Symmetry of C^1 solutions of p-Laplace equations in R^N

Symmetry of C 1 Solutions of p - Laplace equations in IRN Lucio Damascelli∗ Dipartimento di Matematica, Universit`a di Roma “ Tor Vergata ” Via della Ricerca Scientifica 00133 Roma, Italy

Mythily Ramaswamy T.I.F.R. Centre I.I.Sc. Campus Bangalore - 560 012 India

Abstract The moving plane method is used in this paper to get the radial symmetry of the positive C 1 ground state solutions of the equation − div(|Du|p−2 Du) = f (u) in IRN , in the case 1 < p < 2, for a locally Lipschitz continuous function f on (0, ∞) under two different sets of assumptions on the behaviour of f near zero, namely f is either nonincreasing or f (u) = O(uα+1 ). In the  latter  case, we assume further that the 1 solution u = O |x|1m and Du(x) = O |x|m+1 at infinity with m(α + 2 − p) > p, to derive our symmetry results. We do not require any condition on the set where Du (x) = 0. These results extend to p-Laplace equations, 1 < p < 2, earlier results available for the case p = 2.

1

Introduction

In this paper, we study symmetry properties of positive solutions of p-Laplace equations in IRN , N ≥ 2, with the ground state condition at infinity, namely (

for 1 < p < 2,

−∆p u(x) = f (u(x)), u > 0 in IRN u(x) → 0 as |x| → ∞

(1.1)

with ∆p u = div(|Du|p−2 Du).

In the case p = 2, this problem for C 2 solutions was first considered by Gidas, Ni and Nirenberg in [GNN], using the method of moving planes of Alexandrov and Serrin [S]. For a ∗

Supported by MURST, Project “Metodi Variazionali ed Equazioni Differenziali Non Lineari”

1

C 1 function f , satisfying f (0) = 0, f 0 (0) < 0 and f ∈ C 1+ν near zero, the radial symmetry of positive solutions is proved there. Furthermore, under another set of assumptions involving the decay rate of u at infinity and the growth rate of f near 0, symmetry results are derived. Later, these results were extended by Li ( see [L]) to fully nonlinear elliptic equations under milder conditions and by Li and Ni (see [LN]) for the case f in C 1 and f 0 (s) ≤ 0 for s small. For the p-Laplace operator, the coefficient (|Du|p−2 ) is vanishing or singular at the critical points of u for p > 2 or p < 2 respectively and hence strict comparison principles are available only in a restricted form outside the set of the common critical points of the functions being compared. That is why many earlier symmetry results for (1.1) were proved under some sort of assumptions on Z, the critical set of u. Symmetry results for positive solutions of p-Laplace equations, without any assumptions on Z, were derived in [DP1], [DP2] for bounded domains and in [DPR] for IRN by the moving plane method, as well as in [B1], [B2] with a different method. Positive weak solutions in W 1,p (IRN ) ∩ C 1 (IRN ) of (1.1) are proved in [DPR] to be radially symmetric under the assumption that f is locally Lipschitz on (0, ∞) and f is nonincreasing near 0. In [SZ] the authors study the symmetry of nonnegative C 1 ground states of quasilinear elliptic equations, which include the p-Laplacian, for 1 < p < 2, as a special case. Under the same assumptions on f as above, radial symmetry of nonnegative ground states of (1.1) is proved in [SZ], under the extra assumption that the set where |Du(x)| > 0 is connected, but requiring u to belong only to C 1 (IRN ) . They also consider the case p > 2 under the assumption that the solution has only one critical point. We use here the moving plane method to prove that C 1 weak solutions u of (1.1) are radially symmetric about a point in IRN , under two different sets of hypotheses on the nonlinearity f , without requiring any conditions on the set where Du(x) = 0. To be more precise, we assume that (H1) f is locally Lipschitz continuous in (0, ∞). and either one of the following conditions holds: 2

(H2) there exists s0 > 0 such that f is nonincreasing on (0, s0 ). (H3) There exists s0 > 0 and α > p − 2 such that for 0 < u < v < s0 , f (v) − f (u) (v − u)

(

≤

Cv α Cuα

if if

α≥0 α<0

.

Our main results are the following theorems: Theorem 1.1. Under the assumptions (H1) and (H2), if u ∈ C 1 (IRN ) is a weak solution of (1.1), for 1 < p < 2, then u is radially symmetric about some point x0 ∈ IRN , i.e. u = u(r), with r = |x − x0 | and u0 (r) < 0 for all r > 0. Theorem 1.2. Under the assumptions (H1) and (H3) let u ∈ C 1 (IRN ) be a weak solution of (1.1) for 1 < p < 2 satisfying further, for some m > 0, as |x| → ∞, u(x) = 0( |x|1m ) C ( u(x) ≥ |x|m as |x| → ∞ when α < 0 1 Du(x) = 0( |x|m+1 ) as |x| → ∞ .

),

(1.2)

If m(α + 2 − p) > p,

(1.3)

then u is radially symmetric about some point x0 ∈ IRN and is strictly radially decreasing. The proof of the above theorems are based on weak comparison theorems for solutions of p-Laplace equations in unbounded domains as in [DPR]. Theorem 1.1 is an improvement of the one proved in [DPR] in the sense that we do not require u ∈ W 1,p (IRN ) and the proof is simplified here. The main new ingredients here are a Poincar´e type inequality for functions on half balls, vanishing on the boundary hyperplane and the weighted Hardy-Sobolev inequality. It is this kind of Poincar´e inequality which allows us to drop the assumption that u ∈ W 1,p (IRN ) in Theorem 1.1, which was necessary in [DPR]. In Theorem 1.2 we use instead both inequalities. The hypothesis (H2) was used in [LN] and (H3) in [L], for a Lipschitz function f to derive symmetry results for positive solutions of nondegenerate quasilinear and fully nonlinear 3

elliptic equations. Here we assume only local Lipschitz property on (0, ∞) for f and extend in a “natural way ” the condition in [L] in Theorem 1.2 to the degenerate case 1 < p < 2. More precisely C. Li derived the symmetry results in [L], in particular, for the case of the Laplacian, under the assumptions f (v) − f (u) v−u

≤

C(|u| + |v|)α

for small u, v and some α > 0, with mα > 2 (see Theorem 3 in [L]). Clearly this condition on f is equivalent to (H3) when α > 0 and (1.3) reduces to mα > 2 when p = 2. In particular, if f (0) = 0, Li’s conditions imply f (u) u

≤

C|u|α

and mα > 2 .

For the same nonlinearity f , our conditions can be rewritten as f (u) ≤ Cuα1 up−1

for u > 0 and mα1 > p ,

where α1 = α + 2 − p. Thus our conditions are a natural generalizations of Li’s conditions. Let us stress here that the use of the weighted Hardy-Sobolev inequality in Theorem 1.2 is essential to get the “right” assumptions on the decay rate of u (right assumptions in the sense that are natural generalizations of those for the case p = 2). Example. Some model nonlinearities that we can treat are the functions that behave like the powers uα+1 , p − 1 < α + 1, near zero. More precisely if α > p − 2, f ∈ C 1 (0, ∞) and f 0 (s) = O(sα ) as s → 0, then using the mean value theorem it is easy to see that (H3) is satisfied. Our theorems apply to positive solutions. However the conditions on f which force a nonnegative ground state to be a positive one can be derived using the strong maximum principle in Vazquez [V]. Let us also remark that when f satisfies (H1) and (H2) and f (0) = 0, the strict maximum principle has been studied in [PSZ] for general quasilinear operators. For the case of p-Laplacian, their result is that if u is a nonnegative ground state of (1.1) under assumptions (H1) and (H2) and f (0) = 0, then u is positive on IRN if Z

0

dt |F (t)|1/p 4

=

∞,

(1.4)

where F is the primitive of the nonlinearity f . For the sake of reader’s convenience we give in Proposition 2.3 some conditions including (1.4), which ensure the positivity of the nonnegative solutions when (H1) and either (H2) or (H3) hold. The existence and uniqueness of radial ground state solutions of quasilinear equations have been studied in [FLS]. For a continuous f , with f (0) = 0 and f negative near 0 and positive later on, existence and uniqueness of nonnegative radial ground states have been proved under some additional hypotheses (see Corollary 1 in [FLS]). For those cases, where nonnegative radial ground states are necessarily positive and unique in the radial class, our symmetry result (Theorem 1.1) shows that there is a unique nonnegative ground state for (1.1) . In [GV] and [AGP], positive radial solutions of (1.1) with f (u) = uα+1 are considered. They prove that for p − 1 < α + 1 < p∗ − 1, the problem (

−(rN −1 |u0 (r)|p−2 u0 (r))0 = rN −1 |u|α u, 0 < r < ∞, u0 (0) = 0, u(r) ≥ 0 ,

(1.5)

has only trivial solutions (see corollary 3.4 in [AGP]). A nonnegative solution of (1.1) with f (u) = uα+1 for the above range of α’s, is necessarily positive everywhere because (1.4) holds for f here (see Proposition 2.3 (b) below). Then theorem 1.2 implies that (1.1) with f (u) = uα+1 ,

p ∈ (1, 2), p − 1 < α + 1 < p∗ − 1, has only trivial solutions in the class of

C 1 functions which decay like ( |x|1m ) at ∞ with m > p/(α + 2 − p). This extends partially the well known result of Gidas-Spruck [GS] regarding subcritical power nonlinearities, to p-Laplace equation. Let us remark that to our knowledge it is not known the precise decay rate of the solutions of (1.1) at infinity. For the critical case α = p∗ − 2, Guedda and Veron [GV] prove that the only positive radial solutions are ya (x) = (N a(

p−N N − p p−1 Np−p ) ) 2 (a + |x|p/p−1 ) p p−1

where a is any positive number. Then Theorem 1.2 with α = p∗ − 2 and m = (N − p)/(p − 1) p−N

gives the uniqueness of positive solution which decay like |x| p−1 at ∞, as it happens for the ya , and more generally with decay like

1 |x|m

with m > 5

p α+2−p

=

p p∗ −p

=

N −p p

.

As mentioned before, F. Brock (see [B1], [B2]) has obtained recently symmetry results by an entirely different method using continuous Steiner symmetrization for nonnegative solutions of quasilinear equations, which include p-Laplacian for 1 < p < ∞, in balls as well as in IRN . He assumes u ∈ W 1,p (IRN ) and f (u(.)) ∈ L1 (IRN ) but he needs less regularity on f , namely continuity and a certain growth condition in the neighbourhood of the zeros of f . His symmetry results concern either the case when f satisfy (H2) or the case of solutions with prescribed decay rate at infinity, but different from (1.2) (see Theorem 2, (i) and (ii) in [B1]). The paper is organized as follows. In section 2, we prove a Poincar´e type inequality, a weighted Hardy-Sobolev type inequality and some weak comparison theorems. In section 3 we give the proof of the symmetry theorems.

2

Preliminary Results

We start with some notations. Let ν be a direction in IRN , i.e |ν| = 1. Define for λ ∈ IR, the half space Σνλ = {x ∈ IRN : x · ν < λ}, and the hyperplane Tλν = {x ∈ IRN : x · ν = λ}. Let Rλν be the reflection through Tλν , i.e. Rλν (x) = xνλ = x + 2(λ − x · ν)ν , for any x ∈ IRN . We also set (Σνλ )0 = Rλν (Σνλ ). For a function u ∈ C 1 (IRN ), we define the reflected function uνλ (x) = u(xνλ ) ∀ x ∈ IRN .

6

We need weak comparison principles in unbounded domains, for positive weak solution u ∈ C 1 (IRN ) of (1.1). These will rely on the Poincar´e type inequality proved below. Let us consider a ball B(P, R), centered at P ∈ Tλν having radius R. Define Bλν = B(P, R) ∩ Σνλ . For any subset A of Σνλ , define ˜ = A ∪ A0 ∪ (Tνλ ∩ A). A0 = Rλν (A) and A The following lemma applies to functions on Bλν , vanishing on Tλν . 1,q Lemma 2.1 (Poincar´e type inequality). Let w ∈ Wloc (IRN ), 1 ≤ q < ∞, vanish on Tλν

and let (supp w) ∩ Bλν = A1 ∪ A2 , for some disjoint measurable sets Ai . Then there exists a constant C = C(N ) such that 1

1

1

kwkq,Bλν ≤ C|Bλν ∩ supp w| N q (|A1 | N q0 kDwkq,A1 + |A2 | N q0 kDwkq,A2 ).

(2.1)

˜ ν , and exploit the fact that wˆ has Proof : The idea is to take an odd extension wˆ of w to B λ ˜λν to derive the Poincar´e inequality. Define zero average over B w(x) ˆ = w(x) on Bλν = −w(2λ − x1 , x2 , . . . xn ) for x = (x1 . . . xn ) ∈ (Bλν )0 , so that

R

˜ν B λ

w(x)dx ˆ = 0.

˜λν ). Since w ∈ W 1,q , it follows that w ˜λν ). Further We now verify that wˆ ∈ W 1,1 (B ˆ ∈ L1 (B loc one can check easily that the distributional derivatives of wˆ for 2 ≤ i ≤ N are ∂ wˆ (x) = ∂xi

(

∂w (x) ∂xi ∂w − ∂xi (2λ

∂ wˆ (x) = ∂x1

(

∂w (x) ∂x1 ∂w (2λ ∂x1

− x 1 , x 2 , . . . xn )

on Bλν for x ∈ (Bλν )0

and on Bλν for x ∈ (Bλν )0 .

− x1 , x2 , . . . xn )

Hence we have for 1 ≤ q < ∞, 1

kwk ˆ q,B˜ ν = 2 q kwkq,Bλν , λ

1

kDwk ˆ q,B˜ ν = 2 q kDwkq,Bλν . λ

7

(2.2)

˜ ν = B(P, R), applying lemma 7.16 from [GT] we have Since wˆ has zero mean in B λ |w(x)| ˆ ≤ C(N ) where C(N ) =

2N . N wN

Z

˜ν B λ

|Dw(y)| ˆ dy |x − y|N −1

a.e.

˜λν \ supp w, Since Dw(y) ˆ = 0 a.e. in B ˆ 



|w(x)| ˆ ≤ C(N ) VA˜1 (|Dw|(x)) ˆ + VA˜2 (|Dw|(x)) ˆ ,

(2.3)

where VA f (x) =

Z

A

f (y) dy, |x − y|N −1

x ∈ IRN .

Recall that (see [D], lemma 2.2) kVA f kq,K ≤ N ωN (

|A| N1q0 |K| N1q ) ( ) kf kq,A . ωN ωN

Using this and (2.2) in (2.3), the lemma follows. In the proof of the weak comparison principles we will use the following standard estimate whose proof can be found, for example, in Lemma 2.1 of [D]: (|η|p−2 η − |η 0 |p−2 η 0 ) · (η − η 0 ) ≥ c(|η| + |η 0 |)p−2 |η − η 0 |2

(2.4)

for any η, η 0 ∈ IRN , where c is a constant depending on N and p. Now we prove a weak comparison principle, when u solves (1.1) weakly and f satisfies (H1) and (H2). Since u(x) → 0 as |x| → ∞ we can find a ball BR0 , centered at the origin and of radius R0 such that max u(x) < s0 .

RN \BR0

(2.5)

Let us put v = uνλ and for any set T , MT = MTν,λ (u) = supT (|Du| + |Dv|) . Proposition 2.1. Let N ≥ 2, 1 < p < 2 and let u in C 1 (IRN ) be a weak solution of (1.1), with f satisfying (H1) and (H2). Let R0 be such that (2.5) holds for u. Then (a) If Σνλ ∩ Rλν (BR0 ) = φ, then v ≤ u on Σνλ .

8

(b) Let Σνλ ∩ Rλν (BR0 ) be nonempty and let P ∈ Tλν and R > 0 be such that for the ball B(P, R), with center P and radius R, we have Σνλ ∩ Rλν (BR0 ) ⊂ B(P, R). ˆ ≥ R, there exist positive numbers δ and M > 0 depending on p, N, R, ˆ f Then for any R ˆ such that whenever and the L∞ norms of u, v, |Du| and |Dv| on K := B(P, R) supp (v − u)+ ∩ Bλν = A ∪ B with Bλν = K ∩ Σνλ , |A ∩ B| = 0 , MB < M and |A| < δ , it follows that v≤u

in Σνλ .

Proof : As u, v → 0 at infinity and v − u = 0 on Tλν we have that (v − u − ε)+ has compact o

n

support Kε ⊂ Σνλ for any ε > 0 . Using ρ1/n ∗ (v − u − ε)+ as nonnegative test functions in (1.1), with the usual mollifiers ρ1/n and then passing to the limit as n → ∞, we get Z

Σνλ ∩{v≥u+ε}

(|Dv|p−2 Dv − |Du|p−2 Du) · (D(v − u)+ ) dx Z

≤

Σνλ ∩{v≥u+ε}

(f (v) − f (u))(v − u − ε)+ dx

Here we have used the fact that D(v − u − ε)+ = D(v − u)+ χ{v≥u+ε} . Further by (2.4), there exists c = c(N, p) such that (|Dv|p−2 Dv − |Du|p−2 Du)(D(v − u)+ ) ≥ c(|Du| + |Dv|)p−2 |D(v − u)+ |2 . so that c

Z

≤

Σνλ ∩{v≥u+ε}

Z

(|Du| + |Dv|)p−2 |D(v − u)+ |2 dx

Σνλ ∩{v≥u+ε}

(f (v) − f (u))(v − u − ε)+ dx 9

(2.6)

(a) Assume that Σνλ ∩ Rλν (BR0 ) = φ. Then v(x) < s0 on Σνλ and by (H2) we have f (v) ≤ f (u) on {v ≥ u + ε} since on this set u < v < s0 . So we get c

Z

Σνλ ∩{v≥u+ε}

(|Dv| + |Du|)p−2 |D(v − u)+ |2 dx ≤ 0

As ε → 0 , the sequence of nonnegative functions (|Du| + |Dv|)p−2 |D(v − u)+ |2 χ{v≥u+ε} ) increases monotonically to the function (|Du| + |Dv|)p−2 |D(v − u)+ |2 χ{v≥u} ). Hence by Lebesgue’s monotone convergence theorem Z

p−2

Σνλ

(Du| + |Dv|)

+ 2

|D(v − u) | dx = lim

Z

ε→0 Σν ∩{v≥u+ε} λ

(|Dv| + |Dv|)p−2 |D(v − u)+ |2 dx (2.7)

Passing to the limit in ε we get 0≥

Z

Σνλ

(|Dv| + |Du|)p−2 |D(v − u)+ |2 dx .

Hence D(v − u)+ = 0 a.e. on Σνλ and since (v − u)+ = 0 on ∂Σνλ , (v − u)+ = 0 on Σνλ . (b) Let L denote the Lipschitz constant of f on [inf K u , supK v] and as before let us put Bλν = K ∩ Σνλ . Since outside K we have v > u ⇒ f (v) − f (u) ≤ 0, we get by (2.6), c

Z

p−2

Σνλ ∩{v≥u+ε}

(|Du| + |Dv|)

+ 2

|D(v − u) | dx ≤ L

Z

ν ∩{v≥u+ε} Bλ

(v − u) (v − u − ε)+ dx.

As ε → 0, we get c

Z

(|Dv| + |Du|)p−2 |D(v − u)+ |2 dx ≤ L ν

Σλ

Z

ν Bλ

|(v − u)+ |2 dx

and the integrands being nonnegative we have c

Z

p−2

ν Bλ

(|Dv| + |Du|)

+ 2

|D(v − u) | dx ≤ L

Z

ν Bλ

|(v − u)+ |2 dx

1,2 Since (v − u)+ vanishes on Tλν and is in Wloc (IRN ), lemma 2.1 can be used to get

c

Z

A∪B

≤

(|Dv| + |Du|)p−2 |D(v − u)+ |2 dx 1



1

LC|Bλν | N |A| N

Z

A

1

|D(v − u)+ |2 dx + |B| N 10

Z

B



|D(v − u)+ |2 dx .

It then follows that c

Z

MK2−p

|D(v − u)+ |2 dx +

A



1 LC|Bλν | N

≤

1 ν| |Bλ

Now choose δ =



|A|

1 N

Z

MB2−p

Z

B

|D(v − u)+ |2 dx 1 |Bλν | N

+ 2

|D(v − u) | dx +

A

c 2−p LCMK

c

N

and M p−2 =

Z

ν |(2/N ) LC|Bλ . c

B

(2.8) + 2



|D(v − u) | dx .

With these choices of δ and

M , it is easy to see from (2.8) that under the assumptions as in (b), |D(v − u)+ | = 0 a.e in Σνλ and hence v ≤ u in Σνλ . Now we derive the weighted Hardy-Sobolev inequality which is needed for the next comparison principle. Let us recall first the Hardy’s inequality ([Z], Lemma 1.8.11) for a measurable function h : (0, ∞) → (0, ∞) T

Z

ρr+1

0

for 0 ≤ T ≤ ∞,

1

q

ρ

Z

h(t)dt

0

dρ ≤

 q Z

q r

T

(h(t))q tq−r−1 dt

0

1 ≤ q < ∞ and r > 0. We need the following modified version of this

inequality. For the convenience of the reader, we include its derivation here. Lemma 2.2. Let h : (0, ∞) → (0, ∞) be a measurable function. Then ∞

Z

ρ

r−1

Z

T

for 0 ≤ T < ∞,

∞

q

h(t)dt

ρ

dρ ≤

 q Z

q r

∞

h(t)q tq+r−1 dt,

(2.9)

T

1 ≤ q < ∞ and r > 0.

Proof: Rewriting h(t) as (h(t)t1+r/q )(t−1−r/q) ) and applying H¨ older’s inequality with respect r

to the measure (t−1− q dt), we get ∞

Z

q

h(t)dt

ρ

≤

∞

Z

q q+r−1− rq

h(t) t

ρ

  q−1

dt

q r

ρ

−r (q−1) q

Now integrating in ρ from T to ∞ and using Fubini’s theorem, Z

∞

T

ρr−1

Z ρ

∞

q

h(t)dt

 q−1 Z

∞ ∞ r r q ρr−1 ρ−r+ q h(t)q tq+r−1− q dt dρ r T ρ  q−1 Z ∞ Z t  q q q+r−1− rq −1+ rq (h(t)) t ρ dρ dt = r T T  q−1 Z ∞ Z t  q ≤ (h(t))q tq+r−1−r/q ρ−1+r/q dρ dt r T 0  q Z ∞ q q q+r−1 = h(t) t dt r T

dρ ≤

11

Z



Lemma 2.3 (weighted Hardy-Sobolev inequality). Let u ∈ C01 (IRN ) = {u ∈ C 1 (IRN ) : u(x) → 0 as |x| → ∞} and let q ≥ 1. Then for any s > q − N and T ≥ 0 we have Z

IRN \BT (0)

|u|q |x|s−q dx ≤

q N −q+s

!q Z

IRN \BT (0)

|Du|q |x|s dx

(2.10)

The same inequality holds if we substitute u with its positive (negative) part. Proof: Let u ∈ C 1 (IRN ). We have for w ∈ S N −1 and ρ ≥ 0, |u(ρw)| = | − = |

Z

Z

∞

ρ ∞

ρ

d (u(tw))dt| dt

Du(tw).wdt| ≤

∞

Z

ρ

|Du(tw)|dt.

Calling h(t) = |Du(tw)| and r = N − q + s > 0 and using (2.9), we get Z

∞

T

ρN −1 |u(ρw)|q ρs−q dρ ≤ ≤

Z

∞

ρN −1+s−q

∞

Z

T

ρ

q N −q+s

!q Z

∞

T

q

|Du(tw)|dt

dρ

|Du(tw)|q tN −1 ts dt

Now integrating with respect to w in S N −1 yields (2.10). To prove that the inequality holds for the positive part u+ of u , we consider the 

functions Gn : IR → IR with Gn (s) = 0 if s < 0, Gn (s) = s2 +

1 n2

1 2

− n1 if s ≥ 0, n ∈ IN and

define un (x) = Gn (u(x)). We have that un ∈ C01 (IRN ) for each n ∈ IN and for any x ∈ IRN un (x) → u+ (x) as n → ∞ and |Dun (x)| ≤ |Du(x)|, so that Z

q

IRN \BT (0)

≤

s−q

|u| |x|

dx ≤ lim inf n→∞

q N −q+s ≤

!q

lim sup n→∞

q N −q+s

Z

IRN \BT (0)

Z

IRN \BT (0)

!q Z

|un |q |x|s−q dx |Dun |q |x|s dx

IRN \BT (0)

|Du|q |x|s dx

Remark 2.1. Of course the inequality (2.10) holds for a larger class of measurable functions. 1,1 In fact suppose that u ∈ Wloc (IRN ) with |Du| ∈ Lq (|x|s dx; IRN ) (otherwise there is nothing

to prove). Proceeding as in the last part of the previous proof it is immediate to see that a 12

sufficient condition for the validity of (2.10) is the existence of a sequence un of functions in C01 (IRN ) such that un → u a. e. and |Dun | → |Du| in Lq (|x|s dx; IRN ). In particular if s = 0, q < N and u ∈ Lr (IRN ) for some r ≥ 1, then the usual mollifiers uε = ρε ∗ u tend to zero at infinity and for a sequence εn → 0 the sequence un = uεn satisfies the requirements, so that (2.10) holds. This happens also for every u ∈ W 1,q (IRN ), 1 < q < N . Now we prove a weak comparison principle, when u ∈ C 1 is a weak solution of (1.1) satifying (1.2), f satisfies (H1), (H3), and (1.3) holds. We choose a ball BRˆ0 , centered at the origin and of radius Rˆ0 such that for x 6∈ BRˆ0 , we have |u(x)| ≤

C0 ≤ s0 , |x|m

C00 ≤ |u(x)| ≤ s0 if α < 0 ), |x|m C0 |Du(x)| ≤ . |x|m+1 (

Proposition 2.2. Let N ≥ 2, 1 < p < 2 and u in C 1 (IRN ) be a weak solution of (1.1) satisfying (1.2) . Let f satisfy (H1) and (H3) and assume that (1.3) holds. Let v be uνλ . (a) There exists R0 > Rˆ0 , R0 depending on N, p, f, Rˆ0 , such that if Σνλ ∩ Rλν (BR0 ) = φ, then v ≤ u on Σνλ . ˜ depending on R0 , λ and ν, be such that (b) Let Σνλ ∩ Rλν (BR0 ) be nonempty and let R, ˜ where Pλν is the projection of the origin on Tλν , and that Σνλ ∩ Rλν (BR0 ) ⊂ B(Pλν , R), after shifting the origin to Pλν , we have |u(x)| ≤ C(λ,ν) ≤ so , |x|m C1 (λ,ν) ( |x|m ≤ |u(x)| ≤ s0 if α < 0 ), C(λ,ν) |Du(x)| ≤ |x| . m+1 13

        

(2.11)

˜ Then there exists R > R ˜ , depending on R, ˜ such that for any for x 6∈ B(Pλν , R). ˆ ≥ R there exist positive numbers δ and M , depending on R, ˆ λ, p, N, f and the L∞ R ˆ with the property that whenever norms of u, v, |Du| and |Dv| on B(Pλν , R) ˆ = A∪B supp (v − u)+ ∩ B(Pλν , R) with |A ∩ B| = 0 , MB < M and |A| < δ , we have v≤u

in Σνλ .

Proof : Let us take [(v − u − ε)+ ]t , t ≥ 1, ε > 0, as a test function. Proceeding as in the deduction of (2.6) we get c

Z

Σνλ ∩{v≥u+ε}

(|Du| + |Dv|)p−2 t [(v − u − ε)+ ]t−1 |D(v − u)+ |2 dx

Z

≤

Σνλ ∩{v≥u+ε}

(f (v) − f (u))[(v − u − ε)+ ]t dx

and passing to the limit as ε → 0 we have c

Z

≤

Σνλ ∩{v≥u}

(|Du| + |Dv|)p−2 t[(v − u)+ ]t−1 |D(v − u)+ |2 dx

lim inf ε→0

Z

Σνλ ∩{v≥u+ε}

(f (v) − f (u))[(v − u − ε)+ ]t dx

(a) Let R > Rˆ0 , to be fixed later. There exists R0 = R0 (R) > R such that inf u ≥

sup u

BR

IRN \BR0

If Σνλ ∩ Rλν (BR0 ) = φ we have then that u(x) ≥ v(x) = u(xνλ ) in BR , so we get supp (v − u)+ ⊂ Σνλ \ BR ,

u(x) ≤ v(x) ≤ s0 in supp (v − u)+ .

Then by (H3) and (1.2), Z

Σνλ

+ t

(f (v) − f (u))[(v − u − ε) ] dx ≤ C

where z = u if α < 0, while z = v if α ≥ 0. 14

Z

Σνλ \BR

z α |(v − u)+ |t+1 dx

(2.12)

In the first case u(x) ≥

C00 |x|m

in IRN \ BR by hypothesis, while in the second case we have v(x) = u(xνλ ) ≤

C0 C0 ≤ m ν m |xλ | |x|

because the hypothesis Σνλ ∩ Rλν (BR0 ) = φ implies that λ > 0, so that |x| ≤ |xνλ |. In any case we have that if Σνλ ∩ Rλν (BR0 ) = φ, R0 = R0 (R), then Z

(f (v) − f (u))[(v − u − ε)+ ]t dx ≤ C ν

Σλ

Z

Σνλ \BR

1 |(v − u)+ |t+1 dx. |x|mα

Then by (2.12) we obtain, with w = w(λ, ν, t) = [(v − u)+ ] Z

p−2

Σνλ

(|Du| + |Dv|)

2

|Dw| dx ≤ C

Z

t+1 2

Σνλ \BR

,

1 |w|2 dx . |x|mα

(2.13)

Let us now fix t ≥ 1 so that the integrals in (2.13) are finite and observe that by (1.3) (|Du(x)| +

|Duνλ (x)|)

1 1 + ν m+1 ≤C m+1 |x| |xλ |

!

≤

2C |x|m+1

and hence (|Du(x) + |Duνλ (x)|)p−2 ≥ C1 |x|(m+1)(2−p) . This yields C

Z

Σνλ

(m+1)(2−p)

|x|

2

|Dw| dx ≤

Z

Σνλ

(|Du| + |Dv|)p−2 |Dw|2 dx

On the other hand, using lemma 2.3, with q = 2,

(2.14)

T = 0 and s = (m + 1)(2 − p), after

extending (v − u)+ = 0 outside Σνλ , we get Z 1 1 2 |w| dx = ν |w|2 |x|(m+1)(2−p)−2 dx mα mα−2+(m+1)(2−p) ν Σλ \BR |x| Σλ \BR |x| !Z 1 |Dw|2 |x|(m+1)(2−p) dx. ≤ C sup mα−2+(m+1)(2−p) ν ν |x| Σ Σλ \BR λ Z

Notice that m(α + 2 − p) > p if and only if mα > 2 − (m + 1)(2 − p). Then sup Σνλ \BR

1 |x|mα−2+(m+1)(2−p)

!

≤

with G(R) → 0 as R → ∞. 15

1 Rmα−2+(m+1)(2−p)

:= G(R)

(2.15)

In view of (2.14) and (2.15), (2.13) reduces to Z

Σνλ

|x|(m+1)(2−p) |Dw|2 dx ≤ C G(R)

Z

Σνλ

|x|(m+1)(2−p) |Dw|2 dx

(2.16)

ˆ 0 , such that G(R) C < 1 , since G(R) → 0 as R → ∞ and then We can now choose R > R take R0 = R0 (R). Then (2.16) will imply that |Dw| = 0 a.e and hence w = 0 a.e on Σνλ i. e. u ≥ v on Σνλ . (b)

˜ to be fixed later. For R ˆ ≥ R define Now suppose that Σνλ ∩Rλν (BR0 ) 6= φ and let R > R,

ˆ Let P ν ∈ T ν be the new origin. Observe that in Σν ∩ K c ∩ supp (v − u)+ we K = B(Pλν , R). λ λ λ have u ≤ v ≤ s0 and the estimate (2.11) holds. Further |x| = |xνλ |. Hence we can proceed as in (a) to obtain estimates analogous to (2.14) and (2.15) outside K, namely C

Z

Σνλ \K

|x|(m+1)(2−p) |Dw|2 dx ≤

Z

Σνλ \K

(|Du| + |Dv|)p−2 |Dw|2 dx

ˆ and, using Lemma 2.3 with q = 2, s = (m + 1)(2 − p) and T = R, Z

Σνλ \K

≤

h

(f (v) − f (u)) (v − u − ε)+

Z

Σνλ \K

where w = [(v − u)+ ]

t+1 2

it

dx

Z 1 2 |w| dx ≤ C G(R) ν |Dw|2 |x|(m+1)(2−p) dx mα |x| Σλ \K

, G(R) → 0 as R → ∞ and the constants now depend on λ and ν.

Let L be the Lipschitz constant of f on [inf K u(x), ||u||∞ ] and as before Bλν = K ∩ Σνλ . Then Z

(f (v) − f (u))[(v − u − ε)+ ]t dx ≤ L ν

≤

LC|Bλν |1/N

Z

|w|2 dx

1/N

Z

ν Bλ

Bλ



1/N

|A|

Z

A

2

|Dw| dx + |B|

B

(2.17) 2

|Dw| dx



1,2 by lemma 2.1, since w vanishes on Tλν and belongs to Wloc (IRN ).

The L.H.S of (2.12) can be bounded below as follows: Z

(|Du| + |Dv|)p−2 |Dw|2 dx ≥ ν

Σλ

1 Z 1 Z 2 |Dw| dx + |Dw|2 dx 2−p 2−p A B MK MB

+ C

Z

16

Σνλ \K

|x|(m+1)(2−p) |Dw|2 dx .

(2.18)

Combining this inequality with (2.17) and the one for the integral on Σνλ \ K we get c MK2−p

Z

≤

c

2

A

|Dw| dx +

(LC|Bλν |1/N )

MB2−p 

Z

1/N

|A|

+ C3 (λ, ν) G(R)

Z

2

|Dw| dx + C2 (λ, ν)

B

Z

A

Σνλ \K

|Dw| dx +

|Bλν |1/N

Σνλ \K

Z

B

|x|(m+1)(2−p) |Dw|2 dx 2

|Dw| dx



(2.19)

|x|(m+1)(2−p) |Dw|2 dx.

ˆ ≤ G(R) < We take now R so that G(R) 1 δ= ν |Bλ |

2

Z

c LCMk2−p

C2 (λ,ν) C3 (λ,ν)

!N

ˆ ≥ R. Then it is enough to choose for each R

and M

p−2

=

LC|Bλν |2/N c

!

With these choices of δ and M , it is easy to see that whenever supp (v − u)+ ∩ K = A ∪ B ,

|A ∩ B| = 0 , |A| < δ , MB < M ,

(2.19) implies that |D(v − u)+ | = 0 a.e and hence v ≤ u on Σνλ . Now we introduce the set Λ(ν) = {λ ∈ IR : u ≥ uνµ

in Σνµ

∀ µ > λ} .

Then if Λ(ν) 6= φ, we consider λ0 (ν) = inf Λ(ν). We further define Z = {x ∈ IRN : Du(x) = 0} , Zλν = {x ∈ Σνλ : Du(x) = Duνλ (x) = 0} . We will denote by Pλν the projection of the origin on Tλν . ˆ can be chosen large so that Remark 2.2. Observe that for a given λ0 and ν0 , R ˆ) B( Pλν , R ) ⊂ B( Pλν00 , R for all λ in a small neighbourhood of λ0 , say [λ0 − θ0 , λ0 + θ0 ], and for all ν in Iθ0 (ν0 ) := {ν : |ν| = 1 , |ν − ν0 | ≤ θ0 } 17

where R = R(λ, ν) is the number given by Proposition 2.1 (b) or 2.2 (b), corresponding to (H2) or (H3) respectively. Further δ and M can also be chosen uniformly for all these λ and ˆ )| on the R.H.S of (2.8) or (2.19). ν, by replacing |Bλν | by |B( Pλν00 , R The next lemma gives some sufficient conditions under which the procedure of moving the hyperplanes can be continued, using the comparison principles. Lemma 2.4.

Let u ∈ C 1 (IRN ) be a weak solution of (1.1) for 1 < p < 2. Assume that

ˆ be as in either (H1) and (H2) or (H1), (H3), (1.2) and (1.3) hold. Let λ0 ∈ Λ(ν0 ). Let R Remark 2.2 corresponding to a θ0 -neighbourhood of (λ0 , ν0 ). If we have u > uνλ00

in

ˆ (Σνλ00 \ Zλν00 ) ∩ B(Pλν00 , R),

then there exists θ1 ≤ θ0 such that (a)

u ≥ uνλ0 in Σνλ0

(b)

u ≥ uνλ in Σνλ

∀ λ ∈ (λ0 − θ1 , λ0 + θ1 ) , ∀ λ ∈ (λ0 − θ1 , λ0 + θ1 ), ∀ ν ∈ Iθ1 (ν0 ),

where Iθ1 (ν0 ) := {ν : |ν| = 1 , |ν − ν0 | < θ1 }. ˆ and let δ and M be chosen as in Remark 2.2 uniformly for all Proof : Let K = B(Pλν00 , R) (λ, ν), in a θ0 -neighbourhood of (λ0 , ν0 ). Now choose a small neighbourhood A of K ∩ Tλν00 with |A| < δ/2 and another neighbourhood O of Zλν00 ∩ K with MOν0 ,λ0 < M/2 where MOν,λ : = supO {|Du| + |Duνλ |}. Now K0 = K ∩ Σνλ00 \ (A ∪ O) is a compact set and by our assumption u − uνλ00 > m > 0 on K0 . Then continuity with respect to λ and ν imply that there exists θ1 depending on λ0 , ν0 and K with θ1 ≤ θ0 , such that u − uνλ > 18

m 2

on K0 ,

λ,ν MO∩Σ < M ν λ

|A ∩ B(Pλν , R) ∩ Σνλ | < δ , B(Pλν , R) ∩ Σνλ ⊂ (B(Pλν00 , R) ∩ Σνλ00 ) ∪ A , for all λ ∈ (λ0 − θ1 , λ0 + θ1 ) and ν ∈ Iθ1 (ν0 ). Here R = R(λ) is the number given by Proposition 2.1 (b) or 2.2 (b), corresponding to (H2) or (H3) respectively. Then clearly for such λ0 s near λ0 supp (χKλ (uνλ0 − u)+ ) ⊂ (A ∪ O) ∩ Kλ , where Kλ = B(Pλν0 , R). Further Kλ and supp (χKλ (uνλ0 − u)+ ) = ((Kλ ∩ A) ∪ (Kλ ∩ O)) ∩ Σνλ0 satisfy the conditions of Proposition 2.1 (b) or 2.2 (b). Hence we can conclude that uνλ0 ≤ u on Σνλ0 for all λ ∈ (λ0 − θ1 , λ0 + θ1 ). (b) For this part argue as before taking Kλν = B(Pλν , R) . The following corollary can be deduced easily by using a similar reasoning as in the above theorem. Corollary 2.1. Under the same assumptions on u and f as in Lemma 2.4, suppose that for ˆ and some λ0 ∈ Λ(ν0 ) there exist open neighbourhoods A and O of the sets Tλν00 ∩ B(Pλν00 , R) Zλν00 respectively such that u > uνλ00 with |A| < δ/2 and MOν0 ,λ0 <

M 2

in





ˆ \ (A ∪ O) Σνλ00 ∩ B(Pλν00 , R)

, where δ and M are chosen as in Remark 2.2 uniformly

for all (λ, ν) in a θ0 -neighbourhood of (λ0 , ν0 ). Then there exists θ1 small enough such that (a) and (b) of lemma 2.4 hold. The last proposition recalls some sufficient conditions which will help us to conclude that a nonnegative solution is necessarily a positive solution. In [PSZ] this subject has

19

been treated in a very general and exhaustive way, including also the case of solutions with compact support. For the reader’s convenience we give here the proof in our particular case. Proposition 2.3. Let u be a nonnegative ground state solution of (1.1), for 1 < p < 2. Let f satisfy (H1) and any one of the following set of conditions: (a) either f satisfies (H2), f (0) ≥ 0 and for F (s) = Z

0

ds (F (s))1/p

Rs

0 [f (0)

=

− f (t)] dt,

∞

(b) or there exists some c > 0 and δ > 0 such that f (u) + cup−1 ≥ 0 ∀s ∈ [0, δ). Then u > 0 in IRN . Proof : Let us recall the strict maximum principle, proved in [V]: Let u ∈ L1loc (Ω), ∆p u ∈ L1loc (Ω) and u ≥ 0 a.e. in Ω and −∆p u ≥ −β(u) a.e. where β : [0, ∞) → IR is a continuous nondecreasing function with β(0) = 0 and • either β(s) = 0 for some s > 0 • or β(s) > 0 for s > 0 and

R

ds 0

Rs

(

0

1/p

β(t)dt)

= ∞

Then we have either u ≡ 0 or u > 0 in Ω. Hence it is enough to find a function β satisfying the above conditions such that f (u) ≥ −β(u), so that the strict maximum principle applies. Case 1: Since f is decreasing in (0, s0 ), we can take β(u) = −[f (u) − f (0)]. Then β(0) = 0 and β is nondecreasing. Further −∆p u = f (u) = (f (u) − f (0)) + f (0) = −β(u) + f (0) ≥ −β(u) since f (0) ≥ 0. 20

Case 2: In the second case, by (b) f (u) ≥ −cup−1

∀ u ∈ [0, δ)

Calling β(u) = cup−1 , we have Z

δ

0

ds Rs

(

0

β(u)du)1/p

=

∞.

Thus in both the cases, Vazquez strict maximum principle is applicable and hence u > 0 on IRN .

3

Proof of the Symmetry Result

Here we prove the theorems 1.1 and 1.2. The idea of the proof is to show that for any direction ν, the hyperplanes Tλν , orthogonal to ν, can be moved from infinity upto a critical λ0 (ν) so that Λ(ν) = [λ0 (ν), ∞) and u ≡ uνλ0 (ν) at the critical position for every ν. Proceeding as in [DP1] and [DPR], we will assume that u 6≡ uνλ0 (ν) for some ν and construct a subset Γ of Z on which u ≡ constant and whose projection on Tλν0 (ν) contains an open subset of that hyperplane. The fact that such a set cannot exist has been proved already in [DP1] for bounded domain and in [DPR] for unbounded domain. This contradiction will help us to conclude. Step - 1: Let u ∈ C 1 (IRN ) be a weak solution of (1.1) with 1 < p < 2 and let either (H1) and (H2) or (H1), (H3), (1.2), (1.3) hold. Then for any direction ν we have Λ(ν) 6= φ. Proof : We prove that Λ(ν) actually contains an interval (R0 , +∞) where R0 is defined as in Proposition 2.1 (a) or 2.2 (a) corresponding to f satisfying (H2) or (H3) respectively. If λ ≥ R0 , Σνλ ∩ Rλν (BR0 ) = φ. Hence applying the weak comparison principle, (part (a) of Proposition 2.1 or 2.2) we get u ≥ v in Σνλ so that λ ∈ Λ(ν) for any λ ≥ R0 . Step - 2: We have u > uνλ

in

Σνλ \ Zλν , 21

∀ λ, λ0 (ν) < λ ,

(3.1)

∂u (x) < 0 ∀ x ∈ (Σνλ0 (ν) )0 \ Z . ∂ν

(3.2)

The proof of this step follows as in [DPR] ( see the proof of Theorem 4.1 in [DPR]). ν ˆ be a ball centered at P ν ˆ Step - 3: Let K = B(Pλν0 (ν) , R), λ0 (ν) ∈ Tλ0 (ν) , having radius R, where ν ˆ is chosen as in Remark 2.2 corresponding to λ0 (ν) and ν. Then either u ≡ uν R λ0 (ν) on Σλ0 (ν)

or there exists at least one connected component C ν of Σνλ0 (ν) \ Zλν0 (ν) such that C ν ∩ K 6= φ and u ≡ uνλ0 (ν) in C ν . In the latter case, we also have Du(x) 6= 0 ∀ x ∈ C ν and Du(x) = 0 ∀ x ∈ ∂C ν \ Tλν0 (ν) .

(3.3)

Proof: At the minimal position λ0 (ν), either u ≡ uνλ0 (ν) on Σνλ0 (ν) or there exists a point x0 ∈ Σνλ0 (ν) such that u(x0 ) > uνλ0 (ν) (x0 ). Let us assume that the latter happens. Arguing by contradiction, we suppose that there does not exist any component C ν of Σνλ0 (ν) \Zλν0 (ν) such that C ν ∩ K 6= φ and u ≡ uνλ0 (ν) on C ν . Then by the strong comparison principle (Theorem 1.4 in [D]) we get u > uνλ0 (ν) on Bλν0 (ν) \Zλν0 (ν) with Bλν0 (ν) = K ∩ Σνλ0 (ν) . Then by Lemma 2.4, we see that u ≥ uνλ holds for λ ∈ (λ0 (ν) − θ1 , λ0 (ν)), contradicting the minimality of λ0 (ν). Thus the first part of the assertion is proved. The rest of the assertions can be proved as in [DPR] Theorem 4.1. As in [DPR], we remark here that if the critical set Z of u were regular, then u would be constant on the boundary ∂C ν of the component C ν constructed above. This would give us the “bad” set Γ indicated in the beginning of the proof. But in general Z need not be regular. (See the discussion in Remark 3.1 in [DP1]). So we construct such a set Γ in the next step, by moving simultaneously the hyperplanes orthogonal to the directions ν in a neighbourhood of ν0 and picking a set which is symmetric with respect to Tλν0 (ν) for all these ν’s simultaneously. ˆ as Let us now fix a direction ν0 such that u 6≡ uνλ00 (ν0 ) in Σνλ00 (ν0 ) and choose θ0 , R, in Remark 2.2 corresponding to λ0 (ν0 ), ν0 . Let Fν0 be the collection of the connected 22

components C ν0 of (Σνλ00 (ν0 ) \ Zλν00 (ν0 ) ) such that C ν0 ∩ K 6= φ and u ≡ uνλ00 (ν0 ) in C ν0 , and let {Ciν0 }i∈I⊂IN be an enumeration of the sets in Fν0 . ˆ and let us put Step - 4: Let K = B(Pλν00(ν0 ) , R) F˜ν0 = {C˜i : Ci ∈ Fν0 }. where C˜i = Ci ∪ Rλν00 (ν0 ) (Ci ) ∪ (∂ Ci ∩ Tλν00 (ν0 ) \ Z). Then there exists a direction ν1 near ν0 and a set C˜iν1 ∈ F˜ν1 such that the set C˜iν1 ∈ F˜ν , for every ν in a small neighbourhood of ν1 .

Proof : We first prove that λ0 (ν) is a continuous function of ν at ν0 . Let us fix ε > 0, ε < θ0 . By the definition of λ0 (ν0 ), there exist λ ∈ (λ0 (ν0 ) − ε, λ0 (ν0 )) and x ∈ Σνλ0 such that u(x) < uνλ0 (x). By continuity of u with respect to ν , there exists δ1 > 0 such that x ∈ Σνλ and u(x) < uνλ (x) for every ν ∈ Iδ1 (ν0 ). Hence for every ν ∈ Iδ1 (ν0 ) we have λ0 (ν0 ) − ε < λ < λ0 (ν). Now we claim that there exists δ2 > 0 such that λ0 (ν) < λ0 (ν0 ) + ε for any ν ∈ Iδ2 (ν0 ). If this is not true, then there exists a sequence {νn } of directions such that νn → ν0 and λ0 (νn ) ≥ λ0 (ν0 ) + ε ∀ n. By Step -1 we have that λ0 (ν) < R0 for any direction ν. Thus the sequence λ0 (νn ) is bounded and hence, up to a subsequence, it converges to a number λ ≥ λ0 (ν0 ) + ε. Then by (3.1) we have u > uνλ0

in

Σνλ0 \ Zλν0 .

Now by Lemma 2.4 we have that u ≥ uνλ in Ωνλ for any λ close to λ and any ν close to ν0 . In particular, this will hold for νn and λ0 (νn ) − γ, for some n large and γ small, contradicting the definition of λ0 (νn ). Thus it follows that for each ε > 0 there exists η = η(ε) > 0 such that if ν ∈ Iη (ν0 ), λ0 (ν0 ) − ε < λ0 (ν) < λ0 (ν0 ) + ε. 23

Let M and δ corresponding to K be chosen as in Remark 2.2, uniformly for all (λ, ν) in a θ0 –neighbourhood of (λ0 , ν0 ). Choose a bounded neighbourhood O of Zλν00 (ν0 ) ∩ K such that ν ,λ0 (ν0 )

MO0

<

M . 4

Choose a neighbourhood A of Tλν00(ν0 ) ∩ K such that |A ∩ K| <

δ . 4

Notice that u − uνλ00 (ν0 ) ≥ γ > 0 in K ∩ Σνλ00 (ν0 ) \ (A ∪ O ∪

S

i∈I

Ciν0 ).

By continuity there exists η0 ≤ θ0 such that for each ν ∈ Iη0 (ν0 ) we have λ0 (ν0 ) − θ0 < λ0 (ν) < λ0 (ν0 ) + θ0 ˆ ∩ Σν , and, putting Bλν0 (ν) = B(Pλν0 (ν) , R) λ0 (ν) Bλν0 (ν) ⊂ Bλν00 (ν0 ) ∪ A,

Bλν0 (ν) ∩ Zλν0 (ν) ⊂ O , u−

uνλ0 (ν)

> 0 in

Bλν0 (ν)

\ A∪O∪

[

Ciν0

!

(3.4) (3.5)

i∈I ν,λ (ν)

MO∩Σ0 ν

λ0 (ν)

|A ∩ Bλν0 (ν) | < δ/2.

< M/2 ,

(3.6)

Our second claim is that for each ν ∈ Iη0 (ν0 ), there exists some C˜i ∈ F˜ν ∩ F˜ν0 . Define for each Ci ∈ Fν0 , Si = Ci ∩ K \ (A ∪ O). Notice that Si is compact for each i and there must be some Si which are nonempty. In fact, if all the Si were empty, then Ci ∩ K ⊂ A ∪ O for each Ci ∈ Fν0 and hence u > uνλ00 (ν0 ) would hold in K \ (A ∪ O). Then by Corollary 2.1, λ0 (ν0 ) would not be the critical position, which is absurd. Another crucial remark is that if ν1 and ν2 are two directions and C ν1 ∈ Fν1 , and C ν2 ∈ Fν2 , then either C˜ν1 ∩ C˜ν2 = φ or C˜ν1 ≡ C˜ν2 . ( For a proof of this, see the arguments in Remark 4.1 in [DP1]). Let us fix ν ∈ Iη0 (ν0 ). We can have 24

(i) either u ≡ uνλ0 (ν) on some Si , (ii) or u > uνλ0 (ν)

on all Si .

In case (i), by the strong comparison principle, Si is contained in some component C ν of Σνλ0 (ν) \Zλν0 (ν) . Hence u ≡ uνλ0 (ν) in C ν . Then Si ⊂ C ν ∩ Ci and hence C˜ν = C˜i ∈ F˜ν ∩ F˜ν0 . In case (ii) happens, u > uνλ0 (ν) holds in every Si and hence in Bλν0 (ν) \ (A ∪ O) because of (3.4) and (3.5). Then using (3.6), the minimality of λ0 (ν) is contradicted by Corollary 2.1. Hence case (ii) cannot arise and our claim is proved. Next we claim that there exists a direction ν1 ∈ Iη0 (ν0 ) and a neighbourhood Iη1 (ν1 ) and a fixed C˜i ∈ F˜ν1 such that for any ν ∈ Iη1 (ν1 ), the set C˜i ∈ F˜ν . Let {C˜ ν0 }i∈I be an enumeration of the sets in F˜ν0 . If C˜1ν0 ∈ F˜ν for all ν ∈ Iη0 (ν0 ), then we stop. If not, there exists ν1 ∈ Iη0 (ν0 ) such that C˜1ν0 6∈ F˜ν1 i.e. u > uνλ10 (ν1 ) on S1 . Since S1 is compact we have that u − uνλ10 (ν1 ) ≥ γ > 0 in S1 , so that u > uνλ0 (ν) holds in S1 for the directions close to ν1 and by the previous argument C1ν0 6∈ Fν for all ν ∈ Iη1 (ν1 ) for some η1 < η0 − |ν1 − ν0 |. Now we check if C˜2ν0 ∈ F˜ν for all ν ∈ Iη1 (ν1 ). If not, we find a direction ν2 ∈ Iη1 (ν1 ) and a neighbourhood Iη2 (ν2 ) such that η2 < η1 − |ν2 − ν1 | and C˜2ν0 6∈ F˜ν for all ν ∈ Iη2 (ν2 ). Proceeding in this way, ν0 (i) either we stop at some k’th stage where C˜k+1 ∈ F˜ν ∀ ν ∈ Iηk (νk )

(ii) or the process does not stop at all and I = IN . Now we claim that (ii) cannot arise. In case (ii), we obtain a sequence of nested compact sets {Iηi (νi )}i∈I with the finite intersection property. Then by Cantor’s intersection theorem ∩i∈I Iηi (νi ) 6= φ. For the direction ν in this intersection, C˜iν0 6∈ F˜ν for all i. This contradicts our second claim. Thus step - 4 now follows. 

Step - 5: Let C˜iν1 be as in step - 4 and let (Ciν1 )0 = Rλν10 (ν1 ) (Ciν1 ). Then ∂Ciν1 ∩ Σνλ10 (ν1 )

0

contains a subset Γ on which u is a constant and Du = 0 and whose projection on Tλν01(ν1 ) contains an open subset of that hyperplane. This step, can be proved exactly as in [DPR].(Refer the proof of step-3 in [DPR]). 25

Step - 6: u is radially symmetric about some point x0 ∈ IRN and is radially strictly decreasing. Proof : The set Γ constructed in step -5 leads to a contradiction, in view of the following proposition proved mainly using Hopf’s lemma in [DP1] for bounded domain and in [DPR] for unbounded domain. (See Proposition 5.1 in [DPR]) : Suppose that u ∈ C 1 (IRN ) is a weak solution of (1.1) for 1 < p < 2. For any direction ν, the half space (Σνλ0 (ν) )0 cannot contain a subset Γ of Z on which u = constant and whose projection on the hyperplane Tλν0 (ν) contains a set, open in Tλν0 (ν) . Using this proposition we conclude that u ≡ uνλ0 (ν) at the critical position for every ν. Further we can conclude using the same proposition again that Z ∩ Σνλ0 (ν) = φ for any direction ν (see the proof of Step-4 in [DPR]). Thus u is strictly decreasing in every direction. Now by considering N linearly independent directions in IRN , the final symmetry result follows.

REFERENCES [AGP] A. Ambrosetti, J. Garcia Azorero and I. Peral, Multiplicity results for some nonlinear elliptic equations, J.Funct.Anal. 137 (1996), 219-242 [B1] F. Brock, Radial symmetry for nonnegative solutions of semilinear elliptic equations involving the p-Laplacian, Proceedings of the conference ”Calculus of variations, applications and computations, Pont - a - Mousson (1997). [B2] F. Brock, Continous rearrangement and symmetry of solutions of elliptic problems, Habilitation Thesis, Leipzig (1998). [D] L. Damascelli, Comparison theorems for some quasilinear degenerate elliptic operators and applications to symmetry and monotonicity results, Ann. Inst. H. Poincare, 15, no. 4 (1998), 493 - 516. [DP1] L. Damascelli and F. Pacella, Monotonicity and symmetry of solutions of p - Laplace equations, 1 < p < 2, via the moving plane method, Ann. Sc. Norm. Sup. Pisa Cl. Sci. (4) Vol. XXVI (1998), 689–707. 26

[DP2] L. Damascelli and F. Pacella, Monotonicity and symmetry results for p-Laplace equations and applications, Preprint (1999). [DPR] L. Damascelli , F. Pacella and M. Ramaswamy,

Symmetry of Ground States of

p-Laplace equations via the moving plane method, to appear in Arch. Rat. Mech. Anal. [FLS] B.Franchi, E.Lanconelli and J.Serrin, Existence and Uniqueness of Nonnegative Solutions of Quasilinear Equations in IRn , Advances in Math. 118, (1996), 117 - 243. [GNN] B. Gidas, W.M.Ni and L. Nirenberg, Symmetry of positive solutions of nonlinear elliptic equations in IRN , Math. Anal. Appl., Part A, Advances in Math. Suppl. Studies, 7A (1981), 369 - 403. [GS] B. Gidas and J. Spruck, Global and local behavior of positive solutions of nonlinear elliptic equations, Comm.Pure Appl.Math 34 (1981), 525 - 598. [GT] D. Gilbarg and N.S. Trudinger, Elliptic Partial Differential Equations, (Springer Verlag), 1983. [GV] M. Guedda and L. Veron, Local and Global Properties of solutions of Quasilinear Elliptic Equations, J. Diff. Eq. 76, No.1. (1988), 159 - 189. [Li] C. M. Li, Monotonicity and symmetry of solutions of fully nonlinear elliptic equations on unbounded domains, Comm. P.D.E. 16 (1991), 585 - 615. [LN] Y. Li and W.M. Ni, Radial symmetry of positive solutions of nonlinear elliptic equations in IRN , Comm. P.D.E. 18 (5 & 6), (1993), 1043 - 1054. [PSZ] P. Pucci, J. Serrin and H. Zou, A strong maximum principle for quasilinear elliptic equations , to appear. [S] J. Serrin, A symmetry problem in potential theory, Arch. Rat. Mech. Anal. 43 (1971), 304 - 318. [SZ] J. Serrin and H. Zou, Symmetry of Ground States of Quasilinear Elliptic Equations, to appear in Arch. Rat. Mech. Anal. [V] J.L. Vazquez, A Strong maximum principle for some quasilinear elliptic equations, Appl. Math. Optim, Vol 12, (1984), 191 - 202. [Z] W.P. Zeimer, Weakly differentiable functions, Springer-Verlag, 1989.

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