Symmetry results for p-Laplacian systems involving a first-order term

In this paper we obtain symmetry and monotonicity results for positive solutions to some p-Laplacian cooperative systems in bounded domains involving first-order terms and under zero Dirichlet boundary condition.


Introduction
The aim of this work is to get some symmetry and monotonicity results for nontrivial solutions (u 1 , u 2 , … , u m ) ∈ C 1 (Ω) × C 1 (Ω) … × C 1 (Ω) to the following quasilinear elliptic system F. Esposito  where i = 1, … , m , p i > 1 , q i = max{1, p i − 1} , Ω is a smooth bounded domain (connected open set) of ℝ N , N ≥ 2 , Δ p i u i ∶= div(|∇u i | p i −2 ∇u i ) is the p-Laplace operator and a i , f i are problem data that obey to the set of assumptions (hp * ) . The solution (u 1 , u 2 , … , u m ) has to be understood in the weak distributional meaning. Our result will be obtained by means of the moving plane method, which goes back to the papers of Alexandrov [1] and Serrin [27]. In this work we use a nice variant of this technique: in particular the one of the celebrated papers of Berestycki-Nirenberg [3] and Gidas-Ni-Nirenberg [16], where the authors used, as essential ingredient, the maximum principle by comparing the values of the solution of the equation at two different points after a suitable reflection. Such a technique can be performed in general convex domains providing partial monotonicity results near the boundary and symmetry properties when the domain is convex and symmetric. For simplicity of exposition and without loss of generality, since the system ( S ) is invariant with respect to translations and rotations, we assume directly in all the paper that Ω is a convex domain in the x 1 -direction and symmetric with respect to the hyperplane {x 1 = 0} . When m = 1 the system ( S ) is reduced to a scalar equation, that was already studied in [15] in the case of Ω = ℝ N + and 1 < p < 2. The moving plane procedure was applied to investigate symmetry properties of solutions of cooperative semilinear elliptic systems in bounded domains, firstly by Troy [28] (see also [11,12,26]): in this paper, the author considers the case p i = 2 and a i = 0 of (S ). This technique is very powerful and was adapted also in the case of cooperative semilinear systems in the half-space ℝ N + by Dancer [10] and in the entire space ℝ N by Busca and Sirakov [4]. For other results regarding semilinear elliptic systems in bounded or unbounded domains, involving also critical nonlinearities, we refer to [13].
The moving plane method for quasilinear elliptic equations in bounded domains was developed in several papers by Damascelli, Pacella and Sciunzi [7][8][9] and in [14,18] for quasilinear elliptic equations involving the Hardy-Leray potential and other more general singular nonlinearities. For the case of quasilinear elliptic systems in bounded domains we refer to [23,24], where the authors considered the case m = 2 and a 1 = a 2 = 0 of ( S ). Moreover, for other questions regarding existence, non-existence and Liouville type results, in the case of (pure, i.e., a i = 0 in ( S )) p-Laplace systems, we refer the readers to the papers (and references therein) [2,5,6,20,21].
In this work we consider the general case of m p-Laplace equations with first-order terms.
To deal with the study of the qualitative properties of solutions to ( S ), first we point out some regularity properties of the solutions to ( S ), see Sect. 2. Indeed the fact that solutions to p-Laplace equations are not in general C 2 (Ω) , leads to the study of the summability properties of the second derivatives of the solutions. Thanks to these regularity results, we are able to prove a weak comparison principle in small domains, i.e., Proposition 2.5, that is a first crucial step in the proof of the main result of the paper, namely Theorem 1.1. Moreover, we also get some comparison and maximum principles that we will exploit in the Proof of Theorem 1.1.
Through all the paper, we assume that the following hypotheses (denoted by (hp * ) in the sequel) hold: , and assume that for all t i > 0 . Moreover, the functions f i satisfy The monotonicity conditions (1.1) are also known as cooperativity conditions, see [10,24,26,28].
Finally we have the following The paper is organized as follows: In Section 2 we recall some preliminary results and we prove Proposition 2.5. The Proof of the Theorem 1.1 is contained in Sect. 3.

Preliminaries
In this section, we are going to give some results for p-Laplace equations involving a firstorder term. Through all the paper, generic fixed and numerical constants will be denoted by C (with subscript or superscript in some case) and it will be allowed to vary within a single line or formula. Moreover, by L(Ω) we will denote the Lebesgue measure of a measurable set Ω.

3
Firstly, we recall the following inequalities (see, for example, [7]) that we are going to use along the paper: For all , � ∈ ℝ N with | | + | � | > 0 there exist two positive constants C,C depending on p such that In the following two theorems we give some regularity results and comparison/maximum principles for the solutions to ( S). Theorem 2.1 (See [19,22]). Let Ω a bounded smooth domain of ℝ N , N ≥ 2 , 1 < p < ∞ , q ≥ max{p − 1, 1} and consider u ∈ C 1 (Ω) a positive weak solution to with (i) a ∶ ℝ → ℝ a locally Lipschitz continuous function; Denoting u x i = u∕ x i and setting ∇u x i = 0 on Z u , for any Ω � ⊂ Ω �� ⊂⊂ Ω , we have uniformly for any y ∈ Ω � , with Moreover, if f (x, ⋅) is positive in Ω �� , then it follows that uniformly for any y ∈ Ω � , with for any r < 1 and In particular, these regularity results apply to the solutions u i to ( S ) with Proof The proof follows exploiting and adapting some arguments contained in [19,22] to (2.4)-type nonlinearities. This would imply some technicalities which we rather avoid here. ◻ , with respect to the norm (2.5). We refer to [9] for more details about weighted Sobolev spaces and also to [17, Chapter 1] and the references therein. Theorem 2.1 provides also the right summability of the weight |∇u(x)| p−2 in order to obtain a weighted Poincaré-Sobolev type inequality that will be useful in the sequel. For the proof we refer to [9, Section 3].

Theorem 2.2 (Weighted Poincaré-Sobolev type inequality). Assume that hypotheses (hp * ) hold and let
The following theorem collects some comparison and maximum principles for solutions to the system ( S ). We have Theorem 2.3 (See [19,22]). Let Ω a bounded smooth domain of ℝ N , N ≥ 2, a solution to (S ) and let us assume that assumptions (hp * ) hold.
(1) Then, for i = 1, 2, … , m , any connected domain Ω � ⊆ Ω and for some constant Λ > 0 , such that in the weak distributional meaning, it follows that To prove the part (2) we need to define the linearized equations to the system ( S ). In order to do this, since (u 1 , is a weak solution of (S ), then we set where for p i > 1, for any 1 , … , m ∈ C 1 0 (Ω) . Moreover, using the regularity results contained in Theorem 2.1 (see [22]), the following equation holds Since f i are locally C 1 functions and ‖u i ‖ L ∞ (Ω) ≤ C for any i ∈ {1, … , m} , there exists a positive constant Θ such that Moreover, in light of (1.1) we have for i ≠ k . Therefore, using (2.9) and (2.10) and taking into account (2.8), it follows, for all j = 1, … , N and for all i = 1, … , m , that x j u i are nonnegative functions solving the inequalities for all nonnegative test functions i ≥ 0. Therefore, we can apply [22, Theorem 3.1] to each x j u i separately obtaining that, for every s > 1 sufficiently close to 1 and some positive sufficiently small, there exists a positive constant C such that Then, the sets {x ∈ Ω � ∶ x j u i = 0} are both closed (by continuity) and open (via inequality (2.11)) in the domain Ω � . This yields the assertion. ◻ Remark 2. 4 We point out that Theorem 2.3 holds without any a priori assumption on the critical set of the solution (u 1 , u 2 , … , u m ) , that is, the set where the gradients ∇u i vanish.
On the other hand, though, condition (2.7) can be removed when we work in connected domain Ω � such that ∇u i ≠ 0 for all x ∈ Ω � and for all i ∈ {1, … , m} . Indeed, the statements (1) and (2) of Theorem 2.3 hold in the whole range p i > 1.
Note that the positivity of f (x, ⋅) , is actually needed to obtain (2.3). Furthermore, by (2.3) it follows that the critical set {x ∈ Ω ∶ ∇u(x) = 0} has zero Lebesgue measure.
An essential tool in the Proof of Theorem 1.1 is Proposition 2.5, i.e., a weak comparison principle in small domains. To prove it, we start giving the following assumptions: (2.10) x j u i . The second term on the left-hand side of (2.14) can be estimated as follows L(Ω 0 ) ≤ and u i ≤ũ i on Ω 0 for every i ∈ {1, … , m}, (2.14) Since a i is a locally Lipschitz continuous function (see (hp * ) ), it follows that there exists a positive constant Moreover, denoting by L a i = L a i (‖u i ‖ L ∞ (Ω) ) the Lipschitz constant of a i , we obtain By the mean value's theorem and taking into account that q i ≥ 1 , it follows that The last term (recall that q i ≥ max{1, p i − 1} ) can be written as follows, ) is a positive constant. Exploiting Young's inequality in the right-hand side of (2.16) we finally obtain Therefore, collecting the previous estimates, from (2.15), we obtain Finally, using (2.1) and fixing sufficiently small, from (2.14) we get is a positive constant. The first term on the right-hand side of (2.17) can be arranged as follows (2.17) Using the fact that f i are C In the case p j ≥ 2 , a weighted Poincaré inequality holds true on the right-hand side of (2.21), see Theorem 2.2. Indeed, Eq. (2.6) yields where the Poincaré constant C P,j (Ω 0 ) → 0 , when the Lebesgue measure L(Ω 0 ) → 0 . Actually, we used the fact that, since p j ≥ 2, In the case p j < 2 , we use the standard Poincaré inequality on the right-hand side of (2.21), namely (2.24) we obtain Now, we choose > 0 sufficiently small such that the condition L(Ω 0 ) ≤ implies Therefore, from (2.27) we get the desired contradiction, namely for all i = 1, … , m. ◻

Symmetry results for solutions to ( S ): Proof of Theorem 1.1
In this section, we prove our main result. As we said in the introduction, without loss of generality and for the sake of simplicity, since the problem is invariant with respect to translations, reflections and rotations, we suppose that Ω is a bounded smooth domain which is convex in the x 1 -direction and symmetric with respect to {x 1 = 0} . Let us now recall the main ingredients of the moving plane method. We set Given x ∈ ℝ N and < 0 , we define and the reflected functions We also set Finally, for i = 1, … , m , we define the critical sets for all t ∈ (a, ] and for all i = 1, 2, … , m ̄= sup Λ.

3
Proof of Theorem 1.1 For a < < 0 [see (3.1)] and sufficiently close to a, we assume that L(Ω ) is as small as we need. In particular, we may assume that Proposition 2.5 works with Ω 1 = Ω, Ω 2 = R (Ω), Ω 0 = Ω and ũ i = u i, . Therefore, we set and we observe that, by construction, we have By Proposition 2.5, it follows that Hence, the set Λ [see (3.2)] is not empty and ̄∈ (a, 0] . Note that, by continuity, it follows u i ≤ u i,̄ . We have to show that, actually ̄= 0 . Hence, we assume by contradiction that ̄< 0 and we argue as follows. In the case 1 < p i < 2 , we prove first the following Claim: The case u i ≡ u i,̄ in some connected component C of Ω̄⧵Z u i , such that C ⊂ Ω , is not possible.
We proceed by contradiction. Let us assume that such component exists, namely For all > 0 , let us define G ∶ ℝ + 0 → ℝ by setting Let A be the characteristic function of a set A . We define where C is the reflected set of C with respect to the hyperplane T̄ and where a + i ∶= max{0, a i } ( a − i ∶= − min{0, a i } ) and Ĉ i denotes some positive constant to be chosen later.
We point out that suppΨ ⊂ C ∪ C , which implies Ψ ∈ W 1,p 0 (C ∪ C ) . Indeed by definition of C we have that ∇u i = 0 on (C ∪ C ) . Moreover, using the test function Ψ defined in (3.7), we are able to integrate on the boundary (C ∪ C ) which could be not regular.
Hence, we obtain It is easy to see that for every x ∈ [0, M] and for every l, q ≥ 1 and > 0 , there exists a positive constant C = C(l, q, , M) such that Therefore, (3.9) and (3.10) imply: By (hp * ) − (ii) , since C ∪ C ⊂ Ω we have that there exists i > 0 such that (3.10) Hence, we can choose i in (3.10), say ̄i , small enough such that so that We have: where ‖D 2 u i ‖ denotes the Hessian norm and C i a positive constant. We let → 0 . To this aim, let us first show that Let us prove (i). By Hölder's inequality it follows with 0 ≤ i < 1 and C i a positive constant.
(3.14) Finally, by the Lebesgue's dominate convergence theorem, passing to the limit for → 0 in (3.13) we obtain This gives a contradiction, hence the Claim holds.
Then, using also Hopf's boundary lemma (see [25,Theorem 5.5.1]) for u i > 0 in Ω and u i = 0 on Ω , we deduce that the set Ω̄⧵Z u i is connected. Indeed, thanks to Hopf's lemma, Z u i lies far from the boundary Ω . Moreover, we also remark that since Ω is convex in the x 1 -direction, we have that the boundary Ω is connected. Consequently, for any i = 1, 2, … , m we get in Ω̄⧵Z u i . Consider now a compact set K in Ω̄ such that L(Ω̄⧵K) is sufficiently small so that Proposition 2.5 can be applied. By what we proved before, for any i ∈ {1, … , m} , it holds that u i < u i,̄ in K⧵A , which is compact. Then, by (uniform) continuity, we find > 0 such that, ̄+ < 0 and for ̄< <̄+ we have that L(Ω ⧵(K⧵A)) is small enough as before, and u i, − u i > 0 in K⧵A for any i. In particular, u i, − u i > 0 on (K⧵A) . Consequently, u i ≤ u i, on (Ω ⧵(K⧵A)) . By Proposition 2.5 it follows u i ≤ u i, in Ω ⧵(K⧵A) and, consequently in Ω , which contradicts the assumption ̄< 0 . Therefore ̄= 0 and the thesis is proved. Finally, (1.2) follows by the monotonicity of the solution that is implicit in the moving plane method.
Finally, if Ω is a ball, repeating this argument along any direction, it follows that u i , i = 1, … , m , are radially symmetric. The fact that u i r (r) < 0 for r ≠ 0 , follows by the Hopf's boundary lemma which works in this case since the level sets are balls and, therefore, fulfill the interior sphere condition.  (2)) and the Dirichlet boundary condition of ( S ). ◻ Funding Open access funding provided by Università della Calabria within the CRUI-CARE Agreement.
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