Existence and Non-Existence of Radial Solutions for
Elliptic Equations with Critical Exponent in E2
D. G. DE FIGUEIREDO
IMECC- UNICAMe Campinas
AND
B. RUF
Universita degli Studi, Milano
1. Introduction
Elliptic equations with critical growth have drawn much interest in recent years.
Recall that a nonlinearity f in an elliptic equation on a bounded domain R C
RN, N 2 3,
(1.1)
-Au
=
f(x,u)
u
=
o
in c RN
on a0 ,
is said to have “critical growth,” resp. “subcritical growth,” provided that
where 2* = 2 N / ( N - 2) is the critical Sobolev exponent for the embedding
H’(52) C L2*(0).Criticality and subcriticality play important rBles concerning
the solvability of equation (1.1). Using (by now classical) variational methods one
sees that equation (1.1) is solvable i f f has subcritical growth (and satisfies some
additional hypotheses), while for f(u) = IuIP-~u the well-known Pohoiaev identity shows that the solvability of equation (1.1) is lost if p reaches 2* (and R is
starshaped). In their important paper, [6], Brezis-Nirenberg showed that for N E 4
solvability is regained if one adds a lower order (linear) term to the critical term.
More precisely they showed that for f ( r ) = Ar + lr12*-2rone has ( A , denotes the
1st eigenvalue of -A on HA(R)):
-
if N 2 4 : (1.1) has no solution if A 5 0 and R is starshaped
(1.1) has a positive solution if 0 < A < A , .
In dimension N = 3 the situation is more delicate. To regain solvability, a sufficiently large linear term has to be added, as is most clearly seen in the case of the
ball, 52 = B I ( 0 ) :
-
if N
=
2
3 : (1.1) has no (positive) solution if 0 < A 5
(1.1) has a positive solution if
< A < Al .
2
The case N = 2 is different: “critical growth” is not given by the Sobolev
imbedding, but by the Trudinger-Moserinequality (see [131 and [18]), which says
Communications on Pure and Applied Mathematics, Vol. XLVIII, 63M.55 (1995)
0 1995 John Wiley & Sons, Inc.
CCC 0010-3640/95/06063Sr17
640
D.G . FIGUEIREDO AND B. RUF
that for a 5 47r
sup Leu'' 5 c(a) 5 c(47r) =: c,
(1.2)
Ill11
IHl, 5 I
Note that while the best Sobolev-imbedding constant in the critical case for N 2 3
sup
114lH;)
Id2*= S N
51
is not assumed, the value c, in (1.2) is attained; see Carleson-Chang, 171, Struwe,
[ 171, and Flucher, [ 101.
This paper is concerned with a phenomenon similar to the one described above
for the equation in R2. More precisely, we consider nonlinearities with critical
growth and are interested in conditions on lower order terms to have existence or
non-existence of solutions.
Motivated by inequality (1.2) we say that the nonlinearity f has critical growth,
resp. subcritical growth, i f
One has again (as for N 2 3) that (1.1) has a solution if f has subcritical growth
(and satisfies some additional conditions); see, e.g., [9]. If the nonlinearity f has
critical growth, we write equation (1.1) as
(1.3)
-Au = h(u)eau2= Sgn(h(u))eau2+l"
Ih(u)I ,
u = 0,
in R c R2
on a R .
Here h is a "lower order term" with respect to euu2,i.e., we assume that
We are interested in determining the dividing line for the solvability resp. nonsolvability of (1.3) in terms of the asymptotic growth of the lower order term
h(d.
First note that since the constant c, in (1.2) is assumed, there exists (for some
constant c > 0) a solution for
-Au
=
c2ueau- on R ,
u = 0 on a R
In [2] Adimurthi-Yadava have shown that (besides some additional conditions) a
sufficient growth condition for solvability is
lim h(r)r = +cc ,
IrI-m
641
RADIAL SOLUTIONS
and recently it has been shown in [9] by Miyagaki and the authors that there exists
a constant K I > 0 such that if
lim h(r)r > K I
Isl-c=
(and some additional conditions are satisfied), then equation (1.3) has a solution.
We will show here that this growth is the critical growth of the lower order term
in regard to the solvability of (1.3). Indeed, we will see that if we take R = B1(0),
then there exists a constant KO > 0 such that if
K
r
h(r) =
-
for r > rl ,
with K < K O
(and h(r) satisfies some conditions near 0), then equation (1.1) has no radial solution. Note that by [ l l ] any positive solution of (1.3) on R = Bl(0) is radial; this
implies that there exists no positive solution under these assumptions.
Recall that the mentioned non-existence results for N 2 3 are based on the
Pohoiaev identity. This method of proof does not seem applicable in two dimensions. We follow here another road: We restrict ourselves to the ball R = Bl and
consider only radial solutions, i.e., we reduce (1.1) to the radial equation
(1.4)
Urr
1
+ ; U r + f(u)
U'(0) = U(1)
=
0,
=
0.
O<r<l
This equation can be transformed into an equation on the half-line by the substiI
tution t = -2ln(Tr),
y(t) = ~ ( r ) :
-y"
y(21n2)
=
=
f(y)e-' ,
t > 21n2
0,
y'(+cm) = O .
There have been numerous studies of such equations for superlinear functions
f ; see, e.g., Coffman, [8], Nehari, [14], Hempel, [12], and Ni-Nussbaum, [15]. In
particular, Atkinson-Peletier, [4], have studied this equation for general superlinear
Iw with the aim of proving the existence of so-called ground
functions f : Iw+
state solutions, i.e., positive solutions in all of R2. For this, they considered the
initial value problem
-
(1.6)
-f'
=
f(J+-'
=
y'(+cc)
=
0,
y(+cc)=y
and showed that the first zero from +cc, T ( y ) ,of y(t)
lim T ( y )= -cc
y-o+
provided that f ( 0 )= f'(0) = 0, and
lim T ( y ) =
y-+m
+GO,
= y ( t ,y )
satisfies
642
D. G . FIGUEIREDO AND B. RUF
if g(u) := ln(f(u)) satisfies limsup,-,{g(u) - iug’(u)} > 1 + InM, where -M =
inf{f(u); u > O}. Note that in the critical case we have (cf. (1.3)) that g(u) =
au2 + In h(u).If we assume that h(u)is of the form h(u) = U P ,
for u large, we see
that the Atkinson-Peletier condition is satisfied for a < 0. In this paper we discuss
a > 0 (for u large), i.e., g(u) = u2 - a In u, for u large (note
the case h(u) = U P ,
that we may restrict ourselves to the case cy = 1; see Section 2). More precisely,
we study the equation
-Y”
y(21n2)
(1.7)
=
ev?+lnhoJ)-r
=
0,
t > 21112
y’(+m) = 0 .
with the following hypotheses:
-
Let h : R
R be C2 and suppose that there exist some rl > 0 and a
such that for some constants K > 0, c > 0
Al. h(r) = K/r” ,
for r 2 rl , a > 0 ,
A2. 0 5 h (r) 5 cKrIfU,
for 0 5 r 5 rl
(T
>0
.
We will see that the solvability of equation (1.7) depends on the parameter
a > 0 in h(r).To prove our result, we consider (as Atkinson-Peletier) the shooting
problem from +m
-y//
(1.8)
y’(+m)
=
evz+lnh(y)-t
=
0,
y(+m) = y
3
T ( y )on the exponent a > 0 in h(y).We
and analyze the dependence of lim,,,,
will prove
-
THEOREM
1.1. Suppose that h : R R satisjies A l and A2. Then thejirst zero
T ( y ) (from + m) of the solution y(t, y ) of (13 ) satisfies:
(i) i f a > 1 : limy-+m T ( y ) = -m
(ii) i f a = 1 :
T ( y )E [1/2 + InK, 5/2
(iii) i f a < 1 : 1im7-+&T(y) = +m .
+ In K ]
With these estimates we can give existence, non-existence, as well as nonuniqueness results for equation ( I .3) on R = B I for the nonlinearities considered
in Theorem 1.1.
THEOREM
1.2. Assume that h E C2(R) satisjies assumptions A1 and A2, and
suppose that R = B I(0) and a = 1 in equation ( I .3). Then:
1. l f a < 1, equation (1.3) has a (positive) radial solution;
2. I f a = 1, equation (1.3) has a (positive) radial solution for K > K l := 4/e’I2,
and there exists a constant Ko with 0 < K O S K I such that (1.3) has no radial
solution for 0 < K < KO;
643
RADIAL SOLUTIONS
3. I f a > 1, there exists a constant K2 > 0 such that i f K < K2 then equation
(1.3) has no radial solution, and i f K > K2 then (1.3) has at least two (positive)
radial solutions.
Example. As an example for the "critical" case 2, one may take
Finally, the following theorem contains an existence result for so-called ground
state solutions for equations with critical growth in R2, i.e. solutions of the equation
-AM = f ( u ) ,
u > 0,
(1.9)
4x1
-
in I W ~
0 as 1x1
-
00
We make the following assumptions:
A2*. Ih(r)l 5 krl'", for some k > 0 and o > 0, and for 0 5 r 5 rl.
A3. f(0) = 0 and there exists a number 6 > 0 such that f ( u ) > 0 for u B 6, and
the primitive of f,F(u) = J; f(s)ds, satisfies F(u) < 0, for 0 < u < 6 and F(6) =
0.
THEOREM
1.3. Assume that h f C2(R) satisfies A1 with a < 1, A2* and A3.
Then equation (1.9) has a ground state solution.
GENERALIZATIONS.
One checks that with minor changes in the proofs Theorems 1.1-1.3 can be generalized to nonlinearities g(r) = r2+ln h(r)with g E C2(R),
g' > 0 and g" 2 0 on [rl,+m), and with h satisfying A2, resp. A2*, and A3, and
the following hypothesis
Al'.
lim h(r)ra= K > 0 ,h'(r) = 0
r-tcc
and
This allows us to include nonlinearities h of the form h(r) = $ + f , with b > a,
for r 2 rl.
2. Estimates
In this section we estimate the behavior of solutions of equation (1.6), with
g(u) = In f ( u )of the form g(u) = au2+In h(u).First note that by the transformation
y(t) = f i u ( t ) equation (1.7) becomes -y" = f i h ( y / f i ) e Y 2 - ' . Since h ( r / f i ) =
aa'2K/ra, for r h f i r ] ,we may restrict attention to functions g(u) of the form
644
D.G . FIGUEIREDO AND B. RUF
g(u) = u2 + In h(u), with the corresponding change of K to C Y ( ’ + ~ )in/ ~the
K term
h; hence, we consider (1.71, resp. (1.8).
Next, we recall the following uniqueness and continuous dependence result;
see, e.g., [ 11 and [ 121:
-
LEMMA
2.1. Given any number y > 0 there exists exactly one solution u(t,y )
of (1.8) such that u(t)
y as t
+m. Furthermore, let 0 S c < y and denote
with t,(y) thejirst t (from +GO) at which u(t,y ) = c (note that such a t,(y) exists
always, since u(t, y ) is concave); then t,(y) depends continuously on y.
-
We start with estimates concerning the solutions of (1.6), taken from [4],
Lemma 4.1: Let us denote with T I = Tl(y) the value t for which the solution
y(t, y ) (with y(+ 00, y ) = y, y sufficiently large) reaches the value rl.
As remarked in [4], these bounds are exact if g is linear.
Note that the function g(r) = r2 + In(K/ra) satisfies the hypotheses of Lemma
2.2 on [(a/2)1/2,
00).
We recall the observation from [4] that the solutions y(t, y ) of (1.6) have a
peculiar structure: coming from t = +m, they are nearly straight lines parallel to
the t-axis at level y ( + m , y ) = y until they arrive near the point
Then they rapidly change direction and become nearly straight lines again, with
slope y’(t) 2/g’(y)).
To obtain a good quantitative estimate of the slope after T , (looking from
infinity), we set
-
Setting
we prove
645
RADIAL SOLUTIONS
LEMMA
2.3. Suppose that g(r) = r2 + ln(K/ra), with a > 0, for r 2 rl. Then
the solution y ( t ) of (1.6) satisjiesfor t 2 T5
(2.1)
g(y) - 2 In G(t) S g(y(t)) S g(y ) - 2 In G(t) + ln2G(t) + O ( ? ) .
Y2
~
Proof First note that for t 2 T5 2 T I we have by the monotonicity of y that
y ( t ) > rl, for all t 2 T5 and y sufficiently large. Applying g-’ to (ii) we obtain
y(t) 2
y
1
+ (g-’)‘(g(y))(-2 ln(1 + -g’(y)eg(y)-‘))
2
1
+ (g”)”(g)2ln2(1 + -g’(y)@(+‘)
2
,
where g depends on y and 2 , and
g(y) - 21nG(t) 5 ,ij 5 g(y)
.
Let 7 be the unique point where g(7) = 2. If t 2 T5 we have g(y) 2 g(y) 2 In G(T5) and hence as above 7 2 rl . Now we estimate the second derivative of
g-l at g which is given by
From the expressions of g(r) and g‘(r) we obtain
2
g‘(r) 2 -g(r) for r 2 e1/2K‘/a.
r
So we obtain the estimate
2
2
g’(7) 2 rg 2 - [ g ( y ) - 2lnG(t)] .
Y
Y
Since for t 2 T5 we have In G(t) S 6 In y. it follows that
[
g’(7) 2
y2
+ In
Y
51
- 24-In Y
Y
or
g’(7) 2 2y - (24 + 2 4 In y +0(;).
Y
We use this estimate to obtain
646
D. G . FIGUEIREDO AND B. RUF
Hence
Note that g ' ( y ) = 2 y - a/y, and therefore
Hence, by (2.2) and (i)
Y - - -In G(t)
Y
(2.3)
5
In G(t)
Y-----------Y
a In G(t) + ln2 G(t) + 0
2Y3
(y)
5 y(t)
(y),
y)+ +)+ (y))*
+ $)+ (y))
a In
2y'G(t)
t2TS.
+
Inequality (2.3) implies by monotonicity
g(y(t)) 5
(.
-
(1
0
+lnK - a h ( y -
(I
=
+
y2 - *lnG(t). (1
$)
0
ln2 G(t)
+2
+0
Y
+ 1nK - a l n y + a In G(t) + O ( ? )
~
Y2
Together with (ii) we have (2.1).
LEMMA
2.4.
Suppose that g is as in Lemma 2.3. Then
f o r T I 5 t 5 T s , where as before T I is such that y(T1) = rl.
Proof:
Since y'(t)
=
J+OOeK(~y(s))-sds,
we get by Lemma 2.3, for t B Ts:
647
RADIAL SOLUTIONS
With the substitution x = ig’(y)efi(Y)-swe obtain
Now we set t
expression by
=
T5 and estimate the exponential at the right side of the above
where we have used that ln(1 + x), which is equal to In G(t),is of the order of In y.
Since
we get
The integrals in the above inequality can be calculated and estimated for large
z = ( g ’ ( ~ ) / 2 )One
~ . has
Z
Zln2(1+x)dx
Recalling that
=y
22
2ln(l+z) - ln2(1+d
1+ z
1+ z
=,+,($)
+ O ( $ )we then get
To extend the estimate to the interval [ T I T5]
, we proceed as in [4],
Lemma 10.
Integrating equation (1.6) over [t,T=J,with T I 5 t 5 T5, we have
648
D. G . FIGUEIREDO AND B. RUF
Note that from Lemma 2.2 (i), we get
and since by (2.1)
and by (2.2) and (i)
we get
It follows then that for y sufficiently large and for t E [ T I ,T s ]
Note that
where we have used the monotonicity of y ’ ( t ) , (i), and (2.4). Hence, using once
more the convexity of y, we have
RADIAL SOLUTIONS
649
This completes the proof of the lemma.
Next we estimate the value of TI (where y(T1) = rl).
LEMMA
2.5. Let g be as in Lemma 2.3. Then
where [b, c] denotes lower and upper estimate.
Proof
By (i), (2.2), and Lemma 2.4, we have
=
g ( y ) - y2
=
rly
+ In g'2o + rl y + [1/2,
+ [1/2,
5/21 -(a - 1)ln y
5/21
+0
(?)
+ InK + 0
We are now in the position to complete the
Proof of Theorem 1.1: Integrating once equation (1.6) from t2 to
we obtain
(2.5)
t3.
y'(t3) - y'(t2) = 0+(l)(e-'3 - e-fz)
and integrating it twice we have
y(t3) - y(t2) = y'(td(t3 - t2)+ O+(l){e-'z - e-"(t3 - t2 + 1)) .
Observing that the term in brackets is
eC2(1- er2-'3(t3 - t2 + 1)) = O+(l)e-'z
we have
(2.6)
y(t3) - y(t2) = y'(t3)(t3 - t 2 )
+ O+(l)e-'Z.
t2 < t3,
650
D. G. FIGUEIREDO AND B. RUF
In (2.5) and (2.6) O+(1) denotes a positive bounded term, which comes from the
fact that f(y) is bounded for y in any interval [O,yo]. If t 2 and t 3 are such that
y(td, y(td 5 yo 5 r l , this estimate can be improved using Assumption A2. We
obtain
Proof of (iii):
a < 1. Supposing that T ( y ) S c we get
c
z
-(a - 1)lny
+ O(1)+ O+(l)ye-‘
,
a contradiction, since a - 1 < 0.
To prove (i) and (ii) we proceed in several steps:
1. Claim. There exist yo > 0 such that
and if a = 1 then T ( y ) 2 -c for y 2 yo.
Proof Supposing that T ( y ) 2 41n y we obtain from (2.8) 4 In y s O(1) +
O+(l)$, a contradiction. Furthermore, if a = 1, then we conclude that T ( y ) 2
O(1).
2. Claim. y’(4Iny) = y’(T1) + O(l/y4).
Proof Since for y sufficiently large we have 4 In y < T I = 1-1 y -(a - 1) In y +
0(1),we can apply (2.5) to obtain
TI)
=
=
y’(4Iny) + O+(l)(e-TI - e - 4 1 n y1
y’(4 In y ) + 0+(
l)(e+’Y - Y - ~ ).
65 1
RADIAL SOLUTIONS
3. Claim. y(41ny) =
Proof
([$, 31 - (a - 1)ln y + 1nK) + 0 (9).
-
Using (2.6) and Lemmas 2.4 and 2.5 we get
TI) - y(41ny)
=
~ ' ( T I ) (T I41ny) + O+(l)eP4'"y
This implies
r-1 - y(41ny) = rl
+-
With this we can give the final estimate of T ( y ) .Using (2.7) and Claim 3 we
have
Using again Claim 3 and Claim 2 we obtain
-41ny
-Y- - - ( [Y1' ~ ]2- '(2a - l ) l n y + l n ~
=
(++ (5))
0
(41ny - T ( y ) )+ 0
((y ) )
I +u
e-T(y).
This implies
Proof of (i): a > 1. Suppose that T ( y ) 2 -c, as y
expression we get
-c 5 -(a - 1)lny + O(1) ,
-
+m. From the above
a contradiction.
Proof of (ii): a = 1. By Claim 1 we have T ( y ) B -c, and hence we get
652
D. G . FIGUEIREDO AND B. RUF
3. Radial Solutions on the Disk and Ground States in R2
In this section we use Theorem 1.1 to prove the existence, non-existence, and
non-uniqueness results for radial solutions of equation (1.1) on the disk as stated
in Theorem 1.2, as well as the existence of the ground state solutions given in
Theorem 1.3.
Proof of Theorem 1.2: First we recall (see [ 5 ] , and [12]) that Assumption A2
0'. In fact, from (2.7) with t 2 = T ( y ) ,t 3 = +cc
implies that T ( y ) -cc for y
and yo = y with y S rl we obtain
-
-
y = O(yl+o)e-'(Y)
.
Hence
-
-
0.
i.e., T ( y ) -m as y
1. Since R ( y ) = 2e-'(Y)/* we have by the above argument that R ( y ) +cc for
y o+.
If a < 1 we have by Theorem 1.1, (i), that R ( y ) 0' for y
+m. Since T ( y )
and therefore R ( y ) depend continuously on y , it follows that equation (1.4) has a
positive solution, and equation (1.3) has a positive radial solution on s2 = Bl(0).
2. If a = 1 then we conclude as above that equation ( I .4) has a positive solution
T ( y ) > 21n2. By Theorem 1.1, (ii),
provided limy-+m R ( y ) < 1, i.e., if
this holds if 1/2 + In K > 2 In 2; thus the first part of 2 is proved.
-
-
-
-
-
-
Next, using again that T ( y ) - 0 0 , for y
O+, and that by Theorem 1, (ii),
T ( y ) 5 5/2 + lnK, we have that T ( y ) 5 c, for all y > 0. This implies
that R ( y ) 2 2/ecl2, for all y > 0, and so equation (1.3) has no radial solution on
the disk BE with R < 2/e''12.
Note that R ( y ) depends on the constant K in Assumptions A l , A2 (assuming
that all the other constants remain fixed); we therefore write R ( y , K ) , and derive
a relation between R ( y ,K ) and R ( y , K/d), for d > 0:
R(y,K/d)
= R(y,K)&,
for all y > 0 .
In fact, note that with the change of variable s = r - l n d and setting w(s)
we have
- w ~ ~ (=
s )- u ~ ~ (=t )h(u)eu2-t = h(W)ew'-(.s+lnd)
= u(r)
that is,
7
1
= -h(w)e"--"
d
Denoting with S ( y ) the first zero (from +cc) of w(s) we have S ( y ) = T ( y )- lnd,
for all y > 0; furthermore, h(r)/d satisfies A l , A2 for K = K/d, and so
R ( y , K / d ) = 2e-s(Y)/2 = 2,-T(T')/2+(1nd/2 = ,/&(?,K)
.
RADIAL SOLUTIONS
653
Therefore, in order to have R ( y ,K / d ) > 1, for all y > 0, it suffices to choose
d > i e c , since then a R ( y ,K ) 2 d22/e'/2 > 1, for all y > 0.
3. If a > 1 we have l i m y 4 ~ m
T ( y ) = -aby Theorem 1, (i). Then we have as
in part 2 that T ( y ) 5 c for all y > 0, and for any fixed K > 0. Choose K = 1, and
for the corresponding solutions let c,, = maxy>oT ( y ) .Proceeding as in part 2 we
1
obtain ford, = iecin
that R ( y , l/&) Z 1, for all y > 0. Set K2 = l/d,. We have
for K < K2 that R ( y ,K ) > 1, for all y > 0, i.e., there exists no radial solution
of (1.3) on Bl(0). On the other hand, if K > K2 then R, = miny,O R ( y ,K ) < 1.
Let ym such that R(y,, K ) = R,. Then, by continuity, there exists a y~ < ym and
a 7 2 > ym such that R(y1,K ) = R(y2,K ) = 1, i.e., we have found two positive
radial solutions of equation (1.3) on R = B 1(0).
Remarks
1. The above existence result can be compared to the existence result which
was proved (by variational methods) in [9]. In fact, Theorem 1.3 in [9] gives for
R = BI(O)the existence of a solution provided that
lim h(t)t > 2 .
t-+cc
Thus the existence result in [9] (besides being more general) requires a slightly
weaker asymptotic condition than the one stated here in Theorem 1.2.
2. We mention that in [3] Adimurthi-Srikanth-Yadava have obtained with similar methods a non-existence result provided that f(y) = ymeY2-f, with 0 < p < 1
andm > 1.
3. Uniqueness and non-uniqueness for such equations have been considered
by Ni-Nussbaum in [15]; see also Srikanth, [16]. The range of these uniqueness
results covers nonlinearities of polynomial growth.
Ni-Nussbaum prove for equation (1.2) the existence of a nonlinearity f for
which there is non-uniqueness. Their proof is based on a theorem by Hempel (see
[12], Theorem 1.3); it uses the existence of a solution of type C (cf. [12]), i.e., a
solution u,(r) with u,(a) = 0 for some a E [w and limr++ccuc(r) = +co,in addition
to some growth hypotheses on the nonlinearity f . In particular, the imposed growth
condition (f3) in [15] (resp. Theorem 1.3, (iii), in [12]) is satisfied by nonlinearities
with supercritical growth (but it is not satisfied by the nonlinearities considered
in the present paper). Ni-Nussbaum give an example of a nonlinearity f which
satisfies all these conditions; it is based on the observation (cf. [12]) that In(r)
satisfies the equation u" +f(u)e-' = 0 for f(u) = exp(exp(u)-2u). Incidentally, we
remark that the function r 1 / 2solves this equation for f(u) = $eu2. Redefining this
f near zero (such that it satisfies A2) and continuing the solution r 1 / 2backwards
(as solution of a suitable initial value problem), we see that this problem has a
solution of type C. Since by Theorem 1.2 we obtain for this nonlinearity at least
two solutions on sufficiently large disks, one may conjecture a general connection
between multiplicity and the presence of solutions of type C.
654
D. G. FIGUEIREDO AND B. RUF
Finally, we consider the relation between the Dirichlet problem (1.3) and the
existence of so-called ground state solutions in R2, i.e., solutions of problem (1.9).
It has been shown in [4] and [5] that under certain conditions solutions of (1.9)
can be obtained as the limit of solutions U R of (1.1) on R = BR as R
+00. In
particular, one has the following (see [5] and [4], Lemma 0)
-
PKOPOSITION
3.1. Suppose that f is locally Lipschitz continuous on [O, + m )
and that it satisfies
A3. f ( 0 ) = 0
and there exists a number > 0 such that f (u ) > 0 for u 2 and the primitive
o f f , F(u) = J
: f ( s )ds, satisfies
<
<
F(u) < 0 ,
for 0 < u <
< and F(<)
=0
.
Then, ifthe set S = { y E R + ; T ( y ) > -00) (where T ( y )denotes as before thejrst
zero from +00 of the solution y(t,y ) of (1.6)) is non-empty, problem (1.9) has a
solution y ( t ) with
lim y ( t ) 5 inf S .
/-+m
Based on this proposition, we prove Theorem 1.3, i.e., the existence of ground
state solutions for certain nonlinearities with critical growth.
Proof of Theorem 1.3: By Theorem 1.1 we have for a < 1 l i m y ~ + oTo( y ) =
that in Theorem 3 we replace condition A 2 by the weaker condition
A2 * . Going through the proof of Theorem 1.1 we see that the only changes occur
in formulae (2.5)-(2.8), where the term 0+(1) has to be replaced by a term O(1)
(i.e., without sign). It is easily seen that for the case a < 1 this does not matter;
i.e., we have the same conclusion.
Hence the set S defined in the above proposition is non-empty. Thus equation
(1.9) has a ground state solution by the above proposition.
+m. Note
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Received March 1994.
Revised August 1994.