EXISTENCE RESULTS FOR SOME FUNCTIONAL ELLIPTIC
EQUATIONS
M. CHIPOT AND P. ROY
ABSTRACT. We consider nonlocal elliptic boundary value problems of the form
−A(x, u) 4 u = λf (u)
with Dirichlet boundary conditions. We show that if f has n loops then the problem
has at least n distinct solutions, by using very simple comparison principles. Also
we study the asymptotic behavior of such solutions as the parameter λ tends to ∞.
1. Introduction
Let Ω be a bounded, open subset of Rd . We denote by A a functional from
Ω × Lp (Ω), p ≥ 1, with values in R. We suppose that
(1.1)
x 7→ A(x, u) is measurable ∀u ∈ Lp (Ω)
and that the mapping
(1.2)
u 7→ A(x, u) is continuous from Lp (Ω) into R, a.e. x ∈ Ω.
We make in addition the following ellipticity assumption, namely we assume that
for some positive constants a0 , a∞ one has
(1.3)
0 < a0 ≤ A(x, u) ≤ a∞ a.e. x ∈ Ω, ∀u ∈ Lp (Ω).
We are interested in finding solutions of the following problem
(1.4)
(
−A(x, u) 4 u = λf (u) in Ω,
u=0
on ∂Ω.
Here ∂Ω denotes the boundary of Ω, f some function which will be described later
and λ is a positive parameter. Such a problem has been studied in [4] under some
assumptions on f . The key argument in [4] is the Schauder fixed point theorem
applied on a convex set involving the first eigenfunction of the Dirichlet problem in
Ω. Here we replace the first eigenfunction by the minimizer of a functional related
to the problem. This allows to relax some of the assumptions of [4] and to discover
new solutions. Problems of this type in local framework were considered in [3],
[13]-[17].
The paper is divided as follows. The next section is devoted to our main results.
A third section explains their range of applications and some asymptotic behavior.
Date: September 15, 2013.
2010 Mathematics Subject Classification. 35J15, 35J20, 35J25.
Key words and phrases. Nonlocal problems, Maximum principles, Variational problems.
1
2
M. CHIPOT AND P. ROY
2. The main existence results
Let us denote by f a Lipschitz continuous function from [0, ∞) into itself (i.e.
f ≥ 0) and suppose that there exists two nonnegative numbers 0 ≤ θ < θ0 such
that
(2.1)
f (θ) = f (θ0 ) = 0,
f > 0 on (θ, θ0 ).
We will denote by f˜ the function defined as
(
f (u) for u ∈ (θ, θ0 ),
(2.2)
f˜(u) =
0
for u ∈
/ (θ, θ0 ).
Let us set
v
Z
(2.3)
f˜(s)ds
F (v) =
0
1
(2.4)
Jλ [u] = J[u] =
2
Then we have:
Z
2
Z
|∇u| dx − λ
Ω
F (u)dx,
∀u ∈ H01 (Ω).
Ω
Lemma 2.1. For λ sufficiently large the minimization problem
(
(2.5)
u ∈ H01 (Ω),
J[u] ≤ J[v]
∀v ∈ H01 (Ω),
admits a solution ψ = ψλ such that
(2.6)
0 ≤ ψ ≤ θ0 , |ψ|∞ > θ.
(Here |ψ|∞ denotes the usual L∞ (Ω)- norm of ψ). Moreover ψ is also solution to
(
− 4 u = λf˜(u) in Ω,
(2.7)
u=0
on ∂Ω.
Proof. Claim 1 : J admits a global minimizer ψ on H01 (Ω).
Indeed, due to f ≥ 0 and (2.2) one has
(2.8)
F (v) ≤ F (θ0 ) ∀v ∈ R.
It follows that
(2.9)
J[v] ≥ −λ|Ω|F (θ0 ) ∀v ∈ H01 (Ω)
where |Ω| denotes the Lebesgue measure of Ω, i.e. J is bounded from below. The
usual direct method of calculus of variations allows then to conclude to the existence
of a ψ ∈ H01 (Ω) minimizing J on H01 (Ω). Note at this point that ψ might not be
unique. In what follows ψ is such a minimizer.
Claim 2 : |ψ|∞ > θ for λ large enough.
Suppose on the contrary that |ψ|∞ ≤ θ. Then clearly
F (ψ) = 0
and
(2.10)
J[ψ] ≥ 0.
3
Consider a function approximating the constant function equal to θ0 , for instance
dist(x, ∂Ω)
0
(2.11)
wδ (x) := θ 1 ∧
δ
(δ > 0, dist(x, ∂Ω) is the euclidean distance from x to ∂Ω, ∧ denotes the minimum
of two numbers).
It is clear that wδ ∈ H01 (Ω). Moreover if Ωδ = x ∈ Ω dist(x, ∂Ω) < δ one has
wδ = θ0 on Ω \ Ωδ . Let us fix δ such that |Ωδ | < |Ω| that is
F (θ0 )|Ωδ | − F (θ0 )|Ω| < 0.
(2.12)
Then we have
J[wδ ]
=
=
≤
=
(2.13)
<
Z
Z
1
|∇wδ |2 dx − λ
F (wδ )dx
2 Ω
Ω
Z
Z
Z
1
|∇wδ |2 dx − λ
F (θ0 )dx − λ
F (wδ )dx
2 Ω
Ω\Ωδ
Ωδ
Z
Z
1
|∇wδ |2 dx − λ
F (θ0 )dx
2 Ω
Ω\Ωδ
Z
1
|∇wδ |2 dx + λ{F (θ0 )|Ωδ | − F (θ0 )|Ω|}
2 Ω
0
for λ large enough (see (2.12)). This contradicts (2.10) and completes the proof of
the claim.
Claim 3 : 0 ≤ ψ ≤ θ0 .
Note first that (2.7) is the Euler equation of the minimizing problem of J on H01 (Ω).
Thus ψ is a weak solution to (2.7) and thus is nonnegative. Let us suppose that
ψ > θ0 on a set of positive measure. Set
v := ψ ∧ θ0 ∈ H01 (Ω)
where as above ∧ denotes the minimum of two numbers. Then one has
Z
Z
Z
1
|∇ψ|2 dx − λ
F (ψ)dx − λ
F (θ0 )dx
J[v] =
2 {ψ≤θ0 }
{ψ≤θ 0 }
{ψ>θ 0 }
where {ψ ≤ θ0 } = {x ∈ ω | ψ(x) ≤ θ0 } and {ψ > θ0 } is defined in a similar way.
It follows
Z
Z
Z
1
1
J[v] =
|∇ψ|2 dx − λ
F (ψ)dx −
|∇ψ|2 dx
2 Ω
2 {ψ>θ0 }
Ω
Z
+λ
{F (ψ) − F (θ0 )} dx.
{ψ>θ 0 }
Due to (2.2), (2.3) the last integral above vanishes and one gets
Z
1
J[ψ] ≤ J[v] ≤ J[ψ] −
|∇ψ|2 dx.
2 {ψ<θ0 }
This implies that
Z
|∇ψ|2 dx = 0
{ψ<θ 0 }
and thus ψ ∧ θ0 = ψ. This completes the proof of the lemma.
4
M. CHIPOT AND P. ROY
Remark 2.1. Due to the strong maximum principle one has in fact
ψ > 0 in Ω.
We can now establish our main result:
Theorem 2.1. Under the assumptions (1.1)-(1.3), (2.1), for λ0 sufficiently large
there exists a weak solution u to
(
f (u)
− 4 u = λ0 A(x,u)
in Ω,
(2.14)
u=0
on ∂Ω,
satisfying
0 < u ≤ θ0 , |u|∞ > θ.
(2.15)
Proof. Let us denote by ψ a function satisfying (2.6), (2.7). For any w ∈ L2 (Ω)
one has
(2.16)
− 4ψ = λf˜(ψ) = λA(x, w)
λ0 f (ψ)
f˜(ψ)
f˜(ψ)
≤ λa∞
≤
A(x, w)
A(x, w)
A(x, w)
where we have set λ0 = λa∞ .
Consider the function
g(t) = λ0 f (t) + µt.
If L denotes the Lipschitz constant of f . For t > t0 ≥ 0 one has
g(t) − g(t0 ) = µ(t − t0 ) + λ0 {f (t) − f (t0 )} ≥ µ(t − t0 ) − Lλ0 (t − t0 )
and thus for µ > Lλ0 the function g is increasing. We fix µ in such a way that g is
increasing and set
(2.17)
K := w ∈ L2 (Ω) | ψ ≤ w ≤ θ0 a.e. in Ω .
It is clear that K is a closed convex subset of L2 (Ω). For w ∈ K let us denote by
u = T (w) the unique weak solution to
(
g(w)
µu
= A(x,w)
in Ω,
− 4 u + A(x,w)
(2.18)
u=0
on ∂Ω.
It is clear that T is a map from K into H01 (Ω). Moreover a fixed point for T in K
is clearly a solution to (2.14), (2.15). Let us first prove:
1. T maps K to itself.
Indeed let w ∈ K. One has by the monotonicity of g and (2.16)
−4u+
µu
g(w)
g(ψ)
µψ
=
≥
≥−4ψ+
A(x, w)
A(x, w)
A(x, w)
A(x, w)
and
µu
g(w)
g(θ0 )
µθ0
≤
≤
≤ − 4 θ0 +
.
A(x, w)
A(x, w)
A(x, w)
A(x, w)
Since ψ ≤ u ≤ θ0 on ∂Ω, it follows from the weak maximum principle
−4u+
ψ ≤ u ≤ θ0 a.e in Ω,
that is u ∈ K.
2. T : K → K is compact.
5
For w ∈ K if u is a solution to (2.18) one has clearly
g(w) g(θ0 )
≤
A(x, w)
a0
and u remains in a fixed ball of H01 (Ω). Due to the compactness of the embedding
of H01 (Ω) in L2 (Ω) it is then enough to show that T is continuous from K into itself.
Thus let wn ∈ K with
wn → w in L2 (Ω).
We are going to show that
un := T (wn ) → T (w) := u in H01 (Ω).
Indeed, due to the definition of un and u (see, (2.18)) one has
− 4 (u − un ) +
µun
g(w)
g(wn )
µu
−
=
−
.
A(x, w) A(x, wn )
A(x, w) A(x, wn )
This can be written as
(2.19)
µ(u − un )
1
1
− 4(u − un ) +
= µun
−
A(x, w)
A(x, wn ) A(x, w)
g(w)
g(wn )
+
−
A(x, w) A(x, wn )
1
1
g(w) − g(wn )
= (µun − g(w))
−
+
A(x, wn ) A(x, w)
A(x, wn )
µun − g(w)
g(w) − g(wn )
=
{A(x, wn ) − A(x, w)} +
= fn .
A(x, wn )A(x, w)
A(x, wn )
One notices that un , w are uniformly bounded, A is bounded from below, g is
Lipschitz continuous. Thus for some constant C one has
|fn | ≤ C|A(x, w) − A(x, wn )| + C|w − wn |.
2
Since wn → w in L (Ω), up to a subsequence we have
wn → w a.e. in Ω
and thus by the Lebesgue theorem (recall the definition of K)
wn → w in Lp (Ω), ∀p ≥ 1.
It follows from (1.2) that
A(x, wn ) → A(x, w) a.e. in Ω
and by Lebesgue’s theorem again
A(x, wn ) → A(x, w) in L2 (Ω).
Thus the only possible limit of fn in L2 (Ω) is 0 i.e.
fn → 0 in L2 (Ω).
This shows (see (2.19)) that un → u in H01 (Ω). This completes the proof of the
theorem.
As a corollary consider a Lipschitz continuous function f , nonnegative and having
n bumps- see the figure below:
6
M. CHIPOT AND P. ROY
x2
6
p
0 θ0
p
θ1
p
θ2
p
θ3
p
θn−1
-
x1
θn
Figure 1
Theorem 2.2. Under the assumptions (1.1)-(1.3) and if f is Lipschitz continuous
function which graph is depicted in the figure above, then for λ large enough the
problem
(
−A(x, u) 4 u = λf (u) in Ω,
(2.20)
u=0
on ∂Ω,
admits at least n non trivial solutions.
Proof. It is enough to apply repeatedly Theorem (2.1).
Remark 2.2. If in addition f admits n negative bumps for x < 0 then the problem
(2.20) possesses at least 2n non trivial solutions. It is indeed enough to apply the
theorem above with the function −f (−x).
3. Asymptotic behavior and applications
We will use the same notation as in the previous section. In particular f˜ is the
function defined as in (2.2).
Recall that
Z v
(3.1)
F (v) =
f˜(s)ds
0
(3.2)
J[u] = Jλ [u] =
1
2
Z
|∇u|2 dx − λ
Ω
Z
F (u)dx,
∀u ∈ H01 (Ω).
Ω
Then we have:
Theorem 3.1. Let ψλ be a minimizer of Jλ [u] on H01 (Ω). One has when λ → ∞
(3.3)
ψλ → θ0 in Lp (Ω), ∀p ≥ 1.
Proof. We assume that ψλ satisfies
(3.4)
Jλ [ψλ ] ≤ Jλ [u] ∀u ∈ H01 (Ω).
R
Claim 1 : λ 7→ Ω F (ψλ )dx is nondecreasing.
Indeed, let us suppose that λ > λ0 . By (3.2) and (3.4) one has
Z
Z
Z
Z
1
1
2
2
|∇ψλ | dx − λ
F (ψλ )dx ≤
|∇ψλ0 | dx − λ
F (ψλ0 )dx
2 Ω
2 Ω
Ω
Ω
7
and
Z
Z
Z
Z
1
1
2
0
2
0
|∇ψλ0 | dx − λ
F (ψλ0 )dx ≤
|∇ψλ | dx − λ
F (ψλ )dx.
2 Ω
2 Ω
Ω
Ω
By adding these two inequalities it comes
Z
Z
Z
Z
0
0
0
0
−λ
F (ψλ )dx − λ
F (ψλ )dx ≤ −λ
F (ψλ )dx − λ
F (ψλ )dx
Ω
Ω
⇔ (λ − λ0 )
Ω
Z
Ω
F (ψλ0 )dx ≤ (λ − λ0 )
Ω
Z
F (ψλ )dx
Ω
and the claim 1 follows.
One should note that since ψλ ≤ θ0 , F nondecreasing one has
Z
Z
(3.5)
F (ψλ )dx ≤
F (θ0 )dx
Ω
Ω
and the left hand side of this inequality converges when λ → ∞. We will set
Z
F (ψλ )dx.
(3.6)
` = lim
λ→∞
Ω
Claim 2 : One has f˜(ψλ ) → 0 in L1 (Ω).
As minimizer of Jλ of H01 (Ω), ψλ satisfies the Euler equation
(3.7)
− 4ψλ = λf˜(ψλ ) in Ω, ψλ ∈ H 1 (Ω).
0
Thus for φ ∈ D(Ω), if | |2,Ω denotes the usual L2 (Ω)-norm, one has then
Z
Z
1
ψλ (− 4 φ)dx → 0
(3.8)
f˜(ψλ )φdx =
λ Ω
Ω
when λ → ∞ since 0 ≤ ψλ ≤ θ0 . For any φ ∈ D(Ω) one has
Z
Z
Z
˜
˜
0≤
f (ψλ )dx =
f φdx +
f˜(ψλ )(1 − φ)dx
Ω
Ω
Ω
Z
≤ |
f˜(ψλ )φdx| + |f˜(ψλ )|2,Ω |1 − φ|2,Ω
Ω
Z
1
≤ |
f˜(ψλ )φdx| + |f˜|∞ |Ω| 2 |1 − φ|2,Ω .
(3.9)
Ω
Choosing φ such that
1
|f˜|∞ |Ω| 2 |1 − φ|2,Ω ≤
and then λ such that
2
Z
f˜(ψλ )φdx ≤ ,
2
Ω
we get
Z
0≤
Ω
f˜(ψλ )dx ≤ + = 2 2
and the claim 2 follows.
R
Claim 3 : One has ` = Ω F (θ0 )dx.
Consider the function wδ = θ0 1 ∧ dist(x,∂Ω)
. One has
δ
Z
Z
Z
Z
1
1
2
2
|∇ψλ | dx − λ
F (ψλ )dx ≤
|∇wδ | dx − λ
F (wδ )dx.
2 Ω
2 Ω
Ω
Ω
8
M. CHIPOT AND P. ROY
Dividing by λ we get
Z
Z
Z
Z
1
|∇ψλ |2
1
|∇wδ |2
(3.10)
dx −
F (ψλ )dx ≤
dx −
F (wδ )dx.
2 Ω
λ
2 Ω
λ
Ω
Ω
Due to (3.7) one has
Z
Z
Z
1
|∇ψλ |2
dx =
f˜(ψλ )ψλ ≤ θ0
f˜(ψλ )dx → 0
2 Ω
λ
Ω
Ω
when λ → ∞. Passing to the limit in (3.10) we obtain
Z
−` ≤ −
F (wδ )dx.
Ω
Letting δ → 0 it comes
Z
F (θ0 )dx
`≥
Ω
and the claim 3 follows (see (3.5)).
End of the proof.
Since F (ψλ ) ≤ F (θ) we have
F (θ0 ) − F (ψλ ) → 0 in L1 (Ω).
(3.11)
It follows that for any η > 0 small enough one has, if “| |” denotes the measure of
sets
|{x | ψλ (x) ≤ θ0 − η}| → 0
(3.12)
when λ → ∞. Indeed with an obvious notation for {ψλ ≤ θ0 − η} one has
Z
Z
F (θ0 ) − F (ψλ )dx ≥
F (θ0 ) − F (ψλ )dx ≥
{ψλ ≤θ 0 −η}
Ω
(F (θ0 ) − F (θ0 − η)) |{ψλ ≤ θ0 − η}|
since F is nondecreasing -i.e, ψλ ≤ θ0 − η ⇒ −F (ψλ ) ≥ −F (θ0 − η). (3.12) is then
a consequence of (3.11). To conclude one remarks that for any p ≥ 1
Z
Z
Z
0
p
0
p
|θ − ψλ | dx =
|θ − ψλ | dx +
|θ0 − ψλ |p dx
Ω
≤
{ψλ ≤θ 0 −η}
(θ0 )p |{ψλ ≤
{ψλ >θ 0 −η}
0
p
θ − η}| + η |Ω|.
One fixes first η in such a way that
η p |Ω| ≤
2
then for λ large enough one has by (3.12)
(θ0 )p |{ψλ ≤ θ0 − η}| ≤
2
and this completes the proof of the theorem.
Let u = uλ denote a solution of the problem (2.14) obtained in Theorem 2.1.
Then we have the following asymptotic behavior for uλ as λ → ∞.
Theorem 3.2. Under the assumptions of Theorem 2.1, for every p ≥ 1 one has
(3.13)
uλ → θ0 in Lp (Ω).
9
Proof. From the construction of uλ (see (2.17)) one has
ψλ ≤ uλ ≤ θ0
where ψλ is as in the last Theorem. (3.13) is now a trivial consequence of the
Theorem 3.1.
Remark 3.1. In the case of n loops each solution constructed in the way above
converges toward θi , i = 1, ..., n (see Figure 1).
The range of applications of the results above is analyzed in details in [4] where
various examples of functionals A fulfilling our assumptions are given. For the sake
of completeness let us recall one of them.
Suppose for instance that a is a Carathéodory function from Ω × R into R i.e.
that one has
x 7→ a(x, v) is measurable
v 7→ a(x, v) is continuous
∀v ∈ R,
a.e. x ∈ Ω.
Assume in addition that there exists some positive constants a0 , a∞ such that
0 < a0 ≤ a(x, v) ≤ a∞
Then if we set
a.e. x ∈ Ω, ∀v ∈ R.
Z
A(x, u) = a x,
u(x)dx
Ω0
where Ω0 denotes a subdomain of Ω it is clear that our assumptions (1.1)-(1.3) are
fulfilled for any p ≥ 1.
This kind of functional plays a role in diffusion of population (see [6], [10]). The
quantity u stands for the density of some population which diffusion coefficient
depends on the total population in some domain Ω0 . The source term f in Theorem
2.1 has the property to vanish as soon the population density reaches a certain
threshold.
Acknowledgements: The research of the first author was funded by the
Lithuanian-Swiss cooperation program under the project agreement No. CH-SMM01/01 and by the Swiss National Science Foundation under the contracts # 200021129807/1 and 200021-146620. The research of the second author was supported by
the Swiss National Science Foundation under the contract # 200021-129807/1.
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(Michel Chipot)
Institut für Mathematik, Universität Zurich,
Winterthurerstr. 190, CH-8057 Zürich, Switzerland.
E-mail address: [email protected]
(Prosenjit Roy)
Institut für Mathematik, Universität Zurich,
Winterthurerstr. 190, CH-8057 Zürich, Switzerland.
E-mail address: [email protected]