Elliptic equations involving the $p$-Laplacian and a gradient term having natural growth

0.25 MB·

by  Djairo G. de Figueiredo

ELLIPTIC EQUATIONS INVOLVING THE p-LAPLACIAN AND A
GRADIENT TERM HAVING NATURAL GROWTH
7
1 D.G.DEFIGUEIREDO,J-P.GOSSEZ,H.RAMOSQUOIRIN,ANDP.UBILLA
0
2
Abstract. Weinvestigatetheproblem

n −∆pu=g(u)|∇u|p+f(x,u) in Ω,
a (P)  u>0 in Ω,

J
u=0 on ∂Ω,

9 
in a bounded smooth domain Ω ⊂ RN. Using a Kazdan-Kramer change of

variable we reduce this problem to a quasilinear one without gradient term
]
P and therefore approachable by variational methods. In this way we come to
A somenewandinterestingproblemsforquasilinearellipticequationswhichare
motivated by the need to solve (P). Among other results, we investigate the
h. validity of the Ambrosetti-Rabinowitz condition according to the behavior of
t g and f. Existence and multiplicity results for (P) are established in several
a situations.
m
[

1
1. Introduction
v
8
4 This paper is concerned with existence, non-existence and multiplicity of
1 solutions for the problem
2
0
. −∆pu=g(u)|∇u|p+f(x,u) in Ω,
1
(P)  u>0 in Ω,
0
 u=0 on ∂Ω.
7
1 Here ∆ u := div |∇u|p−2∇u is the p-Laplacian operator with 1 < p < ∞ and
p
:
v Ω⊂RN is a smoo(cid:0)th bounded(cid:1)domain.
Xi Equations like (P) have attracted a considerable interest since the well-known
worksofKazdan-Kramer[20]andSerrin[29]. In[29]itwasobserved,alsoforp=2,
r
a thatthe quadraticgrowthwithrespecttothe gradientplaysacriticalroleasfaras
existence is concerned. Since then, a largeliterature has been devotedto problems
wherethepowerinthegradientisstrictlylessthanp. Weshallfocushereontheso-
called natural growth of the gradient for the p-Laplacian, which is given precisely
by |∇u|p. In [20] it was observed, for p = 2, that the equation in (P) enjoys
some invariance property and can be transformed through a suitable change of
variableintoanequationwithoutgradientterm. Althoughthetransformedproblem
has no gradient term, variational methods do not apply in a straightforward way.
The difficulty lies in establishing a compactness condition (e.g. the Palais-Smale
1991 Mathematics Subject Classification. 35J20,35J25, 35J62.
Keywordsandphrases. quasilinearellipticproblem,naturalgrowthinthegradient,variational
methods,p-Laplacian.
D.GdeFigueiredowassupported byCNPq. J-P.Gossezwassupported byFNRS.H.Ramos
QuoirinandP.UbillaweresupportedbyFondecyt 1161635.
1
2 D.G.DEFIGUEIREDO,J-P.GOSSEZ,H.RAMOSQUOIRIN,ANDP.UBILLA
condition) for the functional associated to the differential equation. Many authors
have studied (P) by different methods like degree theory, sub-super solutions, a
priori estimates, etc. We refer, for instance, to [5, 7, 9, 16, 21, 25, 26, 30]. Some
special cases have been studied by using variational arguments and we refer, for
instance, to [1, 13, 18, 19] for the case p=2 and [15, 17] for p>1.
One of difficulties in the variational approach related to (P) is the fact that
in many cases the nonlinearity in the transformed equation does not satisfy the
Ambrosetti-Rabinowitz condition. For instance, when g ≡ 1, the nonlinearity in
thetransformedequation(seebelow)satisfiestheAmbrosetti-Rabinowitzcondition
s
only if F(x,s) := f(x,t)dt has at least an exponential growth (see Remark
0
4.2). There are howRever a certain number of situations where it does satisfy this
condition. Oneofourpurposesinthepresentpaperistoinvestigatethesesituations
in a rather systematic way.
We will thus follow the approach initiated for p = 2 in [20], transforming (P)
into a problem of the form
−∆ v =h(x,v) in Ω,
p
(Q)  v >0 in Ω,
 v =0 on ∂Ω.

The suitable change of variables v =A(u) turns out to be
s
G(t)
(1.1) A(s):= ep−1 dt,
Z
0
s
where G(s)= g(t) dt and
0
R
(1.2) h(x,s):=eG(A−1(s))f(x,A−1(s)).
Details are given at the beginning of Section 3.
We will consider two cases: h p-superlinear at zero (Subsection 2.1) and h p-
sublinear at zero (Subsection 2.2). As we will see in Lemma 5.3, this classification
corresponds to f being p-superlinear at zero and f being p-sublinear at zero,
respectively.
Inourfirstresult(Theorem2.1), h(x,s)is p-superlinearatzeroandsatisfiesthe
Ambrosetti-Rabinowitz condition. Here are a few equations which can be handled
by Theorem 2.1:
(i) −∆pu=C1|∇u|p+uqeC2u, where C1 >0, 0<C2 <C1pp∗−−1p and q >p−1
(cf. Example 2.3);
(ii) −∆ u = C(1 + u)−α|∇u|p + µup−1eβu1−α, where C > 0, 0 < α < 1,
p
0<β < p∗−p and 0<µ<λ (cf. Example 2.5);
(p−1)(1−α) 1
(iii) −∆ u=C(1+u)−α|∇u|p+ur−1, where C >0, α≥1, p<r <p∗ and, in
p
addition, C <r−p if α=1 (cf. Example 2.7);
(iv) −∆ u = uq|∇u|p +µup−1eβuq+1, where q > 0, 0 < β < p∗−p and
p (p−1)(q+1)
0<µ<λ (cf. Example 2.9).
1
ELLIPTIC EQUATIONS INVOLVING THE p-LAPLACIAN 3
Wehaveusedabovethestandardnotation: p∗ istheSobolevconjugateexponent
given by 1 = 1 − 1 with p∗ = ∞ when p ≥ N, and λ is the first eigenvalue of
p∗ p N 1
−∆ in W1,p(Ω).
p 0
Our second result (Theorem 2.10) concerns once again the case where h(x,s) is
p-superlinear at zero but does not necessarily satisfy the Ambrosetti-Rabinowitz
condition. The approach here relies instead on a monotonicity condition which
enables the verification of the Cerami condition. This monotonicity condition has
been used recently by Liu [23] and Iturriaga-Lorca-Ubilla [17] (see also Miyagaki-
Souto [24] for p=2). Here are a few equations which can be handled by Theorem
2.10:
(v) −∆ u= p−1|∇u|p+up−1(log(u+1))q, where q >0 (cf. Example 2.12);
p u+1
(vi) −∆ u=C|∇u|p+ur−1, where C >0, and p<r (cf. Example 2.13);
p
(vii) −∆ u = C|∇u|p+(log(u+1))r−1, where C > 0, and p < r (cf. Example
p
2.13).
Weremark,inequation(vi),thatifC =0andr =p∗−1then(P)reducestothe
Pohozaevproblem, which has no solutionwhen Ω is starshaped. In the case p=2,
the famous resultof Brezis-Nirenberg[6] states that the existence ofa solutioncan
be recoveredif one adds a perturbation such as λu. This result was generalized to
the case 1 < p2 ≤ N by Garcia-Peral [10], Egnell [8] and Guedda-Veron [11]. It
followsfromourTheorem2.10thattheexistenceofasolutioncanalsoberecovered
in the spirit of Brezis-Nirenberg with a perturbation such as C|∇u|p.
It is also worthwhile to compare examples (i) and (vi), which differ only by the
presence of an exponential term: the Ambrosetti-Rabinowitz condition holds for
(i), but not for (vi).
Theorems 2.1 and 2.10 are stated in Subsection 2.1.
Our third result concerns the case where h(x,s) is p-sublinear at zero and is
stated in Subsection 2.2. We introduce a parameter λ>0 in (P):
−∆ u=g(u)|∇u|p+λf(x,u) in Ω,
p
(Pλ)  u>0 in Ω,
 u=0 on ∂Ω.

The transformed problem thus reads
−∆ v =λh(x,v) in Ω,
p
(Qλ)  v >0 in Ω,
 v =0 on ∂Ω.

For a nonlinearity of concave-convextype we obtain for (P ) a result in the line
λ
of the classical one by Ambrosetti-Brezis-Cerami [2]: for some 0 < Λ < ∞, (P )
λ
admits atleasttwosolutionsfor 0<λ<Λ, atleastone solutionfor λ=Λ,andno
solution for λ>Λ. Precise statements are given in Theorems 2.16 and 2.17, which
areobtainedbyapplyingsomeoftheresultsof[12]. Hereareafewequationswhich
can be handled by Theorems 2.16 and 2.17:
4 D.G.DEFIGUEIREDO,J-P.GOSSEZ,H.RAMOSQUOIRIN,ANDP.UBILLA
(viii) −∆pu = C1|∇u|p + λuqeC2u, where C1 > 0, 0 < C2 < C1pp∗−−1p and
0≤q <p−1;
(ix) −∆ u = C(1 + u)−α|∇u|p + λ ur−1+uq−1 , where C > 0, α ≥ 1,
p
1<q <p<r <p∗ and, in additio(cid:0)n, C <r−p(cid:1)if α=1;
It is valuable to compare examples (i), (iii) with examples (viii), (ix),
respectively: at least one solution for the first ones, at least two solutions for the
second ones.
To conclude this introduction, let us comment on some related works. The
change of variables introduced for p = 2 by Kazdan-Kramer [20] has been used,
for instance, by Montenegro-Montenegro [25], Iturriaga- Lorca-Ubilla [17], and
Iturriaga-Lorca-Sa´nchez [16]. In [25] the authors obtain existence of solutions for
some specific functions g, for instance g constant or g such that lim g(s) = 0, cf.
s→∞
Examples2.3,3.1and4.1in[25]. In[17]theauthorsprovetheexistenceofasolution
when g is a constant and f(x,s) has at most an exponential growth(compare with
ourexample(i)). In[16]the authorsassumethatg is aconstantandthe modelfor
f(x,s) is a power(compare with our example (vii)).
SomesituationswhereitisnotpossibletousethechangeofvariablesofKazdan-
Kramer have also been considered, for instance, by Li-Yen-Ke [21] (see also the
referencestherein),whereasimilarproblemto(P)isstudied. However,thefunction
f mayalsodependonthegradient. Theyconsiderthecaseg(s)= c ,withapower
s+1
as a typical model for f (this is related with our example (iii)).
Let us finally observe that the results in this paper are new even in the case
p=2.
2. Statements of results
Throughoutthispaper,thefunctionsf andgareassumedtosatisfythefollowing
conditions:
(H ) g :[0,∞)→[0,∞) is continuous.
g
(H ) f :Ω×[0,∞)→[0,∞)isaCarath´eodoryfunctionsuchthatf(x,s)remains
f
bounded when s remains bounded.
By asolutionof(P)we meanu∈C1(Ω)suchthatu>0inΩ, u=0on∂Ω,and
u satisfies the equation in (P) in the weak sense.
Our first result involves the following four assumptions on g and f:
(H ) There exists r<p∗ such that
SC
f(x,s)eG(s)
lim =0
s→∞ s G(t) r−1
ep−1dt
0
(cid:16)R (cid:17)
uniformly with respect to x∈Ω.
(H ) There exist s ≥0 and θ >p such that
AR 1 0
s s
θ ep−p1G(t)f(x,t)dt≤eG(s)f(x,s) eGp−(t1)dt
Z Z
0 0
ELLIPTIC EQUATIONS INVOLVING THE p-LAPLACIAN 5
for a.e. x∈Ω and all s≥s .
0
(H ) There exist s ≥0, δ >0 and a non-empty subdomain Ω ⊂Ω such that
AR 2 0 1
s
ep−p1G(t)f(x,t)dt≥δ
Z
0
for a.e. x∈Ω and all s≥s .
1 0
As will be seen in the next section, (H ) corresponds to a subcritical growth
SC
condition for the transformed problem (Q) (cf. Lemma 3.2), (H ) and (H )
AR 1 AR 2
correspondto the Ambrosetti-Rabinowitzconditionfor (Q)(cf. Lemma 3.3). Note
that (H ), (H ) and (H ) concern the behavior of f at infinity. Note also
SC AR 1 AR 2
that if g ≡0, then (H ) reduces to the standard subcritical growth condition for
SC
f(x,s) while (H ) and (H ) reduce to the standard Ambrosetti-Rabinowitz
AR 1 AR 2
condition for f(x,s).
2.1. The case f(x,s) p-superlinear at zero.
We recall that f(x,s) is p-superlinear at zero when it satisfies the following
condition:
f(x,s)
(H ) limsup <λ uniformly with respect to x∈Ω.
λ1 s→0 sp−1 1
We shallsee thatunder this conditionh(x,s) is p-superlinearatzeroas well(cf.
Lemma 3.4). Our first result in this case is:
Theorem 2.1 (Existence with the Ambrosetti-Rabinowitz condition). Assume
(H ), (H ) , (H ) and (H ). Then problem (P) has at least one solution.
SC AR 1 AR 2 λ1
Theorem 2.1 will be proved in Section 5, and its corollariesin Section 4.
ToillustrateTheorem2.1,letusindicateafewtypicalsituationswhereitapplies.
We will distinguish a number of cases accordinglyto the behaviour of g at infinity.
The corollaries and examples below provide various concrete situations where
problem (P) can be handled in a variational way, with the standard Ambrosetti-
Rabinowitz condition being satisfied.
To simplify our statements, we assume a differentiability condition on f:
(H )′ There exist s ≥ 0 and ε > 0 such that f(x,s) ≥ ε and ∂f(x,s) exists for
f 0 ∂s
for a.e. x∈Ω and all s≥s . This derivative will be denoted by f′(x,s).
0
Corollary 2.2. Assume g(s)→g∞ as s→∞ with 0<g∞ <∞, as well as (Hf)′
and (H ). Then (P) has at least one solution if, for some p<r <p∗,
λ1
f(x,s) f′(x,s)
lim =0 and lim >0,
s→∞erp−−p1G(s) s→∞ f(x,s)
uniformly with respect to x∈Ω.
Example 2.3. Corollary 2.2 applies, for instance, to g(s)≡C1 and f(s)=sqeC2s
with C > 0, 0 < C < C p∗−p and q > p−1. This corresponds to example (i)
1 2 1 p−1
from the Introduction.
6 D.G.DEFIGUEIREDO,J-P.GOSSEZ,H.RAMOSQUOIRIN,ANDP.UBILLA
Corollary 2.4. Assume g(s)→0 as s→∞, as well as (H )′ and (H ). Assume
also that there exists s > 0 such that g(s) > 0 and g′(s)fexists for sλ1≥ s , with
0 0
moreover
g′(s)
(2.3) →0 as s→∞.
g(s)2
Then (P) has at least one solution if, for some r ∈(p,p∗),
f(x,s)g(s)r−1 f′(x,s)
(2.4) lim =0 and lim >0
s→∞ erp−−p1G(s) s→∞f(x,s)g(s)
uniformly with respect to x∈Ω.
Example 2.5. Corollary 2.4 applies, for instance, to g(s) = C(1 + s)−α and
f(x,s)=λsp−1eβs1−α, with C >0, 0<α<1, 0<β < p∗−p 1 andλ<λ . This
p−1 1−α 1
corresponds to example (ii) of the Introduction.
Note that (2.3) implies sg(s)→∞ as s→∞. We are thus dealing in Corollary
2.4 with a situation where g(s) → 0 but sg(s) → ∞ as s → ∞. In Corollary 2.6
below, we will consider a situation where g(s)→0 but sg(s)→c with 0 ≤c <∞
as s→∞.
Corollary 2.6. Assume g(s) → 0 and sg(s) → c as s → ∞, with 0 ≤ c < ∞.
Assume also (H )′ and (H ). Then (P) has at least one solution if, for some
p<r <p∗, f λ1
f(x,s) sf′(x,s)
(2.5) lim =0 and lim >p−1+c
s→∞ sr−1 s→∞ f(x,s)
uniformly with respect to x∈Ω.
Example 2.7. Corollary 2.6 applies, for instance, to g(s) = C(1 + s)−α and
f(x,s) = sr−1, where C > 0, α ≥ 1, p < r < p∗ with C < r − p if α = 1.
This corresponds to example (iii) of the Introduction.
Corollary 2.8. Assume g(s)→∞ as s→∞ as well as (H )′ and (H ). Assume
also the existence of s ≥ 0 and C < ∞ such that g(s)f> 0, g′(sλ)1exists and
0
g′(s) ≤C for s≥s . Then (P) has at least one solution if, for some p<r<p∗,
g(s) 0
(cid:12) (cid:12)
(cid:12)(cid:12) (cid:12)(cid:12) f(x,s) f′(x,s)
(2.6) lim =0 and lim >0
s→∞erp−−p1G(s) s→∞f(x,s)g(s)
uniformly with respect to x∈Ω.
Example 2.9. Corollary 2.8 applies, for instance, to g(s) = sq and f(x,s) =
λsp−1eβsq+1, where q > 0, 0 < β < p∗−p , and 0 < λ < λ e−β. This
(p−1)(q+1) 1
corresponds to example (iv) of the Introduction.
Our second result in the case where f(x,s) is p-superlinear at zero involves the
following two assumptions:
(H ) There exist s ≥0 such that for a.e. x∈Ω the function
m 0
eG(s)f(x,s)
s7→
s G(t) p−1
ep−1dt
0
(cid:16) (cid:17)
R
is nondecreasing on [s ,∞).
0
ELLIPTIC EQUATIONS INVOLVING THE p-LAPLACIAN 7
eG(s)f(x,s)
(H∞) sl→im∞ s G(t) p−1 =∞ uniformly with respect to x∈Ω.
ep−1dt
0
(cid:16)R (cid:17)
Aswewillseefromformulas(3.7)and(3.8),thequotient eG(s)f(x,s) is,upto
G(t) p−1
sep−1dt
(cid:18) 0 (cid:19)
R
a change of variable, equal to h(x,s), so that (H ) corresponds to a monotonicity
sp−1 m
condition for hs(px−,s1), while (H∞) corresponds to a p-superlinearity condition at
infinity for h(x,s).
Theorem2.10. (Existencewithamonotonicitycondition)Assume(H ), (H ),
SC λ1
(Hm) and (H∞). Then problem (P) has at least one solution.
Theorem 2.10 as well as its corollary below are proved in Section 5.
Corollary 2.11. Assume (Hf)′, (Hλ1), (HSC), (H∞) and
(H )′ lim f′(x,s) 0seGp−(t1)dt > 1 uniformly with respect to
m s→∞f(x,s) (p−1)eGp−(Rs1) −g(s) seGp−(t1)dt
0
x∈Ω. (cid:16) R (cid:17)
Then problem (P) has at least one solution.
Example 2.12. Let g(s) = p−1. Then Corollary 2.11 applies, for instance, to
1+s
f(x,s) = sp−1(log(s+1))q with q > 0. This corresponds to example (v) from the
Introduction.
Example 2.13. Let g(s)≡C, with C >0. Then (H )′ reduces to
m
f′(x,s)
Cs
lim (ep−1 −1)>C.
s→∞ f(x,s)
Thus Corollary 2.11 applies, for instance, to f(x,s) = sr−1 or f(x,s) = (log(s+
1))r−1 with p<r (no restriction from above is needed on r in view of Proposition
4.3). Thiscorrespondsrespectivelytoexamples(vi)and(vii)fromtheIntroduction.
Corollary 2.11 also applies to f(x,s) = λsp−1 or f(x,s) = λ(log(s+1))p−1 with
0<λ<λ .
1
Example 2.14. Let g(s)=(p−1)s+2. Then (H )′ reduces to
s+1 m
f′(x,s)
lim s(s+1)>p−1.
s→∞ f(x,s)
Thus Corollary 2.11 applies, for instance, to f(x,s) = sr−1 or f(x,s) = (log(s+
1))r−1 with p<r (no restriction from above is needed on r in view of Proposition
4.3). Italsoappliestof(x,s)=λsp−1orf(x,s)=λ(log(s+1))p−1with0<λ<λ .
1
Example 2.15. Let g(s) = sq with q > 0. Then (p−1)eGp−(s1) −g(s) seGp−(t1)dt is
0
decreasing on [0,∞) and consequently (Hm)′ is implied by the conditioRn
f′(x,s) seGp−(t1)dt
lim 0 >1.
s→∞ f(Rx,s)
Thus Corollary 2.11 applies, for instance, to f(x,s) = sr−1 or f(x,s) = (log(s+
1))r−1 with p<r (no restriction from above is needed on r in view of Proposition
4.6).
8 D.G.DEFIGUEIREDO,J-P.GOSSEZ,H.RAMOSQUOIRIN,ANDP.UBILLA
2.2. The case f(x,s) p-sublinear at zero.
Weconsidernowtheparametrizedproblem(P )andstillassume(H )and(H ).
λ g f
Our first result involves the following assumptions on f:
(H ) There exists a non-empty smooth domain Ω ⊂Ω such that
1 1
f(x,s)
lim =+∞
s→0 sp−1
uniformly with respect to x∈Ω .
1
(H ) There exist a non-empty smooth domain Ω ⊂ Ω and n ∈ L∞(Ω ) with
2 2 2
n≥0, n6≡0 such that
f(x,s)≥n(x)sp−1
for a.e. x∈Ω and all s≥0.
2
(H )isa(local)p-sublinearityconditionatzeroand(H )isrelatedtothetrivial
1 2
sufficient condition of non-existence for −∆u = l(u) in Ω, u > 0 in Ω, and u = 0
on ∂Ω, namely, inf{l(s) :s>0}>λ (Ω), where λ (Ω) denotes the first eigenvalue
s 1 1
of −∆ on H1(Ω).
0
Theorem 2.16 (Existence of one solution).
(1) If (H ) holds then there exists 0 < Λ ≤ ∞ such that (P ) has at least one
1 λ
solution for 0<λ<Λ and no solution for λ>Λ.
(2) If (H ) and (H ) hold then Λ<∞.
1 2
(3) If (H ), (H ), (H ) and (H ) hold then (P ) has at least one solution
1 2 SC AR 1 λ
for λ=Λ.
Our purpose now is to derive a multiplicity result for (P ) when 0 < λ < Λ.
λ
More assumptions on f will be needed:
(H ) For any s > 0 there exists B = B ≥ 0 such that for a.e. x ∈ Ω the
3 0 f,s0
function
s7→f(x,s)+Bsp−1
is nondecreasing on [0,s ].
0
(H ) For any u ∈ C1(Ω) with u > 0 in Ω, the function f(x,u(x)) is positive in
4 0
Ω, in the sense that for any compact K ⊂ Ω there exists ǫ > 0 such that
f(x,u(x))≥ε for a.e. x∈K.
These two assumptionsarerelatedto the use ofthe strongcomparisonprinciple
for the p-Laplacian (cf. Proposition 3.4 in [12]). (H ) is clearly satisfied if f(x,s)
3
is nondecreasingwith respectto s. (H ) is satisfied, for instance, if f is continuous
4
and positive in Ω×[0,∞).
Theorem 2.17 (Existence of two solutions). Assume (H ), (H ), (H ) ,
1 SC AR 1
(H ) , (H ), and (H ). Then problem (P ) has at least two solutions u,v for
AR 2 3 4 λ
0<λ<Λ, with u≤v, u6≡v.
The examples (viii) and (ix) from the introduction illustrate Theorem 2.17.
Theorems 2.16 and 2.17 are proved in Section 5.
ELLIPTIC EQUATIONS INVOLVING THE p-LAPLACIAN 9
3. Preliminaries
We first discuss the change of variable which will transform problem (P) into
problem (Q).
It wasobservedin [20] that, inthe case p=2 and g ≡1,the changeof variables
v =eu−1 transforms the quasilinear problem (P) into the semilinear one
−∆v =(1+v)f(x,log(1+v)) in Ω,
 v >0 in Ω,
 v =0 on ∂Ω.
This canbe extendedto the generalcaseof(P) inthe followingway. Considerany
change of variable
v =A(u),
where A : [0,∞) → [0,∞) is a C2 diffeomorphism with A(0) = 0, A(∞) = ∞ and
A′ > 0. Clearly u ∈ C1(Ω) with u = 0 on ∂Ω if and only if v ∈ C1(Ω) with v = 0
on ∂Ω. A simple computation yields
∆ v =(p−1)A′(u)p−2A′′(u)|∇u|p+A′(u)p−1∆ u
p p
in the distributional sense. It follows that u solves (P) if and only if v satisfies
∆ v = (p−1)A′(u)p−2A′′(u)−g(u)A′(u)p−1 |∇u|p−A′(u)p−1f(x,u).
p
(cid:2) (cid:3)
The gradient term will disappear in the above expression if A satisfies
′′ ′
(p−1)A (u)=g(u)A(u),
which will be the case if one takes
s
G(t)
(3.7) A(s):= ep−1 dt.
Z
0
With this choice for A, problem (P) for u is equivalent to problem (Q) for v:
−∆ v =h(x,v) in Ω,
p
(Q)  v >0 in Ω,
 v =0 on ∂Ω,
where 
(3.8) h(x,s):=eG(A−1(s))f(x,A−1(s))
Note that formulae (3.7) and (3.8) were already derived in [15, 25].
Wenowinvestigatehowourassumptionsonf andg transformintoassumptions
on h.
Hypothesis (H ) and (H ) clearly imply in particular that h : Ω ×[0,∞) →
g f
[0,∞) is a Carath´eodoryfunction with h(x,s) remaining bounded when s remains
bounded.
Werecallthatinthecontextofaproblemlike(Q),thefunctionhissaidtohave
subcritical growth if
(SC) There exists r<p∗ such that
h(x,s)
lim =0
s→∞ sr−1
uniformly with respect to x∈Ω.
10 D.G.DEFIGUEIREDO,J-P.GOSSEZ,H.RAMOSQUOIRIN,ANDP.UBILLA
It is said to satisfy the Ambrosetti-Rabinowitz condition if, denoting H(x,s) :=
s
h(x,t)dt,
0
R
(AR) There exist θ >p and s ≥0 such that
1 0
θH(x,s)≤sh(x,s)
for a.e. x∈Ω and all s≥s .
0
(AR) There exist a non-empty smooth subdomain Ω ⊂ Ω, δ > 0 and s ≥ 0
2 1 0
such that
H(x,s)≥δ
for a.e. x∈Ω and all s≥s .
0
Remark 3.1. The most usual version of the Ambrosetti-Rabinowitz condition
[3] in the present p-Laplacian context deals with a continuous function h(x,s) on
Ω×[0,∞) and requires the existence of θ >p and s ≥0 such that
0
(3.9) 0<θH(x,s)≤sh(x,s)
for all x ∈ Ω and all s ≥ s . Condition (3.9) clearly implies (AR) and (AR) .
0 1 2
Some caremusthoweverbe takenwhen dealingwith a continuous(and a fortioria
Carath´eodory)function on Ω×[0,∞), as wasobservedrecently in [27]. This is the
reason for the present formulation of (AR) . Note also that (AR) can be seen as
2 2
a localized version of the first inequality in (3.9).
Lemma 3.2. The function h from (3.8) satisfies (SC) if and only if the functions
f and g satisfy (H ).
SC
Proof. Writing A−1(s) = t in (3.8), replacing in (SC), and using (3.7), the
equivalence follows immediately. (cid:3)
Lemma 3.3. The function h from (3.8) satisfies (AR) and (AR) if and only if
1 2
the functions f and g satisfy (H ) and (H ) .
AR 1 AR 2
Proof. WritingA−1(s)=tin(3.8),replacingin(AR) and(AR) ,andusing (3.7),
1 2
the equivalence follows immediately. (cid:3)
Lemma 3.4. The function h from (3.8) satisfies
h(x,s) f(x,t)
limsup =limsup .
s→0 sp−1 t→0 tp−1
The same conclusion holds with limsup replaced on both sides by either liminf or
lim.
Proof. By L’Hospital rule, we have
tA′(t) teGp−(t1)
lim = lim
t→0 A(t) t→0 teGp−(τ1) dτ
0
ReGp−(t1) +(p−1)−1tg(t)eGp−(t1)
= lim
t→0 eGp−(t1)
= lim(1+(p−1)−1tg(t))
t→0
= 1.