Singular nonlinear equations on an unbounded interval

JOURNAL

OF MATHEMATICAL

ANALYSIS

AND

Singular Nonlinear on an Unbounded S. N. CHOW* Department

of Mathematics,

Michigan

77, 642-653 (1980)

APPLICATIONS

Equations Interval

AND J. C. KURTZ State University,

East Lansing, Michigan

48824

Submitted by J.P. LaSalle

Variational methods and the critical point theory of Lusternik and Schnirelman are used to prove two abstract theorems. These theorems are utilized to obtain the existence of multiple solutions for a class of nonlinear, singular second order equations on (0, co).

1. INTRODUCTION A number of authors have considered the nonlinear equation --Au + u = N(u)

(l-1)

on IR”. Since the Sobolev embeddings are not compact in this case, primary attention has been focused on the class of radial functions u = U(T), where r = 1x1(see, for example, Berger and Schechter [ 11, Strauss [ 81). The radial functions are sufficiently small at infinity so that the Sobolev embeddings are again compact. For radial functions Eq. (1.1) leads to the singular ODE -r l-n(r”-‘~,),

+ 24= N(u),

r E (0, co).

(1.2)

This equation, with n = 3 and N(u) = uk, has been studied in detail by Nehari [6], Ryder [7], and others. In the case n = 2, Zakharov ef al. [lo] have done extensive computations for the equation -( l/r)(ru,),

+ 24= u3.

Defining the “energy functional” E(r) = +(u” - 224’+ 24) * This author’s research supported in part by NSF Grant MCS 76-06739. 0022-247X/80/100329-15$02.00/0 Copyright All rights

t 1980 by Academic Press, Inc. of reproduction in any form reserved.

642

(1.3)

SINGULAR

NONLINEAR

EQUATIONS

643

we see that any solution of (1.3) satisfies E’(r) = -uf/r, so that the solution trajectories in the phase plane have decreasing energy. The phase plane -analysis of Fig. 1 suggests the existence of a sequence of solutions {u,(r)}, Iz = 0, 1,2 ,..., each vanishing at infinity, with a”(r) having precisely n zeros in (0, co). This result has been proved by Kurtz [4], and the initial values u,(O) have been computed by Zakharov, et al. [lo] for 0 < n < 15. Motivated by (1.2), we are led to consider the equation -(llP)(PU’)’

+ 24= N(u),

f E (0, co>,

(1.4)

where p(O) = 0 and p(t)+ co monotonically as t + co. One of the main results of this paper is that, for fairly general p and N, Eq. (1.4) has infinitely many non-trivial solutions u,(t), each vanishing at infinity, and one of which is strictly positive. The existence of this positive solution is of particular importance. Indeed, the technique first introduced by Nehari [5] can be used on equations of type (1.4) to show that the existence of a positive solution on an arbitrary subinterval of (0, co) implies the existence of a solution on (0, co) having any prescribed number of zeros (see also Kurtz [4] and Ryder [7]). In Section 2 we prove two abstract theorems which are used in Section 3 to prove the existence of infinitely many distinct solutions of (1.4), one of which is strictly positive. In Section 4 we apply our techniques to a nonsingular fourth order equation of the type considered by Wong [9]. Section 5 contains some technical lemmas and the proofs of the main theorems.

FIGURE

1

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CHOW

AND

KURT2

2. THE MAIN THEOREMS

Let E be a (real) reflexive Banach space. Suppose a:ExE+IR is a symmetric, bounded bilinear form for which there exists C > 0 such that

4~Jd>a412 for all u E E. Let g be a C* functional

(2.1)

on E for which

g: E + IR is w-continuous, (g’(

),

(2.2)

): E + R is w-continuous, g(0) = 0,

g”(u)@, u) - (g’(u), u> > 0

(2.3) (2.4)

if

24# 0,

(2.5)

and (2.6) for each u E E. We define

J(u)= 44 u>- g(u), I(u)= a(u,u)- f( g’(u),u), M= THEOREM

1.

(2.7)

{uEEJu#O,I(u)=O}.

Suppose that, in addition to the above, J(u) > c > 0

forall

uE M

(2.8)

and {U,} GM,

II%zII+aJ“J&J-+ a*

(2.9)

Then there exists u E M for which J(U) = infueM J(v), and J’(u) = 0. Two important points should be made in regard to Theorem 1. Although neither of the functionals 1, J are weakly continuous, their difference is, and this fact is exploited in the proof. In this manner we can avoid the smoothing process used by Ryder [7] (Green’s function), whereby a strongly convergent minimizing sequence is extracted from a weakly convergent minimizing sequence. Secondly, the Lagrange mutiplier disappears; hence there is no

SINGULARNONLINEAREQUATIONS

645

need for a homogeneity assumption on the nonlinearity, scaling. Replacing (2.2) and (2.3) by the stronger hypothesis u, -

u

implies

and no need for

g’(u,) + g(u)

(2.10)

and (2.5) by the weaker hypothesis g”(u)@, u) - (g’(u), u) z 0

(2.11)

(u + 0)

we have THEOREM 2. If g is even, then there exist infinitely many distinct pairs of critical points { fu,} of J on M for which J’(uk) = 0.

3. SINGULAR EQUATIONS In this section we consider Eq. (1.4) where the nonlinear term has the form

N(u)= f(u')u

(3.1)

and f satisfies

f (0) = 0,

f E C’(O, a),

If @)I< c IsK scsf (s) is increasing

f@> > 0

for

s > 0,

O
(3.2) (3.3)

6 > 0,

(3.4)

sf'(s> < Cf(s).

(3.5)

g(u) = ja F(u2)p dt,

(3.6)

We define 0

where F(u) =

’ f(s) ds. I0

(3.7)

For the function p(t) we require p E C’(O, co), t* = O(p(t))

P(0) = 0, as

t+O+,

p(t) strictly increasing, p(t) + 00

as

t+m.

(3-g) (3.10)

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CHOWANDKURTZ

We define E = {u E D’(0, 00) 1p”‘u,

p”‘ri E L’(O, 00))

(3.10)

with the weighted Sobolev norm

1)u1)= 1“fm(22+ zqp dt/ li2.

(3.11)

0

THEOREM 3. Under the above hypotheses, Eq. (1.4) positive solution which vanishes at infinity.

has a strictly

THEOREM 4. Under the hypotheses of Theorem 3, there exist infinitely many distinct pairs of solutions {fuk} of Eq. (1.4).

4. A NONSINGULAR EQUATION The theorems of Section 2 may of course be applied to nonsingular equations as well. We consider the example ZP”) = f(u2)u,

tE [a,b],

u(a) = u(a) = u(b) = u(b) = 0.

(4-l) (4.2)

Equations of this type have been considered by Wong [9]. In this case we take E = @‘*‘(a, b), the usual Sobolev space. We define g(u) as in (3.6), (3.7) and set b

a(u, v) =

Icl 12’ dt

for u, v E E. By Poincare’s inequality we can consider E with the norm 11u/J = a(u, u)“‘. THEOREM 5.

Suppose f satisfies (3.2), (3.4), (3.5), and also

If@)1GKlsl”

for some

fs > 0.

(4.3)

Then there exists a positive solution of (4.1), (4.2). THEOREM 6. Under the hypotheses of Theorem 5, there exist infinitely many pairs of solutions (fu,} of (4. l), (4.2).

SINGULARNONLINEAREQUATIONS

647

5. PROOF OF THE THEOREMS

Proof of Theorem 1. Choose {u,} E M so that .Z(u,) t inf{.Z(v) 1v E M} = P > 0. BY (2.91, llqll < C, so there exists a subsequence, again call it {u,}, with u, - u. Since Z(u,) = 0 we have

By (2.2)-(2.4), noting that ,D> 0, we see that u # 0. Since Z is weakly lower semicontinuous we have Z(u) < lim Z(u,) = 0. n’m Set

h(a) = $Z(au) = a(u, u) - -$ (g’(au), au), and suppose Z(u) = h(1) < 0. By (2.6), we have h(O+) = a(u, u) > 0, and h E C(0, 1) by (2.3), so there exists a, E (0, 1) such that h(a,) = 0; hence a,u E M. Now let b(a) = f( g’(au>, au> - g(au>,

a E (0, 1).

Using (2.5) we find

b’(a) = & [ g”(au)(au, au) - (g’(au), au)] > 0, so that b(a) is strictly increasing. Hence

J(a, u) = b(a,) < b( 1) = P, and this contradiction shows that a, = 1 and u E M. It follows that there exists 1 E R such that

V’(u), 0) = W’(u), u) for all u E E. Setting u = u gives (J’(u), u> = W)

= 4W)

which implies A = 0 since u EM J’(u) = 0.

- ~[g”@)(~~u) - (g’(u), u)]}, and because of (2.5). It follows

that

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CHOWANDKURTZ

Proof of Theorem 2. Applying the results of [2], we need only verify the Palais-Smale condition C. To this end suppose {a,) c M, .Z(U,)< C, and IIJatll+ 0. htl is bounded, so taking a subsequence we may assume u,, - u. Since (J’(U), u) = 0 for u E M we have

IIJiA4 = IIJ’@“)ll+ 07 in other words,

II2U” - g’o4I>ll+0. Since g’(u,) converges strongly by (2.10) and U, - U, it follows that U, -+ U. Thus there exist {uk} GM and {A,} E R such that J’(Q) = &I’@,). But Lk = 0 as in the proof of Theorem 1. In the following lemmas we assume (3.2)-(3.11). We define

J(u) = IIu II2- g(u), Z(u) = 1)u II* - ,fm f(u’) u’p dt, 0

M= {uEEIu#O,Z(u)=O}.

LEMMA

1.

Zf u E E then p(t) u2(t) < IIu 11’.

ProoJ u’(t)=-2

m utids< OD(u* + li’) ds iI If

< p(t)-’ lo: (u’ + li*)p dt Q p(t)-’ llu[12. t LEMMA

2. There exists a constant C such that

ImuZdt
(5-l)

0

for all u E E. Proof. Choose cpE C’(0, co) such that 0 < P(C)< 1 and p(t) = 1 for t E [0, 11, p(t) = 0 for t E [2, a~). Set v = vu, so that

649

SINGULARNONLINEAREQUATIONS

Also.

<

‘v’dt+1^ 0

1

*

u’p dt.

P(l) I I

(5.3)

By Theorem 330 of [3] and (3.9), we have cc

co

v2dt<4 10

cl2

h2t2 dt < C I

0

I0

h’p dt.

Using (5.2) and (5.3) we obtain cc i0

u2 dt < C, o~(lirl+CIUI)IPdt+CzjmU2pdt i 0 < C m (u’ + zi’)p dt. s0

LEMMA 3. The family any finite interval [0, T].

Proof.

{p(t) u2(t) 1u E E, 1)u 11< C) is equicontinuous on

For 0 < t, < t, < T we have

u2@dt,
(5.4)

making use of Lemma 1. Next we use Lemmas 1 and 2 and (3.8) to obtain 12 u2$dt<2jtt)util jt;@dtdr+2j)A jtt;Ddtdr I t1 112 <2 I’ - pk)) dr u’(P@) - p(b)) dz

I

+ 2(P(tz) - PQl)) ( j t~u2df’2(--&~ti2pd$“2 < Cl02 - t,Y2

+ C,(P(f,)

The lemma follows from (5.4) and (5.5).

- P(tl))“‘.

(5.5)

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CHOW

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LEMMA4. p(t) u*(t) + 0 as t + O+, uniformly for u E {u E E ) I/u 11< C}. Proof:

For 0 < t < s < 1 we have U*(t)_
IfO
then p(t) U*(t) < ~(4 u*(l) + 2 1‘ f

IuziIP dr

2P(4 + &S

’

I ui IP d7.

Since by Lemma 1 ~~Slutilpd7<

([u”df”

([ti$dr)

“*
given E > 0 there exists s < 1 such that for 0 < t < s,

Choose 6 such that 0 < 6 < s and P(& u*(l) < ;9

P(6) l po j, Iuilpd7 < ;.

Then 0 < t < 6 implies p(t) u2(t) < E.

LEMMA 5. The mapping g’ takes weakly convergent sequences into strongly convergent sequences. Proof:

Suppose u, -

u in E. Then

IIg’(u,) - g’@)llE*= ,g$ IKm) un- f(u2h VI = sup O” Lf(u;) u, - f(u’h4 Ilull< I j 0 4supIj;/

VP dt

+suPIjnr~ +supl jy. T

By (3.3) and (3.8), we have mf(u:)uxvpdt~

~~j~~~.~li2u~v~p1+udt
mlu yip&< 1* n ’

’ P(T)“’

(5.6)

SINGULAR

NONLINEAR

EQUATIONS

651

Next we use Lemmas 1, 2, and 4 to get

= o(1)

as

q-+0+.

(5.7)

Lemmas 1 and 3 imply that a subsequence {u,,} converges uniformly to u on [Q T]. It follows from (5.6) and (5.7) that

Hence, by the same argument, every subsequence of { g’(u,)} subsequence converging to g’(u), so we must have g’(u,J+ g’(u).

has a

Proof c$ Theorem 3. Conditions (2.1)-(2.4) are satisfied by virtue of Lemma 5 and (3.6). Condition (2.6) follows from the estimate O”f (a’~‘) u’p dt < Ca2”jD u*+*Op dt s 0

0

< Ca2”I(u(12”(r

u*p’-“dt 0

< CaZU11 u 1120 ( j p,l,+jomu2pdt)

Similarly, if u E M we have /Iu II* = jm f(u’)

u’p dt

< ii l,u,,*+*“, so that

IIuII2 u/w2u.

(5.8)

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CHOW

AND

KURT2

Next, for u E M we use (3.4) to get

ss ds dt

and it follows that J(U)>&1

+6)-‘]lU]j2.

(5.9)

Conditions (2.8) and (2.9) follow from (5.8) and (5.9). A calculation shows that g”(u)@, u) - (g’(u), u) = 4 Irn f’(u’)

u”p dt,

0

which verifies (2.5). Since the positive cone in E is weakly closed, we may now apply Theorem 1, considering only the positive functions in A4. The fact that our solution u(t) is a classical one follows by the usual bootstrap argument. The standard argument also shows that since u(t) > 0, u(t) 10, u(t) must in fact be strictly positive. Theorem 4 follows from Theorem 2 in a similar manner. In Theorems 5 and 6 the argument is the same as in the above, except that the Sobolev embedding is used to verify (2.10).

6. A FINAL EXAMPLE

We conclude with the observation that the theorems of Section 2 can also be applied to the nonlinear Legendre equation -(( 1 - t2)u’)’ + u ==f(u2)u,

t E (-LO].

In this case we set E = {u E D’(-1,0)

with

I U, (1 - t’)“’

ti E L2}

(6.1)

SINGULAR NONLINEAR EQUATIONS

653

Assuming condition (3.2)--(3.7), Eq. (6.1) can be shown to have infinitely many solutions, one of which is strictly positive on (-l,O). All of these will satisfy ti(0) = 0, and hence may be extended by symmetry to give solutions on (-1, 1). The technical details very closely follow those given in Section 5 for Eq. (1.4). REFERENCES 1. M. BERGERAND M. SCHECHTER,Embedding theorems and quasi-linear elliptic boundary value problems for unbounded domains, Trans. Amer. Math. Sot. 172 (1972), 261-278. 2. F. BROWDER,Infinite dimensional manifolds and nonlinear elliptic eigenvalue problems, Ann. ofMath. 82 (1965), 459477. 3. G. HARDY, L. LITTLEWOOD,AND G. POLYA, “Inequalities,” Cambridge Univ. Press, London/New York, 1952. 4. J. KURTZ, A singular nonlinear boundary value problem, Rocky Mtn. J. Math., in press. 5. Z. NEHARI, Characteristic values associated with a class of nonlinear second-order differential equations, Acta Math. 105 (1961), 141-175. 6. Z. NEHARI, On a nonlinear differential equation arising in nuclear physics, Proc. Roy. Irish Acad. Sect. A 62 (1963), 117-135. 7. G. RYDER, Boundary value problems for a class of nonlinear differential equations, Pacific J. Math. 22 (1967), 477-503. 8. W. STRAUSS,Existence of solitary waves in higher dimensions, Commun. Math. Phys. 55 (1977), 149-162. 9. P. WONG, On a class of nonlinear fourth order differential equations, Ann. Mat. Puru Appl. 81 (1969), 331-346. 10. V. ZAKHAROV,V. SOBOLEV,AND V. SYNAKH,Behavior of light beams in nonlinear media, Zh. E/q. Teor, Fiz. 60 (1971), 136-145.