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Symmetry and Non-Uniformly Elliptic Operators Symmetry and non-uniformly elliptic operators Jean Dolbeault ∗ Ceremade, UMR no. 7534 CNRS, Universit´e Paris IX-Dauphine, Place de Lattre de Tassigny, 75775 Paris C´edex 16, France E-mail: [email protected] Internet: http://www.ceremade.dauphine.fr/∼dolbeaul Patricio Felmer ∗; y Departamento de Ingenier´ıa Matem´atica and Centro de Modelamiento Matem´atico, UMR no. 2071 CNRS-UChile Universidad de Chile, Casilla 170 Correo 3, Santiago, Chile E-mail: [email protected] R´egis Monneau ∗ CERMICS, Ecole Nationale des Ponts et Chauss´ees, 6 et 8 avenue Blaise Pascal, Cit´e Descartes, Champs-sur-Marne, 77455 Marne-La-Vall´ee, France E-mail: [email protected] Internet: http://cermics.enpc.fr/∼monneau/home.html Abstract. The goal of this paper is to study the symmetry properties of nonnegative solutions of elliptic equations involving a non uniformly elliptic operator. We consider on a ball the solutions of ∆pu + f(u) = 0 with zero Dirichlet boundary conditions, for p > 2, where ∆p is the p-Laplace operator and f a continuous nonlinearity. The main tools are a comparison result for weak solutions and a local moving plane method which has been previously used in the p = 2 case. We prove local and global symmetry results when u is of class C 1,γ for γ large enough, under some additional technical assumptions. Keywords. Elliptic equations, non uniformly elliptic operators, p-Laplace operator, scalar field equations, monotonicity, symmetry, local symmetry, positivity, non Lipschitz nonlin- earities, comparison techniques, weak solutions, maximum principle, Hopf's lemma, local moving plane method, C 1,α regularity AMS classification (2000). 35B50, 35B65, 35B05 ∗Partially supported by ECOS-Conicyt. yPartially supported by Fondecyt Grant # 1030929 and FONDAP de Matem´aticas Aplicadas. 1 2 March 21, 2004 | J. Dolbeault, P. Felmer & R. Monneau 1 Introduction and main results The goal of this paper is to prove local and global symmetry results for the nonnegative solutions of ∆pu + f(u) = 0 in B 8 u ≥ 0 in B (1) < u = 0 on @B p−2 : where ∆pu = div(jruj ru) is the p-Laplace operator or p-Laplacian, and B = B(0; 1) is the unit ball in Rd, d ≥ 1. We will focus on the non uniformly elliptic case: p > 2, and consider nonlinearities f which are continuous on R+. In earlier works of two of the authors [14, 15, 16], it has been noted that for p = 2 the C2 regularity of u is apparently a threshold in order to apply the moving plane technique. Here we will see that the threshold for p > 2 is rather C 1;1=(p−1). As in [16], additional properties of f are required at least close to its zeros. The main tools of this paper are the maximum principle when the nonlinearity is nonincreasing or, based on a local inversion, when ru 6= 0, and a local moving plane technique. Our results are not entirely satisfactory, since the statements are rather technical, but on the other hand, they are a first achievement in obtaining symmetry results by comparison techniques in the case of non uniformly elliptic operators. Let us start with a simple case, where we assume the following additional conditions on f. (H1) Let f be a real function defined on R+ such that: a) f 2 Cα(R+) for some α 2 (0; 1), b) 9 θ > 0 such that f > 0 on (0; θ), f < 0 on (θ; +1), c) f is nonincreasing on (θ − η; θ + η) for some η > 0. Theorem 1 (Symmetry result for the p-Laplacian with p > 2) Let p > 2, γ 2 (1=(p − 1); 1). Consider a function f satisfying (H1) and a solution u 2 C1,γ(B) of (1). Then u is positive and radially symmetric on B. Remark 1 A consequence of the C 1,γ regularity of the solution on the whole ball is that the maximum of this solution needs to be reached at the level u = θ (see Lemma 8). Does such a solution exist ? Given a function f and using a shooting method, we obtain a solution on a ball of radius large enough. Then a rescaling gives a solution on B for an appropriately rescaled nonlinearity f, which provides an example of existence of a solution. A result of symmetry similar to that of Theorem 1 also holds under weaker regularity assumptions on f. (H2) f 2 C0(R+) and f > 0 on (0; +1). Theorem 2 (Symmetry result with f only continuous) Let p > 2, γ 2 (1=(p − 1); 1). Assume that (H2) holds and consider a solution u 2 1;p 1 W (B) \ L (B) of (1) such that, with M = kukL1(B), u 2 C2 B n (ru)−1(0) \ C1,γ u−1([0; M)) : If f is nonincreasing in a neighborhood of M, then u is positive and radially symmetric on B. March 21, 2004 | Symmetry and non-uniformly elliptic operators 3 Remark 2 If we assume that u belongs to C 1,γ(B) with γ > 1=(p−1), then f(M) = 0. See Lemma 8. Reciprocally if f(M) 6= 0, as we shall see in Example 1, a solution like the one of Theorem 2 has a regularity which is not better than C 1;1=(p−1) at the points x such that u(x) = M. This is the reason why the C 1,γ regularity cannot be assumed at such points. The starting point of the moving plane technique is the celebrated paper by Gidas, Ni and Nirenberg [20] based on [1, 2, 26]. An improvement was made in [19, 5, 27] using the monotonicity properties of the nonlinearity. This lead to a local moving plane technique [14, 15, 16] in the case of the Laplacian, which is well suited for nonlinearities with low regularity. The corresponding notion of local symmetry also seems appropriate in the case of the p-Laplacian. Further interesting results were obtained in [17, 18]. For nonlinear elliptic operators, the story is shorter [8, 9, 10]. It strongly relies on comparison methods [29, 6, 7] but covers the case of the p-Laplace operator only when p < 2. For completeness, let us mention that related results have been obtained by rearrangement techniques, for instance in [23, 21], and that in [3, 4], F. Brock has been able to get symmetry results for the p-Laplace operator by a completely different approach based on a continuous Steiner symmetrization. Our goal is to get local symmetry results for p > 2 using an adapted version of the maximum principle [11, 24, 25] and regularity properties [12, 28, 13, 22], with a local moving plane method which is essentially adapted from [15, 16]. The paper is organized as follows. In the next section, we state a local symmetry result (Theorem 3). Theorem 2 is a simple corollary of Theorem 3. In the third section we start with two examples, which motivate the choice of the class of regularity of the solutions and then state three useful lemmas. The fourth section is devoted to the proof of Theorem 3 by the local moving plane method. Proofs of Theorems 1 and 2 are contained in the proof of Theorem 3. 2 A general result In this section we present a local symmetry result, generalizing Theorem 2 to sign-changing functions f. (H) The function f 2 C0(R+) has a finite number of zeros and f(0) ≥ 0. Moreover, if f(u0) = 0 for some u0 ≥ 0, then there exists η > 0 such that: + (a) either f is nonincreasing on (u0 − η; u0 + η) \ R , (b) or f(u) ≥ 0 on (u0; u0 + η), −1 We say that u0 2 f (0) belongs to Za in case (a) and to Zb in case (b). Note that −1 f (0) = Za [ Zb. Theorem 3 (Local symmetry result with f only continuous) Let p > 2, γ 2 (1=(p − 1); 1). Assume that f satisfies (H) and consider a C 1(B) solution u of (1) such that, with M = kukL1(B), u 2 C2 B n (ru)−1(0) \ C1,γ u−1([0; M)) : −1 −1 If f is nonincreasing in a neighborhood of M and if the set J := u (Zb) \ (ru) (0) is empty, then u is locally radially symmetric and the set u−1([0; M))\(ru)−1(0) is contained in Za. 4 March 21, 2004 | J. Dolbeault, P. Felmer & R. Monneau Here locally radially symmetric [4] means that there exists an at most countable family (ui)i2I of radial functions with supports in balls Bi, i 2 I, such that uijBi is a nonnegative radial nonincreasing solution of ∆pui +f(ui +Ci) = 0 on Bi, where uij(BnBi) ≡ 0 and Ci ≥ 0 is a constant satisfying f(Ci) = 0, and such that u = ui : Xi2I Remark 3 In the case of Theorem 3, I is finite. Monotonicity holds on balls or annuli. As a consequence, in Theorems 1 and 2, monotonicity holds along any radius. The mono- tonicity is even strict in case of Theorem 2, while there might be a plateau at level θ in case of Theorem 1. The regularity of locally radially symmetric solutions can be studied by elementary meth- ods, which, under an additional condition on f in case (a) of (H), allows to give a sufficient condition under which all sufficiently smooth solutions are globally radially symmetric. Corollary 4 Let α 2 (0; (p − 2)=2) and consider f such that for any u0 2 Za, jf(u)j 0 < lim inf α < +1 : u!u0 ju − u0j α+1 Under the same assumptions as in Theorem 3, if p−α−1 < γ < 1, then u is globally radially symmetric and decreasing along any radius.
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