solucionesradiales - alfonso castro

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Infinitely Many Radially Symmetric Solutions to a Superlinear Dirichlet Problem in a BallAuthor(s): Alfonso Castro and Alexandra KurepaSource: Proceedings of the American Mathematical Society, Vol. 101, No. 1 (Sep., 1987), pp. 57-64

Published by: American Mathematical SocietyStable URL: http://www.jstor.org/stable/2046550Accessed: 17/06/2009 17:44

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PROCEEDINGS OF THEAMERICAN MATHEMATICAL SOCIETYVolume 101, Number 1, September 1987

INFINITELY MANY RADIALLY SYMMETRIC SOLUTIONS TO A

SUPERLINEAR DIRICHLET PROBLEM IN A BALLALFONSO CASTRO AND ALEXANDRA KUREPA

ABSTRACT. In this paper we show that a radially symmetric superlinearDirichlet problem in a ball has infinitely many solutions. This result is ob-tained even in cases of rapidly growing nonlinearities, that is, when the growthof the nonlinearity surpasses the critical exponent of the Sobolev embeddingtheorem. Our methods rely on the energy analysis and the phase-plane an-gle analysis of the solutions for the associated singular ordinary differentialequation.

1. Introduction. Over the last two decades considerable progress has beenmade in the study of superlinear boundary value problems, such as:

(1.1) { Au + (u) = q(x), x E Q,u = O, x E 6Q,

where Q is a bounded region in RN, A is the Laplacian operator, g: R - R is acontinuous function, q E L2(FQ), nd

(1.2) lim (g(u)/u) = 00.1u140?

The main goal has been to identify the conditions on Q, g, q under which (1.1)-(1.2) has infinitely many solutions. Most of the results have been obtained usingvariational methods. In order to get solutions of (1.1) as critical points of a certain

functional J defined in the Sobolev space H'(Q) (see Adams [1]), it has becomecustomary to assume that g satisfies the growth condition that ensures J to bewell defined. For general bounded regions, if g is odd and satisfies suitable growthconditions, the existence of infinitely many solutions was obtained simultaneouslyand independently by Bahri and Berestycki [2] and Struwe [9]. Their results were

later extended by Rabinowitz [8] and Bahri and Lions [3] . The only cases known tous where infinitely many solutions are obtained for g growing up to, but excluding,the limit exponent in the Sobolev space are those of Struwe [10]. In this paper wenot only get an extension of those results, but we also get cases where the growth ofg surpasses the Sobolev inequality growth. Indeed, our proofs show that for radiallysymmetric solutions of (1.1) the growth condition depends on the signl of the valueof the solution at 0. Actually our results suggest that it should be sufficient to havethe growth conditions only for u > 0 or for u < 0.

Received by the editors May 1, 1986. Presented to the AMS at the 791 meeting in Denton,Texas.

1980 Mathematics Subject Classification (1985 Revision). Primary 35J65, 34A10.Key words and phrases. Superlinear Dirichlet problem, radially symmetric solution, singular or-

dinary differential equations, phase-plane analysis, growth condition, rapidly growing nonlinearities.

? 1987 American Mathematical Society0002-9939/87 $1.00 + $.25 per page

57


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58 ALFONSO CASTRO AND ALEXANDRA KUREPA

We study problem (1.1) when Q is the ball of radius T in RN, centered at theorigin, and

(1.3) q(x) = p(lIxIl)is a radially symmetric function. For the sake of simplicity of the proofs we assume,without loss of generality, that

(1.4) p = L?? Q), g(0) = 0, and g is strictly increasing.In order to state our main result we introduce the following notation:

(1.5) L(k, u) = NG(ku) - ((N - 2)/2)u(g(u) + 5JJpJJ00sgn(u)),(1.6) L?(k) = lim L(k,u)(u/g(u))N/2,

uo +00

(1.7) F(m, u) = (u/g(u))N+p1 . G(ku) + m,

where G(u) = foug(v) dv, k E (0, 1], p > 0, and m E R.Our main result is

THEOREM A. Suppose g is locally Lipschitzian and (1.2)-(1.3) hold. If(i) L(1,u) is bounded below and L+(k) = oo (respectively L_(k) = oo) for some

k E (O,1), or(ii) F(m, u) -+ oo as u -- oo (respectively F(m, u) -x oo as u -- -oo) for any

m E R,then (1.1) has infinitely many radially symmetric solutions with u(O) > 0 (respec-tively u(O) < 0)).

Theorem A includes the results of Struwe [10] since hypothesis (i) is satisfiedfor ig(u)I < Alulw + B, A,B E R, 1 < w < (N + 2)/(N - 2). Not only that,condition (i) allows the case g(u) = u(N+2)/(N-2) + uw for either u < 0 or u > 0.Also, if Jg(u)l < Auo', A E R, 1 < a < N/(N - 2), then condition (ii) holdswithout any growth restriction on g(u) for u < 0. Hence, cases such as g(u) =u- eu3 (N = 2, r, odd) are encompassed by hypothesis (i). These results, to thebest of our knowledge, have not been obtained before. Moreover, since our methodis not variational we do not have to use compactness arguments of Palais-Smaletype. Therefore cases such as g(u) = u log(jul), Jul > 1, satisfy the hypothesis ofTheorem A, but not the hypothesis required in the work of others (see for example[2, (6.3); 3; 8, (ps); 10, (4)]). Another advantage of our method is that p can beallowed to depend on (IIxIj, , u') as long as p(IlxIj, , u') is uniformly bounded. Thisis not possible when using variational arguments (see [2, 3, 8, 9, 10]).

Our proofs rely on the study of the singular initial value problem

u +-u' + g(u) = p(t), t E [0,T],t(1.8) u(0) = di

u'(0) = O,

where n = N - 1, and d is an arbitrary real numnber. A simple argument based on

the contraction mapping principle and Lemma 2.1 shows that problem (1.8) has aunique solution u(t, d) on the interval [0, oo) depending continuously on d. In ?2we analyze the energy of the corresponding solutions, i.e. we analyze the function

(1.9) E(t, d) = (u'(t, d))2/2 + G(u(t, d)).


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SOLUTIONS TO A SUPERLINEAR DIRECHLET PROBLEM 59

Since radially symmetric solutions of (1.1) are the solutions of (1.8) satisfying

(1.10) u(T, d) = O,

we use the shooting method. In order to do so we estimate from below the valueof to > 0 such that the solution of (1.8) satisfies

(1.11) u(to, d) = k d, k E (0, 1).

Combining this estimate, hypotheses (1.2)-(1.3), and an argument resembling theone used in the proof of Pohozaev's identity we show that

(1.12) lim E(t, d) = oo (respectively lim E(t, d) = oo)d *+oo d --*Coo

uniformly for t E [0, T].

The rest of the proof is based on the phase plane analysis. From (1.12) it followsthat in the (u, u') plane for d sufficiently large a continuous argument function0(t, d) can be defined so that 0(0, d) = 0. Imitating the proof used by Castro andLazer [4] in ?3 we show that

(1.13) lim 0(T, d) = oo.d- oo

Thus, by the intermediate value theorem we have the existence of infinitely manysolutions for (1.10), therefore infinitely many radially symmetric solutions for (1.1)

In a forthcoming paper we treat the jumping nonlinearity case extending thework of Castro and Shivaji [5].

2. Energy analysis. Throughout this paper we assume that (1.4) holds. Thecontinuous dependence of u and u' in (t, d) can be obtained by considering theoperator

(u, d) -- d + r-n j sn(p(s) - g(u(s))) ds dr

in the complete metric space (C[0, ?]) x [d' - ', d' + e], and showing that for eachd E R this operator defines a contraction in u.

LEMMA 2. 1. Every solution to (1.8) is defined for all t E [0, 00).

PROOF. Suppose that for some increasing sequence tn -+ t E R we have

lim (u2(tn,id) + (u/ tnd))2) x 00.n-*ooIf (U'(tn, d))2 does not tend to infinity, then by the mean value theorem a newincreasing sequence t' t- can be found so that (u'(t', d))2 x-+ 0. Thus, withoutloss of generality, we can assume (u'(tn, d))2 - 0. Also, since (1.4) holds, we haveG > 0. Hence

(2.1) lim E(tn, d) =oo.n-*oo

On the other hand we haveEl(t, d) = u/(t, d)p(t) - (u'(t, d))2t

? ju'(t, d)jjpfl00o ? \2jjpHjoo E(t,-d).


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Hence

E(t,d) < (VE+(0d) ? (X/'-/2) Iploot)I

which contradicts (2.1) and thus the lemma is proven.For k E (0,1) and d > 0 with Ig(d)l > IIpIKOOet to := to(k,d) be such that

d > u(t, d) > kd for all t E [0, to) and u(to, d) = kd. Since we are assuming g to bean increasing function we have

rtu'(t, d) = t-n j rn(p(r) - g(u(r, d))) dr > (-IpIIOOg(d))t/(n + 1).

Hence, integrating over [0, to] we obtain

(2.2) to > (2(n + 1)(1 - k) d1(jjpjjoo (d)))?/2

Similar arguments show that if d.< 0 and Ig(d)i > jjpjjIO hen (2.2) also holds. Nowwe are ready to prove

LEMMA 2.2. If L(1, u) is bounded below and for some k E (0, 1), L+(k) = oo(respectively L_(k) = oo), then

lim E(t,d)=-oo respectively lim E(t,d)=ood--+oo d-oo_0


PROOF. Let v(x) = u(llxll), x E Q. Since u satisfies (1.8) we have

(2.3) AV + g(v) = P(IIXII), X E Q,(2.4) v(O) = d, Vv(0) = 0.

The arguments that follow are inspired by Pohozaev's identity. Throughout theproof c denotes various positive constants depending on (N, jjpjjIO,). An elemen-tary calculation shows that

(Av)(x Vv) = 211|v112 + ENjVU )i=l j=l X,

N N\

(2.5) )xjv2j=l i=l X3

(g(v) - p(jjxjj))(x.

Vv) -E(jG(v)),- NG(v) - p(llxll)(x Vv),

where the subindex xk denotes the partial derivative with respect to the variableXk.


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Multiplying (2.3) by x Vv, integrating over Q, = {x E Q; e14 ,}, to < e < T(see (2.2), (2.5)), and using the divergence theorem we have

[N2 1lVvl 2-NG(v)-p(jIxIj)(x.Vv) dx

(2.6) = ' 22N j, xjG(v)) - j (xiv xivx3)j da

= j= j=j i=1EZVt+jGV)- - (1v - xjv2 + xjG(v)) jdcir

where v = (V1 .VN) denotes the outward unit normal to Q, at x, and we have

used the radial symmetry of v.On the other hand, multiplying (2.3) by v and integrating over Q, we have

(2.7) j flVvfl2dx = (g(v) - p(lIxII))v x ? v(v Vv)du.

Replacing (2.7) in (2.6) we obtain

f ( -NG(v) (N- 2)g(v)v ) dx

(2.8) = f [ i,vi112 + eG(v) + 2 (v(v. Vv))] du

-fp(IIxII) ((x Vv) + N2) dx.

Thus, from (2.2), (2.8), and the assumption that L(1, u) is bounded below we have

d N/2 NGk)-(N - 2)g(d)ci\?d ~ NBc()(2.9) NG(kd) + (6N _ tN)B < E(e)d)2 ,'where B is a constant, possibly negative, independent of d. Since, in addition,L+ (k) = oo, from (2.9) we have

lim E(t,d)=cod-oo

uniformly for t E [0, T], and this proves Lemma 2.2.

LEMMA 2.3. If for any m E R F(m, u) -- oo as u -+ oo (respectively F(m, u)oo as u -+ -oo), then

lim E(t, d) = x (respectively lim E(t,d) = oodi Eoo d(-t,)


PROOF. From the definition of E and (1.8) we have

(t2(n+P)E(t, d))' > 2(n + p)t2(n+p)-'G(u(t, d)) -(t2(n+p)+l jjPj12 )/4p.Integrating over [to, t], we infer


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Suppose that t > (T/4) and 0(t, d) belongs to an interval of the form [(j7r/2) -6, (jir/2) + 8], where j is a nonnegative odd integer. From (3.3) we have

O'(t, d) > sin 20(t, d) + g(u(t, d))u(t, d) _4nIu(t, d) u'(t, d) I r)pO)0(3.10) r2 (t, d) Tr2 (t, d)

r

> 028 -4ntan8 _ jpjjo >1cos 6- T - > 4'T ro

where we have used (3.6), (3.7), and that g(u) . u > 0 since (1.4) holds.On the other hand, if 0(t, d) belongs to an interval of the form [(j7r/2) + 6,

(j + 2) (7r/2) - 8] we have

(3.11) O'(t, d) > (g(u(t, d)) sin2 6)/(u(t, d)) - (2n/T) - (liploo/ro)> w(ro, 6)/2 > 0

(see (3.9)).By Remark 3.1 we see that 0(T/4,d) > 0 for all d > do. Now, by (3.10) and

(3.11) we infer that 0(t, d) is an increasing function of t on the interval [T/4, T].Next we estimate O(T, d). Suppose that IO(T/4, d) - (ji/2)1 < 6 for some odd

positive integer j. Since by (3.10) 0(t, d) cannot remain in the interval [(j7r/2) -6, (j7r/2) + 8] longer than 88, there exists t1 E (T/4, (T/4) + 86) such that

(3.12) 0(t1,d) = (jir/2) + 6.

Now, for t > t1 with 0(s,d) E [(jir/2) + 6, (j + 2)(ir/2) -8] and for all s E [t1, t]from (3.11) we see that t - t1 < (r - 26)/w(ro, 8). Thus, there exists t2 E (t1, ti +(2(7r - 26)/w(ro, 6))) and

(3.13) 0 (t2, d) = (j + 2) (ir/2) - 8.

Imitating the argument used in establishing the existence of t1, we see that thereexists t3 E (t2, t2 + 86) such that 0(t3, d) = (j + 2)(Xr/2) + 8. Using the propertiesof t1 and t2 we obtain

(3.14) T/4 < t3 < (T/4) + 166 + (2(7r 26)/w(ro, 6)),(3.14)O~~(t3,) - O(t,d) > rr.

Repeating this argument J times we see that

(3.15) 0(t,d) > Jir

for some t [T/4, (T/4) + J(166 + (2(-r - 26)/w(ro, 8)))]. From (3.15) using (3.8)we get t < T. Since 0(t, d) is increasing we have proved that

(3.16) O(T,d) > J7r.

On the other hand if I0(T/4, d) (jir/2)1 > 6, similar arguments lead to the existenceof t E [T/4, (T/4) + J(86 + (4(7r 26)/w(ro, 8)))] such that 0(t, d) > J7r. Hence,

(3.17) 0(T,d) ? J7r.

From (3.16) and (3.17) follows (3.4). Since the case d < 0 follows the identicalpattern, the theorem is proven.


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REFERENCES

1. R. Adams, Sobolev spaces, Academic Press, New York, 1975.2. A. Bahri and H. Berestycki, A perturbation method in critical point theory and applications,

Trans. Amer. Math. Soc. 267 (1981), 1-32.3. A. Bahri and P. L. Lions, Remarques sur la theorie variationnelle des points critiques et appli-

cations, C. R. Acad. Sci. Paris 301 (1985), 145-149.4. A. Castro and A. C. Lazer, On periodic solutions of weakly coupled systems of differential equa-

tions, Boll. Un. Mat. Ital. B (5) 18 (1981), 733-742.5. A. Castro and R. Shivaji, Multiple solutions for a Dirichlet problem with jumping nonlinearities.

II, J. Math. Anal. Appl. (to appear).6. M. J. Esteban, Multiple solutions of semilinear elliptic problems in a ball, J. Differential Equa-

tions 57 (1985), 112-137.7. S. Fucik and V. Lovicar, Periodic solutions of the equation x (t) + g(x(t)) = p(t), Nasopis

P6st. Mat. 100 (1975), 160-175.8. P. H. Rabinowitz, Multiple critical points of perturbed symmetric functionals, Trans. Amer.

Math. Soc. 272 (1982), 753-769.9. M. Struwe, Infinitely many critical points for functionals which are not even and applications to

superlinear boundary value problems, Manuscripta Math. 32 (1980), 335-364.10. _ , Superlinear elliptic boundary value problems with rotational symmetry, Arch. Math. 39

(1982), 233-240.

DEPARTMENT OF MATHEMATICS, NORTH TEXAS STATE UNIVERSITY, DENTON, TEXAS76203

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