Existence of positive weak solutions with a prescribed ...troy/ed/chenandli.pdf · The Journal of...

The Journal of Geometric Analysis Volume 9, Number 2, 1999

Existence of Positive Weak Solutions with a Prescribed Singular Set of Semilinear

Elliptic Equations B y C h i u n - C h u a n C h e n a n d C h a n g - S h o u L i n

ABSTRACT. In this paper, we consider the problem of the existence of non-negative weak solution u of

A u + u P = O i n ~ u = 0 on Of~

n n + 2 ~ / n - 1 having a given closed set S as its singular set. We prove that when < p < _ _ and

n - 2 n - 4 + 2 nv/'n-L--1 S is a closed subset of f2, then there are infinite many positive weak solutions with S as their singular set.

Applying this method to the conformal scalar curvature equation for n > 9, we construct a weak solution n+2 n+2

u ~ L ~ (S n) ofLou + L n-2 = 0 such that S n is the singular set ofu where L 0 is the conformal Laplacian

with respect to the standard metric of S n. When n = 4 or 6, this kind of solution has been constructed by

Pacard.

1. Introduction

In this paper we are concerned with the existence of positive weak solutions with a prescribed singular set of

A u + u P = O inf2, (1.1) u = 0 on 0f2,

where f2 is a smooth open set in R n with n > 3. We also consider the same problem for the equation

Au + K ( x ) u p : 0 in f2, (1.2)

u > 0 i n f 2 a n d u = 0 o n 0 ~ 2 ,

where K ( x ) ~ C1((2) and 0 < a < K ( x ) < b for two positive constants a, b. u ~ L P ( ~ ) is called a weak solution of (1.2) i f the equality,

f f i A ~ o u d x = f ~ K ( x ) u P g o ( x ) d x ,

Math Subject Classifications. 35J60, 35J20. Key Words and Phrases. semilinear elliptic equation, singular solution, moutain pass lemma, conformal scalar curvature equation.

�9 1999 The JournaI of Geometric Analysis ISSN 1050-6926

222 Chiun-Chuan Chen and Chang-Shou Lin

holds for any q) 6 C2((2) and q) = 0 on 0fa. Let M = {x ~ g2 I there exists a neighborhood o f x such that u(x) is bounded in that neighborhood}, and call the complement S = f 2 \ M the singular

n set of u. It is easy to see that S is a closed set in f2. For 1 - - . The - n - 2

n study of the existence of a weak solution of (1.1) for p > - - was initiated by Schoen [7] for the

n - 2 n

conformal scalar curvature equation, and by Pacard [5] for p = . We define Pl as the unique n - 2

2p 2 n - 2 2 n n + 2 root of p - 1 (n - 2 - P - 1 ) = ( - - ~ ) in the interval ( n _ 2 , n - - - - - 2 ) ' namely,

n + 2 4 % - - 1 Pl = (1.3)

n - 4 + 2 ~ - 1

First we consider the case with K(x) = 1.

n Theorem 1.4. Let f2 be a bounded smooth open domain o f R n. Suppose that ~ < p < Pl,

n - 2 and S is a closed set o f f2. Then there exist infinitely many positive weak solutions o f (1.1) having S as their singular set. Solutions obtained here belong to Lq (f2) for any q < p* where p , = n ( p - 1)

2

n Theorem 1.5. Suppose that - - < p < p l , ~2 is a bounded smooth open set in R n, and S is

n - 2 closed in f2. Then there exist two distinct sequences o f solutions of(1.1) having S as their singular set such that one sequence converges to 0 in Lq (f2), and the other sequence converges to a smooth positive solution of(1.1)in Lq (f2) for q < p*.

n R e m a r k 1.6. For p -- n - 9 ' Theorem 1.4 and Theorem 1.5 have been obtained by Pacard [5].

r/ c For p > n - 2 ' the linearized equation at an approximate solution has the form of A + I - ~ near 0.

1 The difficulty in studying this kind of linear operator comes from the fact that ~ is not compact with

respect to the H i norm. Fortunately, when p < Pl , the linearized operator is still positive-definite.

This is the reason why we restrict on p < Pl .

n+2 In [8], Schoen and Yau conjectured that all positive weak solutions of A0u + u ~ = 0 with

n+2 n -- 2 Here A0 u 6 L ~=7-2 (S n) have a singular set of Hausdorff dimension less than or equal to - - - ~ .

denotes the conformal Laplacian for the standard metric on S n. This problem can be formulated in n-2

R n as follows. Let d/z be the measure (1 + Ix l2 ) - -Udx on R n. Assume that u 6 L ~ (R n , d/z) is

a positive weak solution of n+2

Au + u,vz~-2 = 0 . (1.7)

Then the conjecture is equivalent to saying that the Hausdorff dimension of the singular set is less n - 2

than or equal to - - . For n = 4, 6, Pacard [6] already found a counter-example to this conjecture. 2

For n > 9, we have the following:

n+2 T h e o r e m 1.8. For n >_ 9, there exists a positive weak solution of (l. 7)in L ,7=7-2 (R n , d#) whose

Existence of Positive Weak Solutions with a Prescribed Singular Set of Semilinear Elliptic Equations 223

singular set is the whole R n.

For the nonconstant K (x), we have similar results.

3n n + 3 Theorem 1.9. Suppose that K ( x ) ~ C 1 0 < a < K < b, and , - - _ 5 ,

n - 2 3 n - 5 n

and - - < p < p l for n = 3, 4. Le t S be closed in g2 where f2 is a bounded smooth domain n - 2

in R n. Then there exists infinitely many positive weak solutions o f (1.I) having S as their singular ( p 1)n

set. Solutions obtained here belong to L q (f2) for any q < p* where p* - 2

Similarly, we have the following:

Theorem 1.10. Suppose that ~2 is bounded smooth in R n and the assumption o f Theorem 1.9

holds. Then there exists two distinct infinite sequences o f positive weak solutions o f (1.2) such that

one sequence converges to 0 in L q (~) and the other sequence converges to a smooth solution in

L q ( ~ ) f o rq < p*.

The paper is organized as follows. In Section 2, we will construct a solution of (1.1) with one isolated singular point. Then we use them to construction approximate solutions for the situation with finite isolated singular points in Section 3. In Section 4, we give complete proofs of Theorem 1.4, Theorem 1.5, and Theorem 1.8. In Section 5, we consider the case with nonconstant K(x ) .

2. Solutions with one singular point

In this section we will use the method of supersolution and subsolution to construct a solution of (1.1) with one isolated singularity. First, we consider

Au + u p = 0 inRn\{0} (2.1)

u > 0, andlimx_+0u(x) = + e e

The following proposition classifies all solutions of (2.1).

Proposition 2.2. 2

u(x) = cplxl p-1

2 Cp = ( ( n - 2 - - -

p - 1

o f (2.1) such that

n n + 2 Suppose that - - 0 such that lim lXln-2u(x) = or, where ]xl-++c~

2 ) ) F~-l . Conversely, for any ol > O, there exists a unique solution u(x )

p - 1

l im lx I~-2u(x) = ~ . [xl--++~

We note that any solution of (2.1) must be radially symmetric. See [1]. The proof of Proposi- tion 2.2 is elementary and straightforward. Hence, we omit the proof. Throughout the paper, u~(r)

2 is denoted as the unique solution of(2.1) with lim Ixln-2u(lxl) = o~, and u ~ ( I x l ) - - cplxl ~-1.

x - - + + ~

Let Pl be

n + 2V'-ff - 1 Pl = (2.3)

n - 4 + 2 ~ / n - 1

224 Chiun-Chuan Chert and Chang-Shou Lin

It is easy to see that

n n + 2 - - < P l < - - and n - 2 n - 2

2p ( 2 ) ( ~ @ _ 2 ) 2

p 7 1 n - 2 P - 1 -<

n f o r - -

n - 2 < P < Pl . Then we have

( ) ( 2 ; p-1 2p 2 r - 2 r - 2 puoo ( r ) - p T 1 n - 2 P - 1 -< " (2.4)

n P r o p o s i t i o n 2.5. Suppose that O. Moreover, lim u~(x) = uoo(x) and lim u~(x) = 0 uniformlyin

a~+oo c~-+O any compact set o f R ~ \{0}.

P r o o f . First, we want to prove u~ (r) 0. Suppose that the claim is not true. Let w(r) = u ~ ( r ) - u~(r). Then there exists R0 > 0 such that w(r) > 0 for R0 < r < +c~ and w(Ro) = 0. We note that w satisfies

A w + c ( r ) w = O for r > R 0

where c(r) - - -

sides, we have

-

< p u P - l ( r ) f o r R0 < r < +oo. Multiplying Ixl ~ and integrating both

f O0 n - 2 1 UR 3~ n - 2 r -s c (r )w(r)rn-Ldr = -- r - ~ - A w ( r ) r n - l d r Ro o

( n - 2 ) 2 f ~ n+2 n - 2 = r - - T w ( r ) r n - l d r + Ro 2 w' (Ro) R8 -1 .

~ 2 ] JRo

Since w'(Ro) > 0, we have, by (2.4),

( _ ~ ) 2 fR~176 n+2w(r)rn_ldr fR ~~ n-2 r < r - - Z - c ( r ) w ( r ) r n - l d r 0 0

"n 2-2 ~ O O n+2

< ( ' - ~ - 2 ~ ) JR r 2 w(r) r n - l d r . o

2 n + 2 n+2 r ~ w(r)r n-1 is integrable in JR0, or Hence, Since w(r) = O(r-T~-~) at oo and p < n - 2 '

the above inequality provides a contradiction. Therefore, the claim is proved. With the claim, the remainder of Proposition 2.5 follows immediately. [ ]

n For - -

n - 2 < P < Pl , we denote kl and )~2, respectively, by

2m - n + 2 - ~/(n - 2 - m) 2 - 8(n - 2 - m)

2


and

where m -

)~2 = 2m - n + 2 + ~ / ( n - 2 - m) 2 - 8(n - 2 - m)

2

2 - - - - . We note that

p - 1

n - 2 - m + ) ~ 2 - - - + 3 o (2.6)

2

for 60 = ~/(n - 2 - m) 2 - 8(n - 2 - m) > O. Then we have the following:

T h e o r e m 2 . 7 . Supposethat u(x) = u(lx[) isaradiallysymmetricpositivesolutionof A u + u p = 0 in 0 < r < ro for some ro > O. Then the following statements hold.

(i) I f k)~l < ~'2 < (k + 1)kl for some positive integer k, then u(r) has the asymptotic expansion near O,

Cp a t a k u(r) = ~-~ + ~ + . . . + rm_kZ-------- T (1 t + ~ + 0 rm_k2_?~ ~

for some 3o > 0 depending only on p and n.

(ii) I f k)~ t = )~ 2 for some positive integer k, then u(r ) has the asymptotic expansion near 0,

Cp al = + . . . +

1 a~_ 1 a~ log r + - -

r m - ( k - 1 ) X 1 ym-k~.l t + ~ + 0 rm_22_~o

for some 3o > 0 depending only p and n.

The coefficients a2 . . . . . ag are functions o f al. Moreover, i f two solutions have the same al and bl, then these two solutions must be identical.

Conversely, given a l , bt , and a2 . . . . . ak are given according to the above, then there exists a unique radially symmetric solution o f Au + up = 0 in 0 < r <_ ro for some ro > O.

The asymptotic expansion of u (r) in Theorem 2.7 was proved by Li [4], and Gui et al. [2]. The existence and uniqueness can be proved along the same reasoning as in [2]. We refer the reader to [2] for the proof of Theorem 2.7.

We also consider

n-1 # ( r ) u l t ( r ) + - - 7 - . . + u p ( r ) = O , O < r < R

u(R) = O, u(r) > O, and l im u(r) = +oc r --->0

(2.8)

As an application of Theorem 2.7, we have the following:

P r o p o s i t i o n 2.9. Fix R > O. Then, for each ot~ (0, ~ ] , there exists a unique solution u of(2.8) such that the following holds.

(i) u(r) < u~(r ) for 0 < r < R.


(ii) lu(r) - u~(r)l = O(r - ~ + ~ ~ as r -+ 0 +, where ~o is the constant in (2.6).

The unique solution is denoted by u~(r; R). Then u,(r; R) is strictly increasing with respect to both ~ and R.

Proo f . The idea of the proof is simple and based on an eigenvalue's comparison. By scaling, we find that coefficients at (~) and bl (~) of u~(r) in the asymptotic expansion satisfy

al (or) = ol-)q/m al (1) (2.10)

and bl (or) = ot-Z2/m b 1 (1) (2.11)

with al((X~) = bl(C~) = 0. We claim that a1(1) < 0. Since u~(r) < uc~(r), we have 11(1) < 0. Suppose that 11(1) = 0, namely, al(ot) - 0 for ot ~ (0, oo]. Let w(r) = u f ( r ) - u,(r) , for 0 < o~ < 13 < +co . Then w(r) > 0 and w(r) satisfies, by (2.6),

A w ( r ) + C ( r ) w = 0 , O < r < +oo

w(r) = O ( r - @ +a~ a t0 (2.12)

w(r) = O (r - ' + 2 ) at oc

ups(r) - uP( r ) n - 2 ,-2 - - < ( )2r-2. Multiplying r 2 on both sides, we have where C(F)

u~(r) - uc~(r) 2

- A r - T w ( r ) r n - l d r = - r T A w r n - l d r

because both the boundary terms at 0 and oo vanish. Hence, we have

( n 2 " 2 - e c n+2 n ' f0 c~ n-2 '-T2 ~ ) ]o r " w(r)r -1dr = c ( r ) r - ~ - w ( r ) r n - t d r

( ;fo < r 2 w ( r ) r n - l d r .

Using the asymptotic behavior of w at 0 and eo, it is obvious that the integral of the above is finite. Therefore, we obtain a contradiction. The claim is proved.

Fix ot 6 (0, c~]. By Theorem 2.7, we can find a solution u(r; b) whose coefficients al, bl in the asymptotic expansion satisfy

al = al(ot) and bl = b .

By Proposition 2.2, we know that, for b # bl (a), there exists R(b) ~ (0, oc) such that u (R(b), b) = 0. For 0 0. It implies that R(b) is strictly increasing in b for - c ~ < b < bl(a). Since lim u(r; b) = u~(r),

b---~ ]31 (a) we have lim R(b) = +oc . We claim that lim R(b) = 0. To see this, let ub(r) =

b-----+ bl (a) b-----+-e~ 2 u - 1

Ibl-m/)~lu(tbl-l/Zlr; b) where m = . Then ub(r) is a solution of u" + u' + u p = 0 p - 1 r

with coefficients al = al(ot)lb] -1 and bl = - 1 . Hence, lim ub(r) = ti(r) wherc fi is also a b - ~ -ec


solution with the coefficients al = 0, bi = - 1 in the asymptotic expansion. Since t~(/?) = 0 for some/~ > 0, we have ub(Rb) = 0 with lira Rb = /~. Thus, R(b) = Ibl -a/z1 gb, and then the

b ------~ -oo claim follows immediately. Then, for any R > 0, there exists b such that R(b) = R. This finishes the proof of the existence part of Proposition 2.9. The uniqueness part and the monotonicity of u , (r; R) in terms of oe, R follows easily from the same reasoning as the above. Therefore, the proof of Proposition 2.9 is regarded complete. [ ]

n Theorem 2.13. Le t f2 be a smooth open domain o f R n, and - - n - 2

and o~ ~ (0, oe], there exists a unique solution o f (1 .1) such that

 0 which depends only on p and n.

Proof. Suppose that/~(x0, r0) C f2. By Proposition 2.9, there exists a solution ua(r; ro) of (2.8). Define u by

/ u.([x-x0l,r0) i f lx -x0]<r0 u_(x) / 0 if I x - x 0 [ > r0 �9

Then Au + u p > 0 in the distributional sense. It is obvious that t~(x) ~ uc~(lx - x0[) can be served as a supersolution. Applying the standard monotone method, we can construct a solution of (1.1) with a singularity at x0, which satisfies (i) ~ (iii). Namely, let u_l = u_ and u k be the solution of

Au_~ + u_k_l = 0 in f2

b t k = O o n 3f2.

By induction on k, we may assume u l < . . . < u_~_ 1 < t~. Since ti ~ L q (g2) for some q > 1. Hence, the existence of u_ k is guaranteed by the elliptic L q theory. By the maximum principle, we have _uk_~ < u~ < ~i. Applying the elliptic theory, t~k converges to a function u in C 2,~ in any

compact set of (2\{0} for 0 < fl < 1. And u is a solution of (1.1) with u(x) _< u(x) <_ 6(x) . Hence, u satisfies (i) ~ (iii). The uniqueness follows immediately from the eigenvalue's comparison. [ ]

3. Approximate solutions

In this section, we construct approximate solutions of (1.1). Following notations in [5], a pair of functions (ti, f ) is called a quasi-solution of (1.1) if

A~ +/~P + f = 0 in f2 (3.1)

where f2 is assumed to be a bounded smooth domain in R n throughout this section. Using the family of solutions u~ in Section 2, we have

lira f uq ( x )dx = O. ce-->0 JRn

for 0 < q < p* where p* is defined by

p , _ n (p - 1) 2 (3.2)

228

When - - n

n - 2

Chiun-Chuan Chen and Chang-Shou Lin

 0

2n ~l L e m m a 3 . 3 . Fix Po, qo such that p < Po < P*, - - < qo < ~, ~ > 0 and {21 . . . . . 2k} C g2.

n + 2 Then a quasi-solution (i~, f~) o[(1.1) can be constructed to satisfy the following.

(i) ?~k is smooth except at x j, 1 ~ j < k. At x j, Ftk(x) has the asymptotic behavior.

1

l imuk(x ) lx-2Jlt@r-1 = ( ~ - 1 ( n - 2 - - - 2 ) ) "--~ x-+~ 5 p 1

(3.3.1)

(ii)

f il~~ < ~/, and f fq~ < ~ . (3.3.2)

(iii) Set

k ) ff2 /~ -p-1 2 Qk(g)) = - l + E N-J eo - e 0 [ v ~ol2 - p j a u k ~o , j=l

for ~o ~ H i (f2). Then Qk is positive detinite and equivalent to Hlo norm in Hcl (f2).

(3.3.3)

Proofi We will construct (ilk, fk) by induction on k. Before the construction of quasi-solution (ilk, fk), we note that for any a c (0, ec], we have, by (2.4),

P- l (x )~o2 (x )dx P fRn u=

< ~ n - 2 - - - p 1 ,,lxl-2~~

~- ~ - 1 ( 2P ) ( 2 ) ( n 2 2 ) - z f . n - - 2 - - - - ~ IVqgl2dx p 1 n

= (1 -- e0) s I v ~o[ 2dx (3.4)

for any ~0 6 C~ a (R n).

1 Let 77(x) = ~(txl) be a smooth cut-off function such that r/(t) = 1 for 0 < t < 2 ' and 0(t) = 0

for t > 1. For any r > 0, we denote 0r(x) - ~(x) . For k = 0, (0, 0) is a trivial quasi-solution r

satisfying (i) ~ (iii) of Lemma 3.3.

Now suppose that conclusions of Lemma 3.3 hold true for {21 . . . . . ~k-1}, and {ilk-l, f~-I } is 1

a quasi-solution satisfying (i) ~ (iii). Let 0 < rk < ~ min{dist (2k, Ofa), Ixe - 2j [, 1 <_ j < k - 1},

and define ilk ~-~ ilk-1 q- r/rk (x -- -~k) u~ (X -- )~k) ~ ilk-1 -}- t/kU~ (3.5)

Existence of Positive Weak Solutions with a Prescribed Singular Set of Semilinear Elliptic Equations

where rk and ol will be chosen later. Then

II~k -- ~-111Z~O --< I l u ~ I I L P O �9

I f ot is small enough, then we have

1 Ilu~llLpO < ~ (11 - I I ~ - l l l L * , o ) �9

Therefore, we have

Set

1 il~llLPo < ~ (11 + ilU~-llltP0) < 11

f k = AFtk + uP - p p

=" fk-I q- ( fftp -- uP_I -- 11k Uol) -}- [( r}ff -- 11k)"P q- 2 V 11k V"~ -}- /kllkUc.~] -= f k - 1 --k gl q - g 2 �9

We note that tTk-1 is smooth in [~(YCk, rk). Hence,

229

(3.6)

(3.7)

(3.8)

f f-p-12 2 = p /~f-1112q)2 + P Uk 112q )

< 1 + ~ 3 - J e 0 -- e0 1V111q)l 2 j= l

Step 1. There exists a constant C > 0 such that

~ - p - 1 2 ~ 3 - j 8 0 q)]2 C q)2 p u k ~o < 1 + - e 0 (3.11) j=l

holds true for any q9 6 Ho ~ , where C is a constant independent of or.

Let 11i G C~~ i = 1, 2, such that 112 + 112 = 1, the support of 111 is disjoint from [~(2k, rk), and the support 112 is disjoint from {2~ . . . . . 2k-1} . Then

It is obvious that (3.3.1) holds for fik at 2k.

We divide the proof of (iii) into two steps.

I[fkllLqo ~ [Ifk-llltqo + IIglllLqo -I- llg21[Lqo < I1.

s L l+u l,qo 39, (~k,r~) (2k,rl,)

2n n which can be small if both rk and u are small, and - - < qo < - .

n + 2 - 2

Also we have IIg2]lz~o < C max {]Vu~(x)l + lu=(x)l} (3.10)

~_<lxl_<rk

which is small if ot is small. Therefore, if ot is small, then, by (3.9) and (3.10), we have


+(1-~o)f 1~702~ol2 + C1 / ~o 2

< 1 q- E 3 - J g 0 -- e 0 I V~o12 +C2 ~o 2 j = l

where we utilize the induction step k - 1, and fik-I is smooth in the support ~2.

Step 2. Fix a small e > 0. We can find a finite dimensional subspace N of H~ such that

c f ~o 2 <_ ~Qk_l(~O)

for all ~o ~ H i which is orthorgonal to N with respect to the quadratic form Qk- 1 and C is the constant

stated in (3.1 1). For example, let N be the complete linear subspace spanned by kth eigenfunction of A for k > l with I large. Then N can be chosen as the orthorgonal complimentary subspace of

with respect to Qk-I in H01(fl). For anY ~o 6 H~(f2), decompose ~o = q91 + ~02, ~01 ~ N and

go2 ~ N • (with respect to Qk-1). Let ~k = t~k-1 + vk, and Bk = B(Yq,, rk). Since/it, = ilk-1 outside Bk and tTk-i is smooth in/~k, we have, for x ~ Bk,

Then

-p -1 [ p-1 \ (x) = (v~(x) + ~k_~(x)) p-~ < c t l + v~ (x)j u k D

\ /

p f = p / ~-~ (~o~ + ~02) 2

= P Uk q92 + P Uk W1 + 2p u k ~OW2

< P "k v ' 2 + P uk_ 1 % + 2 p Uk_ lqgtq92

f,-, ] "q'-Cl V k q91 "q- V k l~ol~o2l + I~Pl + ko1~o21 , k

(3.12)

To estimate the sum of the first three terms, we utilize step 1 and the induction step and obtain

p u k ~o 2 + p Uk_ 1 ~o 1 + 2p uk_ 1 ~o1~o2

(f ) < 8 0 IsTq)112 -}- Ivq)2] 2 + 2 p Uk_l~Ol@2-'[-C ~02,

k-1

where 80 denotes 1 + E 3-J8~ - Go. Since ~o 1 is orthorgonal to ~o2, we have j = l

f f - p - I 80 Vqgl �9 Vq92 = p Uk_l (Plq92 .


Therefore we have, by the choice of N,

f ~ , 2 f ~ , ~ f ~ P Uk q92 + P uk-1 r + 2p Uk_ 1 q)1~02

< (ao + 3-(~+1)8o) f I v ~,1 z ,

provided that 8 is small. And, for any 8 > 0 and by (3.4), we have

f " - ' f "-'2 i "-~ v k kol~o21 < e v k ~o + C e v k ~o 1

I / < C18 Ivq)l 2+C~ v k ~o 1

_< 2- ,3 - (k+i ) ,o f lv~o l2+C~f p-12 I)k 'F1 ,

provided that 8 is small.

For the last two terms, we have, for any 8 > 0,

<

k

< 2--13--(k+l)eOi iV~Ol2+Ce fBk~O ~.

Therefore, (3.12) becomes

i [( )2 ]i -p -1 2 ~ Z 3 - J s 0 "~- p u k ~0 1 "+" - 80 " 3-k80 IV ~0[ 2 -[- C1 t J vk ~01 ~- ~~ j= l k

where C1 is a positive constant independent of a and rk. Since N is a finite dimensional, rk can be chosen so small such that

C1 [_ q)2 < 2-13-(k+l)eoQk_1 (~ol) JB k

_< 2-13-(k+l)e0 Q(~o)

-< 2-13-(k+l)e0 J I V ~ol 2 �9

After fixing rk, we may choose ot so small such that the/ef t -hand side of (3.10) is small and, by Proposition 2.5 and (3.4), we have

f p-1 2 C1 v k Pl ~ 2-13-(k+l)80Qk-1 (~ol)

< 2-13-(k+l)eoQk_l(~o)

-< 2-13-(k+t)80 / J V 99[ 2 �9 J


Therefore, we have completed the proof of Lemma 3.3. []

Let 07, f ) be a quasi-solution of (1.1) as stated in Lemma 3.3. Suppose that a solution u of (1.1) can be written as u = t~ + v, then v satisfies

A v + ( ( t + v ) P - - f t P = f

v i a l = 0 , and f i + v > 0

Define

in f2 (3.13)

in S2.

l f ~ f~ f~ E(fp) m~ ~ I V q)[2 _ F(fi, q)) + fq)

for ~o ~ Ho t, where F(s, t) =

Lemma 3.14.

1 p + 1 {Is + tlP(s q- t) - s p+I - (p + 1)sPt}.

E ~ Cl(Hol(g2); R) and any critical point v of E satisfies

{ A v + l F t + v l P - g t P = f i n ~ ,

v + ~ > O in g2, and v = O on O~ .

Moreover, E satisfies the Palais-Smale condition.

Proof. The first part of Lemma 3.14 is standard. We leave the details of the proof to the reader. To prove the second part of Lemma 3.14, suppose that 3vj E HOt (~) such that

as j --+ +cx~. derivative of E can be computed as

E (vj) -+ C, and (3.15)

E ' ( v j ) --+ 0 in HOt (3.16)

We want to show that there exists a strongly convergent subsequence of l)j. The

Case 1. p < 2. A direct computation shows that

-(2 p 1+1) f IWJl2--- Since p < 2, the inequality,

1 E (vj) - p +---5 ( e ' vj)

p + l

IV j q_/~IP/~ -- /~p+l __ p~tPuj < p ( p ~ - - 1) ~tP_lv 2

holds. By (3.15) and (3.16), we have

( p - 1) 12 ~P-~v~] (1 IlvvJllL2 ) .

(3.17)

P p + 1 f fv~.


By Lemma 3.3, the boundedness of II V vj IlL2 follows immediately. Furthermore, by (3.17), we have

(=' (~ ) - e' (~:), ~,- ~j)

= f lv(oi-oj)l=- f ooi +~,:-l~J+~Ip)(~i-~)dx. <3.18)

Since

[ ll 4- x l P - II + ylP l < pmax {ll 4- lx[lP-l, ll 4-1yl[P-l} l x - y ]

< p[x - Y[ 4- C (Ixl p-1 4- lyl p - I ) Ix - Yt

for all x, y 6 R,

f p f . ' ] Ui --VJl 2<-~C f ( [Ui]p-I 4- ]l)J] p - l ) (l;i--UJ)24-O(I Ui --Uj[H1 ) "

By H61der inequality, the first term of the right-hand side can be estimated by

(n-2)(p-1)

f Ouilp-14- l'Ojl p-l) (Ui--/)j)2 < ( f ,1)i,n2@~2+ [,ojln2--_n=) 2~'1 ( f Ioi - vjl2q) ~ ,

1 (n 2 ) ( p 1) ( n - 2 ) 4 - - ~ n - 2 2n where - = 1 - > 1 - , namely, 2q < - - . Hence, there

q 2n 2n n n - 2 exists a subsequence of vj (still denoted by v j) which is convergent in L 2q. Then, by the above inequality, we conclude that vj is strongly convergent in H 1 (f2).

C a s e 2. p > 2.Let Vj ~ max(0, --/)j), (3.17) gives

Since 1 - 11 - x] p < px for all x > 0, we have

/ , / , ] Iv~,jl 2 - p J ~p-~ (vj) = < c IIv~J I1~= �9

Lemma 3.3 implies that I1 V v./IlL2 is bounded.

Let vj = v + - fij, we have

It is easy to see that

F ({t,-fJj < C1 Ifljl p+I + up- lv j

__< C 1 ([[ VJ[IL2 4- [IVf)j _< C2.


Therefore, we have

(~,/ = ~ (~) + o(1).na

(E' (1)j), oJ-) = f ~ToJ - 2 - f (. + ~-.~)~; + f cv~. 2 x)p+ 1 Since (1 + x)Px - x - 1 ((1 + - 1 - (p + 1)x) > p - 1 xP+l for x > O, we have

p + - p + l -

p-If v+ P+I < 2E (vj) _ (E, (vj) < ) - f f o ~ + o(1) . p T 1 - '

For any e > 0, there exists Ce > 0 such that

(U -}-OJ-) p Of -- /~Po~ ~ (p -I- 8)/~P-IoJ -2 -[- Cgo~ p+I .

By (3.17),

(3.19)

L e m m a 3 . 2 0 . {xl . . . . . xk } be any set of finite points in S'2. Then there exists at least two distinct solutions of(1.1)having {Xl . . . . . Xk } as its singular set.

Proof . We claim that there exist positive numbers t/0 , p, 0 > 0 such that if (fi, f ) isaquasi-solution of (1.1) as stated in Lemma 3.3 with 0 < ~ < rl0, then

E(u) > 0 > 0 (3.21)

for u in H I such that p < [1V u II ~ 2p. After the claim, the existence's part of Lemma 3.20 follows immediately. Because one solution can be obtained from the minimizing: rain E(u) < E(O) =

Ilvull_ 0, there exists Ce > 0 such that IF(s, t)l < P(1 + 8)sp-lt 2 q- Cet p+I. Thus,

f f f f 2~ 2E(v)_> I V v l 2 - p ( l + e ) f i p - l v 2 - C ~ v p+I-2~/0( v~7-2)~

By Sobolev's embedding and Lemma 3.3, for small e > 0,

[(f v,~)"+'~ 2E(v) > C1 f a [ V vl 2 - C2 I V

f vvy2-(p+~)fa;"~+2<-(E'(vJ),vY)-ffvJ+cff ~ Together with Lemma 3.3 and (3.19), we have, for small e > 0,

+ 2 _ 1) Vvj L 2 < C ( Vv + L2 + "

After establishing boundedness of 11 V oj II L2, we can utilize (3.18) to obtain a strongly convergent subsequence in H 1 (fa). Therefore, the Palais-Smale condition is satisfied and the proof of Lemma 3.14 is complete. [ ]


Then the claim follows easy.

n + 2 then Suppose that u = 17 + v is any solution of (1.1) with v 6 H01([2). Since p < h--~-2, a bootstrap argument can show that v ~ C ~ ( f 2 \ { x l . . . . . Xk}). And if we assume that x j is a

2 removable singular point, by (i) of Lemma 3.3, v(x ) > clx - x j l -p-:-r- in a neighborhood o f x j which implies that v fL L p+l ([2), a contradiction to v 6 Hd (f2). Therefore, the proof of Lemma 3.20 is finished. [ ]

To complete the proof of Theorem 1.5, we need another lemma.

L e m m a 3.22. L e t v b e a s o l u t i o n o f O . 1 3 ) a n d v ~ H l ( f~ ) . Then Iv[ ~ ~ 1tlo f o r s o m e ot > 1. The constant ot depends only on p, qo and the dimension n.

Proof. By Lemma 3.20, we know that v 6 C ~ except at xi 1 0 depending on p, qo, and n such that Ix - x i l - S v E L "--:~-, 1 < i < k .

Let O(x) - (Ix - xi 12 + X2) -~ , 1 < i < k. Multiplying (3.13) by r/2v, we have

f I v , o l 2 - f , v,12o2 --- f(117+vlp-ap), o-ff,2v <_ (p+e)f17p- ,2oz+Gf,72vp+l-ff,2v

S i n c e I V t/I 2 ~ ~2r] 2 " Ix - xi] -2 , w e h a v e

f lv,v12-a2 f nzlxl-ZvZ-(p+e) f17p-ln2v 2

< Ce ?l 2q ~ v ~2-3~-" + fr l2 2+, { t" 2, \ - - ~

2n where -1 = 1 - (P + 1)(n - 2) Since f E L q~ with q0 > ~ 2 ' 8 can be chosen so that the

q 2n n + right-hand side of the above is bounded. If both e and 8 are small, then, by Lemma 3.3, we have

f l V < C~. ( . v ) l 2

Passing to the limit L --+ 0, then the claim is proved.

Since u --- 17 + v is a weak solution of Au + u p = 0 with an isolated singularity at xi, by Theorem 1.3 of [1],

1

l im u (x ) l x - x i l p-~-r- = [ ~ - 1 ( n - 2 - - - 2 )]p-ZT- x---+xi p - 1

2 By (3.3.1), we have lira v ( x ) l x - xil -y:7-1 = O. Hence, step 1 implies that v 6 L .~zu~ for some

x-- - - -+x i

or0 > 1 which depends only on p, q0, and n. To estimate lip =0 Ilad, we multiply (3.13) by M2=0-2v, then

2or0 - 1


Hence, if both or0 - 1 and e are small, then

f l V < C1 1)~0 12

where C1 is a constant depending on I[ V v ILL2. Therefore, the proof of Lemma 3.22 is finished. [ ]

4. P r o o f o f m a i n t h e o r e m s

Now we are in the position to complete the proof of Theorem 1.4 and Theorem 1.5. Since Theorem 1.4 is a part of Theorem 1.5, we only need to present the proof of Theorem 1.5.

i

P r o o f o f T h e o r e m 1.5. First we give a proof of the existence of weak solutions with a prescribed singular set. If the singular set S is a set of finite points, then the existence of two distinct solutions is proved in Lemma 3.20. Let {xl . . . . . Xk ...} be a countable dense subset of S. Pk is an increasing sequence, lim p~ = p*. For any ~ > 0, by Lemma 3.3, we can construct a sequence of quasi-

k-++oo solution (ilk, f~) with singular set {xi . . . . . Xk} such that

and

Ifik+l - gtk[ p~ dx < ~-s

f~ I f k + l - fk] qO dx < 2- 7

where q0 is a fixed constant such that 2n n - - < qo < Hence, fik converges strongly to fi in n + 2 2"

L q (f2) for any q < p*. When rl is small, by Lemma 3.20, we can find two sequences of solutions �9 i = Ftk + vi~, i = 1, 2 such that u~ of (1.1) such that u k

f 2 s w22 Vv I _< P0 < < Pl

where Pl > Po is two constants independent of k. Let v~ be one of the two solutions 0btained in Lemma 3.20. Then, by Lemma 3.22, we have

Off V v k L2 <_ C1

for some ~i > 1 where o~ I and C1 are independent of k. By Sobolev's embedding and the H61der 2net

inequality, we may further assume that Vk converges in L 7=7-2 and weakly converges to v in H01 (f2) for

any 1 < ot < ot ~. We want to prove that Vk strongly converges to v in H01 (fa). By elliptic estimates,

it suffices to show that Iftk + Vkl p - Ft p converges strongly in L ~ . To prove this statement, we need the following two steps.

- p - 1 Step 1. u~ vk strongly converges to t i p - i v in Ln@2.

f Uk 2ff~2 -P-lvk -- F* p- lv

<_ C1 -p -1 _ ~ v ~ d x q- ~(P- 2ta47, Irk - v174~ dx

where

Since - -


[(s, 2nq' ~7 2ng~_~ \ n-2 _< C l - p - 1 _ ~ p - 1 V/~ 2 a(n+2)

(i , -] . l . 2nq" - - V[ n2r<~ "~ ~ + ~cp- )~77 q'

1 n - 2 (n + 2 ) o r - n + 2 4 - - - - 1 ~ > - - q' (n + 2)or " (n + 2)o~ n + 2

2 n q ~ n ( p - 1)n < and tik converges to ~ in L q for q < p* -- we have

n + 2 2 2 '

l im - p - I __ + 2nc~ k-++c~ y Irk -- vl z~-2 = O .

Thus, step 1 is proved.

237

S t e p 2. Since

. p+l [~tk - p - I -7- I - p - 1 2 v P + I ' ~

"-k Vkl p - L-t; -- p u k Vk < C tUk Vk + k ) '

by step 1 and Lebesgue 's dominated theorem, we have

p+l

l i m f lYlk -Jr- Vk[ p -- uP -p -1 ~p pFtP- lv ~-- k ~ + ~ J k -- PUk Vk -- [Yt + vl p + + d x = O .

1 n - 2 2 n [P _p Since p + 1 _ 1 + - > 1 + - - - and ] t k + Vk can be written as the sum of

p p n + 2 n + 2 - u k -P -P ]P -P strongly converges in [Ytk + Vk[ p -- u k -- p Y t ; _ l V k and PUk_ 1Vk, we conclude that [Ytk + vk -- u k

2n - L ~ (f2). By elliptic estimates, Vk converges to v strongly in H l ( f i ) . Since v~, i = 1, 2, strongly

converges to v i, i = 1, 2, in H I (f2), we have v 1 # v 2.

Set u = fi + v where v is one of v i obtained above. Then u is a solution of (1.1). It is obvious that u E C c ~ ( ~ \ S ) . For any x0 ~ S, any open neighborhood of x0 contains Xk for some k. ff u is

-2 bounded in this neighborhood, ]vr > Yt - c > colx - XklP -7 - -- c by (i) of Lemrna 3.3. Then it is a contradiction to v ~ L p+I . Therefore, the singular set of u is exactly equal tO S.

1 Let 0 = ~ where k is any large positive integer. By the above, we can construct two sequence

of soautions u~, i -- 1, 2, o f (1.~) such that u~ = fi~ + v~, with v~ ~ H I (f2) which satisfy

f f2 i _ i ~ pk 1 ~Uk) < ~, for i = 1 , 2 , and

5 Pk < PO 5 < P t

where pk is an increasing sequence converging to p*, lim Pk = 0, both P0 and Pl are two constants k--++c~

independent of k. Thus, uk 1 converges to zero in L q (~) for any q < p*. As in the proofs of step 1

and step 2 above, v~, after passing to a subsequence, converges to v in H01 (f2). Hence, v is a weak

238 Chiun-Chuan Chert and Chang-Shou Lin

solution of (1.1) with II V rilL2 >_ PO. Since v ~ H01 (f2), by a bootstrap argument, we can show that v ~ C~((2) . Therefore, the proof of Theorem 1.5 is finished. [ ]

P r o o f o f T h e o r e m 1.8, To prove Theorem 1.8, we can follow the same procedure as the proof of previous theorems. Hence, we sketch the main ideas and omit details of the arguments. Let m be a positive integer such that the following holds.

and

( n + 2 ) 2 < ( ~ - 2 2 ) 2 (4.1)

n + 2 m > - - (4.2)

2

n + 3 It is not difficult to check that for n > 9 if m is chosen as m - when n is odd, and

- 2 n + 4

m - when n is even, then m satisfies (4.1) and (4.2). 2

Let S n-m be a (n - m)-dimensional sphere in R n, and Po ~ S n-'n. By using solutions of (2.1) and the Kelvin transformation, we can construct a family of solution uc~(x), ot 6 (0, oo], of

Aua(x) + uu p = 0 in R n (4.3)

n + 2 where p -

n - 2 below.

- - - in the remainder of this section. The family of solutions u~ satisfies (4.4) ~ (4.7)

4

lim uc~(x)d(x)~- = 2 m - n 2 x--,s .-m ~- (4.4)

uniformly in any compact set of S n-m \{P0} where d(x) denotes the distance between x and S n-m .

u~(x) is strictly increasing in or, and lim~--,0 ua(x) = 0 (4.5)

uniformly in any compact set of R n U {oo}\S n-re. Moreover, there exists a constant c independent o f u such that u~(x) < c}xl 2-n for Ixl large. Therefore, we have

lim [ uP(x)dx = 0 . (4.6) u~0 JR"

For ~o ~ H 1 (Rn), we have by part (iii) of Lemma 3.3

JRfn uP-1(x)q92(x)dx ~ ( 1 - e0 ) , tRf lv , ~o12dx P

for some positive constant e0 depending only on n and m.

To see this, let fi~(x, y) = fia([xl) where fi~(r) is the solution of

n+2 m-l i l t (r~ ~--~ O, 0 r < +oe, / ~ ( r ) - l - r ot" " q - = <

fi(r) > 0 and limr---++~ fi~(r)r m-2 = a

and x E R n,

solution of Aft= + fi~=-~

(4.7)

n + 2 n + 2 m y ~ R n-re. Since m > ~ , we have - - n - 2 > - - ' m - 2 Hence, flu(x) is a weak

n+2 = 0 in R m (see Lemma 2.1 in [1]) and then fi~(x, y) is a weak solution


n+2 1 x - xo X0 in_2 U~ ( ' l~ - - - - - - - of A ~ + ~ = 0 in R n. Let e = (1, 0 . . . . . 0) and ua(x) = Ix - - x0[ 2 + e) for

x 6 R n with x0 r R n-re. Then u~(x) is a weak solution of

n + 2

Auto(x) + u~Z-~(x) = 0 in R n

u~(x) = O (ix~,_ ) at c~ .

X - - X 0

L e t S n-m be the image of R n-m under the mapping x ~ [x - x 0 [ 2 + e . I t i s easy to seethat S n-m

is an (n - m)-dimensional sphere in R n and u~(x) is a family of weak solutions of Au + u ~-+: = 0 in R n satisfying (4.4) ~ (4.7). For (4.7), it is easy to check that both sides of (4.7) are invariant under the Kelvin transformation. Since ~n-m is congruent to any (n - m)-sphere, the above claim follows immediately.

Step 1 . L e t S1 . . . . . Sk . . . . be a sequence of disjoint (n - m)-dimensional spheres. Fix a small positive number 0 > 0. 0 will be chosen later. A sequence of positive approximate solutions tik is constructed to satisfy (4.8) ~ (4.11).

Fix Pk ~ Sk, k = 1, 2 . . . . . We have

4

n-: (n_~__2 ( 2 m - n - 2 ) ) ~-~-z l im gtk(x)d j (x )~-= (4.8)

x~S~ 2

uniformly in any compact set of S j \ { p j }, j -- 1, 2 . . . . . k, where d j (x ) denotes the distance between x and Sj.

-P We have Denote 3~ by 3~ = A~Tk + u k .

~Pdx < O, fS4-~dx < t/, (4.9) n n

ti~(x) converges to zi in LP(Rn), (4.10)

support of 3~ C mJ~=l B (Sj, rj) ,

where B(Sj; r j ) = {x e R n I dj(x) < rj}, and lira rj = O. j ~ + ~

The quatratic form

/ _ _ _ _ - p - 1 ,2 (4.11) 1 + Z 3-JE0 60 I ~7 (p[2 p ak ~' j= l

is positive definite and equivalent to the H 1 norm in H 1 (Rn).

The construction of tik is exactly the same as before except the cut off function 0~(x - 2t) is replaced by 0k (d~ (x)). To prove (4.11), it suffices to note that (3.11) becomes

fR~.p -1 _z < 1 q- 3-J60 - I ~Pl 2 ~p2 P . Uk ~0 _ ( 4 . 1 2 )

\ j = l

where c is a positive constant independent of a and K is a bounded set independent of or. Then the proof of Lemma 3.3 can also be applied to prove (4.11) without any modification.


Let

E(qg)=~ ,[V~~ 2 - ,F(ilk'~~ ,fkq9

for ~o 6 Ha(R n) where F(s, t) = _ _ . 1 [Is q- tlP(s + t) - s p+I - (p + 1)sPt}. It is, not difficult to p + l

see that E(~o) is continuous in the strong topology of HI(Rn). For any s > 0, there exists Cs > 0 such that

P (1 + 8)sP- l t 2 + Celt[ p+I IF(s, t)l _< -~

Thus,

2E(~o) > f l V~0l 2 p(1 + s ) f 2 _ c flqq p+I '~ - n+2 1

) - 2 f k n+2 [~olP+ 1 T+~

Fix Sl > 0 so that, by (4.12),

f f I V ~ol 2 - P (1 + el) u k q) >__ 81 I V ~ol 2

By Sobolev's imbedding theorem, we have

I(f, ( f ) 2E(~o) > el f a I V qPl 2 cl V - + 2~ I V qg[ 2

Therefore, there exists small p = p(r/) such that

with lira p(0) = 0. ~/--+0

8 1 p 2 inf E(~o) > - - > 0

Ilwll=p 4

lj (4.13)

Step 2. We claim that there exists vo c H a(R n) with 11 vvoll < p suchthat E(vo) = inf' E(v) < Ilvvlb_<p

0. Letvj 6 H I (R n) withl]Vvjl[ < pand lim E(vj)----- inf E(v) . Since vj is bounded in j--+ + (~ [Ivvl[_<p

HI(Rn), we can assume that vj --~ vo weakly for some vo ~ HI(Rn). If vj is strongly convergent to vo, we are done. Hence, we may assume that ~oj =- vj - vo is weakly convergent', to 0 and 0 < lim ]] V (flj ][ = /). Without loss of generality, we may assume that (pj is weakly convergent

j--++~ 2n

to 0 in L,7~-2 (R n) also. To obtain a contradiction, we compute

E (v0 + ~s) - E (vo)

if / = ~ IV(flj[ 2 - [ F ( i , v o + ~ o j ) - F ( F t , v o ) ] d x + o ( 1 ) ,

2n n . . . . because f~ 6 L ~ ( R ) , where, for slmphclty u denotes the approximate solution ik. We decompose the second term into two terms,

v ( i , vo + ~i) - F ( i , vo)


1 - {I ~ + vo + ~ j l ~ (~ + vo + ~ y ) - I~ + v01P (~ + vo)

P + 1

- (p + 1)1~ + ~ o : ~:} + {l~ + ~o : - ~P} ~:

=- gl + g2qgj �9

241

For gt , we have, ge > 0, there Cs > 0,

Igll < P +---~ I~ + ~0:-1 r + c~ I~jl p+l - 2

_. + +

Therefore,

f lg, Jax / / . 0. By Sobolev's imbedding, we have

Therefore,

I~ + vo: - ~" - y - 1 vol ~ L @ (R n)

f g2qgj

= o ( 1 ) .

Combining these two estimates, we have

E (vO + r - E (vo)

1 2s > ~ f lwil' P + - - ,

sl el Then we have Choose s = ~- el in (4.13), and let O be small enough such that Csp p-1 < --~.

lira E (v0 + ~oj) > E (v0) j--++~

which is a contradiction. Hence, the proof of step 2 is finished.

Let Vk be the solution of E (vk) = inf E(v ) . Then, by the maximum principle, we can show Ifvvl]<_p

that u/r = t/k + v~ is a positive solution of (4.3). Since o/: is bounded in HI (Rn) , we can assume that, after passing to a subsequence, vk converges to v ~ H 1 (R n) in L P ( K ) for any compact set K of R n. Hence, u = t / + v is a non-negative weak solution of (1.7). Since v r H01(Rn), we have


n+2 n+2 v e L;:7- (d/z). Hence, u e L '7~-2 (d/z). We claim that the singular set of u must include U~=tS j. If p e S j \ {p j } , and p r singular set of u, then there exists a neighborhood U o f p such that u(x) < c in U. This implies

- f i (x) < v(x) < c - t/(x), namely, 2n 2n

Mx)l >__ ;~(x) - c >__ rq(x) - c. However, v e L~-=~- (U) implies that fij e L~--~- (U) which is impossible. Therefore, Oj=tS J c the singular set of u. Suppose that UT=IS j is dense in R n, and because the singular set of u is closed, we conclude that the singular set of u is the whole space R n. Hence, the proof of Theorem 1.8 is finished. [ ]

5. The case with nonconstant K(x)

In this section, we consider a positive weak solution with a prescribed singular set of

Au 4-K(x)u p = 0 and u > 0 in [2 u 10a-- 0 (5.1)

Similar to Section 2 and Section 3, we first study the existence of an entire positive singular solution of

Au + K (x)u p : 0 in Rn\ {x0}

u(x) = o (Ixl - '+2) at c~, (5.2)

limx~x0 u(x) = + ~ and u e C 2 (Rn\ {xo})

and its linearized equation. Throughout this section, K(x) is always assumed in C I (Rn). We also assume that lim K(x) exists and 0 < a < K(x ) < b for two positive constants. When

Ixl--*+oo K(x ) -- K(Ix[) is radially symmetric about 0, we consider u(x, or) = u(Ixh o0 to be a positive singular solution of

u ' ( r ) + n~rlu'(r) + K(r)uP(r) = 0 (5.3)

limr----~+c~ rn-2u(r) = ot > 0

By using the Kelvin transformation v(r) = r2-nu(1) , (5.3) becomes F

v ' ( r )q - n~rlV'(r)q- K ( ~ ) r p ( n - Z ) - n - 2 v P ( r ) = 0 ,

v ( 0 ) = ~ .

Since p(n - 2) - (n + 2) > -2 , the existence and uniqueness of solutions v can be proved by the same way of the proof of corresponding theorems in the theory of ordinary differential equations. Therefore, for each o~ > 0, u(r, a) exists in a maximal interval (R0, oo) for R0 > 0, where either u(Ro, a) = 0 or R0 = 0. In the latter case, u(r, oO might have a removable singularity at 0.

Proposition 5.4. Suppose that n - - < p < pl and Lisanycons tantsuchthat 2 P - ( n - 2 - n - 2 p - 1

2 n - 2 2 1) < L < (-----~) . Then there exists a positive oto such that, for 0 < a < eto, the solutioll

p - u(r, or) o f (5.3) exists for the whole interval (0, oo) and satisfies the following:

(i) u(r, a) is singular at O,

(ii) p K ( r ) u p - l ( r , or) < - - and r2'


OJi) u(r, or) is increasing with respect to a.

P r o o f . The proof is based on an eigenvalue comparison theorem. Let ~0(r) = Ix[ - r where r (n - n - 2

2 - r ) = )~ and r < ~ . Then ~o(x) satisfies

A~o + ~.[xl-2~o = 0 . (5.5)

Without loss of generality, we may assume that K(0) = 1 and K(r) > 1 - 8 for 0 < r < r0. Let v be the entire singular solution of

Av + ( 1 - 8 ) v p = 0 inRn\{0}

limlxl~+~ [x[n-21) ( l x[ ) = 1, and v > O.

By Proposition 2.5, we know that

( p(1 - 8)vP-l(r) < n - 2 - - - - p - 1

2 ) )~ (5.6) p 1 r - 2 < Ixl 2"

Step 1. For ot is small, then u(r, or) < v(r) for r > 0. Set

w(r) = v(r) -- u(r; or),

then w(x) satisfies

A w + c(x)w = (K(r) - 1 + 8)uP(r; or) (5.7)

where c(x) (1 8) vp - up = - ~ v - u < p(1 - 8)v p-1 < ~-~ whenever w(r) > 0. Suppose there exists

/~ > 0 such that w(R) = 0 and w(r) > 0 for r > R. It is obvious that k ~ 0 as ot --~ 0. We multiply (5.7) by ~0(x) and integrate over lxl _> k and then obtain

()~ p~12P ( n _ 2 _ _2__~)) flxl>_k ~ o ( x ) w ( x ) d

_< [ ((1 - 8) - K(x))uP(x; ot)~o(x)dx JIx I>_k

< [ [K(x) - 1 + 8]uP(x; ot)~o(x)dx alx I_>r0

Since u(x; or) <_ v(x) for Ix[ > ro and small or, by Lebesgue's dominated convergent theorem, we obtain

f x r-2~~ < ] ]K(x) - + 8]uP(x; ot)go(x)dx = 0 lira 1 l>_1 u-+~ Jlxl>_ro

a contradiction. Hence, the proof of step 1 is finished.

S t e p 2. For small ot > 0, pu p-1 (r, a)K(r) < ~ - ~ . Let 8 in (5.6) be small, then 3rl > 0 such that

pK(r )vp - l ( r ) < -~ for 0 < r < rl . By step 1, for small or, we have pup-l( lx[, ot)K(lxD < ixlZ for all x ~ Rn\{0} such that u(Ixl, o~) exists.


F i n a l s tep . By the comparision between (5.3) and (5.5), and the conclusion of stcp 2, it is easy to

see that, for small ~ such that p K ( r ) u P - l ( r ; or) < - - u(r, or) satisfies (i) and (iii). Therefore, the r 2 '

proof of Proposition 5.4 is finished. [ ]

n n 3n + 3 P r o p o s i t i o n 5.8. Suppose - - 5,

n - 2 n - 2 3n - 5 - 0 < a __< Kz(lxl) _< Kl([X[) < b, and Kl([X[) - Kz(lxl) - O(IxD o. Let u (x ) = u(r, &)

be a solution stated in Proposition 5.4 with K = K t and & is small. Then there exists a posi t ive

& such t h a t f o r a n y s o l u t i o n v = v( lx l ,~) o f ( 5 . 1 ) with K = K2 and 0 < a < 6t, w e h a v e

v(lxl; ~) _< u(Ixl).

P r o o f . As in the proof of the previous proposition, we set w ( r ) = u(r) - v(r; a) . Suppose that the conclusion of Proposition 5.8 is not true. Then there exists/~ > 0 such that w(R) = 0, w ( r ) > 0

f o r r >_ R, and l im/~ = 0. w(Ixl) satisfies c~-+0

A w + c ( x ) w = (K2(r) - K t ( r ) ) vP (r; c~) ,

u p -- v p L where c(x ) = K l ( X ) - - u - v 0. As before in the

proof of Proposition 5.4, we have

_ _ _ 7~qg(x )w(x )dx p- - -1 p _

< [ (Kl ( r ) - K2(r))vP(r)~o(x)dx , (5.9) Jlx i>_k

n - 2 where ~o(x) = Ixl -T , )~ = r (n - 2 - ~:), and r < - -

- 2

2 F o r n > 5 , 1 e t z - - - -

p - 1

4 / ~ > ~ , -

p - 1

p 1

n 3 n + 3 - - n - 2 - - - . (5.11) p - 1 p - 1 p - I

Since - - 3n - 5 ' it is a straightforword computation to show that there exists a fi -2p

satisfying both (5.10) and (5.11). By Proposition 5.4, v p (r; o0 < clx[ ~=r-j for some posilive constant c and 0 < Ix[ < 1. Therefore,

IKl( r ) - K2(r)[ vP(r)qg(r) < Clxl - p2-~-p~+I- pz-~-I+z

for Ix[ < 1. By (5.10), the exponent

2p 1 + 2 /3 4 - - - - - + l - f l < n . p - 1 p - 1 p - 1

Therefore, we have

g / . l im / ~o(x)u(x)dx < C lim / (K l ( r ) - K2(r)) vP(r, ot)~o(x)dx = 0 , ~-+0 JIx ]>_1 cr JRn


a contradiction. et--2

Forn = 3 o r n = 4, w e u s e g ( x ) = I x l - ~ - i n s t e a d o f l x l - r . Then I K l ( r ) - K 2 ( r ) l v P ( r ) 9 ( r ) <

c x - . The exponent

2p n - 2 - - + - - - l < n , p - 1 2

because p > 2 for n = 4 and p > 3 for n = 3. And (5.9) again leads to a contradiction. Thus, the proof of Proposition 5.8 is complete. [ ]

n 3n + 3 n Theorem 5.12. Suppose that 5, and < P < Pl for

n - 2 3 n - 5 - n - 2 n = 3 or 4. Then there exists infintely many entire positive singular solutions ui o f (5.2) such that

6i ui(x) < Ix - xol n - 2 ' x ~ Rnk{x0} and i-~+oalim ei = 0. Furthermore, we have ui(x) j > 0 and p K ( x ) u P - l ( x ) <_ )~[x - x o [ -2 forsome )~ < ( - - - ~ )

P r o o f . Without loss of generality, we may assume that x0 = 0. Let Kl(r) =- sup K(x) , and Ix j=,.

K2(r) = inf K(x) . Since K ~ C 1, we have Kl(r ) - K2(r) = O(r). Let u(r; ~) and v(r; o~) fx[=r

denote solutions of (5.1) with K = K1 and K = / ( 2 , respectively. For any small 8 > 0, we can find > 0 such that v(r; 6t) < u(r; (~) for r > 0 by Proposition 5.8. Since Kl ( r ) > K(x) , u(r; g~) is

a supersolution of (5.2). Also v(r; ~) is a subsolution of (5.2). Then by the well-known monotone scheme, we can find a solution u of(5.2) such that v(r; ~) < u(x) < u(r; ~). Also pu p-1 (x )K (x) <

puP- l ( r ; ~ )K l ( r ) < ~ i~ . Again by similar argument as in Proposition 5.8, we can find oe* <

such that

u (r; u*) _< v (r; 6) < u ( x ) .

Repeat the above once more, and we can find a sequence of solutions of (5.2) to satisfy the conclusion of Theorem 5.12. [ ]

Proof of Theorem 1.9 and Theorem 1.10. Before the prooL we should remark that suppose that u is a positive solution of Au + K(x )u p = 0 in B~ (x0)\{x0} with a nonremovable singularity at

2 2 2 J )] T =r . Applying Theorem 5.12, x0, then limx___+xoU (x)Ix -x0 [ TzI-I = [K(x0) - 1 P - - i (n - 2 -

p - 1 we can construct approximate solutions as Lemma 3.3 except that (3.3.1) and (3.3.3) should be read as

1

lim Ftk(X) l X - - ~ j l p 2-~< = K ( Y j ) -1 n - 2 - (5.13) x---~j p - l '

and

_ _ - p - 1 2 Qk(9) = 1 + Z 3 - J e 0 e0 [ V g l 2 P K(x)u~: ~o . (5.14) j = l

Once we can construct approximate solutions, we can follow the same procedure in Section 3 and Section 4 to prove Theorem 1.9 and Theorem 1.10. [ ]


References

[1] Caffarelli, L.A., Gidas, B., and Spruck, J. Asymptotic symmetry and local behavior of semilinear elliptic equations with critical Sobolev exponent, Comm. PureAppl. Math., 42, 271-297, (1989).

[2] Gui, C., Ni, W.M., and Wang, X. On the stability and instability of positive steady states of a semilinear heat equation in R n, Comm. PureAppl. Math., 45(9), 1153-1181, (1992).

[3] Gidas, B. and Spruck, J. Global and local behavior of positive solutions of nonlinear elliptic equations, Comm. Pure Appl. Math., 34, 525-598, (1981).

[4] Li, Y. Asymptotic behavior of positive solutions of equation of Au + K(x)u p = 0 in R n, J. Diff. Eqns., 95, 304-330, (1992).

n

[5] Pacard, E Existence and convergence of positive weak solutions of - A u = u ~ in bounded domains of R n, n > 3, Calculus of Variations and P.D.E., 3, 243-265, (1993).

[6] Pacard, F. Solutions with high dimensional singular set to a conformally invariant elliptic equation in R 4 and in R 6, Comm. Math. Phys., 159(2), 423--432, (1994).

[7] Schoen, R. The existence of weak solutions with prescribed singular behavior for a conformally invariant scalar equation, Comm. PureAppl. Math., 41, 317-392, (1988).

[8] Schoen, R. and Yau, S.T. Conformally flat manifolds, Kleinian groups and scalar curvature, Inv. Math.. 92, 47-72, (1988).

Received March 22, 1995

Department of Mathematics, National Taiwan University, Taipei, Taiwan, R.O.C. 10617 e-mail: chchchen@ math.ntu.edu.tw

Department of Mathematics, National Taiwan University, Taipei, Taiwan, R.O.C. 10617

Communicated by Joel Spruck

Date post:	26-Aug-2020
Category:	Documents
Upload:	others
View:	0 times
Download:	0 times

Existence of positive weak solutions with a prescribed ...troy/ed/chenandli.pdf · The Journal of...

Documents