LE MATEMATICHE
Vol. LXVI (2011) – Fasc. I, pp. 163–171
doi: 10.4418/2011.66.1.16
EXISTENCE RESULTS FOR A QUASI-LINEAR
DIFFERENTIAL PROBLEM
PASQUALE F. PIZZIMENTI - ANGELA SCIAMMETTA
The aim of this paper is to establish the existence of at least one
non-trivial solution for Neumann quasi-linear problems. Our approach
is based on variational methods.
1.
Introduction
The aim of this paper is to ensure the existence of at least one non-trivial solution
for the following Neumann boundary value problem
−u00 + uh(u0 ) = λ α(x) f (u)h(u0 )
(Nλ )
u0 (a) = u0 (b) = 0,
where α : [a, b] → R is a positive continuous function, f : R → R and h : R → R
are continuous functions and λ is a positive real parameter.
Existence and multiplicity of solutions for Neumann boundary value problems have been investigated by several authors and, for an overview on this
subject, we refer to [1], [3] - [6], [8], [9], [11] - [14].
The main result of this paper is Theorem 3.1, which generalizes [6, Theorem 3.1] to the case where the nonlinear term is not constant with respect to u0 .
Two relevant consequences of Theorem 3.1 (that is, Corollary 3.2 and Theorem
Entrato in redazione: 20 dicembre 2010
AMS 2010 Subject Classification: 34B15; 34B18.
Keywords: Neumann problem, Positive solutions, Critical points.
164
PASQUALE F. PIZZIMENTI - ANGELA SCIAMMETTA
3.3) are also pointed out. Here, as an example, we presented a special case of
our main result.
Theorem 1.1. Let α : [a, b] → R be a nonnegative continuous function and
f , h : R → R be continuous functions. Suppose that h is bounded and strictly
positive, and that
f (ξ )
lim
= +∞.
+
ξ
ξ →0
Then, there exists λ ∗ such that, for each λ ∈ ]0, λ ∗ [, the problem (Nλ ) admits at
least one positive classical solution.
Our approach is based on a critical point theorem obtained in [2] (see Theorem 2.1).
The paper is arranged as follows: in Section 2, we recall some basic definitions and our main tool, while Section 3 is devoted to our main results.
2.
Preliminaries and basic notations
Our main tool is the Ricceri variational principle [10, Theorem 2.5] as given in
[2, Theorem 5.1] which is below recalled (see also [2, Proposition 2.1] and [7,
Theorem 2.1]). First, given Φ, Ψ : X → R, put
Ψ(u) − Ψ(v)
sup
β (r1 , r2 ) =
inf
u∈Φ−1 (]r1 ,r2 [)
v∈Φ−1 (]r1 ,r2 [)
and
Ψ(v) −
ρ2 (r1 , r2 ) =
sup
v∈Φ−1 (]r1 ,r2 [)
(1)
r2 − Φ(v)
sup
u∈Φ−1 (]−∞,r1 ])
Φ(v) − r1
Ψ(u)
,
(2)
for all r1 , r2 ∈ R, with r1 < r2 .
Theorem 2.1. ([2, Theorem 5.1]) Let X be a reflexive real Banach space; Φ :
X → R be a sequentially weakly lower semicontinuous, coercive and continuously Gâteaux differentiable function whose Gâteaux derivative admits a continuous inverse on X ∗ ; Ψ : X → R be a continuously Gâteaux differentiable
function whose Gâteaux derivative is compact. Put Iλ = Φ − λ Ψ and assume
that there are r1 , r2 ∈ R, with r1 < r2 , such that
β (r1 , r2 ) < ρ2 (r1 , r2 ),
(3)
where β and ρ2 aregiven by (1) and (2).
1
1
,
there is u0,λ ∈ Φ−1 (]r1 , r2 [) such that
Then, for each λ ∈
ρ2 (r1 , r2 ) β (r1 , r2 )
Iλ (u0,λ ) ≤ Iλ (u) for all u ∈ Φ−1 (]r1 , r2 [) and Iλ0 (u0,λ ) = 0.
EXISTENCE RESULTS FOR A QUASI-LINEAR DIFFERENTIAL PROBLEM
165
Let X be the Sobolev space W 1,2 ([a, b]) endowed with the norm
kuk :=
b
Z
0
2
|u (x)| dx +
Z b
a
2
21
|u(x)| dx
.
a
Throughout the sequel, f : R → R is a continuous function, h : R → R is a
positive continuous function, α : [a, b] → R is a sommable function and λ is a
positive real parameter. Put
Z t
F(t) =
f (ξ )dξ ,
0
for all t ∈ R,
Z t
F1 (x,t) =
α(x) f (ξ )dξ = α(x)F(t),
0
for all (x,t) ∈ [a, b] × R,
and, put
Z y Z σ
H(y) =
(
0
0
1
dτ)dσ ,
h(τ)
for all y ∈ R.
We recall that u : [a, b] → R is called weak solution of Problem (Nλ ) if u ∈
W 1,2 ([a, b]) and
Z b
a
H 0 (u0 (x))v0 (x)dx +
Z b
Z b
u(x)v(x)dx = λ
a
α(x) f (u(x))v(x)dx,
a
for all v ∈ W 1,2 ([a, b]).
We also recall that a weak solution is a generalized solution, that is,
u ∈ C1 ([a, b]), u0 ∈ AC([a, b]), −u00 (x) + u(x)h(u0 (x)) = λ α(x) f (u(x))h(u0 (x)),
for a.e. x ∈ [a, b], and u0 (a) = u0 (b) = 0.
Moreover, if α is continuous, each weak solution is a classical solution, that is,
u ∈ C2 ([a, b]), −u00 (x) + u(x)h(u0 (x)) = λ α(x) f (u(x))h(u0 (x)) for all x ∈ [a, b],
and u0 (a) = u0 (b) = 0. Finally, put
12
2
γ = max 2(b − a);
,
b−a
we recall the following inequality which we use in the sequel
max |u(x)| ≤ γkuk,
x∈[a,b]
for all u ∈ X and for all x ∈ [a, b].
(4)
166
3.
PASQUALE F. PIZZIMENTI - ANGELA SCIAMMETTA
Main Results
In this Section, we establish existence results for the Neumann boundary value
problem (Nλ ).
Given two positive constants m, M, with m ≤ M, put
δ1 = min
1
1
;
M(b − a) b − a
12
, δ2 = max
1
1
;
m(b − a) b − a
21
.
Moreover, given three nonnegative constants c1 , c2 , d, with δ1 c1 < γd < δ2 c2 ,
put
max F(t) − F(d)
|t|≤c2
a(c2 , d) :=
δ22 c22 − γ 2 d 2
and
F(d) − max F(t)
|t|≤c1
2
2
γ d − δ12 c21
b(c1 , d) :=
.
We give our main result.
Theorem 3.1. Let α : [a, b] → R be a nonnegative function and let f , h : R → R
be continuous functions. Assume that there exist two positive constants m, M,
such that
(i) m ≤ h(y) ≤ M,
for all y ∈ R,
and, assume that there exist three nonnegative constants c1 , c2 , d, with δ1 c1 <
γd < δ2 c2 , such that
a(c2 , d) < b(c1 , d).
(5)
b−a
b−a
Then, for each λ ∈
, 2
, the problem (Nλ ) ad2
2γ kαk1 b(c1 , d) 2γ kαk1 a(c2 , d)
c1
c2
mits at least one weak solution u, such that
< kuk < .
γ
γ
Proof. Put
Φ(u) :=
1
2
Z b
|u(x)|2 dx +
a
Z b
H(u0 (x))dx,
a
Z b
Ψ(u) :=
a
F1 (x, u(x))dx,
for all u ∈ X.
It is well known that Φ and Ψ satisfy all regularity assumptions requested in
EXISTENCE RESULTS FOR A QUASI-LINEAR DIFFERENTIAL PROBLEM
167
Theorem 2.1 and that the critical points in X of the functional Φ − λ Ψ are exactly the weak solutions of the problem (Nλ ). By using (i), one has
1 1
1 1
min
;
kuk2 ≤ Φ(u) ≤ max
;
kuk2 ,
2M 2
2m 2
for every u ∈ X. Our aim is to apply Theorem 2.1. To this end, put
r1 =
b − a δ12 2
c ,
2 γ2 1
r2 =
b − a δ22 2
c
2 γ2 2
and
for all x ∈ [a, b].
u0 (x) = d,
Clearly, u0 ∈ X and one has
1
Φ(u0 ) =
2
Z b
a
2
|u0 | dx +
Z b
Ψ(u0 ) =
a
Z b
a
1
H(u00 )dx = d 2 (b − a),
2
F1 (x, u0 (x))dx = kαk1 F(d),
where
kαk1 :=
Z b
|α(x)|dx.
a
From δ1 c1 < γd < δ2 c2 , one has r1 < Φ(u0 ) < r2 . Moreover, for all u ∈ X such
that Φ(u) < r2 , taking (4) into account, one has
|u(x)| < c2 ,
and
Z b
a
F1 (x, u(x))dx ≤
Z b
for all x ∈ [a, b],
max F1 (x,t)dx = kαk1 max F(t).
a |t|≤c2
|t|≤c2
Therefore
Ψ(u) ≤ kαk1 max F(t).
sup
|t|≤c2
u∈Φ−1 (]−∞,r2 [)
Arguing as before, we obtain
Ψ(u) ≤ kαk1 max F(t).
sup
|t|≤c1
u∈Φ−1 (]−∞,r1 ])
Therefore, one has
Ψ(u) − Ψ(u0 )
sup
β (r1 , r2 ) ≤
≤
u∈Φ−1 (]−∞,r2 [)
r2 − Φ(u0 )
max F(t) − F(d)
2γ 2 kαk1 |t|≤c2
b−a
δ22 c22 − γ 2 d 2
≤
=
2γ 2 kαk1
a(c2 , d).
b−a
(6)
168
PASQUALE F. PIZZIMENTI - ANGELA SCIAMMETTA
On the other hand, one has
Ψ(u0 ) −
ρ2 (r1 , r2 ) ≥
≥
sup
Ψ(u)
u∈Φ−1 (]−∞,r1 ])
Φ(u0 ) − r1
F(d) − max F(t)
2γ 2 kαk1
b−a
|t|≤c1
2
2
γ d − δ12 c21
≥
=
2γ 2 kαk1
b(c1 , d).
b−a
(7)
Hence, from (5) one has
β (r1 , r2 ) < ρ2 (r1 , r2 ).
Therefore, owing to Theorem 2.1, for each
b−a
b−a
λ∈
,
,
2γ 2 kαk1 b(c1 , d) 2γ 2 kαk1 a(c2 , d)
Φ − λ Ψ admits at least one critical point ū such that
r1 < Φ(ū) < r2 ,
that is
c2
c1
< kūk < .
γ
γ
Hence, the proof is complete.
Now, we point out the following consequence of Theorem 3.1.
Corollary 3.2. Let α : [a, b] → R be a nonnegative function, h : R →]0, +∞[ and
f : R → [0, +∞[ be continuous functions. Assume that (i) holds and that there
γ
exist two positive constants c, d, with c > d, such that
δ2
2
F(d)
F(c)
δ2
<
.
(8)
2
c
γ
d2
#
"
b − a d 2 b − a δ2 2 c2
Then, for each λ ∈
,
, the problem (Nλ ) ad2kαk1 F(d) 2kαk1 γ
F(c)
c
mits at least one nontrivial weak solution u such that kuk < .
γ
Proof. Our aim is to apply Theorem 3.1. To this end, we pick c1 = 0 and c2 = c.
From (8) one has
2 2
γ d
F(c)
max F(t) − F(d) F(c) −
δ22 c2
F(c)
|t|≤c
a(c2 , d) =
≤
= 2 2.
δ22 c2 − γ 2 d 2
δ22 c2 − γ 2 d 2
δ2 c
EXISTENCE RESULTS FOR A QUASI-LINEAR DIFFERENTIAL PROBLEM
On the other hand, one has b(c1 , d) =
169
F(d)
. Hence, owing to (8), Theorem 3.1
γ 2d2
ensures the conclusion.
Now, we point out the following relevant consequence of Corollary 3.2.
Theorem 3.3. Let α : [a, b] → R be a nonnegative function and f , h : R → R be
continuous functions. Assume that (i) holds. Assume that
lim
ξ →0+
f (ξ )
= +∞,
ξ
(9)
b − a δ2 2
c2
. Then, for each λ ∈ ]0, λ ∗ [, the problem
and put
=
sup
2kαk1 γ
c>0 F(c)
(Nλ ) admits at least one positive weak solution.
b − a δ2 2 c2
∗
Proof. Fix λ ∈ ]0, λ [. Then, there is c > 0 such that λ <
.
2kαk1 γ
F(c)
1
δ2
2kαk1 F(d)
> . Hence, Corollary 3.2
From (9) there is d < c such that
2
γ
b−a d
λ
ensures the conclusion.
λ∗
Remark 3.4. Taking (9) into account, fix ρ > 0 such that f (ξ ) > 0 for all
b − a δ2 2
c2
ξ ∈ ]0, ρ[. Then, put λ̄ =
supc∈]0,ρ[
. Clearly, λ̄ ≤ λ ∗ . Now,
2kαk
γ
F(c)
1
fixed λ ∈ 0, λ̄ and arguing as in the proof of Theorem 3.3, there are c ∈ ]0, ρ[
b − a d2
b − a δ2 2 c2
δ2
<λ <
. Hence, Coroland d < c such that
γ
2kαk1 F(d)
F(c)
2kαk1 γ
lary 3.2 ensures that, for each λ ∈ 0, λ̄ , the problem (Nλ ) admits at least one
positive weak solution uλ such that
|uλ (x)| <
for all x ∈ [a, b].
ρ
,
γ
170
4.
PASQUALE F. PIZZIMENTI - ANGELA SCIAMMETTA
Acknowledgements
The authors wish to thank Professor Gabriele Bonanno for his remarks and suggestions during the writing of this paper.
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EXISTENCE RESULTS FOR A QUASI-LINEAR DIFFERENTIAL PROBLEM
171
PASQUALE F. PIZZIMENTI
Department of Mathematics
University of Messina
98166 - Messina (Italy)
e-mail: ppizzimenti@unime.it
ANGELA SCIAMMETTA
Department of Mathematics
University of Messina
98166 - Messina (Italy)
e-mail: asciammetta@unime.it