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A SYMMETRY RESULT FOR AN OVERDETERMINED ELLIPTIC PROBLEM USING CONTINUOUS,...

375

RENDICONTI DEL CIRCOLO MATEMATICO DI PALERMO Serie II, Tomo LI (2002), pp. 375-390

A SYMMETRY RESULT FOR AN OVERDETERMINED ELLIPTIC PROBLEM USING CONTINUOUS REARRANGEMENT AND DOMAIN DERIVATIVE F. BROCK - A. HENROT

∗

We develop a new method to prove symmetry results for overdetermined boundary value problems. This method is based on the use of continuous Steiner symmetrization together with derivative with respect to the domain. It allows to consider nonlinear equations in divergence form with dependance in r = |x| in the nonlinearity. By using the notion of “local symmetry” introduced by the first author, we prove that the domain is necessarily a ball. We also give an example where the solution of the overdetermined problem is not radially symmetric.

1. Introduction. Following the seminal papers of J. Serrin [22] and H. F. Weinberger [27], considerable attention has been given these last twenty-five years to the study of overdetermined boundary value prolems. To prove symmetry results, three different approaches have been essentially used. First of all, the Alexandrov-Serrin moving plane method together with a boundary point lemma, for example in [22], [8], [26], [20]. A second approach is based on some maximum and comparison principle (often together with some Rellich’s identity), see e.g. [27], [13], [1], [19]. Another idea, introduced e.g. in [2], [18], is to use an integral identity equivalent to the overdetermined problem under consideration. The aim of this paper is to give an alternative approach based on the continuous Steiner symmetrization together with the derivative with respect to the domain. For other symmetry results using these two tools, we refer to some recent papers: [7], [6], [15]. As an application of our method, we investigate the following overdetermined boundary value problem: (1) ∗

− div (g(|∇u|)|∇u|−1 ∇u) = f (x, u) in , The second author wants to thank the University of Köln for a visiting appointment.

376 (2) (3)

F. BROCK - A. HENROT

u = 0 on ∂, |∇u| = constant on ∂,

where is a connected regular open subset of R N , f , g are given functions satisfying some regularity or monotony assumptions which will be given later. Our aim is to prove the following symmetry result: If is such that there exists a function u satisfying (1)-(3), then is a ball. Note that the problem (1)-(3) seems to fall out of the scope of the classical methods recalled just above because of the dependance on x in the equation. We also mention that, if the differential operator in (1) is not uniformly elliptic (e.g. in the case of the p- Laplacian for p = 2) and if f is independent of x , then to prove the symmetry of the solution, the moving plane technique works only under additional geometric assumptions on the solutions (compare [8], [9], [10], [11], [12], [23]). In section 2, we collect some material which will be used in the following. In particular, we recall the definitions and main properties of the continuous Steiner symmetrization and of the derivative with respect to the domain. Section 3 is devoted to the statement and the proof of the main result of the paper. In section 4, we give an example of [4] which shows that it can happen, for a problem like (1)-(3), that the domain is a ball while the solution u of the problem is not radially symmetric. 2. Prerequesites. In this paper, we will denote by x = (x 1 , . . . , xn ) the points in the Euclidean space Rn , n ≥ 2, and by r = |x| := (x 12 + · · · + xn2 )1/2 its norm. In general will denote a bounded domain in R n , and we will always assume that its boundary ∂ is Lipschitz. For a function u or a vector-valued function U , ∇u will denote the gradient of u and div(U ) the divergence of U (both in the distributional sense). We will work in the framework of the classical Lebesgue and Sobolev spaces: L p (), 1 ≤ p < +∞ is the class of Lebesgue measurable functions which are p th power integrable over and W 1, p () is the class of functions in L p () such that all the first derivatives (in the distributional sense) are also in L p (). Let W01, p () be the closure in W 1, p () of the space of infinitely differentiable functions with compact support in . As usual, we will denote ∗ by p ∗ the critical exponent of the Sobolev embedding of W 1, p () into L p () ∗ ∗ ( p = np/(n − p) if n > p, p = +∞ else). For any function space the subscript ” + ” denotes the corresponding subset of nonnegative functions, e.g. p 1, p L + (), W0+ ().


377

2.1 The continuous Steiner symmetrization. In this and in the following subsection we give the definition of continuous symmetrization and some of its properties which were investigated in [3], [4]. This material is relatively new, and we will describe it in more detail. For x ∈ Rn we write x = (x , y)

x = (x1 , . . . , xn−1), y = xn .

Let M(Rn ) be the set of Lebesgue measurable - measurable in short - sets in Rn with finite measure. If M ∈ M(Rk ) then we denote by |M| its k-dimensional measure, (1 ≤ k ≤ n). Below we will treat measurable sets and functions in a.e. sense. Let S+ (Rn ) denote the class of real, nonnegative measurable functions u satisfying |{x ∈ Rn : u(x) > c}| < +∞

∀c > 0.

Note that L +p (Rn ) and W+1, p (Rn ), 1 ≤ p < +∞, are subspaces of S+ (Rn ). For the convenience of the reader we include the definition of the wellknown Steiner symmetrization (see e.g. [21]). 1) For any set M ∈ M(R) let M ∗ := (−(1/2)|M|, +(1/2)|M|) denote the (one-dimensional) symmetrization of M . 2) Let M ∈ M(Rn ). For every x ∈ Rn−1 let M(x ) := {y ∈ R : (x , y) ∈ M}. The set (4)

M ∗ := {x = (x , y) : y ∈ (M(x ))∗ , x ∈ Rn−1 }

is called the Steiner symmetrization of M with respect to y. Note that M ∗ is symmetric and convex with respect to the hyperplane {y = 0}. 3) If u ∈ S+ (Rn ) then the function (5)

u ∗ (x , y) := sup{c > 0 : y ∈ {u(x , ·) > c}∗ },

x ∈ Rn ,

is called the Steiner symmetrization of u. Note that u ∗ (x , y) is symmetric with respect to {y = 0} and nonincreasing in y for y > 0. DEFINITION 2.1. (Continuous symmetrization of sets in M(R))

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A family of set transformations E λ : M(R) → M(R),

0 ≤ λ ≤ +∞,

satisfying the properties, (M, n ∈ M(R), 0 ≤ μ, λ ≤ +∞) (i) |E λ (M)| = |M|, (equimeasurability), (ii) If M ⊂ N , then E λ (M) ⊂ E λ (N ), (monotonicity), (iii) E λ (E μ (M)) = E μ+λ (M), (semigroup property) (iv) If I = [y1 , y2 ] is a bounded closed interval, then E λ (I ) = [y1λ , y2λ ], where: (6)

y1λ = (1/2)(y1 − y2 + e−λ (y1 + y2 )), y2λ = (1/2)(y2 − y1 + e−λ (y1 + y2 )),

is called a continuous symmetrization. We will write M λ := E λ (M) for the symmetrized sets. The existence, uniqueness and some further properties of the family {E λ }, 0 ≤ λ ≤ +∞, are proved in [4], Theorem 2.1. In particular we have for every M ∈ M(R) M 0 = M,

(7)

M ∞ = M ∗.

DEFINITION 2.2. (Continuous Steiner symmetrization) 1) Let M ∈ M(Rn ). The family of sets (8)

M λ := {x = (x , y) : y ∈ (M(x ))λ , x ∈ Rn−1 }, 0 ≤ λ ≤ +∞,

is called the continuous Steiner symmetrization. 2) Let u ∈ S(Rn ). The family of functions u λ , 0 ≤ λ ≤ +∞, defined by (9)

u λ (x) := sup{c > inf u : x ∈ {u > c}λ },

x ∈ Rn ,

is called continuous Steiner symmetrization of u with respect to y. Remark 2.1. Let us summarize some basic properties of the continuous symmetrization which are proved in [3], [4] (M ∈ M(R n ), u, v ∈ S+ (Rn ), λ ∈ [0, +∞]). 1) Equimeasurability From the very definition we have (10)

|M| = |M λ |

and

{u λ > c} = {u > c}λ

2) Cavalieri’s principle (see [3], Theorem 8) (11) F(u)d x = F(u λ )d x, Rn

Rn

∀c > 0.


379

if F is Borel measurable and the left-hand side of (11) converges. 3) Homotopy We have by 1) (12)

u 0 = u,

u∞ = u∗.

Furthermore, if λm → λ as m → +∞ and u is a.e. finite, then (see [3], Theorem 7), u λm → u λ

(13)

in measure.

4) If M is open, then M λ has an open representative (see [4], Lemma 2.1). Moreover, it is possible to give a pointwise definition of open sets, thus managing, that all the sets M λ , 0 ≤ λ ≤ +∞, are open (see [4], section 2). 5) If u is continuous, then u λ has a (unique) continuous representative (see [3], Theorem 11). Let us agree that if we speak about the continuous symmetrization of open sets or of continuous functions, then we always mean their precise representatives. 6) If u is Lipschitz continuous with Lipschitz constant L , then u λ is Lipschitz continuous too with Lipschitz constant less or equal to L , (see [3], Theorem 7). In contrast, if u ∈ C 1 , then we do not have u λ ∈ C 1 in general (see [3], Remark 8). Remark 2.2. Let us assume that is a bounded open and convex set in

Rn . Then, for an arbitrary system of coordinates x = (x , y), admits a

representation (14)

= {x = (x , y) : y1 (x ) < y < y2 (x ), x ∈ }

where ⊂ Rn−1 is bounded, open and convex, yi ∈ C( ), i = 1, 2, y1 is convex and y2 is concave. In view of Definition 2.1, 2.2, we see that the continuous symmetrizations of can be represented as follows. (15)

λ = {x = (x , y) : y1λ (x ) < y < y2λ (x ), x ∈ },

0 ≤ λ ≤ +∞,

where the functions yiλ , i = 1, 2, are computed according to (6). It is easy to see that the domains λ , 0 ≤ λ ≤ +∞, are convex too. PROPOSITION 2.1. (see [4], Theorem 3.2, Corollary 3.3 and Theorem 4.1) 1, p Let be a bounded domain and u ∈ W 0+ () for some p ∈ [1, +∞). Then, 1, p (λ ) and for every λ ∈ [0, +∞], u λ ∈ W0+ (16)

∇u λ p ≤ ∇u p .

380


Furthermore, uλ → u

(17)

in

W 1, p

as λ 0.

1,1 Finally, if u ∈ W 0+ () and G is some positive increasing and convex function on [0, +∞[, then for every λ ∈ [0, +∞], λ (18) G(|∇u |)d x ≤ G(|∇u|)d x, λ

provided that the right integral in (18) converges. PROPOSITION 2.2. (see [4], Theorem 5.1, Remark 5.1) n Let u ∈ L ∞ + (R ) and suppose that u vanishes outside some ball B R , R > 0. Suppose that F = F(x, v) is bounded and measurable on B R × [0, sup u], continuously differentiable in v and satisfies (19)

F(x, 0) = 0, |F(x, v)| ≤ A(x) ∀(x, v) ∈ BR × [0, sup u],

where A ∈ L 1+ (B R ). Furthermore, assume that (20)

(∂/∂v)F(x, v)

is even in y and nonincreasing in y for y > 0.

Then for every λ ∈ [0, +∞], (21) F(x, u)d x ≤ BR

F(x, u λ )d x. BR

2.2 Local symmetry. In this subsection we introduce the notion of local symmetry and give an analytic description in terms of the continuous Steiner symmetrization. DEFINITION 2.3. (see [4], Definition 6.1) Let be a bounded domain, u ∈ C 1 () ∩ C(), u > 0 in and u = 0 on ∂. Furthermore, suppose that u has the following property. If x 1 = (x0 , y1 ) ∈ Rn with (22)

0 < u(x 1 ) < sup u,

∂u 1 (x ) > 0, ∂y

and x 2 is the (unique !) point satisfying (23) x 2 = (x0 , y2 ),

y2 > y1 ,

u(x 1 ) = u(x 2 ) < u(x 0 , y) ∀y ∈ (y1 , y2 ),


381

then (24)

∂u 1 ∂u 2 (x ) = (x ), i = 1, . . . , n − 1, and ∂ xi ∂ xi ∂u 1 ∂u 2 (x ) = − (x ). ∂y ∂y

Then u is called locally symmetric in the direction y. u is called locally symmetric in every direction, if for every rotation of the cartesian coordinate system x → ξ = (ξ , η), (ξ ∈ R n−1 , η ∈ R), the function v(ξ ) := u(x) is locally symmetric with respect to η. Surprisingly it can be proved that functions which are locally symmetric in every direction are “locally” radially symmetric. PROPOSITION 2.3. (see [4], Theorem 6.1) Let u be locally symmetric in every direction. Then we have the following decomposition (25)

{0 < u < sup u} =

m

B Rk (z k ) \ Brk (z k ) ∪ S,

k=1

where

(26)

Rk > rk ≥ 0, z k ∈ {0 < u < sup u}, ∂u < 0 in B Rk (z k ) \ Brk (z k ), (ρ : radial distance from z k ), ∂ρ min { u(x) : x ∈ Brk (z k )} = min {u(x) : x ∈ ∂ Brk (z k )}, 1 ≤ k ≤ m, ∇u = 0

and

in S.

The sets on the right-hand side of (25) are disjoint and there can be a countable number of annuli, i.e. m = +∞. Next we give a purely analytic description of local symmetry in terms of the continuous symmetrization. PROPOSITION 2.4. (see [4], Theorem 6.2) Let be a bounded open set, u ∈ C 1 () ∩ C(), u > 0 in , u = 0 on ∂. Furthermore, let G be a positive increasing and strictly convex function on [0, +∞[ and suppose that 1 λ G(|∇u|)d x − G(|∇u |)d x = 0. (27) lim λ0 λ λ

382


Then u is locally symmetric in direction y. 2.3 Derivative with respect to the domain. In this subsection, we want to recall the definition of the derivative with respect to the domain and some well-known formulas which are known as Hadamard-Schiffer’s formulae. We refer for more details, proofs and a comprehensive mathematical study of this subject to the classical papers of MuratSimon [17], Simon [24] or the book of Sokolowski-Zolesio [25]. 1) Let J be a “domain functional”, that is to say a functional which assigns a numerical value to every (regular) domain ⊂ Rn . Let θ be any Lipschitz vector field from Rn to Rn (θ can be understood as a field of deformation for ) and set t = {x + tθ (x), x ∈ }, t > 0. The application I d + tθ is a pertubation of the identity which is a Lipschitz diffeomorphism for t small enough. By definition, the derivative of J at in the direction θ is (28)

d J (, θ ) := lim t→0

J (t ) − J () . t

2) Generally, we are faced with some functional J which depends on the domain through the solution of some boundary value problem, say u . So, we need to define also the domain derivative of u . We will denote by u (; θ ), or, more simply by u , the function which is defined on every compact subset of (and then on the whole ) by u − u (29) u := lim t . t→0 t As an example, let us consider a function u which is a solution of the following Dirichlet problem (30)

− div (g(|∇u|)|∇u|−1∇u) = f (x, u) in , u = 0 on ∂.

Then, if g and f are differentiable, we can prove ([24], [25]) that u is the solution of the linear Dirichlet problem

(31)

∇u .∇u − div (|∇u |g (|∇u |) − g(|∇u |)) ∇u + |∇u |3 g(|∇u |)|∇u |−1 ∇u = f u (x, u )u in , ∂u u = − θ.n on ∂, (n : exterior normal). ∂n

Let us notice that in the following we will only use the boundary condition of (31).


383

We also need to know how to compute the derivative of an integral over . This is the classical Hadamard formulae (see [24] or [25]). For J () = h(u )d x , we have

(32)

d J (, θ ) =

h

(u )u d x

+

∂

h(u )θ.ndσ.

3. The main result. Throughout this section, let p ∈ (1, +∞) be fixed and let G denote a positive increasing and strictly convex function of class C 2 on [0, +∞[. We will denote by g = G the derivative of G and we will assume that g satisfies (33) c1 (z p − 1) ≤ zg(z) ≤ c2 (z p + 1),

c3 ≤ zg (z)/g(z) ≤ c4

∀z ≥ 0,

where c1 , . . . , c4 are positive constants. Let f (x, v) be measurable on Rn × [0, +∞[ and continuous and nonincreasing in v. Furthermore, we suppose that f satisfies f = f (r, v),

(34)

f

(r = |x|),

is nonincreasing in r

and (35)

| f (x, v)| ≤ a0 (x)v p

∗ −1

+ a1 (x)

q∗

n n ∗ where a0 ∈ L ∞ + (R ) and a1 ∈ L + (R ). (Here q is the conjugate exponent of ∗ p .) Let F be the following primitive of f , v f (x, w)dw, (x, v) ∈ Rn × [0, +∞[. F(x, v) = 0

Note that F(x, v) is concave in v. Then, it is well known (see e.g. [16]) that the boundary value problem (36)

− div (g(|∇u|)|∇u|−1∇u) = f (x, u) in , u = 0 on ∂,

has a unique solution u which can be characterized as the minimum on 1, p W0 () of the convex functional (37) J (v) := (G(|∇v(x)|) − F(x, v(x)))d x.

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We can now state the main result of this paper. THEOREM 3.1. Let be a bounded convex domain in R n . Assume that the solution u of (36) is in C 1 () and that it satisfies the overdetermined condition (38)

|∇u | = constant = c > 0

on

∂.

Then is a ball. Roughly speaking, the proof of this theorem consists in proving that the function u satisfies (27). Then the assertion follows from Proposition 2.3. To be more precise, we divide the proof in four steps, each of them being of its own interest. Step 1. Computing some domain derivatives. We will need the domain derivative of the functional J (u ) where u is the solution of (36). PROPOSITION 3.1. Let j () := J (u ) is the solution of (36) and J is given by (37) and let θ be some deformation field. Then the derivative of j in the direction θ is given by (39) d j (; θ ) = {G(|∇u |) − g(|∇u |)|∇u |}θ · ndσ. ∂

Proof. By using the Hadamard formula (32), we obtain, since u = 0 on the boundary and F(x, 0) = 0, d j (; θ ) = (g(|∇u |)|∇u |−1 ∇u · ∇u − f (x, u )u )d x (40) + G(|∇u |)θ · ndσ, ∂

where u is defined as the solution of (31). By multiplying both sides of the equation (30) by u and integrating by parts on we infer that −1 ∂u dσ + g(|∇u |)|∇u |−1 ∇u · ∇u d x − u g(|∇u |)|∇u | ∂n ∂ (41) f (x, u )u d x. =


385

Now, by using the boundary condition in (31), we obtain −1 g(|∇u |)|∇u | ∇u · ∇u d x − f (x, u )u d x = (42) =− g(|∇u |)|∇u |θ · ndσ. ∂

This, together with (40) implies (39). Step 2. The “Steiner symmetrization field”. Now we construct a particular deformation field θ ∗ generated by the continuous Steiner symmetrization. Let H be an arbitrary (n − 1)-hyperplane containing the origin and select Cartesian coordinates such that y = x n = 0 on H . By using the notation of Remark 2.2, we set (43)

y(x ) :=

1 [y1 (x ) + y2 (x )], 2

and we define the Steiner deformation field θ ∗ by (44)

θ ∗ (x , y) := (0, −y(x )) ∀x ∈ .

Note that θ ∗ is constant along every straight line orthogonal to H and by (15) we have (45)

t = λ

and t = 1 − e−λ > 0 (0 ≤ λ ≤ +∞).

Furthermore, θ ∗ is Lipschitz since is convex, so that the derivative d j (; θ ∗ ) may be computed from (28) and (39). Since the continuous Steiner symmetrization preserves the volume, the derivative of the volume with respect to θ ∗ vanishes, i.e. ∗ θ ∗ · ndσ = 0. (46) dVol(; θ ) = ∂

Step 3. Using the variational formulation. Here we introduce three different functions: u is the solution of the problem (36) on , u t is the solution of the same problem (36), but posed on t and (47)

u t := (u )λ ,

(t = 1 − e−λ , 0 ≤ λ ≤ +∞),

is the continuous Steiner symmetrization of u (see section 2.1). Since u t 1, p belongs to the Sobolev space W0 (t ) (see Proposition 2.1), it is admissible

386


for the variational formulation of (36). Thus we have that (G(|∇u t (x)|)−F(x, u t (x)))d x t (48) ≤ (G(|∇u t (x)|) − F(x, u t (x)))d x. t

By substracting J (u ) from both sides of (48) we get G(|∇u t (x)|) − G(|∇u (x)|)d x t F(x, u t (x))d x − F(x, u (x))d x − t (49) G(|∇u t (x)|)d x − G(|∇u (x)|)d x ≤ t F(x, u t (x))d x − F(x, u (x))d x . − t

In view of (34) and (35) the function F satisfies (19) and (20). (In particular, (20) is valid in every rotated coordinate system.) According to Proposition 2.2, we infer that (50) F(x, u t (x))d x − F(x, u (x))d x ≤ 0, t

which means that the right-hand side of (49) is less than G(|∇u t (x)|)d x − G(|∇u (x)|)d x ≤ 0 (51) t

Now we multiply both sides of (49) with (1/t), (t > 0), and we let t tend to zero. By the definition of the domain derivative, the left-hand side of (49) tends to d j (; θ ∗ ) (see (39)). Since |∇u | is constant on the boundary, (39) and (46) yield d j (; θ ∗ ) = 0.

(52) This finally gives (53)

1 0 ≤ lim t→0 t

t

G(|∇u t (x)|)d x −

G(|∇u (x)|)d x

≤ 0,

the last inequality in (53) coming from (51) and (18). But (53) means that u is locally symmetric in direction y by Proposition 2.4, and since the hyperplane H was arbitrary, we have proved that u is locally symmetric in every direction.


387

Step 4. Conclusion. To complete the proof of Theorem 3.1, we use Proposition 2.3. First we observe that the set S in (25) does not reach the boundary ∂ since we have |∇u | > 0 on ∂ by (38). Furthermore, the union of annuli in (25) is locally finite near the boundary. Indeed, let z ∈ ∂ and assume that there exists a sequence of annuli {B Rk (z k ) \ Brk (z k )} such that z k → z, rk → 0, Rk → 0 as k → ∞. By selecting points x k ∈ ∂ Brk (z k ) we see that ∇u (xk ) = 0. Since u ∈ C 1 () we infer from this (54)

∇u (x) = lim ∇u (xk ) = 0, k→∞

in contradition to (38). Since is convex it is clear that ∂ coincides with the exterior boundary of a single ringshaped subset in (25). This achieves the proof of Theorem 3.1. Remark 3.1. It seems likely that the convexity assumption in Theorem 3.1 can be relaxed. Indeed the main difficulty when is not convex is to define the continuous Steiner symmetrization field θ ∗ in such a way that the Hadamard formulas (28), (39) remain valid. t can always be defined (see [3]) but the generated deformation field θ ∗ is neither Lipschitz nor continuous in general. (It is easy to give an example of a connected C ∞ domain such that t is not Lipschitz for every t ∈ (0, 1)). Nevertheless, is seems to be possible to come back to the definition of the derivative and to find a direct proof (including some careful study of the singular integrals) of the formulae (28), (39). 4. A counter-example. The following example is taken from [4]. On one hand, it shows that it is not easy to sharpen the conclusion of Theorem 3.1. There are solutions of the problem (1)-(2) in a ball which are not radially symmetric. On the other hand, it is an illustration of the geometry described in Proposition 2.3 and by this, it also justifies the concept of local symmetry. We emphasize that examples analogous to the one given below have also been found by other authors independently (see [14] and [23]). Furthermore, H . Berestycki told us in a discussion that he had used similar constructions in his lectures in the early nineteenth. Let p ≥ 2, s > 2,

(1 − |x|2 )s if |x| ≤ 1 w(x) = , 0 if |x| > 1 and

v(x) =

1 s 1 − (|x|2 − 25)/11

if |x| < 5 if 5 ≤ |x| ≤ 6 .

388


We choose x 1 , x 2 ∈ B4 with |x 1 − x 2 | ≥ 2 and set u(x) := v(x) + w(x − x 1 ) + w(x − x 2 ),

x ∈ B6 .

The graph of u is built up by three radially symmetric “mountains”, one of them having a “plateau” at height 1 while the other two are congruent to each other their “feet” lying on the plateau. After a short computation we see that u satisfies − p u ≡ − div (|∇n| p−2 ∇u) = f (u), u > 0 in

B6 ,

u = 0 on ∂ B6 , |∇u| = where

f (u) :=

constant on ∂ B6 ,

⎧ (2s/11) p−1 (25 + 11(1 − u) 1/s )( p/2)−1 (1 − u) p−( p/s)−1 × ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ×{(50/11)( p − 1)(s − 1) + (2 ps − 2s − p + n)(1 − u) 1/s }

if 0 ≤ u ≤ 1,

⎪ ⎪ (2s) p−1 (1 − (u − 1)1/s )( p/2)−1 (u − 1) p−( p/s)−1 × ⎪ ⎪ ⎪ ⎩ ×{−2(s − 1)( p − 1) + (2 ps − 2s − p + n)(u − 1) 1/s }

if 1 ≤ u ≤ 2.

If p = 2 and s > 2 then we have f ∈ C ∞ ([0, 2] \ {1}) ∩ C 1−(2/s) ([0, 2]). The difference quotient of f is not bounded below near u = 1, i.e. f ∈ C 1 ([0, 2]). In contrast, if p > 2 and s > p/( p − 2), then we have f ∈ C 1 ([0, 2]). Remark 4.1. (i) Assume that u is a solution of the boundary value problem (55) (56)

− p u = f (u), u > 0 in

B,

u ∈ W01, p (B) ∩ C 1 (B),

where B is a ball and f is continuous. Then u is locally symmetric in every direction (see [5], Theorem 7.2 and Corollary 7.6). Furthermore, it can be proved under various additional assumptions on f that the solution u is radially symmetric (see [9], [10],[11], [12], [23] and [5]). (ii) Unfortunately, the nonlinearity f in the counterexamples above and in [23] have two positive zeros. Therefore it is natural to raise the following problem: Is there a non-radial solution of (55), (56) where f has only one positive zero? The answer is no. This can be seen as follows: Suppose that f has only one positive zero and that u is not radially symmetric. Using Proposition 2.3 we find a ball B ⊂⊂ B and a number c > 0 such that u = c, ∇u = 0 on ∂ B and u > c in B . Then, arguing as in the proof


389

of Theorem 1 of [5] we obtain that f (c) = 0. On the other hand, the divergence theorem yields f (u)d x = |∇u| p−1 d S = 0, B

∂ B

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Friedmann Brock Dept. of Mathematics University of Missouri-Columbia Columbia MO 65211 U.S.A. e-mail: [email protected] Antoine Henrot Institut Elie Cartan Nancy Universite Henri Poincare Nancy 1, B.P. 239 54506 Vandoeuvre les Nancy Cedex, France e-mail: [email protected]