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A Necessary and Sufficient Condition for Existence of Large Solutions to Semilinear Elliptic Equations
Journal of Mathematical Analysis and Applications 240, 205᎐218 Ž1999. Article ID jmaa.1999.6609, available online at http:rrwww.idealibrary.com on
A Necessary and Sufficient Condition for Existence of Large Solutions to Semilinear Elliptic Equations Alan V. Lair Department of Mathematics and Statistics, Air Force Institute of TechnologyrENC, 2950 P Street, Wright-Patterson Air Force Base, Ohio 45433-7765 E-mail: [email protected] Submitted by John La¨ ery Received November 9, 1998
We consider the semilinear equation ⌬ u s pŽ x . f Ž u. on a domain ⍀ : R n , n G 3, where f is a nonnegative, nondecreasing continuous function which vanishes at the origin, and p is a nonnegative continuous function with the property that any zero of p is contained in a bounded domain in ⍀ such that p is positive on its boundary. For ⍀ bounded, we show that a nonnegative solution u satisfying uŽ x . ª ⬁ as x ª ⭸ ⍀ exists if and only if the function Ž s . ' H0s f Ž t . dt satisfies H1⬁Ž Ž s ..y1 r2 ds - ⬁. For ⍀ unbounded Žincluding ⍀ s R n ., we show that a similar result holds where uŽ x . ª ⬁ as < x < ª ⬁ within ⍀ and uŽ x . ª ⬁ as x ª ⭸ ⍀ if pŽ x . decays to zero rapidly as < x < ª ⬁. 䊚 1999 Academic Press Key Words: large solution; elliptic equation; semilinear equation.
1. INTRODUCTION We consider the semilinear equation ⌬ u s p Ž x . f Ž u. ,
x g ⍀ : R n , n G 3,
Ž 1.
where the function f is continuous and nondecreasing on w0, ⬁. with f Ž0. s 0 and f Ž s . ) 0 if s ) 0 while the function p is nonnegative and continuous on ⍀. We give necessary and sufficient conditions on the function f which insure that Eq. Ž1. has a nonnegative solution u for which uŽ x . ª ⬁ as x ª ⭸ ⍀. We call these large solutions of Ž1. on ⍀. If ⍀ is unbounded, we also require uŽ x . ª ⬁ as < x < ª ⬁ within ⍀. If ⍀ s R n, we call these entire large solutions. Such problems arise in the study of the subsonic motion of a gas w13x, the electric potential in some bodies w10x, and Riemannian geometry w4x. 205 0022-247Xr99 $30.00 Copyright 䊚 1999 by Academic Press All rights of reproduction in any form reserved.
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Several authors have studied special cases of Ž1.. ŽSee w3, 4, 6, 9, 10᎐13x, and their references. . For ⍀ bounded and pŽ x . s 1, Keller w6x and Osserman w12x show that a necessary and sufficient condition for Ž1. to have a large solution on a bounded domain ⍀ is that the function f satisfy ⬁
y1 r2
s
H1 H0 f Ž t . dt
ds - ⬁.
Ž 2.
Consistent with this, Bandle and Marcus w1x have shown that f satisfying Ž2. is sufficient to insure that Ž1. has a positive large solution provided the continuous function p is strictly positive on ⍀. Marcus w11x extends this to require p to be positive only on ⭸ ⍀. We show that these same results hold if p is allowed to vanish on large parts of ⍀ including its boundary. If f Ž s . s s␥ , then Condition Ž2. is equivalent to ␥ ) 1, and the problem of finding nonnegative large solutions of Eq. Ž1. for this special case, for both ⍀ bounded and ⍀ s R n, were considered in w4, 9x. Our results contain these as special cases. For ⍀ unbounded, all other results known to us Žexcept w6, 12x. require f to have the special forms f Ž s . s s␥ , or f Ž s . s e c s. ŽSee, e.g., w3, 4, 9, 10x.. Our existence results contain these as special cases. We show that, if p decays to zero rapidly as < x < ª ⬁, then, as with ⍀ bounded, Condition Ž2. is both necessary and sufficient to guarantee that Eq. Ž1. has a nonnegative large solution on ⍀. We note that Marcus w11x considers the existence of solutions to Ž1. on unbounded domains but restricts his attention only to those which decay to zero at infinity while blowing up on ⭸ ⍀.
2. MAIN RESULTS In this section we state and prove our results. Throughout this article, we require the nonnegative function p to satisfy the following condition: ŽA. For any x 0 g ⍀ satisfying pŽ x 0 . s 0, there exists a domain ⍀ 0 such that x 0 g ⍀ 0 , ⍀ 0; ⍀, and pŽ x . ) 0 for all x g ⭸ ⍀ 0 . Before proving the main results, we need to establish an important proposition that will be referred to later. PROPOSITION 1. Suppose ⍀ is a bounded domain in R n with C 2 boundary and p is a nonnegati¨ e C Ž ⍀ . function satisfying Condition ŽA.. Then, for any nonnegati¨ e constant c, the boundary ¨ alue problem ⌬¨ s p Ž x . f Ž ¨ . , ¨ Ž x . s c,
x g ⭸⍀
x g ⍀,
Ž 3. Ž 4.
EXISTENCE OF LARGE SOLUTIONS
207
has a unique nonnegati¨ e classical solution. Furthermore, if w is the solution to the same equation but ha¨ ing boundary ¨ alue c 0 G c, then w G ¨ on ⍀. Proof. Extend f as an odd function on R, and let f k 4 be a sequence of nondecreasing C⬁ŽR. functions which converges uniformly on compact sets to f, satisfies sf k Ž s . ) 0 for s / 0, f k Ž0. s 0, and f Ž s . q f Ž s y 1rk . F 2 f k Ž s . F f Ž s . q f Ž s q 1rk . .
Ž 5.
Such a sequence is easy to obtain using mollifiers Žsee w2, p. 145x.. We let pk 4 be a decreasing sequence of positive C ␣ Ž ⍀ . functions which converges uniformly on ⍀ to p. ŽSee Lemma 5 in w8x.. We now show that the boundary value problem ⌬¨ k s pk Ž x . f k Ž ¨ . , ¨ k Ž x . s c,
x g ⭸⍀
x g ⍀,
Ž 6. Ž 7.
has a unique nonnegative classical solution. It is clear Žsee Theorem 4.3 in w5x. that the boundary value problem ⌬ z s MK on ⍀ and z s c on ⭸ ⍀ has a unique solution where M ' 5 p1 5 ⬁ G 5 pk 5 ⬁ G 5 p 5 ⬁ and K ' 1 q sup k f k Ž c .. Furthermore, the function z is a lower solution of Ž6. and Ž7. since z F c on ⍀ so that MK G pf Ž c . G pf Ž z .. ŽSee w14, pp. 23᎐24x.. Also it is clear that Z ' c is an upper solution of Ž6. and Ž7.. Therefore, there exists a classical solution ¨ k of Ž6. and Ž7. such that z F ¨ k F Z on ⍀. ŽSee Theorem 2.3.1 in w14x.. We now show that, in fact, ¨ k must be nonnegative for each k so that we actually have 0 F ¨ k F Z. Thus, for fixed k, suppose min x g ⍀ ¨ k Ž x . - 0. Now let x 0 g R n _ ⍀ and define V Ž x . s ¨ k Ž x . q hŽ r . where hŽ r . ' Ž1 q r 2 .y1 r2 , r Ž x . ' < x y x 0 <, and the positive number is chosen small enough that min x g ⍀ V Ž x . - 0. Obviously this minimum must occur in ⍀, and at this point we have 0 F ⌬V s ⌬¨ k q ⌬ hŽ r . s pk f Ž ¨ k . q ⌬ hŽ r . F ⌬ hŽ r . - 0, a contradiction. Thus min ⍀ ¨ k G 0. To complete the existence proof, we must now prove that the sequence ¨ k 4 converges to a classical solution of Ž3. since each ¨ k already satisfies the boundary condition Ž4.. This can be done easily using a standard bootstrap argument. ŽSee Lemma 3 in w8x.. In fact, the argument here is almost identical to that of Lemma 3 in w8x Žin which h k s yf k .. Apparently the only difference in the argument lies in obtaining an upper bound on < ⌬¨ k < which is independent of k. ŽSee paragraph following Eq. Ž7. in w8x.. In the present case, this estimate becomes < ⌬¨ k < s pk f k Ž ¨ k . F p1 f k Ž ¨ k . F p1 f k Ž Z . F p1 sup k f k Ž c .. Since uniqueness is an easy consequence of the relationship ¨ F w whenever ¨ s c F c 0 F w on ⭸ ⍀, we prove only this relationship. Thus suppose min ⍀ Ž w y ¨ . - 0. Define x 0 , r, and hŽ r . as above, and choose ) 0 small so that min ⍀ Ž w y ¨ q hŽ r .. - 0. Then clearly this minimum
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ALAN V. LAIR
must occur in ⍀ and at this point we have 0 F ⌬Ž w y ¨ q hŽ r .. s pw f Ž w . y f Ž ¨ .x q ⌬ hŽ r . F ⌬ hŽ r . - 0, a contradiction. Thus ¨ F w and the proof is complete. THEOREM 1. Suppose ⍀ is a bounded domain in R n with C 2 boundary and p is a nonnegati¨ e C Ž ⍀ . function satisfying condition ŽA.. Then a necessary and sufficient condition for Eq. Ž1. to ha¨ e a nonnegati¨ e large solution in ⍀ is that f satisfy Ž2.. Proof Ž Necessity .. We prove this by contradiction. Thus suppose u is a nonnegative large solution of Ž1. and ⬁
y1 r2
s
ds s ⬁.
H1 H0 f Ž t . dt
Ž 8.
Since p g C Ž ⍀ ., there exists a positive constant M0 such that pŽ x . F M0 on ⍀. Let ¨ k 4 be the unique nonnegative classical solution of ⌬¨ k s M0 f Ž ¨ k . , ¨ k Ž x . s k,
x g ⍀,
x g ⭸⍀,
which exists by virtue of Proposition 1. Then the sequence ¨ k 4 is nondecreasing and ¨ k F u so that ¨ k 4 converges to a nonnegative function, ¨ . Using the monotonicity of the sequence ¨ k 4 and the fact that ¨ k Ž x . s k on ⭸ ⍀, it is easy to show that ¨ Ž x . ª ⬁ as x ª ⭸ ⍀. Furthermore, a standard bootstrap argument Žsee, e.g., w8x. can be used to show that ¨ is a classical solution of ⌬¨ s M0 f Ž ¨ .
Ž 9.
on ⍀. That is, ¨ is a nonnegative large solution of Ž9. in ⍀. Now we show that this, in fact, cannot happen. We may assume that 0 g ⍀, and we choose any number a such that ¨ Ž0. - a. Now consider the radial solution, w, of Ž9. which satisfies w Ž0. s a, w⬘Ž0. s 0. By Keller w6x w can be extended to the whole space as a nonnegative solution of Ž9.. Hence w is bounded on ⍀ and by the maximum principle, w F ¨ in ⍀. This contradicts ¨ Ž0. - w Ž0.. Thus, if inequality Ž2. fails to hold, then Eq. Ž1. cannot have a nonnegative large solution. Ž Sufficiency .. Suppose f satisfies Ž2.. Let u k be the unique nonnegative solution of ⌬ uk s pŽ x . f Ž uk . , uk Ž x . s k ,
x g ⭸ ⍀.
x g ⍀,
EXISTENCE OF LARGE SOLUTIONS
209
Then the sequence u k 4 is nondecreasing. ŽSee Proposition 1.. Now let x 0 be an arbitrary point in ⍀. Suppose we can show that there is an open set G ; ⍀ containing x 0 and a positive constant, MG , such that u k Ž x . F MG on G. Then our proof will be complete, for then it will be clear that u k 4 converges on ⍀ to some function u. Furthermore, as in the proof of necessity, a standard bootstrap argument will show that u is a classical solution of Eq. Ž1. while the monotonicity of u k 4 and the fact that u k s k on ⭸ ⍀ can be used to prove easily that u blows up on ⭸ ⍀. To prove the existence of a set G and a bound MG , we consider two cases: Ž1. pŽ x 0 . ) 0 and Ž2. pŽ x 0 . s 0. Case Ž1. Ž pŽ x 0 . ) 0.. Since p is continuous, there exists a ball Ž B x 0 , R . : ⍀ centered at x 0 with radius R such that pŽ x . ) 0 on B Ž x 0 , R . . Let m 0 s min pŽ x . : x g B Ž x 0 , R . 4 , and let w be a positive solution of Žsee w6x for the existence proof., ⌬w s m 0 f Ž w . wŽ x. ª ⬁
on B Ž x 0 , R . ,
as x ª ⭸ B Ž x 0 , R . .
It is easy to prove that u k F w on B Ž x 0 , R .. Furthermore, w is bounded on B Ž x 0 , Rr2.. Thus G s B Ž x 0 , Rr2. and MG s max G w. Case Ž2. Ž pŽ x 0 . s 0.. By the hypothesis, there exists a domain ⍀ 0 such that ⍀ 0: ⍀ and pŽ x . ) 0 for all x g ⭸ ⍀ 0 . From Case Ž1. above, we know that for every x g ⭸ ⍀ 0 , there exists a ball B Ž x, r x . and a positive constant M x such that u k Ž x . F M x on B Ž x, r x .. Since ⍀ is bounded Žand hence ⍀ 0 is bounded. ⭸ ⍀ 0 is compact. Thus there exists a finite number of the balls B Ž x, r x . , x g ⭸ ⍀ 0 , which cover ⭸ ⍀ 0 . Let M s max M x 1 , M x 2 , . . . , M x k 4 where the balls B Ž x i , r x i ., i s 1, 2, . . . , k cover ⭸ ⍀ 0 . Clearly then u k F M on ⭸ ⍀ 0 . Applying a maximum principle argument on ⍀ 0 will yield u k F M on ⍀ 0 . Thus G s ⍀ 0 and MG s M. This completes the proof. Remark. The condition on the function p is nearly optimal in the following sense: Let D be a domain such that D : ⍀, and let p be a function which vanishes on ⍀ _ D. Then p does not satisfy the hypothesis of the theorem, and it is rather easy to show that Ž1. has no positive large solution on ⍀. ŽSee w9x.. We now consider the case where ⍀ is unbounded. We begin by considering ⍀ s R n. It is evident from earlier work Žsee, e.g., w3, 4, 9, 12x. that the function p cannot behave arbitrarily as < x < ª ⬁ if we expect Eq. Ž1. to have an entire large solution. Therefore we add an asymptotic condition for the function p and establish results for unbounded ⍀ similar
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ALAN V. LAIR
to those of Theorem 1. Before that, however, we need to prove an inequality that will be needed later. LEMMA 1.
Suppose f satisfies inequality Ž2.. Then ds
⬁
H1
f Ž s.
- ⬁.
Ž 10 .
Proof. If we can prove that there exist positive numbers ␦ and M such that f Ž s. s
G␦2
for s G M,
Ž 11 .
then we will be done since
Ž s. '
s
H0 f Ž t . dt F sf Ž s . F f
2
Ž s . r␦ 2
for s G M,
which, in turn, yields w Ž s .xy1 r2 G ␦rf Ž s . for s G M so that Ž2. implies Ž10.. To prove Ž11., we assume it is false. That is, we assume there exists an increasing sequence s j 4 of real numbers such that lim jª⬁ s j s ⬁ and f Ž s j .rs j - 1rj for all j. Since f is increasing, we have f Ž s . F f Ž s j . for all s g w0, s j x, which, in turn, produces Ž s . F sf Ž s . F sf Ž s j . for s g w0, s j x. Hence,
Hs
sj
1
Ž s.
y1r2
ds G
Hs
sj
sf Ž s j .
y1 r2
1
G Ž jrs j .
1r2
sj
y1r2
Hs s
ds s 2 j 1r2 1 y Ž s1rs j .
1r2
ª ⬁,
1
as j ª ⬁, contradicting Ž2.. Thus Ž11. must be true. This completes the proof. THEOREM 2. Suppose p is a nonnegati¨ e C ŽR n . function which satisfies Condition ŽA. with ⍀ s R n and ⬁
H0 r Ž r . dr - ⬁,
Ž 12 .
where Ž r . ' max < x
211
EXISTENCE OF LARGE SOLUTIONS
Proof. Let ¨ k be any nonnegative solution of < x < - k,
⌬¨ k s p Ž x . f Ž ¨ k . , < x < ª k.
¨ k Ž x . ª ⬁,
ŽThe solution exists because of Theorem 1.. It is easy to prove Žletting
¨ k Ž x . s ⬁ for < x < G k . that ¨ kq1Ž x . F ¨ k Ž x . for all x g R n and for all k. Hence the sequence ¨ k 4 converges on R n to some function ¨ . If we can show that there is a positive function w such that w F ¨ k for all k and w Ž x . ª ⬁ as < x < ª ⬁, then we will have 0 - w F ¨ on R n and the proof
will be complete since another standard bootstrap argument will show that ¨ is a solution of Ž1. on R n.
To obtain such a function w, we observe that, as a result of Ž12., the function Ž r ' < x <., r
zŽ r. ' c y 2
H0 t
1y n
t ny1
H0 s
Ž Ž s . q  Ž s . . ds dt,
where  Ž r . ' Ž1 q r 2 .y2 and ⬁
c'2
H0
t 1y n
t ny1
H0 s
Ž Ž s . q  Ž s . . ds dt
is the unique positive solution of ⌬ z s z⬙ q z⬘ Ž 0 . s 0,
ny1 r
z⬘ s y2 Ž Ž r . q  Ž r . . ,
r)0
z Ž r . ª 0 as r ª ⬁.
From the sequence f k 4 in the proof of Proposition 1, we choose J large so that f Ž s y 1rJ . G y2 for all s G 0 Žin which f has been extended to be an odd function on R.. Then using Ž5. we get 2 1 q fJ Ž s. G f Ž s. ,
s G 0.
Ž 13 .
Now let F be the positive-valued smooth function defined on w0, ⬁. by F Ž s. s
⬁
Hs
dt 1 q fJ Ž t .
,
which exists because of Ž2., Ž13., and Lemma 1. Notice that F⬘ - 0 so that F is strictly decreasing and thus Fy1 has the same property. We observe also that ⌬ FŽ ¨k . G
y⌬¨ k 1 q fJ Ž ¨k .
s
yp Ž x . f Ž ¨ k . 1 q fJ Ž ¨k .
G y2 p Ž x . ,
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ALAN V. LAIR
where the last inequality follows from Ž13.. We now prove that F Ž ¨ k Ž x . . F z Ž < x <. ,
< x < F k.
Ž 14 .
First, since F Ž ¨ k Ž x .. ª 0 as < x < ª k and z G 0, inequality Ž14. is clearly true on < x < s k. Thus suppose max < x < F k w F Ž ¨ k Ž x .. y z Ž< x <.x ) 0. Then this maximum must occur in the open ball < x < - k. At that point, we get 0 G ⌬ Ž F Ž ¨ k . y z . G y2 p Ž x . q 2 Ž Ž r . q  Ž r . . ) 0, which is absurd. Thus Ž14. holds. Furthermore, F Ž ¨ k . F F Ž0. so that F Ž¨kŽ x. . F ˆ z Ž < x <. ,
Ž 15 .
where ˆ z Ž< x <. ' min z Ž< x <., F Ž0.4 ) 0 so that ˆ z Ž< x <. is in the range of F for all x. However, since F is strictly decreasing, we get from Ž15. that ¨ k G Fy1 Ž ˆ z . for < x < F k and thus ¨ Ž x . G Fy1 Ž ˆ z Ž< x <.. on R n. Since ˆ z Ž< x <. ) 0 n y1 Ž Ž< <.. for all x g R , we get F ˆz x ) 0 for all x g R n. Furthermore, we have Fy1 Ž ˆ z Ž< x <.. ª ⬁ as < x < ª ⬁ since ˆ z Ž< x <. ª 0 as < x < ª ⬁. Thus w Ž x . s Fy1 Ž ˆ z Ž< x <.. and the proof is complete. This result is now easily extended to somewhat arbitrary unbounded domains as the following theorem demonstrates. THEOREM 3. Suppose ⍀ is an unbounded domain in R n, n G 3, with compact C 2 boundary and suppose there exists a sequence of bounded domains ⍀ k 4 , each with smooth boundary, such that ⍀ k : ⍀ kq1 for all k s 1, 2, . . . and ⍀ s D ⬁ks 1 ⍀ k . Suppose p is a nonnegati¨ e C Ž ⍀ . function which satisfies condition ŽA.. Let
Ž r . s max p Ž x . : < x < s r , x g ⍀ 4 , and assume that satisfies inequality Ž12.. Then Eq. Ž1. has a positi¨ e large solution pro¨ ided f satisfies Condition Ž2.. Proof. We replace the functions ¨ k given in the proof of Theorem 2 with the solutions to ⌬¨ k s p Ž x . f Ž ¨ k . , ¨ k Ž x . ª ⬁,
x g ⍀k ,
x ª ⭸ ⍀k
for each k. The proof now proceeds very much like the proof of Theorem 2. We omit the details. We would like to establish the converse of Theorem 3. That is, assuming the same hypothesis as Theorem 3 Žespecially concerning the function p ., we would like to prove that f satisfies Condition Ž2. whenever Ž1. has a
213
EXISTENCE OF LARGE SOLUTIONS
nonnegative large solution on ⍀. Although this result is true for f Ž s . s s␥ and ⍀ s R n Žsee w9x., we have been unable to establish it for general f except for functions p which have rather specific decay rates. We prove this in the corollary below. However, before doing that, we establish Theorems 4 and 5, two important ‘‘partial converses’’ to Theorem 3. Theorem 4 specifically is important in that it, combined with Theorem 2, demonstrates that, for functions f such as f Ž u. s u␥ , Eq. Ž1. has a nonnegative entire large solution if and only if ␥ ) 1 Žprovided inequality Ž12. holds.. This extends the results of w9x. More generally, however, for functions f for which Condition Ž10. is equivalent to Ž2. Že.g., f Ž s . s s␥ ., this establishes the desired converse to Theorem 3 when ⍀ s R n. THEOREM 4. Suppose the function p satisfies the hypothesis of Theorem 2 including inequality Ž12.. If Eq. Ž1. has a nonnegati¨ e entire large solution, then f satisfies inequality Ž10.. Proof. Let u be a nonnegative entire large solution of Ž1. and define the sequence f k 4 of nondecreasing C⬁ functions as in the proof of the proposition. We define the nonnegative functions w k and w on R n by wk Ž x . s
H0u x
Ž .
ds
wŽ x. s
1 q fk Ž s .
H0u x
ds
Ž .
1 q f Ž s.
,
and note that w k 4 converges uniformly on compact sets to w. Let w k Ž r . ' Ž n r ny 1 .y1H< x
⌬u 1 q fk Ž u. ⌬u 1 q fk Ž u.
< ⵜu < 2 f kX Ž u .
y s
1 q fk Ž u. p Ž x . f Ž u. 1 q fk Ž u.
2
'ˆ pk .
Thus ⌬w k Ž r . F H< x
r
1y n
H0 t
t ny1
H0 s
H< x
k
d ds dt.
Letting k ª ⬁ in this expression and observing that ˆ p F 2 Ž r ., we get w Ž r . F w Ž 0. q 2
r
H0 t
1y n
t ny1
H0 s
Ž s . ds dt.
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ALAN V. LAIR
Integrating by parts we obtain the estimate w Ž r . F w Ž 0. q 2 Ž n y 2.
y1
r
H0 s Ž s . ds
⬁
H0 s Ž s . ds ' K .
F w Ž 0. q 2
Thus lim inf < x < ª⬁ w Ž x . F K. However, since lim < x < ª⬁ uŽ x . s ⬁, we must have u x H < x <ª⬁ 0
< x <ª⬁
ds
Ž .
K G lim inf w Ž x . s lim
1 q f Ž s.
s
ds
⬁
H0
1 q f Ž s.
.
Therefore Ž10. holds, and the proof is complete. LEMMA 2. Suppose there exists a nonnegati¨ e function h continuous on w0, ⬁. and differentiable on Ž0, ⬁. such that 0 F Ž r . F h 2 Ž r . for all r G 0 and h satisfies one of the following: Ža. there exists a constant C such that r ny 1 hŽ r . F C for all r G 0; or Žb. lim r ª⬁ r ny 1 hŽ r . s ⬁ and H0⬁ hŽ r . dr - ⬁. If u is a nonnegati¨ e entire large radial solution of ⌬ u s Ž r . f Ž u., then f satisfies inequality Ž2.. Proof. Following Osserman w12x, we note that u satisfies Ž r ny 1 u⬘.⬘ s r Ž r . f Ž u. and multiply this expression by r ny 1 u⬘. Since u⬘ G 0, we get ny 1
Ž r ny 1 u⬘ .
2
⬘
s 2 r 2 ny2 Ž r .
d
u r f Ž s . ds H dr 0
F 2 r 2 ny2 h2 Ž r .
Ž .
d
u r f Ž s . ds. H dr 0 Ž .
If h satisfies Ža., then Ž16. produces
Ž r ny 1 u⬘ .
2
⬘
F 2C 2
d
u r f Ž s . ds. H dr 0 Ž .
Integrating this over w0, r x yields Ž Ž s . ' H0s f Ž t . dt .: uŽ r .
2
Ž r ny1 u⬘ . F 2C 2H
u Ž0 .
f Ž s . ds F 2C 2 Ž u Ž r . . .
Taking the square root of both sides and rearranging terms gives d dr
HuuŽ1.r
Ž .
Ž s.
y1 r2
ds F 2Cr 1yn .
Ž 16 .
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EXISTENCE OF LARGE SOLUTIONS
Integrating this over w1, R x produces
HuuŽ1.R
Ž .
Ž s.
y1 r2
ds F
2C ny2
Ž 1 y R 2yn . F
2C ny2
.
Letting R ª ⬁, we get Ž2.. If h satisfies Žb., then we integrate Ž16. directly. After integrating by parts on the right side and dividing by r 2 ny2 , we get uŽ r .
2 Ž u⬘ . s 2 h2 Ž r . H
f Ž s . ds y 2 r 2y2 n
0
r
H0 Ž s
2 ny2
h2 Ž s . . ⬘ Ž u Ž s . . ds. Ž 17 .
The second term on the right side of this equation may be rewritten as 2
2h Ž r.
yH0r Ž s 2 ny2 h2 Ž s . . ⬘ Ž u Ž s . . ds r 2 ny2 h2 Ž r .
.
Ž 18 .
For this ratio, we apply L’Hospital’s rule Žwhich is allowed since the denominator diverges to infinity. to get lim
yH0r Ž s 2 ny2 h 2 Ž s . . ⬘ Ž u Ž s . . ds r 2 ny2 h2 Ž r .
rª⬁
s lim y Ž u Ž r . . s y⬁, rª⬁
since uŽ r . ª ⬁ as r ª ⬁. Thus the expression Ž18. is negative for large r. Using this fact in Ž17. produces Žfor sufficiently large R .: 2 Ž u⬘ . F 2 h2 Ž r . Ž u Ž r . .
for r G R.
Hence, r
HR
Ž uŽ s . .
y1 r2
u⬘ Ž s . ds F 2
r
HR h Ž s . ds,
from which we get
HuuŽ Rr.
Ž .
Ž s.
y1 r2
ds F 2
r
HR h Ž s . ds.
Letting r ª ⬁ and observing that the right side, by hypothesis, converges to a real number, we obtain Ž2.. This completes the proof. THEOREM 5. Let p be a nonnegati¨ e C Ž ⍀ . function satisfying Condition ŽA. of ⍀, an unbounded domain satisfying the hypothesis of Theorem 3. Let be defined as in Theorem 3 and satisfy the hypothesis of Lemma 2. If Eq. Ž1. has a nonnegati¨ e large solution on ⍀, then f satisfies inequality Ž2..
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ALAN V. LAIR
Proof. We assume, with no loss of generality, that 0 g ⍀ and use an argument inspired by Osserman w12x. First, we show that for any a G 0 the equation ⌬¨ s Ž r . f Ž ¨ .
Ž 19 .
has a nonnegative solution with initial values ¨ Ž0. s a and ¨ ⬘Ž0. s 0 valid in some interval 0 F r F r 0 . ŽIn w12x this result follows easily from having f ⬘ continuous everywhere.. We write Ž19. in the integral form ¨Ž r. s a q
r
s
1y n
H0 s H0 t
Ž t . f Ž ¨ Ž t . . dt ds,
ny1
Ž 20 .
and momentarily replace the function f by f M defined by f M Ž s . s f Ž s . for s g w0, M x and f M Ž s . s f Ž M . for s G M. The number M will be chosen momentarily. We now apply a standard iteration procedure by letting ¨ 0 s a and generating a nondecreasing sequence ¨ k 4 in which ¨ k is calculated from ¨ ky 1 by ¨kŽ r . s a q
r
s
1y n
H0 s H0 t
Ž t . f M Ž ¨ ky1 Ž t . . dt ds.
ny1
Ž 21 .
Since X
¨ k Ž r . s r 1yn
r
H0 t
Ž t . f M Ž ¨ ky1 Ž t . . dt G 0,
ny1
we get ¨ Xk Ž r . G 0 for all k s 1, 2, . . . and r G 0 so that Ž21. gives 0 F ¨ k Ž r . F a q ⌽ Ž r . f M Ž ¨ ky1Ž r .. where ⌽Ž r . s
r
s
1y n
H0 s H0 t
Ž t . dt ds.
ny1
Thus, noting that lim r ª 0 ⌽ Ž r . s 0, we choose M to be any number larger than a and choose r 0 ) 0 near zero so that a q ⌽ Ž r 0 . f Ž M . F M. An induction argument may now be used to show that ¨ k Ž r . F M for all k s 1, 2, . . . and 0 F r F r 0 . Therefore ¨ k 4 is nondecreasing and bounded above by M and thus converges on w0, r 0 x to a function ¨ . The function ¨ is a solution of Ž20. for 0 F r F r 0 where f is replaced by f M . However, since each ¨ k is bounded above by M, we have ¨ F M and hence ¨ is a solution of Ž20. on w0, r 0 x. Now let w0, R . be the maximum interval in which ¨ exists. Since ¨ ⬘ Ž r . s r 1y n
r
H0 s
Ž s . f Ž ¨ Ž s . . ds G 0,
ny1
EXISTENCE OF LARGE SOLUTIONS
217
we get ¨ ⬘Ž r . G 0 for each r G 0, and hence if R - ⬁, we must have lim r ª R ¨ Ž r . s ⬁. Then Theorem 1 implies that f must satisfy Ž2. since ¨ would be a nonnegative large solution of Ž19. on a bounded set. Thus suppose R s ⬁. Again, if lim r ª⬁ ¨ Ž r . s ⬁, we have that f must satisfy Ž2. because of Lemma 2. Hence suppose R s ⬁ and lim r ª⬁ ¨ Ž r . s M0 - ⬁. We show that this is impossible. Indeed, if this were the case, then the maximum principle would imply that ¨ Ž< x <. F uŽ x . for all x g ⍀. This would imply, in particular, that a s ¨ Ž0. F uŽ0. for any a ) 0, which is absurd. This completes the proof. As mentioned earlier, if the function p satisfies, in addition to the hypothesis of Theorem 3, a sufficiently rapid decay condition at infinity, then the condition on the function f given by Ž2. is both necessary and sufficient to ensure existence of a large solution of Ž1. on ⍀. We now establish this result. COROLLARY. Suppose ⍀ is an unbounded domain that satisfies the hypothesis of Theorem 3. Let p be a nonnegati¨ e C Ž ⍀ . function which satisfies condition ŽA., and assume that there exists a constant K such that for < x < large and x g ⍀, pŽ x . F K < x
ACKNOWLEDGMENT The author thanks the Žanonymous. referees for their careful reading of the manuscript and numerous suggestions for its improvement.
REFERENCES 1. C. Bandle and M. Marcus, ‘‘Large’’ solutions of semilinear elliptic equations: existence, uniqueness and asymptotic behavior, J. Analyse Math. 58 Ž1992., 9᎐24. 2. L. Bers, F. John, and M. Schechter, ‘‘Partial Differential Equations,’’ Lectures in Applied Mathematics, Vol. 3, Am. Math. Soc., Providence, RI, 1971. 3. K.-S. Cheng and J. Lin, On the elliptic equations ⌬ u s K Ž x . u and ⌬ u s K Ž x . e 2 u , Trans. Amer. Math. Soc. 304 Ž1987., 639᎐668.
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4. K.-S. Cheng and W.-M. Ni, On the structure of the conformal scalar curvature equation on R N , Indiana Uni¨ . Math. J. 41 Ž1992., 261᎐278. 5. D. Gilbarg and N. S. Trudinger, ‘‘Elliptic Partial Differential Equations of Second Order,’’ Springer-Verlag, BerlinrHeidelbergrNew York, 1977. 6. J. B. Keller, On solution of ⌬ u s f Ž u., Comm. Pure Appl. Math. 10 Ž1957., 503᎐510. 7. O. A. Ladyzhenskaya and N. N. Ural’tseva, ‘‘Linear and Quasilinear Elliptic Equations,’’ Academic Press, New YorkrLondon, 1968. 8. A. V. Lair and A. W. Shaker, Classical and weak solutions of a singular semilinear elliptic problem, J. Math. Anal. Appl. 211 Ž1997., 371᎐385. 9. A. V. Lair and A. W. Wood, Large solutions of semilinear elliptic problems, Nonlinear Anal. 37 Ž1999., 805᎐812. 10. A. C. Lazer and P. J. McKenna, On a problem of Bieberbach and Rademacher, Nonlinear Anal. 21 Ž1993., 327᎐335. 11. M. Marcus, On solutions with blow-up at the boundary for a class of semilinear elliptic equations, in ‘‘Developments in Partial Differential Equations and Applications to Mathematical Physics’’ ŽG. Buttazzo, G. Galdi, and L. Zanghirati, Eds.., pp. 65᎐77, Plenum, New York, 1992. 12. R. Osserman, On the inequality ⌬ u G f Ž u., Pacific J. Math. 7 Ž1957., 1641᎐1647. 13. S. L. Pohozaev, The Dirichlet problem for the equation ⌬ u s u 2 , So¨ iet Math. Dokl. 1 Ž1961., 1143᎐1146. 14. D. Sattinger, ‘‘Topics in Stability and Bifurcation Theory,’’ Springer-Verlag, BerlinrHeidelbergrNew York, 1973.
A Necessary and Sufficient Condition for Existence of Large Solutions to Semilinear Elliptic Equations Alan V. Lair Department of Mathematics and Statistics, Air Force Institute of TechnologyrENC, 2950 P Street, Wright-Patterson Air Force Base, Ohio 45433-7765 E-mail: [email protected] Submitted by John La¨ ery Received November 9, 1998
We consider the semilinear equation ⌬ u s pŽ x . f Ž u. on a domain ⍀ : R n , n G 3, where f is a nonnegative, nondecreasing continuous function which vanishes at the origin, and p is a nonnegative continuous function with the property that any zero of p is contained in a bounded domain in ⍀ such that p is positive on its boundary. For ⍀ bounded, we show that a nonnegative solution u satisfying uŽ x . ª ⬁ as x ª ⭸ ⍀ exists if and only if the function Ž s . ' H0s f Ž t . dt satisfies H1⬁Ž Ž s ..y1 r2 ds - ⬁. For ⍀ unbounded Žincluding ⍀ s R n ., we show that a similar result holds where uŽ x . ª ⬁ as < x < ª ⬁ within ⍀ and uŽ x . ª ⬁ as x ª ⭸ ⍀ if pŽ x . decays to zero rapidly as < x < ª ⬁. 䊚 1999 Academic Press Key Words: large solution; elliptic equation; semilinear equation.
1. INTRODUCTION We consider the semilinear equation ⌬ u s p Ž x . f Ž u. ,
x g ⍀ : R n , n G 3,
Ž 1.
where the function f is continuous and nondecreasing on w0, ⬁. with f Ž0. s 0 and f Ž s . ) 0 if s ) 0 while the function p is nonnegative and continuous on ⍀. We give necessary and sufficient conditions on the function f which insure that Eq. Ž1. has a nonnegative solution u for which uŽ x . ª ⬁ as x ª ⭸ ⍀. We call these large solutions of Ž1. on ⍀. If ⍀ is unbounded, we also require uŽ x . ª ⬁ as < x < ª ⬁ within ⍀. If ⍀ s R n, we call these entire large solutions. Such problems arise in the study of the subsonic motion of a gas w13x, the electric potential in some bodies w10x, and Riemannian geometry w4x. 205 0022-247Xr99 $30.00 Copyright 䊚 1999 by Academic Press All rights of reproduction in any form reserved.
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Several authors have studied special cases of Ž1.. ŽSee w3, 4, 6, 9, 10᎐13x, and their references. . For ⍀ bounded and pŽ x . s 1, Keller w6x and Osserman w12x show that a necessary and sufficient condition for Ž1. to have a large solution on a bounded domain ⍀ is that the function f satisfy ⬁
y1 r2
s
H1 H0 f Ž t . dt
ds - ⬁.
Ž 2.
Consistent with this, Bandle and Marcus w1x have shown that f satisfying Ž2. is sufficient to insure that Ž1. has a positive large solution provided the continuous function p is strictly positive on ⍀. Marcus w11x extends this to require p to be positive only on ⭸ ⍀. We show that these same results hold if p is allowed to vanish on large parts of ⍀ including its boundary. If f Ž s . s s␥ , then Condition Ž2. is equivalent to ␥ ) 1, and the problem of finding nonnegative large solutions of Eq. Ž1. for this special case, for both ⍀ bounded and ⍀ s R n, were considered in w4, 9x. Our results contain these as special cases. For ⍀ unbounded, all other results known to us Žexcept w6, 12x. require f to have the special forms f Ž s . s s␥ , or f Ž s . s e c s. ŽSee, e.g., w3, 4, 9, 10x.. Our existence results contain these as special cases. We show that, if p decays to zero rapidly as < x < ª ⬁, then, as with ⍀ bounded, Condition Ž2. is both necessary and sufficient to guarantee that Eq. Ž1. has a nonnegative large solution on ⍀. We note that Marcus w11x considers the existence of solutions to Ž1. on unbounded domains but restricts his attention only to those which decay to zero at infinity while blowing up on ⭸ ⍀.
2. MAIN RESULTS In this section we state and prove our results. Throughout this article, we require the nonnegative function p to satisfy the following condition: ŽA. For any x 0 g ⍀ satisfying pŽ x 0 . s 0, there exists a domain ⍀ 0 such that x 0 g ⍀ 0 , ⍀ 0; ⍀, and pŽ x . ) 0 for all x g ⭸ ⍀ 0 . Before proving the main results, we need to establish an important proposition that will be referred to later. PROPOSITION 1. Suppose ⍀ is a bounded domain in R n with C 2 boundary and p is a nonnegati¨ e C Ž ⍀ . function satisfying Condition ŽA.. Then, for any nonnegati¨ e constant c, the boundary ¨ alue problem ⌬¨ s p Ž x . f Ž ¨ . , ¨ Ž x . s c,
x g ⭸⍀
x g ⍀,
Ž 3. Ž 4.
EXISTENCE OF LARGE SOLUTIONS
207
has a unique nonnegati¨ e classical solution. Furthermore, if w is the solution to the same equation but ha¨ ing boundary ¨ alue c 0 G c, then w G ¨ on ⍀. Proof. Extend f as an odd function on R, and let f k 4 be a sequence of nondecreasing C⬁ŽR. functions which converges uniformly on compact sets to f, satisfies sf k Ž s . ) 0 for s / 0, f k Ž0. s 0, and f Ž s . q f Ž s y 1rk . F 2 f k Ž s . F f Ž s . q f Ž s q 1rk . .
Ž 5.
Such a sequence is easy to obtain using mollifiers Žsee w2, p. 145x.. We let pk 4 be a decreasing sequence of positive C ␣ Ž ⍀ . functions which converges uniformly on ⍀ to p. ŽSee Lemma 5 in w8x.. We now show that the boundary value problem ⌬¨ k s pk Ž x . f k Ž ¨ . , ¨ k Ž x . s c,
x g ⭸⍀
x g ⍀,
Ž 6. Ž 7.
has a unique nonnegative classical solution. It is clear Žsee Theorem 4.3 in w5x. that the boundary value problem ⌬ z s MK on ⍀ and z s c on ⭸ ⍀ has a unique solution where M ' 5 p1 5 ⬁ G 5 pk 5 ⬁ G 5 p 5 ⬁ and K ' 1 q sup k f k Ž c .. Furthermore, the function z is a lower solution of Ž6. and Ž7. since z F c on ⍀ so that MK G pf Ž c . G pf Ž z .. ŽSee w14, pp. 23᎐24x.. Also it is clear that Z ' c is an upper solution of Ž6. and Ž7.. Therefore, there exists a classical solution ¨ k of Ž6. and Ž7. such that z F ¨ k F Z on ⍀. ŽSee Theorem 2.3.1 in w14x.. We now show that, in fact, ¨ k must be nonnegative for each k so that we actually have 0 F ¨ k F Z. Thus, for fixed k, suppose min x g ⍀ ¨ k Ž x . - 0. Now let x 0 g R n _ ⍀ and define V Ž x . s ¨ k Ž x . q hŽ r . where hŽ r . ' Ž1 q r 2 .y1 r2 , r Ž x . ' < x y x 0 <, and the positive number is chosen small enough that min x g ⍀ V Ž x . - 0. Obviously this minimum must occur in ⍀, and at this point we have 0 F ⌬V s ⌬¨ k q ⌬ hŽ r . s pk f Ž ¨ k . q ⌬ hŽ r . F ⌬ hŽ r . - 0, a contradiction. Thus min ⍀ ¨ k G 0. To complete the existence proof, we must now prove that the sequence ¨ k 4 converges to a classical solution of Ž3. since each ¨ k already satisfies the boundary condition Ž4.. This can be done easily using a standard bootstrap argument. ŽSee Lemma 3 in w8x.. In fact, the argument here is almost identical to that of Lemma 3 in w8x Žin which h k s yf k .. Apparently the only difference in the argument lies in obtaining an upper bound on < ⌬¨ k < which is independent of k. ŽSee paragraph following Eq. Ž7. in w8x.. In the present case, this estimate becomes < ⌬¨ k < s pk f k Ž ¨ k . F p1 f k Ž ¨ k . F p1 f k Ž Z . F p1 sup k f k Ž c .. Since uniqueness is an easy consequence of the relationship ¨ F w whenever ¨ s c F c 0 F w on ⭸ ⍀, we prove only this relationship. Thus suppose min ⍀ Ž w y ¨ . - 0. Define x 0 , r, and hŽ r . as above, and choose ) 0 small so that min ⍀ Ž w y ¨ q hŽ r .. - 0. Then clearly this minimum
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ALAN V. LAIR
must occur in ⍀ and at this point we have 0 F ⌬Ž w y ¨ q hŽ r .. s pw f Ž w . y f Ž ¨ .x q ⌬ hŽ r . F ⌬ hŽ r . - 0, a contradiction. Thus ¨ F w and the proof is complete. THEOREM 1. Suppose ⍀ is a bounded domain in R n with C 2 boundary and p is a nonnegati¨ e C Ž ⍀ . function satisfying condition ŽA.. Then a necessary and sufficient condition for Eq. Ž1. to ha¨ e a nonnegati¨ e large solution in ⍀ is that f satisfy Ž2.. Proof Ž Necessity .. We prove this by contradiction. Thus suppose u is a nonnegative large solution of Ž1. and ⬁
y1 r2
s
ds s ⬁.
H1 H0 f Ž t . dt
Ž 8.
Since p g C Ž ⍀ ., there exists a positive constant M0 such that pŽ x . F M0 on ⍀. Let ¨ k 4 be the unique nonnegative classical solution of ⌬¨ k s M0 f Ž ¨ k . , ¨ k Ž x . s k,
x g ⍀,
x g ⭸⍀,
which exists by virtue of Proposition 1. Then the sequence ¨ k 4 is nondecreasing and ¨ k F u so that ¨ k 4 converges to a nonnegative function, ¨ . Using the monotonicity of the sequence ¨ k 4 and the fact that ¨ k Ž x . s k on ⭸ ⍀, it is easy to show that ¨ Ž x . ª ⬁ as x ª ⭸ ⍀. Furthermore, a standard bootstrap argument Žsee, e.g., w8x. can be used to show that ¨ is a classical solution of ⌬¨ s M0 f Ž ¨ .
Ž 9.
on ⍀. That is, ¨ is a nonnegative large solution of Ž9. in ⍀. Now we show that this, in fact, cannot happen. We may assume that 0 g ⍀, and we choose any number a such that ¨ Ž0. - a. Now consider the radial solution, w, of Ž9. which satisfies w Ž0. s a, w⬘Ž0. s 0. By Keller w6x w can be extended to the whole space as a nonnegative solution of Ž9.. Hence w is bounded on ⍀ and by the maximum principle, w F ¨ in ⍀. This contradicts ¨ Ž0. - w Ž0.. Thus, if inequality Ž2. fails to hold, then Eq. Ž1. cannot have a nonnegative large solution. Ž Sufficiency .. Suppose f satisfies Ž2.. Let u k be the unique nonnegative solution of ⌬ uk s pŽ x . f Ž uk . , uk Ž x . s k ,
x g ⭸ ⍀.
x g ⍀,
EXISTENCE OF LARGE SOLUTIONS
209
Then the sequence u k 4 is nondecreasing. ŽSee Proposition 1.. Now let x 0 be an arbitrary point in ⍀. Suppose we can show that there is an open set G ; ⍀ containing x 0 and a positive constant, MG , such that u k Ž x . F MG on G. Then our proof will be complete, for then it will be clear that u k 4 converges on ⍀ to some function u. Furthermore, as in the proof of necessity, a standard bootstrap argument will show that u is a classical solution of Eq. Ž1. while the monotonicity of u k 4 and the fact that u k s k on ⭸ ⍀ can be used to prove easily that u blows up on ⭸ ⍀. To prove the existence of a set G and a bound MG , we consider two cases: Ž1. pŽ x 0 . ) 0 and Ž2. pŽ x 0 . s 0. Case Ž1. Ž pŽ x 0 . ) 0.. Since p is continuous, there exists a ball Ž B x 0 , R . : ⍀ centered at x 0 with radius R such that pŽ x . ) 0 on B Ž x 0 , R . . Let m 0 s min pŽ x . : x g B Ž x 0 , R . 4 , and let w be a positive solution of Žsee w6x for the existence proof., ⌬w s m 0 f Ž w . wŽ x. ª ⬁
on B Ž x 0 , R . ,
as x ª ⭸ B Ž x 0 , R . .
It is easy to prove that u k F w on B Ž x 0 , R .. Furthermore, w is bounded on B Ž x 0 , Rr2.. Thus G s B Ž x 0 , Rr2. and MG s max G w. Case Ž2. Ž pŽ x 0 . s 0.. By the hypothesis, there exists a domain ⍀ 0 such that ⍀ 0: ⍀ and pŽ x . ) 0 for all x g ⭸ ⍀ 0 . From Case Ž1. above, we know that for every x g ⭸ ⍀ 0 , there exists a ball B Ž x, r x . and a positive constant M x such that u k Ž x . F M x on B Ž x, r x .. Since ⍀ is bounded Žand hence ⍀ 0 is bounded. ⭸ ⍀ 0 is compact. Thus there exists a finite number of the balls B Ž x, r x . , x g ⭸ ⍀ 0 , which cover ⭸ ⍀ 0 . Let M s max M x 1 , M x 2 , . . . , M x k 4 where the balls B Ž x i , r x i ., i s 1, 2, . . . , k cover ⭸ ⍀ 0 . Clearly then u k F M on ⭸ ⍀ 0 . Applying a maximum principle argument on ⍀ 0 will yield u k F M on ⍀ 0 . Thus G s ⍀ 0 and MG s M. This completes the proof. Remark. The condition on the function p is nearly optimal in the following sense: Let D be a domain such that D : ⍀, and let p be a function which vanishes on ⍀ _ D. Then p does not satisfy the hypothesis of the theorem, and it is rather easy to show that Ž1. has no positive large solution on ⍀. ŽSee w9x.. We now consider the case where ⍀ is unbounded. We begin by considering ⍀ s R n. It is evident from earlier work Žsee, e.g., w3, 4, 9, 12x. that the function p cannot behave arbitrarily as < x < ª ⬁ if we expect Eq. Ž1. to have an entire large solution. Therefore we add an asymptotic condition for the function p and establish results for unbounded ⍀ similar
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ALAN V. LAIR
to those of Theorem 1. Before that, however, we need to prove an inequality that will be needed later. LEMMA 1.
Suppose f satisfies inequality Ž2.. Then ds
⬁
H1
f Ž s.
- ⬁.
Ž 10 .
Proof. If we can prove that there exist positive numbers ␦ and M such that f Ž s. s
G␦2
for s G M,
Ž 11 .
then we will be done since
Ž s. '
s
H0 f Ž t . dt F sf Ž s . F f
2
Ž s . r␦ 2
for s G M,
which, in turn, yields w Ž s .xy1 r2 G ␦rf Ž s . for s G M so that Ž2. implies Ž10.. To prove Ž11., we assume it is false. That is, we assume there exists an increasing sequence s j 4 of real numbers such that lim jª⬁ s j s ⬁ and f Ž s j .rs j - 1rj for all j. Since f is increasing, we have f Ž s . F f Ž s j . for all s g w0, s j x, which, in turn, produces Ž s . F sf Ž s . F sf Ž s j . for s g w0, s j x. Hence,
Hs
sj
1
Ž s.
y1r2
ds G
Hs
sj
sf Ž s j .
y1 r2
1
G Ž jrs j .
1r2
sj
y1r2
Hs s
ds s 2 j 1r2 1 y Ž s1rs j .
1r2
ª ⬁,
1
as j ª ⬁, contradicting Ž2.. Thus Ž11. must be true. This completes the proof. THEOREM 2. Suppose p is a nonnegati¨ e C ŽR n . function which satisfies Condition ŽA. with ⍀ s R n and ⬁
H0 r Ž r . dr - ⬁,
Ž 12 .
where Ž r . ' max < x
211
EXISTENCE OF LARGE SOLUTIONS
Proof. Let ¨ k be any nonnegative solution of < x < - k,
⌬¨ k s p Ž x . f Ž ¨ k . , < x < ª k.
¨ k Ž x . ª ⬁,
ŽThe solution exists because of Theorem 1.. It is easy to prove Žletting
¨ k Ž x . s ⬁ for < x < G k . that ¨ kq1Ž x . F ¨ k Ž x . for all x g R n and for all k. Hence the sequence ¨ k 4 converges on R n to some function ¨ . If we can show that there is a positive function w such that w F ¨ k for all k and w Ž x . ª ⬁ as < x < ª ⬁, then we will have 0 - w F ¨ on R n and the proof
will be complete since another standard bootstrap argument will show that ¨ is a solution of Ž1. on R n.
To obtain such a function w, we observe that, as a result of Ž12., the function Ž r ' < x <., r
zŽ r. ' c y 2
H0 t
1y n
t ny1
H0 s
Ž Ž s . q  Ž s . . ds dt,
where  Ž r . ' Ž1 q r 2 .y2 and ⬁
c'2
H0
t 1y n
t ny1
H0 s
Ž Ž s . q  Ž s . . ds dt
is the unique positive solution of ⌬ z s z⬙ q z⬘ Ž 0 . s 0,
ny1 r
z⬘ s y2 Ž Ž r . q  Ž r . . ,
r)0
z Ž r . ª 0 as r ª ⬁.
From the sequence f k 4 in the proof of Proposition 1, we choose J large so that f Ž s y 1rJ . G y2 for all s G 0 Žin which f has been extended to be an odd function on R.. Then using Ž5. we get 2 1 q fJ Ž s. G f Ž s. ,
s G 0.
Ž 13 .
Now let F be the positive-valued smooth function defined on w0, ⬁. by F Ž s. s
⬁
Hs
dt 1 q fJ Ž t .
,
which exists because of Ž2., Ž13., and Lemma 1. Notice that F⬘ - 0 so that F is strictly decreasing and thus Fy1 has the same property. We observe also that ⌬ FŽ ¨k . G
y⌬¨ k 1 q fJ Ž ¨k .
s
yp Ž x . f Ž ¨ k . 1 q fJ Ž ¨k .
G y2 p Ž x . ,
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ALAN V. LAIR
where the last inequality follows from Ž13.. We now prove that F Ž ¨ k Ž x . . F z Ž < x <. ,
< x < F k.
Ž 14 .
First, since F Ž ¨ k Ž x .. ª 0 as < x < ª k and z G 0, inequality Ž14. is clearly true on < x < s k. Thus suppose max < x < F k w F Ž ¨ k Ž x .. y z Ž< x <.x ) 0. Then this maximum must occur in the open ball < x < - k. At that point, we get 0 G ⌬ Ž F Ž ¨ k . y z . G y2 p Ž x . q 2 Ž Ž r . q  Ž r . . ) 0, which is absurd. Thus Ž14. holds. Furthermore, F Ž ¨ k . F F Ž0. so that F Ž¨kŽ x. . F ˆ z Ž < x <. ,
Ž 15 .
where ˆ z Ž< x <. ' min z Ž< x <., F Ž0.4 ) 0 so that ˆ z Ž< x <. is in the range of F for all x. However, since F is strictly decreasing, we get from Ž15. that ¨ k G Fy1 Ž ˆ z . for < x < F k and thus ¨ Ž x . G Fy1 Ž ˆ z Ž< x <.. on R n. Since ˆ z Ž< x <. ) 0 n y1 Ž Ž< <.. for all x g R , we get F ˆz x ) 0 for all x g R n. Furthermore, we have Fy1 Ž ˆ z Ž< x <.. ª ⬁ as < x < ª ⬁ since ˆ z Ž< x <. ª 0 as < x < ª ⬁. Thus w Ž x . s Fy1 Ž ˆ z Ž< x <.. and the proof is complete. This result is now easily extended to somewhat arbitrary unbounded domains as the following theorem demonstrates. THEOREM 3. Suppose ⍀ is an unbounded domain in R n, n G 3, with compact C 2 boundary and suppose there exists a sequence of bounded domains ⍀ k 4 , each with smooth boundary, such that ⍀ k : ⍀ kq1 for all k s 1, 2, . . . and ⍀ s D ⬁ks 1 ⍀ k . Suppose p is a nonnegati¨ e C Ž ⍀ . function which satisfies condition ŽA.. Let
Ž r . s max p Ž x . : < x < s r , x g ⍀ 4 , and assume that satisfies inequality Ž12.. Then Eq. Ž1. has a positi¨ e large solution pro¨ ided f satisfies Condition Ž2.. Proof. We replace the functions ¨ k given in the proof of Theorem 2 with the solutions to ⌬¨ k s p Ž x . f Ž ¨ k . , ¨ k Ž x . ª ⬁,
x g ⍀k ,
x ª ⭸ ⍀k
for each k. The proof now proceeds very much like the proof of Theorem 2. We omit the details. We would like to establish the converse of Theorem 3. That is, assuming the same hypothesis as Theorem 3 Žespecially concerning the function p ., we would like to prove that f satisfies Condition Ž2. whenever Ž1. has a
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EXISTENCE OF LARGE SOLUTIONS
nonnegative large solution on ⍀. Although this result is true for f Ž s . s s␥ and ⍀ s R n Žsee w9x., we have been unable to establish it for general f except for functions p which have rather specific decay rates. We prove this in the corollary below. However, before doing that, we establish Theorems 4 and 5, two important ‘‘partial converses’’ to Theorem 3. Theorem 4 specifically is important in that it, combined with Theorem 2, demonstrates that, for functions f such as f Ž u. s u␥ , Eq. Ž1. has a nonnegative entire large solution if and only if ␥ ) 1 Žprovided inequality Ž12. holds.. This extends the results of w9x. More generally, however, for functions f for which Condition Ž10. is equivalent to Ž2. Že.g., f Ž s . s s␥ ., this establishes the desired converse to Theorem 3 when ⍀ s R n. THEOREM 4. Suppose the function p satisfies the hypothesis of Theorem 2 including inequality Ž12.. If Eq. Ž1. has a nonnegati¨ e entire large solution, then f satisfies inequality Ž10.. Proof. Let u be a nonnegative entire large solution of Ž1. and define the sequence f k 4 of nondecreasing C⬁ functions as in the proof of the proposition. We define the nonnegative functions w k and w on R n by wk Ž x . s
H0u x
Ž .
ds
wŽ x. s
1 q fk Ž s .
H0u x
ds
Ž .
1 q f Ž s.
,
and note that w k 4 converges uniformly on compact sets to w. Let w k Ž r . ' Ž n r ny 1 .y1H< x
⌬u 1 q fk Ž u. ⌬u 1 q fk Ž u.
< ⵜu < 2 f kX Ž u .
y s
1 q fk Ž u. p Ž x . f Ž u. 1 q fk Ž u.
2
'ˆ pk .
Thus ⌬w k Ž r . F H< x
r
1y n
H0 t
t ny1
H0 s
H< x
k
d ds dt.
Letting k ª ⬁ in this expression and observing that ˆ p F 2 Ž r ., we get w Ž r . F w Ž 0. q 2
r
H0 t
1y n
t ny1
H0 s
Ž s . ds dt.
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ALAN V. LAIR
Integrating by parts we obtain the estimate w Ž r . F w Ž 0. q 2 Ž n y 2.
y1
r
H0 s Ž s . ds
⬁
H0 s Ž s . ds ' K .
F w Ž 0. q 2
Thus lim inf < x < ª⬁ w Ž x . F K. However, since lim < x < ª⬁ uŽ x . s ⬁, we must have u x H < x <ª⬁ 0
< x <ª⬁
ds
Ž .
K G lim inf w Ž x . s lim
1 q f Ž s.
s
ds
⬁
H0
1 q f Ž s.
.
Therefore Ž10. holds, and the proof is complete. LEMMA 2. Suppose there exists a nonnegati¨ e function h continuous on w0, ⬁. and differentiable on Ž0, ⬁. such that 0 F Ž r . F h 2 Ž r . for all r G 0 and h satisfies one of the following: Ža. there exists a constant C such that r ny 1 hŽ r . F C for all r G 0; or Žb. lim r ª⬁ r ny 1 hŽ r . s ⬁ and H0⬁ hŽ r . dr - ⬁. If u is a nonnegati¨ e entire large radial solution of ⌬ u s Ž r . f Ž u., then f satisfies inequality Ž2.. Proof. Following Osserman w12x, we note that u satisfies Ž r ny 1 u⬘.⬘ s r Ž r . f Ž u. and multiply this expression by r ny 1 u⬘. Since u⬘ G 0, we get ny 1
Ž r ny 1 u⬘ .
2
⬘
s 2 r 2 ny2 Ž r .
d
u r f Ž s . ds H dr 0
F 2 r 2 ny2 h2 Ž r .
Ž .
d
u r f Ž s . ds. H dr 0 Ž .
If h satisfies Ža., then Ž16. produces
Ž r ny 1 u⬘ .
2
⬘
F 2C 2
d
u r f Ž s . ds. H dr 0 Ž .
Integrating this over w0, r x yields Ž Ž s . ' H0s f Ž t . dt .: uŽ r .
2
Ž r ny1 u⬘ . F 2C 2H
u Ž0 .
f Ž s . ds F 2C 2 Ž u Ž r . . .
Taking the square root of both sides and rearranging terms gives d dr
HuuŽ1.r
Ž .
Ž s.
y1 r2
ds F 2Cr 1yn .
Ž 16 .
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EXISTENCE OF LARGE SOLUTIONS
Integrating this over w1, R x produces
HuuŽ1.R
Ž .
Ž s.
y1 r2
ds F
2C ny2
Ž 1 y R 2yn . F
2C ny2
.
Letting R ª ⬁, we get Ž2.. If h satisfies Žb., then we integrate Ž16. directly. After integrating by parts on the right side and dividing by r 2 ny2 , we get uŽ r .
2 Ž u⬘ . s 2 h2 Ž r . H
f Ž s . ds y 2 r 2y2 n
0
r
H0 Ž s
2 ny2
h2 Ž s . . ⬘ Ž u Ž s . . ds. Ž 17 .
The second term on the right side of this equation may be rewritten as 2
2h Ž r.
yH0r Ž s 2 ny2 h2 Ž s . . ⬘ Ž u Ž s . . ds r 2 ny2 h2 Ž r .
.
Ž 18 .
For this ratio, we apply L’Hospital’s rule Žwhich is allowed since the denominator diverges to infinity. to get lim
yH0r Ž s 2 ny2 h 2 Ž s . . ⬘ Ž u Ž s . . ds r 2 ny2 h2 Ž r .
rª⬁
s lim y Ž u Ž r . . s y⬁, rª⬁
since uŽ r . ª ⬁ as r ª ⬁. Thus the expression Ž18. is negative for large r. Using this fact in Ž17. produces Žfor sufficiently large R .: 2 Ž u⬘ . F 2 h2 Ž r . Ž u Ž r . .
for r G R.
Hence, r
HR
Ž uŽ s . .
y1 r2
u⬘ Ž s . ds F 2
r
HR h Ž s . ds,
from which we get
HuuŽ Rr.
Ž .
Ž s.
y1 r2
ds F 2
r
HR h Ž s . ds.
Letting r ª ⬁ and observing that the right side, by hypothesis, converges to a real number, we obtain Ž2.. This completes the proof. THEOREM 5. Let p be a nonnegati¨ e C Ž ⍀ . function satisfying Condition ŽA. of ⍀, an unbounded domain satisfying the hypothesis of Theorem 3. Let be defined as in Theorem 3 and satisfy the hypothesis of Lemma 2. If Eq. Ž1. has a nonnegati¨ e large solution on ⍀, then f satisfies inequality Ž2..
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ALAN V. LAIR
Proof. We assume, with no loss of generality, that 0 g ⍀ and use an argument inspired by Osserman w12x. First, we show that for any a G 0 the equation ⌬¨ s Ž r . f Ž ¨ .
Ž 19 .
has a nonnegative solution with initial values ¨ Ž0. s a and ¨ ⬘Ž0. s 0 valid in some interval 0 F r F r 0 . ŽIn w12x this result follows easily from having f ⬘ continuous everywhere.. We write Ž19. in the integral form ¨Ž r. s a q
r
s
1y n
H0 s H0 t
Ž t . f Ž ¨ Ž t . . dt ds,
ny1
Ž 20 .
and momentarily replace the function f by f M defined by f M Ž s . s f Ž s . for s g w0, M x and f M Ž s . s f Ž M . for s G M. The number M will be chosen momentarily. We now apply a standard iteration procedure by letting ¨ 0 s a and generating a nondecreasing sequence ¨ k 4 in which ¨ k is calculated from ¨ ky 1 by ¨kŽ r . s a q
r
s
1y n
H0 s H0 t
Ž t . f M Ž ¨ ky1 Ž t . . dt ds.
ny1
Ž 21 .
Since X
¨ k Ž r . s r 1yn
r
H0 t
Ž t . f M Ž ¨ ky1 Ž t . . dt G 0,
ny1
we get ¨ Xk Ž r . G 0 for all k s 1, 2, . . . and r G 0 so that Ž21. gives 0 F ¨ k Ž r . F a q ⌽ Ž r . f M Ž ¨ ky1Ž r .. where ⌽Ž r . s
r
s
1y n
H0 s H0 t
Ž t . dt ds.
ny1
Thus, noting that lim r ª 0 ⌽ Ž r . s 0, we choose M to be any number larger than a and choose r 0 ) 0 near zero so that a q ⌽ Ž r 0 . f Ž M . F M. An induction argument may now be used to show that ¨ k Ž r . F M for all k s 1, 2, . . . and 0 F r F r 0 . Therefore ¨ k 4 is nondecreasing and bounded above by M and thus converges on w0, r 0 x to a function ¨ . The function ¨ is a solution of Ž20. for 0 F r F r 0 where f is replaced by f M . However, since each ¨ k is bounded above by M, we have ¨ F M and hence ¨ is a solution of Ž20. on w0, r 0 x. Now let w0, R . be the maximum interval in which ¨ exists. Since ¨ ⬘ Ž r . s r 1y n
r
H0 s
Ž s . f Ž ¨ Ž s . . ds G 0,
ny1
EXISTENCE OF LARGE SOLUTIONS
217
we get ¨ ⬘Ž r . G 0 for each r G 0, and hence if R - ⬁, we must have lim r ª R ¨ Ž r . s ⬁. Then Theorem 1 implies that f must satisfy Ž2. since ¨ would be a nonnegative large solution of Ž19. on a bounded set. Thus suppose R s ⬁. Again, if lim r ª⬁ ¨ Ž r . s ⬁, we have that f must satisfy Ž2. because of Lemma 2. Hence suppose R s ⬁ and lim r ª⬁ ¨ Ž r . s M0 - ⬁. We show that this is impossible. Indeed, if this were the case, then the maximum principle would imply that ¨ Ž< x <. F uŽ x . for all x g ⍀. This would imply, in particular, that a s ¨ Ž0. F uŽ0. for any a ) 0, which is absurd. This completes the proof. As mentioned earlier, if the function p satisfies, in addition to the hypothesis of Theorem 3, a sufficiently rapid decay condition at infinity, then the condition on the function f given by Ž2. is both necessary and sufficient to ensure existence of a large solution of Ž1. on ⍀. We now establish this result. COROLLARY. Suppose ⍀ is an unbounded domain that satisfies the hypothesis of Theorem 3. Let p be a nonnegati¨ e C Ž ⍀ . function which satisfies condition ŽA., and assume that there exists a constant K such that for < x < large and x g ⍀, pŽ x . F K < x
ACKNOWLEDGMENT The author thanks the Žanonymous. referees for their careful reading of the manuscript and numerous suggestions for its improvement.
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