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SOME INDEFINITE NONLINEAR EIGENVALUE
PROBLEMS
A. SU´
AREZ∗
Dp o. Ecuaciones Di e enciales y An´alisis Num´e ico,
Fac. Ma em´a icas, C/ Ta ia s/n, C.P. 41012,
Uni . Se illa, Spain
E-mail: sua [email p o ec ed]
Dedica ed o P o . Jean Mawhin o his i s 60 yea s o Nonlinea Analysis
In his wo k we s udy he s uc u e o he se o posi i e solu ions o a nonlinea
eigen alue p oblem wi h a weigh changing sign. Speci ically, he eac ion e m
a ises om a popula ion dynamic model. We use mainly bi u ca ion me hods o
ob ain ou esul s.
1. In oduc ion
The aim o his wo k is o s udy some nonlinea inde ini e eigen alue p ob-
lems o he o m
½−∆u=λm(x) (u) in Ω,
u= 0 on ∂Ω, (1)
whe e Ω ⊂IRNis a bounded domain wi h a egula bounda y ∂Ω, m∈C(Ω)
changes sign, is a egula unc ion and λplays he ole o eal pa ame e .
We ocus ou a en ion on he case (0) = 0 and λ > 0; simila esul s can
be ob ained o nega i e alues o λ.
Depending o he shape o , Eq. (1) models di e en si ua ions: pop-
ula ion dynamics, popula ion gene ics, combus ion heo y,... see [10].
In he linea case, i.e., (u) = u, (1) is he eigen alue p oblem
½−∆u=λm(x)uin Ω,
u= 0 on ∂Ω. (2)
∗Suppo ed by he Spanish Minis y o Science and Technology unde g an s BFM2000-
0797 and BFM2003-06446.
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I is well known (see o ins ance [19] and [23]) ha he e exis wo alues
o λ,λ−(m)<0< λ+(m), called p incipal eigen alues because hey ha e
associa ed posi i e eigen unc ions. In he p esen wo k, gi en q∈L∞(Ω)
we deno e by σΩ
1[−∆ + q] (we dele e he supe sc ip Ω when no con usion
a ises) he p incipal eigen alue o he p oblem
−∆u+q(x)u=λu in Ω, u= 0 on ∂Ω.
When in (1) he weigh does no appea , i.e., m≡1, he nonlinea
p oblem
½−∆u=λ (u) in Ω,
u= 0 on ∂Ω, (3)
has been ex ensi ely s udied. Classical e e ences a e [2] and [21], bu many
o he s can be gi en whe e, as well as exis ence esul s, uniqueness ones a e
shown: [4], [14], [26], [20], [22] and e e ences he ein.
Much less is known o p oblem (1). In [19], assuming o example ha
0(0) >0, he au ho s showed ha he e exis s an unbounded con inuum o
posi i e solu ions bi u ca ing om he i ial solu ion a λ=λ+(m)/ 0(0).
In [8] he au ho s assumed ha :I7→ IR+,I⊂IR, and 00 <0 and
showed ha e e y posi i e solu ion o (1) is s able. I , mo eo e , I= [0,1],
(1) = 0 and 0(0) >0 hey p o ed ha he e exis s a posi i e solu ion
i , and only i , λ>λ+(m)/ 0(0), and in his case he solu ion is unique.
Simila esul was shown in [13], al hough he au ho s’ mo i a ion was o
s udy he p oblem in he whole space. Ve y ecen ly, in [9] he au ho s
analyze he pa icula cases (u) = gi(u), i= 1,2 wi h
g1(u) = u−u2, g2(u) = u+u2.(4)
Obse e ha he esul o [8] can only be applied o g1. In [9], wi hou
he assump ion ha akes only alues in [0,1], he main esul o [8] was
imp o ed showing (by a ia ional me hod) ha , assuming some es ic ion
in he space dimension, he e exis s posi i e solu ion i λ∈(0, λ+(m)).
Fo he case, =g2, hey also p o ed he exis ence o posi i e solu ion
o λ∈(0, λ+(m)) and ha he e does no exis posi i e solu ion a λ=
λ+(m). In [16] hese esul s ha e been again comple ed. We p o e o
=g1 ha he e exis a leas wo posi i e solu ions in λ∈(λ+(m),∞),
one o hem linea ly asymp o ically s able and ha o =g2 he e exis s
posi i e solu ion i , and only i , λ∈(0, λ+(m)).
In his wo k, we a e going o analyze he ollowing nonlinea i ies
1(u) = u−u2−Ku
1 + u, 2(u) = u+u2−Ku
1 + u,(5)
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whe e K∈IR. Obse e ha he unc ions in (4) a e included in (5). These
las nonlinea i ies a ise in popula ion dynamics. Indeed, when K= 0, 1
is he classical logis ic eac ion e m and o K6= 0 he p eda ion one
Ku/(1 + u) is called he Holling-Tanne e m, see o example [7] o an
ecological in e p e a ion.
In o de o s a e ou main esul s we need some no a ions. Speci ically,
assume ha
M±:= {x∈Ω : m±>0}
a e open and egula se s, whe e m± ep esen he posi i e and nega i e
pa o m espec i ely; and suppose ha m±(x)≈[dis (x, ∂M±)]γ± o x
close o ∂M±and some γ±≥0. The ollowing condi ion will p o ide us
wi h a p io i bounds o he solu ions
2<min ½N+1+γ±
N−1,N+ 2
N−2¾.(6)
Finally, we de ine o K6= 1 he alues
λ+:= λ+(m)
1−Kλ−:= λ−(m)
1−K,
and Π : IR ×C(Ω) 7→ IR he p ojec ion map on o IR, i.e. Π(µ, u) = µ. The
main esul s a e:
Theo em 1.1. Assume ha K6= 1 and (6).
(1) The e exis s an unbounded con inuum Co posi i e solu ions o (1)
bi u ca ing om he i ial solu ion a λ=λ+i K < 1and λ=λ−
i K > 1.
(2) The bi u ca ion is supe c i ical o = 1and o = 2and
K < −1o K > 1and subc i ical o = 2and K∈[−1,1).
(3) I = 1and K < 1( esp. = 2and K > 1), hen Π(C) =
(λ+,∞)( esp. (λ−,∞)). Mo eo e , i (λ, uλ)∈ C, hen uλis
linea ly asymp o ically and such ha uλ≤√1−K( esp. √K−1).
Fu he mo e, he e exis s ano he posi i e solu ion λ o all λ > 0.
(4) I = 1and K > 1( esp. = 2and K < −1) hen Π(C) =
(0, λ∗] o λ∗> λ−( esp. λ+). Mo eo e , he e exis λ0and λ∗
wi h λ0< λ∗such ha o λ≥λ∗ he p oblem (1) does no admi
posi i e solu ions and i possesses a leas wo posi i e solu ions o
λ∈(λ−, λ0)( esp. (λ+, λ0)).
(5) I = 2and K∈[−1,1) he e exis s posi i e solu ion o λ∈
(0, λ+)and (1) does no admi posi i e solu ions o λ≥λ∗.
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(6) In any case, i he e exis s a solu ion λ o λ > 0, hen
limλ→0k λk∞= +∞.
Theo em 1.2. Assume K= 1 and (6). Then he e exis s a leas a solu-
ion uλ o λ > 0and limλ→0kuλk∞= +∞.
Rema k 1.1.
(1) The exis ence o Cis ue wi hou assuming (6). In he cases (4)
and (5) o Theo em 1.1, Ccould “go o in ini y” in a alue λ0.
(2) In he pa icula case = 2and K= 0, in [16] i was p o ed using
a Picone inequali y ha (1) possesses a posi i e solu ion i , and only
i , λ∈(0, λ+).
In Figs. 1 and 2 we ha e summa ized hese esul s ( he case = 2and
K= 1 is simila o = 1and K= 1).
λ
|| . ||
|| . || || . ||
a) b) c)
+λ−λ λ λ
Figu e 1. Bi u ca ion diag ams o = 1: a) K < 1; b) K= 1; c) K > 1.
The es o he pape is o ganized as ollows: Secs. 2 and 3 a e de o ed
o p o e Theo ems 1.1 and 1.2, espec i ely.
2. P oo o Theo em 1.1
2.1. Local bi u ca ion
In his subsec ion we show he di ec ion o bi u ca ion om he i ial
solu ion o bo h cases 1and 2. Fo ha , we w i e he nonlinea i y o he
ollowing manne
(u) = u∓u2−Ku
1 + u=u(1 −K) + u2(K
1 + u∓1).
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λ λ λ λλλ
a) b) c)
+ + −
|| . || || . || || . ||
Figu e 2. Bi u ca ion diag ams o = 2: a) K < −1; b) K∈[−1,1); c) K > 1.
I is clea ha o s udy (1) is equi alen o ind ze os o L(λ)u−N(λ, u) = 0,
whe e
L(λ)u:= u−λ(−∆)−1m(x)(1 −K)u,
N(λ, u) := λ(−∆)−1m(x)u2(K
1 + u∓1).
We can p o e ha
N(L(λ+)) = Span < ϕ+>and d
dλL(λ+)ϕ+/∈R(L(λ+)) (7)
whe e, gi en any linea con inuous ope a o L,N[L] and R[L] s and o he
null space and he ange o L, espec i ely, and
−∆ϕ+=λ+(m)m(x)ϕ+in Ω, ϕ+= 0 on ∂Ω.(8)
The i s equali y o (7) is i ial, o he second exp ession we need he
ollowing esul .
Lemma 2.1. Fo any p≥2we ha e ha
ZΩ
m(x)(ϕ+)p>0.
P oo : Mul iplying (8) by (ϕ+)p−1we ge
λ+(m)ZΩ
m(x)(ϕ+)p=ZΩ
(−∆ϕ+)(ϕ+)p−1= (p−1) ZΩ|∇ϕ+|2(ϕ+)p−2>0.
¦
Now, we show (7). Assume ha he e exis s usuch ha
d
dλL(λ+)ϕ+=−(−∆)−1m(x)(1 −K)ϕ+=u−(−∆)−1m(x)λ+(1 −K)u,

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hen
(−∆−λ+(m)m(x))u=−(1 −K)m(x)ϕ+,
and so, mul iplying by ϕ+we ge a con adic ion using Lemma 2.1.
Now, we can apply he C andall-Rabinowi z Theo em [15] and conclude
ha he e exis s δ > 0 such ha in a neighbo hood o (λ+,0) he non i ial
solu ions o (1) a e o he o m
u(s) = sϕ++s2ϕ2+s3ϕ3+o(s3),
λ(s) = λ++sλ1+s2λ2+o(s2).
In oducing hese e ms in (1), using (8) and a Taylo exp ession o he
unc ion 1/(1 + u(s)), we ge
(−∆−λ+(m)m(x))ϕ2=λ+m(x)(ϕ+)2(K∓1) + λ1m(x)(1 −K)ϕ+,
and so,
λ1=−λ+(K∓1)
1−K
ZΩ
m(x)(ϕ+)3
ZΩ
m(x)(ϕ+)2
.(9)
Obse e ha in he pa icula case = 2and K=−1, λ1= 0, and so we
ha e o calcula e λ2. I can be p o ed ha
λ2=−λ+
2
ZΩ
m(x)(ϕ+)4
ZΩ
m(x)(ϕ+)2
.(10)
F om (9) and (10), we conclude he pa ag aph (2) o Theo em 1.1. Analo-
gously i can be ea ed he case λ−.
2.2. Non-exis ence esul s
Lemma 2.2. Assume = 1and K > 1o = 2and K < 1. Then,
he e exis s λ∗>0such ha o λ≥λ∗(1) does no ha e posi i e solu ions.
P oo : Assume = 1and K > 1. Fi s ly obse e ha
h(x) := x(K
1 + x−1) ≤(√K−1)2,∀x≥0.(11)
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Le ube a posi i e solu ion o (1). Then, using he mono ony o he p in-
cipal eigen alue wi h espec o he domain and (11) we ge
0 = σ1[−∆−λm(x)(1 −K)−λm(x)u(K
1 + u−1)] <
< σM−
1[−∆−λm(x)((1 −K)+(√K−1)2)] =
=σM−
1[−∆−λm(x)2(1 −√K)],
which is an absu dum o λla ge.
Now, assume = 2and K < 1. In his case,
x(K
1 + x+ 1) ≥0,i K≥ −1, ∀x≥0,
x(K
1 + x+ 1) ≥ −(√−K−1)2,i K < −1, ∀x≥0.
So, i −1≤K < 1 we ha e
0 = σ1[−∆−λm(x)(1−K)−λm(x)u(K
1 + u+1)] < σM+
1[−∆−λm(x)(1−K)];
on he o he hand, o K < −1,
0 = σ1[−∆−λm(x)(1−K)−λm(x)u(K
1 + u+1)] < σM+
1[−∆−λm(x)2√−K],
in bo h cases a con adic ion o la ge λ.¦
2.3. Mul iplici y esul s
To ob ain mul iplici y esul s, we include (1) in he mo e gene al equa ion
½−∆u=µm(x)(1 −K)u+λm(x)g(u) in Ω,
u= 0 on ∂Ω, (12)
whe e gsa is ies
(Hg)g(0) = g0(0) = 0, g00(u)<0,lim
s→+∞
g(s)
s2=β < 0.
P oblem (12) has a ac ed a g ea deal o a en ion du ing las yea s (see
o example [1], [3], [5], [6], [18] and [24]) when m≡1 in he i s e m on
he igh -hand side o (12) and in [11], [12] and [13] wi h he igh -hand side
o he o m µh(x)u+g(x)upand es ic i e condi ions on hand gwhich
a e no sa is ied in ou case. In [16] was p o ed (see Fig. 3):
P oposi ion 2.1. Assume ha gsa is ies (Hg),(6),K6= 1 and ix λ > 0.
Deno e by
Λ+:= λ+(m(x)(1 −K)),Λ−:= λ−(m(x)(1 −K)).
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Then, (12) possesses a posi i e solu ion i µ > Λ−. Mo eo e , om he i -
ial solu ion u= 0 emana e wo unbounded in IR×C(Ω) con inua o posi i e
solu ions C+:= {(µ, uµ)}and C−:= {(µ, wµ)}a µ= Λ+and µ= Λ−,
espec i ely. Bo h con inua bi u ca e o he igh and Π(C−)⊃(Λ−,+∞),
Π(C+) = (Λ+,+∞). Finally, o µ > Λ+,uµis linea ly asymp o ically
s able and uµ6=wµ.
Rema k 2.1. Obse e ha o K < 1,
Λ+=λ+and Λ−=λ−,
and o K > 1,
Λ+=λ−and Λ−=λ+.
Indeed, o example o K > 1, i ollows ha
Λ+=λ+(m(x)(1 −K)) = λ+(−m(x))
K−1=−λ−(m(x))
K−1=λ−(m(x))
1−K=λ−.
µλ
λ
C
C
|| . ||
+
+
−
−
Figu e 3. Bi u ca ion diag am o (12) and K < 1.
2.4. P oo o Theo em 1.1:
Be o e p o ing he esul , we gene alize a well-known esul o m≡1. The
p oo is coming om [8].
Lemma 2.3. Assume ha is a egula unc ion and (0) = 0. Le u0be
a posi i e solu ion o (1) such ha (u0)>0, i holds:
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(1) I 00(u0)<0, hen u0is linea ly asymp o ically s able.
(2) I 00(u0)>0, hen u0is uns able.
P oo : We ha e o calcula e he sign o he eigen alue σ1[−∆−
λm(x) 0(u0)]. Take ψ:= (u0)>0, hen
(−∆−λm(x) 0(u0))ψ=− 00(u0)|∇u0|2.
So, i is conca e ( esp. con ex) he unc ion ψis a supe solu ion
( esp. subsolu ion) o −∆−λm(x) 0(u0), and hen (see [23]) σ1[−∆−
λm(x) 0(u0)] >0 ( esp. <0). ¦
The ollowing esul is p o ed in Theo em 3.4 o [3] and p o ides us wi h
a p io i bounds o he posi i e solu ions o (1).
Lemma 2.4. Assume (6). I (λ, u)is a posi i e solu ion o (1) and λ∈J,
whe e Jis a compac subse such ha J⊂(0,∞), hen he e exis s a
posi i e cons an C(independen om λ) such ha
kuk∞≤C.
Finally, he ollowing esul is p o ed in [17].
Lemma 2.5. Assume ha Σ⊂I×C2
0(Ω),I⊂IR an in e al, is a con-
nec ed se o posi i e solu ions o (1). Conside u:I7→ C2
0(Ω) a con inuous
map o supe solu ion o each λ∈I, bu no a solu ion. I u0< u(λ0) o
some (λ0, u0)∈Σ, hen u < u(λ) o all (λ, u)∈Σ.
We a e eady o p o e he esul . By subsec. 2.1 we know ha he e
exis s bi u ca ion om he i ial solu ion a λ=λ+o λ=λ−when K < 1
o K > 1, espec i ely. Mo eo e , we can apply Theo em 6.4.3 o [25], and
conclude ha om λ=λ+o λ=λ−bi u ca es an unbounded con inuum
Co posi i e solu ions o (1). We would like o ema k ha he a de ailed
p oo ha Cis unbounded and i does no sa is y he o he al e na i es o
he abo e men ioned esul will be p esen ed elsewhe e.
Now assume = 1and K < 1. I is clea ha
u:= √1−K
is a supe solu ion o (1). So, we can apply Lemma 2.5 ( aking λ0=λ+)
and conclude ha
o all (λ, uλ)∈ C, we ha e ha uλ<√1−K. (13)
Mo eo e , 1(uλ)>0 and 00
1(uλ)<0, and so by Lemma 2.3 we ge ha
uλis linea ly asymp o ically s able.