Solutions for Indefinite Semilinear Elliptic Equations in Exterior Domains

Journal of Mathematical Analysis and Applications 255, 308᎐318 Ž2001. doi:10.1006rjmaa.2000.7262, available online at http:rrwww.idealibrary.com on

Solutions for Indefinite Semilinear Elliptic Equations in Exterior Domains Hossein Tehrani Department of Mathematical Sciences, Uni¨ ersity of Ne¨ ada, Las Vegas, 4505 Maryland Parkway, Box 454020, Las Vegas, Ne¨ ada 89154-4020 Submitted by J. McKenna Received February 1, 2000

INTRODUCTION In recent years, motivated by applications in physics and geometry, a large number of works have been devoted to existence results for semilinear elliptic equations of the form

½

y⌬u s f Ž x, u . us0

in ⍀ on ⭸ ⍀

Ž G.

in unbounded domains. Depending on the nature of the nonlinearity, variational and topological methods have been used and some existence and multiplicity results have been obtained. Most of these works, however, consider the case of the whole space ⍀ s ⺢ N Žsee w2, 3, 5, 8᎐10x. In this paper we are concerned with the problem

½

y⌬u s a Ž x . g Ž u . us0

in ⍀ on ⭸ ⍀

Ž P.

in an exterior domain ⍀. Our work was motivated in part by a recent paper of Munyamarere and Willem w10x. In this paper the authors study Eq. ŽP. in the whole space ⺢ N under the assumption that a g C Ž⺢ N , ⺢. is radial and positive, and prove the existence of a positive radial solution as well as infinitely many nodal solutions under additional symmetry assumptions. Their approach is variational and, as with all equations set in unbounded domains, compactness or the Palais᎐Smale ŽPS. condition for the corresponding energy functional is the main difficulty in applying critical point 308 0022-247Xr01 $35.00 Copyright 䊚 2001 by Academic Press All rights of reproduction in any form reserved.

SEMILINEAR ELLIPTIC EQUATIONS

309

theories. To overcome this difficulty, in w10x the authors work in the space of radial functions Žhence the radial symmetry assumption on a. and the ŽPS. condition is done by applying a basic inequality of Rother w12x for this class of functions and comparing the asymptotic behavior of a and g at zero and infinity. In addition to taking ⍀ to be an exterior domain, here we consider a general sign-changing function a g C Ž ⍀, ⺢.. Then the semilinear equation ŽP. is called indefinite. In this case if aŽ x . is positive at infinity, that is, lim < x < ª⬁ aŽ x . ) 0, it can be seen that a simple Pohozaev-type identity provides nonexistence results under rather mild conditions. So we further assume that aŽ x . is nonpositive at infinity, that is, aŽ x . F 0

if < x < G R 0 for some R 0 ) 0.

In addition we assume a Ž x . s O Ž < x
< x < ª ⬁.

Regarding g g C Ž⺢, ⺢., we assume the usual subcritical growth at infinity Žsee precise assumptions below.. Under these assumptions we prove the existence of a positive solution for ŽP. as well as infinitely many solutions in case g is odd. An important role is played by a Hardy-type inequality for C0⬁Ž ⍀ .. Using this inequality and the asymptotic behavior of a at infinity, we will handle the ŽPS. condition for the energy functional in Section 1. Section 2 is devoted to existence and multiplicity results.

1. VARIATIONAL FRAMEWORK AND THE PALAIS᎐SMALE CONDITION We consider the equation

½

y⌬u s a Ž x . g Ž u . us0

in ⍀ on ⭸ ⍀ ,

Ž P.

where ⍀ is an interior domain in ⺢ N , i.e., ⍀ s ⺢ N _ O with O a nonempty open bounded neighborhood of the origin. In addition we assume ŽA1. ŽA2. ŽA3.

aŽ x . g C Ž ⍀, ⺢. changes sign in ⍀; aŽ x . F 0 if < x < G R 0 for some R 0 ) 0; sup < aŽ x .5 x < 2 - ⬁;

ŽG1.

g g C Ž⺢, ⺢.;

xg⍀

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HOSSEIN TEHRANI

ŽG2.

g Ž s . s oŽ< s <. as s ª 0;

ŽG3.

< g Ž s .< F C Ž1 q < s < py 1 . for some C ) 0, 2 - p - 2* s

ŽG4. 0 - ␯ GŽ s . F s g Ž s . for < s < G s0 , with ␯ ) 2, s0 ) 0,

2N Ny2

;

where G Ž s . s H0s g Ž t . dt. We will also use the notation ⍀qs  x g ⺢ < aŽ x . ) 04 , ⍀ys  x g ⺢ < aŽ x . - 04 . Furthermore the norm in the space Lt Ž ⍀ ., 1 F t - ⬁, will be denoted by < u < t s Ž H⍀ < uŽ x .< t dx .1r t. Next we consider the corresponding functional I Ž u. s

1

H⍀ 2 < ⵜu <

2

y a Ž x . G Ž u . dx

in the space D 1,0 2 Ž ⍀ . which is the completion of C0⬁Ž ⍀ . under the norm 5 u 5 s Ž H⍀ < ⵜu < 2 dx .1r2 . The fact that I defines a C 1 functional on this space is a simple consequence of Sobolev’s embedding of D 1,0 2 Ž ⍀ . in L2* Ž ⍀ . and the following Hardy-type inequality Žsee w13x.. PROPOSITION 1.1. Let N G 2 and ⍀ s ⺢ N _ O with O a nonempty open bounded neighborhood of the origin. Then there exists a constant C s C Ž N . such that for all u g D 1,0 2 Ž ⍀ .

H⍀

< uŽ x . < 2 < x<

2

dx F C < ⵜu < 22 .

Ž 1.1.

In fact by Ž G 1 . ᎐ Ž G 3 ., there exists C1 , C2 ) 0 such that < G Ž s . < F C1 < s < 2 q C2 < s < 2*

for all s g ⺢.

Ž 1.2.

So we have

H⍀aŽ x . G Ž u . dx

FC

žH

< a Ž x . < u 2 dx q

⍀

F sup < a Ž x . 5 x <

ž

xg⍀

2

/H

⍀

H⍀< u <

2*

< uŽ x . < 2 < x<2

dx

/

dx q < ⵜu < 22

F C < ⵜu < 22 . Clearly critical points of I are weak solutions of ŽP.. Of course, elliptic regularity theory Žsee w6x. then implies that such weak D 1,0 2 Ž ⍀ . solutions are in fact in C 1, ␣ Ž ⍀ . Ž0 - ␣ - 1. and assuming a little more regularity on aŽ x . we will get classical solutions. To show the existence of critical points of I we use the Mountain Pass Lemma, which we now recall Žsee w1x..

SEMILINEAR ELLIPTIC EQUATIONS

311

PROPOSITION 1.2. Let E be a real Hilbert space and consider a functional I g C 1 Ž E, ⺢. satisfying the following conditions: Ža. For any number c, e¨ ery sequence  u n4 ; E such that I Ž u n . ª c and I⬘Ž u n . ª 0 in E* as n ª ⬁, possesses a con¨ ergent subsequence. Žb. There exists ␳ ) 0 and ␣ ) 0 such that I Ž u. G ␣ for all 5 u 5 s ␳ . Žc. I Ž0. s 0 and there exists e g E, 5 e 5 ) ␳ such that I Ž e . F 0. Then I has a critical point ¨ 0 with I Ž ¨ 0 . G ␣ . To obtain solutions of ŽP. we will show that Proposition 1.2 can be applied to the functional I. We recall that a sequence  u n4 ; E such that I Ž u n . ª c and I⬘Ž u n . ª 0 in E* as n ª ⬁ is called a Palais᎐Smale sequence and a functional I g C 1 Ž E, ⺢. satisfying Ža. above is said to satisfy the Palais᎐Smale compactness condition Žhenceforth denoted by ŽPS... First a preliminary result. LEMMA 1.1.

Assume Ž A1 . ᎐ Ž A 3 ., Ž G 1 . ᎐ Ž G4 .. In addition assume either

q

Ž A 4 . ⍀ l ⍀y s ␾ , or Ž G5 . g Ž s . s s < s < qy 2 q O Ž< s < ␣ . as < s < ª ⬁ for some 2 - q - 2*, 0 F ␣ - 1. Then any Ž PS . sequence  u n4 ; D 1,0 2 Ž ⍀ . is bounded in D 1,0 2 Ž ⍀ . norm. Proof. We argue by contradiction to show that < ⵜu n < 22 is bounded. Indeed assume that < ⵜu n < 2 s t n ª ⬁ and define ¨ n s u nrt n . Then we have 1

H < ⵜ¨ 2 ⍀ H⍀ⵜ¨ ⵜ␸ dx y H⍀aŽ x . n

n

< 2 dx y

g Ž un . tn

H⍀aŽ x .

␸ dx s

G Ž un . t n2

o Ž 1. tn

dx s o Ž 1 . ,

5␸5

Ž 1.3.

2 ᭙␸ g D 1, 0 Ž ⍀ . . Ž 1.4 .

Since 5 ¨ n 5 s 1 there exists ¨ g D 1,0 2 Ž ⍀ . such that ¨ n ª ¨ weakly in D 1,0 2 Ž ⍀ .. In addition ¨ nŽ x . ª ¨ Ž x . a.e. in ⍀ and in Lt ŽU . for any bounded subset U of ⍀ and 1 F t - 2*. Claim. ¨ Ž x . ' 0. Assuming this for now, we take the limit in Ž1.3. to obtain

H aŽ x . nª⬁ ⍀ lim

G Ž un . t n2

dx s

1 2

.

Ž 1.5.

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HOSSEIN TEHRANI

Next taking ␸ s ¨ n ␨ in Ž1.4. where ␨ g C0⬁Ž⺢ N ., we get

H⍀< ⵜ¨

n

< 2␨ dx y

g Ž un . un

H⍀aŽ x .

t n2

␨ dx s y ¨ nⵜ¨ nⵜ␨ dx q

H⍀

o Ž 1. tn

5¨n ␨ 5

s o Ž 1. .

Ž 1.6.

Now we consider each of the two cases corresponding to Ž A 4 . or Ž G5 ., respectively. Case 1. If Ž A 4 . is satisfied then there exists ␨ g C0⬁Ž⺢ N . such that ␨ Ž x . s 1 on ⍀q and ␨ Ž x . s 0 on ⍀y. Using Ž G4 . and Ž1.6. we have

H⍀aŽ x .

G Ž un .

␨ dx s

t n2

H< u
t n2

0

F o Ž 1. q F

G Ž un .

1

H aŽ x . ␯ ⍀

1

H < ⵜ¨ ␯ ⍀

n

␨ dx q

H< u <)s aŽ x . n

g Ž un . un t n2

< 2␨ q o Ž 1 . F

1

G Ž un . t n2

0

␨ dx

␨ dx

q o Ž 1.

␯

which contradicts Ž1.5.. Case 2. If Ž G 2 . holds then we have < qG Ž s . y g Ž s . s < s O Ž < s < ␣q1 .

as < s < ª ⬁.

Ž 1.7.

Therefore choosing ␨ g C0⬁Ž⺢ N . such that 0 F ␨ F 1 and ␨ Ž x . s 1 if < x < F R 0 , with R 0 ) 0 as in Ž A 2 . we get

H⍀aŽ x .

qG Ž u n . t n2

␨ dx s

H⍀aŽ x .

g Ž un . un t n2

␨ dx q o Ž 1 . .

Ž 1.8.

Indeed, by Ž1.7., for M large

Hw < u
< qG Ž u n . y g Ž u n . u n < t n2

n

< ␨ < dx F C

HsuppŽ ␨ .

< u n < ␣q1 t n2

dx s o Ž 1 . .

Furthermore since < qGŽ s . y g Ž s . s < F C s C Ž M . for < s < F M and since t n ª ⬁, we clearly have

Hw < u
< qG Ž u n . y g Ž u n . u n < t n2

< ␨ < dx s o Ž 1 . ,

313

SEMILINEAR ELLIPTIC EQUATIONS

which proves Ž1.8.. Now using Ž1.5., Ž1.6., and Ž1.8. we obtain

H⍀aŽ x .

lim

G Ž un . t n2

Ž 1 y ␨ . dx s s

1 2 1 2

y y

1 q

H⍀aŽ x .

lim

1

H < ⵜ¨ q ⍀

n

g Ž un . un t n2

␨ dx

< 2␨ dx

Hence

H⍀aŽ x .

lim

G Ž un . t n2

Ž 1 y ␨ . dx G

1

y

2

1 q

)0

which again contradicts Ž1.5. by our choise of ␨ . So we only need to prove the claim. First we show that ¨ Ž x . s 0 a.e. in q ⍀ . Again we argue by contradiction and assume that there exists a ball Br Ž x 0 . ; ⍀q such that, for some ␣ , ␤ ) 0,

␮ Ž E . [  x g B Ž x0 . : < ¨ Ž x . < G ␣ 4 G ␤ ,

Ž 1.9.

r 2

where ␮ denotes Lebesgue’s measure in ⺢ N. Now since ¨ nŽ x . ª ¨ Ž x . a.e. in Br Ž x 0 ., there exists a set S ; Br Ž x 0 . with measureŽ S . - ␤2 where ¨ nŽ x . ª ¨ Ž x . uniformly in E _ S s F and measureŽ F . G ␤2 . Let ␰ g C0⬁Ž ⍀ . with ␰ Ž x . s 1 on Br r2 Ž x 0 . and ␰ Ž x . s 0 on Brc Ž x 0 .. We take ␸ s ¨ n ␰ in Ž1.4. and obtain

H⍀< ⵜ¨

n

< 2␰ dx q

H⍀¨ ⵜ¨ ⵜ␰ dx s H⍀aŽ x . n

g Ž un . ¨n

n

tn

␰ dx q o Ž 1 . . Ž 1.10.

Furthermore we note that by Ž G4 . there exists A ) 0 such that g Ž s . s G A< s < ␯

for < s < G s0 ,

Ž 1.11.

where without loss of generality we may assume ␯ - 2*. Next using Ž1.10. we have

H⍀< ⵜ¨

n

< 2␰ dx q

H⍀¨ ⵜ¨ ⵜ␰ dx s t n

n

␯ y2 n

G t n␯y2

½H ½H

⍀

F

aŽ x . < ¨ n < ␯

aŽ x . < ¨ n < ␯

g Ž t n¨ n . t n¨ n < t n¨ n < ␯ g Ž t n¨ n . t n¨ n < t n¨ n < ␯

5

␰ dx q o Ž 1 .

5

dx q o Ž 1 . .

Ž 1.12.

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HOSSEIN TEHRANI

Since < ¨ Ž x .< G ␣ on E and ¨ n ª ¨ uniformly in F ; E, for n large we have

HFaŽ x . < ¨

n

g Ž t n¨ n . t n¨ n

<␯

dx G A a Ž x . < ¨ n < ␯ dx

HF

< t n¨ n < ␯

and taking the limit we obtain lim a Ž x . < ¨ n < ␯

HF

g Ž t n¨ n . t n¨ n

dx G A a Ž x . < ␯ < ␯ dx ) 0.

HF

< t n¨ n < ␯

Ž 1.13.

This implies that the right hand side of Ž1.12. goes to infinity as n ª ⬁, which is a contradiction since the left hand side remains bounded. So we have ¨ Ž x . s 0 a.e. in ⍀q. Similarly we get ¨ Ž x . s 0 a.e. in ⍀y. Now let ␸ s ¨ in Ž1.4.; taking the limit we get H⍀ < ⵜ¨ < 2 dx s 0 or ¨ ' 0. This completes the proof of the claim and the proposition. PROPOSITION 1.3. the Ž PS . condition.

Assume the conditions in Lemma 1.1. Then I satisfies

Proof. Suppose  u n4 is a Ž PS . sequence. Then 5 u n 5 is bounded and for some u g D 1,0 2 Ž ⍀ . we have u n ª u weakly in D 1,0 2 Ž ⍀ .. But

Ž I⬘ Ž u n . , ␸ . s H ⵜu nⵜ␸ y H aŽ x . g Ž u n . ␸ dx s o Ž 1. 5 ␾ 5 ⍀

⍀

2 ᭙␸ g D 1, 0 Ž ⍀. .

Ž 1.14.

So 5 u n 5 2 s Ž I⬘ Ž u n . , u n . q 5 u 5 2 s Ž I⬘ Ž u n . , u . q

H⍀aŽ x . g Ž u

H⍀aŽ x . g Ž u

n

n

. u n dx s H aŽ x . g Ž u n . u n dx q o Ž 1 . ⍀

. u dx s H aŽ x . g Ž u n . u dx q o Ž 1 . . ⍀

Next we show lim

H aŽ x . g Ž u

lim

H aŽ x . g Ž u

nª⬁ ⍀

nª⬁ ⍀

n

. u n dx s H aŽ x . g Ž u . udx

Ž 1.15.

. udx s H aŽ x . g Ž u . udx.

Ž 1.16.

n

⍀

⍀

Here we prove Ž1.15. since the proof of Ž1.16. is similar and in fact somewhat easier.

315

SEMILINEAR ELLIPTIC EQUATIONS

Proof of Ž1.15.. Since u n ª u in Lt ŽU . for 2 F t - 2* and U ; ⍀ bounded, if we denote ⍀ R s ⍀ l BR Ž0. then using Ž G 1 . ᎐ Ž G 3 ., for any R ) 0 we clearly have lim

H

nª⬁ ⍀ R

a Ž x . g Ž u n . u n dx s

H⍀ aŽ x . g Ž u . udx. R

So it is enough to show that lim

< a Ž x . g Ž u n . u n < dx s 0

H

uniformly in n.

Rª⬁ ⍀_⍀ R

Ž 1.17.

But by Ž G 1 . ᎐ Ž G 3 . for any ⑀ ) 0, p F t F 2* there exists C s C Ž ⑀ , t . ) 0 such that < g Ž s . < F ⑀ < s < q C < s < ty1

for all s g ⺢.

Ž 1.18.

So for any U ; ⍀, using Ž A 3 . and Holder’s inequality, we get

H⍀< aŽ x . g Ž u

n

2 . u n < dx F ⑀H < aŽ x . < Ž u n . dx q CH < aŽ x . 5 u n < p dx

U

U

F ⑀ Ž sup x g ⍀ < a Ž x . 5 x < 2 . qC

žH

< a Ž x . < r dx

U

F C1 ⑀ 5 u n 5 2 q C

žH

1 r

/

HU

< unŽ x . < < x<2

dx

p < u n < 2*

< a Ž x . < r dx

U

1 r

/

5 un 5 p ,

Ž 1.19.

where r s 2 NrŽ2 N y Ž N y 2. p .. Now Ž A 3 . implies aŽ x . g L␣ Ž ⍀ . for all N ␣) and since 2 - p F 2*, we have N2 - r - ⬁. So there exists R s RŽ ⑀ . 2 such that H⍀ _ ⍀ R < aŽ x .< r dx - ⑀ . Finally taking U s ⍀ _ ⍀ R in Ž1.19. and recalling that 5 u n 5 is uniformly bounded we get Ž1.17.. The limits Ž1.15. and Ž1.16. yield 5 u n 5 ª 5 u 5 which of course implies u n ª u strongly in Ž ⍀ .. D 1,2 0 2. EXISTENCE RESULTS Now we are ready to prove some existence results for ŽP.. Our first result concerns positi¨ e solutions of ŽP..

316

HOSSEIN TEHRANI

THEOREM 2.1. Assume g g C Ž⺢q, ⺢. satisfies Ž G 2 . ᎐ Ž G4 . for s G 0. Furthermore assume Ž A1 . ᎐ Ž A 3 . and Ž A 4 . or Ž G5 .. Then Eq. ŽP. has a positi¨ e solution. Proof. First we extend the definition of g to all s g ⺢ as an odd function. The extension, still denoted by g, satisfies Ž G 1 . ᎐ Ž G4 . Žas well as Ž G5 . if need be.. We will verify that the energy functional I Ž u. s

1

H⍀ 2 < ⵜu <

2

y a Ž x . G Ž u . dx ,

2 u g D 1, 0 Ž ⍀.

satisfies conditions Ža. ᎐ Žc. of Proposition 1.2. Proposition 1.3 provides the ŽPS. condition. Below we first prove that for some ␳ 0 ) 0 for 0 - ␳ F ␳ 0 .

inf I Ž u . ) 0,

5 u 5s ␳

Ž 2.1.

In fact using Ž1.18., for 5 u 5 s ␳ we have I Ž u. G G

G

< ⵜu < 22 2 < ⵜu < 22 2 < ⵜu < 22 2

H⍀< aŽ x . 5 u < dx y H⍀< aŽ x . 5 u 5

y⑀

2

½H

yC ⑀

⍀

< u< 2 < x<2

dx y

H⍀< u <

2*

dx

2*

dx

5

y C Ž ⑀ < ⵜu < 22 q < ⵜu < 2* 2 .

G C0 < ⵜu < 22 s C0 ␳ with the last inequality satisfies for some C0 ) 0 if ␳ F ␳ 0 , and ␳ 0 sufficiently small. Finally since ⍀q/ ␾ , by taking 0 F ␺ g C0⬁Ž ⍀q. and using Ž1.11. we have I Ž t␺ . s

t2 2

H⍀< ⵜ␺ <

2

dx y

H⍀aŽ x . G Ž t␺ . dx ª y⬁

as t ª ⬁.

So conditions of Proposition 1.2 are satisfied. Next we define

␤ [ inf

sup I Ž ␥ Ž t . . ,

␥g⌫ 0FtF1

where 2 ⌫ s  ␥ g C Ž w 0, 1 x , D 1, 0 Ž ⍀ . . : ␥ Ž 0 . s 0, ␥ Ž 1 . s ␺ 4 .

SEMILINEAR ELLIPTIC EQUATIONS

317

Then ␤ ) 0 and by the Mountain Pass Lemma it is a critical value of I, with a corresponding critical point u 0 . Furthermore since Žthe extended. g is odd, G Ž s . is even and we have I Ž< u <. s I Ž u.. Then by a result of Brezis and Nirenberg Žsee w4x. we may assume that u 0 G 0. Now an application of elliptic regularity theory and the maximum principle implies that u 0 g C 1, ␣ Ž ⍀ . and u 0 Ž x . ) 0 in ⍀. Next we recall a critical point theorem for even functionals that will be used in the proof of our next result. Assume E is an infinite dimensional Banach space, I g C 1 Ž E, ⺢., an even functional satisfying the ŽPS. condition. In addition assume Ži. I ) 0 in B␳ _  04 , I G ␣ on ⭸ B␳ for some ␳ , ␣ ) 0. Žii. There exists a k dimensional subspace X k of E, such that X k l A 0 is bounded and sup I Ž u . - ⬁, ugX k

where A0 [  u g E : I Ž u . G 04 . Let ⌫ [  h : h is an odd homeomorphism of E onto E, h Ž 0 . s 0, h Ž B1 . ; A 0 4 with B1 the unit ball of E. Furthermore let ⌫m [  K ; E : K is compact symmetric with respect to 0, and for all h g ⌫, ␥ Ž K l h Ž S . . G m4 , where S s ⭸ B1 and ␥ Ž K . is the classical genus of a closed, symmetric subset K ; E Žsee w11x.. PROPOSITION 2.2. and Žii. abo¨ e. Let

Let E be a Banach space, I g C 1 Ž E, ⺢. satisfying Ži.

bm s inf sup I Ž u . , Kg ␥m ugK

m s 1, 2, . . . , k.

Then we ha¨ e Ž1. 0 - ␣ F b1 F ⭈⭈⭈ F bk - ⬁ and b1 , . . . , bk are critical ¨ alues of I. Ž2. If bm s bmq1 for some m g  1, . . . , k 4 , then I has infinitely many critical points corresponding to bm . Now we can state our multiplicity result for ŽP..

318

HOSSEIN TEHRANI

THEOREM 2.3. Assume Ž A1 . ᎐ Ž A 3 . and Ž G 1 . ᎐ Ž G4 . and either Ž A 4 . or Ž G5 .. Furthermore assume that g is odd, i.e., g Žys . s yg Ž s ., s g ⺢. Then ŽP. has infinitely many solutions. Proof. Since g is odd the functional I Ž u. s

1

H⍀ 2 < ⵜu <

2

y a Ž x . G Ž u . dx ,

2 u g D 1, 0 Ž ⍀.

is even and by Ž2.1. satisfies condition Ži. above. In addition for any k g N given, we can choose ␰ 1 , . . . , ␰ k with 0 F ␰ i g C0⬁Ž ⍀q. , i s 1, . . . , k, Supp ␰ i l Supp ␰ j s ␾ for i / j. Then as was noted in the proof of Theorem 2.1 above, I Ž t ␰ i . ª y⬁ as t ª ⬁ for every i. Next we take X k s Span ␰ 1 , . . . , ␰ k 4 . Since the ␰ i ’s have disjoint support, we have I Ž ⌺ 1k t i ␰ i . s Ý1k I Ž t i ␰ i . which readily implies that condition Žii. is satisfied for this choice of X k . So Proposition 2.2 can be applied and we get at least k solutions of ŽP.. Since k was arbitrary, this proves the existence of infinitely many solutions.

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