SYSTEMS OF QUASILINEAR ELLIPTIC EQUATIONS WITH DEPENDENCE ON THE GRADIENT VIA SUBSOLUTION-SUPERSOLUTION METHOD

. For the homogeneous Dirichlet problem involving a system of equations driven by ( p,q )-Laplacian operators and general gradient dependence we prove the existence of solutions in the ordered rectangle determined by a subsolution-supersolution. This extends the preceding results based on the method of subsolution-supersolution for systems of elliptic equations. Positive and negative solutions are obtained.

A relevant feature of our paper is the fact that the nonlinearities in the righthand side of the equations in (P µ1,µ2 ) depend on the solution and its gradient, which is a serious difficulty to be overcome and is rarely handled in the literature, especially in the system setting due to the interaction between different equations. More precisely, the nonlinearities in the right-hand side of the elliptic equations (called often convection terms) are introduced by means of functions f i : Ω × R × R × R N × R N → R, i = 1, 2, which are Carathéodory meaning that x → f i (x, s 1 , s 2 , ξ 1 , ξ 2 ) is measurable for all (s 1 , s 2 , ξ 1 , ξ 2 ) ∈ R × R × R N × R N and (s 1 , s 2 , ξ 1 , ξ 2 ) → f i (x, s 1 , s 2 , ξ 1 , ξ 2 ) is continuous for a.a. x ∈ Ω. The variables (s 1 , s 2 ) correspond to the solution (u 1 , u 2 ), whereas the vector variables (ξ 1 , ξ 2 ) figure out for the gradients (∇u 1 , ∇u 2 ). The variational methods are not applicable to this setting.
In order to compose with the state of the art for our problem, we mention that some results on existence, uniqueness and asymptotic properties with respect to (µ 1 , µ 2 ) for the general system (P µ1,µ2 ) have been recently obtained in [11] and for the equation version in [1] without location and enclosure properties as provided by subsolution-supersolution approach. The case where in (P µ1,µ2 ) one takes µ 1 = µ 2 = 0 was investigated in [3] through the method of subsolution-supersolution. For the study of equations driven by p-Laplacian and exhibiting gradient dependence in lower order terms we refer to [5], [6], [7], [8], [13], [14].
By a solution to system (P µ1,µ2 ) we mean a weak solution, that is a pair (u 1 , u 2 ) ∈ W 1,p1 (Ω). In the present paper, the approach for studying problem (P µ1,µ2 ) relies on the method of subsolution-supersolution for systems as presented in [2]. We recall that (u 1 , in Ω, i = 1, 2. In the rest of the paper we suppose that N > max{p 1 , p 2 }, which implies that the Sobolev critical exponent corresponding to p i is p * i = N pi N −pi , i = 1, 2. The case N ≤ max{p 1 , p 2 } is omitted in order to avoid developing some arguments basically along the same lines. Subsequently, we make the convention that for every r ∈ [1, +∞], its Hölder conjugate is denoted by, r , i.e., 1 r + 1 r = 1. Our main abstract result is stated as Theorem 3.1 below. It requires the existence of a subsolution-supersolution (u 1 , u 2 ), (u 1 , u 2 ) ∈ W 1,p1 (Ω) × W 1,p2 (Ω) of problem (P µ1,µ2 ) such that the following condition is satisfied: Under hypothesis (H), the integrals in the preceding definitions of solution and subsolution-supersolution exist. It is worth pointing out that assuming condition (H) we cannot apply the known results as available in [2], [3], [11], which hold under more restrictive hypotheses. Indeed, in [3] the driving differential operator is more particular and for the nonlinearities in the reaction terms it is supposed the following growth: there exist constants c i ≥ 0 and functions ρ i ∈ L p i (Ω) such that for a.a. x ∈ Ω and all s = (s 1 , s 2 ) ∈ [u 1 (x), u 1 (x)] × [u 2 (x), u 2 (x)], ξ 1 , ξ 2 ∈ R N , which is more restrictive than hypothesis (H) because p i − 1 = pi In [2] the setting is even more restrictive than in [3]. In [11], due to a completely different approach, there are imposed assumptions of other type, for instance a generalized sign condition.
Weakening the preceding assumptions, Theorem 3.1 below establishes the existence and location properties for the solutions of problem (P µ1,µ2 ) under hypothesis (H). Like in [2] and [3], the proof relies on the study of an associated auxiliary problem (see Theorem 2.1 below). In comparison with what was done before, the main contribution consists in the use of a cut-off function adapted to the general growth condition in assumption (H). Finally, we show in Theorem 4.1 below the existence of a positive solution (u 1 , u 2 ) of system (P µ1,µ2 ) meaning that both components are positive. Proceeding along the same lines, one can obtain a negative solution (u 1 , u 2 ), i.e., both components are negative, as well as hybrid solutions (u 1 , u 2 ), in the sense that the components u 1 and u 2 are of opposite constant sign. The essential point for obtaining these solutions is the construction of an appropriate subsolution-supersolution of problem (P µ1,µ2 ) permitting to apply Theorem 3.1. Contrary to the previous works where the driving operator was the p-Laplacian, the subsolution-supersolution in the case of the (p, q)-Laplacian as driving operator cannot be achieved through the first eigenvalue, which simply does not exist for the negative (p, q)-Laplacian (see [10]). However, we are able to find a verifiable criterion for getting the desired subsolution-supersolution. A major part in these arguments is played by nonlinear regularity theory and strong maximum principle.
2. Auxiliary problem. This section is devoted to solve an auxiliary problem that will permit to establish the existence and location of solutions for the original problem (P µ1,µ2 ). Our goal is to show that the method of subsolution-supersolution can be worked out for the general hypothesis (H) by changing appropriately the cut-off functions. Fix (Ω) be a subsolutionsupersolution of problem (P µ1,µ2 ) as required in condition (H). For i = 1, 2, we consider the truncation operators T i : The operators T i are continuous and bounded.
We introduce the cut-off functions related to the given subsolution-supersolution to match the growth condition in hypothesis (H). Namely, with the constants β i in (H), we set for a.a. x ∈ Ω, all s ∈ R, i = 1, 2: It is clear that π i is a Carathéodory function satisfying the estimate by the Sobolev embedding theorem, and β i < pi (p * i ) by hypothesis (H). Moreover, the same reasoning shows the existence of positive constants r (i) 1 and r From (2) we infer that the corresponding Nemytskij operators Π i : ) are completely continuous due to the compact embedding of W 1,pi (Ω) into L pi (Ω).
For any λ > 0, consider the auxiliary problem The solutions of problem (4) are understood in the weak sense as for the original problem (P µ1,µ2 ) (see Section 1). Their existence is given in the next statement.

DUMITRU MOTREANU, CALOGERO VETRO AND FRANCESCA VETRO
Indeed, writing explicitly the expression of the truncation operator, it turns out whence (10) follows.
Summing up the last two inequalities, we notice that (15) holds true because p 1 , p 2 > 1 and by choosing ε > 0 sufficiently small and λ > 0 sufficiently large. Therefore the operator A : exists. According to the definition of the operator A, equation (16) reads as system (4). We can thus conclude that the solution (u 1 , u 2 ) of (16) is a solution to the auxiliary problem (4), which completes the proof.
3. Existence and enclosure result. Here we present our abstract result on the existence of a solution to problem (P µ1,µ2 ) within the ordered rectangle determined by a subsolution-supersolution provided hypothesis (H) is verified. The proof relies on Theorem 2.1 dealing with the auxiliary problem (4) and on the cut-off functions in (1), which are suitable for comparison with the given subsolution-supersolution for problem (P µ1,µ2 ). (Ω) of problem (P µ1,µ2 ) satisfying the enclosure property u i ≤ u i ≤ u i a.e. in Ω, i = 1, 2.
We start by checking that u 1 ≤ u 1 a.e. in Ω. For this, let us utilize as test (Ω) in the definitions of solution to problem (4) and of supersolution to problem (P µ1,µ2 ), which gives and whenever w 2 ∈ W 1,p2 (Ω) with u 2 ≤ w 2 ≤ u 2 a.e. in Ω. Thanks to the definition of the truncation operator T 2 there holds u 2 ≤ T 2 u 2 ≤ u 2 , so we are able to insert w 2 = T 2 u 2 in (18) resulting in where the last equality follows from the fact that T 1 u 1 = u 1 on the set {u 1 > u 1 }, in view of the definition of the truncation operator T 1 . This amounts to saying that Then, the inequalities for all ζ, η ∈ R N with ζ = η , ensure that u 1 ≤ u 1 a.e. in Ω.
On the same pattern, making use of adequate test functions for needed comparison, we can show that u 1 ≤ u 1 and u 2 ≤ u 2 ≤ u 2 a.e. in Ω. Hence the claim On the basis of the above claim, we see that T i u i = u i and Π i u i = 0 for i = 1, 2. Therefore problem (4) reduces to (P µ1,µ2 ), which renders that (u 1 , u 2 ) solves system (P µ1,µ2 ). The proof is thus complete.

Constant-sign solutions.
We illustrate the application of Theorem 3.1 by showing the existence of at least one positive solution of system (P µ1,µ2 ). Suppose that: (H1) For i = 1, 2, there exist a i ∈ L αi (Ω), with α i > N and a i (x) > 0 on a set of positive measure, and a Carathéodory function g i : for a.a.
Proof. In view of (H1), for i = 1, 2 it holds that a i ∈ W −1,p i (Ω), hence the Dirichlet The assumption a i ∈ L αi (Ω) allows us to invoke [4,Theorem 3.1] for deducing that u i is bounded. Then, through the nonlinear regularity theory, it follows that u i ∈ C 1 (Ω).
Let us prove that To this end, we act with the test function −u − i = − max{−u i , 0}, that is We arrive at in Ω. Taking into account that u i is nontrivial, the strong maximum principle in [12,Theorem 5.4.1] implies (21). We further claim that We only show that u 1 ≤ u 1 because the other inequality can be verified analogously.
Symmetrically to (H1), we can formulate the condition: (H2) For i = 1, 2, there exist a i ∈ L αi (Ω), with α i > N and a i (x) < 0 on a set of positive measure, and a Carathéodory function g i : Ω×R×R N → R satisfying for a.a.
Overall, the proof can be done using the ideas in the proof of Theorem 4.1.