LOCATION OF NODAL SOLUTIONS FOR QUASILINEAR ELLIPTIC EQUATIONS WITH GRADIENT DEPENDENCE

. Existence and regularity results for quasilinear elliptic equations driven by ( p,q )-Laplacian and with gradient dependence are presented. A location principle for nodal (i.e., sign-changing) solutions is obtained by means of constant-sign solutions whose existence is also derived. Criteria for the existence of extremal solutions are ﬁnally established.

1. Introduction. We consider the following quasilinear elliptic problem with gradient dependence −∆ p u − µ∆ q u = f (x, u, ∇u) in Ω, u = 0 on ∂Ω on a bounded domain Ω in R N (N ≥ 2) with C 2 -boundary ∂Ω, where ∆ p and ∆ q stand for the p-Laplacian and q-Laplacian on W 1,p 0 (Ω) and W 1,q 0 (Ω), respectively, with 1 < q < p < +∞, and a constant µ ≥ 0. If µ = 0, the equation in (1) is driven by the p-Laplacian, whereas for µ = 1 the leading operator is the (p, q)-Laplacian ∆ p + ∆ q , which is a nonhomogeneous operator. We suppose the C 2 smoothness for the boundary ∂Ω in order to make use of appropriate regularity results that will be needed later on. For the sake of brevity and in order to get more insight on the main ideas, we suppose that N > p. The case N ≤ p is actually simpler and consequently is omitted.
Here we denote p * := N p N −p , which is the Sobolev critical exponent. Furthermore for every r ≥ 1 we set r := r r−1 , that is the conjugate exponent of r. For a later use, we pose an additional, this time unilateral condition: H(f ) 2 there exist constants b 1 , b 2 ≥ 0 such that b 1 λ −1 1,p + b 2 < 1, and a function τ ∈ L 1 (Ω) such that f (x, s, ξ)s ≤ τ (x) + b 1 |s| p + b 2 |ξ| p for a.e. x ∈ Ω, all s ∈ R, ξ ∈ R N .
The notation λ 1,p stands for the first eigenvalue of −∆ p on W 1,p 0 (Ω), which is expressed by where · p denotes the usual L p -norm. We point out that condition H(f ) 2 is independent of condition H(f ) 1 . For instance, H(f ) 2 is fulfilled if the sign requirement f (x, s, ξ)s ≤ 0 holds, without any growth restriction for f (x, s, ξ).
A solution u ∈ W 1,p 0 (Ω) of problem (1) is understood in the weak sense meaning that for all h ∈ W 1,p 0 (Ω). Hereafter, (·, ·) R N stands for the standard Euclidean scalar product in R N . The definition makes sense in view of condition H(f ) 1 .
We emphasize that due to the presence of the gradient ∇u in the right-hand side of the equation, the variational methods are not applicable to problem (1). This difficulty is overcome by using the theory of pseudomonotone operators. In further results we also employ the method of sub-supersolutions for quasilinear elliptic equations combined with comparison arguments.
In the present paper we first note that, under assumptions H(f ) 1 and H(f ) 2 , a general existence result and a priori estimates for weak solutions to problem (1) hold. Then we turn to study the regularity up to the boundary for the solutions to problem (1). In this respect we need to strengthen the conditions H(f ) 1 and H(f ) 2 (see Lemma 2.1) as follows: H(f ) f : Ω×R×R N → R is a Carathéodory function for which there exist constants Hypothesis H(f ) enables us to apply recent estimates due to Cianchi and Maz'ya [4] ensuring the global boundedness of the gradient of any solution u to (1). Owing to the homogeneous boundary condition, it follows that u ∈ L ∞ (Ω) and u ∞ is bounded above by a constant independent of u. This is actually a substitute for the Moser iteration technique (see, e.g., [8,Theorem C]) and has to be regarded in conjunction with Lieberman's estimates [6,7] for the regularity up to the boundary. In particular, every solution of problem (1) belongs to the space C 1 0 (Ω) = {u ∈ C 1 (Ω) : u = 0 on ∂Ω}. Notice that the strong maximum principle (see [11], [10]) is closely related to the positive cone C 1 0 (Ω) + = {u ∈ C 1 0 (Ω) : u(x) ≥ 0 for all x ∈ Ω} of C 1 0 (Ω), which has a nonempty interior given by where ν(x) stands for the unit outward normal to ∂Ω. Our first main objective is the location of nodal (that is, sign-changing) solutions for problem (1). Specifically, we prove that if assumptions H(f ) 1 , H(f ) 2 and f (x, 0, 0) = 0 for a.e. x ∈ Ω are fulfilled, then every nodal (i.e., sign-changing) solution of problem (1) should be between two opposite constant-sign solutions. In particular, this provides the powerful fact that the existence of a nodal solution implies under the stated hypotheses that two opposite constant-sign solutions must exist. Moreover, this phenomenon occurs even unilaterally: if we only have f (x, 0, 0) ≥ 0 (resp. f (x, 0, 0) ≤ 0) for a.e. x ∈ Ω, then every nodal solution to problem (1) is bounded above by a positive solution (resp. bounded below by a negative solution).
Our second main objective is to establish the existence of extremal (or barrier) solutions to problem (1), that is a biggest solution and a smallest solution. Taking into account what was said before about the location of nodal solutions, the biggest nontrivial solution is positive and the smallest nontrivial solution is negative, and they belong to C 1 0 (Ω). Our approach for obtaining the extremal solutions relies on the method of sub-supersolutions related to problem (1). We investigate separately the cases µ = 0 and µ > 0 because there are relevant differences in the construction of subsolutions and supersolutions corresponding to the two cases. It is worth mentioning that the problem with µ = 0: is a limiting case of problem (1) as µ → 0 + (see [1]). When µ > 0 we have to impose an additional condition describing the behavior of f (x, s, 0) as |s| → +∞. Related results on extremal solutions for variational inequalities, but without gradient dependence, can be found in [2]. The rest of the paper is organized as follows. Section 2 is devoted to preliminary results regarding the existence and regularity of solutions to problem (1). Section 3 sets forth our principle for locating nodal solutions. Section 4 discusses the presence of a positive solution and a negative solution accompanying every nodal solution. Section 5 presents our results on extremal solutions when µ = 0, whereas Section 6 focuses on the case µ > 0 in problem (1).

Preliminary results.
In the sequel, the Banach spaces W 1,p (Ω) and L p (Ω) are equipped with the usual norms · 1,p and · p , respectively, whereas the space The continuity of the embedding of W 1,p 0 (Ω) in L r (Ω) for 1 ≤ r ≤ p * guarantees the existence of a constant c r > 0 such that In view of assumption H(f ) 1 , we can consider the Nemytskii operator N f : Then problem (1) can be equivalently written as First, we point out that condition H(f ) is stronger than both H(f ) 1 and H(f ) 2 .
Proof. By the choice of r in H(f ), we have that p * r < p * − 1 and p Consequently, H(f ) yields H(f ) 1 by taking σ(x) = ω(x), α = p * r and β = p r . From H(f ) and Young's inequality we obtain that x ∈ Ω and all s ∈ R, ξ ∈ R N , with an arbitrary ε > 0 and a corresponding constant c(ε). We note that the choice of r in H(f ) results in p * r + 1 < p and r r−1 < p. A further application of Young's inequality yields x ∈ Ω and all s ∈ R, ξ ∈ R N , with δ > 0 arbitrary and some constant c(δ). Therefore, choosing δ, ε sufficiently small, we can infer that H(f ) 2 holds true, which completes the proof.
For easy reference we recall the notions of supersolution and of subsolution for problem (1). A function u ∈ W 1,p (Ω) is called a supersolution of problem (1) if it satisfies u ≥ 0 on ∂Ω and in Ω. Owing to the growth condition H(f ) 1 , the notions of supersolution and subsolution of problem (1) are correctly defined. The following useful property of sub-supersolutions is shown in [3,Theorem 3.20].
Later on we shall need the L p -normalized, positive eigenfunction φ 1,p of −∆ p on W 1,p 0 (Ω) corresponding to the eigenvalue λ 1,p (see (2)), so that By the nonlinear regularity theory and strong maximum principle, we know that φ 1,p , φ 1,q ∈ int (C 1 0 (Ω) + ) and, for positive constants c 1 and c 2 , there holds We also recall that an operator A : Recall also that an operator A : We quote from [1, Theorem 1] the following existence result for problem (1). 1 and H(f ) 2 hold. Then problem (1) has at least a (weak) solution u ∈ W 1,p 0 (Ω). Next we set forth a result addressing the regularity of solutions and a priori estimates for problem (1).
Lemma 2.4. Assume that condition H(f ) holds. Then there are constants R > 0 and γ ∈ (0, 1) such that every solution u ∈ W 1,p 0 (Ω) of problem (1) belongs to C 1,γ (Ω) and satisfies the estimate Proof. The fact that the solution set of problem (1) is nonempty is ensured by Lemmas 2.1 and 2.3. Let u ∈ W 1,p 0 (Ω) be any solution of (1). Inserting h = u in (3) and using H(f ) 2 and (2), we have Since µ ≥ 0 and b 1 λ −1 1,p + b 2 < 1, there exists a constant c > 0 independent of the solution u such that u ≤ c. (7) Then by (5) and (7) we infer that u p * ≤ c p * c and ∇u p ≤ c for every solution u of problem (1). Using these estimates through the growth condition H(f ), it follows that f (·, u, ∇u) r ≤ c 0 , with a constant c 0 > 0 independent of the solution u. At this point we make use of the assumption that r > N in H(f ), which enables us to refer to the gradient bound in Cianchi-Maz'ya [4, Theorem 3.1; Remark 3.3]. Consequently, from (8) we infer that whenever u is a solution of (1), where the constant C depends only on p and Ω. In particular, the solution set of (1) is uniformly bounded, that is there exists a constant C 0 > 0 independent of any solution u such that Due to (9), (10), the nonlinear regularity up to the boundary in Lieberman [6,7] (see also [10,Theorem 8.10]) applies, which leads to the desired conclusion.
Remark 2.5. A careful reading of the above proof shows that the constant c in (7) is explicitly given by c = Proof. (i) Let u 0 be a nodal solution of problem (1). The assumption f (x, 0, 0) ≥ 0 for a.e. x ∈ Ω ensures that 0 is a subsolution of problem (1). By Lemma 2.2 (a) we infer that u := max{0, u 0 } is a subsolution of problem (1).
Let T : W 1,p 0 (Ω) → W 1,p 0 (Ω) be the truncation operator defined by for a.e. x ∈ Ω. It is clear that the operator T : W 1,p 0 (Ω) → W 1,p 0 (Ω) is bounded and continuous. We also consider the following cut-off function by setting for a.e.

Now we state the auxiliary problem
Using the Nemytskii operator N f : W 1,p 0 (Ω) → W −1,p (Ω) given by (6), problem (13) can be equivalently expressed as By We claim that the operator −∆ p − µ∆ q + B − N f • T is pseudomonotone on W 1,p 0 (Ω). In order to show this, let a sequence (u n ) n≥1 ⊂ W 1,p 0 (Ω) be such that u n u in W 1,p 0 (Ω) and lim sup By H(f ) 1 we have that p * p * −α < p * . Invoking the boundedness of the operator T and Rellich-Kondrachov compact embedding theorem, it follows that the sequence Also, by H(f ) 1 we have that p p−β < p * . Using again the boundedness of the operator T and Rellich-Kondrachov compact embedding theorem, we infer that the sequence (|∇(T u n )| β ) n≥1 is bounded in L ( p p−β ) (Ω) = L p β (Ω) and that u n → u in L p p−β (Ω). Consequently, Hölder's inequality yields Therefore, thanks to assumption H(f ) 1 and (16), (17), we derive that Combining (15), (18), and the complete continuity of B, it turns out that lim sup n→∞ −∆ p u n − µ∆ q u n , u n − u ≤ 0.

DUMITRU MOTREANU, VIORICA V. MOTREANU AND ABDELKRIM MOUSSAOUI
inequality and the definition of T , we get the estimate with positive constants 1 , 2 . In view of the condition b 1 λ −1 1,p + b 2 < 1 supposed in H(f ) 2 , we obtain that the operator −∆ p − µ∆ q + B − N f • T is coercive. According to the properties above, we are in a position to apply the main theorem for pseudomonotone operators (see [3,Theorem 2.99]) to the operator −∆ p − µ∆ q + B − N f • T . It entails the existence of u + ∈ W 1,p 0 (Ω) solving (14). Therefore u + is a solution of the auxiliary problem (13).
Since the solution u 0 of (1) is nodal, its positive part u + 0 = max{0, u 0 } is strictly positive on a subset of Ω of positive measure, so from (19) and (11) we deduce that u + ≥ 0 and u + ≡ 0.
(23) In order to prove the claim we act on (22) with the test function (u − − u) + = max{0, u − − u} ∈ W 1,p 0 (Ω) and make use of (20) and (21). We are led to Following the reasoning in part (a) we can show that the claim in (23) is valid. From (22), (21) and (23) we infer that u − ∈ W 1,p 0 (Ω) is a solution of problem (1). Recalling that the solution u 0 of (1) is nodal, for its negative part u − 0 = max{0, −u 0 } we have that −u − 0 is strictly negative on a subset of Ω of positive measure. Then (23) and (20) enable us to conclude that u − ≤ 0 and u − ≡ 0.
(iii) If f (x, 0, 0) = 0 for a.e. x ∈ Ω, then we can apply simultaneously assertions (i) and (ii), which gives rise to two nontrivial opposite constant-sign solutions u + and u − of problem (1) with the properties required in the statement. 4. Positive and negative solutions generated by nodal solutions. The goal of this section is to provide a criterion for having in Theorem 3.1 the enclosure of the nodal solution u 0 with a positive solution u + and a negative solution u − . This is achieved by requiring certain behavior near (0, 0) for f (x, ·, ·) and through the strong maximum principle adapted to the presence of the gradient of solution in the right-hand side of equation (1). The approach is closely related to the regularity of the solutions as discussed in Lemma 2.4.

We note that
The function H is increasing on [0, +∞) and we have  [11,Remark 3,p. 117]), thereby u + ∈ int (C 1 0 (Ω) + ). (ii) Observe that hypothesis (25) implies that f (x, 0, 0) ≤ 0 for a.e. x ∈ Ω. Hence, by means of Lemma 2.1, we can refer to Theorem 3.1 (ii) for obtaining a nontrivial nonpositive solution u − of (1) such that u 0 ≥ u − on Ω. Consequently, it is sufficient to show that the solution u − has the properties required in the statement.
First, through assumption H(f ) and Lemma 2.4, we know that u − ∈ C 1 0 (Ω). Then we utilize the strong maximum principle in [11,Theorem 5.4.1] and Hopf's boundary point lemma in [11,Theorem 5.5.1] applied to the nonnegative function wheref (x, s, ξ) := −f (x, −s, −ξ). From now on the proof proceeds in a manner similar to part (i) with −u − in place of u + . (iii) We note that assumption (26) implies both (24) and (25). Then part (iii) can be readily obtained from the assertions (i) and (ii), which completes the proof.

5.
Extremal solutions for problem (4). The aim of this section is to prove the existence of extremal solutions, i.e., the biggest and smallest solutions for problem (4). This will permit to bound the whole set of nodal solutions by a positive solution and a negative solution, thus complementing the location principle in Theorem 3.1 where the bounds with constant-sign solutions depend on the fixed nodal solution.
The result below describes a continuity property of λ 1,p with respect to the domain.
In view of (28), the proof is complete.
The next result provides subsolutions and supersolutions for problem (4). the proof of Theorem 5.3, so we omit it. Similar arguments lead to the existence of the smallest solution of problem (1). Part (i) of the statement is established. Finally, combining with Theorem 4.1, we deduce part (ii) of the statement of the theorem. The proof is thus complete. Remark 6.2. Theorem 6.1 is also valid if, in place of (30), we assume the existence of constants c, c such that c, 0) for a.e. x ∈ Ω.