Electronic Journal of Differential Equations, Vol. 2014 (2014), No. 200, pp. 1–7.
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
ftp ejde.math.txstate.edu
EXISTENCE AND MULTIPLICITY OF SOLUTIONS FOR
DIRICHLET PROBLEMS INVOLVING NONLINEARITIES WITH
ARBITRARY GROWTH
GIOVANNI ANELLO, FRANCESCO TULONE
Abstract. In this article we study the existence and multiplicity of solutions
for the Dirichlet problem
−∆p u = λf (x, u) + µg(x, u)
u=0
in Ω,
on ∂Ω
where Ω is a bounded domain in RN , f, g : Ω × R → R are Carathèodory functions, and λ, µ are nonnegative parameters. We impose no growth condition at
∞ on the nonlinearities f, g. A corollary to our main result improves an existence result recently obtained by Bonanno via a critical point theorem for C 1
functionals which do not satisfy the usual sequential weak lower semicontinuity
property.
1. Introduction
In this article we study the Dirichlet problem
−∆p u = λf (x, u)
u=0
in Ω
on ∂Ω,
(1.1)
where p ∈]1, +∞[, ∆p (·) := div(|∇(·)|p−2 ∇(·)) is the p-laplacian operator, Ω is a
bounded smooth domain in RN , λ is a positive parameter, and f : Ω × R → R is a
Carathèodory function. We will establish some existence and multiplicity results for
problem (1.1) for small values of the parameter λ by imposing only local conditions
on the nonlinearity f , allowing this latter to be of arbitrary growth at ∞. In
particular, our existence result improves and extends a recent result by Bonanno [5,
Theorem 8.1] obtained as application of a critical point theorem for C 1 functionals,
which may fail to be sequentially weakly lower semicontinuous, established by the
same author. Here, we will apply classical variational methods, regularity theory
and truncation arguments. To establish the multiplicity of solutions, we will make
use of a Mountain Pass Theorem by Pucci-Serrin [12] which applies in the case
in which the energy functional possesses at least two (not necessarily strict) local
minima. In our case, the energy functional associated to problem (1.1), with f
2000 Mathematics Subject Classification. 35J20, 35J25.
Key words and phrases. Existence and multiplicity of solutions; Dirichlet problem;
growth condition; critical point theorem.
c
2014
Texas State University - San Marcos.
Submitted May 20, 2014. Published September 26, 2014.
1
2
G. ANELLO, F. TULONE
EJDE-2014/200
suitably truncated, admits a global minimum with negative energy, and a local
minimum at 0. Our multiplicity result extends to more general nonlinearities [4,
Theorem 1]. We refer the reader to [1, 2, 3, 7, 10] for other existence and multiplicity
results for problem (1.1) involving nonlinearities with arbitrary growth.
2. Main results
Throughout this section, Ω is a bounded smooth domain in RN , and f : Ω×R →
R is a Carathèodory function. The solutions of problem (1.1) will be understood in
the weak sense. Therefore, a function u ∈ W01,p (Ω) is a (weak) solution of problem
(1.1) if and only if, for every v ∈ W01,p (Ω):
(1) Rthe function x ∈ Ω → f (x, u(x))v(x)
is summable in Ω;
R
(2) Ω |∇u(x)|p−2 ∇u(x)∇v(x)dx − λ Ω f (x, u(x))v(x)dx = 0.
2.1. Existence of solutions. The next Lemma follows by applying the well known
Moser’s iterative scheme ([6, 11]) and standard regularity results ([9]).
Lemma 2.1. Let γ > max{1, Np }. For each h ∈ Lγ (Ω) (resp. h ∈ L∞ (Ω)) denote
by uh ∈ W01,p (Ω) the (unique) weak solution of the problem
−∆p u = h(x)
u=0
in Ω
on ∂Ω.
1
Then uh ∈ C (Ω) and
Cγ :=
sup
h∈Lγ (Ω)\{0}
max Ω |uh |
1
p−1
(resp. C∞ :=
khkγ
sup
maxΩ |uh |
h∈L∞ (Ω)\{0}
1
)
p−1
khk∞
is a positive finite constant.
Our existence result reads as follows:
Theorem 2.2. Assume that the following conditions hold:
(i) there exist C > 0 and γ ∈] max{1, Np }, +∞] such that sup|t|≤C |f (·, t)| ∈
Lγ (Ω).
(ii) there exist a closed ball Br (x0 ) ⊂ Ω and η ∈ R \ {0}, with |η| ≤ C, such
that
Z
r p 1
Λ1 (η) := p
(1 − t)N −1 ess inf x∈Br (x0 ) f (x, ηt)dt
|η|
0
Cγ p−1
>
k sup |f (·, t)|kγ =: Λ2 .
C
|t|≤C
Then, for each λ ∈]Λ1 (η)−1 , Λ−1
2 ], problem (1.1) admits at least a weak solution
uλ ∈ W01,p (Ω) ∩ C 1 (Ω) such that
Z Z uλ (x)
1
p
kuλ k < λ
f (x, t)dt dx.
(2.1)
p
Ω
0
Proof. Let C > 0 be as in the hypotheses and define


f (x, −C) if (x, t) ∈ Ω×] − ∞, −C[,
fC (x, t) = f (x, t)
if (x, t) ∈ Ω × [−C, C],


f (x, C)
if (x, t) ∈ Ω×]C, +∞[.
(2.2)
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EXISTENCE AND MULTIPLICITY OF SOLUTIONS
3
Moreover, for each λ > 0, put
Ψλ (u) =
1
kukp − λ
p
Z Z
Ω
u(x)
fC (x, t)dt dx
(2.3)
0
for every u ∈ W01,p (Ω). From i) and the definition of fC , we have that Ψλ is of class
C 1 in W01,p (Ω), sequentially weakly lower semicontinuous and coercive. Hence, it
admits a global minimum uλ ∈ W01,p (Ω) which is a weak solution of the problem
−∆p u = λfC (x, u)
u=0
in Ω,
on ∂Ω.
From assumption (i) and Lemma 2.1 we have uλ ∈ C 1 (Ω) and
1
1
kuλ k∞ ≤ Cγ λ p−1 k sup |f (·, t)|kγp−1 .
|t|≤C
In particular, if λ ≤ Λ−1
we obtain kuλ k∞ ≤ C. Consequently, uλ is a weak
2
solution of problem (1.1). Now, let η and Br (x0 ) be as in the hypotheses. Let us
to show that, if λ > Λ−1
1 , then inequality (2.1) holds. To this end, it is sufficient to
show that Ψλ (ϕ) < 0 for some ϕ ∈ W01,p (Ω). Define
(
η
(r − |x − x0 |) if x ∈ Br (x0 ),
ϕ(x) = r
0
if x ∈ Ω \ Br (x0 ).
Observe that ϕ(x) ∈ [0, C] for all x ∈ Ω. Thus, if we denote by ωN the volume
of the unit ball in RN and use the polar coordinates and the integration by parts
formula, we can compute Ψλ (ϕ) as follows
Ψλ (ϕ)
=
1
ωN rN −p |η|p − λ
p
Z
Z
Br (x0 )
1
≤ ωN rN −p |η|p − λN ωN
p
Z
ϕ(x)
f (x, t)dt dx
0
r
Z
η(1− ρ
r)
ess inf x∈Br (x0 ) f (x, t)dt ρN −1 dρ
0
0
ηρ
1
ωN rN −p |η|p − λN ωN rN
ess inf x∈Br (x0 ) f (x, t)dt (1 − ρ)N −1 dρ
p
0
0
Z 1
1
= ωN rN −p |η|p − λωN rN
(1 − t)N ess inf x∈Br (x0 ) f (x, ηt)dt
p
0
Z
=
1
Z
From λ > Λ−1
1 , we promptly obtain Ψλ (ϕ) < 0.
Remark 2.3. Observe that the key inequality Λ1 > Λ2 in Theorem 2.2 is automatically satisfied if lim supη→0 Λ1 (η) = +∞. This is true, for instance, if
Rξ
ess inf x∈Br (x0 ) f (x, t)dt
lim 0
= +∞.
(2.4)
ξ→0+
|ξ|p
Rξ
Indeed, putting F (ξ) = 0 ess inf x∈Br (x0 ) f (x, t)dt for short, we have
R1
Rη
(1 − t)N ess inf x∈Br (x0 ) f (x, ηt)dt
(η − ξ)N −1 F (ξ)dξ
0
0
=
N
.
(2.5)
|η|p
|η|N +p
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G. ANELLO, F. TULONE
EJDE-2014/200
Moreover, one has
Z η
Z η
di
N −1
(η − ξ)
F (ξ)dξ = (N − 1) · · · (N − i)
(η − ξ)N −i−1 F (ξ)dξ
dη i 0
0
for all i = 1, . . . , N − 1, and
Z η
dN
(η − ξ)N −1 F (ξ)dξ = (N − 1)!F (η).
dη N 0
Therefore, using (2.4), (2.5) and the de L’Hopital rule, we easily obtain
R1
(1 − t)N ess inf x∈Br (x0 ) f (x, ηt)dt
= +∞,
lim 0
η→0
|η|p
that is to say limη→0 Λ1 (η) = +∞.
If f is nonnegative, i.e., if F is nondecreasing (and so nonnegative in [0, +∞[
and non-positive in ] − ∞, 0]), then to guarantee the limit lim supη→0 Λ1 (η) = +∞
it is sufficient requiring that
lim sup
ξ→0
F (ξ)
= +∞.
|ξ|p
(2.6)
Indeed, let {ξn } ⊂ R \ {0} be a sequence such that ξn → 0 and
F (ξn )
→ +∞.
|ξn |p
(2.7)
Without loss of generality, we can suppose ξn > 0, for all n ∈ N. Then, we have
R 2ξn
R 2ξn
(2ξn − ξ)N −1 F (ξ)dξ
(2ξn − ξ)N −1 F (ξ)dξ
ξn
0
≥
(2ξn )N +p
(2ξn )N +p
R 2ξn
N −1
dξ
F (ξn ) ξn (2ξn − ξ)
≥
·
p
N
+p
N
(ξn )
2
ξn
F (ξn )
1
=
N 2N +p (ξn )p
for all n ∈ N. Hence, in view of (2.5) and (2.7), we have
R1
(1 − t)N ess inf x∈Br (x0 ) f (x, 2ξn t)dt
lim 0
= +∞,
n→+∞
|2ξn |p
that is to say lim supη→0 Λ1 (η) = +∞.
Remark 2.4. For applications of Theorem 2.2, it is useful to have upper estimates
of the constant Cγ (γ ∈] max{1, Np }, +∞]). For the constant C∞ an upper estimate
is easy to find. Indeed, let x̄ ∈ Rn and R > 0 such that BR (x̄) ⊇ Ω and define
p
p
uR (x) = R p−1 − |x − x̄| p−1 ,
for all x ∈ BR (x̄).
Then, uR ∈ C01 (BR (x̄)) and a simple computation shows that
p p−1
−∆p uR (x) = N
for all x ∈ BR (x̄).
p−1
Now, let h ∈ L∞ (Ω) and put M = ess supΩ |h| = khk∞ . Also, let uh be the unique
solution of the problem
−∆p u = h(x)
in Ω
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EXISTENCE AND MULTIPLICITY OF SOLUTIONS
u=0
5
on ∂Ω .
Then, we have
u (x) h(x)
p−1
1 p − 1 p−1
h
−∆p
(−∆
u
(x))
=
−∆
=
≤
1
=
u
(x)
,
p
R
p
R
1
1
M
N
p
M p−1
pN p−1
for all x ∈ Ω. Since
uh (x)
M
1
p−1
≤
p−1
1
pN p−1
uR (x),
for all x ∈ ∂Ω,
by the comparison principle for the p-Laplacian, one has
1
p
1
p−1
p − 1 p−1
p−1
p−1 u (x) ≤
uh (x) ≤
khk∞
,
R
1 M
1 R
pN p−1
pN p−1
It follows that
C∞ ≤
p−1
pN
1
p−1
for all x ∈ Ω.
p
R p−1 .
Remark 2.5. Note that, if f (x, t) = 0 for all (x, t) ∈ Ω×] − ∞, 0] and f (x, t) ≥ 0
for all (x, t) ∈ Ω×]0, +∞], the nonzero solutions of problem (1.1) are positive in Ω
by the Strong Maximum Principle. Thus, if f satisfies the above condition, we can
compare Theorem 2.2 with [5, Theorem 8.1]. In our case, differently to [5], where a
polynomial growth up to the critical exponent on f was imposed (being the same
function independent of x ∈ Ω), to guarantee the existence of a positive solution
for small λ0 s, besides (2.6) and the summability condition i), no other condition is
required on f .
2.2. Multiplicity of solutions. We now state and proof our multiplicity result.
Theorem 2.6. Assume that f satisfies (i) and (ii) of Theorem 2.2. Moreover,
suppose that there exists δ > 0 such that
Z ξ
ess supx∈Ω
f (x, t)dt ≤ 0, for all ξ ∈ [−δ, δ].
(2.8)
0
Then, for each λ ∈]Λ1 (η) , Λ−1
2 ], problem (1.1) admits at least two weak solutions
uλ , vλ ∈ W01,p (Ω) ∩ C 1 (Ω) such that
Z Z uλ (x)
Z Z vλ (x)
1
1
kuλ kp < λ
kvλ kp > λ
f (x, t)dt dx,
f (x, t)dt dx.
p
p
Ω
Ω
0
0
−1
Proof. Let fC be as in (2.2) and, for λ ∈]Λ1 (η)−1 , Λ−1
2 ], let Ψλ be as in (2.3). From
the proof of Theorem 2.2, we know that Ψλ is a C 1 -functional that admits a global
minimum uλ ∈ W01,p (Ω) such that Ψλ (uλ ) < 0. Moreover, again from the proof of
Theorem 2.2, we have that every critical point of Ψλ is a weak solution of problem
(1.1). Thus, if we show that u = 0 is a local minimum for Ψλ , conclusion follows by
the mountain pass theorem of Pucci-Serrin [12]. To this end, it is sufficient to show
that u = 0 is a local minimum for Ψλ in the C01 (Ω) topology (see [8, Theorem 3.1]).
Indeed, for each sequence {un }n∈N in C01 (Ω) such that limn→+∞ kun kC 1 (Ω) = 0,
we have, thanks to (2.8), Ψλ (un ) ≥ 0 for n ∈ N large enough. Hence, 0 is a local
minimum for Ψλ .
Here is a consequence of Theorem 2.6.
6
G. ANELLO, F. TULONE
EJDE-2014/200
Corollary 2.7. Let R > 0 be the radius of the smallest ball containing Ω and let
h, g : [0, +∞[→ R be two continuous functions such that h(0) = g(0) = 0 and
Rξ
h(t)dt
lim+ 0 p
= +∞,
(2.9)
ξ
ξ→0
Rξ
g(t)dt
lim 0 s
= +∞, for some s ∈]0, p[.
(2.10)
ξ
ξ→0+
Finally, let
N Cp p−1
p p−1
C>0 R
M = sup
−1 sup |h(t)|
.
0≤t≤C
Then, for each λ ∈]0, M [, there exists µλ > 0 such that, for each µ ∈]0, µλ [, the
problem
−∆p u = λ(h(u) − µg(u))
u=0
in Ω,
on ∂Ω
admits at least two nonzero and nonnegative solutions.
Proof. Let λ ∈]0, M [ and let C > 0 be such that
−1
N Cp p−1 .
sup |h(t)|
λ< p
R p−1
0≤t≤C
Put f (x, t) = h(t) for each (x, t) ∈ Ω × [0, +∞[ and f (x, t) = 0 for each (x, t) ∈
Ω × [−∞, 0[. Let Br (x0 ) be a closed ball contained in Ω. Thanks to (2.9) and
Remark 2.3, we have
Z
r p 1
lim Λ1 (η) = lim+ p
(1 − t)N −1 h(ηt)dt = +∞.
|η|
η→0+
η→0
0
Therefore, we can find η0 ∈]0, C[ and µλ > 0 such that
h r Z 1
i−1
p
p
(1 − t)N −1 (h(η0 t) − µg(η0 t))dt
η0
0
−1
N Cp p−1 sup |h(t) − µg(t)|
.
<λ< p
R p−1
0≤t≤C
for all µ ∈]0, µλ [. From Remark 2.4, it turns out that
−1 h C i−1
N Cp p−1 ∞ p−1
sup
|h(t)
−
µg(t)|
<
sup
|h(t)
−
µg(t)|
.
Rp p − 1
C
0≤t≤C
0≤t≤C
Moreover, from (2.9) and (2.10), for each µ ∈]0, µλ [, there exists δµ > 0 such that
Z ξ
(h(t) − µg(t))dt ≤ 0,
0
for each ξ ∈ [0, δµ ]. Conclusion now follows from Theorem 2.6 applied to the
function h(t) − µg(t), extended by continuity to the whole real axis by putting
h(t) − µg(t) = 0 for all t ∈] − ∞, 0[, and from the maximum principle.
Example 2.8. Let R > 0 be as in Corollary 2.7. Moreover, let s ∈]1, p[ and
r ∈]1, s[. Then, Corollary 2.7 can be applied to the functions h(t) = ts−1 et and
EJDE-2014/200
EXISTENCE AND MULTIPLICITY OF SOLUTIONS
7
g(t) = tr−1 et . In this case, the constant M can be explicitly computed and one
has:
n N Cp −1 o
N
p p−1 p − s p−s
p−1
M = sup
sup
|h(t)|
= p
.
p p−1
R
R
p
−
1
e
C>0
0≤t≤C
We conclude that, for each λ ∈]0, M [, there exists µλ > 0 such that for each
µ ∈]0, µλ [, the problem
−∆p u = λ(us−1 − µur−1 )eu
u=0
in Ω,
on ∂Ω
admits at least two nonzero and nonnegative solutions.
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Giovanni Anello
Department of Mathematics and Computer Science, Messina University, Viale F. Stagno
D’Alcontres 31, 98166, Messina, Italy
E-mail address: ganello@unime.it
Francesco Tulone
Department of Mathematics and Computer Science, Palermo University, Via Archirafi
34, 90123, Palermo, Italy
E-mail address: francesco.tulone@unipa.it