Semilinear Poisson problems in Sobolev-Besov spaces on Lipschitz domains

Publ. Ma . 46 (2002), 353–403
SEMILINEAR POISSON PROBLEMS IN
SOBOLEV-BESOV SPACES ON LIPSCHITZ DOMAINS
Ma in Dindoˇ
s∗and Ma ius Mi ea†
Abs ac
Ex ending ecen wo k o he linea Poisson p oblem o he
Laplacian in he amewo k o Sobole -Beso spaces on Lipschi z
domains by Je ison and Kenig [16], Fabes, Mendez and Mi ea [9],
and Mi ea and Taylo [30], he e we ake up he ask o de-
eloping a simila sha p heo y o semilinea p oblems o he
ype ∆u−N(x, u)=F(x), equipped wi h Di ichle and Neumann
bounda y condi ions.
1. In oduc ion
As e idenced by he la ge body o wo ks (c ., e.g., he mono-
g aphs [23], [13], [10], [11], [4], [12], [36], [1], [14] and he e e ences
he ein) nonlinea ellip ic bounda y alue p oblems ha e become la ely
he cen e o conside able a en ion, especially due o hei pi o al ole
in such di e se disciplines as spec al and sca e ing heo y, diffe en ial
geome y, ma hema ical physics, e c. Meanwhile, and pa icula ly mo e
so in he las decade, his field o esea ch has subs an ially p ofi ed om
basic p og ess in many o he ela ed a eas in which some o i s me hods
and echniques a e oo ed.
In [28], [29], [30], [31], [27], he au ho s ha e ini ia ed a p og am
aimed a ex ending he Euclidean, cons an coefficien heo y om [39],
[6], [16], [9] o he case o Lipschi z domains in Riemannian mani olds.
One no able achie emen in [30] is de eloping a sha p linea heo y o
he Poisson p oblem wi h Di ichle and Neumann bounda y condi ions
in Lipschi z domains o he Laplace-Bel ami ope a o , hus ex ending
2000 Ma hema ics Subjec Classifica ion. P ima y: 35J65, 35B65; Seconda y: 42B20,
46E35.
Key wo ds. Nonlinea equa ions, Lipschi z domains, ellip ic PDE’s, bounda y alue
p oblems.
∗This esea ch was pa ially conduc ed by he au ho o he Clay Ma hema ics In-
s i u e.
†Au ho pa ly suppo ed by NSF.
354 M. Dindoˇ
s, M. Mi ea
he main esul s in [16], [9]. The aim o he p esen pape is o con inue
his line o wo k and p oduce a semilinea e sion o he a o emen ioned
esul s, which is in he na u e o bes possible. A ask simila in spi i
bu in a diffe en unc ional analy ic se ing has been accomplished in [7],
[8].
In o de o be mo e specific, we momen a ily dig ess o he pu -
pose o in oducing some no a ion and making some defini ions. Le M
be a smoo h, connec ed, compac , bounda yless mani old, o eal di-
mension dim M=n(unless explici ly men ioned o he wise, we shall
assume ha n≥3), equipped wi h a Riemannian me ic enso g=
j,k gjk dxj⊗dxkwhose coefficien s sa is y
gjk ∈C1+γ,some γ>0.(1.1)
The Laplace-Bel ami ope a o on Mis hen gi en in local coo dina es
by
∆u:= di (g ad u)=g−1/2∂j(gjkg1/2∂ku),(1.2)
whe e we use he summa ion con en ion, ake (gjk) o be he ma ix
in e se o (gjk), and se g:= de (gjk). Recall ha Ω ⊂Mis called
aLipschi z domain p o ided ∂Ω can be desc ibed in app op ia e local
coo dina es by means o g aphs o Lipschi z unc ions. Also, he Sobole
scale Lp
s(M), 1 <p<∞,s≥0, is ob ained by li ing Lp
s(Rn):=
{(I−∆)s/2 ; ∈Lp(Rn)} o M. We deno e by Lp
s(Ω) he es ic ion
o elemen s in Lp
s(M) o he Lipschi z domain Ω. As is cus oma y, we
se Lp
s,0(Ω) o he subspace consis ing o es ic ions o Ω o elemen s
om Lp
s(M) wi h suppo con ained in ¯
Ω. Fo s>0 and 1 <p,q<∞
wi h 1/p +1/q = 1, we se Lp
−s(Ω) := (Lq
s,0(Ω))∗. As is well-known, i
Bp,q
s(∂Ω), 1 ≤p, q ≤∞,0<|s|<1, s ands o he usual class o Beso
spaces on ∂Ω, hen he ace ope a o T is well-defined om Lp
s(Ω) on o
Bp,p
s−1/p(∂Ω) o each 1 <p<∞and 1/p<s<1+1/p. Fo a mo e
de ailed exposi ion, he in e es ed eade is e e ed o [32], [37], [3], [2],
and especially [16] o he con ex o Lipschi z domains.
Re u ning o he mains eam discussion, he e we shall be conce ned
wi h he semilinea ellip ic PDE
∆u−N(x, u)=F(x)inΩ⊂M(1.3)
equipped wi h ei he Di ichle o Neumann bounda y condi ions. Le
us poin ou ha when he nonlinea i y N(x, u) is o class C1in u hen
(1.3) can be eph ased as
∆u−a(x, u)u= in Ω,(1.4)
Semilinea Poisson P oblems 355
whe e
a(x, u):=1
0
∂N
∂u (x, u)d , and (x):=F(x)+N(x, 0).(1.5)
Unde a ious g ow h and smoo hness assump ions on Ω, N,Fand u,
he PDE (1.3) has ecei ed a lo o a en ion in he li e a u e la ely.
Ex ending some ea lie wo k in [38], T. Runs and W. Sickel ha e con-
side ed in [32, Chap e 6] he case o (1.3) when Ω is smoo h, he non-
linea i ies a e o powe ype, and u,Fbelong o Sobole -Beso -T iebel-
Lizo kin spaces. In [15], V. Isako and A. I. Nachman ha e ea ed (1.3)
equipped wi h a Di ichle bounda y condi ion in he case o a wo di-
mensional Euclidean Lipschi z domain Ω and when u∈L2
1(Ω) ∩C0(¯
Ω).
In [20], J. Johnsen and T. Runs ha e add essed (1.3) in he amewo k
o Sobole -Beso -T iebel-Lizo kin spaces in he case o smoo h domains
and nonlinea i ies o composi ion ype (i.e., when N(x, u)=N(u)). Re-
la ed wo k can also be ound in [18], [13], [5], [22].
In he case o Di ichle bounda y condi ions, ou main esul s (c . The-
o em 3.1 and Theo em 3.2) deal wi h he ollowing si ua ion: Ω is an a -
bi a y Lipschi z domain, u∈Lp
s+1/p(Ω), F∈Lp
s+1/p−2(Ω), 0 <s<1,
1<p<∞, and ei he N(x, u) has sublinea g ow h in uo a(x, u)
( om (1.5)) has an admissible polynomial g ow h (including a limi ing
case o exponen ial beha io ). Simila (albei echnically somewha less
efined) esul s hold in he case o Neumann bounda y condi ions, e en
in he p esence o (sublinea , powe ype) nonlinea i ies in he bounda y
condi ions; c . Theo em 4.1 o a p ecise s a emen .
We a e in e es ed in he maximal ange o indices s,p, o which (1.3)
is sol able in he con ex o Lipschi z domains on he Sobole -Beso
scales. In his espec , we would like o s ess ha ou esul s a e sha p
in he sense ha hey educe o an op imal linea heo y in he absence
o nonlinea i ies. Tha is, we sol e
∆u−a(x, u)u= ∈Lp
s+1/p−2(Ω),
T u=g∈Bp,p
s(∂Ω),u∈Lp
s+1/p(Ω)
(1.6)
unde admissible g ow h condi ions on a(x, u) (o polynomial and expo-
nen ial na u e) and o he same ange o indices s,pas in [30].
The emphasis in he app oach we de elop is on unde s anding how
he solu ion o he linea Poisson p oblem depends on lowe o de pe -
u ba ions o he Laplace-Bel ami ope a o . The main achie emen in
his ega d is o p o e ha he nonlinea mapping V→ (∆ −V)−1has
356 M. Dindoˇ
s, M. Mi ea
sublinea g ow h in V. Tha is, in a sui able unc ional analy ic con ex ,
(1.7) (∆ −V)−1≤C(1 + V)θ,
whe e θ∈[0,1) and Cis independen o V≥0.
See (2.6)–(2.7) o an exac o mula ion. In u n, his key es ima e allows
o a con enien implemen a ion o he Schaude fixed-poin heo em
in he amewo k o Lebesgue spaces. The p oo o (1.7) is a delica e
a gumen which elies on he maximum p inciple, es ima es o ac ional
powe s o he Di ichle Laplacian and in e pola ion. In he p ocess, we
also show ha i V∈Ln/2is nonnega i e, hen he Sch ¨odinge ope a o
∆−V:Lp
s+1/p,0(Ω) −→ Lp
s+1/p−2(Ω)(1.8)
has he same op imal in e ibili y ange as he unpe u bed Laplacian ∆.
The in eg abili y exponen n/2 is na u al, gi en he sha p unique con-
inua ion esul s om [17].
The layou o he pape is as ollows. In Sec ion 2 we efine he ea -
men o he linea Di ichle Poisson p oblem om [30] by allowing less
egula lowe o de e ms and by de i ing mo e p ecise es ima es. This
is hen used in Sec ion 3 o p o e, among o he hings, he sol abili y
o (1.6). Neumann bounda y condi ions a e conside ed in Sec ion 4. The
case o o nonlinea i ies N(x, u) wi h sublinea g ow h in uis ea ed in
Sec ion 5. This sec ion also con ains a mo e de ailed analysis o se e al
ele an examples. Finally, in Sec ion 6, sufficien condi ions on he da a
a e p oduced so ha a solu ion o (1.6) obeying non angen ial maximal
unc ion es ima es can be ound. He e we also analyze he case when
bounda y nonlinea i ies a e allowed.
Acknowledgemen s. I is a pleasu e o hank Michael Taylo o b ing-
ing us oge he and o his cons an suppo and in e es in ou wo k.
The second named au ho would also like o hank Win ied Sickel and
Jon Johnsen o some help ul con e sa ions.
2. The linea heo y e isi ed
We shall e ain as much as possible he no a ion in oduced in Sec-
ion 1. The aim o his sec ion is o p o e a efined e sion o he
well-posedness esul o he Di ichle Poisson p oblem o ∆ on Sobole -
Beso spaces om [30].
Semilinea Poisson P oblems 357
Theo em 2.1. Le Ω⊂Mbe an a bi a y Lipschi z domain and o a
unc ion Vsa is ying
V≥0and V∈L (M) o some ≥n/2,(2.1)
se L:= ∆ −V. Then he e exis s ε=ε(Ω) >0wi h he ollowing
significance. Le 0<s<1and 1<p<∞sa is y a leas one o he
ollowing condi ions:
2
1+ε<p< 2
1−εand 0<s<1;
1≤p< 2
1+εand 2
p−1−ε<s<1;
2
1−ε<p≤∞ and 0<s<2
p+ε,
(2.2)
and, in addi ion,
1
−2
n<n−1
np −s
n<1−1
.(2.3)
Then he Di ichle Poisson p oblem
(DP)








Lu = ∈Lp
s+1
p−2(Ω),
T u=g∈Bp,p
s(∂Ω),
u∈Lp
s+1
p
(Ω),
(2.4)
has a unique solu ion, which also sa isfies
uLp
s+1
p
(Ω) ≤C Lp
s+1
p−2(Ω) +gBp,p
s(∂Ω)
(2.5)
o some C=C(Ω,V,p,s)>0.
In oduce τ:= np/(n−1−sp)i sp < n−1and τ:= ∞i sp > n−1.
Then, i 1/p ≤sand sp =n−1, he e exis s a cons an C=C(Ω,p,s)>
0, independen o V, such ha
uLτ(Ω) ≤C( Lp
s+1
p−2(Ω) +gBp,p
s(∂Ω)).(2.6)

358 M. Dindoˇ
s, M. Mi ea
I sp =n−1 hen o some C>0independen o Vand any τ∈[2,∞),
uLτ(Ω) ≤Cτ1−1/p( Lp
s+1
p−2(Ω) +gBp,p
s(∂Ω)).(2.7)
On he o he hand, i s<1/p, hen he es ima e
uLτ(Ω) ≤C(1 + VL (Ω))θ( Lp
s+1
p−2(Ω) +gBp,p
s(∂Ω)),(2.8)
holds wi h
θ:=


ε+
2( −1) (1
p−s),i ≥1+(1−sp)/(2p−2),
ε+1
2−n/ ((n−1)/p−s−n+2),i <1+(1−sp)/(2p−2),
(2.9)
whe e ε>0is a bi a ily small and C=C(Ω,s,p,ε)>0is independen
o V.
In dimension n=2, esul s simila in spi i hold p o ided (2.2) is
eplaced by
1
2−ε<1
p<1
2+εand 0<s<1;
1
2+ε≤1
p<1and 1
p−1
2−ε<s<1;
0<1
p≤1
2−εand 0<s<1
p+1
2+ε,
(2.10)
whe e ε=ε(∂Ω) ∈(0,1
2]. Mo e specifically, he es ima es (2.5)–(2.7)
hold unchanged, whe eas (2.8) holds i also >4/3.
Finally, when ∂Ω∈C1, ins ead o (2.2) when n≥3,o (2.10) when
n=2, we may simply allow s∈(0,1),p∈(1,∞).
Pa en he ically, we no e ha when >n hen (2.3) is sa isfied o
any s∈(0,1), p∈(1,∞). Ano he simple ye use ul ema k is ha , i
ε>0 is small enough, hen
n≥3, >n
2,s,pas in (2.2) wi h ε>0 small =⇒(2.3) holds.(2.11)
Fu he mo e, in he wo dimensional con ex ,
(2.12) >4
3,s,pas in (2.10)
wi h ε>0 small =⇒(2.3) holds au oma ically.
Semilinea Poisson P oblems 359
1
p
sε
1−ε
1
2
1+ε
2
1−ε
2
1
1
✦✦✦✦✦✦
✦
✦✦✦✦✦✦
✦
0
Figu e 1. The hexagon desc ibing he well posedness
egion (2.2).
Be o e p esen ing he p oo o he Theo em 2.1, we need o discuss
se e al auxilia y esul s.
Lemma 2.2. Le Ωbe a Lipschi z domain (o dimension ≥2) and fix
1<p<∞,0<s<1,n/2≤ ≤∞, so ha (2.3) holds. Then he
inclusion
L (Ω) ·Lp
s+1/p(Ω) !→Lp
s+1/p−2(Ω)(2.13)
is compac .
P oo : Conside fi s he case when sp < n −1 and >n/2. In his
scena io, we shall show ha he e exis s ε=ε(n, p, s, )>0 such ha
L (Ω) ·Lp
s+1/p(Ω) !→Lp
s+ε+1/p−2(Ω).(2.14)
In conce wi h Rellich’s selec ion lemma, his p o es ha , unde he
cu en assump ions, mul iplica ion by an elemen om L (Ω) is a com-
pac ope a o om Lp
s+1/p(Ω) in o Lp
s+1/p−2(Ω).
To see (2.14), i 1/p0:= 1/p −(s+1/p)/n i ollows ha 1 <p
0<∞
and Lp
s+1/p(Ω) !→Lp0(Ω). Going u he , i 1/p1:= 1/ +1/p0 hen
1<p
1<∞, hanks o (2.3), and we ha e L (Ω) ·Lp0(Ω) !→Lp1(Ω).
Now, since Lp
s+ε+1/p−2(Ω)=(Lp
2−s−ε−1/p,0(Ω))∗wi h 1/p +1/p=1,i
360 M. Dindoˇ
s, M. Mi ea
suffices o p o e ha Lp
2−s−ε−1/p(Ω) !→Lp2(Ω), whe e 1/p2+1/p1=1.
This, howe e , is gua an eed by he es ima e 1/p2>1/p−(2−s−1/p)/n.
Some simple algeb a now shows ha his is always he case p o ided
>n/2. When sp ≥n−1, >n/2, one p oceeds analogously, choosing
p0∞.
Finally, when =n/2, he well-defini eness o he inclusion (2.13)
is p o ed simila ly. Then, based on his, (2.14) and an app oxima ion
a gumen , i ollows ha (2.13) is compac as well.
Lemma 2.3. Le Ωbe a Lipschi z domain and assume ha s,psa is y
a leas one o he condi ions in (2.2). Also, fix a nonnega i e unc-
ion V∈L (M), wi h ≥n/2, and suppose ha (2.3) holds. Then
u∈Lp
s+1/p(Ω),(∆ −V)u≤0,T u≥0=⇒u≥0.(2.15)
In pa icula , i Vj∈L (M), ≥n/2, a e such ha 0≤V1≤V2, hen
(2.16) uj∈Lp
s+1/p(Ω),T u1≥T u2≥0,
(∆ −V1)u1≤(∆ −V2)u2≤0=⇒u1≥u2.
Analogous esul s a e alid o n=2, p o ided ha s,psa is y a leas
one o he condi ions in (2.10). Finally, when ∂Ω∈C1, any s∈(0,1),
p∈(1,∞)will do (in all dimensions) as long as (2.3) is sa isfied.
P oo : Fo an a bi a y, fixed u∈Lp
s+1/p(Ω), le us se := (∆ −V)u∈
Lp
s+1/p−2(Ω) and g:= T u∈Bp,p
s(∂Ω). Thus, by assump ion, ≤0
and g≥0.
Fo s a e s, we make he claim ha he e exis j∈C∞
comp(Ω), gj∈
Lip(∂Ω), such ha j≤0, gj≥0, and j→ in Lp
s+1/p−2(Ω), gj→gin
Bp,p
s(∂Ω). Indeed, he app oxima ing sequence {gj}jcan be p oduced
ia a s anda d localiza ion and molli ying p ocedu e. The e emains
o p o e ha belongs o he closu e o he con ex se C:= {ψ∈
C∞
comp(Ω); ψ≤0}in Lp
s+1/p−2(Ω). Assuming he opposi e (and seeking
a con adic ion), he Hahn-Banach heo em ensu es he exis ence o some
Φ∈Lp
s+1/p−2(Ω)∗=Lp
2−s−1/p,0(Ω), 1/p+1/p= 1, such ha Φ,φ≤
0<Φ,  o each φ∈C. (S ic ly speaking, wha he Hahn-Banach
heo em o iginally gi es is he p e ious double inequali y wi h ‘ze o’
eplaced by some eal numbe λ. Howe e , gi en ha Cis in ac a cone,
i is easy o see ha we can always assume ha λ= 0.) Now, he fi s
inequali y implies Φ ≥0 which, in u n, con adic s he second, gi en
ha ≤0. This finishes he p oo o he claim made a he beginning
o he pa ag aph.
Semilinea Poisson P oblems 361
Nex , g an ed he cu en assump ions, we will show in he cou se o
he p oo o Theo em 2.1 ha u(is uniquely de e mined by, and) de-
pends con inuously on ,g. Consequen ly, hanks o a limi ing a gumen
(whose implemen a ion is ou ine, gi en he claim abo e), i suffices o
p o e (2.15) in he case when ∈L∞(Ω) and g∈C0(∂Ω), which we will
assume he ea e . I is no difficul o see ha hese ex a assump ions
en ail u∈C0(¯
Ω) ∩C1
loc(Ω).
Wi h an eye on (2.15) and seeking a con adic ion, assume nex ha
he e exis s x0∈Ω such ha u(x0)<0. I we le Obe he connec ed
componen o he se {x∈Ω; u(x)<0}which con ains x0and ecall
ha u|∂Ω>0, i ollows ha O⊂⊂Ω.
Le ψ∈C∞(R) be a Lipschi z, non-dec easing, odd unc ion such ha
ψ( )=0 o | |≤1 and ψ( )= o | |≥2. Se ui(x):=i−1ψ(iu(x)),
x∈¯
O,i=1,2,... . Then, so we claim,
(2.17) ui∈C1
comp(O),u
i≤0,
and u−uiL∞(O),∇u−∇uiL2(O)−→ 0.
Indeed, ha uiis compac ly suppo ed in Oand ∇u−∇uiL2(O)→0
a e consequences o u|∂O= 0 and defini ions. As o ∇u−∇uiL2(O)→
0, no e ha ∇ui(x)=ψ(iu(x))∇u(x)→∇u(x)asi→∞, o each
x∈O. Since ψis Lipschi z and u∈C1(¯
O), Lebesgue’s domina ed
con e gence heo em hen yields he desi ed conclusion. Going u he ,
O
|∇u|2+V|u|2= lim O
∇u, ∇ui+Vuu
i
=−lim O
ui(∆ −V)u≤0,
(2.18)
which con adic s he ac ha uis no iden ically ze o in O. This
finishes he p oo o he implica ion (2.15).
Finally, o jus i y (2.16), i suffices o apply (2.15) o he diffe -
ence u:= u1−u2(wi h he choice V:= V1).
Lemma 2.4. Fo each Lipschi z domain Ω⊂Mand 1<p<∞ he e
exis s a cons an C=C(M,Ω,p)>0such ha o any u∈Lp
n/p(Ω) and
τ∈[2,∞)
uLτ(Ω) ≤Cτ1−1/puLp
n/p(Ω).(2.19)
Fu he mo e, i n≥3and n−1<p<∞, hen
 Lτ
2/τ,0(Ω) ≤Cτ1−1/p Lp
n/p,0(Ω), o all τ∈[p, ∞),(2.20)
whe e C=C(M,Ω,n,p)>0is independen o τ.
368 M. Dindoˇ
s, M. Mi ea
hence u(x∗)>1
2 (x∗)>0. Obse e ha he unc ion uis a supe solu-
ion o he Laplace-Bel ami ope a o on Ω (since ∆u=V ≥0inΩ)so
ha , when conside ed on O, i mus a ain i s maximum a a bounda y
poin . Thus, he e exis s z∈∂O⊂Ω a which u(z) = maxx∈O u(z)≥
u(x∗)>1
2 (x∗). On he o he hand, he defini ion o Oensu es ha
≡0on∂O. Consequen ly, ˙
T−1
V (z)= (z)−u(z)<−1
2 (x∗)<0,
so ha |˙
T−1
V (z)|>1
2 L∞(Ω). This con adic s (2.51), and concludes
he p oo o (2.50).
Tu ning now o he ask o p o ing (2.49), ecall ha o each
/( −1) ≤p≤∞, he e exis s a cons an κ(V)>0, independen
o p, such ha
˙
TV Lp(Ω) ≤κ(V) Lp(Ω) and ˙
TV L∞(Ω) ≤2 L∞(Ω),(2.53)
uni o mly o ∈C0(Ω). A s anda d in e pola ion inequali y (c ., e.g.,
[13, p. 41]) hen yields, o ∈C0(Ω) and p∈[ /( −1),∞],
˙
TV Lp(Ω) ≤˙
TV θ
L /( −1)(Ω)˙
TV 1−θ
L∞(Ω)
≤κ(V)θ21−θ θ
L /( −1)(Ω) 1−θ
L∞(Ω),
(2.54)
whe e θ:= /(p( −1)) ∈(0,1). In pa icula , θ!0asp∞.
Now, gi en any ∈L∞(Ω), we can find a sequence o unc ions ( j)j
in C0(Ω) such ha  jL∞(Ω) ≤ L∞(Ω) and j→ in Lq(Ω) o any
q<∞. Then, by i ue o (2.54), we ge
TV L∞(Ω) = lim
p→∞ TV Lp(Ω) = lim
p→∞ lim
j→∞ ˙
TV jLp(Ω)
≤lim
p→∞ lim
j→∞ κ(V)θ21−θ jθ
L /( −1)(Ω) j1−θ
L∞(Ω)
≤2 L∞(Ω).
(2.55)
This jus ifies (2.49) and shows ha TVL(L∞(Ω)) ≤2, independen ly
o V, hus p o ing (2.41).
Wi h (2.41) in hand and elying on Lemma 2.5, we see ha (2.40)
holds o each 2 ≤p<∞, uni o mly in V.
The e emains he case 1 <p<2 which we ea nex . Ou s a egy
is based on duali y and equi es analyzing he ac ion o he ope a o
KV,p :=(−∆)1/p
DTV(−∆)−1/p
D=−(−∆)1/p
D(∆ −V)−1
D(−∆)1−1/p
D
(2.56)
on Lpspaces. Conc e ely, om P oposi ion 2.6 (wi h µ=2/p) i suffices
o show ha KV,pL(Lp(Ω)) is bounded uni o mly in Vo , equi alen ly,

Semilinea Poisson P oblems 369
ha K∗
V,pL(Lp(Ω)) ≤C(p) uni o mly in V, whe e 1/p+1/p= 1. Since
K∗
V,p =−(−∆)1−1/p
D(∆ −V)−1
D(−∆)1/p
D
=(−∆)1/p
DTV(−∆)−1/p
D=KV,p
(2.57)
by elying once again on P oposi ion 2.6, we see ha ma e s a e u he
educed o p o ing ha TVL(Lp
2/p,0(Ω)) ≤Cuni o mly in V. Howe e ,
ha ing p∈(1,2) en ails p∈(2,∞) and his is p ecisely he case al eady
add essed. The p oo o he lemma is he e o e finished.
In ou nex lemma we conside he ac ion o he ope a o TVon
Lpspaces.
Lemma 2.8. Assume ha n≥3. Then, i V∈L (Ω), >n/2,is
nonnega i e, he e holds
TVL(Lp(Ω)) ≤C(1 + VL (Ω)) o each p> /( −1),(2.58)
whe e C=C(Ω, ,p)>0is independen o V. The es ima e (2.58) also
holds when n=2p o ided >4/3.
P oo : Assume fi s ha n≥3. To p o e (2.58), we ely on he ac ha
TV = +(∆−V)−1
D(V ) o ∈C∞
comp(Ω), and analyze he ac ions o
he mul iplica i e ope a o MV, defined by MV := V· , and (∆−V)−1
D
sepa a ely. In his scena io, he c ux o he ma e is es ablishing he
es ima e
(∆ −V)−1
DL(Lq(Ω),L
qn/(n−2q)(Ω)) ≤C, q ∈(1,n/2),(2.59)
o some C=C(q)>0 independen o V. Indeed, (2.59) in conce wi h
he elemen a y es ima e MVL(Lp(Ω),Lp /(p+ )(Ω)) ≤VL (Ω), eadily
yields (2.58) p o ided ha one can choose q∈(1,n/2) so ha 1/ +1/p ≤
1/q ≤1/p +2/n. As his la e condi ion is easily checked, (2.58) will
ollow as soon as we jus i y (2.59).
In his ega d, a simple ye use ul obse a ion (whose p oo amoun s
o an algeb a exe cise) is ha
(2.60) ∀q∈(1,n/2),
∃(s, 1/p) as in (2.2) such ha Lq(Ω) !→Lp
s+1/p−2(Ω).
Consequen ly, one can always employ a ac o iza ion o he ype
(2.61) (∆ −V)−1
D:Lq(Ω) ι
−→ Lp
s+1/p−2(Ω) (∆−V)−1
D
−→ Lp
s+1/p,0(Ω)
ι
−→ Lqn/(n−2q)(Ω)
370 M. Dindoˇ
s, M. Mi ea
in o de o jus i y ha (∆ −V)−1
D∈L(Lq(Ω),L
qn/(n−2q)(Ω)) o each
q∈(1,n/2).
Tu ning now o he ac ual ask o p o ing (2.59), fix an a bi a y
∈Lq(Ω) which we w i e as = +− −, whe e 0 ≤ +, −≤| |.
Posi i i y o (−∆)−1
Dgi es
0≤ ±≤V(−∆)−1
D ±+ ±.(2.62)
Applying (V−∆)−1
D o he abo e inequali ies and in oking Lemma 2.3
yields
0≤(V−∆)−1
D ±≤(−∆)−1
D ±.(2.63)
Toge he wi h he decomposi ion = +− −and (2.61) wi h V=0,
his allows us o w i e
(V−∆)−1
D Lqn/(n−2q)(Ω) ≤(−∆)−1
D +Lqn/(n−2q)(Ω)
+(−∆)−1
D −Lqn/(n−2q)(Ω)
≤2C Lq(Ω),
(2.64)
as desi ed. This es ablishes (2.59) and finishes he p oo o (2.58) when
n≥3.
When n= 2 one can p oceed in a simila ashion, he mos no able
diffe ence being ha (2.59) now becomes (∆ −V)−1
DL(Lq(Ω),L
∞(Ω)) ≤
C, o q>1.
A med wi h Lemma 2.7 and Lemma 2.8 we now u n o he ask o
es ablishing he ollowing impo an es ima es.
Lemma 2.9. Assume ha V∈L (Ω), >n/2, is an a bi a y non-
nega i e unc ion, and ha 0<s<1,1<p<∞.
(i) I s/(n−1) <1/p ≤s, hen
TVL(Lp
s+1/p,0(Ω),Lnp/(n−1−sp)(Ω)) ≤C,(2.65)
whe e C=C(Ω,s,p)>0is a fini e cons an independen o V.
(ii) I sp>n−1 hen, o some cons an C=C(Ω, ,p,s)>0inde-
penden o he unc ion V,
TVL(Lp
s+1/p,0(Ω),L∞(Ω)) ≤C.(2.66)
(iii) I sp =n−1, he e exis s C=C(Ω, ,p)>0independen o V
such ha
TVL(Lp
s+1/p,0(Ω),Lτ(Ω)) ≤Cτ1−1/p o any τ∈[2,∞).(2.67)
Semilinea Poisson P oblems 371
(i ) I s<1
p<s
n−1+n( −1)
(n−1) ,n≥3, hen
(2.68) TVL(Lp
s+1/p,0(Ω),Lnp/(n−1−sp)(Ω))
≤


C(1+VL (Ω)
)ε+
2( −1) (1
p−s),i ≤1+(1−sp)/(2p−2),
C(1+VL (Ω)
)ε+1
2−n/ ((n−1)/p−s−n+2),i <1+(1−sp)/(2p−2),
o each ε>0, whe e C=C(Ω,s,p,ε)>0is independen o V.
When n=2, he same holds p o ided >4/3.
P oo : Conside fi s he si ua ion when 0 <s/(n−1) <1/p ≤s<1
(which au oma ically en ails n≥3). The case sp =1,1<p<∞has
been deal wi h in Lemma 2.7. The la ge ange we in end o add essed
he e is hen a simple consequence o his and he ac o iza ion
TV:Lp
s+1/p,0(Ω)!→Lp∗
2/p∗,0(Ω) TV
−→ Lp∗
2/p∗,0(Ω)!→Lnp/(n−1−sp)(Ω),(2.69)
whe e 1/p∗:= (n−1)(1/p−s/(n−1))/(n−2), and bo h inclusions abo e
a e classical embedding esul s. As o (ii), i.e. he case sp > n−1, n≥2,
we use (2.41) and he ac o iza ion
TV:Lp
s+1/p,0(Ω) !→L∞(Ω) TV
−→ L∞(Ω).(2.70)
Nex we add ess he case sp =n−1 (which o ces p>n−1).
When n= 2, he conclusion we seek is an immedia e consequence o
Lemma 2.7 and Lemma 2.4. Simila ing edien s can be used o handle
he case n≥3 as well. The idea is o use he ac o iza ion
TV:Lp
n/p,0(Ω) ι1
!→Lτ
2/τ,0(Ω) TV
−→ Lτ
2/τ,0(Ω) ι2
!→Lτ(Ω),(2.71)
in conce wi h ι1≤Cτ1−1/p, (2.40), and he ac ha ι2≤C,
uni o mly in τ∈[p, ∞). This finishes he analysis o he poin (iii) in
ou lemma.
A his s age, i emains o deal wi h he si ua ion desc ibed in he
poin (i ) o he lemma, and ou in en ion is o e en ually in e pola e
be ween (2.40) and (2.58). To his end, ecall fi s ha he wo classes,
Lp
s+1/p,0(Ω) and Lnp/(n−1−sp)(Ω), wi h 1 <p<∞and −1/p ≤s, a e
complex in e pola ion scales, in he sense ha o each 1 <p
j<∞,
372 M. Dindoˇ
s, M. Mi ea
sj≥−1/pj,j=0,1,
Lp0
s0+1/p0,0(Ω),L
p1
s1+1/p1,0(Ω)θ=Lp∗
s∗+1/p∗,0(Ω),
Lnp0/(n−1−s0p0)(Ω),L
np1/(n−1−s1p1)(Ω)θ=Lnp∗/(n−1−s∗p∗)(Ω),
i 1/p∗:= (1 −θ)/p0+θ/p1,s
∗:= (1 −θ)s0+θs1,0<θ<1.
(2.72)
This obse a ion and con exi y a gumen s eadily en ail he ollowing
p inciple: i 1 <p
i<∞,−1/pi≤si,τi:= npi/(n−1−sipi)>1, and
TVL(Lpi
si+1/pi(Ω),Lτi(Ω)) ≤Mi,i=1,2,3, hen
TVL(Lp
s+1/p,0(Ω),Lnp/(n−1−sp)(Ω)) ≤Mλ1
1Mλ2
2Mλ3
3,(2.73)
p o ided he poin (s, 1/p) has ba icen ical coo dina es (λ1,λ
2,λ
3),
λi≥0, ela i e o he iangle wi h e ices (si,1/pi), i=1,2,3.
We shall use he abo e ema k wice, fi s o he iangle wi h e -
ices a (ε, ε), (1 −ε, 1−ε), (1/( +ε)−1,1−1/( +ε)) wi h ε>0
sufficien ly small. In his scena io, assuming ha n≥3, M1and M2a e
con olled by Cεand M3≤Cε(1+VL (Ω)). Also, λ3is a O(ε) a ia ion
o (1/p −s)/(2( −1)). This yields he fi s line in (2.68). The con-
di ion >1+(1−sp)/(2p−2) gua an ees ha he poin (s, 1/p) lies
inside he iangle unde discussion.
The second applica ion o he a o emen ioned ema k is simila in
spi i and equi es a p epa a o y s ep. Specifically, as a esul o (2.58)
and he ac o iza ion
TV:Lp
s+1/p,0(Ω) !→Lnp/(n−1−sp)(Ω) TV
−→ Lnp/(n−1−sp)(Ω)(2.74)
i ollows ha
(2.75) TVL(Lp
s+1/p,0(Ω),Lnp/(n−1−sp)(Ω)) ≤C(1 + VL (Ω))
i 0 <1
p−s
n−1<n( −1)
(n−1).
No e ha he in e sec ion be ween 1
p−s
n−1=n( −1)
(n−1) wi h 1/p = 1 is he
poin wi h coo dina es (n/ −1,1). Inspi ed by his obse a ion, we w i e
(2.73) o he iangle wi h e ices a (1−ε, 1−ε), (1/( +ε)−1,1−1/( +
ε)), (n/( +ε)−ε(n−1)−1,1−ε) o some sufficien ly small ε>0. This
ime, assuming ha n≥3, we ge M1≤Cε,M2,M
3≤Cε(1+VL (Ω)),
whe eas λ2+λ3is a O(ε) a ia ion o [(n−1)/p −s−n+2]/(2 −n/ ).
A ailing ou sel es o (2.58), (2.40), he second line o (2.68) ollows. The
Semilinea Poisson P oblems 373
ole o ≤1+(1−sp)/(2p−2) and 1
p−s
n−1<n( −1)
(n−1) is o ensu e ha
he poin (s, 1/p) lies inside he iangle we a e cu en ly conside ing.
The subcase o (i ) co esponding o n= 2 is handled simila ly, and
equi es ha >4/3 (c . Lemma 2.8).
Finally, we ha e all he necessa y ing edien s in o de o ackle he
P oo o Theo em 2.1: Pa II: He e we p esen he final a gumen s in
he p oo o (2.6), (2.8), (2.7), and also ea he case =n/2.
In wha ollows, i his an a bi a y eal- alued unc ion, we se
h±:= max{±h, 0}=1
2(|h|±h),(2.76)
so ha h±≥0 and h=h+−h−. Also, |h±(x)−h±(y)|≤|h(x)−h(y)|
so ha
(·)±:Bp,p
s(∂Ω) −→ Bp,p
s(∂Ω) a e bounded, 1<p<∞,0<s<1.(2.77)
Fo he ime being, we con inue o assume >n/2. I we now deno e
by (±)
V∈Lp
s+1/p(Ω) he solu ions o
L (±)
V=(∆−V) (±)
V= 0 in Ω,T (±)
V=g±,(2.78)
hen, clea ly, V:= (+)
V− (−)
Vsol es
L V=(∆−V) V= 0 in Ω,T V=g.(2.79)
Se also (±)
0 o he solu ions o (2.78) wi h V= 0. Acco ding o
Lemma 2.3 we ha e
0≤ (+)
V≤ (+)
0and 0 ≤ (−)
V≤ (−)
0.(2.80)
Assuming sp < n −1, i ollows ha
 VLnp/(n−1−sp)(Ω) ≤ (+)
VLnp/(n−1−sp)(Ω) + (−)
VLnp/(n−1−sp)(Ω)
≤ (+)
0Lnp/(n−1−sp)(Ω) + (−)
0Lnp/(n−1−sp)(Ω)
≤C (+)
0Lp
s+1/p(Ω) +C (−)
0Lp
s+1/p(Ω)
≤2CgBp,p
s(∂Ω),
(2.81)
whe e he las cons an Cin (2.81) is ha appea ing in he es ima e (2.5)
o V= 0. The case when sp > n−1 is simila , while he case sp =n−1
is p o ed wi h he help o Lemma 2.4.
Simila ly, by wVwe deno e he solu ion o
LwV=(∆−V)wV= in Ω,T wV=0.(2.82)

374 M. Dindoˇ
s, M. Mi ea
Clea ly, w0=∆
−1
D and wV=(∆−V)−1
D , hence
wVLτ(Ω) =(∆ −V)−1
D∆w0Lτ(Ω)
≤TVL(Lp
s+1/p(Ω),Lτ(Ω))w0Lp
s+1/p(Ω)
≤CTVL(Lp
s+1/p(Ω),Lτ(Ω)) Lp
s+1/p−2(Ω).
(2.83)
The cons an Cappea ing in he las line is again he cons an in he
es ima e (2.5) o V=0. Now,uV= V+wVsol es he desi ed equa-
ion (2.4) and he es ima es (2.6)–(2.7) ollow om (2.81), (2.83) and
Lemma 2.9.
In he wo dimensional si ua ion, he well posedness ange (2.10) is
handled analogously, g an ed he wo k in [25], [26]. He e we only wan
o ema k ha , in o de o jus i y (2.30) o u∈Lp
s+1/p,0(Ω) and V∈
L (Ω), >(1−|1/(2p)−s/2|)−1, obse e fi s ha o some ε>0 small,
Vu∈L1+ε(Ω) !→Lq
2/q−2+ε(Ω), i q>1. In u n, his u he en ails u=
p 1[Di −1(Vu)] ∈Lq
2/q+ε(Ω) !→L2
1(Ω), hanks o he a o emen ioned
e e ences. Since, unde he cu en assump ions, V|u|2∈L1(Ω) is also
eadily e ified, his akes ca e o (2.30) when n=2.
Finally, we a e le wi h he analyzing he case =n/2, a ask which
we ake up nex . Fo s a e s, we claim ha i suffices o deal wi h he
si ua ion when he da um is ac ually selec ed om Lq(Ω), whe e qis
gi en by
1
q:= 2
n+n−1−sp
np .(2.84)
Indeed, gi en an a bi a y ∈Lp
s+1/p−2(Ω), se w:=∆−1
D ∈Lp
s+1/p,0(Ω).
Then u=u0+wsol es (2.1), p o ided u0is a solu ion o









(∆ −V)u0=Vw in Ω,
T u0=g∈Bp,p
s(∂Ω),
u0∈Lp
s+1
p
(Ω).
(2.85)
Now, w∈Lnp/(n−1−sp)(Ω) by s anda d embedding esul s and, u he ,
Vw ∈Lq(Ω) by H¨olde ’s inequali y (no e ha (2.3) gua an ees ha
q>1). This jus ifies he claim made a he beginning o he pa ag aph.
Assuming nex ha ∈Lq(Ω), i ollows ha  ±Lq(Ω) ≤ Lq(Ω).
Conside an app oxima ing sequence Vj→Vin Ln/2(Ω), so ha Vj≥0,
Semilinea Poisson P oblems 375
Vj∈L∞(Ω), o each j, and le u±
jbe he unique solu ion o









(∆ −Vj)u±
j=− ∓∈Lq(Ω),
T u±
j=g±∈Bp,p
s(∂Ω),
u±
j∈Lp
s+1
p
(Ω).
(2.86)
As be o e, we ge ha u±
jLnp/(n−1−sp)(Ω) ≤u±
0Lnp/(n−1−sp)(Ω), whe e
u±
0sol es (2.86) wi h V0:= 0. F om his and he decomposi ion uj=
u+
j−u−
j, we conclude ha he e exis s some cons an C>0 independen
o Vsuch ha
ujLnp/(n−1−sp)(Ω) ≤C( Lq(Ω) +gBp,p
s(Ω)).(2.87)
We now make he claim ha (uj)j∈Nis a Cauchy sequence in
Lnp/(n−1−sp)(Ω). To see his, o j, k ∈N, we no e ha w:= uj−uk
sol es he p oblem
(∆ −Vj)w=(Vj−Vk)uk∈Lq(Ω),w∈Lp
s+1
p,0(Ω).(2.88)
Hence, by i ue o (2.87), we ha e
uj−ukLnp/(n−1−sp)(Ω) ≤C(Vj−Vk)ukLq(Ω)
≤CVj−VkLn/2(Ω)ukLnp/(n−1−sp)(Ω)
≤CVj−VkLn/2(Ω)( Lq(Ω) +gBp,p
s(Ω)).
(2.89)
F om his, ou claim ollows.
In o de o con inue, le ube he limi o he sequence (uj)j∈Nin
Lnp/(n−1−sp)(Ω). We now in end o show ha usol es (2.1). Deno e
by u=Tg( ) he solu ion ope a o o he Poisson p oblem (2.1) co e-
sponding o V:= 0. In pa icula , uj=Tg( +Vjuj). Since Vjuj→Vu
in Lq(Ω) !→Lp
s+1/p−2(Ω), we may conclude ha uj=Tg( +Vjuj)→
Tg( +Vu)inLp
s+1/p(Ω). F om his we see ha u=Tg( +Vu), i.e., u
sol es (2.1).
To finish he p oo o he heo em we need o show ha he solu ion
we ha e jus cons uc ed is unique. By linea i y, his comes down o
p o ing ha he ope a o
∆−V:Lp
s+1
p,0(Ω)−→ Lp
s+1
p−2(Ω)=Lp
2−s−1
p
(Ω)∗,1
p+1
p=1,(2.90)
is one- o-one. F om wha we ha e p o ed so a , his ope a o is on o
so, gi en he in a iance o he condi ions (2.2)–(2.3) o he ans o ma-
ion (s, 1/p)→ (1 −s, 1−1/p), he desi ed esul ollows by duali y.
376 M. Dindoˇ
s, M. Mi ea
Pa en he ically, le us no e ha we could ha e eached he same conclu-
sion using he on oness o (2.90) plus he ac ha , much as in (2.28),
he ope a o (2.90) is F edholm wi h index ze o.
3. Nonlinea i ies wi h polynomial g ow h
In his sec ion we s udy he semilinea e sion o (2.4) in he case
when he nonlinea i y is allowed o ha e supe linea g ow h. In o de o
s a e ou fi s esul , ecall ha a unc ion a(x, u) is called Ca a h´eodo y
i i is measu able in xand con inuous in u.
Theo em 3.1. Assume ha Ω⊂Mis a connec ed Lipschi z domain
in M, whe e dim M=n≥4. Also, suppose ha a:Ω×R→Ris a
Ca a h´eodo y unc ion such ha
0≤a(x, u)≤k1(x)+k2(x)|u|m,(3.1)
whe e 0≤kj∈Lqj(Ω) wi h qj≥1,j=1,2.
Fo 1<p<∞,0<s<1, conside he ollowing semilinea Poisson
p oblem wi h Di ichle bounda y condi ion:
∆u−a(x, u)u= ∈Lp
s+1/p−2(Ω),
T u=g∈Bp,p
s(∂Ω),u∈Lp
s+1/p(Ω).
(3.2)
Then he e exis s ε=ε(Ω) >0such ha he p oblem (3.2) has a leas
one solu ion p o ided he ollowing is ue: The pai (p, s)sa isfies a
leas one o he condi ions in (2.2),sp < n −1, and
0≤1
q1
≤2
nand 1
q1
<1−n−1−sp
np ,(3.3)
0≤m< np
n−1−sp min 2
n,1−n−1−sp
np −1
q2.(3.4)
Fo sp ≥n−1 he numbe mcould be aken a bi a ily la ge as long as
q1,q
2>(2
n+n−1
np −s
n)−1. Mo eo e ,
(3.5) ∀M>0∃K>0
so ha  Lp
s+1/p−2(Ω) +gBp,p
s(∂Ω) ≤M=⇒uLp
s+1/p(Ω) ≤K.
I sp ≥1 he solu ion usa isfies he es ima e
uLnp/(n−1−sp)(Ω) ≤C( Lp
s+1/p−2(Ω) +gBp,p
s(∂Ω))(3.6)
o some C=C(∂Ω,s,p,a)>0.
Semilinea Poisson P oblems 377
I , in addi ion, he nonlinea i y b(x, u):=a(x, u)usa isfies
0≤∂
∂ub(x, u)≤˜
k1(x)+˜
k2(x)|u|m,(3.7)
wi h 0≤˜
kj∈Lqj(Ω),j=1,2,qj’s and mas be o e, hen he solu ion u
o he bounda y p oblem (3.2) is also unique.
Analogous esul s a e alid in dimensions n=2and n=3.I
dim M=n=3 he esul s s a ed abo e emain ue wi hou change,
p o ided sp ≥1/2and p(1 + s)≥2. O he wise
0≤m< 3p
2−sp 1
σ−1
q2,0≤1
q1
<1−2−sp
3p,(3.8)
will suffice, whe e
σ:= 


3
2+1−2sp
2(2−sp),i 2sp < 1and p(3 + s)≥5,
3
21+2−p(1+s)
2−sp ,i p(1 + s)<2and p(3 + s)<5.
(3.9)
I n=2 he p e ious esul s a e alid p o ided (2.10) is used in place
o (2.2), and
q1>2,0≤m< p(q2−2)
2q2(1 −sp) o sp < 1,
q1,q
2>1−
1
2p−s
2−1
,m≥0, o sp ≥1.
(3.10)
By way o con as , 1<p<∞,0<s<1, will do in all dimensions
when ∂Ω∈C1.
A ew commen s a e in o de he e.
(i) As a as he ange o alidi y (desc ibed by means o (2.2), (2.10))
is conce ned, ou heo em is in he na u e o bes possible. This
is because he a o emen ioned ange is op imal o he associa ed
linea p oblems.
(ii) In some ins ances min (3.4) can be allowed o a ain he alue
np
n−1−sp 2
n−1
q2and he s a emen o he heo em emains ue.
In o de o a oid u he complica ions we decided no o include
his case in he heo em gi en abo e. In e es ed eade can analyze
his bounda y case by echniques ou lined in he p oo ha ollows.
(iii) Simila esul s a e alid a he le el o 2nd-o de , o mally sel -
adjoin , non-posi i e, s ongly ellip ic sys ems, a leas when n=2
o n=3.
384 M. Dindoˇ
s, M. Mi ea
(a) sp =n−1and o each A>0 he e exis s ˜
k∈L (Ω), >n/2,
such ha
0≤∂
∂ub(x, u)≤˜
k(x) exp(A|u|γ),wi h 0≤γ=γ(p)=p/(p−1),(3.44)
(b) sp > n −1and
(3.45) 0 ≤∂
∂ub(x, u),
∀M>0 sup
u∈[−M,M]
∂
∂ub(x, u)∈L (Ω) o some >σ,
hen he solu ion uo he bounda y p oblem (3.38) is also unique.
Finally, analogous esul s a e alid in dimension n=2, p o ided
(2.10) is used in place o (2.2). Mo eo e , 1<p<∞,0<s<1,
sp ≥n−1, will do in all dimensions when ∂Ω∈C1.
P oo : Conside fi s he somewha simple case sp > n −1. I ollows
om Theo em 2.1 ha any solu ion o he equa ion (2.4) sa isfies
uL∞(Ω) ≤C( Lp
s+1
p−2(Ω) +gBp,p
s(∂Ω)),(3.46)
wi h C>0 independen o V≥0. In oduce N:= C( Lp
s+1
p−2(Ω) +
gBp,p
s(∂Ω)), hen define
ψN(u):=








u, o |u|≤N,
N, o u>N,
−N, o u<−N,
(3.47)
and, finally, conside he Poisson p oblem:
∆u−a(x, ψN(u))u= ∈Lp
s+1/p−2(Ω),
T u=g∈Bp,p
s(∂Ω),u∈Lp
s+1/p(Ω).
(3.48)
The p oblem (3.48) sa isfies all he assump ions in Theo em 3.1 and,
hence, has a leas one solu ion u. Mo eo e , ma e s can be a anged
so ha he solu ion also sa isfies he es ima e (3.46). This o ces
a(x, u(x)) = a(x, ψN(u(x))), hus usol es (3.38) as well. The p oo
o he uniqueness pa pa allels i s coun e pa in Theo em 3.1.

Semilinea Poisson P oblems 385
Nex , we u n ou a en ion o he mo e in e es ing case sp =n−1.
To begin wi h, define he ec o space Xpby se ing
Xp={h:Ω−→ R; sup
τ≥2
τ−1+1/phLτ(Ω) <∞},(3.49)
and equip i wi h he no m
hXp:= sup
τ≥2
τ−1+1/phLτ(Ω).(3.50)
I is no e y difficul o check ha Xpis indeed a Banach space. An
essen ial p ope y enjoyed by unc ions in Xpis ha he e exis s κ=
κ(M,Ω) >0 such ha
h∈Xp,α:= κ
hp/(p−1)
Xp
=⇒exp(α|h|p/(p−1))∈L1(Ω).(3.51)
The p oo o his ac can be ca ied ou exac ly as he p oo o T udin-
ge ’s inequali y; c . [36, Chap e 13] o mo e de ails. Taking ano he
look a he es ima e (2.7), we lea n ha he solu ion u o he equa-
ion (2.4) belongs o Xpand he e exis s C>0 independen o Vsuch
ha
uXp≤C( Lp
s+1
p−2(Ω) +gBp,p
s(∂Ω)).(3.52)
To p oceed om he e, conside fi s he case when ∈L∞(Ω). Deno e
by u(±)
V he solu ion o he bounda y p oblem:
(∆ −V)u(±)
V= ∓in Ω,T u(±)
V=g±.(3.53)
In his si ua ion, elying on Lemma 2.3 we in e ha 0 ≤u(±)
V≤u(±)
0,
hence he e exis s h=u(+)
0+u(−)
0∈X
psuch ha o any V≥0 he
solu ion uVo (2.4) sa isfies
|uV(x)|≤h(x), o any x∈Ω.(3.54)
Le us now define
O:= {u∈L1(Ω); |u|≤h},(3.55)
so ha , clea ly, O⊂X
p. Ano he simple ye use ul obse a ion is ha
o any τ∈[2,∞) he se Ois closed and con ex in Lτ(Ω). Finally,
conside he ope a o T:O→Odefined by ag eeing ha := Tu is he
(unique) solu ion o
∆ −a(x, u) = in Ω,T =g, ∈Lp
s+1/p(Ω).(3.56)
386 M. Dindoˇ
s, M. Mi ea
In o de o see ha is well defined, i is sufficien o obse e ha o
any u∈Owe ha e a(x, u(x)) ∈L (Ω) o some >n/2. This, howe e ,
is a consequence o
0≤a(x, u)≤k(x) exp(A|u|γ)≤k(x) exp(A|h|γ).(3.57)
Indeed, since h∈Xp, one can selec Asmall enough so ha exp(A|h|γ)∈
Lq(Ω). Thus, he desi ed conclusion ollows by choosing qsufficien ly
la ge, so ha 1/q +1/ <2/n. Also =Tu∈Oby (3.54). Finally, Tis
a con inuous and compac map in he Lτ(Ω) opology o any τ∈[2,∞).
Con inui y ollows om he ac ha i ui→upoin wise on Ω and all
ui∈O, hen also a(., ui(.)) →a(., u(.)) poin wise. As (3.57) gi es
a common majo an in L (Ω) ( o some fixed >n/2), we see ha
a(., ui(.)) →a(., u(.)) in L (Ω). Wi h his a hand, Theo em 2.1 eadily
finishes he p oo o he con inui y o T. The p oo o he compac ness
o Tgoes exac ly as in Theo em 3.1. Hence, by Schaude ’s fixed-poin
heo em, he map Thas a fixed poin u∈Owhich, in u n, is he
solu ion o he Poisson p oblem we seek. Mo eo e , usa isfies (3.52).
The e emains o show how o dispense off he ex a hypo hesis ∈
L∞(Ω). The idea is o app oxima e in Lp
s+1/p−2(Ω) by a sequence o
unc ions j∈L∞(Ω). G an ed wha we ha e p o ed so a , a each
s ep j, we hen sol e
∆uj−a(x, uj)uj= jin Ω,T uj=g, uj∈Lp
s+1/p(Ω).(3.58)
By (3.52), (uj)jis a bounded sequence in Xp. Going u he , his en ails
ha Vj(x):=a(x, uj(x)) a e uni o mly bounded in L (Ω), o some
>n/2. Since we ha e ha uj=Tg( +Vjuj) (he e Tghas he same
meaning as a he end o Sec ion 2), an applica ion o Lemma 2.2 shows
ha he sequence (uj)j≥1is bounded in Lp
s+1/p(Ω) and ha he e exis s
a subsequence which con e ges o some uin he Lp
s+1/p(Ω) no m. In
u n, his eadily yields ha usol es (3.38), as desi ed. Finally, he
p oo o uniqueness is essen ially he same as be o e.
4. Neumann bounda y condi ions
Building on he sha p linea heo y om [9], [30], in his sec ion
we analyze he semilinea Poisson p oblem wi h nonlinea Neumann
bounda y condi ions.
Semilinea Poisson P oblems 387
Theo em 4.1. Le Ω⊂Mbe an a bi a y Lipschi z domain wi h ou -
wa d uni cono mal ν∈T∗M. Also, fix a nonnega i e unc ion a(x, u)∈
L∞(Ω ×R)and conside L=∆−V, whe e V∈L (Ω) o some >n,
V≥0and V>0on a se o posi i e measu e in Ω. Then he e ex-
is s ε=ε(Ω,a)∈(0,1] wi h he ollowing significance. I s∈(0,1),
p∈(1,∞)sa is y ei he one o he h ee condi ions in (2.2) (o , espec-
i ely, (2.10) i n=2) and i q:= (1 −1/p)−1,λ∈R,0<δ<1, hen
he Neumann Poisson p oblem
(NP)








Lu −a(x, u)u= ∈Lq
1
q−s−1,0(Ω),
∂νu+λ|u|δ=g∈Bq,q
−s(∂Ω),
u∈Lq
1−s+1
q
(Ω),
(4.1)
has a leas a solu ion which sa isfies
(4.2) ∀M>0∃K>0
so ha  Lq
1
q−s−1,0(Ω) +gBq,q
−s(∂Ω) ≤M=⇒uLq
1
q−s+1(Ω) ≤K.
In he case o linea bounda y condi ions, i.e., when λ=0, and when
he nonlinea i y b(x, u):=a(x, u)ualso sa isfies 0≤∂ub(x, u)∈L∞(Ω×
R), he solu ion is also unique.
Fu he mo e, when ∂Ω∈C1, we can ake any s∈(0,1) and p∈
(1,∞).
P oo : I u∈Lq
1−s+1
q
(Ω), hen T u∈Bq,q
1−s(∂Ω) and ecall om [33]
ha
|·|
δ:Bq,q
1−s(∂Ω) −→ Bq/δ,q/δ
(1−s)δ(∂Ω)(4.3)
is well-defined and bounded. Fu he mo e, by allowing an a bi a y
small de ec o smoo hness ( o he a ge space), his ope a o also be-
comes con inuous and compac ; c . [32, Rema k 5, p. 377].
Nex , wi h uas abo e, we le := T ,g(u) be he unique solu ion o
he linea Poisson p oblem









L −a(x, u) = ∈Lp
1
q−s−1,0(Ω),
∂ν =g−λ|u|δ∈Bq,q
−s(∂Ω),
∈Lq
1−s+1
q
(Ω).
(4.4)
388 M. Dindoˇ
s, M. Mi ea
Tha he la e is well posed, g an ed he cu en hypo heses, is gua an-
eed by he esul s in [30]. We aim a showing ha
T ,g :Lq
1−s+1
q
(Ω) −→ Lq
1−s+1
q
(Ω)(4.5)
is a con inuous, compac mapping.
Fi s , le uj→u0in Lq
1−s+1
q
(Ω) and se j:= T ,g(uj), 0:= T ,g(u0).
We will show ha
j−→ 0in Lq
1−s+1
q
(Ω).(4.6)
Clea ly, i suffices o p o e he con e gence in (4.6) o a subsequence
(s ill deno ed ( j)j≥1). Since, by he es ima es o he linea heo y and
(4.3), he e exis s C=CaL∞,Ω,s,q,λ
>0 such ha
 jLq
1−s+1
q
(Ω) ≤C Lq
1
q−s−1(Ω) +gBq,q
−s(∂Ω) +ujδ
Lq
1
q−s−1(Ω),(4.7)
an easy applica ion o Rellich’s selec ion lemma, in conce wi h he
uniqueness in he linea heo y, gi es ha o each small ε>0 (and
a e possibly es ic ing o a subsequence),
j−→ 0in Lq
1−s+1
q−ε(Ω).(4.8)
Nex , obse e ha j− k∈Lq
1−s+1
q
(Ω) sa isfies
(4.9) L( j− k)=a(x, uj) j−a(x, uk) k−→ 0inLq
1−s+1
q
(Ω),
and ∂ν( j− k)−→ 0inBq,q
−s(Ω).
Hence, once again by i ue o he es ima es in he linea heo y,
 j− kLq
1−s+1
q
(Ω) −→ 0.(4.10)
Clea ly, his and (4.8) yield (4.6), hence he ope a o in (4.5) is con in-
uous.
Tu ning a en ion o he compac ness o T ,g in (4.5), assume ha
uj∈Lq
1−s+1
q
(Ω) is an a bi a y bounded sequence and se
j:= T ,g(uj),
j:= +a(x, uj) j∈Lq
1
q−s−1,0(Ω),
gj:= g−λ|uj|δ∈Bq,q
−s(∂Ω).
(4.11)
Semilinea Poisson P oblems 389
Since a(x, uj)∈L∞, jis a bounded sequence in Lq
1−s+1
q
(Ω) (c . (4.7))
and he embedding
L∞(Ω) ·Lq
1−s+1
q
(Ω) !→Lq
1
q−s−1,0(Ω)(4.12)
is compac , he e is no loss o gene ali y (c . also he discussion pe aining
o (4.3)) assuming ha
(4.13) jcon e ges o some 0in Lq
1
q−s−1,0(Ω),
and gjcon e ges o some g0in Bq,q
−s(∂Ω).
G an ed his and obse ing ha









L j= j∈Lq
1
q−s−1,0(Ω),
∂ν j=gj∈Bq,q
−s(∂Ω),
j∈Lq
1−s+1
q
(Ω),
(4.14)
i ollows om he well-posedness o he linea p oblem ha j→ 0in
Lq
1−s+1
q
(Ω), whe e 0is he unique solu ion o (4.14) wi h j,gj eplaced,
espec i ely, by 0and g0. This p o es ha he ope a o in (4.5) is also
compac .
No e ha i we le BRs and o he closed ball (cen e ed a he
o igin) o adius Rin he space Lq
1−s+1
q
(Ω) hen, by (4.7), he ope a-
o T ,g :BR→BRis well-defined, con inuous and compac , p o ided
Ris la ge enough. A his s age, Schaude ’s fixed-poin heo em applies
and akes ca e o he sol abili y o he Neumann Poisson p oblem (4.1),
as well as, he accompanying es ima e (4.2).
Finally, g an ed ha λ= 0 and he ex a condi ion 0 ≤∂u[a(x, u)u]∈
L∞(Ω ×R), uniqueness o he p oblem (4.1) can be eadily educed o
i s linea e sion, i.e., when a(x, u) is independen o u. The easoning
when ∂Ω∈C1is simila and his finishes he p oo .
5. Nonlinea i ies wi h sublinea g ow h
We e ain ou s anda d hypo heses on Mand Ω made in Theo em 3.1.
As usual, se L=∆−V. The main esul o his sec ion is he ollowing.

390 M. Dindoˇ
s, M. Mi ea
Theo em 5.1. Le N(x, u)be a Ca a h´eodo y unc ion such ha
|N(x, u)|≤k1(x)+k2(x)|u(x)|δ, o some 0≤δ<1,(5.1)
whe e 0≤kj∈Lqj(Ω) wi h qj>n/2,j=1,2. Also, assume ha V
sa isfies (2.1). Then, i s,psa is y ei he one o he h ee condi ions
in (2.2) and i (2.3) holds, he Poisson p oblem wi h Di ichle bounda y
condi ion
Lu −N(x, u)= ∈Lp
s+1/p−2(Ω),
T u=g∈Bp,p
s(∂Ω),u∈Lp
s+1/p(Ω),
(5.2)
has a leas one solu ion.
On he o he hand, i q:= (1−1/p)−1and 0≤V∈L (M)wi h >n,
V>0on a se o posi i e measu e in Ω, hen he Poisson p oblem wi h
Neumann bounda y condi ion
Lu −N(x, u)= ∈Lq
1/q−1−s,0(Ω),
∂νu=g∈Bq,q
−s(∂Ω),u∈Lq
1−s+1/q(Ω),
(5.3)
has a leas one solu ion.
Simila esul s hold in dimension n=2g an ed ha (2.10) is used
in lieu o (2.2) and q1,q
2>(1 −|1
2p−s
2|)−1in he case o Di ichle
bounda y condi ions.
Finally, when ∂Ω∈C1, one can simply ake 1<p<∞and 0<s<1
in place o (2.2),(2.10).
P oo : We shall only deal wi h he case o (5.2), since he case o Neu-
mann bounda y condi ions is simila . To his effec , conside fi s he
case when sp < n −1, and define he map T:Lnp/(n−1−sp)(Ω) →
Lnp/(n−1−sp)(Ω) by aking := Tu o be he unique solu ion o
L = +N(x, u)inΩ,
T =g∈Bp,p
s(∂Ω), ∈Lp
s+1/p(Ω),
(5.4)
o each u∈Lnp/(n−1−sp)(Ω). Since by (5.1) and ( he p oo o ) Lem-
ma 2.2, N(x, u(x)) belongs o Lp
s+1/p−2(Ω), Theo em 2.1 gi es us ha T
is well defined. Fu he mo e, he e exis s C>0 such ha
(5.5) TuLnp/(n−1−sp)(Ω)
≤C( Lp
s+1/p−2(Ω) +N(x, u)Lp
s+1/p−2(Ω) +gBp,p
s(∂Ω)).
The es ima e (5.1) also gi es us
N(x, u(x))Lp
s+1/p−2(Ω) ≤K(1 + uδ
Lnp/(n−1−sp)(Ω))(5.6)
Semilinea Poisson P oblems 391
o some K=K(k1,k
2)>0 independen o u. In pa icula , i we ake
R>0 big enough such ha
C( Lp
s+1/p−2(Ω) +gBp,p
s(∂Ω) +K(1 + Rδ)) ≤R,(5.7)
hen Tmaps he ball {h∈Lnp/(n−1−sp)(Ω); hLnp/(n−1−sp)(Ω) ≤R}
in o i sel .
Nex , he ac ha Tis con inuous and compac is seen essen ially as
in Theo em 3.1. Hence, by Schaude ’s fixed-poin heo em, he map T
has a fixed poin Tu =u. Since Tu sol es (5.4) we also ha e ha
u=Tu ∈Lp
s+1/p(Ω) and T u=T (Tu)=g. This concludes he p oo
o he case sp < n −1.
The emaining cases discussed in he s a emen o he heo em a e
deal wi h simila ly and a e somewha easie ; we omi he s aigh o -
wa d de ails.
Example 5.1. Ou fi s example illus a ing Theo ems 3.1, 4.1 is he
bounda y p oblem
(DP)±








∆u±|u|qu= ∈Lp
s+1
p−2(Ω),
T u=g∈Bp,p
s(∂Ω),
u∈Lp
s+1
p
(Ω).
(5.8)
Theo em 3.1 gi es ha o he choice o he nega i e sign (5.8) is sol able
o all p,ssa is ying (2.2) and qsuch ha
0≤q< np
n−1−sp ·min 2
n,1−n−1−sp
np i sp < n −1,(5.9)
and q>0i sp ≥n−1, g an ed ha n≥4. Explici condi ions, modeled
upon (3.8)–(3.10), can be also gi en o n=2,3.
A case which is no di ec ly amenable o he analysis we ha e de el-
oped so a co esponds o he choice o a posi i e sign in (5.8). In his
si ua ion, a pa ial answe can be ob ained by elying on Theo em 2.1
and p oceeding much as in [32, Chap e 6] (pa en he ically, i should be
poin ed ou ha his app oach wo ks only o small da a).
Finally, he ange −1<q<0 is co e ed by Theo em 4.1 o ei-
he choice o he sign. Fu he mo e, o (DP)−one can also es ablish
uniqueness in his ange. This can be p o ed by adap ing he a gumen
used in [8, Example 4.5].
Nex we discuss a wo dimensional cu a u e equa ion (c . also [7], [8]
o a diffe en con ex ).
392 M. Dindoˇ
s, M. Mi ea
Example 5.2. Le Ω ⊂Mbe a connec ed Lipschi z domain on a wo
dimensional compac mani old M, equipped wi h a Riemannian me ic g,
whose Gauss cu a u e is k(x). The p oblem o be add essed is ha o
con o mally al e ing g o a new me ic ˜gin Ω wi h a p esc ibed Gaussian
cu a u e ˜
k(x)≤0 on Ω, and such ha ˜g|∂Ω=g|∂Ω.
A well known o mula, whose p oo can be ound in, e.g., [36, Appen-
dix C], s a es ha i gand ˜ga e con o mally ela ed, i.e.,
˜g=e2ug,(5.10)
hen he cu a u es ˜
kand ksa is y
˜
k(x)=e−2u(−∆u+k(x)),(5.11)
whe e ∆ = ∆gis he Laplace-Bel ami ope a o associa ed wi h he
o iginal me ic g. Thus, in iew o (5.10)–(5.11), ma e s come down o
sol ing he nonlinea PDE
∆u=k(x)−˜
k(x)e2u,u|∂Ω=0.(5.12)
Nex we s udy condi ions gua an eeing ha he nonlinea Di ichle
p oblem jus o mula ed sa isfies he assump ions o Theo em 3.2. Fi s ,
obse e ha we can ew i e (5.12) as
∆u−−˜
k(x)e2u−1
uu=k(x)−˜
k(x),(5.13)
which, in he no a ion employed in Theo em 3.2, ansla es in o
a(x, u):=−˜
k(x)e2u−1
u,and (x):=k(x)−˜
k(x).(5.14)
Clea ly, a(x, u)≥0 since, by assump ion, ˜
k(x)≤0. Also, bo h (3.39)
and (3.40) hold p o ided ˜
k∈L (Ω) o some >(1 + 1
2p−s
2)−1. Hence,
i we assume ha he o iginal me ic enso gsa isfies (1.1), as well as,
g∈Lp
s+1
p
(Ω),(5.15)
hen k∈Lp
s+1
p−2(Ω), since i has he same smoo hness as ∇2g. Thus,
g an ed hese condi ions, he exis ence pa o Theo em 3.2 is applica-
ble. In ac , he uniqueness condi ion in Theo em 3.2 also holds, since
b(x, u):=−˜
k(x)(e2u−1), and he e o e
∂
∂ub(x, u)=−2˜
k(x)e2u≥0.(5.16)
Semilinea Poisson P oblems 393
The conclusion is ha o any s,psa is ying (2.10), and sp ≥1,
we can uniquely ex end gsa is ying (5.15) and (1.1) con o mally in-
side Ω o a new me ic ˜gwi h a p esc ibed cu a u e ˜
k≤0, ˜
k∈L (Ω),
>(1 + 1
2p−s
2)−1.
A discussion o he case when k,˜
ka e locally bounded in R2and
en i e solu ions (wi h p esc ibed asymp o ic beha io a ∞) a e sough ,
is con ained in [22].
6. O he ypes o es ima es and unc ion spaces
In his sec ion we conside he egula i y o he solu ion o Poisson
ype p oblems in e ms o he so called non angen ial maximal ope a o .
Recall ha o a unc ion u∈L∞
loc(Ω), he la e is defined by
u∗(x) := sup
y∈γ(x)
|u(x)|, o each x∈∂Ω.(6.1)
He e γ(x) is he non angen ial app oach egion wi h e ex a he bound-
a y poin x; c . [27], [7]. In wha ollows, we make he assump ion ha
∪x∈∂Ωγ(x)=Ω.
An issue which a ises na u ally in his con ex is ha o p o iding an
in insic desc ip ion o he space
{∆u;u∗∈Lp(∂Ω)}.(6.2)
While a he p esen ime his ques ion emains open, ou nex defini ion
iden ifies a ich linea subspace o (6.2).
Specifically, le Ω be a Lipschi z subdomain o he Riemannian man-
i old Mand fix n/2<ρ≤∞,1≤q≤∞. We se
(6.3) Lq
ρ(Ω) := 

 =
j
uj j;u∗
j∈Lq(∂Ω),
j∈Lρ(Ω),

j
u∗
jLq(∂Ω) jLρ(Ω) <∞


and equip i wi h he no m
(6.4)  Lp
ρ(Ω) :=  L1(Ω)
+ in 


j
u∗
jLq(∂Ω) jLρ(Ω); =
j
uj ja.e. on Ω

.
400 M. Dindoˇ
s, M. Mi ea
Re e ences
[1] T. Aubin,“Some nonlinea p oblems in Riemannian geome y”,
Sp inge Monog aphs in Ma hema ics, Sp inge -Ve lag, Be lin,
1998.
[2] C. Benne and R. Sha pley,“In e pola ion o ope a o s”, Pu e
and Applied Ma hema ics 129, Academic P ess, Inc., Bos on, MA,
1988.
[3] J. Be gh and J. L¨
o s ¨
om,“In e pola ion spaces. An in o-
duc ion”, G undleh en de Ma hema ischen Wissenscha en 223,
Sp inge -Ve lag, Be lin-New Yo k, 1976.
[4] L. A. Ca a elli and X. Cab ´
e,“Fully nonlinea ellip ic
equa ions”, Ame ican Ma hema ical Socie y Colloquium Publica-
ions 43, Ame ican Ma hema ical Socie y, P o idence, RI, 1995.
[5] Z.-Q. Chen, R. J. Williams and Z. Zhao, On he exis ence
o posi i e solu ions o semilinea ellip ic equa ions wi h singula
lowe o de coefficien s and Di ichle bounda y condi ions, Ma h.
Ann. 315(4) (1999), 735–769.
[6] B. E. Dahlbe g and C. E. Kenig, Ha dy spaces and he Neu-
mann p oblem in Lp o Laplace’s equa ion in Lipschi z domains,
Ann. o Ma h. (2) 125(3) (1987), 437–465.
[7] M. Dindoˇ
s, Exis ence and uniqueness o a semilinea ellip ic p ob-
lem on Lipschi z domains in Riemannian mani olds, Comm. Pa ial
Diffe en ial Equa ions ( o appea ).
[8] M. Dindoˇ
s, Exis ence and uniqueness o a semilinea ellip ic p ob-
lem on Lipschi z domains in Riemannian mani olds. II, P ep in
(2000).
[9] E. Fabes, O. Mendez and M. Mi ea, Bounda y laye s on
Sobole -Beso spaces and Poisson’s equa ion o he Laplacian in
Lipschi z domains, J. Func . Anal. 159(2) (1998), 323–368.
[10] S. Fuˇ
c´
ık,“Sol abili y o nonlinea equa ions and bounda y alue
p oblems”, Ma hema ics and i s Applica ions 4, D. Reidel Publish-
ing Co., Do d ech -Bos on, Mass., 1980.
[11] S. Fuˇ
c´
ık and A. Ku ne ,“Nonlinea diffe en ial equa ions”,
S udies in Applied Mechanics 2, Else ie Scien ific Publishing Co.,
Ams e dam-New Yo k, 1980.
[12] M. Giaquin a,“In oduc ion o egula i y heo y o nonlinea el-
lip ic sys ems”, Lec u es in Ma hema ics ETH Z¨u ich, Bi kh¨ause
Ve lag, Basel, 1993.

Semilinea Poisson P oblems 401
[13] D. Gilba g and N. S. T udinge ,“Ellip ic pa ial diffe en ial
equa ions o second o de ”, Rep in o he 1998 edi ion, Classics in
Ma hema ics, Sp inge -Ve lag, Be lin, 2001.
[14] E. Hebey,“Nonlinea analysis on mani olds: Sobole spaces and
inequali ies”, Cou an Lec u e No es in Ma hema ics 5, New Yo k
Uni e si y, Cou an Ins i u e o Ma hema ical Sciences, New Yo k,
1999.
[15] V. Isako and A. I. Nachman, Global uniqueness o a wo-
dimensional semilinea ellip ic in e se p oblem, T ans. Ame . Ma h.
Soc. 347(9) (1995), 3375–3390.
[16] D. Je ison and C. E. Kenig, The inhomogeneous Di ichle p ob-
lem in Lipschi z domains, J. Func . Anal. 130(1) (1995), 161–219.
[17] D. Je ison and C. E. Kenig, Unique con inua ion and absence
o posi i e eigen alues o Sch ¨odinge ope a o s, Ann. o Ma h. (2)
121(3) (1985), 463–494.
[18] Z. Jin, Sol abili y o Di ichle p oblems o semilinea ellip ic equa-
ions on ce ain domains, Pacific J. Ma h. 176(1) (1996), 117–128.
[19] F. John and L. Ni enbe g, On unc ions o bounded mean os-
cilla ion, Comm. Pu e Appl. Ma h. 14 (1961), 415–426.
[20] J. Johnsen and T. Runs , Semi-linea bounda y p oblems o
composi ion ype in Lp- ela ed spaces, Comm. Pa ial Diffe en ial
Equa ions 22(7–8) (1997), 1283–1324.
[21] C. E. Kenig,“Ha monic analysis echniques o second o de el-
lip ic bounda y alue p oblems”, CBMS Regional Con e ence Se ies
in Ma hema ics 83, Ame ican Ma hema ical Socie y, P o idence,
RI, 1994.
[22] C. E. Kenig and W.-M. Ni, On he ellip ic equa ion Lu −k+
Kexp[2u]=0,Ann. Scuola No m. Sup. Pisa Cl. Sci. (4) 12(2)
(1985), 191–224.
[23] J.-L. Lions,“Con ˆole op imal de sys `emes gou e n´es pa des
´equa ions aux d´e i ´ees pa ielles”, Gau hie -Villa s, Pa is, 1968.
[24] O. Mendez and M. Mi ea, Complex powe s o he Neumann
Laplacian in Lipschi z domains, Ma h. Nach . 223 (2001), 77–88.
[25] D. Mi ea, Laye po en ials and Hodge decomposi ions in wo
dimensional Lipschi z domains, Ma h. Ann. 322(1) (2002), 75–101.
[26] D. Mi ea and M. Mi ea, Gene al second o de , s ongly el-
lip ic sys ems in low dimensional nonsmoo h mani olds, in: “Ha -
monic analysis and bounda y alue p oblems” (Faye e ille, AR,
2000), Con emp. Ma h. 277, Ame . Ma h. Soc., P o idence, RI,
2001, pp. 61–86.
402 M. Dindoˇ
s, M. Mi ea
[27] D. Mi ea, M. Mi ea and M. Taylo , Laye po en ials, he
Hodge Laplacian, and global bounda y p oblems in nonsmoo h
Riemannian mani olds, Mem. Ame . Ma h. Soc. 150(713) (2001),
120 pp.
[28] M. Mi ea and M. Taylo , Bounda y laye me hods o Lip-
schi z domains in Riemannian mani olds, J. Func . Anal. 163(2)
(1999), 181–251.
[29] M. Mi ea and M. Taylo , Po en ial heo y on Lipschi z do-
mains in Riemannian mani olds: LPHa dy, and H¨olde space e-
sul s, Comm. Anal. Geom. 9(2) (2001), 369–421.
[30] M. Mi ea and M. Taylo , Po en ial heo y on Lipschi z do-
mains in Riemannian mani olds: Sobole -Beso space esul s and
he Poisson p oblem, J. Func . Anal. 176(1) (2000), 1–79.
[31] M. Mi ea and M. Taylo , Po en ial heo y on Lipschi z do-
mains in Riemannian mani olds: H¨olde con inuous me ic enso s,
Comm. Pa ial Diffe en ial Equa ions 25(7–8) (2000), 1487–1536.
[32] T. Runs and W. Sickel,“Sobole spaces o ac ional o de ,
Nemy skij ope a o s, and nonlinea pa ial diffe en ial equa ions”,
de G uy e Se ies in Nonlinea Analysis and Applica ions 3, Wal e
de G uy e & Co., Be lin, 1996.
[33] W. Sickel, Boundedness p ope ies o he mapping → | |µ,
0<µ<1 in he amewo k o Beso Spaces, P ep in (2000).
[34] E. M. S ein,“Singula in eg als and diffe en iabili y p ope ies o
unc ions”, P ince on Ma hema ical Se ies 30, P ince on Uni e si y
P ess, P ince on, N.J., 1970.
[35] R. S. S icha z, A no e on T udinge ’s ex ension o Sobole ’s
inequali ies, Indiana Uni . Ma h. J. 21 (1971/72), 841–842.
[36] M. Taylo ,“Pa ial diffe en ial equa ions. Basic heo y”, Tex s in
Applied Ma hema ics 23, Sp inge -Ve lag, New Yo k, 1996.
[37] H. T iebel,“Theo y o unc ion spaces”, Monog aphs in Ma he-
ma ics 78, Bi kh¨ause Ve lag, Basel, 1983.
[38] H. T iebel, Mapping p ope ies o nonlinea ope a o s gene a ed
by Φ(u)=|u|ρand by holomo phic Φ(u) in unc ion spaces o
Beso -Ha dy-Sobole ype. Bounda y alue p oblems o ellip ic di -
e en ial equa ions o ype ∆u= (x)+Φ(u), Ma h. Nach . 117
(1984), 193–213.
[39] G. Ve cho a, Laye po en ials and egula i y o he Di ichle
p oblem o Laplace’s equa ion in Lipschi z domains, J. Func . Anal.
59(3) (1984), 572–611.
Semilinea Poisson P oblems 403
Ma in Dindoˇs:
Depa men o Ma hema ics
Co nell Uni e si y
310 Malo Hall I haca, NY, 14853
U.S.A.
E-mail add ess:[email p o ec ed]
Ma ius Mi ea:
Depa men o Ma hema ics
Uni e si y o Missou i a Columbia
Columbia, MO 65211
U.S.A.
E-mail add ess:[email p o ec ed]
Rebu el 17 de se emb e de 2001.