Positive Solutions for the Neumann pLaplacian with Superdiffusive Reaction
Positive Solutions for the Neumann pLaplacian with Superdiffusive Reaction
Leszek Gasin´ ski 0 1
Nikolaos S. Papageorgiou 0 1
Mathematics Subject Classification 0 1
0 Department of Mathematics, National Technical University , Zografou Campus, 15780 Athens , Greece
1 Faculty of Mathematics and Computer Science, Jagiellonian University , ul. Łojasiewicza 6, 30348 Kraków , Poland
We consider a generalized logistic equation driven by the Neumann pLaplacian and with a reaction that exhibits a superdiffusive kind of behavior. Using variational methods based on the critical point theory, together with truncation and comparison techniques, we show that there exists a critical value λ∗ > 0 of the parameter, such that if λ > λ∗, the problem has at least two positive solutions, if λ = λ∗, the problem has at least one positive solution and it has no positive solution if λ ∈ (0, λ∗). Finally, we show that for all λ λ∗, the problem has a smallest positive solution. Communicated by Rosihan M. Ali.
pLaplacian; Superdiffusive reaction; Local minimizers; Mountain pass theorem; Comparison principle; Bifurcationtype theorem

35J25 · 35J92
1 Introduction
Let ⊆ RN be a bounded domain with a C 2boundary ∂ . In this paper, we study
the following nonlinear parametric Neumann problem:
⎩ ∂n = 0 on ∂ , λ > 0, u > 0,
⎨⎧ −∂u pu(z) + β(z)u(z) p−1 = λg z, u(z) − f z, u(z)
in
,
( P)λ
with β ∈ L∞( )+, β = 0. Here
defined by
p denotes the pLaplace differential operator,
pu = div
∇u p−2∇u
∀u ∈ W 1, p( ),
with p ∈ (1, +∞). Also n(·) denotes the outward unit normal on ∂ . When the
reaction in ( P)λ has the particular form
λζ q−1 − ζ r−1,
with q < r , then the resulting equation is the plogistic equation (or simply the logistic
equation when p = 2). The logistic equation is important in mathematical biology (see
Gurtin and Mac Camy [
21
] and Afrouzi and Brown [
1
]) and describes the dynamics
of biological populations whose mobility is density dependent.
There are three different types of the plogistic equation, depending on the value
of the exponent q with respect to p. More precisely, we have
• the “subdiffusive” type, when q < p < r ;
• the “equidiffusive” type, when q = p < r ;
• the “superdiffusive” type, when p < q < r .
The subdiffusive and equidiffusive cases are similar, but the superdiffusive case
differs essentially and it exhibits bifurcation phenomena (see Takeuchi [
29,30
] and
Filippakis et al. [7], where the Dirichlet problem is studied).
The aim of this work, is to prove a bifurcationtype theorem for the positive solutions
of ( P)λ as the parameter λ > 0 varies in (0, +∞) and the reaction ζ −→ λg(z, ζ ) −
f (z, ζ ) (which is more general than the standard plogistic equation; see Afrouzi and
Brown [
1
]), exhibits a superdiffusive kind of behavior. To the best of our konwledge,
the Neumann plogistic equation has not been studied. There is only the recent work
of MaranoPapageorgiou [
25
], where the equidiffusive case is examined.
Our approach is variational based on the critical point theory, combined with
suitable truncation and comparison techniques. In the next section, for the convenience
of the reader we recall main mathematical tools which we will use in the sequel.
This work is the outgrowth of a remark made by the referee of [
19
]. In that paper,
the authors deal with the parametric equation
− pu(z) = λ f z, u(z)
u∂ = 0 on ∂
in
and some analogous bifurcationtype results were proved. It was pointed out by the
referee that in mathematical biology, the Neumann model is a more realistic one. For
some other recent results on nonlinear Neumann boundary value problems involving
pLaplacian, we refer to Gasin´ski and Papageorgiou [
11–17
].
2 Mathematical Background
Let X be a Banach space and let X ∗ be its topological dual. By ·, · we denote the
duality brackets for the pair (X, X ∗). Let ϕ ∈ C 1(X ). We say that ϕ satisfies the
Palais–Smale condition, if the following holds:
“Every sequence {xn}n 1 ⊆ X , such that ϕ(xn) n 1 ⊆ R is bounded and
ϕ (xn) −→ 0 in X ∗ as n → +∞,
admits a strongly convergent subsequence.”
Using this compactnesstype condition on ϕ, we can state the following theorem,
known in the literature as the “mountain pass theorem”.
Theorem 2.1 If X is a Banach space, ϕ ∈ C 1(X ) satisfies the Palais–Smale condition,
x0, x1 ∈ X , 0 < < x0 − x1 ,
max ϕ(x0), ϕ(x1) < inf ϕ(x ) :
x − x0 =
= η ,
c = γin∈f 0mtax1 ϕ γ (t ) ,
where
=
γ ∈ C [
0, 1
]; X : γ (0) = x0, γ (1) = x1 ,
then c η and c is a critical value of ϕ (i.e., there exists x ∈ X , such that ϕ (x ) = 0
and ϕ(x ) = c).
In the study of problem ( P)λ, we will use the Sobolev space W 1, p( ) and the
ordered Banach space C 1( ). The positive cone of the latter is
C+ =
u ∈ C 1( ) : u(z)
0 for all z ∈
This cone has a nonempty interior, given by
int C+ =
u ∈ C+ : u(z) > 0 for all z ∈
The next result relates local minimizers in W 1, p( ) with local minimizers in the
smaller Banach space C 1( ). A result of this type was first proved for the Dirichlet
Laplacian by Brézis and Nirenberg [
5
] and was later extended to the pLaplacian by
García Azorero et al. [
8
] and Guo and Zhang [
20
] (in the latter, for p 2). Extensions
to the Neumann pLaplacian or Neumann pLaplacianlike operators can be found in
Motreanu et al. [26] and Motreanu and Papageorgiou [28].
So let f0 : ×R −→ R be a Carathéodory function (i.e., for all ζ ∈ R, the function
z −→ f0(z, ζ ) is measurable and for almost all z ∈ , the function ζ −→ f0(z, ζ )
is continuous), which exhibits subcritical growth in ζ ∈ R, i.e.,
f0(z, ζ )
with a ∈ L∞( )+, c > 0 and 1 < r < p∗, where
p∗ =
N p
N − p
+∞
p < N ,
p N .
F0(z, ζ ) ds =
f0(z, s) ds
if
if
0
ζ
We set
and consider the C 1functional ψ0 : W 1, p( ) −→ R, defined by
ψ0(u) =
1
p ∇u pp −
F0 z, u(z) d z ∀u ∈ W 1, p( ).
Theorem 2.2 If u0 ∈ W 1, p( ) is a local C 1( )minimizer of ψ0, i.e., there exists
1 > 0, such that
ψ0(u0)
ψ0(u0 + h) ∀h ∈ C 1( ), h C1( )
1,
then u0 ∈ C 1( ) and it is a local W 1, p( )minimizer of ψ0, i.e., there exists 2 > 0,
such that
ψ0(u0)
ψ0(u0 + h) ∀h ∈ W 1, p( ), h
2.
Remark 2.3 In [
26,28
], the result was stated in terms of Wn1, p( ) = C 1( ) · , where
n
C 1( ) =
n
∂u
u ∈ C 1( ) : ∂n = 0 on ∂
Proposition 2.4 The map A : W 1, p( ) −→ W 1, p( )∗ defined by (2.1) is continuous,
strictly monotone (hence maximal monotone too) and of type (S)+, i.e., if {un}n 1 ⊆
W 1, p( ) is a sequence, such that un −→ u weakly in W 1, p( ) and
lim sup A(un), un − u
n→+∞
0,
then un −→ u in W 1, p( ).
The next simple lemma, will be useful in our estimations and can be found in
Aizicovici et al. [4, Lemma 2]. Recall that by · we denote the norm of the Sobolev
space W 1, p( ), i.e.,
u =
u pp + ∇u pp 1p
∀u ∈ W 1, p( ).
Lemma 2.5 If β ∈ L∞( ), β(z)
exists ξ0 > 0, such that
We conclude this section by fixing some notation. By  · N we denote the Lebesgue
measure on RN . For every u ∈ W 1, p( ), we set u± = max{±u, 0}. We know that
u± ∈ W 1, p( ), u = u+ − u−, u = u+ + u−.
Finally for every measurable function h :
× R −→ R, we define
Nh (u)(·) = h ·, u(·)
∀u ∈ W 1, p( )
(the Nemytskii map corresponding to h).
3 A BifurcationType Theorem
The hypotheses on the data of problem ( P)λ are the following:
Hg g : × R −→ R is a Carathéodory function, such that g(z, 0) = 0 for almost all
z ∈ and
(i) we have
with a ∈ L∞( )+, c > 0 and p < r < p∗;
(ii) there exist ϑ > q > p, such that
0 < ηg
lim inf
ζ →+∞ ζ q−1
uniformly for almost all z ∈
is nonincreasing on (0, +∞);
(iii) we have
and for almost all z ∈
g(z,ζ )
, the function ζ −→ ζ ϑ−1
H0 For every λ > 0 and > 0, we can find γ = γ (λ) > 0, such that for almost all
z ∈ , the function ζ −→ λg(z, ζ ) − f (z, ζ ) + γ ζ ϑ−1 is nondecreasing on [0, ]
(ϑ > q > p as in the hypothesis Hg(i i )).
Remark 3.1 Since we are interested in positive solutions and hypotheses Hg, H f and
H0 concern the positive semiaxis R+ = [0, +∞), we may (and will) assume that
g(z, ζ ) = f (z, ζ ) = 0 for almost all z ∈
Example 3.2 The following functions satisfy hypotheses Hg, H f and H0 (for the sake
of simplicity we drop the zdependence):
(a) g(ζ ) = ζζ qs−−11 iiff ζζ ∈> [
10, 1
], and f (ζ ) = ζ ϑ−1 for all ζ 0, with p <
q < s < ϑ < p∗.
(b) g(ζ ) = ζζ qs−−11 −− ζζϑp−−11 iiff ζζ ∈> [
10, 1
], and f (ζ ) = ζ ϑ−1 − ζ s−1 for all ζ 0,
with p < q < s < ϑ < p∗.
Example (a) corresponds to the standard superdiffusive plogistic reaction (see
Afrouzi and Brown [1]).
By a positive solution of problem ( P)λ, we understand a function u ∈ W 1, p( ),
u = 0, which is a weak solution of ( P)λ. Then u ∈ L∞( ) (see e.g., Gasin´ski
and Papageorgiou [
9,18
] and Hu and Papageorgiou [23]). Invoking Theorem 2 of
Lieberman [
24
], we have that u ∈ C+ \ {0}. Let = u ∞ and let γ = γ (λ) > 0
be as postulated by hypothesis H0. We have
− pu(z) + β(z)u(z) p−1 + γ u(z)ϑ−1
= λg z, u(z) − f z, u(z) + γ u(z)ϑ−1
(see Motreanu and Papageorgiou [
27
]), so
pu(z)
β ∞ + γ
ϑ− p u(z) p−1 for almost all z ∈
and finally
u ∈ int C+
(see Vázquez [
31
]).
So, we see that the positive solutions of problem ( P)λ, if they exist, belong in
int C+.
Let
Y =
λ > 0 : problem ( P)λ has a positive solution.
Proposition 3.3 If hypotheses Hg, H f and H0 hold, then inf Y > 0.
Proof By virtue of hypothesis Hg(i i ), we can find η1 > 0 and M > 0, such that
η1ζ q−1 for almost all z ∈
On the other hand, from hypothesis Hg(i i i ), for a given ε > 0, we can find δ ∈ (0, 1)
small, such that
εζ p−1 for almost all z ∈
, all ζ ∈ [0, δ].
(3.1)
(3.2)
The function ζ −→ σζ1q(−ζ1) is upper semicontinuous on [δ, M ] and so, we can find
ξ ∈ [δ, M ], such that
(see hypothesis Hg(i v)). From (3.1), (3.2) and (3.3), it follows that
(see (2.1) for the definition of A). On (3.7) we act with uλ and obtain
p
∇uλ p +
βuλp d z =
λg(z, uλ) − f (z, uλ) uλ d z,
so using Lemma 2.5 and (3.6), we have
ξ0 uλ p
2ε uλ p.
recalling that ε ∈ 0, ξ20 , we conclude that uλ = 0, a contradiction. Therefore inf Y
λ > 0.
If Y = ∅, then inf Y = +∞. In the next proposition, we establish the nonemptiness
of Y.
Proposition 3.4 If hypotheses Hg, H f and H0 hold, then Y = ∅ and if λ ∈ Y and
τ > λ, then τ ∈ Y.
(3.3)
(3.4)
(3.5)
(3.7)
Proof Let ϕλ : W 1, p( ) −→
by
ϕλ(u) =
R be the energy functional for problem ( P)λ, defined
Since q < ϑ , using Young inequality with ε > 0, from (3.8) we see that for a given
ε > 0, we can find c2 = c2(ε) > 0, such that
ε(ζ +)ϑ + c2 for almost all z ∈
Also, from hypotheses H f (i ), (i i ), we see that we can find ξ2 > 0 and c3 > 0, such
that
F (z, ζ )
ξ2(ζ +)ϑ − c3 for almost all z ∈
ξ0 u p + (ξ2 − λε) u+ ϑϑ − c4 ∀u ∈ W 1, p( ),
p
for some c4 = c4(ε) > 0 (see Lemma 2.5 and (3.9), (3.10)).
We choose ε ∈ 0, ξλ2 . Then, from (3.11), it follows that ϕλ is coercive. Also, it
is easy to see that ϕλ is sequentially weakly lower semicontinuous. Therefore, by the
Weierstrass theorem, we can find uλ ∈ W 1, p( ), such that
ϕλ(uλ) =
inf
u∈W 1,p( )
ϕλ(u) = mλ.
Let u ∈ int C+. Then clearly for λ > 0 big, we have ϕλ(u) < 0. Hence
ϕλ(uλ) = mλ < 0 = ϕλ(0) ∀λ > 0, big
(see (3.12)), so
From (3.12), we have
so
On (3.14) we act with −uλ− ∈ W 1, p( ) and we obtain
(see Lemma 2.5), i.e., uλ
Then (3.14) becomes
0, uλ = 0 (see (3.13)).
uλ solves problem ( P)λ,
sϑ−1g z, uλ(z)
g z, suλ(z)
= g z, u(z)
for almost all z ∈
.
(3.17)
Similarly, using hypothesis H f (i i ), we have
f (z, uλ(z))
uλ(z) p−1
f (z, u(z))
u(z) p−1
f (z, u(z))
= s p−1uλ(z) p−1
,
s p−1 f z, uλ(z)
f z, suλ(z)
= f z, u(z)
i.e., Y = ∅.
Now suppose that λ ∈ Y and τ > λ. We choose s ∈ (0, 1), such that
(Wreecasellttuha=tϑsu>λ ∈p ainntdCλ+<.Tτh)e.nSince λ ∈ Y, problem ( P)λ has a solution uλ ∈ int C+.
−
pu + βu p−1 = s p−1 −
uλ + βuλp−1
= s p−1 λg(z, uλ) − f (z, uλ) . (3.16)
By virtue of hypothesis Hg(i i ) and since s ∈ (0, 1), we have
g(z, uλ(z))
uλ(z)ϑ−1
g(z, u(z))
u(z)ϑ−1
g(z, u(z))
= sϑ−1uλ(z)ϑ−1
,
λ = sϑ−1τ
(3.15)
Returning to (3.16) and using (3.15), (3.17) and (3.18), we have
We consider the following truncation of the reaction in problem ( P)τ :
− pu(z) + β(z)u(z) p−1
= λs p−1g z, uλ(z) − s p−1 f z, uλ(z)
sϑ−1τ g z, uλ(z) − f z, u(z)
τ g z, u(z) − f z, u(z)
and consider the C 1functional ψτ : W 1, p( ) −→ R, defined by
ψτ (u) =
As we did for ϕλ earlier in this proof, we can check that ψτ is coercive and sequentially
weakly lower semicontinuous. So, we can find uτ ∈ W 1, p( ), such that
On (3.21) we act with (u − uτ )+ ∈ W 1, p( ) and obtain
ψτ (uτ ) = u∈Wi n1,fp( ) ψτ (u),
ψτ (uτ ) = 0,
A(uτ ) + βuτ  p−2uτ = Nhτ (uτ ).
(3.21)
A(uτ ), (u − uτ )+ +
βuτ  p−2uτ (u − uτ )+ d z
=
=
hτ (z, uτ )(u − uτ )+ d z
τ g(z, u) − f (z, u) (u − uτ )+ d z
A(u), (u − uτ )+ +
βu p−1(u − uτ )+ d z
(see (3.20) and (3.19)), so
{u>uτ }
+
{u>uτ }
∇uτ p−2∇uτ − ∇u p−2∇u, ∇uτ − ∇u Rd z
β uτ  p−2uτ − u p−1 (uτ − u) d z
We recall the following elementary inequalities (see e.g., Gasin´ski and Papageorgiou
[10, Lemma 6.2.13, p. 740]). If 1 < p 2, then
( p − 1)y − v2(1 + y + v) p−2
≤ y p−2 y − v p−2v, y − v RN ∀y, v ∈ RN
and if 2 < p, then
1
2 p−2 y − v p ≤ y p−2 y − v p−2v, y − v RN ∀y, v ∈ RN .
If 1 < p
2, then from (3.22), (3.23) and since uτ , u ∈ int C+, we have
(3.22)
(3.23)
(3.24)
for some c5 > 0, so
i.e., u uτ .
If 2 < p, then from (3.22) and (3.24), we have
p − 1
c5
{u>uτ }
∇uτ − ∇u 2 d z
0
{u > uτ } N = 0,
1
2 p−2
{u>uτ }
∇uτ − ∇u p d z
0,
{u > uτ } N = 0,
uτ and then (3.21) becomes
A(uτ ) + βuτp−1 = τ Ng(uτ ) − N f (uτ )
so
i.e., u uτ .
So, finally u
(see (3.20)), so uτ ∈ int C+ is a positive solution of ( P)λ, i.e., τ ∈ Y.
Proposition 3.5 If hypotheses H f , Hg and H0 hold and λ > λ∗, then problem ( P)λ
has at least two positive solutions.
Proof Let τ ∈ (λ∗, λ) ∩ Y. Then, we can find uτ ∈ int C+, such that
= 0 on ∂ .
,
Proceeding as in the proof of Proposition 3.4, we introduce the following truncation
of the reaction:
hλ(z, ζ ) =
λg z, uλ(z) − f z, uλ(z)
λg(z, ζ ) − f (z, ζ )
if ζ uτ (z),
if uτ (z) < ζ.
This is a Carathéodory function. We set
Hλ(z, ζ ) =
hλ(z, s) ds
0
ζ
and consider the C 1functional ψ : W 1, p( ) −→ R, defined by
ψλ(u) =
As we did for ϕλ in the proof of Proposition 3.4, we can check that ψλ is coercive and
0
sequentially weakly lower semicontinuous. So, we can find uλ ∈ W 1, p( ), such that
ψλ(u0λ) = u∈Wi n1,fp( ) ψλ(u),
ψλ(u0λ) = 0,
A(u0λ) + βu0λ p−2u0λ = Nhλ (u0λ).
From this, as before, acting with (uτ − u0λ)+ ∈ W 1, p( ) and using (3.25) and (3.26),
we show that uτ u0λ. Hence, we have
A(u0λ) + β(u0λ) p−1 = λNg(u0λ) − N f (u0λ)
(see (3.26)), so u0λ ∈ int C+ is a solution of ( P)λ and u0λ
uτ .
Claim 1 u0λ − uτ ∈ int C+.
Let = u0λ ∞. By hypothesis H0, we can find γ = γ (λ) > 0, such that for all
z ∈ , the function ζ −→ λg(z, ζ ) − f (z, ζ ) + γ ζ ϑ−1 is nondecreasing on [0, ].
For δ > 0, we set
uτ = uτ + δ ∈ int C+.
Then
− puτ + βuτp−1 + γ uτϑ−1
− puτ + βuτp−1 + γ uτϑ−1 + ξ(δ)
= τ g(z, uτ ) − f (z, uτ ) + γ uτϑ−1 + ξ(δ)
= λg(z, uτ ) − f (z, uτ ) + (τ − λ)g(z, uτ ) + γ uτϑ−1 + ξ(δ)
λg(z, uτ ) − f (z, uτ ) − (λ − τ )σ0(uτ ) + γ uτϑ−1 + ξ(δ)
with ξ(δ) → 0 as δ 0 (see hypothesis Hg(i v) and recall that τ < λ).
Since uτ ∈ int C+, the function z −→ σ0 uτ (z) is upper semicontinuous on
(see hypothesis Hg(i v)). So, we can find z0 ∈ , such that
σ0 uτ (z0)
= max σ0 uτ (z)
> 0.
z∈
We use (3.28) in (3.27). Since ξ(δ)
0 and δ
0 and λ > τ , we infer that
− puτ + βuτp−1 + γ uτϑ−1
λg(z, uτ ) − f (z, uτ ) + γ uϑ−1
τ
λg(z, u0λ) − f (z, u0λ) + γ (u0λ)ϑ−1
= − pu0λ + β(u0λ) p−1 + γ (u0λ)ϑ−1 for almost all z ∈
.
for δ > 0 small (see H0 and recall that uτ u0λ). Acting on this inequality with
(uτ − u0λ)+ ∈ W 1, p( ) and using the nonlinear Green’s identity (see e.g., Gasin´ski
and Papageorgiou [
9
]) as above, we obtain
so
This proves Claim 1. Let
[uτ ) =
u ∈ W 1, p( ) : uτ (z)
u(z) for almost all z ∈
.
(3.27)
(3.28)
From (3.26), we see that
ψλ[uτ ) = ϕλ[uτ ) + c,
(3.29)
for some c ∈ R. Then Claim 1 and (3.29) imply that u0 is a local C 1( )minimizer
λ
of ϕλ. From Theorem 2.2, it follows that u0λ is a local W 1, p( )minimizer of ϕ(λ).
By virtue of hypotheses Hg(iii) and H f (iii), for a given ε > 0 we can find δ =
δ(ε) > 0, such that
− εp ζ p for almost all z ∈
So, if u ∈ C 1( ) with u C1( )
δ, then
, all ζ ∈ (0, δ].
(3.30)
ξ0 u p
p −
ξ0 − (λ + 1)ε u p
p
(see Lemma 2.5) and (3.30). Choosing ε ∈ 0, λξ+01 , we infer that
ϕλ(u)
0 = ϕλ(0) ∀u ∈ C 1( ), u C1( )
δ,
so
and thus
u = 0 is a local C 1( )minimizer of ϕλ
u = 0 is a local W 1, p( )minimizer of ϕλ
(see Theorem 2.2).
Without any loss of generality, we may assume that
ϕλ(0) = 0
ϕλ(u0λ)
(the analysis is similar if the opposite inequality is true). Moreover, we may assume
that both local minimizers u = 0 and u = u0λ are isolated (otherwise it is clear that we
have a whole sequence of positive solutions of ( P)λ and so we are done). Reasoning
as in Aizicovici et al. [2, Proposition 29], we can find ∈ 0, u0λ small, such that
ϕλ(0) = 0
ϕλ(u0λ) < inf ϕλ(u) :
0
u − uλ =
= η0λ.
(3.31)
Recall that ϕλ is coercive (see the proof of Proposition 3.4). Hence it satisfies the Palais–
Smale condition. This fact and (3.1) permit the use of the mountain pass theorem (see
Theorem 2.1) and so, we obtain uλ ∈ W 1, p( ), such that
and
η
ϕλ(uλ)
ϕλ(uλ) = 0.
From (3.31) and (3.32), it follows that uλ ∈/ {0, u0λ}. From (3.33), we have
Proposition 3.6 If hypotheses H f , Hg and H0 hold, then λ∗ ∈ Y.
Proof Let λn > λ∗ for n 1 be such that λn λ∗ and let un = uλn ∈ int C+ be
positive solutions for problem ( P)λ for n 1 (see Proposition 3.4). We have
A(un) + βunp−1 = λn Ng(un) − N f (un) ∀n
1.
(3.34)
By virtue of hypothesis Hg(i i ) and since ϑ > q, we have
lim
ζ →+∞ ζ ϑ−1
= 0 uniformly for almost all z ∈
.
This fact combined with hypothesis Hg(i ), implies that for a given ε > 0, we can find
c6 = c6(ε) > 0, such that
g(z, ζ )ζ
(ζ +)ϑ + c6 for almost all z ∈
In a similar fashion, using hypotheses H f (i ) and (i i ), we see that we can find η > 0
and c7 > 0, such that
f (z, ζ )ζ
(ζ +)ϑ − c7 for almost all z ∈
On (3.34) we act with un ∈ W 1, p( ) and obtain
βunp d z = λn
g(z, un)un d z −
f (z, un )un d z
λnε − η
ϑ
un ϑϑ + c8 ∀n
1,
for some c8 > 0 (see (3.35) and (3.36)).
(3.32)
(3.33)
(3.35)
(3.36)
(3.37)
We choose ε ∈ 0, λη1
Lemma 2.5, it follows that
(recall that λn
λ1 for all n
1). Then from (3.37) and
with θ < p∗. On (3.34) we act with un − u∗, pass to the limit as n → +∞ and use
(3.38). We obtain
so
Let
From (3.41) and Theorem 2 of Lieberman [
24
], we know that we can find α ∈ (0, 1)
and M > 0, such that
un ∈ C 1,α( ) and
un C1,α( )
M ∀n
1.
From the compactness of the embedding C 1,α( ) ⊆ C 1( ), we have
un −→ u∗ in C 1( ).
un
un
yn =
∀n
(3.42)
(3.43)
(3.44)
(3.45)
with 0 ζ (z) ζ ∗ for almost all z ∈ . So, if in (3.44) we pass to the limit as
n → +∞ and we use (3.45) and (3.46), we obtain
A(y∗) + β y∗p−1 = −ζ y∗p−1,
So, passing to a subsequence if necessary, we may assume that From (3.34), we have Then so
so
thus
yn
0,
yn
= 1 ∀n
1.
yn −→ y∗
yn −→ y∗ in Lϑ ( ).
From hypotheses Hg(i ), (iii) and H f (i ), (iii), it follows that
the sequences
Ng(un)
un p−1
n 1
,
N f (un)
un p−1
n 1
⊆ L p ( ) are bounded
(where 1p + p1 = 1).
Acting on (3.44) with yn − y∗, passing to the limit as n → +∞ and using (3.42),
we obtain
p
∇ y∗ p +
β y∗p d z
ζ y∗p d z
0,
ξ0 y∗
p
−
0
(see Lemma 2.5) and finally, we have that y∗ = 0, which contradicts to (3.45).
TheTrheifsorperoλv∗e∈s tYha.t u∗ = 0. Hence u∗ ∈ int C+ is a solution of problem ( P)λ∗ .
We show that for every λ
solution.
λ∗, problem ( P)λ has an extremal (smallest) positive
Proposition 3.7 If hypotheses H f , Hg, and H0 hold and λ
has a smallest positive solution u∗λ ∈ int C+.
λ∗, then problem ( P)λ
Proof Let S(λ) be the set of positive solutions for problem ( P)λ. Since λ λ∗,
S(λ) = 0 and S(λ) ⊆ int C+. Let C ⊆ S(λ) be a chain (i.e., a nonempty linearly
ordered subset of S(λ)). From Dunford and Schwartz [6, p.336], we know that we can
find a sequence {un}n 1 ⊆ C , such that
inf un = inf C.
n 1
Moreover, from Lemma 11.5(a) of Heikkilä and Lakshmikantham [22, p. 15], we
know that we may assume that the sequence {un}n 1 is decreasing. We have
A(un) + βunp−1 = λNg(un) − N f (un) ∀n
1,
(3.47)
βunp d z =
λg(z, un) − f (z, un) un d z
M1 ∀n
1,
for some M1 > 0 (see hypotheses Hg(i ), H f (i ) and recall that un
So,
u1 for all n
1).
so
so
(see Proposition 2.4).
(see Lemma 2.5) and thus the sequence {un}n 1 ⊆ W 1, p( ) is bounded.
So, passing to a subsequence if necessary, we may assume that
with θ < p∗. On (3.47) we act with un − u∗, pass to the limit as n → +∞ and use
(3.48). Then
un −→ u∗
un −→ u∗ in Lθ ( ),
weakly in W 1, p( ),
lim
n→+∞
u∗
λ
( P )λ.
Reasoning as in the proof of Proposition 3.6, we show that u∗ = 0 and so u∗ ∈
int C+ is a positive solution of ( P )λ. Hence u∗ = inf C ∈ S(λ) and since C was an
arbitrary chain, from the KuratowskiZorn lemma, we infer that S(λ) has a minimum
element u∗λ ∈ int C+. But S(λ) is downward directed (i.e., if u, v ∈ S(λ), then there
exists y ∈ S(λ), such that y min{u, v}; see Aizicovici et al. [
3
]). So, it follows that
u for all u ∈ S(λ), i.e., u∗λ ∈ int C+ is the smallest positive solution of problem
Summarizing the situation, we have the following bifurcationtype theorem
describing the dependence of positive solutions of ( P )λ on the parameter λ > 0.
Theorem 3.8 If hypotheses H f , Hg and H0 hold, then there exists λ∗ > 0, such that:
(a) for all λ > λ∗, problem ( P )λ has at least two positive solutions
u0, u ∈ int C+;
(b) for λ = λ∗, problem ( P )λ has at least one positive solution u∗ ∈ int C+;
(c) for all λ ∈ (0, λ∗), problem ( P )λ has no positive solution.
Moreover, if λ
λ∗, then problem ( P )λ has a smallest positive solution u∗λ ∈ int C+.
Acknowledgments The authors wish to thank the two referees for their corrections and helpful remarks.
The research was supported by the Marie Curie International Research Staff Exchange Scheme Fellowship
within the 7th European Community Framework Programme under Grant Agreement No. 295118, the
International Project cofinanced by the Ministry of Science and Higher Education of Republic of Poland
under Grant No. W111/7.PR/2012 and the National Science Center of Poland under Maestro Advanced
Project No. DEC2012/06/A/ST1/00262.
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0
International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution,
and reproduction in any medium, provided you give appropriate credit to the original author(s) and the
source, provide a link to the Creative Commons license, and indicate if changes were made.
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