Monatsh Math DOI 10.1007/s00605-015-0874-9

Extremal solutions for nonvariational quasilinear elliptic systems via expanding trapping regions Siegfried Carl1 · Dumitru Motreanu2

Received: 6 December 2015 / Accepted: 28 December 2015 © Springer-Verlag Wien 2016

Abstract We consider the Dirichlet boundary value problem for quasilinear elliptic systems in a bounded domain  ⊂ R N with a diagonal ( p1 , p2 )-Laplacian as leading differential operator of the form − pi u i = f i (x, u 1 , u 2 , ∇u 1 , ∇u 2 ) in , u i = 0 on ∂, where the component functions f i (i = 1, 2) of the lower order vector field may also depend on the gradient of the solution u = (u 1 , u 2 ). The main goal of this paper is twofold. First, we establish an enclosure and existence result by means of the trapping region which is formed by pairs of appropriately defined sub-supersolutions. Second, by a suitable construction of sequences of expanding trapping regions we are able to prove the existence of extremal positive and negative solutions of the system. The theory of pseudomonotone operators, regularity results due to Cianchi-Maz’ya, as well as a strong maximum principle due to Pucci-Serrin are essential tools in the proofs. Keywords Quasilinear elliptic system · Pseudomonotone operator · Trapping region · Enclosure and comparison principle · Minimal and maximal solution

Communicated by A. Constantin.

B

Siegfried Carl [email protected] Dumitru Motreanu [email protected]

1

Institut für Mathematik, Martin-Luther-Universität, Halle-Wittenberg, 06099 Halle, Germany

2

Département de Mathématiques, Université de Perpignan, 52 Avenue de Villeneuve, 66860 Perpignan, France

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S. Carl, D. Motreanu

35J57 · 35J92 · 35B50 · 35B65

Mathematics Subject Classification

1 Introduction Let  ⊂ R N be a bounded domain with smooth boundary ∂, and let 1 < pi < ∞, i = 1, 2. As we are going to make use of regularity results for quasilinear elliptic equations due to Lieberman (see [9]) and Cianchi-Maz’ya (see [6]), as well as of a strong maximum principle due to Pucci-Serrin (see [11, Theorem 5.5.1]), we assume ∂ to be a C 2 -boundary. In this paper we study the following system of quasilinear elliptic equations with homogeneous Dirichlet boundary condition ⎧ ⎨ − p1 u 1 = f 1 (x, u 1 , u 2 , ∇u 1 , ∇u 2 ) − p2 u 2 = f 2 (x, u 1 , u 2 , ∇u 1 , ∇u 2 ) ⎩ u 1 = u 2 = 0,

in  in  on ∂,

(1.1)

where  pi u = div(|∇u| pi −2 ∇u) is the pi -Laplacian operator for i = 1, 2, and the right-hand side lower order vector field ( f 1 , f 2 ) :  × R × R × R N × R N → R2 of (1.1) is supposed to be a Carathéodory vector field, i.e., x → f i (x, s1 , s2 , ξ1 , ξ2 ) is measurable in  for all (s1 , s2 , ξ1 , ξ2 ) ∈ R × R × R N × R N , and (s1 , s2 , ξ1 , ξ2 ) → f i (x, s1 , s2 , ξ1 , ξ2 ) is continuous in R × R × R N × R N for a.a. x ∈ . 1, p Let W 1, pi () denote the usual Sobolev space, and denote by W0 i () the subspace whose elements have generalized homogeneous boundary values. We denote by  · 1, p  L pi () the usual norm of L pi () and by  · 1, pi that of W 1, pi (), and for W0 i () we make use of the equivalent norm given by u1, pi = ∇u L pi () . Further we will use the Banach space C01 () whose positive cone is given by   C01 ()+ = u ∈ C01 () : u(x) ≥ 0 for all x ∈  . It is known that its interior int(C01 ()+ ) is nonempty and is characterized as follows int(C01 ()+ )

  ∂u 1 < 0 on ∂ , = u ∈ C0 ()+ : u(x) > 0 for all x ∈  and ∂ν

pi where ∂u ∂ν stands for the (exterior) normal derivative on ∂. We equip L () with the natural partial ordering defined by the positive cone: p

L +i () = {u ∈ L pi () : u(x) ≥ 0 a.e. in }, which implies a corresponding partial ordering in its subspaces W 1, pi () and 1, p W0 i (), and the positive cone p

p

L+ = L +1 () × L +2 ()

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Extremal solutions for nonvariational quasilinear. . .

induces the componentwise partial ordering in the product space L = L p1 () × L p2 (), with norm uL = u 1  L p1 () + u 2  L p2 () which yields a corresponding componentwise partial ordering in the subspaces 1, p1

V = W 1, p1 () × W 1, p2 (), V0 = W0

1, p2

() × W0

(),

with norm uV = u 1 1, p1 + u 2 1, p2 . Definition 1.1 A weak solution of (1.1) is any pair (u 1 , u 2 ) ∈ V0 such that  

|∇u 1 |

p1 −2

∇u 1 ∇v1 d x =

|∇u 2 | p2 −2 ∇u 2 ∇v2 d x =

 

f 1 (x, u 1 , u 2 , ∇u 1 , ∇u 2 )v1 d x, f 2 (x, u 1 , u 2 , ∇u 1 , ∇u 2 )v2 d x

for all (v1 , v2 ) ∈ V0 . We say that a solution (u 1 , u 2 ) is positive (resp. negative) if u 1 (x) > 0 and u 2 (x) > 0 for a.a. x ∈  (resp. u 1 (x) < 0 and u 2 (x) < 0 for a.a. x ∈ ), and a solution (u 1 , u 2 ) is called nontrivial if (u 1 , u 2 ) = (0, 0). The notion of trapping region based on an appropriately defined notion of subsupersolution for systems plays an important role in what follows. As we don’t impose any further structure conditions on the right-hand side vector field of (1.1), unlike in the scalar case sub-supersolutions for the system are mutually dependent. Definition 1.2 We say that (u 1 , u 2 ), (u 1 , u 2 ) ∈ V form a pair of sub-and supersolution for problem (1.1) if u i ≤ u i a.e. in , u i ≤ 0 ≤ u i a.e. on ∂ for i = 1, 2, and

 |∇u 1 | p1 −2 ∇u 1 ∇v1 − f 1 (x, u 1 , w2 , ∇u 1 , ∇w2 )v1 d x 

 + |∇u 2 | p2 −2 ∇u 2 ∇v2 − f 2 (x, w1 , u 2 , ∇w1 , ∇u 2 )v2 d x ≤ 0, 

and

 |∇u 1 | p1 −2 ∇u 1 ∇v1 − f 1 (x, u 1 , w2 , ∇u 1 , ∇w2 )v1 d x 

 + |∇u 2 | p2 −2 ∇u 2 ∇v2 − f 2 (x, w1 , u 2 , ∇w1 , ∇u 2 )v2 d x ≥ 0 

for all (v1 , v2 ) ∈ V0 ∩ L+ , and all (w1 , w2 ) ∈ V with u i ≤ wi ≤ u i for i = 1, 2. If u = (u 1 , u 2 ), u = (u 1 , u 2 ) is a pair of sub-and supersolution, then the order interval [u, u] = [u 1 , u 1 ] × [u 2 , u 2 ] is called a trapping region. We have denoted [u i , u i ] = {u ∈ W 1, pi () : u i ≤ u ≤ u i a.e. in }.

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Remark 1 The two inequalities of Definition 1.2 are equivalent to the following 4 inequalities (in their respective corresponding weak form): − p1 u 1 − f 1 (x, u 1 , w2 , ∇u 1 , ∇w2 ) ≤ 0, for all w2 ∈ [u 2 , u 2 ], − p2 u 2 − f 2 (x, w1 , u 2 , ∇w1 , ∇u 2 ) ≤ 0, for all w1 ∈ [u 1 , u 1 ], − p1 u 1 − f 1 (x, u 1 , w2 , ∇u 1 , ∇w2 ) ≥ 0, for all w2 ∈ [u 2 , u 2 ], − p2 u 2 − f 2 (x, w1 , u 2 , ∇w1 , ∇u 2 ) ≥ 0, for all w1 ∈ [u 1 , u 1 ].

(1.2)

The main goal of this paper is twofold. First, we establish an enclosure and existence result in terms of trapping regions. We should note that this result, which is proved in Sect. 2, cannot be deduced straightforward from earlier results by the authors (see [4, Chapter 5]), since here the leading operators − pi act on different Sobolev spaces W 1, pi (). Second, by a suitable construction of sequences of expanding trapping regions we are able to prove the existence of multiple nontrivial solutions. More precisely, under conditions on the vector field ( f 1 , f 2 ) to be specified later, we show the existence of a positive minimal and a negative maximal solution, where the notion maximal and minimal refer to the partial ordering of vector-valued functions introp p duced by the order cone L+ = L +1 () × L +2 (). Unlike in earlier papers by the authors (see [1–3], here the right-hand sides f i depend, in addition, on the gradients of u i , and therefore variational methods cannot be applied in the study of system (1.1). The main tools we are using are truncation and differential inequality techniques based on the notion of trapping region as well as the theory of pseudomonotone operators, together with regularity results due to Cianchi-Maz’ya (see [6]), as well as of a strong maximum principle due to Pucci-Serrin (see [11, Theorem 5.5.1]). Finally, we should note that the results obtained here can be extended in a straightforward manner to systems of m > 2 equations, and only for the sake of simplifying notion we have restricted our consideration to system (1.1).

2 Enclosure and existence via trapping region Let (u 1 , u 2 ), (u 1 , u 2 ) ∈ V be a pair of sub-supersolution, i.e. [u, u] forms a trapping region. We impose the following hypothesis on the right-hand side vector field: (H1) For i = 1, 2, the functions f i :  × R × R × R N × R N → R are Carathéodory which satisfy the growth conditions

p2  | f 1 (x, s1 , s2 , ξ1 , ξ2 )| ≤ 1 (x) + c1 |ξ1 | p1 −1 + |ξ2 | q1 ,

 p1 | f 2 (x, s1 , s2 , ξ1 , ξ2 )| ≤ 2 (x) + c2 |ξ1 | q2 + |ξ2 | p2 −1 ,

(2.1) (2.2)

for a.a. x ∈ , for all s = (s1 , s2 ) ∈ [u(x), u(x)], and for all ξi ∈ R N , where ci ≥ 0 q are some constants, and i ∈ L +i () with qi > 1 denoting the Hölder conjugate to pi , i.e., 1/ pi + 1/qi = 1. In preparation for the main result of this section, we introduce the following truncation operators Tk , (k = 1, 2):

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Extremal solutions for nonvariational quasilinear. . .

⎧ ⎨ u k (x) (Tk u k )(x) = u k (x) ⎩ u k (x)

if u k (x) > u k (x), if u k (x) ≤ u k (x) ≤ u k (x), if u k (x) < u k (x).

(2.3)

It is well known that the truncation operators Tk : W 1, pk () → W 1, pk () are continuous and bounded. Further, we introduce cut-off functions bk :  × R → R defined by ⎧ ⎨ (s − u k (x)) pk −1 bk (x, s) = 0 ⎩ −(u k (x) − s) pk −1

if s > u k (x), if u k (x) ≤ s ≤ u k (x), if s < u k (x).

(2.4)

One readily verifies that bk is a Carathéodory function satisfying the growth condition (2.5) |bk (x, s)| ≤ ˜ k (x) + c˜k |s| pk −1 q

for a.a. x ∈ , for all s ∈ R, where c˜k ≥ 0 is some constant and ˜ k ∈ L +k (). Further, there are some positive constants a1(k) and a2(k) such that

(k)



p

(k)

bk (x, u k )u k d x ≥ a1 u k  Lkpk () − a2 , ∀ u k ∈ L pk ().

(2.6)

In view of (2.5) the Nemytskij operator Bk defined by Bk u k (x) = bk (x, u k (x)) is well defined and Bk : L pk () → L qk () is continuous and bounded, and thus the operator B defined by Bu = (B1 u 1 , B2 u 2 ) is well defined too and B : L → L∗ = L q1 () × L q2 () is continuous and bounded as well.

(2.7)

Let λ = (λ1 , λ2 ) with λk ≥ 0, and let λ Bu = (λ1 B1 u 1 , λ2 B2 u 2 ). Further, we set F(u, ∇u) = (F1 (u 1 , u 2 , ∇u 1 , ∇u 2 ), F2 (u 1 , u 2 , ∇u 1 , ∇u 2 )), where Fk is the Nemytskij operator generated by f k , which is well defined provided u belongs to the trapping region [u, u], and in view of the growth condition of (H1) we have (2.8) F : [u, u] ⊂ V → L∗ → V ∗ is continuous and bounded. Let Au = (− p1 u 1 , − p2 u 2 ), then the duality pairing Au, v for all u, v ∈ V is given by Au, v =

2 2 − pk u k , vk  = |∇u k | pk −2 ∇u k ∇vk d x, ∀ u, v ∈ V. k=1

k=1 

(2.9)

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Finally, with F(u, ∇u), v (with u, v ∈ V) given by F(u, ∇u), v =

2 k=1

Fk (u, ∇u), vk  =

2 k=1 

Fk (u 1 , u 2 , ∇u 1 , ∇u 2 )vk d x, (2.10)

an equivalent definition for u being a solution of system (1.1) is given in a compact form as follows: u ∈ V0 : Au, v = F(u, ∇u), v, ∀ v ∈ V0 .

(2.11)

Before proving our enclosure and existence result for (1.1) (resp. (2.11)) we recall the notion of Leray–Lions operators (see, e.g., [12, Chapter 4.2]). Definition 2.1 (Leray–Lions operator) Let X be a real, reflexive Banach space. We say that P : X → X ∗ is a Leray–Lions operator if it is bounded and satisfies Pu = A(u, u), for u ∈ X, where A : X × X → X ∗ has the following properties: (i) For any u ∈ X, the mapping v → A(u, v) is bounded and hemicontinuous from X to its dual X ∗ , with A(u, u) − A(u, v), u − v ≥ 0 for v ∈ X ; (ii) For any v ∈ X, the mapping u → A(u, v) is bounded and hemicontinuous from X to its dual X ∗ ; (iii) For any v ∈ X, A(u n , v) converges weakly to A(u, v) in X ∗ if (u n ) ⊂ X is such that u n u in X and A(u n , u n ) − A(u n , u), u n − u → 0; (iv) For any v ∈ X, A(u n , v), u n  converges to F, u if (u n ) ⊂ X is such that u n u in X, and A(u n , v) F in X ∗ . As for the proof of the next theorem, see, e.g., [12]. Theorem 2.1 Every Leray–Lions operator A : X → X ∗ is pseudomonotone. We are going to make use of Theorem 2.1 in the proof of the main result of this section, which is the following existence and enclosure theorem. Theorem 2.2 If [u, u] is a trapping region of (1.1), then under hypothesis (H1) the quasilinear system (1.1) has a solution u ∈ [u, u]. Proof The proof is divided into two steps. First we prove the existence of solutions of an appropriately constructed auxiliary truncated problem. Then by comparison we

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Extremal solutions for nonvariational quasilinear. . .

are going to verify that any solution of that auxiliary problem is in fact a solution of the original problem (1.1). Step 1: Auxiliary truncated problem Let T u = (T1 u 1 , T2 u 2 ), where Tk are truncation operators defined by (2.3), and let F ◦ T u = F(T u, ∇T u). We consider the following auxiliary, truncated problem associated to (2.11) (resp. to (1.1)): u ∈ V0 : Au + λBu, v = F(T u, ∇T u), v, ∀ v ∈ V0 , (2.12) where λ = (λ1 , λ2 ) with λk > 0 will be specified later. We are going to prove the existence of solutions of (2.12) for λk > 0 appropriately chosen. To this end we make use of the theory of pseudomonotone operators, see e.g. [13]. Since (2.12) is nothing else than (2.13) u ∈ V0 : Au + λBu − F ◦ T u = 0 in V0∗ , we need to verify that the operator A+λB−F ◦T : V0 → V0∗ is a continuous, bounded, pseudomonotone, and coercive operator. By the compact embedding V0 → → L, and taking into account (2.7), it follows that B : V0 → V0∗ is bounded and completely continuous.

(2.14)

Since T : V0 → [u, u] is continuous and bounded, by (2.8) we readily see that

and thus

F ◦ T : V0 → V0∗ is bounded and continuous,

(2.15)

Pu := A − F ◦ T : V0 → V0∗ is bounded and continuous.

(2.16)

By means of Theorem 2.1 we next show that Pu := A − F ◦ T : V0 → V0∗ is pseudomonotone.

(2.17)

To this end we let Pu = A(u, u) where A : V0 × V0 → V0∗ is defined by A(u, v) := Av − F ◦ T u.

(2.18)

Let us check properties (i)–(iv) of Definition 2.1 for Leray–Lions operators. Property (i) is satisfied, because A : V0 → V0∗ is monotone (even strictly monotone). To check (ii), let v ∈ V0 be fixed. Then u → A(u, v) is bounded and continuous due to (2.15), which implies (ii). In (iii) we need to show that for fixed v ∈ V0 we have A(u n , v) A(u, v) in V0∗

(2.19)

u n u in V0 , and A(u n , u n ) − A(u n , u), u n − u → 0.

(2.20)

provided that

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From (2.20) we get Au n − Au, u n − u → 0 which by u n u yields Au n , u n − u → 0. Since A enjoys the (S)+ -property (see e.g.[4, p. 40]), we infer that u n → u (strongly) in V0 , which by means of (2.15) and (2.18) implies (2.19). Finally, let us verify (iv) of Definition 2.1. For fixed v ∈ V0 , we need to show (D ∈ V0∗ ) A(u n , v), u n  → D, u

(2.21)

(u n ) ⊂ V0 , u n u in V0 , and A(u n , v) D in V0∗ .

(2.22)

A(u n , v), u n  = A(u n , v), u + A(u n , v), u n − u.

(2.23)

provided that

Consider

For the first term on the right-hand side of (2.23) we get by means of (2.22) A(u n , v), u → D, u. As for the second term on the right-hand side of (2.23), by definition (2.18) we have A(u n , v) := Av − F ◦ T u n , and thus A(u n , v), u n − u = Av − F ◦ T u n , u n − u → 0, because due to u n u it follows Av, u n − u → 0. Moreover, (u n ) is bounded in V0 and u n → u in L, which yields in view of (2.10) with u = (u 1 , u 2 ) and u n = (u n,1 , u n,2 ) F ◦ T u n , u n − u =

2 k=1 

Fk (T u n , ∇T u n )(u n,k − u k ) d x → 0,

where we have used that (Fk (T u n , ∇T u n )) is bounded in L qk () in view of (H1) and (u n ) bounded in V0 . Therefore (2.21) holds true which completes the proof for P being a pseudomonotone operator. Finally, since the operator A + λB − F ◦ T = P + λB is the sum of a pseudomonotone operator P and a compact operator λ B, it follows that A + λB − F ◦ T is in fact a pseudomonotone operator. To apply the surjectivity result for pseudomonotone operators and thus the existence of solutions for (2.13), it remains to show that A + λB − F ◦ T is coercive, i.e., we need to verify that the following holds true: (A + λB − F ◦ T )u, u → 0 as uV0 → ∞. (2.24) uV0 From (2.9) we readily obtain p

p

Au, u = ∇u 1  L1p1 () + ∇u 2  L2p2 ()

(2.25)

Applying inequality (2.6), we get (1)

p

(2)

p

(1)

(2)

λBu, u ≥ λ1 a1 u 1  L1p1 () + λ2 a1 u 2  L2p2 () − (λ1 a2 + λ2 a2 ).

123

(2.26)

Extremal solutions for nonvariational quasilinear. . .

By means of (H1) and (2.10) and applying Young’s inequality we get for any positive εk the following estimate |F ◦ T u, u| ≤

2 k=1 

|Fk (T1 u 1 , T2 u 2 , ∇T1 u 1 , ∇T2 u 2 )u k |d x

 p2 p1 −1 q 1 |u 1 | d x ≤ (1 + c1 |∇T1 u 1 | + |∇T2 u 2 | 

 p1 p2 −1 q 2 |u 2 | d x (2 + c2 |∇T1 u 1 | + |∇T2 u 2 | +      ≤ 1  L q1 () + c˜1 u 1  L p1 () + 2  L q2 () + c˜2 u 2  L p2 ()

p

p

+ ε1 ∇u 1  L1p1 () + ε2 ∇u 2  L2p2 () p

p

+ C(ε1 )u 1  L1p1 () + C(ε2 )u 2  L2p2 () ,

(2.27)

where C(εk ) are some positive constants depending only on εk > 0. We note that for the estimate (2.27) we have taken into account the following estimate of the terms involving the gradients of the truncated functions: 



|∇T1 u 1 | p1 −1 |u 1 | d x =

{u 1 ≤u 1 ≤u 1 }



+ + ≤



{u 1 >u 1 } {u 1
|∇u 1 | p1 −1 |u 1 | d x

|∇u 1 | p1 −1 |u 1 | d x |∇u 1 | p1 −1 |u 1 | d x

|∇u 1 | p1 −1 |u 1 | d x + c1 u 1  L p1 () + c1 u 1  L p1 () p

p

≤ c˜1 u 1  L p1 () + ε1 ∇u 1  L1p1 () + C(ε1 )u 1  L1p1 () . In a similar way we estimate the terms 

p2

|∇T2 u 2 | q1 |u 1 | d x;





p1

|∇T1 u 1 | q2 |u 2 | d x; and



|∇T2 u 2 | p2 −1 |u 2 | d x.

  Taking into account (2.25), (2.26), and (2.27), and setting Ck = k  L qk () + c˜k , we finally get the following estimate p

p

(A + λB − F ◦ T )u, u ≥ (1 − ε1 )∇u 1  L1p1 () + (1 − ε2 )∇u 2  L2p2 () (1)

p

(2)

p

+ (λ1 a1 − C(ε1 )) u 1  L1p1 () + (λ2 a1 − C(ε2 )) u 2  L2p2 () − (C1 u 1  L p1 () + C2 u 2  L p2 () ) (1)

(2)

− (λ1 a2 + λ2 a2 ).

(2.28)

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Now we specify εk > 0 such that εk < 1, and then we may choose λk > 0 such that (k)

(λk a1 − C(εk )) > 0, k = 1, 2,

(2.29)

which in view of (2.28) implies that the operator A + λB − F ◦ T is in fact coercive. Now we are in the position to apply the main surjectivity result for pseudomonotone operators (see e.g. [4, Theorem 2.99]) to ensure the existence of solutions of the auxiliary truncated problem (2.12) (resp. (2.13)) for λk > 0 large enough. Step 2: Comparison Here we are going to show that any solution u of the auxiliary truncated problem (2.12) (resp. (2.13)) satisfies u ≤ u ≤ u, which will complete the proof of Theorem 2.2, because then T u = u and Bu = 0, and thus (2.12) coincides with the original problem (1.1). Let u be any solution of (2.12), we are going to show that u ≤ u. By Definition 1.2 we have

 |∇u 1 | p1 −2 ∇u 1 ∇v1 − f 1 (x, u 1 , w2 , ∇u 1 , ∇w2 )v1 d x 

 + |∇u 2 | p2 −2 ∇u 2 ∇v2 − f 2 (x, w1 , u 2 , ∇w1 , ∇u 2 )v2 d x ≥ 0 (2.30) 

for all (v1 , v2 ) ∈ V0 ∩ L+ , and all (w1 , w2 ) ∈ V with u i ≤ wi ≤ u i for i = 1, 2. Subtracting (2.30) from (2.13) and specifying the test function v = (v1 , v2 ) = ((u 1 − u 1 )+ , (u 2 − u 2 )+ ) ∈ V0 ∩ L+ yields

 |∇u 1 | p1 −2 ∇u 1 − |∇u 1 | p1 −2 ∇u 1 ∇(u 1 − u 1 )+ d x 0≥ 

 + |∇u 2 | p2 −2 ∇u 2 − |∇u 2 | p2 −2 ∇u 2 ∇(u 2 − u 2 )+ d x  + +λ1 b1 (x, u 1 )(u 1 − u 1 ) d x + λ2 b2 (x, u 2 )(u 2 − u 2 )+ d x  

 − F1 (T1 u 1 , T2 u 2 , ∇T1 u 1 , ∇T2 u 2 ) − F1 (u 1 , w2 , ∇u 1 , ∇w2 ) (u 1 − u 1 )+ d x 

− F2 (T1 u 1 , T2 u 2 , ∇T1 u 1 , ∇T2 u 2 ) − F2 (w1 , u 2 , ∇w1 , ∇u 2 )(u 2 − u 2 )+ d x 

(2.31) which holds true for all (w1 , w2 ) ∈ V with u i ≤ wi ≤ u i for i = 1, 2. In particular, we may specialize (w1 , w2 ) in (2.31) as w1 = T1 u 1 and w2 = T2 u 2 , which results in that the last two integrals of (2.31) are equal to zero. Since the first two integrals of (2.31) are nonnegative, we obtain from (2.31) the inequality λ1



123

+

b1 (x, u 1 )(u 1 − u 1 ) d x + λ2



b2 (x, u 2 )(u 2 − u 2 )+ d x ≤ 0,

(2.32)

Extremal solutions for nonvariational quasilinear. . .

Note, λk > 0. By definition of bk (see (2.4)), the last inequality can be written as λ1



[(u 1 − u 1 )+ ] p1 d x + λ2



[(u 2 − u 2 )+ ] p2 d x ≤ 0,

(2.33)

which implies (u 1 − u 1 )+ = 0 and (u 2 − u 2 )+ = 0, and thus u k ≤ u k , that is, u ≤ u. In an analogous way we show that for any solution u of the auxiliary problem (2.13)   we get u ≤ u, which completes the proof of the theorem.

3 Positive and negative solutions In this section we are going to show the existence of at least one positive and one negative solution of (1.1) under additional hypotheses on the vector field ( f 1 , f 2 ). 1, p First, let us recall some basic facts about the spectrum of − p on W0 () with 1 < p < ∞ that will be used in the sequel. The nonlinear eigenvalue problem 

− p u = λ|u| p−2 u u=0

in  on ∂

(3.1)

has a first eigenvalue λ1, p > 0, which is isolated, its corresponding eigenspace is one dimensional and λ1, p > 0 admits the variational characterization  λ1, p = inf



p

∇u L p () p

u L p ()

: u∈

1, p W0 (),

u = 0 .

By  u 1, p we denote the corresponding L p -normalized positive eigenfunction, i.e. u 1, p > 0 in , and  u 1, p  L p () = 1.  u 1, p solves (3.1) with λ = λ1, p ,  We impose the following hypotheses on the right-hand side vector field ( f 1 , f 2 ): (H2) The functions f k are Carathéodory as in Sect. 2, and there are constants κ1 > 0, κ2 > 0, d1 < 0, d2 < 0 such that f 1 (x, κ1 , s2 , 0, ξ2 ) ≤ 0 for a.a. x ∈  and all

s2 ∈ [0, κ2 ], and all

ξ2 ∈ R N ,

f 1 (x, d1 , s2 , 0, ξ2 ) ≥ 0 for a.a. x ∈  and all f 2 (x, s1 , κ2 , ξ1 , 0) ≤ 0 for a.a. x ∈  and all f 2 (x, s1 , d2 , ξ1 , 0) ≥ 0 for a.a. x ∈  and all

s2 ∈ [d2 , 0], and all s1 ∈ [0, κ1 ], and all s1 ∈ [d1 , 0], and all

ξ2 ∈ R N , ξ1 ∈ R N , ξ1 ∈ R N .

(H3) There are constants α1 ≥ β1 > λ1, p1 and α2 ≥ β2 > λ1, p2 such that α1 ≥

lim sup

s1 →0+ ,ξ1 →0

f 1 (x, s1 , s2 , ξ1 , ξ2 ) p −1 s1 1

≥

lim inf

s1 →0+ ,ξ1 →0

f 1 (x, s1 , s2 , ξ1 , ξ2 ) p −1

s1 1

≥ β1

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uniformly for a.a. x ∈ , for 0 ≤ s2 ≤ κ2 , and for all ξ2 ∈ R N ; α1 ≥

lim sup

s1 →0− ,ξ1 →0

f 1 (x, s1 , s2 , ξ1 , ξ2 ) f 1 (x, s1 , s2 , ξ1 , ξ2 ) ≥ lim inf ≥ β1 p −2 |s1 | 1 s1 |s1 | p1 −2 s1 s1 →0− ,ξ1 →0

uniformly for a.a. x ∈ , for d2 ≤ s2 ≤ 0, and for all ξ2 ∈ R N ; α2 ≥

lim sup

s2 →0+ ,ξ2 →0

f 2 (x, s1 , s2 , ξ1 , ξ2 ) p −1 s2 2

≥

lim inf

s2 →0+ ,ξ2 →0

f 2 (x, s1 , s2 , ξ1 , ξ2 ) p −1

s2 2

≥ β2

uniformly for a.a. x ∈ , for 0 ≤ s1 ≤ κ1 , and for all ξ1 ∈ R N ; α2 ≥

lim sup

s2 →0− ,ξ2 →0

f 2 (x, s1 , s2 , ξ1 , ξ2 ) f 2 (x, s1 , s2 , ξ1 , ξ2 ) ≥ lim inf ≥ β2 |s2 | p2 −2 s2 |s2 | p2 −2 s2 s2 →0− ,ξ2 →0

uniformly for a.a. x ∈ , for d1 ≤ s1 ≤ 0, and for all ξ1 ∈ R N . Remark 2 Hypotheses (H3) implies that system (1.1) has the trivial solution u = (0, 0). By means of (H2) and (H3) we readily verify that the rectangles [(0, 0), (κ1 , κ2 )] and [(d1 , d2 ), (0, 0)], form trapping regions, which by applying Theorem 2.2, implies the existence of solutions within these trapping regions. However, since u = (0, 0) is a solution belonging to both trapping regions, we don’t get any new information. The purpose of the following theorem is to prove the existence of nontrivial solution by means of Theorem 2.2 and by constructing appropriate trapping regions that do not include the trivial solution. Theorem 3.1 Assume hypotheses (H1)–(H3), where the growth conditions (2.1), (2.2) of (H1) are supposed to hold true for all s = (s1 , s2 ) within the constant rectangle [d, κ] = [d1 , κ1 ] × [d2 , κ2 ]. Then, for ε > 0 small, system (1.1) admits a positive u 1, p1 , κ1 ] × [ε u 1, p2 , κ2 ], and a negative solution (v1 , v2 ) ∈ solution (u 1 , u 2 ) ∈ [ε u 1, p1 ] × [d2 , −ε u 1, p2 ] [d1 , −ε Proof By assumption (H2) we get for a.a. x ∈  

− p1 (κ1 ) − f 1 (x, κ1 , s2 , 0, ξ2 ) ≥ 0, − p2 (κ2 ) − f 2 (x, s1 , κ2 , ξ1 , 0) ≥ 0,

s2 ∈ [0, κ2 ], ξ2 ∈ R N , s1 ∈ [0, κ1 ], ξ1 ∈ R N .

(3.2)

Hypothesis (H3) guarantees the existence of a δ ∈ (0, min{κ1 , κ2 }) such that for a.a. x ∈   p −1 f 1 (x, s1 , s2 , ξ1 , ξ2 ) > λ1, p1 s1 1 , s1 , |ξ1 | ∈ (0, δ), s2 ∈ [0, κ2 ], ξ2 ∈ R N p −1 f 2 (x, s1 , s2 , ξ1 , ξ2 ) > λ1, p2 s2 2 , s1 ∈ [0, κ1 ], s2 , |ξ2 | ∈ (0, δ), ξ1 ∈ R N . (3.3) Now we select ε > 0 sufficiently small such that 0 < ε u 1, pi (x) < δ, ε|∇ u 1, pi (x)| < δ for all x ∈ , i = 1, 2.

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Extremal solutions for nonvariational quasilinear. . .

Thus from (3.3) we obtain for a.a. x ∈  

u 1, p1 ) − f 1 (x, ε u 1, p1 , s2 , ε∇ u 1, p1 , ξ2 ) ≤ 0, − p1 (ε − p2 (ε u 1, p2 ) − f 2 (x, s1 , ε u 1, p2 , ξ1 , ε∇ u 1, p2 ) ≤ 0,

s2 ∈ [0, κ2 ], ξ2 ∈ R N s1 ∈ [0, κ1 ], ξ1 ∈ R N . (3.4) u 1, p1 , κ1 ], Since (3.2) and (3.4), in particular, hold true when replacing s1 by w1 ∈ [ε u 1, p2 , κ2 ], and ξ2 by ∇w2 , we see that and ξ1 by ∇w1 as well as s2 by w2 ∈ [ε u 1, p1 , ε u 1, p2 ) and u = (u 1 , u 2 ) := (κ1 , κ2 ) form a pair of sub-and u = (u 1 , u 2 ) := (ε u 1, p1 , ε u 1, p2 ), (κ1 , κ2 )] supersolution for ε > 0 sufficiently small, that is [u, u] = [(ε forms a trapping region. We now may apply Theorem 3.1 to ensure the existence of a (weak) solution u ∈ [u, u], that is, its components satisfy ε u 1, p1 ≤ u 1 ≤ κ1 , ε u 1, p2 ≤ u 2 ≤ κ2 .

(3.5)

By an analogous approach as before we can show that the pair of functions (d1 , d2 ) u 1, p2 ) constitute a pair of sub- and supersolution in the sense of and (−ε u 1, p1 , −ε Definition 1.2 for problem (1.1) provided ε > 0 is sufficiently small. Consequently, we u 1, p1 ] × [d2 , −ε u 1, p2 ], which completes obtain a negative solution (v1 , v2 ) ∈ [d1 , −ε the proof.   Remark 3 We remark that for the proof of Theorem 3.1 we basically used the boundedness of the lim inf below in (H3). By making use of the boundedness above of the corresponding lim sup in (H3), we are even able to show the existence of minimal and maximal constant-sign solutions of (1.1), where the notions minimal and maximal are understood in the order theoretical sense. The proof of this result, which will be done in the following section, requires a stronger growth condition on the right-hand side of (1.1) in order to make use of recent L ∞ -gradient estimates due to [6, Theorem 3.1], see also [7,8], as well as of a strong maximum principle due to Pucci-Serrin (see [11, Theorem 5.5.1]).

4 Extremal constant-sign solutions via expanding trapping regions Theorem 3.1 ensures the existence of a positive solution u = (u 1 , u 2 ) of (1.1) within u 1, p2 , κ2 ] for ε > 0 small enough, and a negative the trapping region [ε u 1, p1 , κ1 ] × [ε u 1, p1 ] × [d2 , −ε u 1, p2 ] for ε > 0 small enough. solutions v = (v1 , v2 ) within [d1 , −ε The following theorem proves the existence of a minimal solution u ε = (u ε1 , u ε2 ) within u 1, p2 , u 2 ], as well as the existence of a maximal the trapping region [ε u 1, p1 , u 1 ] × [ε solution v ε = (v1ε , v2ε ) within [v1 , −ε u 1, p1 ] × [v2 , −ε u 1, p2 ]. More precisely, we have the following result. Proposition 4.1 Assume hypotheses of Theorem 3.1. Then, given a positive solution u 1, p1 , κ1 ] × [ε u 1, p2 , κ2 ] for some ε > 0 small enough, (u 1 , u 2 ) of problem (1.1) in [ε u 1, p1 , κ1 ] × [ε u 1, p2 , κ2 ] such that there exists a minimal solution (u ε1 , u ε2 ) of (1.1) in [ε u iε ≤ u i , i = 1, 2. Similarly, given a negative solution (v1 , v2 ) of problem (1.1) in u 1, p1 ] × [d2 , −ε u 1, p2 ] for some ε > 0 small, there exists a maximal solution [d1 , −ε u 1, p1 ] × [d2 , −ε u 1, p2 ] such that viε ≥ vi , i = 1, 2. (v1ε , v2ε ) of (1.1) in [d1 , −ε

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Proof Let us focus on the proof of the first part of the theorem only, i.e., on the existence of a minimal positive solution u ε = (u ε1 , u ε2 ) satisfying u iε ≤ u i , i = 1, 2, u 1, p1 , κ1 ] × [ε u 1, p2 , κ2 ] where (u 1 , u 2 ) is a given solution of problem (1.1) within [ε for some ε > 0 small enough, whose existence is ensured by Theorem 3.1. The second part of Theorem 4.1 follows basically similar arguments. u 1, p1 , κ1 ] × [ε u 1, p2 , κ2 ] To this end let us denote by Sε the set of all (w1 , w2 ) ∈ [ε that are solutions of (1.1), and which satisfy wi ≤ u i , i = 1, 2. Using Zorn’s lemma, we are going to prove that there is a minimal element of Sε . In order to apply Zorn’s lemma let us consider a chain C in Sε . Then there is a sequence {(u k1 , u k2 )}k≥1 ⊂ C, with u ik+1 ≤ u ik , i = 1, 2, for all k ≥ 1, such that inf C = inf (u k1 , u k2 ). k≥1

u 1, p1 , κ1 ]×[ε u 1, p2 , κ2 ] (which Since (u k1 , u k2 ) are solutions of (1.1) that belong to [ε is a trapping region), we can show by using similar arguments as in Step 1 of the proof 1, p 1, p of Theorem 2.2 that the sequence {(u k1 , u k2 )}k≥1 ∈ V0 = W0 1 () × W0 2 () is 1, p bounded. So, along a subsequence, we may suppose that u ik uˆ i in W0 i (), u ik → uˆ i in L pi (), and u ik (x) → uˆ i (x) a.e. in  as k → ∞, for i = 1, 2. It follows u 1, p2 , κ1 ]×[ε u 1, p2 , κ2 ] and uˆ i ≤ u i , for i = 1, 2. Moreover, through that (uˆ 1 , uˆ 2 ) ∈ [ε the (S)+ -property of the operators − p1 and − p2 , we infer the strong convergence 1, p u ik → uˆ i in W0 i () as k → ∞, for i = 1, 2. Consequently, (uˆ 1 , uˆ 2 ) is a solution of (1.1) that belongs to Sε and inf C = (uˆ 1 , uˆ 2 ) ∈ Sε . Then Zorn’s Lemma can be  applied, which yields the existence of a minimal element (u ε1 , u ε2 ) of Sε as required.  By means of Proposition 4.1 we are now able to show that there is a minimal positive solution (u 1,+ , u 2,+ ) among all positive solutions of (1.1), and there is a maximal negative solution (u 1,− , u 2,− ) among all negative solutions of (1.1). For the proof of the existence of these extremal solutions we need to strengthen the growth assumptions of (H1) on the right-hand side of (1.1) in order to make use of recent global gradient estimates due to [6, Section 3]. For the reader’s convenience, here we recall a result that fits our purpose and which can be deduced as a special case from [6, Theorem 3.1, Remark 3.3]. 1, p

Corollary 4.1 Let u ∈ W0 (), 1 < p < ∞, be a (weak) solution of the Dirichlet problem − p u = f in , u = 0 on ∂, where  ⊂ R N , N ≥ 2, is a bounded domain with C 2 -boundary ∂. Then for f ∈ L q () with q > N , the following gradient bound holds true: 1

∇u L ∞ () ≤ C  f  Lp−1 q () , for some constant C = C( p, ).

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Extremal solutions for nonvariational quasilinear. . .

For the proof of the existence of extremal positive and negative solutions, instead of (H1) we make use of the following stronger hypothesis: (H4) For i = 1, 2, the functions f i :  × R × R × R N × R N → R are Carathéodory which satisfy the growth conditions

p1 p2  | f 1 (x, s1 , s2 , ξ1 , ξ2 )| ≤ 1 (x) + c1 |ξ1 | q + |ξ2 | q ,

p1 p2  | f 2 (x, s1 , s2 , ξ1 , ξ2 )| ≤ 2 (x) + c2 |ξ1 | q + |ξ2 | q ,

(4.1) (4.2)

for a.a. x ∈ , for all s = (s1 , s2 ) ∈ [d1 , κ1 ] × [d2 , κ2 ], and for all ξi ∈ R N , where ci ≥ 0 are some constants, and i ∈ L ∞ + (), and q > max{N , q1 , q2 }. Theorem 4.1 Assume hypotheses (H2)–(H4). Then problem (1.1) admits a positive solution (u 1,+ , u 2,+ ) with u i,+ ≤ κi , for i = 1, 2, which is minimal among all positive solutions (v1 , v2 ) of (1.1). Similarly, there is a negative solution (u 1,− , u 2,− ) with u i,− ≥ di , for i = 1, 2, which is maximal among all negative solutions (w1 , w2 ) of (1.1). Proof As before we only prove the first part of the theorem because the second part can be proved through similar reasoning. By Theorem 3.1 there exists ε0 := n10 sufficiently small (i.e., n 0 ∈ N sufficiently large) such that there is a positive solution (u 1 , u 2 ) of (1.1) with (u 1 , u 2 ) ∈ u 1, p1 , κ1 ] × [ε0 u 1, p2 , κ2 ]. In view of Proposition 4.1, there is a minimal posi[ε0 u 1, p1 , κ1 ] × [ε0 u 1, p2 , κ2 ] such that u in 0 ≤ u i , tive solution (u n1 0 , u n2 0 ) of (1.1) in [ε0 i = 1, 2. For n > n 0 consider the trapping region [ n1  u 1, p1 , κ1 ] × [ n1  u 1, p2 , κ2 ] n0 n0 u 1, p1 , κ1 ] × [ε0 u 1, p2 , κ2 ] and (u 1 , u 2 ). Therefore, we again meet which contains [ε0 the situation of Proposition 4.1, which ensures the existence of a minimal solution u 1, p1 , κ1 ] × [ n1  u 1, p2 , κ2 ] satisfying u in ≤ u in 0 , i = 1, 2. Iteratively (u n1 , u n2 ) ∈ [ n1  applying Proposition 4.1, we get a sequence {(u n1 , u n2 )}n≥n 0 ⊂ V0 of minimal posu 1, p1 , κ1 ] × [ n1  u 1, p2 , κ2 ] satisfying u in+1 ≤ u in , itive solutions with (u n1 , u n2 ) ∈ [ n1  i = 1, 2, for all n ≥ n 0 . In other words, the obtained sequence {(u n1 , u n2 )}n≥n 0 of minimal positive solutions is monotone decreasing, and its members belong to trapping regions that are expanding as n is increasing. The boundedness of this sequence in L ∞ () × L ∞ () → L along with the growth condition (H4) on the nonlinearities f i readily imply the boundedness of {(u n1 , u n2 )}n≥n 0 in V0 , which in view of the monotonicity yields the following convergence properties (for i = 1, 2) 1, pi

u in u i,+ in W0

(), u in → u i,+ in L pi (), and u in ↓ u i,+ pointwise in , (4.3) 1, p1 1, p2 with some (u 1,+ , u 2,+ ) ∈ V0 = W0 () × W0 (). By the same reasoning as in the proof of Proposition 4.1, and based on the (S)+ -property of the operators − p1 and − p2 we even deduce the strong convergence 1, p1

(u n1 , u n2 ) → (u 1,+ , u 2,+ ) in V0 = W0

1, p2

() × W0

(),

(4.4)

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S. Carl, D. Motreanu

which allows us to pass to the limit as n → ∞ to show that (u 1,+ , u 2,+ ) is a solution for (1.1). Regarding the convergence properties of {(u n1 , u n2 )}n≥n 0 we can say even more. This sequence satisfies (i = 1, 2) −  pi u in = f i (x, u n1 , u n2 , ∇u n1 , ∇u n2 ) in , u in = 0 on ∂.

(4.5)

Applying (H4), the right-hand side of (4.5) is bounded in L q () for all n ≥ n 0 due to the boundedness of {(u n1 , u n2 )}n≥n 0 in V0 . Corollary 4.1 then yields 1

p −1 ˆ ∀ n > n0. ∇u in  L ∞ () ≤ C  f i (·, u n1 , u n2 , ∇u n1 , ∇u n2 ) Liq () ≤ C,

(4.6)

In view of (4.6) and again applying (H4), the right-hand sides of (4.5) are in L ∞ (), i.e., ˜ ∀ n > n0. (4.7)  f i (·, u n1 , u n2 , ∇u n1 , ∇u n2 ) L ∞ () ≤ C, Now we may apply a classical regularity result up to the boundary due to Lieberman [9] or Chen-DiBenedetto [5] (which includes also the elliptic case), according to which we obtain u in ∈ C 1,γ (), where γ ∈ (0, 1), and C 1,γ () is the space of C 1 -functions whose derivative is Hölder continuous with exponent γ , and moreover due to (4.7) we get (4.8) u in C 1,γ () ≤ C , ∀ n > n 0 , ˜ By Arzela-Ascoli’s where the C > 0 is a constant depending only on N , pi , , C. theorem, from (4.8) we infer the existence of a subsequence which is again denoted by {(u n1 , u n2 )}n≥n 0 that satisfies u in → u i,+ in C 1 () i = 1, 2.

(4.9)

Apparently, u i,+ ≥ 0. Next, we claim that u i,+ = 0 for i = 1, 2. Arguing by contradiction, let us suppose that u 1,+ = 0. Set u˜ n =

u n1 f 1 (x, u n1 , u n2 , ∇u n1 , ∇u n2 ) , g (x) = . n u n1 1, p1 (u n1 ) p1 −1

Since (u n1 , u n2 ) are positive solutions of problem (1.1), dividing the first equation of p −1 system (1.1) by u n1 1,1p1 we get 

|∇ u˜ n | p1 −2 ∇ u˜ n · ∇ϕ d x =



p −1

gn u˜ n 1

1, p1

ϕ d x, ∀ ϕ ∈ W0

().

(4.10)

Obviously u˜ n 1, p1 = 1, and by using (H3) and (4.9) we see that for n large we have gn  L ∞ () ≤ C (note: u 1,+ = 0 by assumption). Thus, there is a subsequence 1, p of (u˜ n ) that is weakly convergent in W0 1 () and strongly convergent in L p1 (),

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Extremal solutions for nonvariational quasilinear. . .

as well as a subsequence of (gn ) that is weakly ∗ convergent in L ∞ (), i.e., we get (again relabeling the subsequences by n) gn ∗ g in L ∞ (), u˜ n u˜ in W0

1, p1

(), u˜ n → u˜ in L p1 (),

(4.11)

p −1

and thus, in particular, u˜ n 1 → u˜ p1 −1 in L q1 (). Taking the special test function ˜ in (4.10), we get by using (4.11) along with the boundedness of (gn ) ϕ = (u˜ n − u) 

|∇ u˜ n | p1 −2 ∇ u˜ n · ∇(u˜ n − u)d ˜ x=



p −1

gn u˜ n 1

(u˜ n − u)d ˜ x → 0, as n → ∞.

(4.12) 1, p The weak convergence u˜ n u˜ in W0 1 (), and the (S)+ -property of the operator 1, p − p1 implies that u˜ n → u˜ (strongly) in W0 1 (), and thus it follows u˜ = 0. Applying the convergence properties of (4.11) along with the strong convergence of 1, p u˜ n → u˜ in W0 1 (), we may pass to the limit in (4.10) as n → ∞ which yields 

|∇ u| ˜ p1 −2 ∇ u˜ · ∇ϕ d x =



g u˜ p1 −1 ϕ d x, ∀ ϕ ∈ W0

1, p1

(),

(4.13)

where in view of (H3) the g ∈ L ∞ () satisfies 0 < λ1, p1 < β1 ≤ g(x) ≤ α1 . 1, p Consequently, u˜ ∈ W0 1 () with u˜ = 0 and u˜ ≥ 0 is an eigenfunction associated to the eigenvalue 1 for the weighted problem (with the weight g(x) > 0) ⎧ ˜ = g(x)u(x) ˜ p1 −1 ⎨ − p1 u(x) ⎩

in  (4.14)

u˜ = 0

on ∂.

Since the eigenfunction u˜ does not change sign, from (4.14) it follows that 1 = λ1 (g), must be the first eigenvalue for ⎧ ˜ = λg(x)u(x) ˜ p1 −1 ⎨ − p1 u(x) ⎩

u˜ = 0

in  on ∂.

However, on the other hand we know that λ1 (g) ≤ λ1 (c1 ) < λ1 (λ1, p1 ) = 1, which is a contradiction. Therefore u 1,+ = 0. In just the same way one shows that also u 2,+ = 0. So far we have seen that the limits u i,+ ∈ C 1 (), u i,+ = 0, u i,+ ≥ 0, and (u 1,+ , u 2,+ ) is a solutions of (1.1), that is, −  pi u i,+ = f i (x, u 1,+ , u 2,+ , ∇u 1,+ , ∇u 2,+ ) in , u i,+ = 0 on ∂. (4.15) Since the right-hand side of (4.15) is in L ∞ (), the solution u i,+ belongs even to and from Harnack’s inequality we immediately get that u i,+ (x) > 0 for all x ∈ . To complete the proof of the first part, it remains to show that (u 1,+ , u 2,+ ) is a minimal positive solution. To this end let (v1 , v2 ) be any positive solution of (1.1)

C 1,γ (),

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S. Carl, D. Motreanu

with v1 ≤ u 1,+ and v2 ≤ u 2,+ . Again by a bootstrapping argument using Corollary 4.1 and boundary regularity results due to Lieberman [9] or Chen-DiBenedetto [5], we see that vi ∈ C 1,γ (), i = 1, 2. Now we are going to show that vi belong even to int(C01 ()+ ). For this, a strong maximum principle due to Pucci-Serrin (see [11, Theorem 5.5.1]) is our main tool. We recall that (v1 , v2 ) is a positive solution of (1.1). In what follows we focus on the proof of v1 ∈ int(C01 ()+ ), and recall that v1 ∈ C 1,γ (), 0 < γ < 1, and ∃ M > 0 such that |∇v1 (x)| ≤ M, ∀ x ∈ , as well as v1 satisfies −  p1 v1 = f 1 (x, v1 , v2 , ∇v1 , ∇v2 ) in , v1 = 0, on ∂.

(4.16)

Applying (H3), there is a δ > 0 with δ ≤ min{κ1 , κ2 } such that for some positive constants αˆ 1 ≥ βˆ1 > 0 the following holds true for 0 < s1 < δ, 0 ≤ s2 ≤ κ2 , |ξ1 | < δ, ∀ ξ2 ∈ R N p −1 p −1 βˆ1 s1 1 ≤ f 1 (x, s1 , s2 , ξ1 , ξ2 ) ≤ αˆ 1 s1 1 . (4.17) We set

p1 p2  C1 = 1  L ∞ () + c1 M q + M q . Then in view of (H4) we get for a.a. x ∈ , s1 ∈ [δ, κ1 ], s2 ∈ [0, κ2 ], and ξi ∈ R N with |ξi | ≤ M C1 p −1 | f 1 (x, s1 , s2 , ξ1 , ξ2 )| ≤ p −1 s1 1 . (4.18) δ 1 For 0 < s1 < δ, δ ≤ |ξ1 | ≤ M, s2 ∈ [0, κ2 ], and |ξ2 | ≤ M we get | f 1 (x, s1 , s2 , ξ1 , ξ2 )| ≤

C1 |ξ1 | p1 −1 . δ p1 −1

(4.19)

Combining (4.17), (4.18), (4.19) yields p −1

| f 1 (x, s1 , s2 , ξ1 , ξ2 )| ≤ a1 s1 1

+ b1 |ξ1 | p1 −1 , ∀ si ∈ [0, κi ], |ξi | ≤ M,

(4.20)

where a1 , b1 > 0 are some positive constants. Now we are applying the strong maximum principle due to Pucci-Serrin in the form [11, Theorem 5.5.1, p. 120] to the equation (4.16), i.e., to  p1 v1 + f 1 (x, v1 , v2 , ∇v1 , ∇v2 ) = 0 in , v1 = 0, on ∂. Setting B(x, v1 , ∇v1 ) = f 1 (x, v1 , v2 , ∇v1 , ∇v2 )

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Extremal solutions for nonvariational quasilinear. . .

we are going to check the hypotheses of [11, Theorem 5.5.1] (with the notation therein): We have A(s) := |s| p1 −2 ⇒ lim s↓0

s A (s) = p1 − 2 =: c > −1, A(s)

which verifies supposition (5.4.3) of [11, p. 111]. When p1 = 2 (i.e., c = 0), we obtain  √ √ p1 + 2 p1 − 1 (0) 2+c+2 1+c = >1= , φ(c) := |c| | p1 − 2| λ(0) which verifies (5.4.4) of [11, p. 111]. From (4.20) it follows B(x, z, ξ ) = f 1 (x, z, v2 (x), ξ, ∇v2 (x)) ≥ −a1 z p1 −1 − b1 |ξ | p1 −1 , for all z ∈ [0, κ1 ] and |ξ | ≤ M, which verifies assumption (B1) of [11, p. 107], since (by taking M > 1) (|ξ |) = |ξ |A(|ξ |) = |ξ | p1 −1 , and f (z) = a1 z p1 −1 , z ≥ 0, and thus assumption (F2) of [11, p. 107] is fulfilled. In order to finally apply [11, Theorem 5.5.1] it remains to check (1.1.5) of [11, Theorem 1.1.1], i.e., 0+

ds = ∞, H −1 (F(s))

where F(u) = 0

u

f (s) ds, and H (s) := s(s) −

s

(t) dt =

0

p 1 − 1 p1 s p1

which readily verifies (1.1.5) of [11, Theorem 1.1.1]. Now we are in a position to apply [11, Theorem 5.5.1, 120] to the positive solution v1 of (4.16) and get ∂v1 /∂ν < 0 on ∂, which ensures that v1 ∈ int(C01 ()+ ). In just the same way one shows v2 ∈ int(C01 ()+ ). Since vi ∈ int(C01 ()+ ), there is a n sufficiently large such that 1 u 1, pi ≤ vi , i = 1, 2, and by taking into account the construction of the solution n (u 1,+ , u 2,+ ) we get 1  u 1, pi ≤ vi ≤ u i,+ ≤ u in ≤ κi , i = 1, 2, n

(4.21)

u 1, p1 , κ1 ] × [ n1  u 1, p2 , κ2 ], from Because (u n1 , u n2 ) is a minimal solution of (1.1) in [ n1  n (4.21) we see that u i = vi , i = 1, 2, and hence it follows that u 1,+ = v1 and u 2,+ = v2 ,   which shows that (u 1,+ , u 2,+ ) is in fact a minimal positive solution.

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Example 4.1 Let us provide an example of a vector field f = ( f 1 , f 2 ) that satisfies hypotheses (H2)–(H4). To this end let di < 0 and κi > 0 such that κi di < −1, i = 1, 2. Further, let gi :  × R × R × R N × R N → R be Carathéodory functions for i = 1, 2, such that lim

s1 →0, ξ1 →0

g1 (x, s1 , s2 , ξ1 , ξ2 ) = 0

uniformly for a.a. x ∈ , s2 ∈ [d2 , κ2 ], and ξ2 ∈ R N ; lim

s2 →0, ξ2 →0

g2 (x, s1 , s2 , ξ1 , ξ2 ) = 0

uniformly for a.a. x ∈ , s1 ∈ [d1 , κ1 ], and ξ1 ∈ R N ;

p1 p2  |gi (x, s1 , s2 , ξ1 , ξ2 )| ≤ i (x) + ci |ξ1 | q + |ξ2 | q for a.a. x ∈ , ∀ (s1 , s2 ) ∈ [κ1 , d1 ] × [κ2 , d2 ], and ∀ ξi ∈ R N with i ∈ L ∞ (), and constants q > max{N , q1 , q2 }, and ci > 0, i = 1, 2. Then f = ( f 1 , f 2 ) with f i defined as follows satisfies hypotheses (H2)–(H4):   f i (x, s1 , s2 , ξ1 , ξ2 ) = |si | pi −2 si (κi − si )(si − di ) λ1, pi + gi (x, s1 , s2 , ξ1 , ξ2 ) . Notice that lim

si →0, ξi →0

f i (x, s1 , s2 , ξ1 , ξ2 ) = −κi di λ1, pi > λ1, pi , i = 1, 2. |si | pi −2 si

Remark 4 By inspection of the proof of Theorem 4.1, one readily observes that any positive solution (v1 , v2 ) of (1.1) within [0, κ1 ] × [0, κ2 ] belongs to int(C01 ()+ ) × int(C01 ()+ ). Similarly, any negative solution (w1 , w2 ) of (1.1) within [d1 , 0]×[d2 , 0] belongs to −int(C01 ()+ ) × −int(C01 ()+ ).

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