J. Korean Math. Soc. 37 (2000), No. 5, pp. 763-777
EXISTENCE OF GROUP INVARIANT SOLUTIONS OF A SEMILINEAR ELLIPTIC EQUATION
RYUJI KAJIKIYA
ABSTRACT. We investigate the existence of group invariant solu- tions of the Emden-Fowler equation, -.6.u=Ixl"lulp-1uinB,u =0 onaBandu(gx)=u(x) in B for9EG. HereB is the unit ball in JRn,n 2 2, 1<p< (n+2)/(n-2),(J'20 andGis a closed subgroup of the orthogonal group. A solution of the problem is called aGin- variant solution. We prove that there exists aGinvariant non-radial solution if and onlyifGis not transitive on the unit sphere.
1. Introduction
I
We consider the existence of group invariant solutions of the EmdenJ Fowler equation,
(1.1) -~u = Ixl17lujp-1u, X E B,
(1.2) u = 0, x E oB,
(1.3) u(gx) =u(x), x E B, gEG,
where B = {x E ~n : Ixj < I}, n ~ 2, 1 < p < (n+2)j(n - 2), a ~ 0 and G is a closed subgroup of the orthogonal group O(n). We call a solution of (1.1)-(1.3) a G invariant solution. In this paper, we extend the result of [8] with a = 0 to the case a ~ O. It is known that (1.1)- (1.2) has infinitely many radially symmetric solutions. Any radially symmetric solution becomes a G invariant solution. We consider the converse problem: Must a G invariant solution be radially symmetric?
Otherwise, does there exists a G invariant non-radial solution?
Received August 7,1999.
2000 Mathematics Subject Classification: Primary 35J20; Secondary 35J60.
Key words and phrases: semilinear elliptic equation, Emden-Fowler equation, group invariant solution, non-radial solution, variational method.
Since G is a closed subgroup of the orthogonal group D(n), G is a transformation group on the unit spheresn-l,
sn-l ={x EJRn :Ixl = I}.
G is said to be transitive onsn-l if for any two points x,yE sn-l there exists agE G such thatgx=y. Our answer to the problem is as follows.
THEOREM 1. The following two assertions are equivalent.
(i) G is not transitive onsn-l.
(ii) There exists a sequence {Uk}k=l ofsolutions of(1.1)-(1.3) such that each Uk isG invariant, not radially symmetric and
0< !IUl!IHJ(B) < Ilu21IHJ(B) < Ilu31IHJ(B) < ... ? 00,
where 11 . IIHJ(B) denotes the£2 Sobolev norm of the first order.
Itis clear that when G is transitive onsn-l, anyGinvariant solution has radial symmetry. Therefore the assertion (ii) implies (i). Theorem 1 asserts mainly that (i) implies (ii).
COROLLARY 1. If dim G :s; n - 2, then the assertion (ii) of Theorem 1holds.
COROLLARY 2. IfG is a finite subgroup of D(n), then the assertion (ii) ofTheorem 1 remains valid.
By Cartan's theorem [7, p115, Theorem2.3], a closed subgroup G of D(n) is also a Lie subgroup. Therefore dimG denotes the dimension of ,Lie groupG. Ifdim G :s; n - 2<dimsn-l, then G can not be transitive on sn-l. Therefore Corollary 1follows from Theorem 1. Corollary 2 is trivial because a finite group is not transitive.
We consider the question related to Theorem 1: What kind of G is transitive? This problem has already been solved by Montgomery, Samelson and Borel.
THEOREM A ([9], [3]). Let n ~ 2 and G be a connected closed subgroup of SO(n). Then the following are equivalent.
(i) G is transitive onsn-l.
(ii) G is O(n)-conjugate to one of the following subgroups: SO(n);
SU(m), U(m) (n = 2m); Sp(m) , Sp(m)Sp(l), Sp(m)U(l) (n = 4m);
Spin(9) (n = 16); Spin(7) (n= 8); G2 (n = 7).
IfG is not necessarily connected, Theorem A implies the next theo- rem.
THEOREM B. Let n ::::: 2 and G be a closed subgroup of O(n). Then the following are equivalent.
(i) G is transitive onsn-l.
(ii) The connected component ofG which has the unit matrix is O(n)- conjugate to one of the Lie groups listed in (ii) ofTheorem A.
2. G invariant eigenvalues
In this section, we investigate the upper estimate for the G invariant eigenvalues of the problem,
(2.1)
{
-.6.u = Alxlt7u,
u=O, u(gx) =u(x),
x EB, x E aB,
x E B, 9 EG.
This problem has countably infinitely many eigenvalues {Ad, o< Al < A2 :s; A3 :s; ... ? .
Here the eigenvalue is repeated a number of times equal to its multiplic- ity. We define
(2.2) G(x) = {gx : 9E G} for x E sn-l, (2.3) m = max{dimG(x) : x E sn-l}.
For eachx E sn-l,the orbitG(x) is a closed submanifold ofsn-l, and so the number m is well-defined. IfG is not transitive, then 0 :s; m :s; n - 2.
The main assertion in this section is as follows.
PROPOSITION 2.1. Let G be not transitive on sn-l. Then 0:s; m :s;
n - 2 and there existsa constant C >0 such that Ak :s;Ck2/(n-m) for kENo
To prove this proposition, we introduce norms, function spaces and operators.
DEFINITION 2.2. We define
Ilullq,CT (k lulqjxlCTdx) l/q ,
lIullq = Ilullq,o = ( k IU1qdX) l/q .
In case ofq=2, the inner product is defined by (U,Vh,CT kuvlxlCTdX,
(u,vh (u,vh,o = k uvdx.
We set
L~(B)= {u: Ilullq,CT < oo},
L~(B,G) = {u E L~(B) :u(gx) = u(x) (x E B, 9E Gn, HJ(B,G) = {u E HJ(B) :u(gx) =u(x) (x E B,9E Cn, Au= -Ixl-CT~u,
Acu= -Ixl-CT~u,
D(A) = {u E L;AB)nHJ(B) :Au EL;(Bn, D(Ac) =D(A) nL;(B,C).
LEMMA 2.3. The operator Ac is self-adjoint in L;(B,G) and has a compact resolvent.
We will give the proof of the lemma in Appendix. Lemma 2.3 means that the spectrum of Ac consists only of discrete eigenvalues and the system of eigenfunctions is complete in L;(B, G). The eigenvalue Ak is characterized by the minimax: value of the Rayleigh quotient,
(Acu,uh,CT/(U,Uh,CT = - k Ixl-CT~uulxICTdx/ k u21xlCTdx
k IVul2dx / k u21xlCT
dx =R(u).
It is well-known that (see [5, p. 405], [11, p. 76]) Al = inf{R(u): u E HJ(B, C) \ {On,
Ak = sUP{d(Vl,"', Vk-l) : Vb'" ,Vk-l EHJ(B,Cn for k2: 2, where
d(Vl,··· ,Vk-l) = inf{R(u) : uE HJ(B, G) \ {a}, (u,Vih" =afor 1:s; i :s; k - I}.
Proof of Proposition 2.1. We set
D = {x E jRn: 1/2< Ixl < I}.
Let J.lk denote the k-th eigenvalue of the problem (2.1) with D instead of B. From D c B, it follows that Ak :s; J.lk. The Rayleigh quotient associated withJ.lk is
{ l\7ul2dx ( l\7ul2dx
R(u) = lD < 2"lD .
Lu2lxl"dx - Lu2dx
The minimax value of the last quotient is the k-th eigenvalue Vk of the problem,
{ -~~: ~u, ~ ~ giJ,
u(gx) =u'(x), x E D, 9E G.
Therefore Ak :s; J.lk :s; 2"Vk. By [8, Proposition 3.3], there exists a constant C > asuch that Vk :s; Ck2/(n-m) for kEN. This completes
the proof. 0
3. G invariant critical values
Our argument to prove Theorem 1 is based on the variational method of the functional I (u),
I(u) = { (~I\7UI2 - -l-lxl"luIP+l) dx, u E HJ(B,G).
lE 2 p+ 1
The solution of (1.1)- (1. 3) is considered as a critical point of1(·)because I' (u) is calculated as
I'(u)v = L(\7u\7v -lxl"luIP-1uv) dx for V E HJ(B,G).
Ifl'(u)v = afor all v E HJ(B,G), then this identity remains valid for any v E HJ(B). See [8] for the proof. Therefore usatisfies (1.1)-(1.3) in the distribution sense. The elliptic regularity theory proves that u is of
class C2(B). Consequently, to prove Theorem 1, we have only to find a non-radial critical point in HJ(B, G).
In this section, we use the symmetric mountain pass method by Ambrosetti-Rabinowitz [1] to construct Ginvariant critical values.
DEFINITION 3.1. Let K be the closed subset of HJ(B, G) such that
otJ. K and -u E K for u E K. We define the genus "((K) of K by the smallest integer k such that there exists an odd continuous mapping from K to IRk \ {O}. When there does not exist a finite such k, we set
"((K) =00.
In the next proposition, we obtainG invariant critical values.
PROPOSITION 3.2. There exists a sequence {Q;k} of real numbers which satisfies the following conditions.
(i) Each Q;k is a G invariant critical value of 1(·).
(ii) 0 <Q;l ~ Q;2 ~ ••• / ' 00.
(iii) IfQ;k = Q;k+l = ... = Q;k+i == Q;, then it holds that "((KO'.) 2::j +1, where
KO'. = {u E HJ(B,G) :I'(u) = 0 andI(u) = Q;}.
(iv) There isa constant C > 0such that
2(p+l)
Q;k ~ Ck(n m)(p-l) for kEN,
wherem is defined by(2.3).
Proof. This proposition except for (iv) is due to Ambrosetti and Rabinowitz [1]. The proof of (iv) is proved in the same way as [8] by using Proposition 2.1. However, for the reader's convenience, we give the proof. Let )..k be the k-th eigenvalue of (2.1) and cPk the eigenfunction.
We set
Ek =span{cPl' ... ,cPk}.
Recall the definition ofI (u),
(3.1) I(u) = ~IIV'ull~- P: 11Iull~L(T'
where the norm 11·llq,(Tand 1I·llqare defined by Definition 2.2. Since each Ek is a finite dimensional linear space, the HJ(B) norm is equivalent to the .v:l(B) norm, and so we have a sequence {Rk } such that
(3.2) I(u) ~ 0 for IlullHJ 2:: Rk and u EEk .
(3.3)
We may assume RI <R2 < ... /"00 without loss of generality. Set fk {h E C(Dk ,HJ(B, G)) : h is odd on Dk and h(u) =u on 8Dk }, Dk = {u E Ek : IlullHJ :::; Rk},
8Dk = {u E Ek : IlullHJ = R k }.
We define
ak = inf maxI(h(u)).
hErkUEDk
The assertions (i)-(iii) are proved in the same method as [10, pp55-60].
To prove (iv), we take h= identityE f k and obtain (3.4) ak:::; maxI(u) :::; supI(u).
UEDk Ek
It follows from the definition ofEk that II\7ull~:::; Akllull~,l7 for u E Ek •
Since p > 1, there is a C > 0 such that IluI12,l7 :::; Cllullp+l,l7' Therefore there is a constant Co such that CoA;(P+l)/211\7ullr-l
:::; Ilull~1~,l7 forU E Ek .
Hence we have
ak :::;sUP{~II\7ull§ - Co+ 1A;(p+l)/211\7ull~+1} :::; ClAi;+l)!(P-l).
Ek P
Here Cl is independent ofk. This inequality with Proposition 2.1 com-
pletes the proof. D
4. Radially symmetric critical values
In this section, we consider the radially symmetric solutions of (1.1)- (1.2). We set u =u(r), r = Ixl and reduce (1.1)-(1.2) to
n - l
(4.1) u"+- - u '+rl7lulp-lu= 0, 0< r < 1,
r
(4.2) u'(O) = 0, u(l) = O.
PROPOSITION 4.1. For each integerk 2:: 1 there exists a unique solu- tion u of (4.1)-(4.2) such that u(O) > 0 and u(r) has exactly k zeros in [0,1]. Denote it by Uk. Then the set of all solutions of(4.1)-(4.2) consists ofuk, -Uk for all kEN and the zero solution. Set /3k = I(uk) = I( -Uk).
Then it holds that :
(i) 0</31 </32 < ... /" 00.
(ii) I(uk) = /3k and I'(Uk) = O.
(iii) There existsa constantc >°such that
ck2(p+1)/(p-l) :::; f3k for kEN.
Proof. Instead of (4.1)-(4.2), we consider the initial value problem, (4.3) w"+--w'n-1 +ralwIP-1w=0, 0<r < 00,
r
(4.4) w'(O) = 0, w(O) = 1.
This problem has a unique global solutionw(r) and it is oscillatory, i.e., it has an unbounded sequence of zeros. To check this, we consider the energy function,
1 1
E(r) = -w'(r)2r-a+--lw(r)lp-tl.
2 p+1
Then er
E'(r) = -(n+2" - 1)w'(r)2r-a-l :::; 0,
which means that E(r) is bounded above. This shows the global exis- tence ofw(r).
Letn 2: 3. To show thatw is oscillatory, we take a change of variable, t = rn- 2, v(t) = tw(r), which reduces (4.3)-(4.4) to
v"+a(t)lvIP-1v=0, t >0, v'(O) = 1, v(o) =0,
where a(t) = (n - 2)-2t(a+2)/(n-2)-p-l. Sincep < (n +2)/(n - 2) and er 2: 0, t(P+3)/2a(t) is nondecreasing. By [4, Corollary 10], the solution v(t) with a zero at t =°is oscillatory. Hence w(r) is also oscillatory.
Whenn = 2, we sett =logr, v(t) =w(r), which reduces (4.3) to (4.5) v"+a(t)lvIP-1v=0, a(t) =e(a+2)t.
Since foOOta(t)dt = 00, all solutions of (4.5) are oscillatory by [2].
We denote the zeros of the solution w of (4.3)-(4.4) by{tk}, 0< t1<
t2 <t3 < ... /' 00. We set
(4.6) uk(r) = t~a+2)/(p-l)w(tkr).
Itis a solution of (4.1)-(4.2) such thatu(O) >°and it has exactlykzeros in [0,1]. Such a solution is unique because A(a+2)/(p-l)w(Ar) (A > 0) represents any solution of (4.1) with u'(O) = °and u(O) > 0. Itis clear
that the set of all solutions of(4.1)-(4.2) consists ofUk, -Uk, for kEN and the zero solution.
We show the assertion (i) of Proposition 4.1. Multiplying (4.1) by u(r)rn- 1and integl1ating by parts, we get
11u'(r)2r n-1dr = 11luIP+1 r °-+n-1dr.
Therefore we have
(4.7) I(u) = (p - l)wn t luIP+1 r <7+n-1dr,
2(p+1) lo
for any solution U of (4.1)-(4.2). HereW n is the surface area of the unit sphere. This identity with (4.6) gives
(3 = I( ) = (p - 1)Wnt(<7+2l(p+1l/(p- 1l-n-<7 tk I ()lp+1 <7+n-1d
k Uk 2(p+1) k lo W S S s,
which proves the assertion (i) because of(o+2)(p+1)j(p-1)-n-0- > 0.
The assertion (ii) is trivial. We show (iii). Let n 2: 3. We take the change of variable,
(4.8) t = r1/o., v(t) =t{3u(r) ,
where 2(3 = (n - 2)Q +1 and a > 1 will be determined later. Then equation (4,1) is reduced to
(4.9) v"+a2t"YlvI P-1v - (3((3 - 1)r2v = 0, 0< t < 1, whereI = (0-+2)a - (p - 1)(3 - 2. We need the next lemma.
LEMMA 4.2 ([6, p. 346, Corollary 5.2]). Let q(t) be a continuous func- tion on [a, b]. Let v(t) -=t °be a solution of the equation,
v"+q(t)v = 0, t E [a, b].
Assume that v(t) has exactlyk zeros in (a,b]. Then we have
1 ( b ) 1/2
k < 2 (b - a) [ q+(t)dt +1,
where q+(t) == max{q(t) ,O}.
We set q(t) = a2t"Ylv(t)IP-1 - (3((3 - 1)r2• Sincea > 1 will be chosen very large, we see (3> 1, and so we haveq+(t) :S a2Plv(t)jP-1. Letv be a solution of (4.9) corresponding to Uk. Then we choose c > 0 so small
that v has k zeros in [e,l] and q(t) is continuous on [e,I]. Applying Lemma 4.2 on [e,l], we get
where we have used relation (4.8) in the last inequality. We set
{L= 0-+1 - 1/0 - (0-+n - 1)(p - 1)/(p+1)
and rewrite the last integral into 1
1
(luIP+1r lT+n-1 )(P-1)/(P+1)rJ1.dr.
By Holder's inequality with exponents (p+1) /(p - 1) and(p+1) /2, this integral is estimated from above by
(
1 ) (p-1)/{p+1) ( 1 ) 2/{p+1)
1IulP+lrlT+n-1dr 1rJ1.(P+l)/2dr
We choose 0 > 1 so large that {L(P+1) /2 > -1 and the last integral is finite. Therefore we have
or equivalently
k - 1<Cj3(p-1)/2(P+1)
- k ,
where C is independent of k. This shows the assertion (iii) except for k = 1.
We show a priori lower bound of any solution of (1.1)-(1.2). Multi- plying (1.1) byu and integrating by parts, we get
ll\7u j2dX = lluIP+1IxllTdX::; llulP+1dX::; C(ll\7U I2dX){P+1)/2,
where we have used Sobolev's inequality. This gives a constant C > 0 such that
C :S llVU l2 dX = lluIP+1Ixl<TdX
for any solution u =i= 0 of (1.1)-(1.2). This proves (iii) for k = 1.
Let n = 2. The change of variable t = 1/(1 - logr), v(t) = tu(r), transforms (4.1) to
O<t<1.
We set
q(t) =r p-3e(<T+2)(t-1)/tlv(t)IP-1
and apply Lemma 4.2 to obtain
1 ( 1 ) 1/2
k <"2 1q+(t)dt +1
=c (11tu(r)IP-1r<T+1110g(r/e)12dr)1/2+1
(
1 ) (p-1)/2(P+1) ( 1 ) 1/(p+1)
:sC 1IulP+1r<T+1dr 1r<T+11Iog(r/e)\P+1dr +1, by Holder's inequality. Therefore we obtain the assertion (iii) for n =
2. 0
5. Proof of Theorem 1
Our idea to prove Theorem 1 is to make use of the difference be- tween the distribution of G invariant critical values{ad and that of the radially symmetric critical values {,Bd.
Proof of Theorem 1. If G is transitive, any G invariant solution be- comes a radially symmetric solution. That is, the assertion (ii) implies (i) .
We prove the converse by contradiction. Suppose that the condition (i) holds but (ii) does not. Then there exists an integer ko2: 1 such that
{ak :k 2: ko}C {,Bk :k 2: I}.
If<Y.k =<Y.k+1with somek2: ko, then(iii) of Proposition 3.2 gives'Y(K) 2:
2, where
K = {uE HJ(B,G) :I'(u) =0 and I(u) =ak}'
Since<Y.k =/3j with a certainj 2: 1, we seeK = {Uj, -Uj}. HereUjis the radially symmetric solution with exactly j zeros. Since the genus of a finite set is one by definition, it holds that'Y(K) = 1. This contradiction shows that {<Y.k} is strictly increasing for k 2: ko. Therefore we have a sequence VL(k)} of positive integers such that <Y.k+ko =(3p.(k) for k 2: 1.
Since {<Y.k} and {{3k} are strictly increasing, so is {JL(k)}, and it holds that k :::; JL(k) for k 2: 1. Hence (iv) of Proposition 3.2 and (iii) of Proposition 4.1 show
C k1 2(P+1)/(p-l) <_ fJk _Po < ak+ko _< C2(k+k0)2(P+l)/((n-m)(p-l».
This yields a contradiction by letting k ~ 00 because of 0 :::; m :::; n - 2.
The proof is complete. 0
Appendix
Lemma 2.3 can be proved in the same way as [8] with the help of the next lemma.
LEMMA A.I. Let the operator A be defined in Definition 2.2. Then A is a self-adjoint operator in L;(B) and has a compact resolvent.
Proof. ForU E D(A), we have the identity
(Au,U)2,11= - Lb.u· udx = IIVull~·
Poincare's inequality shows
(A.l) IlulI~,11 =Lu2lxll1dx:::; Lu2dx:S ClIVull~ for U EHJ(B).
Here C is a positive constant independent ofu. These two inequalities imply
(A.2) IIVuI12 :::; CIIAUlkl1'
Let AUk = Vk and {Vk} be bounded in L;(B). Then inequality (A.2) means that {Uk} is bounded inHJ(B). Therefore {Uk}has a convergent subsequence inL2(B),and so inL;(B). ThusAhas a compact resolvent.
We show that D(A) is dense in L;(B). To this end, we define X = {u ECgo(B) : u(x) ==constant nearx = O}.
ThenX c D(A) c L;(B). Let u E L;(B) and set
{
U (e < Ixl < 1 - c)
UE = 0 (otherwise).
ThenUEconverges toU inL;(B) as e-+ O. We use a mollifierJ8 *UE EX, which is a convolution of J8 and uE • Here J8(X) = 5-nJ(xlo) with
°<0< e and J satisfies
J E cgo(]Rn), supp J c B, J 2: 0, JJRnrJ(x)dx = 1.
Since J8*UE converges toUEin L;(B) as0-+0, the space X, and D(A) also, is dense inL;(B). Therefore the adjoint operatorA* is well-defined.
We show that A is self-adjoint. For u, v E D(A), the relation (Au,vh,O" = - istluvdx = (u,Avh,O",
implies that A is a symmetric operator, Le., A c A*. Therefore we have only to prove D(A*) C D(A). Let v E D(A*). Then there exists a w E L;(B) such that
(Au, vh,O"= (u, whO" for uE D(A), or equivalently
(A.3) - istluvdx = isuwlxlO"dx for u E D(A).
Let B8= {x E ]Rn : 0< Ixl < I} for 0 < 0< 1. Relation (A.3) for any u E Co(B8 ) c D(A) shows
-~v = IxlO"w a.e. onB8.
Since °< 0 < 1 is arbitrarily, this relation is valid for a.e. on B, and Av= -Ixl-O"tlv = wE L;.
We show that v E HJ(B). We set uE = (1+eA)-Iv for e > 0. This is well-defined because -lie is in the resolvent set of A by (A.1) and (A.2). It holds that
(AA) UE E D(A), IluEI12,0":::::; IlvlkO",
and Uc: converges to v in L;(B) as c: -+ 0+. Substituting u = Uc: into (A.3), we get
(A.5) - isb..uc:vdx =isuc:wlxl<Tdx.
The definition ofUc: gives
Uc: - c:!xl-<Tb..uc: = v.
We multiply both sides by -b..uc: to get -b..uc:uc: ::; -b..uc:v .
SinceUc: E D(A) C H2(B) nHJ(B), the integration by parts yields
Iluc:II~I(E) == { l\7uc:12dx ::; - ( b..uc:vdx ,
o lE lE
which together with (A.5) and (AA) shows
Iluc:II~J(E) ::; (isu;lxl<Tdxy/2 (Lw2lxl<Tdxy/2 ::; IIvlb,<TllwI12,<T.
Therefore there exists a sequence {ck} convergent to 0 such that {uc:J has a weak limit in HJ(B). Since Uc: converges to v in L;(B), v is in HJ(B). Consequently, v E D(A). This completes the proof. 0
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Nagasaki Institute of Applied Science 536 Aba-machi
Nagasaki 851-0193 Japan
E-mail: [email protected]
