On the Number of Positive Solutions of a
Quasilinear Elliptic Problem
G IOVANY M. F IGUEIREDO & M ARCELO F. F URTADO
A BSTRACT. We obtain multiplicity of positive solutions for the
quasilinear equation
− εp div(a(x)|∇u|p−2 ∇u) + up−1 = f (u)
in RN ,
u ∈ W 1,p (RN ),
where ε > 0 is a small parameter, 1 < p < N , f is a subcritical nonlinearity and a is a positive potential such that inf∂Λ a >
infΛ a for some open bounded subset Λ ⊂ RN . We relate the number of positive solutions with the topology of the set where a attains its minimum in Λ. The result is proved by using LjusternikSchnirelmann theory.
1. I NTRODUCTION
Several physical phenomena related to equilibrium of continuous media are modeled by the problem
− div(a(x)∇u) = g(x, u) in Ω,
u = 0 on ∂Ω,
where Ω is a domain of RN , g is a regular function and a is a nonnegative weight
(see [15]). There is an extensive literature about the regularity and spectral theory
of the above problem when g(x, u) ≡ g(u) is a linear function (see [3, 5, 8, 23]
and references there in). Concerning the nonlinear problem we can cite [9, 10,
24–26].
In [11], Chabrowski studied the problem
(1.1)
− div(a(x)∇u) + λu = K(x)|u|q−2 u
in RN ,
1836
G IOVANY M. F IGUEIREDO & M ARCELO F. F URTADO
with λ > 0, 2 < q < 2N/(N − 2), and a ∈ C(RN ) ∩ L∞ (RN ) satisfying
0 ≤ a(x) ≤ lim a(x)
|x|→∞
and being positive in the exterior of some ball BR (0). By using minimization
arguments he obtained a nonzero solution of (1.1) belonging in some appropriated
subspace of W 1,2 (RN ). In his result, it was also supposed an integrability condition
for a(x) and that K ∈ L∞ (RN ) verified either K(x) ≥ lim|x|→∞ K(x) or K is
periodic.
More recently, Lazzo [19] considered the equation (1.1) with K ≡ 1 and the
function a satisfying
0 < a0 := infN a(x) < a∞ := lim inf a(x).
(1.2)
|x|→∞
x∈R
She proved that, for λ sufficiently large, there is an effect of the topology of the set
{x ∈ RN : a(x) = a0 } on the number of positive solutions of (1.1).
Motivated by [19], we are interested in studying the number of positive solutions of a nonhomogeneous quasilinear form of equation (1.1) under a local
condition on the potential a. More precisely, we deal with the problem
(P ε )

−εp div(a(x)|∇u|p−2 ∇u) + up−1 = f (u)
in RN ,

in RN ,
1,α
u ∈ Cloc (RN ) ∩ W 1,p (RN ), u > 0
where ε > 0 is a small real parameter, 1 < p < N , 0 < α < 1, and the potential
a : RN → R is continuous and verifies
(a1 ) a0 := infx∈RN a(x) > 0,
(a2 ) there exists an open bounded set Λ ⊂ RN such that
a0 < min a
∂Λ
and M := {x ∈ Λ : a(x) = a0 } ≠ œ.
Note that the local condition (a2 ) is weaker than (1.2) in the sense that it does
not restrict the behavior of a at infinity. This kind of hypothesis was introduced
by Del Pino and Felmer [16] in the study of a semilinear Schrödinger equation.
We also suppose that f ∈ C 1 (R+ , R) satisfies
(f1 ) f (s) = o(s p−1 ) as s → 0+ ,
(f2 ) there exists p < q < p ∗ := Np/(N −p) such that f (s) = o(s q−1 ) as s → ∞,
(f3 ) there exists p < θ < q such that
Zs
0 < θF (s) := θ
0
f (τ) dτ ≤ sf (s)
for all s > 0,
Positive Solutions for a Quasilinear Problem
1837
(f4 ) there exist p < σ < p ∗ and Cσ > 0 such that
f 0 (s)s − (p − 1)f (s) ≥ Cσ s σ −1
for all s > 0.
If Y is a closed set of a topological space X , we denote by catX (Y ) the LjusternikSchnirelmann category of Y in X , namely the least number of closed and contractible sets in X which cover Y . We are now ready to state the main result of this
paper.
Theorem 1.1. Suppose that the potential a satisfies (a1 )–(a2 ) and the function
f satisfies (f1 )–(f4 ). Then, for any δ > 0 such that
Mδ := {x ∈ RN : dist(x, M) < δ} ⊂ Λ,
there exists εδ > 0 such that, for any ε ∈ (0, εδ ), the problem (Pε ) has at least
catMδ (M) solutions.
The proof of Theorem 1.1 will be done in three main steps. First, we apply
the penalization method introduced by Del Pino and Felmer in [16]. It consists
in modifying the function f (u) outside the set Λ in such a way that the energy
functional of the modified problem satisfies the Palais-Smale condition.
In the second step, by using a technique due to Benci and Cerami [6], we relate the category of the set M with the number of positive solutions of the modified
problem. It is worthwhile to mention that, since we deal with a nonhomogeneous
term f (u), we cannot apply the concentration compactness principle [21] directly as in [6, 13, 19]. This difficulty is overcame by a detailed study of the energy
functional restricted to its Nehari manifold.
In the last step we prove that the solutions obtained in the second one are in
fact solutions of (Pε ). The main problem here is that, since we are dealing with a
quasilinear problem, we cannot argue as in [16,25] to obtain uniform convergence
(on compact sets) of the solutions. Thus, we proceed as in [12] by adapting the
Moser’s interaction method [22] in order to make careful estimates on the behavior
of the solutions obtained in the second step.
To the best of our knowledge, there are no multiplicity results for quasilinear
equations via penalization methods. However, our result seems to be new even in
the semilinear case p = 2. It extends the results in [19] and complements those of
[11, 13, 16, 25].
We end this introduction by quoting some papers which dealt with the nonlinear Schrödinger problem
(1.3)

−ε2 ∆u + V (x)u = K(x)ur −1 + Q(x)ut−1
in RN ,
u ∈ C 2 (RN ) ∩ W 1,2 (RN ), u > 0
in RN ,
with ε > 0, 2 < t < r < 2∗ , V ∈ C(RN ) satisfying V0 := infRN V > 0, K positive
and K , Q ∈ L∞ (RN ). In [13], Cingolani and Lazzo considered K ≡ 1, Q ≡ 0 and
1838
G IOVANY M. F IGUEIREDO & M ARCELO F. F URTADO
V verifying a global condition similar to (1.2). They related the number of solutions of (1.3) with the topology of the set where V attains its minimum. Later, the
same authors [14] supposed that Q could change sign and obtained a multiplicity
result involving the set of global minima of a function which gave the ground state
levels of some autonomous problems related with (1.3). Roughly speaking, this
function provides some kind of global median value between the minimum of V
and the maximum of K and Q. We finally mention the paper of Ambrosetti, Malchiodi and Secchi [4], where the case Q ≡ 0 is studied. Among other results, they
proved that the number of solutions of (1.3) is related with the set of minima of
a function given explicitly in terms of V , K , r , and the dimension N . Since they
used a finite dimensional reduction method, it is supposed that V , K ∈ C 2 (RN )
are bounded, with D2 V also bounded. We emphasize that, unlike the aforementioned works, we make no assumptions on the behavior of a at infinity. In particular, we allow the potential a be unbounded or lim inf|x|→∞ a(x) < a0 .
The paper is organized as follows: in Section 2, we modify the original problem and also prove some results concerning the autonomous problem related with
(Pε ). In Section 3, we present a multiplicity result for the modified problem.
Theorem 1.1 is proved in Section 4.
2. S OME N OTATION AND THE P ENALIZATION S CHEME
Throughout the paper the conditions (a1 )–(a2 ) and (f1 )–(f4 ) will be assumed.
Since we are interested in positive solutions, we extend f to the whole
Z real line by
setting f (s) := 0 for s ≤ 0. To simplify notation, we write only
Z
RN
u instead of
u(x) dx .
In order to overcome the lack of compactness of the problem (Pε ) we make a
slight adaptation of the penalization method introduced by Del Pino and Felmer
in [16]. So, we choose k > θ/(θ − p), where θ is given by (f3 ), and take a > 0
to be the unique number such that f (a)/ap−1 = 1/k. We set
fˆ (s) :=


f (s)
if s ≤ a,

 1 s p−1
k
if s > a.
Let 0 < ta < a < Ta and take a function η ∈ C0∞ (R, R) such that
(η1 ) η(s) ≤ fˆ (s) for all s ∈ [ta , Ta ],
(η2 ) η(ta ) = fˆ (ta ), η(Ta ) = fˆ (Ta ), η0 (ta ) = fˆ 0 (ta ) and η0 (Ta ) = fˆ 0 (Ta ),
(η3 ) the map s , η(s)/s p−1 is increasing for all s ∈ [ta , Ta ].
By using the above functions we can define f˜ ∈ C 1 (R, R) as follows

fˆ (s)
f˜ (s) :=
η(s)
if s 6∈ [ta , Ta ],
if s ∈ [ta , Ta ].
Positive Solutions for a Quasilinear Problem
1839
If χΛ denotes the characteristic function of the set Λ, we introduce the penalized
nonlinearity g : RN × R → R by setting
g(x, s) := χΛ (x)f (s) + (1 − χΛ (x))f˜ (s).
(2.1)
Notice that, using (f1 )–(f4 ) and (η1 )–(η3 ), it is easy to check that g(x, s) satisfies
the following properties:
(g1 ) g(x, s) = o(s p−1 ) as s → 0, uniformly in x ∈ RN ,
(g2 ) g(x, s) = o(s q−1 ) as s → ∞, for some q ∈ (p, p ∗ ),
(g3 ) there exists θ ∈ (p, q) such
Z that
s
(i) 0 < θG(x, s) := θ
0
g(x, τ) dτ < g(x, s)s for all x ∈ Λ, s > 0,
(ii) 0 ≤ pG(x, s) ≤ g(x, s)s ≤ s p /k for all x ∈ RN \ Λ, s > 0,
(g4 ) the function s , g(x, s)/s p−1 is increasing for all x ∈ Λ, s > 0.
Remark 2.1. It is easy to check that, if uε is a positive solution of the equation
−εp div(a(x)|∇u|p−2 ∇u) + up−1 = g(x, u)
(2.2)
in RN
such that uε (x) ≤ ta for all x ∈ RN \ Λ, then g(x, uε ) = f (uε ) and therefore
uε is also a solution of (Pε ).
In view of the remark above, we deal in the sequel with the penalized problem
(P̃ε )

− div(a(εx)|∇u|p−2 ∇u) + up−1 = g(εx, u)
in RN ,

for all x ∈ RN ,
1,α
u ∈ Cloc (RN ) ∩ W 1,p (RN ), u(x) > 0
and we will look for solutions uε of (P̃ε ) verifying
(2.3)
uε (x) ≤ ta
where
for all x ∈ RN \ Λε ,
Λε := {x ∈ RN : εx ∈ Λ}.
For any ε > 0, we denote by Xε the Sobolev space W 1,p (RN ) endowed with
the norm
Z
kukε :=
(a(εx)|∇u|p + |u|p )
1/p
.
The weak solutions of (P̃ε ) are the positive critical points of the C 1 -functional
Eε : Xε → R given by
Eε (u) :=
1
p
Z
(a(εx)|∇u|p + |u|p ) −
Z
G(εx, u).
G IOVANY M. F IGUEIREDO & M ARCELO F. F URTADO
1840
For any given ξ > 0 there exists Cξ > 0 such that
|g(x, s)| ≤ ξ|s|p−1 + Cξ |s|q−1
(2.4)
Thus, if
for all x ∈ RN , s ∈ R.
Nε := {u ∈ Xε \ {0} : hEε0 (u), ui = 0}
denotes the Nehari manifold of Eε , we can use (2.1) and (g3 ) to obtain rε > 0
such that
kukε ≥ rε > 0
(2.5)
for all u ∈ Nε .
In what follows, supp u denotes the support of a function u.
Lemma 2.2. Let u ∈ Xε be a nonnegative function such that supp u ∩ Λε has
positive measure. Then there exists a unique tu > 0 such that tu u ∈ Nε .
Proof. If ψ(t) := Eε (tu) for t ≥ 0, inequality (2.4) and the Sobolev embeddings imply that ψ is positive near t = 0. Moreover,
ψ(t) ≤
tp up −
ε
p
Z
Λε
G(εx, tu).
❐
Since the set {x ∈ Λε : u(x) > 0} has positive measure, the above expression
and (g3 )(i) imply that ψ(t) → −∞ as t → ∞. Hence, there exists tu > 0 such
that ψ0 (tu ) = 0, namely the point where ψ attains its maximum. A direct computation shows that tu u ∈ Nε . The uniqueness follows from the monotonicity
condition (g4 ). The lemma is proved.
Remark 2.3. If u ∈ Nε , then the last inequality in (g3 )(ii) implies that
supp u ∩ Λε has positive measure. Thus, we can argue as above to conclude that
Eε (tu) ≤ Eε (u) for all t ≥ 0.
We make now some comments about the autonomous problem
(2.6)

−a0 div(|∇u|p−2 ∇u) + up−1 = f (u)
in RN ,

for all x ∈ RN ,
1,α
u ∈ Cloc (RN ) ∩ W 1,p (RN ), u(x) > 0
whose solutions are the positive critical points of the C 1 -functional I0 : W 1,p (RN ) →
R given by
Z
1 p
I0 (u) :=
u X − F (u),
p
where
Z
kukX :=
1/p
(a0 |∇u|p + |u|p )
for all u ∈ X := W 1,p (RN ).
Positive Solutions for a Quasilinear Problem
Let
1841
M0 := {u ∈ X \ {0} : hI00 (u), ui = 0}
be the Nehari manifold of I0 and consider the minimization problem
m0 := inf I0 (u).
u∈M0
It can be proved (see [28, Chapter 4]) that m0 is positive and can be characterized
as
(2.7)
m0 =
inf
sup I0 (tu) > 0.
u∈X\{0} t≥0
Moreover, for any u ∈ X \ {0}, there exists a unique tu > 0 such that tu u ∈ M0 .
The maximum of the function t , I0 (tu) for t ≥ 0 is achieved at t = tu .
The following result presents an interesting property of the minimizing sequences of m0 .
Lemma 2.4. Let (un ) ⊂ M0 be such that I0 (un ) → m0 and un * u weakly in
X . Then there exists a sequence (ỹn ) ⊂ RN such that vn := un (· + ỹn ) → v ∈ M0
with I0 (v) = m0 . Moreover, if u ≠ 0, then (ỹn ) can be taken identically zero and
therefore un → u in X .
❐
Proof. The proof is similar to that presented in [2, Theorem 3.1] and it will
be omitted.
We recall that a solution u of (2.6) is called ground state solution if
I0 (u) = min{I0 (v) : v is a solution of (2.6)}.
As an easy consequence of the above lemma we have the following useful result.
Corollary 2.5. The problem (2.6) possesses a ground state solution.
Proof. By the preceding lemma, the number m0 is achieved by some u ∈
M0 . Since u is a critical point of the functional I0 restricted to M0 , we can use
standard arguments (see [2, Proposition 3.2] for example) to get I00 (u) = 0 in
the dual space of X . Hence, u satisfies the equation in (2.6). If we denote by
u− := max{−u, 0} the negative part of u, we get
Z
0 = hI00 (u), u− i = u− pX − f (u)u− = u− pX
❐
and therefore u ≥ 0 in RN . Adapting arguments from [20, Theorem 1.11] we
1,α
conclude that u ∈ L∞ (RN ) ∩ Cloc
(RN ) for some 0 < α < 1, and therefore it
follows from Harnack’s inequality [27] that u(x) > 0 for all x ∈ RN . Since all
critical points of I0 belong to M0 , the solution u is a ground state solution.
1842
G IOVANY M. F IGUEIREDO & M ARCELO F. F URTADO
3. M ULTIPLICITY OF S OLUTIONS FOR THE M ODIFIED P ROBLEM
We devote this section to the proof of the following result.
Theorem 3.1. For any δ > 0 such that Mδ ⊂ Λ, there exists εδ > 0 such that,
for any ε ∈ (0, εδ ), the problem (P̃ε ) has at least catMδ (M) solutions.
Since we are intending to apply critical point theory we need to introduce
some compactness property. So, let V be a Banach space, V be a C 1 -manifold
of V and I : V → R a C 1 -functional. We say that I|V satisfies the Palais-Smale
condition at level c ((PS)c for short) if any sequence (un ) ⊂ V such that I(un ) →
c and kI 0 (un )k∗ → 0 contains a convergent subsequence. Here, we are denoting
by kI 0 (u)k∗ the norm of the derivative of I restricted to V at the point u.
We will use the following Ljusternik-Schnirelmann abstract result for C 1 manifolds (see [18, Corollary 4.17]).
Theorem 3.2. Let I be a C 1 -functional defined on a C 1 -Finsler manifold V . If
I is bounded below and satisfies the Palais-Smale condition, then I has at least cat(V )
distinct critical points.
The proof of Theorem 3.1 is rather long and will be done by applying the
above result to the functional Eε restricted to Nε . Thus, we need first relate
the category of Nε with that of M . This is exactly the content of the next two
subsections. The following result, whose proof is similar to that presented in
[7, Lemma 4.3], will be used.
Lemma 3.3. Let H , Ω+ , Ω− be closed sets with Ω− ⊂ Ω+ . Let β : H → Ω+ ,
Φ : Ω− → H be two continuous maps such that β ◦ Φ is homotopically equivalent to
the embedding ι : Ω− → Ω+ . Then cat(H) ≥ catΩ+ (Ω− ).
3.1. The map Φε Let δ > 0 be such that Mδ ⊂ Λ and choose a cut-off
function ψ ∈ C0∞ (R+ , [0, 1]) such that ψ(s) := 1 if 0 ≤ s ≤ δ/2 and ψ(s) := 0
if s ≥ δ. If ω is a ground-state solution of the problem (2.6), we define for each
y ∈ M the function
Ψε,y (x) := ψ(|εx − y|)ω
εx − y
ε
.
Let Φε : M → Nε be given by
Φε (y) := tε Ψε,y ,
where tε > 0 is the unique number such that tε Ψε,y ∈ Nε . Since ψ(|εx−y|) ≡ 1
in Bδ/2ε (y/ε) and y/ε ∈ Λε , Lemma 2.2 shows that Φε is well defined.
Lemma 3.4. Uniformly for y ∈ M , we have
lim Eε (Φε (y)) = m0 .
ε→0+
Positive Solutions for a Quasilinear Problem
1843
Proof. Arguing by contradiction, we suppose that there exist γ > 0, (yn ) ⊂
M and εn → 0+ such that
|Eεn (Φεn (yn )) − m0 | ≥ γ > 0.
(3.1)
In order to simplify the notation, we write only Φn , Ψn and tn to denote Φεn (yn ),
Ψεn ,yn and tεn , respectively.
Since hEε0n (tn Ψn ), tn Ψn i = 0, with the change of variables z := (εn x −
yn )/εn we get
tn Ψn p =
εn
Z
g(εn x, tn Ψn (x))tn Ψn (x) dx
Z
=
g(εn z + yn , tn ψ(|εn z|)w(z))tn ψ(|εn z|)w(z) dz.
Notice that, if z ∈ Bδ/εn (0), then εn z + yn ∈ Bδ (yn ) ⊂ Mδ ⊂ Λ. Recalling
that g(x, s) = f (s) for any x ∈ Λ and ψ(s) = 0 for s ≥ δ, the above expression
yields
(3.2)
p
Ψn =
εn
Z
f (tn ψ(|εn z|)ω(z))
|ψ(|εn z|)ω(z)|p dz.
(tn ψ(|εn z|)ω(z))p−1
Let α := min{w(z) : |z| ≤ δ/2}. If n0 ∈ N is such that Bδ/2 (0) ⊂ Bδ/(2εn )
for all n ≥ n0 , we obtain
(3.3)
p
Ψn ≥
εn
Z
f (tn ω(z))
|ω(z)|p dz
p−1
(t
Bδ/2 (0)
n ω(z))
Z
f (tn α)
≥
|ω(z)|p dz,
(tn α)p−1 Bδ/2 (0)
for all n ≥ n0 , where we have used that s , f (s)/s p−1 is increasing (see (f4 )).
By using Lebesgue’s theorem we may easily check that
(3.4)
p
Ψn → ωp
εn
X
Z
Z
and
F (Ψn ) →
F (ω).
If |tn | → ∞, we can use (3.3) and (f3 ) to conclude that kΨn kpεn → +∞, contradicting the first assertion above. Thus, up to a subsequence, tn → t0 ≥ 0.
For any given ξ > 0, we can use (2.4) and tn Ψn ∈ Nεn to get
p
Ψn ≤ ξ
εn
Z
p
|Ψn | +
q−p
Cξ tn
Z
|Ψn |q .
Since ξ is arbitrary, if t0 = 0 the above expression and the boundedness of (Ψn )
would imply kΨn kpεn → 0, contradicting (3.4). Hence t0 > 0 and we can take the
1844
G IOVANY M. F IGUEIREDO & M ARCELO F. F URTADO
limit in (3.2) to obtain
Z
(a0 |∇ω|p + |ω|p ) =
Z
f (t0 ω)ω
p−1
t0
,
from which it follows that t0 ω ∈ M0 . Since ω also belongs to M0 , we conclude
that t0 = 1. Thus, letting n → ∞ in
p
tn
Eεn (Φn ) =
p
Z
(a(εn z + yn )|∇(ψ(|εn z|)ω(z))|p + |ψ(|εn z|)ω(z)|p ) dz
Z
− F (tn ψ(|εn z|)ω(z)) dz
and using the second statement in (3.4), we get
lim Eεn (Φεn (yn )) = I0 (ω) = m0 ,
which contradicts (3.1) and proves the lemma.
❐
n→∞
3.2. The map βε Let δ > 0 be such that Mδ ⊂ Λ and choose ρ = ρδ > 0
in such a way that Mδ ⊂ Bρ (0). Let Υ : RN → RN be defined as Υ(x) := x for
|x| < ρ and Υ(x) := ρx/|x| for |x| ≥ ρ . Finally, consider the barycenter map
βε : Nε → RN given by
Z
βε (u) :=
Υ(εx)|u(x)|p dx
Z
.
|u(x)|p dx
Since M ⊂ Bρ (0), we can use the definition of Υ and Lebesgue’s theorem to
conclude that
(3.5)
lim βε (Φε (y)) = y
ε→0
uniformly for y ∈ M.
Lemma 3.5. Let (εn ) ⊂ R+ be such that εn → 0 and (un ) ⊂ Nεn satisfying
Eεn (un ) → m0 . Then there exists a sequence (ỹn ) ⊂ RN such that the sequence
vn := un (· + ỹn ) has a convergent subsequence in W 1,p (RN ). Moreover, up to a
subsequence, (yn ) := (εn ỹn ) satisfies yn → y ∈ M .
Proof. We start by proving that there exists a sequence (ỹn ) ⊂ RN and constants R , γ > 0 such that
Z
(3.6)
lim inf
n→∞
BR (ỹn )
|un |p ≥ γ > 0.
Positive Solutions for a Quasilinear Problem
1845
Indeed, if this is not true, then the boundedness of (un ) in X and a lemma due
to P.-L. Lions [21, Lemma I.1] imply that un → 0 in Ls (RN ) for all p < s < p ∗ .
Given ξ > 0, we can use (2.4) and un ∈ Nεn to get
p
un =
εn
Z
Z
g(εx, un )un ≤ ξ
|un |p + Cξ
Z
|un |q .
q
N
Since un → 0 in
Z L (R ) and ξ is arbitrary, we conclude that
Z kun kεn → 0.
Moreover, since g(εx, un )un → 0, it follows from (g3 ) that G(εx, un ) → 0.
Hence, Eεn (un ) → 0, contradicting m0 > 0. Thus, (3.6) holds and, along a
subsequence,
vn := un (· + ỹn ) * v ≠ 0 weakly in X.
Let (tn ) ⊂ R+ be such that wn := tn vn ∈ M0 . Defining yn := εn ỹn ,
changing variables and using un ∈ Nεn , we get
p
tn
m0 ≤ I0 (wn ) =
p
p
≤
tn
p
Z
Z
p
tn
a0 |∇vn | +
p
p
Z
p
Z
|vn | −
(a(εn x)|∇un |p + |un |p ) −
F (tn vn )
Z
G(εn x, tn un )
= Eεn (tn un ) ≤ Eεn (un ).
Hence limn→∞ I0 (wn ) = m0 , from which it follows that wn 6→ 0 in X .
Since (wn ) and (vn ) are bounded in X and vn 6→ 0 in X , the sequence (tn )
is bounded. Thus, up to a subsequence, tn → t0 ≥ 0. If t0 = 0 then kwn kX → 0,
which does not make sense. Hence t0 > 0, and therefore the sequence (wn )
satisfies
I0 (wn ) → m0 , wn * w := t0 v ≠ 0 weakly in X.
It follows from Lemma 2.4 that wn → w , or equivalently, vn → v in X . This
proves the first part of the lemma.
We claim that (yn ) has a bounded subsequence. Indeed, if this is not the case,
then |yn | → ∞. Consider R > 0 such that Λ ⊂ BR (0). Since we may suppose that
|yn | > 2R , for any z ∈ BR/εn (0) we have
|εn z + yn | ≥ |yn | − |εn z| > R.
If Γn := BR/εn (0), we can use (un ) ⊂ Nεn , (a1 ), the change of variables x ,
z + ỹn , the expression above, and (2.1) to get
1846
Z
G IOVANY M. F IGUEIREDO & M ARCELO F. F URTADO
p
Z
p
(a0 |∇vn | + |vn | ) dz ≤
g(εn z + yn , vn )vn dz
Z
˜
=
f (vn )vn dz +
g(εn z + yn , vn )vn dz
Γ
RN \Γn
Zn
Z
≤
f˜ (vn )vn dz +
f (vn )vn dz.
Z
RN \Γn
Γn
Since f˜ (s) ≤ (1/k)s p−1 , the above expression, vn → v in X , and the
Lebesgue Dominated Convergence theorem imply that
1−
1 k
p
vn X ≤
Z
RN \BR/εn (0)
f (vn )vn dz = on (1).
Letting n → ∞ we conclude that v = 0, which is a contradiction. Thus, up to a
subsequence, yn → y ∈ RN .
It remains to check that y ∈ M . Arguing by contradiction again, we suppose
that a(y) > a0 . Then, recalling that wn → w , we can use Fatou’s lemma to
obtain
(3.7)
Z
Z
1
m0 = I0 (w) <
(a(y)|∇w|p + |w|p ) − F (w)
p
Z
Z
1
(a(εn z + yn )|∇wn |p + |wn |p ) − F (wn )
≤ lim inf
n→∞
p
≤ lim inf Eεn (tn un ) ≤ lim inf Eεn (un ) = m0 ,
n→∞
which does not make sense. The proof is finished.
❐
n→∞
Following [13], we introduce the set
(3.8)
Σε := {u ∈ Nε : Eε (u) ≤ m0 + h(ε)},
where h : R+ → R+ is such that h(ε) → 0 as ε → 0+ . Given y ∈ M , we can use
Lemma 3.4 to conclude that h(ε) = |Eε (Φε (y)) − m0 | is such that h(ε) → 0 as
ε → 0+ . Thus, Φε (y) ∈ Σε and therefore Σε ≠ œ for any ε > 0.
Lemma 3.6. For any δ > 0 we have that
lim sup dist(βε (u), Mδ ) = 0.
ε→0+ u∈Σε
Σε n
Proof. Let (εn ) ⊂ R+ be such that εn → 0. By definition there exists (un ) ⊂
such that
dist(βεn (un ), Mδ ) = sup dist(βεn (u), Mδ ) + on (1),
u∈Σεn
Positive Solutions for a Quasilinear Problem
1847
where hereafter on (1) denotes a quantity which goes to 0 as n → ∞. Thus, it
suffices to find a sequence (yn ) ⊂ Mδ such that
|βεn (un ) − yn | = on (1).
(3.9)
In order to obtain such a sequence, we note that I0 (tun ) ≤ Eεn (tun ) for any
t ≥ 0. Thus, recalling that (un ) ⊂ Σεn ⊂ Nεn , we can use (2.7) to get
m0 ≤ max I0 (tun ) ≤ max Eεn (tun ) = Eεn (un ) ≤ m0 + h(εn ),
t≥0
t≥0
from which it follows that Eεn (un ) → m0 . Thus, we may invoke Lemma 3.5 to
obtain a sequence (ỹn ) ⊂ RN such that (yn ) := (εn ỹn ) ⊂ Mδ for n sufficiently
large. Hence,
Z
Z
p
Υ(εn z + yn )|un (z + ỹn )|p dz
Υ(εn x)|un | dx
Z
Z
=
βεn (un ) =
p
|un | dx
|un (z + ỹn )|p dz
Z
(Υ(εn z + yn ) − yn )|un (z + ỹn )|p dz
Z
= yn +
.
|un (z + ỹn )|p dz
❐
Since εn z + yn → y ∈ M , we have that βεn (un ) = yn + on (1) and therefore the
sequence (yn ) verifies (3.9). The lemma is proved.
3.3. Proof of Theorem 3.1 In view of condition (g3 ), it is standard to check
that Eε satisfies the Palais-Smale condition at any level (see [25, Lemma 3.1] for
example). The next result shows that the same is true for Eε constrained to the
manifold Nε .
Lemma 3.7. The functional Eε restricted to Nε satisfies the Palais-Smale condition.
Proof. Let (un ) ⊂ Nε be such that Eε (un ) → c and kEε0 (un )k∗ → 0. Then
there exists (λn ) ⊂ R such that
Eε0 (un ) = λn Jε0 (un ) + on (1),
(3.10)
where Jε : Xε → R is given by
Z
Jε (u) :=
p
a(εx)|∇u| +
Z
p
|u| −
Z
g(εx, u)u.
1848
G IOVANY M. F IGUEIREDO & M ARCELO F. F URTADO
Since (un ) ⊂ Nε , we can use (2.1), (η3 ) and (f4 ) to get
hJε0 (un ), un i =
Z
{(p − 1)g(εx, un )un − g 0 (εx, un )u2n }
Z
=
Λε ∪{un <ta }
{(p − 1)f (un )un − f 0 (un )u2n }
Z
+
(RN \Λε )∩{ta ≤un ≤Ta }
Z
≤ −Cσ
{(p − 1)η(un )un − η0 (un )u2n }
σ
Λε ∪{un <ta }
|un | ≤ −Cσ
Z
Λε
|un |σ ,
where g 0 (x, s) means the derivative with respect to the second variable and the
numbers ta and Ta were fixed at the beginning of Section 2.
By the above expression, we may suppose that hJε0 (un ), un i → ` ≤ 0. We
claim that ` < 0. If this is the case, it follows from
0 = hEε0 (un ), un i = λn hJε0 (un ), un i + on (1)
that λn → 0. Hence, use can use (3.10) to conclude that Eε0 (un ) → 0 in the dual
space of Xε . Since we already know that the unconstrained functional satisfies
Palais-Smale, we conclude that (un ) has a convergent subsequence.
It remains to prove that ` < 0. ZSuppose, by contradiction, that ` = 0. Then it
follows from |hJε0 (un ), un i| ≥ Cσ
Λε
|un |σ that un → 0 in Lσ (Λε ). By interpo-
lation, un → 0 Zin Lq (Λε ). The same argument employed in the proof of Lemma
g(εx, un )un = on (1). This and (g3 )(ii) provide
3.5 shows that
Λε
Z
Z
RN \Λε
and therefore
g(εx, un )un +
1−
g(εx, un )un ≤
Λε
1
k
Z
RN \Λε
|un |p + on (1),
1 p
un ε = on (1),
k
which contradicts (2.5) and proves the lemma.
❐
p
un =
ε
We are now ready to present the proof of Theorem 3.1.
Proof of Theorem 3.1. Given δ > 0 such that Mδ ⊂ Λ, we can use (3.5),
Lemmas 3.4 and 3.6, and argue as in [13, Section 6] to obtain εδ > 0 such that,
for any ε ∈ (0, εδ ), the diagram
Φε
βε
M --------→
-- Σε -------→
--- Mδ
Positive Solutions for a Quasilinear Problem
1849
❐
is well defined and βε ◦ Φε is homotopically equivalent to the embedding ι : M →
Mδ . It follows from Lemmas 3.7 and 3.3, and from Theorem 3.2 that Eε restricted
to Nε possesses at least catMδ (M) critical points ui . Arguing as in the proof of
Corollary 2.5, we conclude that each ui is positive and it is a solution of (P̃ε ).
The theorem is proved.
4. P ROOF OF T HEOREM 1.1
In order to prove Theorem 1.1 we need to verify that, for ε > 0 small enough, the
solutions given by Theorem 3.1 satisfy the estimate in (2.3). As in [16], the key
step for that is the following.
Proposition 4.1. For any ε > 0, let
bε∗ := sup{max uε : uε ∈ Σε is a solution of (P̃ε )}.
∂Λε
Then bε∗ is finite for ε small enough and limε→0+ bε∗ = 0.
Assuming the proposition for a moment, let us see how Theorem 1.1 follows
from it.
Proof of Theorem 1.1. Given δ > 0 such that Mδ ⊂ Λ, we can invoke Theorem 3.1 to obtain, for any ε ∈ (0, εδ ) fixed, catMδ (M) solution of (P̃ε ). Taking εδ
smaller if necessary, we can use Proposition 4.1 to conclude that, if uε is one of
these solutions, then
uε (x) < ta
(4.1)
for all ∈ ∂Λε .
The proof now can be done as in [16]. We recall the argument for completeness. The function uε ∈ W 1,p (RN ) solves the equation
div(a(εx)|∇u|p−2 ∇u) − |u|p−2 u + g(εx, u) = 0
in RN .
Let vε be defined as vε (x) = max{uε − ta , 0} if x ∈ RN \ Λε , vε (x) = 0
otherwise. In view of (4.1), we can take vε as a test function in the above equation
to get
Z
(4.2)
RN \Λε
a(εx)|∇vε |p + c(x)vε2 + ta c(x)vε = 0,
where
c(x) := |uε (x)|p−2 −
g(εx, uε (x))
.
uε (x)
❐
Condition (g3 )(ii) yields that c(x) ≥ 0 in RN \ Λε , hence all the terms in
(4.2) are zero. In particular, vε ≡ 0. Thus, (2.3) holds and uε is a solution of (Pε ).
The theorem is proved.
G IOVANY M. F IGUEIREDO & M ARCELO F. F URTADO
1850
It remains to prove Proposition 4.1. We commence with a technical convergence
result.
Lemma 4.2. Let (εn ) ⊂ R+ be such that εn → 0 and (xn ) ⊂ Λ̄εn . If uεn is a
solution of (P̃εn ) then, up to a subsequence, vn := uεn (· + xn ) converges uniformly
on compact subsets of RN .
Proof. For each n ∈ N and L > 0, we define
vL,n :=

 vn
if vn ≤ L,
L
if vn ≥ L,
p(β−1)
zL,n := vn vL,n
,
β−1
wL,n := vn vL,n ,
whit β > 1 to be determined later.
Let an (x) := a(εn x + εn xn ). Since uεn is a solution of (P̃εn ) we have that
Z
(an (x)|∇vn |
p−2
∇vn · ∇ϕ +
p−1
vn ϕ)
Z
=
g(εn x + εn xn , vn )ϕ,
for any ϕ ∈ Xε . Taking ϕ = zL,n we obtain
Z
p(β−1)
vL,n
Z
p(β−1)−1
an |∇vn |p = −p(β − 1) vL,n
vn |∇vn |p−2 ∇vn · ∇vL,n
Z
Z
p(β−1)
p(β−1)
+ g(εn x + εn xn , vn )vn vL,n
− |vn |p vL,n
.
Since
Z
p(β−1)−1
vL,n
vn |∇vn |p−2 ∇vn · ∇vL,n =
Z
p(β−1)
{vn ≤L}
vn
|∇vn |p ≥ 0,
we can use (a1 ) to get
Z
a0
p(β−1)
vL,n
|∇vn |p
Z
≤
g(εn x +
p(β−1)
εn xn , vn )vn vL,n
Z
−
p(β−1)
|vn |p vL,n
Thus, by using (2.4) with 0 < ξ < 1, we obtain a constant C1 > 0 such that
Z
(4.3)
p(β−1)
|∇vn |p
vL,n
Z
≤ C1
q
p(β−1)
vn vL,n
.
.
Positive Solutions for a Quasilinear Problem
1851
On the other hand, by the Sobolev embedding we get
wL,n p∗ ≤ C2
p
Z
|∇wL,n | = C2
p
Z
= C3 (β − 1)p
p
Z
β−1
|∇(vn vL,n )|p
p(β−2) p
vn |∇vL,n |p
vL,n
≤ C3 (β − 1)
≤ C4 β
Z
p
Z
p(β−1)
{vn ≤L}
p(β−1)
vL,n
vL,n
Z
p(β−1)
+ C3
|∇vn |p + C3
vL,n
Z
|∇vn |p
p(β−1)
vL,n
|∇vn |p
|∇vn |p .
This, (4.3), Hölder’s inequality and the boundedness of (vn ) imply
wL,n p∗ ≤ C5 βp
p
≤ C5 βp
Z
q
p(β−1)
= C5 βp
vn vL,n
Z
p∗
Z
(q−p)/p∗ Z
vn
q−p
vn
p
wL,n
pp ∗ /(p ∗ −(q−p))
wL,n
(p∗ −(q−p))/p∗
p
≤ C6 βp wL,n α∗ ,
∗
with p < α∗ = pp ∗ / p ∗ − (q − p) < p ∗ , whenever wL,n ∈ Lα (RN ).
∗
∗
Since vL,n ≤ vn , we conclude that wL,n ∈ Lα (RN ), whenever vnβ ∈ Lα (RN ).
If this is the case, it follows from the above inequality that
Z
p ∗ p ∗ (β−1)
vn vL,n
p/p∗
≤ C6 β
p
Z
β−1 ∗
(vn vL,n )α
p/α∗
βp
≤ C6 βp vn βα∗ .
By Fatou’s lemma in the variable L, we get
|vn |βp∗ ≤ (C7 )1/β β1/β |vn |βα∗ < ∞,
(4.4)
∗
whenever vnβα ∈ L1 (RN ).
∗
We now set β := p ∗ /α∗ > 1 and note that, since vn ∈ Lp (RN ), the above
2
∗
inequalities hold for this choice of β. Thus, since β α = βp ∗ , it follows that
(4.4) also holds with β replaced by β2 . Hence,
2
2
2
2
|vn |β2 p∗ ≤ (C7 )1/β β2/β |vn |β2 α∗ ≤ (C7 )1/β+1/β β1/β+2/β |vn |βα∗ .
By iterating this process and using that βα∗ = p ∗ , we obtain
Pm
|vn |βm p∗ ≤ C7
i=1
β−i
β
Pm
i=1
iβ−i
|vn |p∗ .
1852
G IOVANY M. F IGUEIREDO & M ARCELO F. F URTADO
Since β > 1 and (vn ) is bounded, we can take the limit as m → ∞ to get
|vn |∞ ≤ C8 .
Let Ω ⊂ RN be a bounded domain and ξ > 0. The above inequality and (2.4)
imply that
p−1
|vn
p−1
− g(εx + εn xn , vn )| ≤ (1 + ξ)C8
q−1
+ Cξ C8
.
Since uεn is a solution of (P̃εn ), we can use the above expression and a result of Di
Benedetto [17, Theorem 2] to conclude that, for any compact set K ⊂ Ω, there
exists a constant C̄K,Ω , depending only of C8 , N , p , and dist(K, ∂Ω), such that
|vn |C 0,α (Ω) ≤ C̄K,Ω ,
❐
for some 0 < α < 1. It follows from the Schauder embedding theorem [1, Theo0
rem 1.31] that vn possesses a convergent subsequence in Cloc
(RN ). The lemma is
proved.
Proof of Proposition 4.1. Arguing by contradiction, we suppose that there exists (εn ) ⊂ R+ such that εn → 0 and bεn = ∞. Then we can take a sequence
uεn ∈ Σεn of solutions of (P̃εn ) such that uεn (xn ) ≥ b > 0, for some b > 0 and
(xn ) ∈ ∂Λεn .
As in the proof of Lemma 3.6, we have that Eεn (un ) → m0 . If vn := uεn (· +
xn ), then vn * v weakly in X . Since vn (0) = uεn (xn ) ≥ b > 0, Lemma 4.2
implies that v ≠ 0. If tn > 0 is such that wn = tn vn ∈ M0 , we can argue as in
the proof of Lemma 3.5 to conclude that I0 (wn ) → m0 . It follows from Lemma
2.4 that wn → w in X and I0 (w) = m0 .
Since ∂Λ is compact, we may suppose that εn xn → x̄ ∈ ∂Λ. In view of
(a2 ), we have that a(x̄) > a0 . Thus, the same calculations made in (3.7) give a
contradiction and we conclude that bε∗ < ∞ for ε > 0 small enough.
For the second part we argue by contradiction again and suppose that there
exists εn → 0+ and b > 0 with bε∗n ≥ b > 0. For each n ∈ N there exists a
solution uεn ∈ Σεn of the problem (P̃εn ) in such a way that
b
2
=b−
b
2
≤ bε∗n −
b
2
≤ max uεn .
∂Λεn
❐
Hence we can take a sequence (xn ) ⊂ ∂Λεn such that uεn (xn ) ≥ b/2 > 0. The
same argument employed in the first part of the proof gets a contradiction. This
finishes the proof.
A CKNOWLEDGEMENT
The second author was supported by post-doctoral grant no. 04/09232-2 from
FAPESP/Brazil.
Positive Solutions for a Quasilinear Problem
1853
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G IOVANY M. F IGUEIREDO :
Departamento de Matemática
Universidade Federal do Pará
66075-110 Belém-PA, Brazil.
E- MAIL: giovany@ufpa.br
Positive Solutions for a Quasilinear Problem
1855
M ARCELO F. F URTADO :
Departamento de Matemática
Universidade de Brası́lia
70910-900 Brası́lia-DF, Brazil
E- MAIL: mfurtado@unb.br
K EY WORDS AND PHRASES: Ljusternik-Schnirelmann theory; penalization method; quasilinear
problems; positive solutions
2000 M ATHEMATICS S UBJECT C LASSIFICATION: 35J20 (35J60)
Received : July 26th, 2005; revised: September 28th, 2005.