qualitative properties of solutions to some nonlinear ... - Project Euclid

Vol. 71, No. 2

DUKE MATHEMATICAL JOURNAL (C)

August 1993

QUALITATIVE PROPERTIES OF SOLUTIONS TO2 SOME NONLINEAR ELLIPTIC EQUATIONS IN R WENXIONG CHEN

AND

CONGMING LI

O. Introduction. In this paper, we investigate properties of the solutions to the elliptic equations

R(x)e u(x)

-Au

x

R2

(,)

for functions R(x) which are positive near infinity. Equations of this kind arise from a variety of situations, such as from prescribing Gaussian curvature in geometry I-2] and from combustion theory in physics [3]. Recently, a series of works have been sought to understand the existence and the qualitative properties of the solutions of (,). Ni I-5] and Ni & Cheng [4-1 considered the case where R(x)is nonpositive; McOwen [6-1 and Aviles [7] investigated the In our previous paper [1], we situation where R(x) --, 0 in some order as Ixl --* consider a special case where R is a constant. We proved that the solutions are radially symmetric and, hence, classified all the solutions. In this paper, we consider more general functions R(x). First, we obtain the asymptotic behavior of the solution near infinity. Consequently, we prove that all the solutions satisfy an identity, which is somewhat of a generalization of the well-known Kazdan-Warner condition. Finally, using the asymptotic behavior together with the further development of the method employed in our previous paper I-1], we show that all the solutions are radially symmetric provided R is radially symmetric and nonincreasing. This part can be viewed as the completion

.

of[1]. Throughout this paper, we assume that the function R(x) is positive near infinity. In 1, we study the asymptotic behavior of the solution u(x) of (,). Let fl 1/2n R2 R(x)e () dx. Under some appropriate conditions, we show that the solutions approach at the rate -fl In Ixl, and the value of fl depends on the of the function monotonicity R(x) in the radial direction. More precisely, we prove

-

the following theorems.

THEOREM 1. Assume that R(x) is a bounded function and u is a solution of (,) with

f

eU(X) dx < oo.

R

Received 10 September 1992. Revision received 26 January 1993. Chen partially supported by NSF Grant DMS-9116949. Li partially supported by NSF Grant DMS-9003694.

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CHEN AND LI

428

Then

C

-fl ln(Ixl 4- 1) with

< u(x) < -fl ln(Ixl + 1) + C

fl > 2. Furthermore, we have the identity x" VR(x)e utx)

dx

rcfl(fl- 4).

And this immediately implies that

(i) if Rr > 0 but O, then fl > 4; (ii) if Rr < 0 but O, then fl < 4; 4. (iii) if R O, then where r Ixl and R, dR/dr. Remark 1. One can use the stereographic projection to pull back the metric to the sphere S 2. For fl 4 one gets a smooth metric, while for other values of fl, a metric with one singularity. Then the well-known Kazdan-Warner necessary condition for prescribing Gaussian curvature on S 2 can be stated equivalently as follows. If fl 4, then

f

x" VR(x)e tx

dx

O.

R

This is a special case of our identity x" VR(x)e "tx

dx

rr, fl(fl-

4).

We also consider unbounded functions R(x) with some growth restrictions. THEOREM 2. Assume that R(x)

< C(Ixl

4-

1) for some positive number and u is

a solution of(,) with

(IxI + 1)eUdx < Then

-fl In(Ix] 4- 1)

C

u(x)

-fl ln(lxl 4- 1) 4- C

with

more, we have x" VR(x)e

"’ dx

nfl(fl- 4).

fl > 2 +

.

Further-

429

NONLINEAR ELLIPTIC EQUATIONS IN R

THEOREM 3. Assume that fl is finite and that R(x)/R(xo) < C for Ix where C is a constant independent of Xo. Let u be a solution of (,) with

f

e u(x) dx

Xo[

< 1,

2.

xl

In 2, using the method of moving planes [9], we study the symmetry of the solutions of (,) and obtain the following result. THEOREM 4. Assume that u is a solution of (,) with u(x)/ln Ixl --, as Ixl--, o and fl > 2. If the function R(x) is radially symmetric and monotone decreasin9, then u is also radially symmetric and monotone decreasing. Remark 2. The assumptions in Theorems 1, 2, or 3 imply that the first condition in Theorem 4 is satisfied.

In 3, we provide some examples. We construct two families of increasing or decreasing functions R,(r). Associated to each R,(r), we explicitly write down solutions of (,), and those solutions have the desired order of growth as described in Theorems and 2. 1. Asymptotic behavior of the solutions. In this section, we study the asymptotic behavior of the solutions to the equation

Au

R (x)e utx)

x

R 2.

(,)

To prove Theorems and 2, we show the identity x" VR(x)e

’ dx

rtfl(fl- 4)

by using integration by parts on a large ball in R 2. In order to obtain the limits of the boundary terms as the radius of the ball tends to infinity, we need some estimates on the solution u and its first derivatives, as given by the following lemmas.

LEMMA 1.1. Assume that R(x) R2 e "(x) dx < +co. Let

fl

-fl ln(lxl + 1)- C. Now the assumption R2 e"tx) dx < oe implies fl > 2. This proves one side of inequality (1.1). (iv) To prove the other side of inequality (1.1), it suffices to show that w(x) > / 1) C. ln(lxl fl In fact, for Ix Yl > 1, we have

Ixl

Ix- yl(lyl + 1).

Then

lnlxl- 2 ln(lyl + 1) < lnlx yl- ln(ly[ + 1). Consequently,

(lnlxl- 2 ln(lyl + 1))R(y)e "tr) dy

{lnlx Yl- ln(lYl + 1)}R(y)e "’ dy

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CHEN AND LI

fllnlxl

-nl f

lnlxl/(" ln(lyl

+

R(y)e

" dy + n

In Ix

Yl R(Y) eu’ dy

1)R(y)e ut dy

fl lnlxl + 11 + 12 + 13. Taking into account of the fact that

u(x)

and

lnlxl

fl> 2

and by the boundedness of R(x), one can easily see that as

Ixl

,

11, 12

0

and that 13 is finite. Therefore,

ln(Ixl + 1)- C.

w(x) This completes the proof.

LEMMA 1.2. Assume that R(x) < C(Ixl of (,) with

/

1)for some positive number and u is

a solution

(Ixl + 1)e dx
2 + Proof. First, we show that Ixle (x) is bounded. Let Xo be a point in R 2, and Bl(xo) be the ball of radius centered at Xo. Let f(x) u(x) + lnlxol. Then Then

Af

R(x)

e y’x’

From the assumptions of the lemma, one can easily see that there exists a constant C independent of Xo, such that

uniformly as Xo

.

L(B(xo))

and | e stx dx --, 0, da (xo)

NONLINEAR ELLIPTIC EQUATIONS IN R

433

Then similar to the proof of Theorem 2 in [10], one can show that f(x) is bounded from above on the ball B1/2(Xo) and this bound is independent of Xo. It follows that

u(x) < C

ln(Ixl + 1)

for some constant C. Therefore, Ixle "tx) is bounded. Then the rest of the proof is similar to that of Lemma 1.1.

LEMMA 1.3. Under the assumptions of Lemma 1.1 or of Lemma 1.2, we have

-fl and uo O,

urr

where (r, 0) is the polar coordinate in R 2.

c

asr

--

Proof. We present the proof under the assumption of Lemma 1.1. A similar argument works under the assumption of Lemma 1.2. Applying (1.2), we have

rur= xl ul + x2u2

fR i

y’(x--y)

fl

and

fR

uo where y

y’(x-y)

i,

:l

R(y)eUtY) dy

R(y)eUr) dy

(Y2, -Y 1). Hence it suffices to show that I

Ix

lYl R(y)e tr) dy 0 y- - -

as

Ixl--* o.

In fact, I

(lx

-Yl < Ixl/2

xl/2 0 independent of 2, such that if x is a minimum point of x(x) and x(x ) < O, then Ixl < Ro. Proof. (i) There are two cases as Ixl oo. In this case, O(x) +oo. By the assumption that u(x)/lnlxl Case(a): Ixxl -fl, we see wx(x) is bounded. Hence x(x) --. 0 as Ixl oo. Case (b): Ixxl is bounded while Ixzl--, oo. Again by the assumption u(x)/lnlxl

.

-fl, wx(x)

-fl In

x.

-

0

due to the definition of x And it follows that x(x) (ii) To prove this part we first note that

Ag g

0.

-1

(X

2) 2 In(

x + 2)

and (x ) < max{u(x), u(x)} u(x). Then the assumption u(x)/lnlxl with fl > 2 implies that there exists some number Ro, such that

R(x ) exp ff(x ) +

Ag g



Ro.

-fl

NONLINEAR ELLIPTIC EQUATIONS IN R

437

Now conclusion (ii) of Lemma 2.1 follows directly from equation (2.2). Proof of Theorem 4. Step I. Let R o be given in Lemma 2.1. We will show that, if 2 < -Ro, then (x) > 0 for x E. In fact, suppose there is some x Eh, such that h(x) < 0. Then by (i) of Lemma 2.1, liml,+/o h(x) 0 and hence one can find a point x Eh, such that h(x ) min h < 0. This contradicts (ii) of Lemma 2.1. Step 2. Now let 2o be the largest possible value of 2 < 0, such that h(x) > 0 for x Eh and 2 < 20. We finish our proof with the following claims: (a) wh(x) > 0 for x Eh, 2 < 20, and Ou(x)/Oxl > 0 for x Eho. (b) 2o 0 or Wo 0. To prove the first claim, we notice that (i) w h > 0 implies c3u/t?xl > 0 for x < 20, and (ii) wh(x) 0 if and only if Ou/t?x 0 somewhere on Th by the maximum principle and the Hopf lemma. Now suppose there exists a 6 > 0, such that Who_o(x) =_ O. Then U(2o 26, y) u(;o, y). Combining this with the above fact (i), we get Ou(x, y)/Ox 0 for 20 26 < xl < 20. In particular, we have Ou/Ox 0 on Tho-20, and consequently Who-2O 0.

Continuing this way, we can prove that u is independent of x. This is impossible. Thus we proved wh(x) > 0 for x Eh and 2 < 20. The Hopf lemma implies that t?wh/t?xx < 0 on Th for 2 < 20, or equivalently

Ou

Ox

Ow 2

Oxx

>0

Th for 2 < 20, which means cu/t?x > 0 for x < 20. The first claim is proved. Now we prove claim (b). We prove it by contradiction. Suppose that Who 0 and 2o < 0. By (2.1) and the maximum principle and the Hopf Lemma, we have Who > 0 in Eho and cOWho/OX < 0 on Tho. On the other hand, by the definition of 20, there exists a sequence of real numbers 2k 20, such that hk(X)< 0 for some x e Ehk. Let x k be a minimum point of Lemma 2.1 implies h; then h(x k) < 0 and Vh(x k) 0 for k 1, 2, 3, that Ixl < Ro, and hence there is a subsequence of {x k} converging to some point x e R 2. Obviously x e Eho w Tho, ho(X ) < 0 and Vho(X ) 0, which is on

"

impossible. Finally, we show that in fact claims (a) and (b) imply Wo(X) O. To verify this, consider the function (x, x2)= u(-xl, x2). One can see easily that is also a solution of (.) since the function R is radially symmetric. Applying the above

,

arg.ument to we obtain (b) t(x h) t(x) for some

.o .o

or O. Now combining claims (a), (b), and (1), we conclude that Who(X) =_ 0 for some This completes the proof of the theorem.

20.

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CHEN AND LI

3. Examples. From Theorems and 2, we see that the solutions of equation (,) approach -c in the order of -fl ln lxl, and the monotonicity of the function R determines the value of ft. Then there may arise a natural question: Do such solutions exist? To partially answer this, we provide the following examples. We construct two families of increasing or decreasing functions R(r). Associated to each R(r), we find the explicit solutions of (,). One can see that those solutions have the desired order of growth as described in Theorems and 2.

Example 1. Let > 0 be the parameter. Consider a family of functions 20- 12 + (54 + s2(r)

R,(r)

3)s2(r)

where s is a function of r defined implicitly by

s(4 + sZ)-)/.

r

The corresponding solutions of (,) are given by

u,(r)

In

- Z(r)

+ 4)

3a-x)/}

(s2(r) + 4)#

It can easily be seen that R,(r) is bounded from above. One can also verify that

In r C < u(r) < In r + C with < 4; (i) for < 1, c3R/tr < O, and for and 0>1, (ii) -flln(r+l)-C 4; (iii) for 0 1, R(r) 2 and u(r) ln(16/(4 + r2)2), which corresponds to the standard metric of the sphere $2; for (iv) 3/5 < < 3, R(r) > O. Example 2. Let R,(r) (,)is

(2 + )((4 + r2)/4)". Then the corresponding solution of

u,(r)

(2 + )In

4+r 2"

Obviously, R satisfies the conditions in Theorem 2. And one can easily verify that

(i) for e < O, OR,/Or < O, and -flln(r + 1)- C < u,(r) < -ln(r + 1) + C with fl < 4; (ii) for e>O, OR,/Or>O and -flln(r+l)-C