establishes that T ( K ) ⊂ K . Therefore we are in a position toapply Schauder’s fixed point Theorem to the set K and the map T : K → K . Thisensures the existence of ( u, v ) ∈ K satisfying ( u, v ) = T ( u, v ) . Moreover, becausethe solution ( u, v ) ∈ K and (H .f ) , (H .g ) , (1.9) and (1.10) are fulfilled, we concludefrom [3, Lemma 2] that ( u, v ) ∈ C (Ω) × C (Ω). This ends the proof. References [1] E. Acerbi & G. Mingione,
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