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A bifurcation problem for a quasi-linear elliptic boundary value problem
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V
M. NAKAO
252
A, . g is a smooth (C’- class is sufficient), positive and bounded function on [O, to), satisfying 2g’(p2)p -I- g(p’) > 0. We assume g(0) = 1 without loss of generality. In order to get a preciser result (for the case N = 1) we shall make a further assumption. Bifurcation problems of positive solutions for semilinear elliptic equations have been considered by many authors (cf. Lions [5], Nishiura [6], Keller and Cohen [4] etc.). But, there seems to be very little result for quasi-linear equations (see Kaper and Man Kam Kwong (31, Concus and Finn [l], Serrin [8] etc. for related topics). 2. A GENERAL
RESULT
FOR
THE
CASE
Nr
1
Let I, is the least eigenvalue of - A and &,(x) be the positive normalized (I/&, llLz = 1) eigenfunction, i.e. - A&, =
in Q
A,&,
(2.1)
&Ian = 0. We introduce,
for (Y> 0, classical Banach spaces c,‘+*@> = (U E C2f”(Q)lulan c;;“(Q
and
= 0)
= (U E c;+“(Q)l(U, $0) = 0)
equipped with the usual norm, where ( , ) denotes the L2 inner product. Similar notation will be used freely if necessary. Our first result is the following. THEOREM2.1. (Local existence and uniqueness.) Q(s), u(s)) belonging to C’((-s,,
There exists a,, > 0 and a pair of functions
6,)) X C2((-s,,
6,) : C,““@>>,
satisfying = A(s)u(s),
(2.2)
u(s) = GO0 + w(s))
(2.3)
- v]g(lv~(s)lz)v~(~)l 1(O) = 10
and
for - 6, < s < 6,) where w(s) is a function in C2(( - &, 6,) : C$a(n)) w(0) = 0 Moreover, any solution I(A(s)*~(s))I-60
and
(A, u,) of the problem
w(-s)
= w(s).
(lS)-(1.6)
such that (2.4)
near (&,, 0) is on the branch
Proof. The proof is essentially included in Rabinowitz [7]. However, we shall give a slightly simpler proof of the main part by a direct use of the implicit function theorem. Set F(s, w) = s-l] - V(g(lVu J2)VU)- AU) =
-V(g(lVu12)V(~,+ w)> - UC4 + w)
(2.5)
Quasi-linear
elliptic boundary
253
value problem
with U = s(& + w)
and
I = &s, w) = iR
g(lV42)V(&l
+ w)V&l dx.
(2.6)
Then, it is immediately seen that I(0, 0) = A,
and
F(0, 0) = 0.
(2.7)
Further we have FE C2(R x C,$+m(sl)).
(2.8)
Indeed, we see F(s + h, w + W) - F(s, w) = D,F(s,
w)W + D,F(s,
w)h + o (h + I(WIIcz+a),
where D,F(s, w)h = -V(2g’(s21V(& -2
in
+ w)12)slV(+, + w)t’V(&, + w)Jh
g’(lv42m#b + W4J,dxh
(2.9)
and D,F(s,
w)W = - V (2g’(lVu12)?(V(&, + w) * VW)V(&
-v(g(lvUIz)vW) -2
+ w))
- Ilw
g’(lVL412)s2(V(4, + w) * vw(w$,
+ w) * V&J) d.e#Jo + w)
\ .n -
~ng(lv42w--bo~ i
for s, h E R and w, WE Cj,‘“(fi). From this we see easily (2.8). In particular, D,,F(O,O)W=
-AW-
(2.10)
we have by (2.10)
I,Wi .R
= -AW
g(O)V WV&l d-x
(g(O) = 1)
(2.11)
- &W.
Hence, by a standard theory of elliptic equations, D,F(O, 0) is an isomorphism from Ci,‘a(fi) to C,4(n) for any 0 < cx < 1. Thus, applying the implicit function theorem we conclude that there exists a0 > 0 and w E C2(( -a,,, 6,); CiJU(&3)) such that w(0) = 0
and
F(s, w(s)) = 0
for -6,
< s < 6,.
Defining U(S) and A(s) = A(s, w(s)) through (2.6) we obtain the desired set of solutions (4s), NJ. Moreover, we can show that there exists a neighbourhood (2, - E,, ,I, + E,,) x U of (A,,, 0) in R x Ciofa(S1) such that for each A E (A0 - E,,) the problem (2.5)-(2.6) admits a unique positive solution ux E U. This fact is proved quite similarly as in Rabiniwitz [7] and omitted. From this local uniqueness and the fact that A(s) = A( --s) we see - U( --s) = U(S) and consequently w( --s) = w(s). A(s) has the following convexity property (cf. [2]).
M.
254
PROPOSITION 2.1. Let (A(S), U(S)), -6, A’(0) = 0
< s < a,, be the solution
in theorem
2.1. Then,
we have
and
(Note that 2g’(O) = - 1 < 0 for the mean Proof.
NAKAO
curvature
equation.)
A(s) is given by n
A(s) =
I g(~*lw#%+ w(ml*)wo +
w)V&~.
,fl
Therefore,
(2.13)
which implies n
A’(0) =
$
I
Vw(s)V&
dx = 0.
.R
Moreover,
by (2.13), A”(S) = 2
g’(ln412)lV(&
1 .fi
+
w)12V(4, + w). Vdkl~
(2.14)
and hence A”(0) = 2
I .Q
g’(O)lVq&)I4dx +
, !
*fl
a* g(O)V 2 w(s) . v4, d.x
1
.=
!
2gf(o), R I~+, 14dx + g(o)
= 2g’(O)
!’
(VI& I4dx.
$ .fi[
VW(S) * V& dx
w
.fl
Concerning
the global
structure
of the bifurcating
solutions
we have the following.
THEOREM 2.2. Assume that the condition A, is satisfied and let e +(-) be the maximal consolutions of (1.5)-(1.6) containing (I,(S), U(S)), tinuum in R x C*+” (a) of the nontrivial O
Quasi-linear elliptic boundary value problem
255
and s+(-)
= (U E c2+*(n)l(A, U) E ,+(-)].
Then, it holds that (i) A C (0, A0 ;zp, g(h% (ii) S+(S-) (lS)-(1.6).
is unbounded
in C2’“(@
and consists of positive
(negative) solutions
of
Proof. It is clear that (.?- = ((A, u)j(A, -u) E tL?‘J and S- = (u] -U E S+]. For the proof we _ rely on Rabinowitz [7]. Suppose that (3 + were bounded in R x C”“(Q) and
Ilullcz+Y5 M
if (A,u) E (3
with some M > 0. Then, considering a smooth positive function gM on 10, 00) such that if IpI 5 2M gM(P2)
=
if Ipl 2 2M + 1,
(A, U) E (9 satisfies the modified problem - v{g,(pu12)vuJ
= Au
ulan = 0.
i
(2.15)
Since (2.15) is uniformly elliptic equation we can apply Rabinowitz [7] to see that (2.15) has a unbounded maximal continuum C,$ ((3,) in R x C”m(n), hence in R x C2’*(0), of nontrivial solutions bifurcating from &, 0). Moreover, it is shown in [7] that e$ consists of positive solutions. Since (3 + C C,$ by (2.15) and gM((Vul’) = g(lVu12) as long as Iu] I 2Mit must follow that A is unbounded and S’ = S,$ = ]u/(A, U) E CL]. However, since S’ = S,& consists of positive solutions, we have for any (A, ux) E e + , m
1
A = min BE& .a 5 SuPgoT) 720
=
IO
g(lV~,12)lv12~mllt~
i .n
I v4J12 d-dldl:~
sup g(g) < a. 720
This is a contradiction. Thus, S+ must be unbounded in C2+a (a). Moreover, repeating almost the same argument as in the above we see that S+ consists of positive solutions and also 0 < A 5 A, sup g(q) if 1 E A. W n>O Remark.
It is easy to see that if g(p’)
< g(0) = 1 for p > 0, then A C (0, Ao).
M.
256 3. GLOBAL
STRUCTURE
OF
THE
NAKAO
BRANCH
OF N=
In this section
we consider
POSITIVE
SOLUTIONS
FOR
THE
CASE
I
the one dimensional
equation
--j-(g(~~~z)~)=i~
on-l
i U(- 1) = U(1) = 0. We make the following assumptions AZ. A,. g E C’([O, CD)), d/dp[g(p2)p) > 0 on [0, co) and
lim g(p’)p = + a0 (possibily a, = a). p- *cO Let us denote the inverse function of the mapping: p + g(p2)p byf, which is strictly increaslim f(u) = fm. By setting u = g(ju’I’)u’, i.e. U’ =f(u) the ing function on (-a,,~,) and v* *a” problem (3.1) is equivalent to 2 -
-$
u
=
Af(v)
(3.2)
i
(uyThe equivalence
relation
1) = v’(1) = 0.
is given by 1X li= I
To solve (3.2) we consider
-1
(3.3)
f(v (x)) dx*
the initial value problem -
$
u = Af(u)
on -1
1 (3.4)
i
(
and
u(-l)=a>O
t”( - 1) = 0
for 0 < a < a,. Note that by (3.4) we have + S(u) where we set
= const.
sL’ F(u) = , ,f(v) \
It is easy to see that (3.1) admits x0 E (- 1, 1) such that u = g(lu’I’)u’
= H(a)
drl.
a positive solution u if and only if there satisfies (3.2) (see also (3.3)) and
u(x) > 0
exists unique
for - 1 < x < x0,
u(xo) = 0, u(x) < 0
(3.5)
(3.6) forx,
1.
257
Quasi-linear elliptic boundary value problem
Using (3.5) and (3.6), u = l?, f(v(x)) dx is a positive solution of (3.1) if and only if nil
dv
‘I
= AI(x
U(X) JR@ - F(u) “a dv
G(u(x)) =
1o
JF(4
0
-
=%5(x,+
1)
if -1
(3.7)
F(v)
dv
k i v(X)JF(a)
+ 1)
= JZ(x
- x0)
ifx,
1
- F(u)
and the additional condition u’(1) = 0 hold. Since v’(l) = 0 is equivalent to F(u(1)) = F(a) by (3.5), that is, to U(1) = -a we see from (3.7) and (3.8) 0
o JF(a)
a < a, such that a
Ha) =
dv
i
c o JW4
-
= JZ,
and in this case u(x) is determined by (3.7) with x0 = 0 and u(x) = I”_1f(v(x))
PROPOSITION
(3.9)
F(v)
dx. We claim:
3.1. 4(a) is continuous function on (0, a,) and lim 4(a) = Jg(O)/2n
= n/42.
a++0
(3.10)
Proof. The former assertion is almost trivial and the proof is omitted.
Since A, = 7rL/4 in our situation (3.10) follows from (3.9) and theorem 2.1 (see (2.3)). But, we can prove it directly as follows. Setting F(r)/F(a) = x we see JxFW du * ‘(xF(a))
‘(a)= .i:x&h h f(F-
(3.11)
We shall show that dxF(a) O-of(F-l(xF(a)) -
lim
=
g(0) . 2 umformly in x E (0, 1). J-
(3.12)
M. NAKAO
258 For this it suffices
to show
l/v5 which is proved
as f(O) = 1/2f ‘(0) = g(O)/2. = lim e-+0 2f(0)f ‘(0)
/ilof$ It follows
(3.13)
from (3.11) and (3.12) that
o-+0 lim
$0)
Now let us make a further
“1
-1
Jg(0)/2
=
1
du = Jg(o)/2
ON
assumption
TC. n
on g.
A,. 2f’ (WW where we recall that f is the inverse
LEMMA 3.1. The assumption
> f’(u) function
A, is satisfied
on 0 < a < a,, of k(p)
and F(u) = 10” f(q)
= g(p2)p
dq.
if
A;. f”(U) > 0 In particular, Proof.
if g(x’)
= l/J-
the condition
Y(u) = 2f ‘(u)F(u)
Setting
onO~u
= u/d-i-?,
= 3~(1 - u~)-~'~ > 0 if 0 < u < 1.
PROPOSITION 3.2. Under the assumptions decreasing
on (0, uo) and we can define G
Proof. Sincefand that
Fare
decreasing.
= ;ilo@(a)
strictly increasing
n @(a) defined
on (0, q,) it suffices
by the formula
0 < a < a,,
the above
- 2F(u)f ‘WI/f 3(u) < 0
arguments
we arrive at the following
in (3.9) is strictly
(3.14)
= o Jim ““HO).
But, w’(a) = if’(u)
Summarizing
O
A2 and A,, the function ,I* by
w(a) = F(u)/f’(u), is strictly
on (0, uo) by A;,
Y(u) > 0, i.e. A3. = l/m it is easy to see that a, = 1 and f(u)
Thus, f”(u)
holds.
- f2(u), we see Y(0) = 0 and
Y’(u) = 2f “(u)F(u) > 0 which implies When g(g)
and hence A,,
by A,. theorem.
n
(3.11) to prove
Quasi-linear
elliptic boundary
259
value problem
3.1. Under the assumptions A2 and A,, the problem (3.1) has a unique positive solution ux for each J E (A*, &,), 13, = n*/4, and the set (u, Jx*
THEOREM
Fig. 1.
Finally we give a condition which assures I* > 0. 3.3. If lim a/a (1.3)-(1.4) we have a--rao
> 0, then we have I* > 0. In particular,
PROPOSITION
for the problem
57//z -de
(3.15)
> 0.
s0 Proof. Under the assumption we see i0 1 dv lim 4(a) = lim ll-il0 a+(10JJ0 F(a) - F(u) (I
rlim
When g(p*)p = p/m
a-a0 i 0 we see a, = F(u)
=
Hence, @(a) =
-
dv = hm a/JF(a) o-(10
l/JF(;;r 1, f(x)
“f(q)dq i0
= x/m =
LI l/J-
1 -
> 0.
and m.
- &--?dv.
i0
Thus, we have */2
1
(1 - v2)-1’4dU = lim &a) = 0-l s 0 i0 Example with I* = 0. Let f(x) = x/(1 - g)*, - 1 < x off. Define g(q) by g(V) = k(G)/\l;j
ifq > 0,
-de. <
1,
g(0) = 1
Then, g is a smooth function on [0, 00) and for this we have F(u)=
%(A-1)
n
and k(p) be the inverse function ifq = 0.
(3.16)
260
M. NAKAO
and
-a @J(a)= 42 d-i-2
m/md,+O
as a -) 1.
! .0 This is a simple example with A* = 0. Finally, using theorem 3.1 we shall determine Our result reads as follows. THEOREM
positive
the structure
of all nontrivial
solutions
of (3.1).
3.2. Assume A, and A,. Let n21* < A < n2Ao, n = 1, 2, 3, ... , and let uX,,,z be the solution of the problem (3.1) with ,I replaced by I./n’. Define u,,~ on (- 1, 1) by uX,nz(nx + n - 2i +x(x)
1)
5 i I n -
ifiisevenando
1, (3.17)
=
UX,n2( - nx - n + 2i + 1)
ifiisoddand
1 5 i 5 n
for - 1 < x < 1 with - 1 + 2i/n < x < - 1 + 2(i + 1)/n. Then, u,,,(x) is a nontrivial solution of (3.1) with n - 1 zeros on (- 1, 1). Conversely, if (A, U) is a nontrivial solution of (3.1), then it must hold that n2A* < A < n2Ao for some n > 1 and U = fU, x. Proof. It is clear from the definition of ux (see (3.3) and (3.7)) that u,,~ is a solution of the problem (3.1) with n - 1 zeros in (- 1, 1). Suppose that (,I, u) is a nontrivial solution of (3.1). Then, there exist - 1 = x0 < x, < x2 < . . . < x, = 1 such that U(Xj) = 0 Of course,
we assume
and
u(x) +V0 if x f xi. Further, and hence
u’(x0) > 0
i = 0, 1, ... n.
u’(x;) + 0,
we may assume
u(x) > 0
on x0 < x < x, .
We shall show that i = 1, 2, ..., n.
x, - xi-1 = x, - x0 = 2/n, Indeed,
setting
(3.18)
g( lu’12)u’ = u we have - v”(x) = Af(u(x))
onx,
V(X,) = g((u’(From
this we conclude
Since u should
satisfy
1)12)u’(-
and
1) = a > 0
(see (3.4) and (3.7)) that v(x,) -a
1
L\ 0 JF(a)
- F(v)
and
du = (x, - x,,m.
-
v’(x) = Mu(x)),
u(x,) = -a
= -a
U’(Xo) = u’(x,) = 0.
and
x,
u’(xJ = u’(x2) = 0,
(3.20)
Quasi-linear
elliptic boundary
261
value problem
we have again as in (3.4) and (3.7) that u(x,) = CIand -lI
!
1
0 v’F(a) - F(v)
(3.21)
dv = (xz - x,)v%%
(Note that there is nothing to prove if n = 1 and we assume n 5: 2.) It follows from (3.20) and (3.21) that x2 - x1 = x, - x0. Repeating this argument we get (3.18) and consequently u(x) coincides with unVxdefined by (3.17). n Set and e,
= [(A, -&JA*
< A < A,).
Then, by theorem 3.2 (?,‘((I?,) is the maxima1 continuum of the nontrivial solutions which bifurcates from (n21,, 0). COROLLARY 3.1.Let k 5 0 be any integer and take 3Lsuch that n’/E* -C A < (n - k)‘A,, which is possible if n is sufficiently large. Then, the problem (3.1) admits k + 1 pairs of nontrivial solutions f u,,~~ m = n - k, n - k + 1, . . . . n. In particular, if A* = 0 the problem has infinitely many pairs of nontrivial solutions f u,, h (n > [a]) for any A > 0. (See Fig. 2.)
Acknowle&en?enr-The
author
would like to thank Prof.
J. Serrin for useful comments
on the manuscript.
REFERENCES 1. CONCUS P. & FINAN R., On capillary free surfaces in the absence of gravity, 2. CRANDALL M. G. & RAB~NOWITZ P. H., Bifurcation, perturbation of simple
Arch. RationalMeck.
Anal. 43, 161-180 (1973).
Acta Mark. eigenvalues
f32,
177-198
and linearized
(1974).
stability,
262
M. NAKAO
3. KAPER H. G. & MAN KAM KWONG, Uniqueness results for some nonlinear initial and boundary value problems, Arch. Rational Mech. Anal. 102, 45-56 (1988). 4. KELLER H. B. & COHEN D. S., Some positone problems suggested by nonlinear heat generatior, .I. Morh. Mech. 16, 1361-1376 (1967). 5. LIONS P. L., On the existence of positive solutions of semilinear elliptic equations, SIAM Rev. 24, 441-467 (1982). 6. NISHIURA Y ., Global structure of bifurcating solutions of some reaction diffusion systems, SIAM J. Morh. Analysis 13, 555-593 (1982). 7. RABINOWITZ P. H., Some global results for nonlinear eigenvalue problems, J. Functional And. 7, 487-513 (1971). 8. SERRIN J., Positive solution of a prescribed mean curvature problem, Lecrure Nores in Mofhemorics 1340, 248-255. Springer, Berlin (1988).
7.Ju!
IS2
:uo!1~puo3 %U!MOlIO3aq1 %U!~JS!leS UO!lXIn31?Sl8 aJaqM
() = uepl ny = I~L~(,~~A/)~A-I uoymba (‘I
‘C
‘JJaJoWl
[Waua8 aJOm e 1eaJ1[[eqs.3M MoIW
I
‘%I
=S)
‘,@P/4’)
1x2~ +
UI I/VI
&!~oau![uou aql Kq paurJo3aps! ‘fl/zp- JoleJado aq1 30 uoysun3ua%!a 1~3 aql Zyaq O@‘uraIqoJdJeauq aql30([~~]‘~y)asedsuo!lsun3ua8!a aqlMOM s~oqs(I'~u.~aJoaql)l~nsaJ Jng *QaspaJd tUa[qoJdaql adiosuw a~ asw s!q1u! pue uoyenba ~!UI~JCIJJ!~ AJWI~JO ue s! sy~,
‘0
(VI)
(E'I)
‘(I‘I-) U!
ny=
=
(1)n= (I-)n
xp,(xp/nP) + rp xp -I ( nP > P
1
1
uogenba [euo!suauupauo aql 01 saA[asJno ~~!JIS~J sn ia? *llnsaJ ~CJO~X~S!~~?S 1souqt2 UE ia%ue3 aM ‘JaAaMoq ‘ase3p2uo!suau.up au0 JOB *paJ!sap s!uoy@isaAu! JaylJn3pue qsueJqaq13oaJn1~nJ1saq1uoaJowlCueMOUY IOU opaMau1ys~q11~*(0‘~y)Jeau q3ueJqaq130 &JadOJd bl!XaAUO3E MOljS OSIB IIt?l+-aM ‘[L] Zl!MOU!qE~lUOJ3h[p.?llUaSSa SMO[[O31X3 S!qL 'VJoleJadoaq130 ayeAua8!aisea aq1s!Oy aJaqM ‘(~‘Oy)uoyrqosp+~!Jlaq1uroq saieaJn3Iq q3iqM suoytqos(aA!lek?au ~o)aA!l!sOd30 (h ‘y)q~JeJqleqoI830a3ua1yxaaql aAoJd IIeqsaM ‘1s~g .pa!3yles ale suoyenba aq131 (z-I)-(I'I)30 uoyrqos e n se IIaMSE (n ‘7)J!ede (]'e3 IIEqSaM *Ja1aUJeJEd E S!y PUE OQ iiJE?pUnOq q1OOuISe ql!MNxU! U~""Op PapUnOq )?S!0 aJaqM 0 = ueln
(z-1) (1'1)
uoysnba aJnlBAJn3UE?aW aql~o3 uJa[qoJdanIt?A LJepunoq aq101 suoynIos(aAy2%u~o)aA~l!sOd3oa2ua1yxaaqiy pa1saJalu!aJIeaMa1ou SIHlNI N0113I-ICIOXlNI
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‘I
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V
M. NAKAO
252
A, . g is a smooth (C’- class is sufficient), positive and bounded function on [O, to), satisfying 2g’(p2)p -I- g(p’) > 0. We assume g(0) = 1 without loss of generality. In order to get a preciser result (for the case N = 1) we shall make a further assumption. Bifurcation problems of positive solutions for semilinear elliptic equations have been considered by many authors (cf. Lions [5], Nishiura [6], Keller and Cohen [4] etc.). But, there seems to be very little result for quasi-linear equations (see Kaper and Man Kam Kwong (31, Concus and Finn [l], Serrin [8] etc. for related topics). 2. A GENERAL
RESULT
FOR
THE
CASE
Nr
1
Let I, is the least eigenvalue of - A and &,(x) be the positive normalized (I/&, llLz = 1) eigenfunction, i.e. - A&, =
in Q
A,&,
(2.1)
&Ian = 0. We introduce,
for (Y> 0, classical Banach spaces c,‘+*@> = (U E C2f”(Q)lulan c;;“(Q
and
= 0)
= (U E c;+“(Q)l(U, $0) = 0)
equipped with the usual norm, where ( , ) denotes the L2 inner product. Similar notation will be used freely if necessary. Our first result is the following. THEOREM2.1. (Local existence and uniqueness.) Q(s), u(s)) belonging to C’((-s,,
There exists a,, > 0 and a pair of functions
6,)) X C2((-s,,
6,) : C,““@>>,
satisfying = A(s)u(s),
(2.2)
u(s) = GO0 + w(s))
(2.3)
- v]g(lv~(s)lz)v~(~)l 1(O) = 10
and
for - 6, < s < 6,) where w(s) is a function in C2(( - &, 6,) : C$a(n)) w(0) = 0 Moreover, any solution I(A(s)*~(s))I-60
and
(A, u,) of the problem
w(-s)
= w(s).
(lS)-(1.6)
such that (2.4)
near (&,, 0) is on the branch
Proof. The proof is essentially included in Rabinowitz [7]. However, we shall give a slightly simpler proof of the main part by a direct use of the implicit function theorem. Set F(s, w) = s-l] - V(g(lVu J2)VU)- AU) =
-V(g(lVu12)V(~,+ w)> - UC4 + w)
(2.5)
Quasi-linear
elliptic boundary
253
value problem
with U = s(& + w)
and
I = &s, w) = iR
g(lV42)V(&l
+ w)V&l dx.
(2.6)
Then, it is immediately seen that I(0, 0) = A,
and
F(0, 0) = 0.
(2.7)
Further we have FE C2(R x C,$+m(sl)).
(2.8)
Indeed, we see F(s + h, w + W) - F(s, w) = D,F(s,
w)W + D,F(s,
w)h + o (h + I(WIIcz+a),
where D,F(s, w)h = -V(2g’(s21V(& -2
in
+ w)12)slV(+, + w)t’V(&, + w)Jh
g’(lv42m#b + W4J,dxh
(2.9)
and D,F(s,
w)W = - V (2g’(lVu12)?(V(&, + w) * VW)V(&
-v(g(lvUIz)vW) -2
+ w))
- Ilw
g’(lVL412)s2(V(4, + w) * vw(w$,
+ w) * V&J) d.e#Jo + w)
\ .n -
~ng(lv42w--bo~ i
for s, h E R and w, WE Cj,‘“(fi). From this we see easily (2.8). In particular, D,,F(O,O)W=
-AW-
(2.10)
we have by (2.10)
I,Wi .R
= -AW
g(O)V WV&l d-x
(g(O) = 1)
(2.11)
- &W.
Hence, by a standard theory of elliptic equations, D,F(O, 0) is an isomorphism from Ci,‘a(fi) to C,4(n) for any 0 < cx < 1. Thus, applying the implicit function theorem we conclude that there exists a0 > 0 and w E C2(( -a,,, 6,); CiJU(&3)) such that w(0) = 0
and
F(s, w(s)) = 0
for -6,
< s < 6,.
Defining U(S) and A(s) = A(s, w(s)) through (2.6) we obtain the desired set of solutions (4s), NJ. Moreover, we can show that there exists a neighbourhood (2, - E,, ,I, + E,,) x U of (A,,, 0) in R x Ciofa(S1) such that for each A E (A0 - E,,) the problem (2.5)-(2.6) admits a unique positive solution ux E U. This fact is proved quite similarly as in Rabiniwitz [7] and omitted. From this local uniqueness and the fact that A(s) = A( --s) we see - U( --s) = U(S) and consequently w( --s) = w(s). A(s) has the following convexity property (cf. [2]).
M.
254
PROPOSITION 2.1. Let (A(S), U(S)), -6, A’(0) = 0
< s < a,, be the solution
in theorem
2.1. Then,
we have
and
(Note that 2g’(O) = - 1 < 0 for the mean Proof.
NAKAO
curvature
equation.)
A(s) is given by n
A(s) =
I g(~*lw#%+ w(ml*)wo +
w)V&~.
,fl
Therefore,
(2.13)
which implies n
A’(0) =
$
I
Vw(s)V&
dx = 0.
.R
Moreover,
by (2.13), A”(S) = 2
g’(ln412)lV(&
1 .fi
+
w)12V(4, + w). Vdkl~
(2.14)
and hence A”(0) = 2
I .Q
g’(O)lVq&)I4dx +
, !
*fl
a* g(O)V 2 w(s) . v4, d.x
1
.=
!
2gf(o), R I~+, 14dx + g(o)
= 2g’(O)
!’
(VI& I4dx.
$ .fi[
VW(S) * V& dx
w
.fl
Concerning
the global
structure
of the bifurcating
solutions
we have the following.
THEOREM 2.2. Assume that the condition A, is satisfied and let e +(-) be the maximal consolutions of (1.5)-(1.6) containing (I,(S), U(S)), tinuum in R x C*+” (a) of the nontrivial O
Quasi-linear elliptic boundary value problem
255
and s+(-)
= (U E c2+*(n)l(A, U) E ,+(-)].
Then, it holds that (i) A C (0, A0 ;zp, g(h% (ii) S+(S-) (lS)-(1.6).
is unbounded
in C2’“(@
and consists of positive
(negative) solutions
of
Proof. It is clear that (.?- = ((A, u)j(A, -u) E tL?‘J and S- = (u] -U E S+]. For the proof we _ rely on Rabinowitz [7]. Suppose that (3 + were bounded in R x C”“(Q) and
Ilullcz+Y5 M
if (A,u) E (3
with some M > 0. Then, considering a smooth positive function gM on 10, 00) such that if IpI 5 2M gM(P2)
=
if Ipl 2 2M + 1,
(A, U) E (9 satisfies the modified problem - v{g,(pu12)vuJ
= Au
ulan = 0.
i
(2.15)
Since (2.15) is uniformly elliptic equation we can apply Rabinowitz [7] to see that (2.15) has a unbounded maximal continuum C,$ ((3,) in R x C”m(n), hence in R x C2’*(0), of nontrivial solutions bifurcating from &, 0). Moreover, it is shown in [7] that e$ consists of positive solutions. Since (3 + C C,$ by (2.15) and gM((Vul’) = g(lVu12) as long as Iu] I 2Mit must follow that A is unbounded and S’ = S,$ = ]u/(A, U) E CL]. However, since S’ = S,& consists of positive solutions, we have for any (A, ux) E e + , m
1
A = min BE& .a 5 SuPgoT) 720
=
IO
g(lV~,12)lv12~mllt~
i .n
I v4J12 d-dldl:~
sup g(g) < a. 720
This is a contradiction. Thus, S+ must be unbounded in C2+a (a). Moreover, repeating almost the same argument as in the above we see that S+ consists of positive solutions and also 0 < A 5 A, sup g(q) if 1 E A. W n>O Remark.
It is easy to see that if g(p’)
< g(0) = 1 for p > 0, then A C (0, Ao).
M.
256 3. GLOBAL
STRUCTURE
OF
THE
NAKAO
BRANCH
OF N=
In this section
we consider
POSITIVE
SOLUTIONS
FOR
THE
CASE
I
the one dimensional
equation
--j-(g(~~~z)~)=i~
on-l
i U(- 1) = U(1) = 0. We make the following assumptions AZ. A,. g E C’([O, CD)), d/dp[g(p2)p) > 0 on [0, co) and
lim g(p’)p = + a0 (possibily a, = a). p- *cO Let us denote the inverse function of the mapping: p + g(p2)p byf, which is strictly increaslim f(u) = fm. By setting u = g(ju’I’)u’, i.e. U’ =f(u) the ing function on (-a,,~,) and v* *a” problem (3.1) is equivalent to 2 -
-$
u
=
Af(v)
(3.2)
i
(uyThe equivalence
relation
1) = v’(1) = 0.
is given by 1X li= I
To solve (3.2) we consider
-1
(3.3)
f(v (x)) dx*
the initial value problem -
$
u = Af(u)
on -1
1 (3.4)
i
(
and
u(-l)=a>O
t”( - 1) = 0
for 0 < a < a,. Note that by (3.4) we have + S(u) where we set
= const.
sL’ F(u) = , ,f(v) \
It is easy to see that (3.1) admits x0 E (- 1, 1) such that u = g(lu’I’)u’
= H(a)
drl.
a positive solution u if and only if there satisfies (3.2) (see also (3.3)) and
u(x) > 0
exists unique
for - 1 < x < x0,
u(xo) = 0, u(x) < 0
(3.5)
(3.6) forx,
1.
257
Quasi-linear elliptic boundary value problem
Using (3.5) and (3.6), u = l?, f(v(x)) dx is a positive solution of (3.1) if and only if nil
dv
‘I
= AI(x
U(X) JR@ - F(u) “a dv
G(u(x)) =
1o
JF(4
0
-
=%5(x,+
1)
if -1
(3.7)
F(v)
dv
k i v(X)JF(a)
+ 1)
= JZ(x
- x0)
ifx,
1
- F(u)
and the additional condition u’(1) = 0 hold. Since v’(l) = 0 is equivalent to F(u(1)) = F(a) by (3.5), that is, to U(1) = -a we see from (3.7) and (3.8) 0
o JF(a)
a < a, such that a
Ha) =
dv
i
c o JW4
-
= JZ,
and in this case u(x) is determined by (3.7) with x0 = 0 and u(x) = I”_1f(v(x))
PROPOSITION
(3.9)
F(v)
dx. We claim:
3.1. 4(a) is continuous function on (0, a,) and lim 4(a) = Jg(O)/2n
= n/42.
a++0
(3.10)
Proof. The former assertion is almost trivial and the proof is omitted.
Since A, = 7rL/4 in our situation (3.10) follows from (3.9) and theorem 2.1 (see (2.3)). But, we can prove it directly as follows. Setting F(r)/F(a) = x we see JxFW du * ‘(xF(a))
‘(a)= .i:x&h h f(F-
(3.11)
We shall show that dxF(a) O-of(F-l(xF(a)) -
lim
=
g(0) . 2 umformly in x E (0, 1). J-
(3.12)
M. NAKAO
258 For this it suffices
to show
l/v5 which is proved
as f(O) = 1/2f ‘(0) = g(O)/2. = lim e-+0 2f(0)f ‘(0)
/ilof$ It follows
(3.13)
from (3.11) and (3.12) that
o-+0 lim
$0)
Now let us make a further
“1
-1
Jg(0)/2
=
1
du = Jg(o)/2
ON
assumption
TC. n
on g.
A,. 2f’ (WW where we recall that f is the inverse
LEMMA 3.1. The assumption
> f’(u) function
A, is satisfied
on 0 < a < a,, of k(p)
and F(u) = 10” f(q)
= g(p2)p
dq.
if
A;. f”(U) > 0 In particular, Proof.
if g(x’)
= l/J-
the condition
Y(u) = 2f ‘(u)F(u)
Setting
onO~u
= u/d-i-?,
= 3~(1 - u~)-~'~ > 0 if 0 < u < 1.
PROPOSITION 3.2. Under the assumptions decreasing
on (0, uo) and we can define G
Proof. Sincefand that
Fare
decreasing.
= ;ilo@(a)
strictly increasing
n @(a) defined
on (0, q,) it suffices
by the formula
0 < a < a,,
the above
- 2F(u)f ‘WI/f 3(u) < 0
arguments
we arrive at the following
in (3.9) is strictly
(3.14)
= o Jim ““HO).
But, w’(a) = if’(u)
Summarizing
O
A2 and A,, the function ,I* by
w(a) = F(u)/f’(u), is strictly
on (0, uo) by A;,
Y(u) > 0, i.e. A3. = l/m it is easy to see that a, = 1 and f(u)
Thus, f”(u)
holds.
- f2(u), we see Y(0) = 0 and
Y’(u) = 2f “(u)F(u) > 0 which implies When g(g)
and hence A,,
by A,. theorem.
n
(3.11) to prove
Quasi-linear
elliptic boundary
259
value problem
3.1. Under the assumptions A2 and A,, the problem (3.1) has a unique positive solution ux for each J E (A*, &,), 13, = n*/4, and the set (u, Jx*
THEOREM
Fig. 1.
Finally we give a condition which assures I* > 0. 3.3. If lim a/a (1.3)-(1.4) we have a--rao
> 0, then we have I* > 0. In particular,
PROPOSITION
for the problem
57//z -de
(3.15)
> 0.
s0 Proof. Under the assumption we see i0 1 dv lim 4(a) = lim ll-il0 a+(10JJ0 F(a) - F(u) (I
rlim
When g(p*)p = p/m
a-a0 i 0 we see a, = F(u)
=
Hence, @(a) =
-
dv = hm a/JF(a) o-(10
l/JF(;;r 1, f(x)
“f(q)dq i0
= x/m =
LI l/J-
1 -
> 0.
and m.
- &--?dv.
i0
Thus, we have */2
1
(1 - v2)-1’4dU = lim &a) = 0-l s 0 i0 Example with I* = 0. Let f(x) = x/(1 - g)*, - 1 < x off. Define g(q) by g(V) = k(G)/\l;j
ifq > 0,
-de. <
1,
g(0) = 1
Then, g is a smooth function on [0, 00) and for this we have F(u)=
%(A-1)
n
and k(p) be the inverse function ifq = 0.
(3.16)
260
M. NAKAO
and
-a @J(a)= 42 d-i-2
m/md,+O
as a -) 1.
! .0 This is a simple example with A* = 0. Finally, using theorem 3.1 we shall determine Our result reads as follows. THEOREM
positive
the structure
of all nontrivial
solutions
of (3.1).
3.2. Assume A, and A,. Let n21* < A < n2Ao, n = 1, 2, 3, ... , and let uX,,,z be the solution of the problem (3.1) with ,I replaced by I./n’. Define u,,~ on (- 1, 1) by uX,nz(nx + n - 2i +x(x)
1)
5 i I n -
ifiisevenando
1, (3.17)
=
UX,n2( - nx - n + 2i + 1)
ifiisoddand
1 5 i 5 n
for - 1 < x < 1 with - 1 + 2i/n < x < - 1 + 2(i + 1)/n. Then, u,,,(x) is a nontrivial solution of (3.1) with n - 1 zeros on (- 1, 1). Conversely, if (A, U) is a nontrivial solution of (3.1), then it must hold that n2A* < A < n2Ao for some n > 1 and U = fU, x. Proof. It is clear from the definition of ux (see (3.3) and (3.7)) that u,,~ is a solution of the problem (3.1) with n - 1 zeros in (- 1, 1). Suppose that (,I, u) is a nontrivial solution of (3.1). Then, there exist - 1 = x0 < x, < x2 < . . . < x, = 1 such that U(Xj) = 0 Of course,
we assume
and
u(x) +V0 if x f xi. Further, and hence
u’(x0) > 0
i = 0, 1, ... n.
u’(x;) + 0,
we may assume
u(x) > 0
on x0 < x < x, .
We shall show that i = 1, 2, ..., n.
x, - xi-1 = x, - x0 = 2/n, Indeed,
setting
(3.18)
g( lu’12)u’ = u we have - v”(x) = Af(u(x))
onx,
V(X,) = g((u’(From
this we conclude
Since u should
satisfy
1)12)u’(-
and
1) = a > 0
(see (3.4) and (3.7)) that v(x,) -a
1
L\ 0 JF(a)
- F(v)
and
du = (x, - x,,m.
-
v’(x) = Mu(x)),
u(x,) = -a
= -a
U’(Xo) = u’(x,) = 0.
and
x,
u’(xJ = u’(x2) = 0,
(3.20)
Quasi-linear
elliptic boundary
261
value problem
we have again as in (3.4) and (3.7) that u(x,) = CIand -lI
!
1
0 v’F(a) - F(v)
(3.21)
dv = (xz - x,)v%%
(Note that there is nothing to prove if n = 1 and we assume n 5: 2.) It follows from (3.20) and (3.21) that x2 - x1 = x, - x0. Repeating this argument we get (3.18) and consequently u(x) coincides with unVxdefined by (3.17). n Set and e,
= [(A, -&JA*
< A < A,).
Then, by theorem 3.2 (?,‘((I?,) is the maxima1 continuum of the nontrivial solutions which bifurcates from (n21,, 0). COROLLARY 3.1.Let k 5 0 be any integer and take 3Lsuch that n’/E* -C A < (n - k)‘A,, which is possible if n is sufficiently large. Then, the problem (3.1) admits k + 1 pairs of nontrivial solutions f u,,~~ m = n - k, n - k + 1, . . . . n. In particular, if A* = 0 the problem has infinitely many pairs of nontrivial solutions f u,, h (n > [a]) for any A > 0. (See Fig. 2.)
Acknowle&en?enr-The
author
would like to thank Prof.
J. Serrin for useful comments
on the manuscript.
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Anal. 43, 161-180 (1973).
Acta Mark. eigenvalues
f32,
177-198
and linearized
(1974).
stability,
262
M. NAKAO
3. KAPER H. G. & MAN KAM KWONG, Uniqueness results for some nonlinear initial and boundary value problems, Arch. Rational Mech. Anal. 102, 45-56 (1988). 4. KELLER H. B. & COHEN D. S., Some positone problems suggested by nonlinear heat generatior, .I. Morh. Mech. 16, 1361-1376 (1967). 5. LIONS P. L., On the existence of positive solutions of semilinear elliptic equations, SIAM Rev. 24, 441-467 (1982). 6. NISHIURA Y ., Global structure of bifurcating solutions of some reaction diffusion systems, SIAM J. Morh. Analysis 13, 555-593 (1982). 7. RABINOWITZ P. H., Some global results for nonlinear eigenvalue problems, J. Functional And. 7, 487-513 (1971). 8. SERRIN J., Positive solution of a prescribed mean curvature problem, Lecrure Nores in Mofhemorics 1340, 248-255. Springer, Berlin (1988).