2 Mathematical background
In this section, we briefly recall the main mathematical notions and tools which we will use in this work.
Let X be a Banach space.
By X* we denote the topological dual of X and by
〈⋅,⋅〉 the duality brackets for the pair
(X*,X).
Given φ∈C1(X;ℝ) we say that φ satisfies the Cerami condition
if the following property holds:
Cerami Condition.
Every sequence {un}n⩾1⊆X such that
{φ(un)}n⩾1⊆ℝ is bounded and
(1+∥un∥)φ′(un)→0 in X*,
admits a strongly convergent subsequence.
This is a compactness-type condition on the functional φ which leads to a deformation theorem,
describing the changes in the topological structure of the sublevel sets of φ
along a (pseudo)gradient flow.
Using this deformation theorem, one can derive the minimax theory of the critical values of φ.
Prominent in that theory is the so-called mountain pass theorem
of Ambrosetti–Rabinowitz [4]
stated here in a slightly more general form
(see Gasiński–Papageorgiou [10, p. 648]).
Theorem 1.
If X is a Banach space,
φ∈C1(X;R) satisfies the Cerami condition,
u0,u1∈X,
∥u1-u0∥>r>0,
max{φ(u0),φ(u1)}<inf{φ(u):∥u-u0∥=r}=mr,
and
c=infγ∈Γmaxt∈[0,1]φ(γ(t)),
with
Γ={γ∈C([0,1];X):γ(0)=u0,γ(1)=u1},
then c⩾mr and c is a critical value of φ.
In the analysis of problem (($P_{\lambda}$)),
basically we will use the Sobolev space W01.p(Ω),
the Banach space
C1(Ω¯)
and the “boundary” Lebesgue space Ls(∂Ω), 1⩽s⩽+∞.
By ∥⋅∥ we denote the norm of the Sobolev space W1,p(Ω) defined by
∥u∥=(∥u∥pp+∥Du∥pp)1p for all u∈W1,p(Ω).
The Banach space C1(Ω¯) is an ordered Banach space with positive (order) cone
C+={u∈C1(Ω¯):u(z)⩾0 for all z∈Ω¯}.
This cone has a nonempty interior which contains the set
D+={u∈C+:u(z)>0 for all z∈Ω¯}.
On ∂Ω we consider the (N-1)-dimensional Hausdorff (surface) measure σ. Using this measure, we can define in the usual way the boundary Lebesgue
spaces Ls(∂Ω) (1⩽s⩽+∞).
We know that there exists a unique continuous linear map
γ0:W1,p(Ω)→Lp(∂Ω),
known as the trace operator, such that
γ0(u)=u|∂Ω for all u∈W1,p(Ω)∩C(Ω¯).
So, the trace map assigns “boundary values”
to every Sobolev function.
The trace map γ0 is compact into Ls(∂Ω)
for all s∈[1,(N-1)pN-p) if N>p and into Ls(∂Ω) for all s⩾1 if N⩽p.
Also, we have
kerγ0=W01,p(Ω) and imγ0=W1p′,p(∂Ω)
(with 1p+1p′=1).
In the sequel for the sake of notational simplicity,
we will drop the use of the map γ0.
The restrictions of all Sobolev functions on ∂Ω are understood in the sense of traces.
Suppose that f0:Ω×ℝ→ℝ is a Carathéodory function such that
|f0(z,ζ)|⩽a0(z)(1+|ζ|r-1) for a.a. z∈Ω,all ζ∈ℝ
with r⩽p* and a0∈L∞(Ω)+.
We set
F0(z,ζ)=∫0ζf0(z,s)𝑑s
and consider the C1-functional φ0:W1,p(Ω)→ℝ defined by
φ0(u)=1p∥Du∥pp+1p∫∂Ωβ(z)|u|p𝑑σ-∫ΩF0(z,u)𝑑z for all u∈W1,p(Ω).
We assume that β∈C0,α(∂Ω) with α∈(0,1) and β⩾0.
The next result can be found in
Papageorgiou–Rădulescu [22]
(subcritical case, that is, r<p*) and
Papageorgiou–Rădulescu [24]
(critical case, that is, r=p*).
Let Σ0⊆∂Ω be a closed subset,
V=C*1(Ω¯)={v∈C1(Ω¯):v|Σ0=0}
and X=W*1,p(Ω)=C*1(Ω¯)¯∥⋅∥.
We have the following result.
Proposition 2.
If u0∈W1,p(Ω) is a local V-minimizer of φ0,
that is, there exists ϱ1>0 such that
φ0(u0)⩽φ0(u0+h) for all h∈V,∥h∥C1(Ω¯)⩽ϱ0,
then u0∈C1,α(Ω¯)
with α∈(0,1) and u0 is a local X-minimizer of φ0,
that is, there exists ϱ1>0 such that
φ0(u0)⩽φ0(u0+h) for all h∈X,∥h∥⩽ϱ1.
Consider the nonlinear map
A:W1,p(Ω)→W1,p(Ω)* defined by
〈A(u),h〉=∫Ω|Du|p-2(Du,Dh)ℝN𝑑z for all u,h∈W1,p(Ω).
The following properties of A are well known
(see, e.g., Gasiński–Papageorgiou [10, p. 746]
and Gasiński–Papageorgiou [11]
for a more general result).
Proposition 3.
The map A is bounded
(that is, maps bounded sets to bounded sets), continuous, monotone
(hence maximal monotone too) and of type (S)+, that is:
If un→𝑤u in W1,p(Ω) and
lim supn→+∞〈A(un),un-u〉⩽0,
then un→u in W1,p(Ω).
The next proposition provides an equivalent norm for the Sobolev space W1,p(Ω).
It extends a result that can be found in
Zeidler [26, p. 1033] and
Gasiński–Papageorgiou [10, p. 218].
Proposition 4.
If β∈L∞(∂Ω), β(z)⩾0 for all z∈∂Ω and β≢0,
then the map
u↦|u|=(∥Du∥pp+∫∂Ωβ(z)|u|p𝑑σ)1p
is an equivalent norm on the Sobolev space W1,p(Ω).
Proof.
First note that
|u|p⩽∥Du∥pp+∥β∥L∞(∂Ω)∥u∥Lp(∂Ω)p⩽∥Du∥pp+∥β∥L∞(∂Ω)∥γ0∥ℒp∥u∥p
(from the continuity of the trace map γ0), so
(2.1)|u|⩽c1∥u∥
for some c1>0.
Next we show that we can find c2>0 such that
(2.2)∥u∥p⩽c2|u| for all u∈W1,p(Ω).
Arguing by contradiction, suppose that (2.2) is not true.
We can find a sequence {un}n⩾1⊆W1,p(Ω) such that
(2.3)∥un∥p>n|un| for all n⩾1.
Normalizing in Lp(Ω), we have
(2.4)∥un∥p=1 and |un|⩽1n for all n⩾1
(see (2.3)),
so
(2.5)|un|→0.
From (2.5), we have
∥Dun∥p→0,
so the sequence {un}n⩾1⊆W1,p(Ω) is bounded
(see (2.4)).
By passing to a suitable subsequence if necessary,
we may assume that
(2.6)un→𝑤u in W1,p(Ω) and un→u in Lp(Ω)and inLp(∂Ω).
From (2.4) and (2.6) it follows that
(2.7)∥Du∥pp+∫∂Ωβ(z)|u|p𝑑σ⩽0,
so u≡ξ∈ℝ.
If ξ≠0, then from (2.7) and the hypothesis on β, we have
0<|ξ|p∫∂Ωβ(z)𝑑σ⩽0,
a contradiction.
If ξ=0, then from (2.6) we have
1=∥un∥p→0
(see (2.4)),
again a contradiction.
So, (2.2) is true.
From (2.2) it follows that
(2.8)∥u∥⩽c3|u| for all u∈W1,p(Ω)
for some c3>0.
From (2.1) and (2.8),
we conclude that ∥⋅∥ and |⋅| are equivalent norms on the Sobolev space
W1,p(Ω).
∎
Consider the following nonlinear eigenvalue problem:
(2.9){-Δpu(z)=λ^|u(z)|p-2u(z)in Ω,∂u∂np+β(z)|u|p-2u=0on ∂Ω.
Using the Ljusternik–Schnirelmann minimax scheme,
we know that problem (2.9) has a whole sequence
{λ^k}k⩾1⊆[0,+∞)
of distinct eigenvalues such that λ^k→+∞
(recall that β∈L∞(∂Ω), β⩾0).
We do not know if this sequence exhausts the spectrum of (2.9).
Concerning the first eigenvalue, we have the following properties:
λ^1⩾0 and λ^1=0 if β≡0
(Neumann problem),
λ^1>0 if β≢0.
λ^1 is isolated and simple
(that is, if u^1 and u^2 are eigenfunctions corresponding to
λ^1, then u^1=ξu^2 for some ξ∈ℝ∖{0}).
If γ(u)=∥Du∥pp+∫∂Ωβ(z)|u|p𝑑σ for all u∈W1,p(Ω),
then
(2.10)λ^1=inf{γ(u)∥u∥pp:u∈W1,p(Ω),u≠0}.
λ^1 is the only eigenvalue with eigenfunctions of constant sign (see below).
All the other eigenvalues have nodal
(that is, sign changing) eigenfunctions.
The infimum in (2.10) is realized on the corresponding one-dimensional eigenspace.
From the above properties, it follows that the elements of this eigenspace do not change sign.
Let u^1 be the Lp-normalized
(that is, ∥u^1∥p=1)
positive eigenfunction.
Assuming that β∈C0,α(∂Ω) with α∈(0,1),
the nonlinear regularity theory of
Lieberman [19]
and the nonlinear maximum principle,
imply that u^1∈D+.
Finally, let us introduce some notation which will be used throughout this work.
So, for ζ∈ℝ we set
ζ±=max{±ζ,0}.
Then for every u∈W1,p(Ω),
we define
u±(⋅)=u(⋅)±.
We have
u±∈W1,p(Ω),u=u+-u-,|u|=u++u-.
By |⋅|N we denote the Lebesgue measure on ℝN.
Finally, if φ∈C1(X;ℝ), then by Kφ
we denote the critical set of φ, that is,
Kφ={u∈X:φ′(u)=0}.
3 Strong comparison theorems
It is well known that in order to have strong comparison results for the p-Laplacian,
we need stronger conditions on the data compared to the linear case (p=2).
The first such result was proved by
Guedda–Véron [15] for the Dirichlet p-Laplacian
and then extensions were obtained,
again for the Dirichlet problem,
by
Cuesta–Takač [6], and
Arcoya–Ruiz [5].
In this section, we present two such strong comparison properties,
suitable for Robin problems, which extend the results in
[15, 6, 5].
Given h1,h2∈L∞(Ω), we say that h1≺h2 if for every compact set
K⊆Ω we can find ε=ε(K)>0 such that
h1(z)+ε⩽h2(z) for a.a.z∈K.
Clearly, if h1,h2∈C(Ω) and h1(z)<h2(z) for all z∈Ω,
then h≺h2.
Since we believe that the results of this section are of independent interest, we will formulate them for more general differential operators which need not
be homogeneous and include the p-Laplacian as a special case.
So, let ϑ∈C1(0,∞) and assume that
0<c^⩽tϑ′(t)ϑ(t)⩽c0 and c1tp-1⩽ϑ(t)⩽c2(1+tp-1) for all t>0
for some c1,c2>0.
We consider differential operators of the form
u↦div(a(Du)) for all u∈W1,p(Ω),
with a:ℝN→ℝN being a map satisfying the following conditions:
Hypotheses H($\boldsymbol{a}$).
We have
a(y)=a0(∥y∥)y for all y∈ℝN and a0(t)>0 for t>0 and
a0∈C1(0,+∞),
t↦a0(t)t is strictly increasing,
ta0(t)→0+ as t→0+ and
limt→0+ta0′(t)a0(t)>-1,
there exists c4>0 such that
|∇a(y)|⩽c4ϑ(|y|)|y| for all y∈ℝN∖{0},
(∇a(y)ξ,ξ)ℝN⩾ϑ(|y|)|y||ξ|2
for all y∈ℝN∖{0}, all ξ∈ℝN.
Remark 1.
These conditions were motivated by the nonlinear regularity theory of
Lieberman [20].
They extend the conditions in
Lieberman [19]
and they incorporate a large class of nonlinear operators of interest,
such as the p-Laplacian (1<p<+∞)
and the (p,q)-Laplacian, that is, the operator
u↦Δpu+Δqu with 1<q<p<+∞,u∈W1,p(Ω)
(see Gasiński–O’Regan–Papageorgiou [9],
Gasiński–Papageorgiou [14]).
Such differential operators arise in problems of mathematical physics.
Proposition 2.
If Hypotheses H(a) hold,
ξ∈L∞(Ω)+,
h1,h2∈L∞(Ω)
with h1≺h2,
u∈C1(Ω¯),
u≠0, v∈D+,
u⩽v and they satisfy
{-diva(Du(z))+ξ(z)|u(z)|p-2u(z)=h1(z)for a.a.z∈Ω,-diva(Dv(z))+ξ(z)v(z)p-1=h2(z)for a.a.z∈Ω,∂v∂n|∂Ω<0,
then
(v-u)(z)>0 for allz∈Ω 𝑎𝑛𝑑 ∂(v-u)∂n|Σ0<0,
where Σ0={z∈∂Ω:v(z)=u(z)}.
Proof.
Let
D={z∈Ω:u(z)=v(z)},
E={z∈Ω:Du(z)=Dv(z)=0}.
Let z0∈D and let y=v-u.
Then y attains its minimum at z0
(recall that by hypothesis u⩽v).
Hence we have
Dy(z0)=0,
so
Dv(z0)=Du(z0).
If z0∉E, then we can find ϱ>0
small such that
B¯ϱ(z0)={z∈ℝN:|z-z0|⩽ϱ}⊆Ω
and
|Du(z)|>0,|Dv(z)|>0 and (Dv(z),Du(z))ℝN>0 for all z∈B¯ϱ(z0).
By hypothesis, we have
(3.1)-div(a(Dv)-a(Du))=h2(z)-h1(z)-ξ(z)(vp-1-|u|p-2u) a.e. inΩ.
Let a=(ak)k=1N.
For k∈{1,…,N},
by the mean value theorem we have
ak(ξ)-ak(ξ′)=∑i=1N∫01∂ak∂yi(ξ′+t(ξ-ξ′))(ξi-ξi′)𝑑t
for all ξ=(ξi)i=1N,ξ′=(ξi′)i=1N∈ℝN.
On B¯ϱ(z0) we define the following continuous coefficients:
ηk,i(z)=∫01∂ak∂yi(Du(z)+t(Dv(z)-Du(z))dt
and using them we introduce the following linear differential operator:
L(w)=-div(∑i=1Nηk,i(z)∂w∂zi) for all w∈W1,p(Bϱ(z0)).
Hypotheses H(a) imply that by taking ϱ>0 even smaller if necessary,
we can have that the operator L is strictly elliptic on
B¯ϱ(z0).
Also, since ξ∈L∞(Ω)+ and h1≺h2, we can have
(3.2)h2(z)-h1(z)-ξ(z)(v(z)-|u(z)|p-2u(z))>0 for a.a.z∈B¯ϱ(z0).
From (3.1) and (3.2), we have
L(y(z))>0 for a.a.z∈B¯ϱ(z0).
The strong maximum principle implies that
y(z)=(v-u)(z)>0 for all z∈Bϱ(z0),
which contradicts the fact that z0∈D.
So, z0∈E and we have proved the claim.
By hypothesis,
v∈D+ and ∂v∂n|∂Ω<0.
Hence the set E⊆Ω is compact.
Therefore so is D
(see the claim) and we can an find open subset Ω1⊆Ω
with C2-boundary such that
(3.3)D⊆Ω1⊆Ω¯1⊆Ω.
Let ε>0 be such that
u(z)+ε<v(z) for all z∈∂Ω
(see (3.3)) and
(3.4)h1(z)+ε<h2(z) for a.a.z∈Ω¯
(recall that by hypothesis h1≺h2).
We choose δ∈(0,min{ε,1}) such that
for almost all z∈Ω and all s,t∈[-∥u∥∞,∥v∥∞]
with |s-t|⩽2δ, we have
(3.5)ξ(z)||s|p-2s-|t|p-2t|<ε.
Then for this δ>0, we have
-diva(D(u+δ))+ξ(z)|u+δ|p-2(u+δ)=-diva(Du)+ξ(z)|u+δ|p-2(u+δ)
=ξ(z)(|u+δ|p-2(u+δ)-|u|p-2u)+h1
⩽h1+ε<h2
=-diva(Dv)+ξ(z)vp-1 for a.a.z∈Ω1
(see (3.5) and (3.4)), so
u+δ⩽v on Ω1
(by the weak comparison principle; see, e.g.,
Motreanu–Motreanu–Papageorgiou [21, p. 214]),
thus
D=∅ (since D⊆Ω1) and hence
(v-u)(z)>0 for all z∈Ω.
Moreover, from [6, Proposition 2.4], we also have
∂(v-u)∂n|Σ0<0,
with Σ0={z∈∂Ω:v(z)=u(z)}.
∎
Remark 3.
Let Σ0⊆∂Ω be as above and let
C*1(Ω)={h∈C1(Ω¯):h|Σ0=0}.
This is an ordered Banach space with order cone
C^+(Σ0)={h∈C*1(Ω¯):h(z)⩾0for allz∈Ω}.
This cone has a nonempty interior which contains the set
D^+(Σ0)={y∈C1(Ω¯):y(z)>0for allz∈Ω,∂y∂n|Σ0<0}.
So, according to Proposition 2,
we have
v-u∈D^+(Σ0).
Also, we set
W*1,p(Ω)=C*1(Ω¯)¯∥⋅∥.
When ξ=0, we can relax the condition h1≺h2.
More precisely, we have the following strong comparison principle.
Proposition 4.
If Hypotheses H(a) hold,
h1,h2∈L∞(Ω),
h1(z)⩽h2(z) for almost all z∈Ω and the inequality is strict on a set of positive measure and
u1,u2∈C(Ω¯) satisfy u1⩽u2 in Ω and
{-diva(Du1(z))=h1(z) for a.a. z∈Ω,-diva(Du2(z))=h2(z) for a.a. z∈Ω,∂u1∂n|∂Ω<0,∂u2∂n|∂Ω<0,
then
(u2-u1)(z)>0 for allz∈Ω,𝑎𝑛𝑑 ∂(u2-u1)∂n|Σ0<0,
Proof.
The hypotheses on the normal derivatives of u1,u2 imply that we can find
δ>0 small such that
|D((1-t)u1(z)+tu2(z))|⩾ε>0 for all z∈Ω¯δ,t∈[0,1],
where Ωδ={z∈Ω:d(z,∂Ω)<δ}.
We have
(3.6)-div(a(Du2)-a(Du1))⩾0 for a.a.z∈Ω.
As in the proof of Proposition 2,
using the mean value theorem,
equation (3.6) becomes
(3.7)L(u2-u1)(z)⩾0 for a.a.z∈Ωδ,
with L being strictly elliptic on Ωδ, by the choice of δ>0.
We claim that u1≢u2 on Ωδ.
Arguing by contradiction, suppose that u1=u2 on Ωδ.
Then h1=h2 on Ωδ.
Since by hypothesis h1≢h2,
we must have that h1≠h2 on Ω∖Ωδ.
Choose a test function ζ∈W1,p(Ω) such that
(3.8)ζ>0 on Ω and ζ|Ω∖Ωδ=1.
Let 〈⋅,⋅〉∂Ω denote the duality brackets for the pair
(W-1p,p′(∂Ω),W1p,p(∂Ω)).
Using the nonlinear Green’s identity
(see Gasiński–Papageorgiou [10, p. 210]),
we have
∫Ωh1ζ𝑑z=∫Ω(a(Du1),Dζ)ℝN𝑑z-〈∂u1∂na,ζ〉∂Ω
=∫Ωδ(a(Du1),Dζ)ℝN𝑑z-〈∂u1∂na,ζ〉∂Ω
=∫Ωδ(a(Du2),Dζ)ℝN𝑑z-〈∂u2∂na,ζ〉∂Ω
=∫Ωh2ζ𝑑z
(here ∂u1∂na=(a(Du),n)ℝN,
see (3.8) and recall that we have assumed that u1=u2 on Ωδ),
a contradiction, since
∫Ωh1𝑑z<∫Ωh2𝑑z.
So, we have that
u1≢u2 on Ωδ.
Then from (3.7) and the strong maximum principle it follows that
(3.9)u1(z)<u2(z) for all z∈Ωδ.
We introduce the set
D={z∈Ω:u1(z)=u2(z)}.
From (3.9) it follows that the set D⊆Ω is compact and so we can find
an open set Ω^⊆Ω such that
D⊆Ω^⊆Ω^¯⊆Ω.
Evidently on ∂Ω^ we have
u1<u2 and so we can find ε>0 such that
(3.10)u1+ε<u2 on ∂Ω^.
We have
(3.11)-diva(D(u1+ε))=-diva(Du1)⩽-diva(Du2) for a.a.z∈Ω^
(since h1⩽h2).
From (3.10) and (3.11) it follows that
u1+ε⩽u2 on Ω^,
so D=∅,
that is,
(u2-u1)(z)>0 for all z∈Ω.
Moreover, invoking the nonlinear boundary point lemma
(see Pucci–Serrin [25, p. 120]), we have
∂(u2-u1)∂n|∂Ω<0,
as desired.
∎
4 Bifurcation-type theorems
In this section we consider problem (($P_{\lambda}$)) and prove a bifurcation-type theorem describing the set of positive solutions as the
parameter λ>0 varies.
We introduce the following conditions on the data of (($P_{\lambda}$)).
Hypothesis H($\boldsymbol{\beta}$).
We have β∈C0,α(∂Ω) with α∈(0,1) and β(z)>0 for all z∈∂Ω.
Hypotheses H1.
The function f:Ω×ℝ→ℝ is a Carathéodory function such that
f(z,0)=0 for almost all z∈Ω and
f(z,ζ)⩽a(z)(1+ζr-1) for almost all z∈Ω, all ζ∈ℝ,
with a∈L∞(Ω)+, p<r<p*,
if F(z,ζ)=∫0ζf(z,s)𝑑s, then
limζ→+∞F(z,ζ)ζp=+∞ uniformly for a.a.z∈Ω
and there exists τ∈(max{(r-p)Np,1},p*) such that
0<γ0⩽lim infζ→+∞f(z,ζ)ζ-2F(z,ζ)ζτuniformly for a.a.z∈Ω,
we have
limζ→0+f(z,ζ)ζp-1=+∞
uniformly for almost all z∈Ω,
for every s>0, there exists ms>0 such that
0<ms⩽infs⩽ζf(z,ζ) for a.a.z∈Ω
and for every ϱ>0,
there exists ξ^ϱ>0
such that for almost all z∈Ω the function
ζ↦f(z,ζ)+ξ^ϱζp-1
is nondecreasing on [0,ϱ].
Remark 1.
Since we are looking for positive solutions and the above hypotheses concern the positive semiaxis,
without any loss of generality, we may assume that f(z,ζ)=0 for almost all z∈Ω, all ζ⩽0.
Hypothesis H1 (ii) implies that
limζ→+∞f(z,ζ)ζp-1=+∞ uniformly for a.a.z∈Ω.
So, the reaction term f(z,⋅) is (p-1)-superlinear.
However, note that we do not employ the usual for superlinear problems
Ambrosetti–Rabinowitz condition, the unilateral version
(that is, only for the positive semiaxis), which says that there exist η>p and M>0 such that
(4.1)0<ηF(z,ζ)⩽f(z,ζ)ζ for a.a.z∈Ω,allζ⩾M,
and
(4.2)0<ess infΩF(⋅,M).
Integrating (4.1) and using (4.2), we obtain the weaker condition which says
c5ζη⩽F(z,ζ) for a.a.z∈Ω,allζ⩾M,
with c5>0.
Thus the Ambrosetti–Rabinowitz condition implies that
f(z,⋅) eventually has at least
(η-1)-polynomial growth.
Hypothesis H1 (ii) is less restrictive and incorporates
(p-1)-superlinear nonlinearities with slower growth near +∞ which fail to satisfy the
Ambrosetti–Rabinowitz condition
(see Example 2).
Hypothesis H1 (iii) implies that near 0+,
f(z,ζ) is (p-1)-superlinear.
So, we have the competition of concave and convex nonlinearities.
Example 2.
The following functions satisfy Hypotheses H1.
For the sake of simplicity we drop the z-dependence:
f1(ζ)=ζr-1+ζq-1 for all ζ⩾0,
f2(ζ)={-ζq-1lnζif ζ∈[0,1e],ζp-1lnζ+1+ep-qep-1if 1e<ζ,
with 1<q<p<r<p*.
Note that f1 satisfies the Ambrosetti–Rabinowitz condition,
while f2 does not.
We introduce the following sets:
ℒ={λ>0:problem (Pλ) admits a positive solution}
(so ℒ is the set of admissible parameters) and
S(λ), the set of positive solutions for problem (($P_{\lambda}$)).
First we establish a property of the solution set S(λ).
Proposition 3.
If Hypotheses H(β) and H1 hold, then for every λ>0,
S(λ)⊆D+.
Proof.
Suppose that λ∈ℒ
(otherwise S(λ)=∅ and so the conclusion of the proposition is trivially true).
Let u∈S(λ).
We have
〈A(u),h〉+∫∂Ωβ(z)up-1h𝑑σ=λ∫Ωf(z,u)h𝑑z for all h∈W1,p(Ω),
so
(4.3){-Δpu(z)=λf(z,u(z))for a.a.z∈Ω,∂u∂np+β(z)up-1=0on ∂Ω
(see Papageorgiou–Rădulescu [22]).
From (4.3) and
Papageorgiou–Rădulescu [24],
we have that u∈L∞(Ω).
Then we can apply
[19, Theorem 2] and infer that
u∈C+∖{0}.
Let ϱ=∥u∥∞ and let ξ^ϱ>0 be as postulated by Hypothesis H1 (iv).
Then from (4.3) we have
Δpu(z)⩽λξ^ϱu(z)p-1 for a.a.z∈Ω,
so u∈D+
(by the nonlinear maximum principle).
∎
Next we establish the nonemptiness and a structural property of the admissible set ℒ.
Proposition 4.
If Hypotheses H(β) and H1 hold, then
L≠∅ and λ∈L implies that (0,λ]⊆L.
Proof.
We consider the following auxiliary Robin problem:
(4.4){-Δpu(z)=1 in Ω,∂u∂np+β(z)up-1=0 on ∂Ω.
Let ψ:W1,p(Ω)→ℝ be the C1-functional defined by
ψ(u)=1pγ(u)+1p∥u-∥pp-∫Ωu+𝑑z for all u∈W1,p(Ω).
We have
(4.5)ψ(u)=1pγ(u-)+1p∥u-∥pp+1pγ(u+)-∫Ωu+𝑑z
(4.6)⩾1p∥u-∥p+c6p∥u+∥p-c7∥u+∥
for some c6,c7>0
(see Hypothesis H(β) and Proposition 4).
From (4.5) it is clear that ψ is coercive.
Also the Sobolev embedding theorem and the compactness of the trace map
imply that ψ is sequentially weakly lower semicontinuous.
So, by the Weierstrass–Tonelli theorem,
we can find u¯∈W1,p(Ω) such that
(4.7)ψ(u¯)=infu∈W1,p(Ω)ψ(u).
If u∈D+, then for t∈(0,1) we have
ψ(tu)=tppγ(u)-t∫Ωu𝑑z.
Since p>1 choosing t∈(0,1) small, we have
ψ(tu)<0,
so
ψ(u¯)<0=ψ(0)
(see (4.7)), hence u¯≠0.
From (4.7) we have
ψ′(u¯)=0,
so
(4.8)〈A(u¯),h〉+∫∂Ωβ(z)|u¯|p-2u¯h𝑑σ-∫Ω(u¯-)p-1h𝑑z=∫Ωh𝑑z for all h∈W1,p(Ω).
In (4.8) we choose h=-u¯-∈W1,p(Ω).
Then
γ(u¯-)+∥u¯-∥pp⩽0,
so
u¯⩾0,u¯≠0
(see Hypothesis H(β)).
Therefore (4.8) becomes
〈A(u¯),h〉+∫∂Ωβ(z)u¯p-1h𝑑σ=∫Ωh𝑑z for all h∈W1,p(Ω),
so
{-Δpu¯(z)=1 for a.a.z∈Ω,∂u¯∂np+β(z)up-1=0 on ∂Ω,
hence
u¯∈D+
(as before by the nonlinear regularity theory and the nonlinear maximum principle).
In fact, this positive solution is unique.
To see this, let
v¯∈W1,p(Ω) be another positive solution of (4.4).
As above, we have v¯∈D+.
Then
(4.9)∫Ω1u¯p-1(u¯p-v¯p)𝑑z=∫Ω(u¯-v¯pu¯p-1)𝑑z
(4.10)=∫Ω(-Δpu¯)(u¯-v¯pu¯p-1)𝑑z
(4.11)=∫Ω|Du¯|p-2(Du¯,D(u¯-v¯pu¯p-1))ℝN𝑑z+∫∂Ωβ(z)(u¯p-v¯p)𝑑σ
(4.12)=∥Du¯∥pp-∥Dv¯∥pp+∫ΩR(v¯,u¯)𝑑z+∫∂Ωβ(z)(u¯p-v¯p)𝑑σ
(by the nonlinear Green’s identity) with
R(v¯,u¯)(z)=|Dv¯(z)|p-|Du¯(z)|p-2(Du¯(z),D(v¯pu¯p-1)(z)).
Interchanging the roles of u¯ and v¯ in the above argument, we also have
(4.13)∫Ω1v¯p-1(v¯p-u¯p)𝑑z=∥Dv¯∥pp-∥Du¯∥pp+∫ΩR(u¯,v¯)+∫∂Ωβ(z)(v¯p-u¯p)𝑑σ.
Adding formulas (4.9) and (4.13) and using the nonlinear Picone’s identity
(see Allegretto–Huang [2]
and Motreanu–Motreanu–Papageorgiou [21, p. 255]),
we have
0⩽∫Ω(R(u¯,v¯)+R(v¯,u¯))=∫Ω(1u¯p-1-1v¯p-1)(u¯p-v¯p)𝑑z⩽0,
so u¯=v¯.
This proves the uniqueness of the positive solution
u¯∈D+ of (4.4).
Next let
λ~=1∥Nf(u¯)∥∞,
with Nf(u¯)(⋅)=f(⋅,u¯(⋅))∈L∞(Ω)
(see Hypothesis H1 (i)).
Then for all h∈W1,p(Ω), h⩾0
we have
(4.14)〈A(u¯),h〉+∫∂Ωβ(z)u¯p-1h𝑑σ=∫Ωh𝑑z=λ~∥Nf(u¯)∥∞∫Ωh𝑑z⩾λ~∫Ωf(z,u¯)h𝑑z
(recall that h⩾0).
We introduce the following truncation of f(z,⋅):
(4.15)g(z,ζ)={0if ζ<0,f(z,ζ)if 0⩽ζ⩽u¯(z),f(z,u¯(z))if u¯(z)<ζ.
This is a Carathéodory function.
We set
G(z,ζ)=∫0ζg(z,s)𝑑s
and consider the C1-functional
φ^λ~:W1,p(Ω)→ℝ defined by
φ^λ~(u)=1pγ(u)-λ~∫ΩG(z,u)𝑑z for all u∈W1,p(Ω).
Using Proposition 4, we have
φ^λ~(u)⩾c8p∥u∥p-c9 for all u∈W1,p(Ω)
for some c8,c9>0,
so
φ^λ~ is coercive.
Also, it is sequentially weakly lower semicontinuous.
So, we can find u~∈W1,p(Ω) such that
(4.16)φ^λ~(u~)=infu∈W1,p(Ω)φ^λ~(u).
As before (see the first part of the proof)
for u∈D+ and t∈(0,1) small,
we have
φ^λ~(tu)<0,
so
φ^λ~(u~)<0=φ^λ~(0),
hence u~≠0.
From (4.16) we have
φ^λ~′(u~)=0,
so
(4.17)〈A(u~),h〉+∫∂Ωβ(z)|u~|p-2u~h𝑑σ=λ~∫Ωg(z,u~)h𝑑z for all h∈W1,p(Ω).
In (4.17) first we choose
h=-u~-∈W1,p(Ω) and we obtain
γ(u~-)=0
(see (4.15)),
so
c10∥u~-∥⩽0
for some c10>0
(see Proposition 4), thus
u~⩾0,u~≠0.
Next in (4.17) we choose h=(u~-u¯)+∈W1,p(Ω).
Then
〈A(u~),(u~-u¯)+〉+∫∂Ωβ(z)u~p-1(u~-u¯)+𝑑σ=λ~∫Ωf(z,u¯)(u~-u¯)+𝑑z
⩽〈A(u¯),(u~-u¯)+〉+∫∂Ωβ(z)u¯p-1(u~-u¯)+𝑑σ
(see (4.14)), so
(4.18)〈A(u~)-A(u¯),(u~-u¯)+〉+∫∂Ωβ(z)(u~p-1-u¯p-1)(u~-u¯)+𝑑σ⩽0.
If p⩾2, then from (4.18) we have
c11(∥D(u~-u¯)+∥pp+∫∂Ωβ(z)((u~-u¯)+)p𝑑σ)⩽0
for some c11>0
(see García Azorero–Manfredi–Peral Alonso [8, inequalities (3.8)]),
so
c12∥(u~-u¯)+∥p⩽0
for some c12>0
(see Proposition 4) and hence
u~⩽u¯.
If p∈(1,2), then from (4.18) and since u~,u¯∈D+, we have
c13(∥D(u~-u¯)+∥22+∫∂Ωβ(z)((u~-u¯)+)2𝑑σ)⩽0
for some c13>0
(again see [8, inequalities (3.8)]), so
c14∥(u~-u¯)+∥p⩽0
for some c14>0
(see Proposition 4) and hence
u~⩽u¯.
So, finally we have
u~∈[0,u¯],
where [0,u¯]={u∈W1,p(Ω):0⩽u(z)⩽u¯(z)for a.a.z∈Ω}.
Then equation (4.17) becomes
〈A(u~),h〉+∫∂Ωβ(z)u~p-1h𝑑σ=λ~∫Ωf(z,u~)h𝑑z for all h∈W1,p(Ω),
so u~∈S(λ~)⊆D+
(see Proposition 3) and so λ~∈ℒ≠∅.
Next we prove the structural property of ℒ.
So, suppose that λ∈ℒ and let ϑ<λ.
Consider uλ∈S(λ)⊆D+
(see Proposition 3).
Truncating f(z,⋅) at
{0,uλ(z)}
(as in (4.15)) and reasoning as above,
via the direct method of the calculus of variations, we obtain
uϑ∈S(ϑ)⊆D+ and uϑ⩽uλ.
So, ϑ∈ℒ and we conclude that [0,λ]⊆ℒ.
∎
Remark 5.
This proposition implies that ℒ is an interval.
In the last proof we have established the following monotonicity property for the solution set S(λ):
If 0<ϑ<λ, λ∈ℒ and uλ∈S(λ)⊆D+,
then ϑ∈ℒ and we can find uϑ∈S(ϑ)⊆D+ such that uϑ⩽uλ.
We can improve this monotonicity property.
Proposition 6.
If Hypotheses H(β) and H1 hold, λ∈L, ϑ∈(0,λ) and uλ∈S(λ)⊆D+,
then
we can find uϑ∈S(ϑ)⊆D+ such that
uλ-uϑ∈D^+(Σ0),
with Σ0={z∈∂Ω:uλ(z)=uϑ(z)}.
Proof.
From Proposition 4 and its proof, we know that we can find
uϑ∈S(ϑ)⊆D+ such that
(4.19)uϑ⩽uλ.
Let δ>0 and set
uϑδ=uϑ+δ∈D+.
Let ϱ=∥uλ∥∞ and let ξ~ϱ>0 be as postulated by Hypothesis H1 (iv).
We have
-Δpuϑδ+λξ^ϱ(uϑδ)p-1⩽-Δpuϑ+λξϱ^uϑp-1+χ(δ)
=ϑf(z,uϑ)+λξ^ϱuϑp-1+χ(δ)
=λf(z,uϑ)+λξ^ϱuϑp-1-(λ-ϑ)f(z,uϑ)+χ(δ)
⩽λ(f(z,uϑ)+ξ^ϱuϑp-1)-(λ-ϑ)ms0+χ(δ)
⩽λf(z,uλ)+λξ^ϱuλp-1-(λ-ϑ)ms0+χ(δ)
<-Δpuλ+λξ^ϱuλp-1 for a.a.z∈Ωand forδ>0small,
with χ(δ)→0+ as δ→0+,
s0=minΩ¯uϑ>0 and ms0>0 as in the Hypothesis H1 (iv)
(since uϑ∈S(ϑ) and using (4.19) and Hypothesis H1 (iv)).
Invoking Proposition 2, we conclude that
uλ-uϑ∈D^+(Σ0),
with Σ0={z∈∂Ω:uλ(z)=uϑ(z)}.
∎
Remark 7.
Evidently if Σ0=∅, then
uλ-uϑ∈D+.
Let λ*=supℒ.
Proposition 8.
If Hypotheses H(β) and H1 hold,
then
λ*<+∞.
Proof.
First we claim that for λ0>0 big, we have
(4.20)λ0f(z,ζ)⩾λ^1ζp-1 for a.a.z∈Ω,allζ⩾0.
Indeed, Hypotheses H1 (ii) and (iii) imply that given ξ>λ^1
we can find M,δ>0 such that
(4.21)f(z,ζ)⩾ξζp-1 for a.a.z∈Ω,allζ⩾Mand allζ∈[0,δ].
On the other hand for λ0⩾1 big, we have
(4.22)λ0f(z,ζ)⩾λ0mδ⩾λ^1Mp-1⩾λ^1ζp-1 for a.a.z∈Ω,allζ∈[δ,M]
(see Hypothesis H1 (iv)).
From (4.21) and (4.22) it follows that (4.20) is true.
Let λ>λ0 and suppose that λ∈ℒ.
Pick uλ∈S(λ)⊆D+
(see Proposition 3) and let t>0 be the biggest real such that
(4.23)tu^1⩽uλ
(recall that u^1,uλ∈D+).
Let ϱ=∥uλ∥∞ and let ξ^ϑ>0 be as postulated by Hypothesis H1 (iv).
Then we have
-Δp(tu^1)+λξ^ϱ(tu^1)p-1=λ^1(tu^1)p-1+λξ^ϱ(tu^1)p-1
⩽λ0f(z,tu^1)+λξ^ϱ(tu^1)p-1
<λf(z,tu^1)+λξ^ϱ(tu^1)p-1
⩽λf(z,uλ)+λξϱ^uλp-1
=-Δpuλ+λξ^ϱuλp-1 for a.a.z∈Ω
(since λ>λ0 and using (4.20), (4.23), Hypothesis H1 (iv) and the fact that uλ∈S(λ)),
so
uλ-tu^1∈D^+(Σ0),
with Σ0={z∈∂Ω:uλ(z)=tu^1(z)}
(see Proposition 2).
This contradicts the maximality of t>0 (see (4.23)).
Therefore λ∉ℒ and so we have
λ*⩽λ0<+∞,
as desired.
∎
From Propositions 4 and 8, it follows that
(0,λ*)⊆ℒ⊆(0,λ*].
Next we show that λ*∈ℒ and so ℒ=(0,λ*].
Proposition 9.
If Hypotheses H(β) and H1 hold,
then
λ*∈L and so L=(0,λ*].
Proof.
Let {λn}n⩾1⊆ℒ be a sequence such that
λn↗λ*.
We choose un=uλn∈S(λn)⊆D+ for all n⩾1.
Suppose that m>n.
Then for all h∈W1,p(Ω) with h⩾0,
we have
(4.24)〈A(um),h〉+∫∂Ωβ(z)ump-1h𝑑σ=λm∫Ωf(z,um)h𝑑z⩾λn∫Ωf(z,um)h𝑑z
(since m>n).
Truncating f(z,⋅) at {0,um(z)) and reasoning as in the proof of Proposition 4,
via the direct method of the calculus of variations and using (4.24),
we obtain a solution un∈S(λn)⊆D+ such that
φλn(un)<0,
with φλ:W1,p(Ω)→ℝ being the C1-energy (Euler) functional of problem (($P_{\lambda}$))
defined by
φλ(u)=12γ(u)-∫ΩF(z,u)𝑑z for all u∈W1,p(Ω).
So, we may assume that the solutions un∈S(λn)⊆D+ satisfy
φλn(un)<0 for all n⩾1,
so
(4.25)γ(un)-λn∫ΩpF(z,un)𝑑z<0 for all n⩾1.
Since un∈S(λn) for n⩾1,
we have
(4.26)〈A(un),h〉+∫∂Ωβ(z)unp-1h𝑑σ=λn∫Ωf(z,un)h𝑑z for all h∈W1,p(Ω).
In (4.26) we choose h=un∈W1,p(Ω) and we obtain
(4.27)-γ(un)+λn∫Ωf(z,un)un𝑑z=0 for all n⩾1.
We add (4.25) and (4.27) and have
(4.28)∫Ω(f(z,un)un-pF(z,un))𝑑z<0 for all n⩾1
(recall λn>0).
Hypotheses H1 (i)–(ii) imply that we can find
γ1∈(0,γ0) and c15>0 such that
γ1ζτ-c15⩽f(z,ζ)ζ-pF(z,ζ) for a.a.z∈Ω,allζ⩾0.
Using this unilateral growth estimate in (4.28), we obtain that
(4.29)the sequence{un}n⩾1⊆Lτ(Ω)is bounded.
In Hypothesis H1 (ii) without any loss of generality we may assume that
τ<r<p*.
Let t∈(0,1) be such that
(4.30)1r=1-tτ+tp*.
First suppose that N≠p.
Using the interpolation inequality, we have
∥un∥r⩽∥un∥τ1-t∥un∥p*t for all n⩾1,
so
(4.31)∥un∥rr⩽c16∥un∥tr for all n⩾1
for some c16>0.
In this last inequality we have used (4.29) and the Sobolev embedding theorem.
In (4.26) we choose h=un∈W1,p(Ω) and obtain
(4.32)γ(un)=λn∫Ωf(z,un)un𝑑z⩽c17(1+∥un∥rr)⩽c18(1+∥un∥tr) for all n⩾1
for some c17,c18>0
(see (4.31) and Hypothesis H1 (i)).
Hypothesis H1 (ii) and (4.30) imply that tr<p.
So, from (4.32) and Proposition 4 it follows that
(4.33)the sequence{un}n⩾1⊆W1,p(Ω)is bounded.
Now suppose that N=p.
In this case by definition p*=+∞, while the Sobolev embedding theorem says that W1,p(Ω)
is embedded compactly into Ls(Ω) for all s∈[1,+∞).
So, in this case let s>r>τ and as before let t∈(0,1) be such that
1r=1-tτ+ts,
so
tr=s(r-τ)s-τ→r-τ<p as s→p*=+∞.
Then the previous argument works, if we choose s>r big such that
tr<p.
Therefore again we conclude that (4.33) is true.
Because of (4.33), we may assume that
(4.34)un→𝑤u* in W1,p(Ω) and un→u* in Lr(Ω)and inLp(∂Ω).
In (4.26) we choose h=un-u*∈W1,p(Ω),
pass to the limit as n→+∞ and use (4.34).
Then
limn→+∞〈A(un),un-u*〉=0,
so
(4.35)un→u* in W1,p(Ω)
(see Proposition 3).
In (4.26) we pass to the limit as n→+∞ and use (4.35).
Then
〈A(u*),h〉+∫∂Ωβ(z)(u*)p-1h𝑑σ=λ*∫Ωf(z,u*)h𝑑z for all h∈W1,p(Ω),
so u*⩾0 is a solution of (($P_{\lambda}$)).
We need to show that u*≠0 to conclude that u*∈S(λ*)⊆D+ and so
λ*∈ℒ.
Arguing by contradiction, suppose that u*=0.
Choose ξ>0 such that
(4.36)λ1ξ>λ^1.
Hypothesis H1 (iii) implies that we can find δ>0 such that
(4.37)f(z,ζ)⩾ξζp-1 for a.a.z∈Ω,allζ∈[0,δ].
From (4.26) we have
(4.38){-Δpun(z)=λnf(z,un(z))for a.a.z∈Ω,∂un∂np+β(z)unp-1=0on ∂Ω
(see Papageorgiou–Rădulescu [22]).
From (4.38), (4.33) and [24, Proposition 7],
we know that there exists M1>0 such that
∥un∥∞⩽M1 for all n⩾1.
Then invoking [19, Theorem 2],
we can find
α∈(0,1) and M2>0 such that
(4.39)un∈C1,α(Ω¯) and ∥un∥C1,α(Ω¯)⩽M2 for all n⩾1.
Exploiting the compactness of the embedding C1,α(Ω¯)⊆C1(Ω¯),
from (4.39) and (4.35),
we infer that
un→u*=0 in C1(Ω¯).
So, we can find n0⩾1 such that
(4.40)un(z)∈(0,δ] for all z∈Ω¯,n⩾n0
(here δ>0 is as in (4.37)).
From (4.38) we have
(4.41)-Δpun(z)=λnf(z,un(z))⩾λ1f(z,un(z))⩾λ1ξun(z)p-1 for a.a.z∈Ω,alln⩾n0
(see (4.40) and (4.37)).
We consider the following auxiliary Robin problem:
(4.42){-Δpv(z)=λ1ξv(z)p-1in Ω,∂v∂np+β(z)vp-1=0on ∂Ω,v⩾0.
From (4.36) we infer that v=0 is the only solution of (4.42).
On the other hand, fix n⩾n0 and consider the following truncation of the reaction term in (4.42):
(4.43)g(z,ζ)={0if ζ<0,λ1ξζp-1if 0⩽ζ⩽un(z),λ1ξun(z)p-1if un(z)<ζ.
This is a Carathéodory function.
We set
G(z,ζ)=∫0ζg(z,s)𝑑s
and consider the C1-functional e:W1,p(Ω)→ℝ defined by
e(v)=1pγ(v)-∫ΩG(z,v)𝑑z for all v∈W1,p(Ω).
Proposition 4 and (4.33) imply that e is coercive.
Also, e is sequentially weakly lower semicontinuous.
So, we can find v^∈W1,p(Ω) such that
(4.44)e(v^)=infv∈W1,p(Ω)e(v).
Because of (4.36) and (4.37), for t∈(0,1) small we have
e(tu^1)<0,
so
e(v^)<0=e(0)
(see (4.44)), hence v^≠0.
From (4.44) we have
e′(v^)=0,
so
(4.45)〈A(v^),h〉+∫∂Ωβ(z)|v^|p-2v^h𝑑σ=∫Ωg(z,v^)h𝑑z for all h∈W1,p(Ω).
In (4.45) we choose h=-v^-∈W1,p(Ω) and obtain
γ(v^-)=0
(see (4.43)), so
c19∥v^-∥2⩽0
for some c19>0 (see Proposition 4).
Also, in (4.45) we choose h=(v^-un)+∈W1,p(Ω).
Then
〈A(v^),(v^-un)+〉+∫∂Ωβ(z)(v^-un)+𝑑z=∫Ωλ1ξunp-1(v^-un)+𝑑z
⩽∫Ωλnf(z,un)(v^-un)+𝑑z
=〈A(un),(v^-un)+〉+∫∂Ωβ(z)unp-1(v^-un)+𝑑σ
(see (4.43), (4.41) and use the fact that un∈S(λn)),
so
v^⩽un
(see the proof of Proposition 4).
So, we have proved that
v^∈[0,un],v^≠0,
with [0,un]={v∈W1,p(Ω):0⩽v(z)⩽un(z)for a.a.z∈Ω},
so
〈A(v^),h〉+∫∂Ωβ(z)v^p-1h𝑑σ=∫Ωλ1ξv^p-1h𝑑z for all h∈W1,p(Ω)
(see (4.45) and (4.43)), thus
{-Δpv^(z)=λ1ξv^(z)for a.a.z∈Ω,∂v^∂np+β(z)v^p-1=0on ∂Ω,
a contradiction.
Therefore
u*≠0,
so
u*∈S(λ*)⊆D+
and hence we conclude that
λ*∈ℒ, that is, ℒ=(0,λ*].
∎
Using similar arguments, we can show that for every
λ∈ℒ=(0,λ*]
problem (($P_{\lambda}$)) admits a smallest positive solution
(minimal positive solution); see also Gasiński–Papageorgiou [12].
Proposition 10.
If Hypotheses H(β) and H1 hold and λ∈L=(0,λ*],
then
problem (($P_{\lambda}$)) admits a smallest positive solution
u~λ∈D+.
Proof.
As in [7, Lemma 4.1],
we can see that S(λ) is downward directed
(that is, if u1,u2∈S(λ), then we can find u∈S(λ)
such that u⩽u1, u⩽u2).
Invoking [17, Lemma 3.10, p. 178],
we can find a decreasing sequence
{un}n⩾1⊆S(λ) such that
infS(λ)=infn⩾1un.
We have
(4.46)〈A(un),h〉+∫∂Ωβ(z)unp-1h𝑑σ=∫Ωλf(z,un)h𝑑z for all h∈W1,p(Ω),n⩾1.
Since un⩽u1 for all n⩾1, using formula (4.46) with h=un∈W1,p(Ω)
and Hypothesis H1 (i),
we infer that the sequence {un}n⩾1⊆W1,p(Ω) is bounded.
So, as in the proof of Proposition 9,
using the nonlinear regularity theory, we obtain
(at least for a subsequence) that
un→u~λ in C1(Ω¯),
with u~λ≠0.
So, we have
u~λ∈S(λ)⊆D+ and u~λ=infS(λ),
as desired.
∎
We examine the continuity and monotonicity properties of the map
λ↦u~λ
from ℒ=(0,λ*] into C1(Ω¯).
Proposition 11.
If Hypotheses H(β) and H1 hold,
then
the map λ↦u~λ is left continuous
from L=(0,λ*] into C1(Ω¯)
and it is strictly increasing in the sense that,
if ϑ,λ∈(0,λ*), ϑ<λ,
then
u~λ-u~ϑ∈D^+(Σ0),
with Σ0∈{z∈∂Ω:u~λ(z)=u~ϑ(z)}.
Proof.
Suppose that {λn}n⩾1⊆ℒ and λn→λ-.
From the proof of Proposition 4,
we know that
(4.47)the sequence{u~n=u~λn}n⩾1⊆D+is increasing,
while from the proof of Proposition 9,
we have
(4.48)u~n→uλ* in C1(Ω¯).
We claim that uλ*=u~λ.
Arguing by contradiction,
suppose that uλ*=u~λ.
Then we can find z0∈Ω such that
u~λ(z0)<uλ*(z0).
From (4.48) we see that we can find n0⩾1 such that
u~λ(z0)<u~n(z0) for all n⩾1,
which contradicts (4.47).
Therefore uλ*=u~λ and this proves the left continuity of
λ↦u~λ.
The strict monotonicity property is a consequence of Proposition 6.
∎
Next we show that for λ∈(0,λ*),
problem (($P_{\lambda}$)) has at least two positive solutions.
Proposition 12.
If Hypotheses H(β) and H1 hold and λ∈(0,λ*),
then
problem (($P_{\lambda}$)) admits at least two positive solutions
u0,u^∈D+,u0⩽u^,u0≠u^.
Proof.
Since λ∈ℒ,
we have a positive solution u0∈S(λ).
On account of Proposition 10,
we can find u0=u~λ∈D+
(the minimal positive solution of (($P_{\lambda}$))).
We consider the following truncation of f(z,⋅):
(4.49)k(z,ζ)={f(z,u0(z))if ζ⩽u0(z),f(z,ζ)if u0(z)<ζ.
This is a Carathéodory function.
We set
K(z,ζ)=∫0ζk(z,s)𝑑s
and consider the C1-functional
dλ:W1,p(Ω)→ℝ defined by
dλ(u)=1pγ(u)-λ∫ΩF(z,u)𝑑z for all u∈W1,p(Ω).
Claim 1.
The functional dλ satisfies the Cerami condition.
Let {un}n⩾1⊆W1,p(Ω) be a sequence such that
(4.50)|dλ(un)|⩽M3 for all n⩾1,
(4.51)(1+∥un∥)dλ′(un)→0 in W1,p(Ω)*
for some M3>0.
From (4.51) we have
(4.52)|〈A(un),h〉+∫∂Ωβ(z)|un|p-2un𝑑z-∫Ωλk(z,un)h𝑑z|⩽εn∥h∥1+∥un∥ for all h∈W1,p(Ω),
with εn↘0.
In (4.52) we choose h=-un-∈W1,p(Ω).
Then
γ(un-)-λ∫Ωf(z,un)(-un-)𝑑z⩽εn,
so
∥un-∥p⩽c20 for all n⩾1
for some c20=c20(λ)>0
(see Proposition 4 and Hypothesis H1 (i)),
thus
(4.53)the sequence{un-}n⩾1⊆W1,p(Ω)is bounded.
Next in (4.52) we choose h=un+∈W1,p(Ω).
Then
(4.54)-γ(un+)+λ∫Ωk(z,un+)un+𝑑z⩽εn for all n⩾1.
On the other hand from (4.50) and (4.53) we have
(4.55)γ(un+)-λ∫ΩpK(z,un+)𝑑z⩽M4 for all n⩾1
for some M4>0.
Adding (4.54) and (4.55), we obtain
λ∫Ω(k(z,un+)un+-pK(z,un+))𝑑z⩽M5 for all n⩾1
for some M5>0, so
(4.56)λ∫Ω(f(z,un+)un+-pF(z,un+))𝑑z⩽M6 for all n⩾1
for some M6>0
(see (4.49)).
Using (4.56) and reasoning as in the proof of Proposition 3.1
(see the part of the proof after (4.28)), we show that
the sequence{un+}n⩾1⊆W1,p(Ω)is bounded,
thus
(4.57)the sequence{un}n⩾1⊆W1,p(Ω)is bounded.
Because of (4.57) we may assume that
(4.58)un→𝑤u in W1,p(Ω) and un→u in Lr(Ω)and inLp(∂Ω).
In (4.52) we choose h=un-u∈W1,p(Ω),
pass to the limit as n→+∞ and use (4.58).
Then
limn→+∞〈A(un),un-u〉=0,
so
un→u in W1,p(Ω)
(see Proposition 3), thus dλ
satisfied the Cerami condition.
This proves Claim 1.
Let u~λ*∈D+ be the minimal positive solution of the problem
(Pλ*)
(see Proposition 10).
From Proposition 11 we know that
(4.59)u~λ*-u0∈D^+(Σ0),
with Σ0={z∈∂Ω:u~λ*(z)=u0(z)}
(recall that u0=u~λ).
For this Σ0 we consider the space
W*1,p(Ω)=C*1(Ω¯)¯∥⋅∥
and the functional
dλ0:W*1,p(Ω)→ℝ defined by
dλ0(⋅)=dλ(u0+⋅)|W*1,p(Ω).
Claim 2.
We may assume that v=0 is a local minimizer of dλ0.
Consider the Carathéodory function
(4.60)k^0(z,ζ)={k(z,u0(z)+ζ)if ζ⩽u~λ*(z)-u0(z),k(z,u~λ*(z))if u~λ*(z)-u0(z)<ζ
(see (4.59)).
We set
K^0(z,ζ)=∫0ζk^0(z,s)𝑑s
and consider the C1-functional
d^λ0:W*1,p(Ω)→ℝ defined by
d^λ0(v)=1pγ(u0+v)-λ∫ΩK^0(z,v)𝑑z for all v∈W*1,p(Ω).
Evidently, from Proposition 4 and (4.60) we have that
d^λ0 is coercive.
Also, it is sequentially weakly lower semicontinuous.
Hence we can find
v0∈W*1,p(Ω) such that
(4.61)d^λ0(v0)=infv∈W*1,p(Ω)d^λ0(v).
From (4.61) we have
(d^λ0)′(v0)=0,
so
(4.62)〈A(u0+v0),h〉+∫∂Ωβ(z)|u0+v0|p-2(u0+v0)h𝑑σ=λ∫Ωk^0(z,v0)h𝑑z for all h∈W*1,p(Ω).
In (4.62) we choose
h=(u0-(u0+v0))+∈W*1,p(Ω).
Then
〈A(u0+v0),(u0-(u0+v0))+〉+∫∂Ωβ(z)|u0+v0|p-2(u0+v0)(u0-(u0+v0))+𝑑σ
=λ∫Ωf(z,u0)(u0-(u0+v0))+𝑑z
=〈A(u0),(u0-(u0+v0))+〉+∫∂Ωβ(z)u0p-1(u0-(u0+v0))+𝑑σ
(see (4.60), (4.49) and use the fact that u0∈S(λ)),
so
〈A(u0)-A(u0+v0),(u0-(u0+v0))+〉+∫∂Ωβ(z)(u0p-1-|u0+v0|p-2(u0+v0))(u0-(u0+v0))+𝑑σ=0,
thus
u0⩽u0+v0.
Also in (4.62) we choose h=(u0+v0-u~λ*)+∈W*1,p(Ω).
Then
〈A(u0+v0),(u0+v0-u~λ*)+〉+∫∂Ωβ(z)|u0+v0|p-1(u0+v0-u~λ*)+𝑑σ
=λ∫Ωf(z,u~λ*)(u0+v0-u~λ*)+𝑑z
⩽λ*∫Ωf(z,u~λ*)(u0+v0-u~λ*)+𝑑z
=〈A(u~λ*),(u0+v0-u~λ*)+〉+∫∂Ωβ(z)u~λ*p-1(u0+v0-u~λ*)+𝑑σ,
so
〈A(u0+v0)-A(u~λ*),(u0+v0-u~λ*)+〉+∫∂Ωβ(z)((u0+v0)p-1-u~λ*p-1)(u0+v0-u~λ*)+𝑑σ=0,
thus
u0+v0⩽u~λ*.
So, we have proved that
u0+v0∈[u0,u~λ*],
thus
u0+v0∈S(λ)⊆D+
(see (4.60) and (4.49)).
If u0+v0≠u0
(that is, v0≠0),
then this is the desired second positive solution of (($P_{\lambda}$)) and so we are done.
If u0+v0=u0
(that is, v0=0),
then since
dλ0|[0,u~λ*-u0]=d^λ0|[0,u~λ*-u0]
(see (4.60) and (4.49)) and
u~λ*-u0∈D^+(Σ0), we have
v=0is a localC*1(Ω¯)-minimizer ofdλ0,
hence
v=0is a localW*1,p(Ω)-minimizer ofdλ0
(see Proposition 2).
This proves Claim 2.
Note that from Claim 1 it follows that
(4.63)dλ0 satisfies the Cerami condition.
Also, using (4.49) we can easily check that
Kdλ⊆[u0),
where [u0)={u∈W1,p(Ω):u0(z)⩽u(z)for a.a.z∈Ω}.
From this we see that we may assume that Kdλ is finite
(otherwise we already have an infinity of positive solutions all bigger than u0
and smooth by the regularity theory and so we are done).
Hence Kdλ0 is finite.
Thus on account of Claim 2 we can find ϱ∈(0,1) small such that
(4.64)0=dλ0(0)<inf{dλ0(v):∥v∥=ϱ}=mλ0
(see Aizicovici–Papageorgiou–Staicu [1, Proposition 29]).
Hypothesis H1 (ii) implies that for every y∈D^+(Σ0) we have
(4.65)dλ0(ty)→-∞ as t→+∞.
Then (4.63), (4.64), (4.65) permit the use of
the mountain pass theorem (see Theorem 1) and so we can find v^∈W*1,p(Ω), v^≠0
such that
u^=u0+v^∈C1(Ω)
(nonlinear regularity theory) and
u0⩽u^,u0≠u^
(see (4.64)) and
u^ solves (($P_{\lambda}$))
(by the nonlinear Green’s identity in W*1,p(Ω)).
∎
Summarizing our findings thus for,
we can state the following bifurcation-type theorem.
Theorem 13.
If Hypotheses H(β) and H1 hold
then:
There exists λ*∈(0,+∞) such that
for every λ∈(0,λ*) problem (($P_{\lambda}$)) has at least two positive solution
u0,u^∈D+,u0⩽u^,u0≠u^,
for λ=λ* problem (Pλ*) has at least one positive solution
u*∈D+,
for λ>λ* problem (($P_{\lambda}$)) has no positive solutions.
For every λ∈ℒ=(0,λ*], problem (($P_{\lambda}$)) has a smallest positive solution
u~λ∈D+
and the map λ↦u~λ from ℒ=(0,λ*] into C1(Ω¯)
is left continuous and strictly increasing in the sense that
if 0<ϑ<λ,
λ∈ℒ,
then u~λ-u~ϑ∈D^+(Σ0)with Σ0={z∈∂Ω:u~λ(z)=u~ϑ(z)}.
We can have a similar bifurcation-type theorem
if we replace Hypothesis H1 (iv) by a different one,
which strengthens the monotonicity of f(z,⋅) but on the other hand allows for zero cores.
So, the new hypotheses on the nonlinearity f are the following:
Hypotheses H2.
The function
f:Ω×ℝ→ℝ is a Carathéodory function such that
f(z,0)=0 for almost all z∈Ω,
hypotheses (i)–(iii) are the same as the corresponding hypotheses H1 (i)–(iii)
and
for almost all z∈Ω, the map ζ↦f(z,ζ) is nondecreasing.
Example 14.
Let η∈L∞(Ω) and Ω0={z∈Ω:η(z)=0}.
We assume that |Ω0|N is positive and consider the Carathéodory function f defined by
f(z,ζ)=η(z)(ζp-1ln(1+ζ)+ζq-1) for all ζ⩾0,
with 1<q<p.
Then f satisfies Hypotheses H2 but not Hypotheses H1.
From the previous analysis, the only thing that has to be verified is Proposition 6.
Proposition 15.
If Hypotheses H(β) and H2 hold, λ∈L, ϑ∈(0,λ) and uλ∈S(λ)⊆D+,
then
we can find uϑ∈S(ϑ)⊆D+ such that
uλ-uϑ∈D^+(Σ0),
with Σ0={z∈∂Ω:uλ(z)=uϑ(z)}.
Proof.
From Proposition 4 and its proof, we know that we can find
uϑ∈S(ϑ)⊆D+ such that
uϑ⩽uλ and uϑ≠uλ.
We set h1(z)=f(z,uϑ(z)) and h2(z)=f(z,uλ(z)).
Then
h1,h2∈L∞(Ω) and h1(z)⩽h2(z) for a.a.z∈Ωwithh1≠h2
(see Hypothesis H2 (iv) and recall that uϑ≠uλ).
Then we have
-Δpuϑ=ϑf(z,uϑ(z))=ϑh1(z)⩽λh1(z)⩽λh2(z)=λf(z,uλ(z))=-Δpuλ(z) for a.a.z∈Ω.
Invoking Proposition 4, we conclude that
uλ-uϑ∈D^+(Σ0),
with Σ0={z∈∂Ω:uλ(z)=uϑ(z)}.
∎
We can state the following variant of Theorem 13.
Theorem 16.
If Hypotheses H(β) and H2 hold, then:
There exists λ*∈(0,+∞) such that
for every λ∈(0,λ*) problem (($P_{\lambda}$)) has at least two positive solution
u0,u^∈D+,u0⩽u^,u0≠u^,
for λ=λ* problem (Pλ*) has at least one positive solution
u*∈D+,
for λ>λ* problem (($P_{\lambda}$)) has no positive solutions.
For every λ∈ℒ=(0,λ*], problem (($P_{\lambda}$)) has a smallest positive solution
u~λ∈D+
and the map λ↦u~λ from ℒ=(0,λ*] into C1(Ω¯)
is left continuous and strictly increasing in the sense that
if 0<ϑ<λ,
λ∈ℒ,
then u~λ-u~ϑ∈D^+(Σ0)
with Σ0={z∈∂Ω:u~λ(z)=u~ϑ(z)}”.
and
Nikolaos S. Papageorgiou
