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5 Landesman–Lazer Conditions: The Asymmetric Case

5 Landesman–Lazer Conditions: The Asymmetric Case

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146



6 Playing Around Resonance



Theorem 6.5.1 Assume that

g.t; x/ D xC

where



> 0,



x C h.t; x/ ;



> 0 are such that

p



Cp D



T

;

N



and h is a bounded function, i.e., there is a C > 0 such that

jh.t; x/j Ä C ;



for every .t; x/ 2 Œ0; T



R:



If, moreover, for any non-zero v satisfying

.Plim /



v 00 C v C

v D 0;

v.0/ D v.T/ ; v 0 .0/ D v 0 .T/ ;



one has

Z



Z

lim sup h.t; x/v.t/ dt C



fv<0g x! 1



lim inf h.t; x/v.t/ dt > 0 ;



fv>0g x!C1



then problem (P) has a solution.

Proof We fix a small " > 0 and consider, for

.P /



2 Œ 12 ; 1, the problem



x00 C .2 2 /.. C "/xC . C "/x / C .2

x.0/ D x.T/ ; x0 .0/ D x0 .T/ :



1/g.t; x/ D 0 ;



As usual, we assume by contradiction that there are two sequences . n /n in Œ 12 ; 1,

and .xn /n , with xn solution of (P n ), such that kxn k1 ! 1. Then, vn D xn =kxn k1

verifies

8 00

v .t/ C Œ.2 2 n /. C "/ C .2 n 1/  vnC

ˆ

ˆ

< n

h.t; xn /

D 0;

Œ.2 2 n /. C "/ C .2 n 1/  vn C .2 n 1/

ˆ

kxn k1

ˆ

:

0

0

vn .0/ D vn .T/ ; vn .0/ D vn .T/ :

There are some subsequences, which we denote by . n /n , and .vn /n , a real number

N 2 Œ 12 ; 1, and a function v 2 C1 .Œ0; T/ such that n ! N , and vn ! v in

C1 .Œ0; T/. So, kvk1 D 1, and v satisfies

v 00 C Q v C Q v D 0 ;

v.0/ D v.T/ ; v 0 .0/ D v 0 .T/ ;



6.5 Landesman–Lazer Conditions: The Asymmetric Case



147



with Ä Q Ä C ", Ä Q Ä C ". Therefore, it has to be N D 1, Q D , Q D ,

and v is a solution of .Plim /.

Let us write .xn ; x0n / in modified polar coordinates, as follows:

if xn 0,

1

xn D p



n



cos Ân ; x0n D



n



sin Ân ;



1

xn D p



n



cos Ân ; x0n D



n



sin Ân :



if xn Ä 0,



For n large enough, we have that

xn .t/2 C x0n .t/2 > 0 ;



for every t 2 Œ0; T ;



and we can see that

8

p x00n xn .x0n /2

ˆ

ˆ

; if xn > 0 ;

<

x2n C .x0n /2

Ân0 D

00

0

2

p xn xn .xn /

ˆ

ˆ

:

; if xn < 0 :

x2n C .x0n /2

Integrating on fxn > 0g and fxn < 0g, respectively, we obtain

N D



p

p



N D



Z

Z



p



Z



p



Z



Œ.1

fxn >0g



fxn >0g



Â

1C

Œ.1



fxn <0g



fxn <0g



n /.



n h.t; xn /xn

x2n C .x0n /2

n /.



Â

1C



C "/ C

Ã



C "/ C



n h.t; xn /xn

x2n C .x0n /2



n

x2n



Ã



 x2n C n h.t; xn /xn C .x0n /2

C .x0n /2



;

n

x2n



 x2n C n h.t; xn /xn C .x0n /2

C .x0n /2



:



Hence,

Z

Z



fxn >0g



n h.t; xn /xn

x2n C .x0n /2



N

Äp



measfxn > 0g ;



N

Äp



fxn <0g



n h.t; xn /xn

x2n C .x0n /2



measfxn < 0g :



148



6 Playing Around Resonance



Since v has only simple zeros, and vn ! v in C1 .Œ0; T/, also vn has only simple

zeros, for n sufficiently large, so that the set of points where xn vanishes has zero

measure. Therefore,

Z



T

0



N

N

n h.t; xn /xn

Ä p Cp

2 C .x /2 C .x0 /2

.xC

/

n

n

n



T D 0:



Hence also

Z



h.t; xn /vn

Ä 0:

.vnC /2 C .vn /2 C .vn0 /2



T

0



and, by the Fatou Lemma,

Z



h.t; xn /vn

Ä 0:

.vnC /2 C .vn /2 C .vn0 /2



T



lim inf

0



n



Since v C .t/2 C v .t/2 C v 0 .t/2 is constant in t, and

lim. .vnC /2 C .vn /2 C .vn0 /2 / D .v C /2 C .v /2 C .v 0 /2 ;

n



uniformly in Œ0; T, it has to be

Z



T

0



lim inf h.t; xn /vn Ä 0 ;

n



and, therefore,

Z



Z

lim sup h.t; x/v.t/ dt C



fv<0g x! 1



lim inf h.t; x/v.t/ dt Ä 0 ;



fv>0g x!C1



in contradiction with the hypothesis.

For 2 0; 12 , one proceeds as in the proof of Theorem 5.5.1, connecting the

point . C "; C "/ to the diagonal by the use of a curve which does not touch the

set †. The proof is thus completed.

Symmetrically, we also have the following.

Theorem 6.5.2 Assume that

g.t; x/ D xC



x C h.t; x/ ;



6.6 Lazer–Leach Conditions for the Asymmetric Oscillator



> 0,



where



149



> 0 are such that

Cp D



p



T

;

N



and h is a bounded function, i.e., there is a C > 0 such that

jh.t; x/j Ä C ;



for every .t; x/ 2 Œ0; T



R:



If, moreover, for any non-zero v satisfying

v 00 C v C

v D 0;

v.0/ D v.T/ ; v 0 .0/ D v 0 .T/ ;



.Plim /

one has



Z



Z

lim inf h.t; x/v.t/ dt C



fv<0g x! 1



lim sup h.t; x/v.t/ dt < 0 ;



fv>0g x!C1



then problem (P) has a solution.



6.6 Lazer–Leach Conditions for the Asymmetric Oscillator

We consider again problem (Q), and assume that

g.x/ D xC

with



> 0,



x C h.x/ ;



> 0 such that

Cp



D p



D



T

;

N



and that h has finite limits

h. 1/ D lim h.x/ ;

x! 1



h.C1/ D lim h.x/ :

x!C1



As in Sect. 5.4, let



.t/ D



8

1

p

ˆ

ˆ

t/ ;

< p sin.

p

1

ˆ

ˆ

: p sin



p



Ä

if t 2 0 ; p

Ä

ÁÁ

t ; if t 2 p ;



;

;



150



6 Playing Around Resonance



extended by -periodicity to the whole R. Let us define the -periodic continuous

function

à Z T

Â

h.C1/ h. 1/

ˆ.Â/ D 2N

e.t/ .t C Â/ dt :

(6.3)

0



The Lazer–Leach condition is generalized in the following corollary.

Corollary 6.6.1 (Dancer, 1976) If

./ Ô 0 ;



for every 2 0; ;



then problem (Q) has a solution.

Proof Assume, for instance, that ˆ.Â/ > 0 for every  2 Œ0; . Writing v.t/ D

.t C Â/, it can be seen that

Z

fv>0g



vD



2N



Z

;



fv<0g



. v/ D



2N



:



(6.4)



Hence,

Z

lim sup .h.x/



e.t//v.t/ dt D



fv<0g x! 1



D



2N



Z

h. 1/

fv<0g



e.t/ .t C Â/ dt ;



and

Z

lim inf .h.x/



e.t//v.t/ dt D



fv>0g x!C1



D



2N



Z

h.C1/

fv>0g



whence, setting h.t; x/ D h.x/



e.t/ .t C Â/ dt ;



e.t/,

Z



Z

lim sup h.t; x/v.t/ dt C



fv<0g x! 1



Â

D 2N



h.C1/



lim inf h.t; x/v.t/ dt D



fv>0g x!C1



h. 1/



Ã



Z



T

0



e.t/ .t C Â/ dt > 0 ;



so that Theorem 6.5.1 applies.

In the case when ˆ.Â/ < 0 for every  2 Œ0; , one sees analogously that

Theorem 6.5.2 can be applied.



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