Electronic Journal of Differential Equations, Vol. 2012 (2012), No. 26, pp. 1–14.
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
ftp ejde.math.txstate.edu
EXISTENCE OF SOLUTIONS TO NONLOCAL ELLIPTIC
EQUATIONS WITH DISCONTINUOUS TERMS
FRANCISCO JULIO S. A. CORRÊA, RUBIA G. NASCIMENTO
Abstract. In this article, we study the existence of nonnegative solutions for
the elliptic partial differential equation
−[M (kukp1,p )]1,p ∆p u = f (x, u)
u=0
in Ω,
on ∂Ω,
where Ω ⊂ RN is a bounded smooth domain, f : Ω×R+ → R is a discontinuous
nonlinear function.
1. Introduction
This article concerns the existence of solution to the elliptic problem
−[M (kukp1,p )]1,p ∆p u = f (x, u)
u=0
in Ω,
on ∂Ω,
(1.1)
where Ω ⊂ RN is a bounded smooth domain, f : Ω × R+ → R is a discontinuous
function, M : R+ → R, R+ = [0, ∞), ∆p is the p-Laplacian
∆p u = div(|∇u|p−2 ∇u),
p > 1,
and k · k1,p is the usual norm
kukp1,p =
Z
|∇u|p
Ω
in the Sobolev space W01,p (Ω).
The interest of the mathematicians on the so called nonlocal problems like (1.1)
(nonlocal because of the presence of the term M (kukp1,p ), has increased because they
represent a variety of relevant physical and engineering situations and requires a
nontrivial apparatus to solve them.
More precisely, we study the existence of nonnegative nontrivial solutions of the
problem
−[M (kukp1,p )]p−1 ∆p u = λH(u − a)uq + h(x)us in Ω,
(1.2)
u = 0 on ∂Ω,
2000 Mathematics Subject Classification. 35A15, 35J40, 34A36.
Key words and phrases. Variational methods; elliptic problem; discontinuous nonlinearity.
c
2012
Texas State University - San Marcos.
Submitted October 12, 2011. Published February 7, 2012.
F.J.S.A.C. was supported by grants 620150/2008-4 and 303080/2009-4 from CNPq/Brazil.
R.G.N. was supported by grant 505407/2008-6 from CNPq/Brazil.
1
2
F. J. S. A. CORRÊA, R. G. NASCIMENTO
EJDE-2012/26
where M : R+ → R is a continuous function, 1 < q + 1 < p < s + 1 < p∗ = NpN
−P ,
a > 0 and λ > 0 are real parameters, h : Ω → (0, ∞) is a positive measurable
function, h ∈ L∞ (Ω) and H is the Heaviside function
(
0 if t ≤ 0,
H(t) =
1 if t > 0.
We assume the following conditions:
(H1) There exist m1 , t1 > 0 such that M (t) ≥ m1 if 0 ≤ t ≤ t1 ;
(H2) There exist m2 , t2 > 0 such that 0 < M (t) ≤ m2 if t ≥ t2 ;
(H3) limt→∞ [M (tp )]p−1 t(p−1)−q = +∞ ;
(H4) M is non-increasing and M (t) > 0 for all t > 0.
Problems involving discontinuous nonlinearity appears in several physical situations. Among these, we may cite electrical phenomena, plasma physics, free
boundary value problems, etc. The reader may consult Ambrosetti-CalahorranoDobarro [5], Ambrosetti-Turner [6], Arcoya-Calahorrano [7], Arcoya-Diaz-Tello [8],
Badialle [9], [10], and the references therein. Some physical problems are related
to discontinuous surface
Γa (u) = {x ∈ Ω; u(x) = a}
which causes difficulties in analyzing this kind of problems.
When M ≡ 1, (1.2) becomes a local problem, and has been widely studied. In
particular, Alves-Bertone [1] and Alves-Bertone-Goncalves [2] use variants of the
Mountain Pass Theorem (for locally Lipschitz functionals), the Ekeland Variational
Principle, and the Subdifferential Calculus. On the other hand, after the work
by Alves, Correa and Matofu [3] several papers appeared dealing with nonlocal
problems with variational techniques; see for example [13, 14, 15, 16, 17, 20, 21, 24,
25].
This article maybe the the first study of a nonlocal problem with variational
techniques for a non-differentiable functional. We consider the non-differentiable
functional
Z
1
1c
(kukp1,p ) − λψ(u) −
h(x)(u+ )s+1 ,
Iλ,a (u) = M
p
s+1 Ω
defined on W01,p (Ω), where
Z t
Z
c(t) =
M
[M (s)]p−1 ds, ψ(u) =
F (u),
0
Ω
Z u
F (u) =
f (t)dt, f (t) = H(t − a)(t+ )q , t+ = max{0, t}.
0
By a solution to (1.2) we mean a function u ∈ W01,p (Ω) ∩ W p,
−
[M (kukp1,p )]p−1 ∆p u(x)
s
− h(x)u(x) ∈ λ[f (u(x)), f (u(x))]
where f (t) = H(t − a)(t+ )q is nondecreasing,
f (t) = lim f (t + δ),
δ→0+
f (t) = lim f (t − δ).
δ→0+
Let us consider the level set
Γa (u) = {x ∈ Ω; u(x) = a}.
q+1
q
(Ω) satisfying
a.e. in Ω,
(1.3)
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EXISTENCE OF SOLUTIONS
3
Note that if |Γa (u)| = 0, then u satisfies
− [M (kukp1,p )]p−1 ∆p u(x) = λH(u(x) − a)u(x)q + h(x)u(x)s
a.e. in Ω.
(1.4)
Clearly, a solution in the sense of (1.3) is also a solution in the sense of (1.4). The
main result of this work is as follows.
Theorem 1.1. Suppose that M satisfies assumptions (H1)–(H4) and h ∈ L∞ (Ω).
Then there are λ∗ > 0 and a∗ > 0 such that for λ ∈ (0, λ∗ ) and a ∈ (0, a∗ ),
Problem (1.2) possesses at least two nontrivial and nonnegative solutions u1 and
u2 satisfying
(i) |Γa (ui )| = 0, i = 1, 2;
(ii) Iλ,a (u2 ) < 0 < Iλ,a (u1 );
(iii) |{x ∈ Ω; ui (x) > a}| > 0, i = 1, 2.
2. Abstract Framework
In this section we establish some basic results on critical point theory for locally
Lipschitz functionals, developed by Chang [11] based on Convex Analysis and on
the Subdifferential Calculus by Clarke [12].
Definition 2.1. Let X be a Banach space. We say that the functional I : X → R
is locally Lipschitz (I ∈ Liploc (X, R)) if, given u ∈ X, there exist a neighborhood
V ≡ Vu ⊂ X, u ∈ V and a constant k ≡ kV > 0 such that
|I(v2 ) − I(v1 )| ≤ kkv2 − v1 k,
v1 , v2 ∈ V.
Definition 2.2. The directional derivative of the locally Lipschitz functional I :
X → R at u ∈ X in the direction v ∈ X is defined by
I 0 (u; v) = lim sup
h→0,λ→0+
I(u + h + λv) − I(u + h)
.
λ
We may prove that I 0 (u; v) is subadictive and positively homogeneous; that is,
I 0 (u; v1 + v2 ) ≤ I 0 (u; v1 ) + I 0 (u; v2 )
and
I 0 (u; λv) = λI 0 (u; v)
for all u, v1 , v2 ∈ X and λ > 0.
Using these properties, it follows that
|I 0 (u, v1 ) − I 0 (u, v2 )| ≤ K|v1 − v2 |,
K ≡ Ku > 0.
0
Consequently, I (u, ·) is continuous and, because it is also convex, we may consider
its subdifferential at z ∈ X which is given by
∂I 0 (u; z) = {µ ∈ X ∗ ; I 0 (u; v) ≥ I 0 (u; z) + hµ, v − zi, v ∈ X},
where X ∗ is the topological dual of X and h·, ·i is the duality pairing between X ∗
and X.
Definition 2.3. The generalized gradient of I ∈ Liploc (X, R) at u ∈ X is defined
as being the set
∂I(u) = {µ ∈ X ∗ ; I 0 (u; v) ≥ hµ, vi, for all v ∈ X}.
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F. J. S. A. CORRÊA, R. G. NASCIMENTO
EJDE-2012/26
Since I 0 (u; 0) = 0, it follows that ∂I(u) = ∂I 0 (u; 0). Furthermore, for all v ∈ X,
we have
I 0 (u; v) = max{hµ, vi; v ∈ ∂I(u)}.
An important property of the generalized gradient is as follows: if u ∈ X, then ∂I(u)
is a convex, nonempty and weak∗ − compact. In particular, there is ω ∈ ∂I(u) such
that
m(u) = min{kωk∗ ; ω ∈ ∂I(u)}.
The reader may find more properties on this subject in [12] and [18]. We note that
∂I(u) = {I 0 (u)} when I ∈ C 1 (X, R).
Definition 2.4. A sequence (un ) ⊂ X is a Palais-Smale sequence at the level c
((P S)c ), if
I(un ) → c, m(un ) → 0.
Definition 2.5. We say that the functional I ∈ Liploc (X, R) satisfies the PalaisSmale condition at the level c, if any (P S)c sequence possesses a strongly convergent
subsequence.
The proof of our main result rests heavily on the following version of the Mountain Pass Theorem for Liploc functionals whose proof may be found in Chang [11].
Its proof uses an appropriate version of the Deformation Lemma whose proof is
found in [18].
We say that u0 ∈ X is a critical point of I if 0 ∈ ∂I(u0 ). Clearly, every local
minimum (maximum) point is a critical point.
Theorem 2.6. Let I ∈ Liploc (X, R) be a functional such that I(0) = 0 and suppose
that:
(i) There are constants η > 0 and ρ > 0 such that I(u) > η, for kuk = ρ, u ∈ X;
(ii) There is e ∈ X, with kek > ρ, such that I(e) < 0.
If, in addition, I satisfies the Palais-Smale condition at the level
c = inf max I(γ(t)),
γ∈Γ t∈[0,1]
where
Γ = {γ ∈ C([0, 1], X), γ(0) = 0 and γ(1) = e},
then c > 0 is a critical value of I.
3. Preliminary Results
In this section we establish some results for proving the main result of this article.
Lemma 3.1. There is λ∗ > 0 such that the functional Iλ,a satisfies the geometric
conditions (i) and (ii) of Mountain Pass Theorem 2.6, for all a > 0
Proof. (i) Let us consider u ∈ W01,p (Ω) such that 0 < kukp1,p = r < t1 . Then, by
(H1),
Z Z u
Z
1
mp−1
H(t − a)(t+ )q dt dx −
h(x)(u+ )s+1
Iλ,a (u) ≥ 1 kukp1,p − λ
p
s+1 Ω
Ω 0
Noticing that H ≤ 1 and u+ ≤ |u|, we obtain
mp−1
λ
1
Iλ,a (u) ≥ 1 kukp1,p −
|u|q+1 −
p
q + 1 q+1 s + 1
Z
Ω
h(x)(u+ )s+1 .
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EXISTENCE OF SOLUTIONS
5
From the Sobolev immersions and from the fact that h ∈ L∞ (Ω), we obtain
C1 λ q+1
mp−1
1
rp −
r
− C2 rs+1 .
p
q+1
Choosing r > 0 sufficiently small, there exists
Iλ,a (u) ≥
λ∗ =
mp−1
(q + 1)
1
4pC1 r(q+1)−p
such that Iλ,a (u) ≥ η > 0 for kuk1,p = r, for all λ ∈ (0, λ∗ ), and for some η > 0.
(ii) Let ϕ be a function in C0∞ (Ω), ϕ > 0 in Ω. Hence, for t > 0 with ktϕkp1,p > t2 ,
it follows from (H2) and recalling that H ≥ 0, we have
Z
1
1 p−1 p
p
e
h(x)(u+ )s+1 + C
Iλ,a (tϕ) ≤ m2 t kϕk1,p −
p
s+1 Ω
and because 1 < q + 1 < p < s + 1, we have Iλ,a (tϕ) → −∞ as t → +∞. This
completes the proof.
Remark 3.2. Using Lemma 3.1 we may infer, from the Mountain Pass Theorem for
Liploc functionals, the existence of a sequence (un ) ⊂ W01,p (Ω) such that Iλ,a (un ) →
c and m(un ) → 0.
The proof of the following lemma can be found in [11].
Lemma 3.3. If u ∈ W01,p (Ω) and ω ∈ ∂ψ(u), then
ω(x) ∈ [f (u(x)), f (u(x))]
a.e. in Ω.
In what follows, for the functional Iλ,a , we will use notation
Iλ,a (u) = φ(u) − λψ(u) − J(u),
where
φ(u) =
1c
M (kukp1,p ),
p
Z
ψ(u) =
F (u),
J(u) =
Ω
1
s+1
Z
h(x)(u+ )s+1 .
Ω
Lemma 3.4. The functional Iλ,a satisfies the Palais-Smale condition.
Proof. Let (un ) ⊂ W01,p (Ω) be a sequence satisfying Iλ,a (un ) → c and m(un ) → 0.
For the rest of this article, consider (ωn ) ⊂ (W01,p (Ω))∗ be such that m(un ) =
kωn k∗ and
ωn = φ0 (un ) − λρn − J 0 (un )
(3.1)
with (ρn ) ⊂ ∂ψ(un ).
Claim 3.5. The sequence (un ) ⊂ W01,p (Ω) is bounded.
Indeed, from (3.1),
hωn + λρn , un i = [M (kun kp1,p )]p−1 kun kp1,p −
Z
s
h(x)(u+
n ) un
Ω
and so
Iλ,a (un ) −
1
hωn + λρn , un i
s+1
Z
1c
= M
(kun kp1,p ) − λ
p
1
F (un ) −
s
+
1
Ω
Z
Ω
s+1
h(x)(u+
n)
6
F. J. S. A. CORRÊA, R. G. NASCIMENTO
−
1
1
[M (kun kp1,p )]p−1 kun kp1,p +
s+1
s+1
EJDE-2012/26
Z
s
h(x)(u+
n ) un .
Ω
−
Writing un = u+
n − un , we obtain
Z
Z
1
1
s
s+1
h(x)(u+
)
u
=
h(x)(u+
,
n
n
n)
s+1 Ω
s+1 Ω
which implies
Iλ,a (un ) −
=
1
hωn + λρn , un i
s+1
Z
Z
1
1c
s+1
M (kun kp1,p ) − λ
F (un ) −
h(x)(u+
n)
p
s+1 Ω
Ω
Z
1
1
p
p
p−1
s+1
−
[M (kun k1,p )] kun k1,p +
h(x)(u+
,
n)
s+1
s+1 Ω
from which it follows that
1
Iλ,a (un ) −
hωn + λρn , un i
s+1
Z
1
1c
(kun kp1,p ) −
[M (kun kp1,p )]p−1 kun kp1,p − λ
F (un ).
= M
p
s+1
Ω
(3.2)
Since (un ) is a (P S)c sequence, there exists a constant C2 > 0 such that |Iλ,a (un )| ≤
C2 for all n ∈ N. Because
1
1
1
[−hωn , un i] ≤
|hωn , un i| ≤
kωn k∗ kun k1,p ≤ C3 kun k1,p ,
s+1
s+1
s+1
by (3.2), we obtain
1
1
1
Iλ,a (un ) −
hωn + λρn , un i ≤ C2 +
|hωn , un i| +
|hλρn , un i|
s+1
s+1
s+1
λ
|hρn , un i|.
≤ C2 + C3 kun k1,p +
s+1
Since (ρn ) ⊂ ∂ψ(un ), hρn , vi ≤ ψ 0 (un , v) for all v ∈ W01,p (Ω).
Using arguments found in [11] and [12], we can show that
hρn , un i ≤ ψ 0 (un ; un )
Z
Z
≤
f (un )un +
f (un )un
{un <0}
{un >0}
Z
Z
≤
|un |q+1 ≤
|un |q+1
{un >0}
Ω
≤ C4 kun kq+1
1,p .
and
1
l∠ωn + λρn , un i ≤ C2 + C3 kun k1,p + C5 kun kq+1
1,p .
s+1
From inequalities (3.2)-(3.3), it follows that
1c
1
M (kun kp1,p ) −
[M (kun kp1,p )]p−1 kun kp1,p
p
s+1
Z Z un
q+1
≤ C2 + C3 kun k1,p + C5 kun k1,p + λ
(t+ )q dt dx.
Iλ,a (un ) −
Ω
0
(3.3)
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EXISTENCE OF SOLUTIONS
Using the Sobolev immersions,
p
Z
1 kun k1,p
1
[M (s)]p−1 ds −
[M (kun kp1,p )]p−1 kun kp1,p
p 0
s+1
≤ C2 + C3 kun k1,p +
7
(3.4)
C7 kun kq+1
1,p .
From the continuity of M p−1 on [0, kun kp1,p ] and in view of Mean Value Theorem
for integrals, there exists ξn , 0 < ξn < kun kp1,p , such that
p
Z
1 kun k1,p
[M (s)]p−1 ds = [M (ξn )]p−1 kun kp1,p .
p 0
Since M is a nonincreasing function, we obtain
[M (ξn )]p−1 ≥ [M (kun kp1,p )]p−1
(3.5)
and
1
1
[M (kun kp1,p )]p−1 kun kp1,p −
[M (kun kp1,p )]p−1 kun kp1,p
p
s+1
≤ C2 + C3 kun k1,p + C7 kun kq+1
1,p ,
from which
C2
1 C3
1
(p−1)−q
≤
[M (kun kp1,p )]p−1 kun k1,p
−
+
+ C7 .
q
p s+1
ku
kun kq+1
n k1,p
1,p
From (H3), we conclude that (un ) is bounded. Showing the claim.
Since {un } is (P.S) sequence, using standard arguments, we can assume, without
loss of generality, that un ≥ 0 for all x ∈ Ω.
As (un ) is bounded and using the reflexivity of W01,p (Ω) there are u1 ∈ W01,p (Ω)
and ϑ ∈ R such that, up to a subsequence,
kun kp1,p → ϑp ,
u n * u1
in W01,p (Ω).
Consequently, u1 ≥ 0.
Let us now show that un → u1 in W01,p (Ω). From the continuity of M and
kun kp1,p → ϑp , we obtain M (kun kp1,p ) → M (ϑp ) and because M (ϑp ) > 0, there is
K > 0 such that
M (kun kp1,p ) ≥ K > 0
for n large enough.
Using the well known Simon inequality (see [23]), we obtain
Z
K p−1 Cp
|∇un − ∇u1 |p
Ω
Z
≤ [M (kun kp )]p−1 h|∇un |p−2 ∇un − |∇u1 |p−2 ∇u1 , ∇un − ∇u1 i
Ω
and so
K p−1 Cp
Z
|∇un − ∇u1 |p
Ω
Z
≤ [M (kun kp1,p )]p−1 kun kp1,p − [M (kun kp1,p )]p−1
|∇un |p−2 ∇un ∇u1
Ω
Z
p
p−1
− [M (kun k1,p )]
|∇u1 |p−2 ∇u1 ∇un + [M (kun kp1,p )]p−1 ku1 kp1,p .
Ω
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F. J. S. A. CORRÊA, R. G. NASCIMENTO
EJDE-2012/26
Noticing that
[M (kun kp1,p )]p−1 ku1 kp1,p − [M (kun kp1,p )]p−1
Z
|∇u1 |p−2 ∇u1 ∇un = on (1),
Ω
we obtain
K p−1 Cp
Z
|∇un − ∇u1 |p
Ω
≤
[M (kun kp1,p )]p−1 kun kp1,p
−
[M (kun kp1,p )]p−1
Z
|∇un |p−2 ∇un ∇u1 + on (1).
Ω
We point out that
Z
h(x)us+1
n
ZΩ
h(x)us1 u1
Ω
Z
h(x)us+1
1
→
ZΩ
→
h(x)us+1
1
Ω
|hρn , u1 i − hρn , un i| = |hρn , un i − hρn , u1 i|
= |hρn , un − u1 i|
≤ kρn k∗ |un − u1 |p → 0,
From (3.1) and boundedness of {un }, it follows that (ρn ) is bounded in (W01,p (Ω))∗ ,
and since un → u1 in Lα (Ω), 1 ≤ α < p∗ we obtain
λ(hρn , u1 i − hρn , un i) = on (1).
We may write
K p−1 Cp
Z
|∇un − ∇u1 |p
Ω
≤
[M (kun kp1,p )]p−1 kun kp1,p
−
[M (kun kp1,p )]p−1
Z
|∇un |p−2 ∇un ∇u1
Ω
Z
−
h(x)us+1
+
n
Ω
Z
h(x)usn u1 − λhρn , un i + λhρn , u1 i + on (1)
Ω
= hωn , un i−iωn , u1 i = on (1).
Hence, un → u1 in W01,p (Ω) which shows that Iλ,a satisfies the (P S)c condition. 4. Proof of the Theorem 1.1
Part I: Multiplicity of solutions. Using Lemmas 3.1 and 3.4, from the Mountain
Pass Theorem for Liploc functionals, it follows that u1 is a critical point of Iλ,a at
the level c; i.e.,
Iλ,a (u1 ) = c > 0
(4.1)
which implies that u1 6≡ 0.
Since {un } is (P.S) sequence, there are {ωn } ⊂ ∂Iλ,a (un ) and {ρn } ⊂ ∂Ψ(un )
verifying kωn k∗ → 0 and
Z
Z
Z
hωn , φi = [M (kun kp1,p )]p−1
|∇un |p−2 ∇un ∇ϕ −
h(x)usn − λ
ρn ϕ (4.2)
Ω
Ω
Ω
where, by Lemma 3.3,
ρn ∈ [f (un (x)), f (un (x))]
a.e in Ω.
(4.3)
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EXISTENCE OF SOLUTIONS
9
The boundedness of {un } combined with (4.3) implies in particular that {ρn } is
q+1
q+1
bounded in L q (Ω). Thus, there is ρ0 ∈ L q (Ω) such that, up to a subsequence
q+1
ρn * ρ0 in L q (Ω), or equivalently
Z
Z
ρn ϕ →
ρ0 ϕ, ∀ϕ ∈ Lq+1 (Ω)
(4.4)
Ω
Ω
By [4, Lemma 3.3], ρ0 (x) ∈ [f (u1 (x)), f (u1 (x))] in Ω.
Letting n → +∞ in (4.2), and using (4.4), we obtain the identity
Z
Z
Z
[M (ku1 kp1,p )]p−1
|∇u1 |p−2 ∇u1 ∇ϕ −
h(x)us1 ϕ = λ
ρ0 ϕ.
Ω
Ω
Ω
Showing that u1 is a weak solution of the problem
−[M (ku1 kp1,p )]p−1 ∆p u1 − hus1 = λρ0
u1 > 0
By elliptic regularity, once ρ0 ∈ L
q+1
q
in Ω
in Ω.
(Ω), it follows that u1 ∈ W p,
−[M (ku1 kp1,p )]p−1 ∆p u1 (x)
− h(x)us1 (x) = λρ0 (x)
q+1
q
(Ω) and
a.e. in Ω
which implies
−[M (ku1 kp1,p )]p−1 ∆p u1 (x) − h(x)us1 (x) ∈ λ[f (u1 (x)), f (u1 (x))]
a.e. in Ω.
This shows that u1 is a solution of (1.2).
Proof of (i). Let us show that |Γa (u1 )| = 0, where Γa (u1 ) = {x ∈ Ω; u1 (x) = a}.
Let us suppose, by contradiction, that |Γa (u1 )| > 0. From the Morrey-Stampacchia
Theorem [22], −∆p u1 (x) = 0 a.e. in Γa (u1 ), and so
− [M (ku1 kp1,p )]p−1 ∆p u1 (x) = 0
a.e. in Γa (u1 ).
(4.5)
Since u1 is a critical point, it follows that
−[M (ku1 kp1,p )]p−1 ∆p u1 (x) − h(x)us1 (x) ∈ λ[f (u1 (x)), f (u1 (x))]
a.e. in Ω.
From (4.5), we obtain
−h(x)us1 (x) ∈ λ[f (u1 (x)), f (u1 (x))]
a.e. in Ω.
+ q
q
As 0 ≤ H(u1 − a)(u+
1 ) ≤ (u1 ) , it follows from the definition of f (u1 (x)), f (u1 (x))
and from the fact that u1 ≥ 0, that
0 ≤ f (u1 (x)) ≤ f (u1 (x)) ≤ (u1 )q .
Thus, −h(x)us1 (x) ∈ [0, λaq ] which is impossible. Hence |Γa (u1 )| = 0.
Second Solution (Ekeland Variational Principle). By Lemma 3.1, we obtain
Iλ,a (u) ≥ η for 0 < kuk1,p = ρ and so Iλ,a (u) is bounded from below on B r and so
there is inf B r Iλ,a (u).
Claim 4.1. There is a∗ > 0 such that for a ∈ (0, a∗ ) we have inf Br (0) Iλ,a (u) < 0.
Indeed, let us define the auxiliary function


|x| ≤ 21 ,
τ a
ϕτ (x) = 2τ a(1 − |x|) 21 ≤ |x| ≤ 1,


0
|x| ≥ 1.
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F. J. S. A. CORRÊA, R. G. NASCIMENTO
EJDE-2012/26
where τ > (q + 2)1/(q+1) . We point out that ϕτ ∈ W01,p (Ω) and
Z
Z
Z
kϕτ kp1,p =
|∇ϕτ |p =
|∇ϕτ |p = (2τ a)p
|∇|x|)|p .
{|x|≤1}
Ω
{|x|≤1}
N −1
Using the change of variables x = ωτ , with ω ∈ S
implies dx = τ N −1 ds(ω)dτ .
xi
∂τ
Then we obtain |x| = τ . Hence, ∂xi = τ which implies |∇r| = 1. In this way,
kϕτ kp1,p ≤ (2τ a)p αN ,
where αN is √
the volume of the unit ball.
p r
√
If a < 2τ p αN =: a1 , where r is given by the geometry of the Mountain Pass
Theorem, ϕτ ∈ Br .
On the other hand,
Z Z ϕr
Z
Z ϕr
H(t − a)(t+ )q dt dx ≥
H(t − a)(t+ )q dt dx
Ω
{|x|≤1/2}
0
0
aq+1 (τ q+1 − 1)
=
q+1
Z
dx;
{|x|≤1/2}
that is,
Z Z
Ω
ϕ
H(t − a)(t+ )q dt dx ≥
0
aq+1 (τ q+1 − 1)
C1
q+1
and so
Iλ,a (ϕτ ) =
As
ts+1
s+1
R
Ω
1c
M (kϕτ kp1,p ) − λ
p
Z Z
Ω
ϕτ
H(t − a)(t+ )q dt dx −
0
ts+1
s+1
Z
q+1
h(x)(ϕ+
.
τ )
Ω
q+1
h(x)(ϕ+
≥ 0, it follows that
τ )
Iλ,a (ϕτ ) ≥
1c
λC1 aq+1 (τ q+1 − 1)
M (kϕτ kp1,p ) −
.
p
q+1
Because M is a continuous function on [0, kϕτ kp1,p ],
C2
λC1 aq+1 (τ q+1 − 1)
kϕτ kp1,p −
p
q+1
C2 p p p
λC1 aq+1 (τ q+1 − 1)
≤
2 a τ αN −
p
q+1
C 2p ap−(q+1) τ p α
λC1 (τ q+1 − 1) N
2
= aq+1
−
.
p
q+1
We now point out that
Iλ,a (ϕτ ) ≤
C2 2p ap−(q+1) τ p αN
λC1 (τ q+1 − 1)
λC1 (τ q+1 − 1)p
.
−
≤ 0 ⇔ ap−(q+1) ≤
p
q+1
C2 2p (q + 1)αN τ p
Setting
1 λC1 (τ q+1 − 1)p
2 C2 2p (q + 1)αN τ p
and taking a∗ = a∗ (λ) = min{a1 , a2 } it follows that inf Br Iλ,a < 0 for a ∈ (0, a∗ ),
which proves the claim.
a2 =
By the Ekeland Variational Principle, there exists u ∈ Br such that
Iλ,a (u ) < inf Iλ,a + ,
Br
(4.6)
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EXISTENCE OF SOLUTIONS
11
Iλ,a (u ) < Iλ,a (u) + ku − u k1,p , for all u ∈ W01,p (Ω), with u 6= u .
(4.7)
Let us choose > 0 in such a way that
0 < < inf Iλ,a − inf Iλ,a
Br
∂Br
and so u ∈ Br .
Let γ > 0 be small enough and v ∈ W01,p (Ω) with kvk1,p < 1 so that uγ =
u + γv ∈ Br . From (4.7) we have
Iλ,a (u ) < Iλ,a (u + γv) + γkvk1,p
which implies
Iλ,a (u + γv) − Iλ,a (u ) + γkvk1,p ≥ 0.
Consequently,
−kvk1,p ≤
and so
−kvk1,p ≤ lim sup
γ→0
Iλ,a (u + γv) − Iλ,a (u )
γ
Iλ,a (u + γv) − Iλ,a (u )
0
≤ Iλ,a
(u ; v).
γ
Now, since
0
Iλ,a
(u; v) =
max
for all u, v ∈ W01,p (Ω),
hµ, vi,
µ∈∂Iλ,a (u )
it follows that
0
−kvk1,p ≤ Iλ,a
(u ; v) =
hω, vi.
max
ω∈∂Iλ,a (u )
Interchanging v and −v we obtain
−kvk1,p ≤
max
hω, −vi = −
ω∈∂Iλ,a (u )
min
hω, vi.
ω∈∂Iλ,a (u )
Therefore,
min
ω∈∂Iλ,a (u )
for all v ∈ W01,p (Ω),
hω, vi ≤ kvk1,p ,
concluding that
sup
hω, vi ≤ .
min
kvk1,p <1 ω∈∂Iλ,a (u )
By Fan’s Min-max theorem, we obtain
min
sup hω, vi ≤ .
ω∈∂Iλ,a (u ) kvk1,p <1
Which along with (4.6) yields the existence of un ∈ Br such that
Iλ,a (un ) → e
c,
m(un ) =
min
ω∈∂Iλ,a (un )
kωk∗ → 0;
that is, (un ) is a Palais-Smale sequence at the level e
c.
By lemma 3.4, there exists u2 ∈ W01,p (Ω), where, passing to a subsequence if
necessary, we obtain
un → u2
in W01,p (Ω),
Iλ,a (u2 ) = e
c = inf Iλ,a < 0.
(4.8)
(4.9)
Br (0)
Thus, u2 is a local minimum point and consequently is a critical point of Iλ,a .
Hence, following the same arguments made before, we have that u2 is also solution
12
F. J. S. A. CORRÊA, R. G. NASCIMENTO
EJDE-2012/26
of the problem (1.2) and using the same arguments used in (i) with u1 solution, we
obtain also u2 satisfy (i).
Proof of (ii): By (4.1) and (4.9), it follows that Iλ,a (u2 ) < 0 < Iλ,a (u1 ).
Proof of (iii): We will show now that |{x ∈ Ω; ui (x) > a}| > 0, i = 1, 2. We
begin with the solution u1 obtained via the Mountain Pass Theorem. Suppose, by
contradiction, that u1 (x) ≤ a a.e, in Ω. So
Z Z u1
λ
H(t − a)(t+ )q = 0.
Ω
0
By the above equality and since u1 is critical point of Iλ,a , we obtain
Z
[M (ku1 kp1,p )]p−1 ku1 kp1,p =
hus+1
1 .
Ω
Note that
Z
hus+1
≤ |h|∞
1
Ω
Z
up1 us+1−p
1
Ω
and thus
[M (ku1 kp1,p )]p−1 ku1 kp1,p ≤ |h|∞
Z
up1 us+1−p
≤ C|h|∞ as+1−p ku1 kp1,p
1
Ω
which implies
[M (ku1 kp1,p )]p−1 ≤ C|h|∞ as+1−p .
Note that, there exists C > 0 such that ku1 k ≥ C. Hence, using (H4); that is,
e > 0 such that
M (t) > 0 for all t ≥ 0, there exists C
e ≤ [M (ku1 kp )]p−1 ≤ C|h|∞ as+1−p ,
0<C
1,p
for all a > 0, which is impossible.
We now consider the solution u2 obtained via Ekeland Variational Principle.
Suppose, by contradiction, that u2 (x) ≤ a a.e. in Ω. Thus
Z Z u2
λ
H(t − a)(t+ )q = 0.
Ω
0
By the above equality and since u2 is a critical point of Iλ,a , we obtain
Z
[M (ku2 kp1,p )]p−1 ku2 kp1,p =
hus+1
2 .
Ω
We will consider two cases:
Case (1): If 0 < ku2 kp1,p ≤ t1 , from (H1 ) we have
M (ku2 kp1,p ) ≥ m1 > 0.
(4.10)
Hence,
[M (ku2 kp1,p )]p−1 ku2 kp1,p ≤ |h|∞
Z
up2 us+1−p
≤ C|h|∞ as+1−p ku2 kp1,p ;
2
Ω
that is,
[M (ku2 kp1,p )]p−1 ≤ C|h|∞ as+1−p .
So [M (ku2 kp1,p )] → 0 as a → 0 which cannot happen from (4.10).
c(ku2 kp ) ≥ M
c(t1 ) > 0, because M
c is increasing.
Case (2): If ku2 kp1,p ≥ t1 , then M
1,p
Moreover,
Z
0 ≤ [M (ku2 kp1,p )]p−1 ku2 kp1,p =
hus+1
≤ |h|∞ as+1 .
2
Ω
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EXISTENCE OF SOLUTIONS
13
Thus [M (ku2 kp1,p )]p−1 ku2 kp1,p → 0 as a → 0. Since
Z
c(ku2 kp ) − 1
c(t1 ) − [M (ku2 kp )]p−1 ku2 kp ,
Iλ,a (u2 ) = M
hus+1 ≥ M
1,p
1,p
1,p
s+1 Ω 2
we obtain
c(t1 ) ≤ Iλ,a (u2 ) + [M (ku2 kp )]p−1 ku2 kp .
0<M
1,p
1,p
Because Iλ,a (u2 ) < 0, we obtain
c(t1 ) ≤ [M (ku2 kp )]p−1 ku2 kp .
0<M
1,p
1,p
c(t1 ) = 0 for t1 > 0, which is an absurd. With this, we
Hence, as a → 0, we have M
conclude the proof of the theorem.
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Francisco Julio S. A. Corrêa
Universidade Federal de Campina Grande, Unidade Acadêmica de Matemática e Estatı́stica, CEP:58109-970, Campina Grande-PB, Brazil
E-mail address: fjsacorrea@gmail.com
Rubia G. Nascimento
Faculdade de Matemática, Universidade Federal do Pará, CEP:66075-110, Belém -PA,
Brazil
E-mail address: rubia@ufpa.br