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Boundedness of solutions to quasilinear elliptic systems

  • Fang Mi , Xia Liuye , Han Yingxiao and Gao Hongya EMAIL logo
Published/Copyright: September 13, 2025
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Advances in Nonlinear Analysis
From the journal Advances in Nonlinear Analysis Volume 14 Issue 1

Abstract

This article deals with elliptic systems of the form

− ∑ i = 1 n ∂ ∂ x i ∑ β = 1 N ∑ j = 1 n a i , j α , β ( x , u ( x ) ) ∂ u β ( x ) ∂ x j = f α ( x ) , α = 1 , … , N .

Under ellipticity conditions of the diagonal coefficients and proportional conditions of the off-diagonal coefficients, we derive local and global boundedness results. Under ellipticity of all the coefficients and “butterfly” support of off-diagonal coefficients, we derive a global boundedness result. This article also considers regularizing effect of a lower-order term.

Keywords: quasilinear elliptic system; proportional condition; “butterfly” support; regularizing effect; boundedness
MSC 2010: 35J57

1 Introduction

Let n > 2 , N ≥ 2 be integers and Ω an open bounded subset of R n . We consider quasilinear elliptic systems involving N equations of the form

(1.1) − ∑ i = 1 n ∂ ∂ x i ∑ β = 1 N ∑ j = 1 n a i , j α , β ( x , u ( x ) ) ∂ u β ( x ) ∂ x j = f α ( x ) , α = 1 , … , N ,

where α is the equation index and u ( x ) = ( u 1 ( x ) , … , u N ( x ) ) : Ω ⊂ R n → R N . Denote

D u β ( x ) = ∂ u β ( x ) ∂ x 1 , … , ∂ u β ( x ) ∂ x n , β = 1 , … , N ,

which is the β th row of the matrix

D u ( x ) = ∂ u β ( x ) ∂ x j 1 ≤ i ≤ n 1 ≤ β ≤ N ∈ R N × n .

We consider the following two sets of assumptions on the coefficients a i , j α , β ( x , y ) , i , j ∈ { 1 , … , n } , α , β ∈ { 1 , … , N } .

The first set of assumptions is denoted by ( A ): for all i , j ∈ { 1 , … , n } and all α , β ∈ { 1 , … , N } , we consider that a i , j α , β ( x , y ) : Ω × R N → R satisfying the following conditions:

( A 1 ) (Carathéodory condition) x ↦ a i , j α , β ( x , y ) is measurable for all y ∈ R N and y ↦ a i , j α , β ( x , y ) is continuous for almost all x ∈ Ω ;

( A 2 ) (Boundedness of all the coefficients) there exists a positive constant c ˜ such that for almost all x ∈ Ω and all y ∈ R N ,

∣ a i , j α , β ( x , y ) ∣ ≤ c ˜ ;

( A 3 ) (Ellipticity of the diagonal coefficients) there exists a positive constant c 0 such that for almost all x ∈ Ω , all y ∈ R N and all λ ∈ R n ,

c 0 ∣ λ ∣ 2 ≤ ∑ i , j = 1 n a i , j α , α ( x , y ) λ i λ j ;

( A 4 ) (Proportional condition of the off-diagonal coefficients) there exist constants r α , β , α , β ∈ { 1 , … , N } , such that for almost all x ∈ Ω and all y ∈ R N ,

a i , j α , β ( x , y ) = r α , β a i , j β , β ( x , y ) ,

the constants r α , β , α , β ∈ { 1 , … , N } , be such that r α , α = 1 and

det ℛ = det 1 r 2,1 r 3,1 … r N , 1 r 1 , 2 1 r 3,2 … r N , 2 r 1,3 r 2,3 1 … r N , 3 ⋮ ⋮ ⋮ ⋱ ⋮ r 1 , N r 2 , N r 3 , N … 1 ≠ 0 .

The second set of assumptions is denoted by ( A ) ′ : for all i , j ∈ { 1 , … , n } and all α , β ∈ { 1 , … , N } , we consider that a i , j α , β ( x , y ) : Ω × R N → R satisfying ( A 1 ), ( A 2 ), and the following:

( A 3 ) ′ (ellipticity of all the coefficients) there exists a positive constant c ˜ 0 such that for almost all x ∈ Ω , all y ∈ R N and all ξ ∈ R N × n ,

c ˜ 0 ∣ ξ ∣ 2 ≤ ∑ α , β = 1 N ∑ i , j = 1 n a i , j α , β ( x , y ) ξ i α ξ j β ;

( A 4 ) ′ (“butterfly” support of off-diagonal coefficients) there exists Q 0 ∈ ( 0 , + ∞ ) such that for all Q ≥ Q 0 , when α ≠ β ,

( a i , j α , β ( x , y ) ≠ 0 and ∣ y α ∣ > Q ) ⇒ ∣ y β ∣ > Q .

For the figure of “butterfly support,” see [25, Figure 1].

The following example gives the coefficients a i , j α , β ( x , y ) : Ω × R N → R with i , j ∈ { 1 , … , n } and α , β ∈ { 1 , … , N } , which satisfy the set of assumptions ( A ).

Example 1.1

We let δ i j be the Kronecker symbol and Ω = B 1 ( 0 ) , the unit ball in R n . For i , j ∈ { 1 , … , n } and α , β ∈ { 1 , … , N } , we define a i , j α , β ( x , y ) as follows: for α ∈ { 1 , … , N } ,

a i , j α , α ( x , y ) = 1 + ∣ x ∣ + ∣ y α ∣ 1 + ∣ y α ∣ δ i j ,

and for α , β ∈ { 1 , … , N } with α ≠ β ,

(1.2) a i , j α , β ( x , y ) = r α , β a i , j β , β ( x , y ) = r α , β 1 + ∣ x ∣ + ∣ y β ∣ 1 + ∣ y β ∣ δ i j ,

where the real numbers r α , β are such that det ℛ = det ( r α , β ) ≠ 0 ; thus, the condition ( A 4 ) holds true naturally; moreover, the condition ( A 1 ) is satisfied because x ↦ a i , j α , β ( x , y ) is measurable and y ↦ a i , j α , β ( x , y ) is continuous; we note from (1.2) that ∣ a i , j α , β ( x , y ) ∣ ≤ 3 ∣ r α , β ∣ for x ∈ B 1 ( 0 ) , thus, the condition ( A 2 ) is satisfied with c ˜ = 3 ‖ ℛ ‖ = 3 ∑ α , β = 1 N ∣ r α , β ∣ 2 1 ⁄ 2 ; the condition ( A 3 ) is satisfied with c 0 = 1 since

∑ i , j = 1 n a i , j α , α ( x , y ) λ i λ j = ∑ i = 1 n 1 + ∣ x ∣ + ∣ y α ∣ 1 + ∣ y α ∣ λ i 2 ≥ ∣ λ ∣ 2 .

We note that in this article, we consider the case N ≥ 2 , i.e., we deal with elliptic systems. For the case N = 1 , (1.1) is only one single equation, existence and regularity results of distributional solutions u : Ω ⊂ R N → R have been deeply studied, we refer the reader to [5,6,9,10,12,16,34] for existence results and [4,8,14,23,30–33] for regularity results.

For N ≥ 2 , one cannot expect, due to De Giorgi’s counterexample, see [17], that weak solutions of (1.1) are locally and globally bounded if no additional assumptions are proposed. Quasilinear elliptic system (1.1) with the set of assumptions ( A ) has been studied in [28, Theorem 2], where the authors considered the special case N = 2 . The general case N ≥ 2 may be found in [19, Theorem 2.1]. Quasilinear elliptic system (1.1) with the set of assumptions ( A ) ′ has been studied in [25], where the authors give a local boundedness result. We refer the readers to [20,24,28,29] for some regularity results and estimates related to quasilinear elliptic systems under some staircase support conditions of the coefficients, [15,21] for some results related to nonlinear elliptic systems, and [18] for some local boundedness under nonstandard growth conditions.

In the next two sections, we shall give some boundedness results related to system (1.1) under the conditions ( A ) or ( A ) ′ . In the sequel, we shall denote by c a generic constant, whose value, depending on the data, may vary from one line to another.

2 Boundedness under ( A )

This section deals with local and global boundedness for solutions to elliptic systems (1.1) under the set of assumptions ( A ).

2.1 Local boundedness result

In this section, we consider (1.1), i.e.,

(2.1) − ∑ i = 1 n ∂ ∂ x i ∑ β = 1 N ∑ j = 1 n a i , j α , β ( x , u ( x ) ) ∂ u β ( x ) ∂ x j = f α ( x ) , α = 1 , … , N .

Let

(2.2) f = ( f 1 , … , f N ) ∈ L loc ( 2 * ) ′ ( Ω ; R N ) , ( 2 * ) ′ = 2 n n + 2 .

We give the following:

Definition 2.1

A function u ∈ W loc 1 , 2 ( Ω ; R N ) is a local solution to (2.1) if

(2.3) ∫ Ω ∑ α , β = 1 N ∑ i , j = 1 n a i , j α , β ( x , u ( x ) ) D j u β ( x ) D i φ α ( x ) d x = ∫ Ω ∑ α = 1 N f α ( x ) φ α ( x ) d x

holds true for all φ ∈ W 1 , 2 ( Ω ; R N ) with compact support.

We note that (2.2) is added in order to make finite the right-hand integral in (2.3).

The main result of this section is the following theorem.

Theorem 2.1

Let u ∈ W loc 1 , 2 ( Ω ; R N ) be a local solution to (2.1). Under the set of assumptions ( A ), if f ∈ L loc m ( Ω ; R N ) , m > n 2 , then u is locally bounded in Ω .

In order to prove Theorem 2.1, we need the following Caccioppoli inequality.

Lemma 2.1

Let u ∈ W loc 1 , 2 ( Ω ; R N ) be a local solution to (2.1) under the set of assumptions ( A ). Let B R ( x 0 ) ⋐ Ω be the ball centered at x 0 ∈ Ω with radius R , ∣ B R ( x 0 ) ∣ < 1 . For k ≥ 0 , 0 < s < t ≤ R , denote

A k β = { x ∈ Ω : ∣ u β ( x ) ∣ > k } , A k , t β = A k β ∩ B t ( x 0 ) .

If f ∈ L loc m ( Ω ; R N ) , m > ( 2 * ) ′ , then

(2.4) ∑ β = 1 N ∫ A k , s β ∣ D ∣ u β ∣ ∣ 2 d x ≤ c ∑ β = 1 N ∫ A k , t β ∣ u β ∣ − k t − s 2 d x + ∣ A k , t β ∣ θ ,

where c is a constant depending upon n , N , m , ‖ f ‖ L m ( B R ) , c ˜ , c 0 and r α , β , α , β = 1 , … , N , and

θ = 2 1 ( 2 * ) ′ − 1 m > 0 .

Proof

Let u ∈ W loc 1 , 2 ( Ω ; R N ) be a local solution to (2.1). Let B R ( x 0 ) ⋐ Ω with ∣ B R ( x 0 ) ∣ < 1 (which implies R < 1 ). For 0 < s < t ≤ R , let us consider a smooth cut-off function η ∈ C 0 ∞ ( B t ( x 0 ) ) satisfying

0 ≤ η ≤ 1 , η ≡ 1 , in B s ( x 0 ) and ∣ D η ∣ ≤ 2 t − s .

Let us take φ = ( φ 1 , … , φ N ) with

(2.5) φ α = ∑ γ = 1 N C α γ η 2 G k ( u γ ) , α ∈ { 1 , … , N } ,

here and in what follows, for s ∈ R ,

(2.6) G k ( s ) = s − T k ( s ) = s − min 1 , k ∣ s ∣ s ,

and C α γ , α , γ ∈ { 1 , … , N } , are the real constants to be chosen later. (2.5) yields

D i φ α = ∑ γ = 1 N C α γ [ η 2 D i u γ + 2 η D i η G k ( u γ ) ] χ A k γ , i = 1 , … , n ,

where, for a set E , χ E ( x ) is the characteristic function of the set E , i.e., χ E ( x ) = 1 if x ∈ E and χ E ( x ) = 0 otherwise. Such a function φ is admissible for Definition 2.1 since it is obvious that φ ∈ W 1 , 2 ( Ω ; R N ) and supp φ ⋐ Ω . We use such a φ in (2.3), and we have

∫ Ω ∑ α , β = 1 N ∑ i , j = 1 n a i , j α , β ( x , u ) D j u β ∑ γ = 1 N C α γ [ η 2 D i u γ + 2 η D i η G k ( u γ ) ] χ A k γ d x = ∫ Ω ∑ α = 1 N f α ∑ γ = 1 N C α γ η 2 G k ( u γ ) d x ,

from which we derive

(2.7) ∫ Ω ∑ α , β = 1 N ∑ i , j = 1 n a i , j α , β ( x , u ) D j u β ∑ γ = 1 N C α γ η 2 D i u γ χ A k γ d x = − ∫ Ω ∑ α , β = 1 N ∑ i , j = 1 n a i , j α , β ( x , u ) D j u β ∑ γ = 1 N C α γ 2 η D i η G k ( u γ ) χ A k γ d x + ∫ Ω ∑ α = 1 N f α ∑ γ = 1 N C α γ η 2 G k ( u γ ) d x .

For the left-hand side of (2.7), we use the proportional condition ( A 4 ) and we have

(2.8) ∫ Ω ∑ α , β = 1 N ∑ i , j = 1 n a i , j α , β ( x , u ) D j u β ∑ γ = 1 N C α γ η 2 D i u γ χ A k γ d x = ∫ Ω ∑ α = 1 N ∑ i , j = 1 n a i , j α , α ( x , u ) D j u α ∑ γ = 1 N C α γ η 2 D i u γ χ A k γ d x ( terms for β = α ) + ∫ Ω ∑ α = 1 N ∑ β = 1 , β ≠ α N ∑ i , j = 1 n a i , j α , β ( x , u ) D j u β ∑ γ = 1 N C α γ η 2 D i u γ χ A k γ d x ( terms for β ≠ α ) = ∫ Ω ∑ α = 1 N ∑ i , j = 1 n C α α a i , j α , α ( x , u ) D j u α D i u α χ A k α η 2 d x ( terms for γ = β = α ) + ∫ Ω ∑ α = 1 N ∑ i , j = 1 n a i , j α , α ( x , u ) D j u α ∑ γ = 1 , γ ≠ α N C α γ D i u γ χ A k γ η 2 d x ( terms for γ ≠ β = α ) + ∫ Ω ∑ α = 1 N ∑ β = 1 , β ≠ α N ∑ i , j = 1 n r α , β a i , j β , β ( x , u ) D j u β C α β D i u β χ A k β η 2 d x ( terms for γ = β ≠ α ) + ∫ Ω ∑ α = 1 N ∑ β = 1 , β ≠ α N ∑ i , j = 1 n r α , β a i , j β , β ( x , u ) D j u β ∑ γ = 1 , γ ≠ β N C α γ D i u γ χ A k γ η 2 d x ( terms for γ ≠ β , β ≠ α ) = I 1 + I 2 + I 3 + I 4 .

It is obvious that, recalling that r α , α = 1 for α ∈ { 1 , … , N } ,

I 1 + I 3 = ∫ Ω ∑ α , β = 1 N ∑ i , j = 1 n r α , β a i , j β , β ( x , u ) D j u β C α β D i u β χ A k β η 2 d x = ∫ Ω ∑ β = 1 N ∑ i , j = 1 n ∑ α = 1 N r α , β C α β a i , j β , β ( x , u ) D j u β D i u β χ A k β η 2 d x

and

I 2 + I 4 = ∫ Ω ∑ α , β = 1 N ∑ i , j = 1 n r α , β a i , j β , β ( x , u ) D j u β ∑ γ = 1 , γ ≠ β N C α γ D i u γ χ A k γ η 2 d x = ∫ Ω ∑ β , γ = 1 , β ≠ γ N ∑ i , j = 1 n ∑ α = 1 N r α , β C α γ a i , j β , β ( x , u ) D j u β D i u γ χ A k γ η 2 d x .

If one can choose

(2.9) ∑ α = 1 N r α , β C α β = 1 , for β ∈ { 1 , … , N } ,

and

(2.10) ∑ α = 1 N r α , β C α γ = 0 , for β , γ ∈ { 1 , … , N } , β ≠ γ ,

then the assumption ( A 3 ) allows us to estimate

(2.11) ∑ j = 1 4 I j = ∫ Ω ∑ β = 1 N ∑ i , j = 1 n a i , j β , β ( x , u ) D j u β D i u β χ A k β η 2 d x ≥ c 0 ∑ β = 1 N ∫ A k , t β η 2 ∣ D u β ∣ 2 d x .

Now, we prove that equations (2.9) and (2.10) are valid for appropriate choice of the constants C α γ , α , γ ∈ { 1 , … , N } . In fact, (2.9) and (2.10) have the form

(2.12) ∑ α = 1 N r α , β C α γ = δ β γ , for β , γ ∈ { 1 , … , N } .

We note that the aforementioned system has N 2 equations with N 2 unknowns C α γ , α , γ ∈ { 1 , … , N } and can be rewritten as the form

(2.13) ℛ 0 … 0 0 ℛ … 0 ⋮ ⋮ ⋱ ⋮ 0 0 … ℛ C 1 C 2 ⋮ C N = e 1 e 2 ⋮ e N ,

where

ℛ = 1 r 2,1 r 3,1 … r N , 1 r 1 , 2 1 r 3,2 … r N , 2 r 1,3 r 2,3 1 … r N , 3 ⋮ ⋮ ⋮ ⋱ ⋮ r 1 , N r 2 , N r 3 , N … 1 , C j = C 1 j C 2 j C 3 j ⋮ C N j ,

and e j is the unit vector of R N , j ∈ { 1 , … , N } . By assumption ( A 4 ), det ℛ ≠ 0 ; thus, the determinant of the left-hand side square matrix in (2.13) is nonzero, and noting the right-hand side of system (2.13) is nonzero, then there exists a unique nonzero solution to (2.13). We choose C α γ to be the unique nonzero solution to (2.13) and we have (2.9) and (2.10). Note that the values of C α γ rely on the values of r α , β , α , β = 1 , … , N .

We now use (2.12) again, ( A 2 ) and the proportional condition of the off-diagonal coefficients in ( A 4 ) to estimate the first term of the right-hand side of (2.7):

(2.14) − ∫ Ω ∑ α , β = 1 N ∑ i , j = 1 n a i , j α , β ( x , u ) D j u β ∑ γ = 1 N C α γ 2 η D i η G k ( u γ ) χ A k γ d x = − ∫ Ω ∑ α , β = 1 N ∑ i , j = 1 n r α , β a i , j β , β ( x , u ) D j u β ∑ γ = 1 N C α γ 2 η D i η G k ( u γ ) χ A k γ d x = − ∫ Ω ∑ γ , β = 1 N ∑ i , j = 1 n ∑ α = 1 N r α , β C α γ a i , j β , β ( x , u ) D j u β 2 η D i η G k ( u γ ) χ A k γ d x = − ∫ Ω ∑ β = 1 N ∑ i , j = 1 n a i , j β , β ( x , u ) D j u β 2 η D i η G k ( u β ) χ A k β d x ≤ 2 c ˜ n 2 ∫ Ω ∑ β = 1 N ∣ D u β ∣ η ∣ D η ∣ ∣ G k ( u β ) ∣ χ A k β d x ≤ c ˜ n 2 ∫ Ω ∑ β = 1 N ε η 2 ∣ D u β ∣ 2 + ∣ D η ∣ 2 ∣ G k ( u β ) ∣ 2 ε χ A k β d x ≤ ε c ˜ n 2 ∑ β = 1 N ∫ A k , t β η 2 ∣ D u β ∣ 2 d x + 4 c ˜ n 2 ε ∑ β = 1 N ∫ A k , t β ∣ u β ∣ − k t − s 2 d x ,

where we used Young’s inequality

2 a b ≤ ε a 2 + b 2 ⁄ ε , ε > 0 ,

and the fact

∣ G k ( u β ) ∣ = ∣ u β ∣ − k , x ∈ A k β .

We next estimate the second term of the right-hand side of (2.7). Sobolev embedding theorem, Young’s and Hölder’s inequalities allow us to obtain

(2.15) ∫ Ω ∑ α = 1 N f α ∑ γ = 1 N C α γ η 2 G k ( u γ ) d x ≤ c ¯ ∫ A k , t β ∑ α = 1 N ∣ f α ∣ ∑ β = 1 N ∣ η 2 G k ( u β ) ∣ d x ≤ c ¯ ∑ β = 1 N ∫ A k , t β ∑ α = 1 N ∣ f α ∣ ( 2 * ) ′ d x 1 ( 2 * ) ′ ∫ B t ∣ η 2 G k ( u β ) ∣ 2 * d x 1 2 * ≤ c ¯ ∑ β = 1 N ∫ A k , t β ∑ α = 1 N ∣ f α ∣ m d x ( 2 * ) ′ m ∣ A k , t β ∣ 1 − ( 2 * ) ′ m 1 ( 2 * ) ′ c ( n ) ∫ B t ∣ D ( η 2 G k ( u β ) ) ∣ 2 d x 1 2 ≤ c ¯ N ‖ f ‖ L m ( B t ) ∑ β = 1 N ∣ A k , t β ∣ θ 2 ∫ A k , t β ∣ 2 η D η G k ( u β ) + η 2 D u β ∣ 2 d x 1 2 ≤ c ¯ N ‖ f ‖ L m ( B t ) ∑ β = 1 N ∣ A k , t β ∣ θ 2 ∫ A k , t β ∣ 2 η D η G k ( u β ) ∣ 2 d x 1 2 + ∫ A k , t β ∣ η 2 D u β ∣ 2 d x 1 2 ≤ c ¯ N ‖ f ‖ L m ( B t ) ∑ β = 1 N ∣ A k , t β ∣ θ 2 4 ∫ A k , t β ∣ u β ∣ − k t − s 2 d x 1 2 + ∫ A k , t β η 2 ∣ D u β ∣ 2 d x 1 2 ≤ c ¯ N ‖ f ‖ L m ( B t ) ε ∑ β = 1 N ∫ A k , t β η 2 ∣ D u β ∣ 2 d x + 4 ε ∑ β = 1 N ∫ A k , t β ∣ u β ∣ − k t − s 2 d x + c ( ε ) ∑ β = 1 N ∣ A k , t β ∣ θ ,

where c ¯ = c ( n ) ∑ α , γ = 1 N ∣ C α γ ∣ and θ = 2 1 ( 2 * ) ′ − 1 m .

Substituting (2.8), (2.11), (2.14), (2.15) into (2.7), and choosing ε small enough such that

( c ¯ N ‖ f ‖ L m ( B t ) + c ˜ n 2 ) ε = c 0 2 ,

we then derive

∑ β = 1 N ∫ A k , s β ∣ D u β ∣ 2 d x ≤ c ∑ β = 1 N ∫ A k , t β ∣ u β ∣ − k t − s 2 d x + ∑ β = 1 N ∣ A k , t β ∣ θ ,

where c is a constant depending upon n , N , m , ‖ f ‖ L m ( B R ) , c ˜ , c 0 , and c ¯ . Note that

∣ D i ∣ u β ∣ ∣ = ∣ D i u β ∣ ,

then (2.1) reduces to (2.4), completing the proof of Lemma 2.1.□

In the next lemma, we state and prove a general result that holds true for some general vectorial function v ∈ W loc 1 , p ( Ω ; R N ) . Eventually, we will use such a result with v = ( ∣ u 1 ∣ , … , ∣ u N ∣ ) and p = 2 . Note that this lemma is a generalization of [25, Lemma 3.2].

Lemma 2.2

Let v = ( v 1 , … , v N ) ∈ W loc 1 , p ( Ω ; R N ) with 1 < p < n . Suppose that there exist constants c 1 > 0 , L 0 ≥ 0 and θ > 1 − p n , such that

(2.16) ∑ β = 1 N ∫ A L , s β ∣ D v β ∣ p d x ≤ c 1 ∑ β = 1 N ∫ A L , t β v β − L t − s p d x + ∣ A L , t β ∣ θ ,

for every s , t , L , where 0 < s < t , B t ( x 0 ) ⋐ Ω and L > L 0 , where

A L , s β = { x ∈ B s ( x 0 ) : v β ( x ) > L } .

Then, for every β = 1 , … , N , v β is locally bounded from above, and for every r , R with 0 < r < R , B R ( x 0 ) ⋐ Ω and ∣ B R ( x 0 ) ∣ < 1 ,

sup B r ( x 0 ) v β ≤ c ˆ ,

where c ˆ is a constant depending only upon n , p , θ , c 1 , ∣ Ω ∣ , L 0 , 1 R − r and

∑ β = 1 N ∫ B R ( x 0 ) max { v β ; 0 } p d x .

Remark 2.1

Without loss of generality, we assume θ ≤ 1 in (2.16) since otherwise,

∣ A L , t β ∣ θ = ∣ A L , t β ∣ θ − 1 ∣ A L , t β ∣ 1 ≤ ∣ Ω ∣ θ − 1 ∣ A L , t β ∣ ,

then (2.16) holds true with c 1 replaced by c 1 N max { ∣ Ω ∣ θ − 1 , 1 } and θ replaced by 1.

We shall need the following preliminary lemma in order to prove Lemma 2.2, see [22, Lemma 7.1].

Lemma 2.3

Let α > 1 and let { J i } be a sequence of real positive numbers, such that

J i + 1 ≤ C B i J i α ,

with C > 0 and B > 1 . If

J 0 ≤ C − 1 α − 1 B − 1 ( α − 1 ) 2 ,

we have

J i ≤ B − i α − 1 J 0 ,

and hence, in particular, lim i → + ∞ J i = 0 .

Proof of Lemma 2.2

Let us consider balls B r 1 ( x 0 ) and B r 2 ( x 0 ) with 0 < r 1 < r 2 < R , B R ( x 0 ) ⋐ Ω and ∣ R R ( x 0 ) ∣ < 1 . Let η : R n → R be a cut-off function such that

0 ≤ η ≤ 1 , η ∈ C 0 1 ( B r 1 + r 2 2 ( x 0 ) ) , η = 1 in B r 1 ( x 0 ) and ∣ D η ∣ ≤ 4 r 2 − r 1 .

Then, using Hölder’s inequality, Sobolev embedding theorem, and the properties of the cut-off function η , one obtains

∫ A L , r 1 β ( v β − L ) p d x ≤ ∫ A L , r 1 β ( v β − L ) p * d x p p * ∣ A L , r 1 β ∣ 1 − p p * = ∫ A L , r 1 β [ η ( v β − L ) ] p * d x p p * ∣ A L , r 1 β ∣ 1 − p p * = ∫ B r 1 [ η max { v β − L ; 0 } ] p * d x p p * ∣ A L , r 1 β ∣ 1 − p p * ≤ ∫ B r 1 + r 2 2 [ η max { v β − L ; 0 } ] p * d x p p * ∣ A L , r 1 β ∣ 1 − p p * ≤ c ∫ B r 1 + r 2 2 ∣ D [ η max { v β − L ; 0 } ] ∣ p d x ∣ A L , r 1 β ∣ 1 − p p * = c ∫ B r 1 + r 2 2 ∣ max { v β − L ; 0 } D η + η D max { v β − L ; 0 } ∣ p d x ∣ A L , r 1 β ∣ 1 − p p * = c ∫ A L , r 1 + r 2 2 β ∣ ( v β − L ) D η + η D v β ∣ p d x ∣ A L , r 1 β ∣ 1 − p p * ≤ c ∫ A L , r 1 + r 2 2 β ∣ ( v β − L ) D η ∣ p d x + ∫ A L , r 1 + r 2 2 β ∣ η D v β ∣ p d x ∣ A L , r 1 β ∣ 1 − p p * ≤ c ∫ A L , r 1 + r 2 2 β v β − L r 2 − r 1 p d x + ∫ A L , r 1 + r 2 2 β ∣ D v β ∣ p d x ∣ A L , r 1 β ∣ 1 − p p * ,

where c is a constant depending upon n and p . Now, we sum upon β from 1 to N obtaining

∑ β = 1 N ∫ A L , r 1 β ( v β − L ) p d x ≤ c ∑ β = 1 N ∫ A L , r 1 + r 2 2 β v β − L r 2 − r 1 p d x + ∫ A L , r 1 + r 2 2 β ∣ D v β ∣ p d x ∣ A L , r 1 β ∣ 1 − p p * ≤ c ∑ β = 1 N ∫ A L , r 1 + r 2 2 β v β − L r 2 − r 1 p d x + ∑ β = 1 N ∫ A L , r 1 + r 2 2 β ∣ D v β ∣ p d x ∑ β = 1 N ∣ A L , r 1 β ∣ 1 − p p * .

In order to control the second term in the aforementioned bracket, we use our assumption (2.16) with s = r 1 + r 2 2 and t = r 2 . From the aforementioned inequality, one obtains

(2.17) ∑ β = 1 N ∫ A L , r 1 β ( v β − L ) p d x ≤ c ∑ β = 1 N ∫ A L , r 1 + r 2 2 β v β − L r 2 − r 1 p d x + ∑ β = 1 N ∫ A L , r 2 β v β − L r 2 − r 1 p d x + ∑ β = 1 N ∣ A L , r 2 β ∣ θ ∑ β = 1 N ∣ A L , r 1 β ∣ 1 − p p * ≤ c ∑ β = 1 N ∫ A L , r 2 β v β − L r 2 − r 1 p d x ∑ β = 1 N ∣ A L , r 1 β ∣ 1 − p p * + c ∑ β = 1 N ∣ A L , r 2 β ∣ θ ∑ β = 1 N ∣ A L , r 1 β ∣ 1 − p p * ,

where c is a constant depending upon n , p , and c 1 .

We are able to estimate ∣ A L , r 1 β ∣ and ∣ A L , r 2 β ∣ by means of ∫ A L ˜ , r 2 β ( v β − L ˜ ) p d x , where L > L ˜ ≥ L 0 . In fact,

(2.18) ∣ A L , r 1 β ∣ ≤ ∣ A L , r 2 β ∣ = 1 ( L − L ˜ ) p ( L − L ˜ ) p ∣ A L , r 2 β ∣ = 1 ( L − L ˜ ) p ∫ A L , r 2 β ( L − L ˜ ) p d x ≤ 1 ( L − L ˜ ) p ∫ A L , r 2 β ( v β − L ˜ ) p d x ≤ 1 ( L − L ˜ ) p ∫ A L , r 2 ˜ β ( v β − L ˜ ) p d x .

In the mean time,

(2.19) ∫ A L , r 2 β ( v β − L ) p d x ≤ ∫ A L , r 2 β ( v β − L ˜ ) p d x ≤ ∫ A L ˜ , r 2 β ( v β − L ˜ ) p d x .

Substituting (2.18) and (2.19) into (2.17), and noting that 1 − p p * = p n , we then arrive at

(2.20) ∑ β = 1 N ∫ A L , r 1 β ( v β − L ) p d x ≤ c ( r 2 − r 1 ) p ( L − L ˜ ) p p ⁄ n ∑ β = 1 N ∫ A L , r 2 β ( v β − L ) p d x ∑ β = 1 N ∫ A L ˜ , r 2 β ( v β − L ˜ ) p d x p n + c ( L − L ˜ ) p p ⁄ n + p θ ∑ β = 1 N ∫ A L , r 2 ˜ β ( v β − L ˜ ) p d x θ ∑ β = 1 N ∫ A L ˜ , r 2 β ( v β − L ˜ ) p d x p n ≤ c ( r 2 − r 1 ) p ( L − L ˜ ) p p ⁄ n ∑ β = 1 N ∫ A L , r 2 ˜ β ( v β − L ˜ ) p d x 1 + p n + c ( L − L ˜ ) p p ⁄ n + p θ ∑ β = 1 N ∫ A L , r 2 ˜ β ( v β − L ˜ ) p d x θ + p n .

Now, we fix 0 < r < R , with B R ( x 0 ) ⋐ Ω and ∣ B R ( x 0 ) ∣ < 1 , and we take the following sequence of radii:

ρ i = r + R − r 2 i , i = 0 , 1 , 2 , … ,

then ρ 0 = R and

ρ i − ρ i + 1 = R − r 2 i + 1 > 0 ,

so ρ i strictly decreases and r < ρ i ≤ R . Let us fix a level d > max { L 0 , 1 } and we take the following sequence of levels:

k i = 2 d 1 − 1 2 i + 1 , i = 0 , 1 , 2 , … ,

then k 0 = d and k i + 1 − k i = d 2 i + 1 > 0 , so k i strictly increases and L 0 < d ≤ k i < 2 d . We can use (2.20) with levels L = k i + 1 > k i = L ˜ and radii r 1 = ρ i + 1 < ρ i = r 2 :

(2.21) ∑ β = 1 N ∫ A k i + 1 , ρ i + 1 β ( v β − k i + 1 ) p d x ≤ c 2 ( i + 1 ) p ( 1 + p ⁄ n ) ( R − r ) p d p p ⁄ n ∑ β = 1 N ∫ A k i , ρ i β ( v β − k i ) p d x 1 + p n + c 2 ( i + 1 ) ( p p ⁄ n + p θ ) d p p ⁄ n + p θ ∑ β = 1 N ∫ A k i , ρ i β ( v β − k i ) p d x θ + p n .

Let us set

J i = ∑ β = 1 N ∫ A k i , ρ i β ( v β − k i ) p d x , i = 0 , 1 , 2 , … .

Since

J i + 1 = ∑ β = 1 N ∫ A k i + 1 , ρ i + 1 β ( v β − k i + 1 ) p d x ≤ ∑ β = 1 N ∫ A k i , ρ i + 1 β ( v β − k i + 1 ) p d x ≤ ∑ β = 1 N ∫ A k i , ρ i β ( v β − k i + 1 ) p d x ≤ ∑ β = 1 N ∫ A k i , ρ i β ( v β − k i ) p d x = J i ,

{ J i } is a decreasing sequence. Note that d > L 0 ≥ 0 , so when v β > d , we have v β − d ≤ v β = max { v β ; 0 } ; then

(2.22) J 0 = ∑ β = 1 N ∫ A d , R β ( v β − d ) p d x ≤ ∑ β = 1 N ∫ A d , R β ( max { v β ; 0 } ) p d x ≤ ∑ β = 1 N ∫ B R ( x 0 ) max { v β ; 0 } p d x ≔ T .

We use the aforementioned number T , and we keep in mind that θ ≤ 1 , R − r < 1 and d > 1 , then (2.21) yields

J i + 1 ≤ c ( R − r ) p d p p ⁄ n 2 1 + p n p i J i 1 + p n + c d p p ⁄ n + p θ 2 p n + θ p i J i θ + p n ≤ c ( R − r ) p d p p ⁄ n 2 1 + p n p i J i θ + p n T 1 − θ + c ( R − r ) p d p p ⁄ n 2 1 + p n p i J i θ + p n ≤ c ( R − r ) p d p p ⁄ n 2 1 + p n p i J i θ + p n ,

where c is a constant depending upon n , N , p , θ , c 1 , ∣ Ω ∣ , and T . We would like to use Lemma 2.3 to obtain

(2.23) lim i → ∞ J i = 0 ,

this is true provided that

(2.24) θ + p n > 1

and

(2.25) J 0 ≤ c ( R − r ) p d p p ⁄ n − 1 θ + p n − 1 2 1 + p n p − 1 θ + p n − 1 2 .

(2.24) is easy since θ > 1 − p n . Let us try to check (2.25). Since J 0 ≤ T by (2.22), we obtain the following sufficient condition when checking (2.25):

(2.26) ∑ β = 1 N ∫ B R ( x 0 ) max { v β ; 0 } p d x ≤ c ( R − r ) p d p p ⁄ n − 1 θ + p n − 1 2 1 + p n p − 1 θ + p n − 1 2 .

Then, we fix d verifying (2.26) and d > max { L 0 , 1 } ; then, (2.25) is satisfied and (2.23) holds true. It is obvious that such a constant d depends upon n , N , p , θ , c 1 , ∣ Ω ∣ , L 0 , 1 R − r , and T .

We keep in mind that r < ρ i and k i < 2 d , so we can use (2.19) with r 2 = r < ρ i , L = 2 d , and L ˜ = k i :

(2.27) ∫ A 2 d , r β ( v β − 2 d ) p d x ≤ ∫ A k i , r β ( v β − k i ) p d x ≤ ∫ A k i , ρ i β ( v β − k i ) p d x ,

so that

0 ≤ ∑ β = 1 N ∫ A 2 d , r β ( v β − 2 d ) p d x ≤ ∑ β = 1 N ∫ A k i , ρ i β ( v β − k i ) p d x = J i .

Since (2.23) holds true, we have, by Lemma 2.3, lim i → ∞ J i = 0 , so

∑ β = 1 N ∫ A 2 d , r β ( v β − 2 d ) p d x = 0 ,

this mean that ∣ A 2 d , r β ∣ = 0 , so that

v β ≤ 2 d , a.e. in B r ( x 0 ) .

This completes the proof of Lemma 2.2.□

Proof of Theorem 2.1

Caccioppoli inequality proved in Lemma 2.1 with v β = ∣ u β ∣ and p = 2 allows us to use Lemma 2.2 to derive local boundedness of u α (note that θ = 2 1 ( 2 * ) ′ − 1 m > 1 − 2 n since m > n 2 ). The arbitrariness of α implies that u is locally bounded in Ω .□

2.2 Global boundedness result

Let n > 2 , N ≥ 2 be integers and Ω an open bounded subset of R n . Consider Dirichlet problem of the following quasilinear elliptic systems involving N equations:

(2.28) − ∑ i = 1 n ∂ ∂ x i ∑ β = 1 N ∑ j = 1 n a i , j α , β ( x , u ( x ) ) ∂ u β ( x ) ∂ x j = f α ( x ) , in Ω , u ( x ) = 0 , on ∂ Ω .

where α ∈ { 1 , 2 , … , N } is the equation index.

We let the coefficients a i , j α , β ( x , y ) , i , j ∈ { 1 , … , n } , α , β ∈ { 1 , … , N } , satisfying the set of assumptions ( A ) and

(2.29) f ( x ) = ( f 1 ( x ) , … , f N ( x ) ) ∈ L m ( Ω ; R N ) , m ≥ ( 2 * ) ′ .

Definition 2.2

A function u ∈ W 0 1 , 2 ( Ω ; R N ) is a global solution to (2.28), if

(2.30) ∫ Ω ∑ α , β = 1 N ∑ i , j = 1 n a i , j α , β ( x , u ( x ) ) D j u β ( x ) D i φ α ( x ) d x = ∫ Ω ∑ α = 1 N f α ( x ) φ α ( x ) d x

holds true for all φ ∈ W 0 1 , 2 ( Ω ; R N ) .

Next we prove that if the right-hand side function f is good enough, then the global solution to (2.28) is globally bounded.

Theorem 2.2

Suppose that u is a weak solution of (2.28). Under the set of assumptions ( A ), if f ∈ L m ( Ω ; R N ) , m > n 2 , then any global solution u ∈ W 0 1 , 2 ( Ω ; R N ) to (2.28) is globally bounded.

Proof

We take for any k > 0 , φ = ( φ 1 , … , φ N ) with

φ α = ∑ γ = 1 N C α γ G k ( u γ ) , α ∈ { 1 , … , N } ,

where G k ( s ) is defined in (2.6), and C α γ , α , γ ∈ { 1 , … , N } , are the real constants satisfying (2.12). It is obvious that

D i φ α = ∑ γ = 1 N C α γ D i u γ χ A k γ , i = 1 , … , n .

Such a function φ is admissible for Definition 2.2 since it belongs to W 0 1 , 2 ( Ω ; R N ) . We use φ in (2.30) and we have

(2.31) ∫ Ω ∑ α , β = 1 N ∑ i , j = 1 n a i , j α , β ( x , u ( x ) ) D j u β ( x ) ∑ γ = 1 N C α γ D i u γ χ A k γ d x = ∫ Ω ∑ α = 1 N f α ∑ γ = 1 N C α γ G k ( u γ ) d x .

We compare the left-hand side of (2.31) with the left-hand side of (2.7), and we find that the only difference between them is a function η 2 . We use the method from (2.8) to (2.11) and we have that

(2.32) ∫ Ω ∑ α , β = 1 N ∑ i , j = 1 n a i , j α , β ( x , u ( x ) ) D j u β ( x ) ∑ γ = 1 N C α γ D i u γ χ A k γ d x ≥ c 0 ∑ β = 1 N ∫ A k β ∣ D u β ∣ 2 d x .

In order to estimate the right-hand side of (2.31), we use Hölder’s inequality and Sobolev embedding theorem to derive

(2.33) ∫ Ω ∑ α = 1 N f α ∑ γ = 1 N C α γ G k ( u γ ) d x ≤ c ¯ ∫ Ω ∑ α = 1 N ∣ f α ∣ ∑ β = 1 N ∣ G k ( u β ) ∣ d x ≤ c ¯ ∑ β = 1 N ∫ A k β ∑ α = 1 N ∣ f α ∣ ( 2 * ) ′ d x 1 ( 2 * ) ′ ∫ A k β ∣ G k ( u β ) ∣ 2 * d x 1 2 * ≤ c ∑ β = 1 N ∫ A k β ∑ α = 1 N ∣ f α ∣ ( 2 * ) ′ d x 1 ( 2 * ) ′ ∫ A k β ∣ D u β ∣ 2 d x 1 2 ,

where C α γ , α , γ = 1 , … , N , are solutions to (2.13) and c ¯ = ∑ α , γ = 1 N ∣ C α γ ∣ is a constant.

Substituting (2.32) and (2.33) into (2.31), we arrive at

∑ β = 1 N ∫ A k β ∣ D u β ∣ 2 d x ≤ c ∑ β = 1 N ∫ A k β ∑ α = 1 N ∣ f α ∣ ( 2 * ) ′ d x 2 ( 2 * ) ′ .

Hölder’s inequality gives

(2.34) ∑ β = 1 N ∫ A k β ∣ D u β ∣ 2 d x ≤ c ∑ β = 1 N ∫ A k β ∑ α = 1 N ∣ f α ∣ m d x ( 2 * ) ′ m ∣ A k β ∣ 1 − ( 2 * ) ′ m 2 ( 2 * ) ′ ≤ c ∑ β = 1 N ∣ A k β ∣ 2 ( 2 * ) ′ − 2 m ,

where c is a constant depending upon n , N , m , c 0 , c ˜ , ‖ f ‖ L m ( Ω ) and r α , β , α , β = 1 , … , N . The left-hand side of the aforementioned inequality can be estimated by Sobolev embedding theorem, for any L > k ,

(2.35) ∑ β = 1 N ∫ A k β ∣ D u β ∣ 2 d x = ∑ β = 1 N ∫ Ω ∣ D G k ( u β ) ∣ 2 d x ≥ c ∑ β = 1 N ∫ Ω ∣ G k ( u β ) ∣ 2 * d x 2 2 * ≥ c ∑ β = 1 N ∫ A L β ∣ G k ( u β ) ∣ 2 * d x 2 2 * ≥ c ( L − k ) 2 ∑ β = 1 N ∣ A L β ∣ 2 2 * ≥ c ( L − k ) 2 ∑ β = 1 N ∣ A L β ∣ 2 2 * .

(2.34) and (2.35) merge into

∑ β = 1 N ∣ A L β ∣ ≤ c ( L − k ) 2 * ∑ β = 1 N ∣ A k β ∣ 2 ( 2 * ) ′ − 2 m 2 * 2 ,

for every L , k with L > k > 0 . We let

ψ ( t ) = ∑ β = 1 N ∣ A t β ∣ = ∑ β = 1 N ∣ { ∣ u β ( x ) ∣ > t } ∣ .

We use the following Stampacchia Lemma, see [35, Lemma 4.1], which we provide below for the convenience of the reader.

Lemma 2.4

Let ψ ( k ) : [ k 0 , + ∞ ) → [ 0 , + ∞ ) be decreasing. We assume that there exists c , α ∈ ( 0 , + ∞ ) and β ∈ ( 1 , + ∞ ) such that

L > k ≥ k 0 ⇒ ψ ( L ) ≤ c ( L − k ) α [ ψ ( k ) ] β .

Then, it results that ψ ( k 0 + d ) = 0 , where

d = c ( ψ ( k 0 ) ) β − 1 2 α β β − 1 1 α .

Since m > n 2 implies β = 2 ( 2 * ) ′ − 2 m 2 * 2 > 1 , we use Lemma 2.4 and we have

∑ β = 1 N ∣ { ∣ u β ( x ) ∣ > d } ∣ = 0 ,

almost everywhere in Ω , which implies the desired result ∣ u β ( x ) ∣ ≤ d , a.e. Ω , β ∈ { 1 , … , N } .□

3 Boundedness under ( A ) ′

This section deals with global boundedness for solutions to elliptic systems (1.1) under the set of assumptions ( A ) ′ . We also consider regularizing effect of a lower-order term.

3.1 Global boundedness result

In this section, we also consider Dirichlet problem of quasilinear elliptic systems involving N equations of the form (2.28). We let the coefficients a i , j α , β ( x , y ) , i , j ∈ { 1 , … , n } , α , β ∈ { 1 , … , N } , satisfying the set of assumptions ( A ) ′ and f satisfying (2.29). The definition for a global solution u ∈ W 0 1 , 2 ( Ω ; R N ) to (2.28) is the same as Definition 2.2.

Next we prove that, if the right-hand side function f is good enough, then the global solution to (2.28) is globally bounded.

Theorem 3.1

Suppose that u ∈ W 0 1 , 2 ( Ω ; R N ) is a weak solution of (2.28). Under the set of assumptions ( A ) ′ , if f = ( f 1 , … , f N ) ∈ L m ( Ω ; R N ) , m > n 2 , then any global solution u ∈ W 0 1 , 2 ( Ω ; R N ) to (2.28) is globally bounded.

Proof

Let u ∈ W 0 1 , 2 ( Ω ; R N ) be a global solution to (2.28). For every L ≥ Q 0 , we define φ = ( φ 1 , … , φ N ) with

φ α = G L ( u α ) ,

then

D i φ α = D i u α χ A L α .

Such a φ is admissible for Definition 2.2 since φ ∈ W 0 1 , 2 ( Ω ; R N ) . We use φ in (2.30) and we have

∫ Ω ∑ α , β = 1 N ∑ i , j = 1 n a i , j α , β ( x , u ) D j u β ( x ) D i u α χ A L α d x = ∫ Ω ∑ α = 1 N f α G L ( u α ) d x .

Now, assumption ( A 4 ) ′ guarantees that

a i , j α , β ( x , u ) χ A L α = a i , j α , β ( x , u ) χ A L α χ A L β

when β ≠ α and L ≥ Q 0 . It is worthwhile to note that (3.1) holds true when α = β as well; then,

∫ Ω ∑ α , β = 1 N ∑ i , j = 1 n a i , j α , β ( x , u ) D j u β χ A L β D i u α χ A L α d x = ∫ Ω ∑ α = 1 N f α G L ( u α ) d x .

Now, we can use ellipticity assumption ( A 3 ) ′ with ξ i α = D i u α χ A L α and we obtain

(3.1) c ˜ 0 ∑ α = 1 N ∫ A L α ∣ D u α ∣ 2 d x ≤ ∫ Ω ∑ α = 1 N f α G L ( u α ) d x .

In order to estimate the right-hand side of (3.1), we use Hölder’s inequality and Sobolev embedding theorem to derive

(3.2) ∫ Ω ∑ α = 1 N f α G L ( u α ) d x ≤ ∑ α = 1 N ∫ Ω ∣ f α ∣ ∣ G L ( u α ) ∣ d x ≤ ∑ α = 1 N ∫ A L α ∣ f α ∣ ( 2 * ) ′ d x 1 ( 2 * ) ′ ∫ Ω ∣ G L ( u α ) ∣ 2 * d x 1 2 * ≤ c ∑ α = 1 N ∫ A L α ∣ f α ∣ ( 2 * ) ′ d x 1 ( 2 * ) ′ ∫ A L α ∣ D u α ∣ 2 d x 1 2 ≤ c ∑ α = 1 N ∫ A L α ∣ f α ∣ ( 2 * ) ′ d x 1 ( 2 * ) ′ ∑ β = 1 N ∫ A L β ∣ D u β ∣ 2 d x 1 2 .

Substituting (3.2) into (3.1), we arrive at

∑ α = 1 N ∫ A L α ∣ D u α ∣ 2 d x ≤ c ∑ α = 1 N ∫ A L α ∣ f α ∣ ( 2 * ) ′ d x 2 ( 2 * ) ′ .

Hölder’s inequality gives

(3.3) ∑ α = 1 N ∫ A L α ∣ D u α ∣ 2 d x ≤ c ∑ α = 1 N ∫ A L α ∣ f α ∣ m d x ( 2 * ) ′ m ∣ A L α ∣ 1 − ( 2 * ) ′ m 2 ( 2 * ) ′ ≤ c ∑ α = 1 N ∣ A L α ∣ 2 ( 2 * ) ′ − 2 m ,

where c is a constant depending upon n , N , c ˜ 0 , and ‖ f ‖ L m ( Ω ) . The left-hand side of the aforementioned inequality can be estimated by Sobolev embedding theorem: for any L ˜ > L ,

(3.4) ∑ α = 1 N ∫ A L α ∣ D u α ∣ 2 d x = ∑ α = 1 N ∫ Ω ∣ D G L ( u α ) ∣ 2 d x ≥ c ∑ α = 1 N ∫ Ω ∣ G L ( u α ) ∣ 2 * d x 2 2 * ≥ c ∑ α = 1 N ∫ A L ˜ α ∣ G L ( u α ) ∣ 2 * d x 2 2 * ≥ c ( L ˜ − L ) 2 ∑ α = 1 N ∣ A L ˜ α ∣ 2 2 * ≥ c ( L ˜ − L ) 2 ∑ α = 1 N ∣ A L ˜ α ∣ 2 2 * .

(3.3) and (3.4) merge into

∑ α = 1 N ∣ A L ˜ α ∣ ≤ c ( L ˜ − L ) 2 * ∑ α = 1 N ∣ A L α ∣ 2 ( 2 * ) ′ − 2 m 2 * 2 ,

for every L ˜ , L with L ˜ > L ≥ Q 0 . We let

ψ ( t ) = ∑ α = 1 N ∣ A t α ∣ = ∑ α = 1 N ∣ { ∣ u α ( x ) ∣ > t } ∣ .

We use the Stampacchia Lemma 2.4 and we keep in mind that m > n 2 implies β = 2 ( 2 * ) ′ − 2 m 2 * 2 > 1 , then

∑ α = 1 N ∣ { ∣ u α ( x ) ∣ > Q 0 + d } ∣ = 0 ,

which implies the desired result ∣ u α ( x ) ∣ ≤ Q 0 + d , a.e. Ω , α ∈ { 1 , … , N } .□

Remark 3.1

We note that, in [27], the authors considered the elliptic system (2.28) with f α ( x ) = 0 , a.e. Ω , α = 1 , … , N . Under the assumptions ( A 1 ), ( A 2 ), ( A 3 ) ′ and support of off-diagonal coefficients (see [27, Figure 1]), the authors derives a local boundedness result by using De Giorgi’s iterative method. We mention that the support of off-diagonal coefficients in [27] is contained in the “butterfly support” (compare [27, Figure 1] with [27, Figure 1]); thus, the condition ( A ) ′ in this article is weaker than the one proposed in [27]. Of course, generally, dealing with local boundedness requires more skills than global ones. Existence and boundedness results of weak solutions to some vectorial Dirichlet problems of elliptic systems can be found in the recent article [13].

3.2 Regularizing effect

In this section, we concentrate ourselves to regularizing effect of a lower-order term. A good reference in this field is the article [1] by Arcoya and Boccardo, where the authors studied the regularizing effect of the interaction between the coefficient of the zero-order term and the datum in some linear, semilinear and nonlinear Dirichlet problems. For other results related to regularizing effect, we refer to [7, Section 11.8] and the recent articles [2,3,11].

We next show that there is also regularizing effect of a lower-order term for elliptic systems. More precisely, let n > 2 , N ≥ 2 be integers and Ω an open bounded subset of R n . We consider quasilinear elliptic systems involving N equations of the form

(3.5) − ∑ i = 1 n ∂ ∂ x i ∑ β = 1 N ∑ j = 1 n a i , j α , β ( x , u ( x ) ) ∂ u β ( x ) ∂ x j + b α ( x ) u α ( x ) = f α , in Ω , u ( x ) = 0 , on ∂ Ω ,

where α ∈ { 1 , 2 , … , N } is the equation index. The difference between (3.5) and (2.28) is that there is a lower-order term b α ( x ) u α ( x ) in the left-hand side of (3.5).

We assume that the coefficients a i , j α , β satisfy ( A ) ′ . For the functions f α ( x ) and b α ( x ) , α = 1 , … , N , we assume

(3.6) 0 ≤ b α ( x ) ∈ L n 2 ( Ω ) ,

(3.7) ∣ f α ( x ) ∣ ≤ Q b α ( x ) , for some Q ≥ Q 0 .

Definition 3.1

We say that a function u ∈ W 0 1 , 2 ( Ω ; R N ) is a global solution with respect to (3.5), if

(3.8) ∫ Ω ∑ α , β = 1 N ∑ i , j = 1 n a i , j α , β ( x , u ( x ) ) D j u β ( x ) D i φ α ( x ) d x + ∫ Ω ∑ α = 1 N b α ( x ) u α ( x ) φ α ( x ) = ∫ Ω ∑ α = 1 N f α ( x ) φ α ( x ) d x

holds true for all φ ∈ W 0 1 , 2 ( Ω ; R N ) .

We remark that conditions (3.6) and (3.7) guarantee integrability of the second and third integrands in (3.8).

Theorem 3.1 tells us that, in order to guarantee boundedness of solutions to (2.28), we need f ∈ L m ( Ω ) , m > n 2 . From (3.6) and (3.7), we know that, f ∈ L n 2 ( Ω ) . The next theorem shows that there is a regularizing effect of (3.7), which forces global boundedness of solutions to (3.5).

Theorem 3.2

Assume ( A ) ′ , (3.6), and (3.7), then a solution u ∈ W 0 1 , 2 ( Ω ; R N ) of system (3.5) is bounded. Moreover,

‖ u ‖ L ∞ ( Ω ; R N ) ≤ Q N .

Proof

Let u ∈ W 0 1 , 2 ( Ω ; R N ) be a global solution of system (3.5). We take a test function φ = ( φ 1 , … , φ N ) with

(3.9) φ α = G Q ( u α ) , α ∈ { 1 , … , N } .

Such a function φ is admissible for Definition 3.1 since it belongs to W 0 1 , 2 ( Ω ; R N ) . We use such a test function in the weak formulation (3.8) and we have

(3.10) ∫ Ω ∑ α , β = 1 N ∑ i , j = 1 n a i , j α , β ( x , u ) D j u β D i G Q ( u α ) d x + ∑ α = 1 N ∫ Ω b α u α G Q ( u α ) d x = ∑ α = 1 N ∫ Ω f α G Q ( u α ) d x .

For the first integral in the right-hand side of (3.10), we use ( A 4 ) ′ and ( A 3 ) ′ to derive

(3.11) ∫ Ω ∑ α , β = 1 N ∑ i , j = 1 n a i , j α , β ( x , u ) D j u β D i G Q ( u α ) d x = ∫ Ω ∑ α , β = 1 N ∑ i , j = 1 n a i , j α , β ( x , u ) D j u β D i u α χ A Q α d x = ∫ Ω ∑ α , β = 1 N ∑ i , j = 1 n a i , j α , β ( x , u ) D j u β χ A Q β D i u α χ A Q α d x ≥ c 0 ∑ α = 1 N ∫ A Q α ∣ D u α ∣ 2 d x ≥ 0 .

Using (3.7), we obtain

(3.12) ∑ α = 1 N ∫ Ω f α ( x ) G Q ( u α ) d x ≤ ∑ α = 1 N ∫ Ω ∣ f α ( x ) ∣ ∣ G Q ( u α ) ∣ d x ≤ ∑ α = 1 N ∫ Ω Q b α ( x ) ∣ G Q ( u α ) ∣ d x .

Combining (3.10)–(3.12), and noting u α ( x ) G Q ( u α ) = ∣ u α ( x ) ∣ ∣ G Q ( u α ) ∣ , one obtains

(3.13) ∑ α = 1 N ∫ Ω b α ( x ) ∣ G Q ( u α ) ∣ ( ∣ u α ( x ) ∣ − Q ) d x ≤ 0 ,

from which we derive

∣ u α ( x ) ∣ ≤ Q , a.e. Ω ,

we thus have derived that u ∈ L ∞ ( Ω ) and

‖ u ‖ L ∞ ( Ω ) ≤ N Q .

This completes the proof of Theorem 3.2.□

Acknowledgments

All authors would like to thank the anonymous referees for their detailed comments and helpful suggestions.

  1. Funding information: The corresponding author thanks NSFC (Grant No. 12071021), NSF of Hebei Province (Grant No. A2024201024) and the Innovation Capacity Enhancement Program-Science and Technology Platform Project, Hebei Province (Grant No. 22567623H) for the support.

  2. Author contributions: The authors contributed equally to the preparation, the revision, and the writing of the manuscript.

  3. Conflict of interest: The authors state no conflict of interest.

  4. Data availability statement: No data were used for the research described in the article.

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Received: 2024-10-29
Revised: 2025-06-02
Accepted: 2025-07-15
Published Online: 2025-09-13

© 2025 the author(s), published by De Gruyter

This work is licensed under the Creative Commons Attribution 4.0 International License.

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https://doi.org/10.1515/anona-2025-0108
Keywords for this article
quasilinear elliptic system; proportional condition; “butterfly” support; regularizing effect; boundedness
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