Department of Mathematics University of Trento
Via Sommarive 14, 38123 Povo (TN) - Italy alessio.marinelli@unitn.it
Dimitri Mugnai
Department of Mathematics and Computer Sciences University of Perugia
Via Vanvitelli 1, 06123 Perugia - Italy dimitri.mugnai@unipg.it
Abstract
We consider an elliptic problem driven by the square root of the Lapla- cian in presence of a general logistic function having an indefinite weight.
We prove a bifurcation result for the associated Dirichlet problem via reg- ularity estimates of independent interest when the weight belongs only to certain Lebesgue spaces.
Keywords: Indefinite potential, principal eigenfunction, nonlinear regularity, existence and multiplicity theorems, nonlinear maximum principle
2000AMS Subject Classification: 35J20, 35J65, 58E05
1 Introduction
One of the most popular equations in Biology, and in particular in population dynamics, is the logistic equation, which we write as
cut(x, t) = ∆u(x, t) + λ(β(x)u(x, t) − u2(x, t)) x ∈ Ω ⊂ RN, t > 0. (1) Here Ω is a domain in RN where the population having nonnegative density u lives and c, λ are positive constants. However, it is well known that the heat operator may be too rigid to describe the possible interaction of the specie in the whole of Ω (for instance, see [5], [6], [19], [21] and the references therein), and for this fact a nonlocal operator may be more useful than a local one.
Moreover, a first natural step consists in studying steady states for these new equations (like in [5]). For this reason, in this paper, the diffusion is not decribed by classical Laplacian operator (or its generalizations), but by the square root of −Laplacian, √
−∆. This operator can be defined in several ways, and we
1
follow the approach developed in the pioneering works of Caffarelli, Silvestre [10], Caffarelli, Vasseur [11], Cabr´e, Tan [9], to which we refer in Section 2 for the precise mathematical description and properties. Here, we only say that it is becoming more and more popular for its applications in Physics, Probability, Finance and Biology. In particular, we recall the pure periodic logistic fractional equation in RN studied in [7] - where the behaviour of level set of solutions is investigated - , the evolution Fisher-KKP equation in RN driven by the generator of a Feller semigroup (for instance,√
−∆) studied in [8] - where the authors prove that the stable state invades the unstable one with a front position which grows exponentially in time - , and the general fractional porous medium equation in RN studied in [13] - where existence and uniqueness are established.
The problem we are concerned with has homogeneous Dirichlet boundary conditions, so that the population is confined in a bounded domain Ω ⊂ RN, N ≥ 2. Hence, we shall study the following problem:
(√−∆u(x) = λ(β(x)u(x) − g(x, u(x))) in Ω
u(x) = 0 on ∂Ω. (Pλβ)
Here the nonlinear term g(x, u) is a self–limiting factor for the population which generalizes the classical quadratic nonlinearity in (1), and whose properties will be introduced later on; we only emphasize the fact that, unlike similar variational problems, we don’t impose any growth condition at infinity for g.
The weight β : Ω −→ R corresponds to the birth rate of the population and, as main contribution of this paper, we suppose that it is sign changing and (generally) unbounded, so that both contributions to development and limitation of the population are possible. To our best knowledge, no results are known for unbounded sign changing weights, while this situation is clearly possible. Under this setting, we will establishes a relation between the parameter λ > 0 and the existence of a nontrivial stationary solution of the problem, showing that if λ > 0 is large, non trivial steady state are possible, though the birth rate β is mainly negative in the domain (as it happens in regions with natural or artificial difficulties which prevent proliferation); on the other hand if λ is small, the population is not safely protected in regions of disadvantage and steady state cannot exist (see Theorem 4.1). Let us also remark that λ large corresponds to a small diffusion and λ small corresponds to a large diffusion in (1).
We recall that logistic–type equations have been extensively studied and, in addition to the papers cited above, any bibliography would be incomplete. We only add [20], where steady states of equation (1) where found when Ω = RN and β is sign–changing. Hence our result, is the counterpart of the results therein in the case of problem (Pλβ).
The paper is organized as follows. Section 2 introduces the mathematical background we shall need in the following sections. In particular we recall some basic facts on the square root of the Laplacian and some fundamental inequalities.
In Section 3 we will introduce the main character of this paper (i.e. the bifurcation parameter λ∗), and we will prove a general maximum principles (see Proposition 3.2) and a regularity result (see Proposition 3.3) of independent interest. In the last section we will prove a bifurcation theorem for problem (Pλβ): via critical
point theory and truncation techniques we will show that problem (Pλ) admits a nontrivial solution if and only if λ > λ∗, and, with additional but natural conditions, we will show that nontrivial solutions bifurcate from λ∗(see Theorems 4.1 and 4.2).
2 Mathematical Background
In this section we briefly recollect some definitions and inequalities we will use throughout the paper.
From now on, Ω ⊆ RN will denote a bounded smooth domain, and we denote by C = Ω × (0, +∞) the half cylinder with base Ω × {0} - which we shall identify with Ω - , and lateral boundary ∂LC = ∂Ω × [0, +∞).
Then, we introduce the Sobolev Space
H0,L1 (C) =v ∈ H1(C) | v = 0 a.e. on ∂LC , equipped with the norm
kvkH1 0,L(C) =
Z
C
|Dv(x, y)|2dxdy
12 , induced by the scalar product
Z
C
Dv · Dz dxdy, v, z ∈ H0,L1 (C).
As usual, a crucial rˆole is played by trace operator
T rΩ: H0,L1 (C) −→ H12(Ω), (2) which is a continuous map (see [9, Lemma 2.6]), and which gives several infor- mation, which we recall in the following. In particular, we will repeatedly use the following
Lemma 2.1 (Sobolev trace inequality, Lemma 2.3, [9]). Let N ≥ 2 and 2] =
2N
N −1. Then there exists a constant C = C(N ), such that, for all v ∈ H0,L1 (C)
Z
Ω
|v(x, 0)|2]dx
1
2] ≤ C
Z
C
|Dv(x, y)|2dxdy
12
. (3)
From Lemma 2.1 it is immediate to prove
Lemma 2.2 (Lemma 2.4, [9]). Let N ≥ 2 and 1 ≤ q ≤ 2]= N −12N . Then there exists a positive C = C(N, q, |Ω|), such that, for all v ∈ H0,L1 (C)
Z
Ω
|v(x, 0)|qdx
1q
≤ C
Z
C
|Dv(x, y)|2dxdy
12
. (4)
If N = 1, inequality (4) holds for all q ∈ [1, ∞).
Moreover, a stronger useful relation, which will be frequently used, is given by the following compact embedding:
Lemma 2.3 (Lemma 2.5,[9]). Let 1 ≤ q < 2] if N ≥ 2 and q ∈ [1, ∞) if N = 1;
then
T rΩ(H0,L1 (C)) ,→,→ Lq(Ω).
The trace operator in (2) permits to introduce the trace space on Ω × {0} of functions in H0,L1 (C) as
V0(Ω) :=u = T rΩv | v ∈ H0,L1 (C) ⊂ H12(Ω), equipped with the norm
kukV
0(Ω)=
kuk2
H12 + Z
Ω
u2 d(x)
12 ,
where d(x) = dist(x, ∂Ω). However, there exists another characterization of V0(Ω):
Lemma 2.4 ( Lemma 2, 7, [9]). The following identity holds:
V0(Ω) =
u ∈ H12(Ω) | Z
Ω
u2
d(x)dx < ∞
.
Here, and in the following, we have denoted by x a generic point of Ω and by y a generic point in (0, ∞).
Next we introduce the following minimizing problem: fix u ∈ V0(Ω) and consider
inf
Z
C
|Dv|2dxdy | v ∈ H0,L1 (C), v(., 0) = u in Ω
. (5)
Proposition 2.1 (Lemma 2.8, [9]). For every u ∈ V0(Ω) there exists a unique minimizer v of (5), called the harmonic extension of u to C which vanishes on ∂LC.
Such a function is denoted by v =h-ext(u).
Remark 2.1. If {ek}k is the orthonormal frame of L2(Ω) obtained by the eigen- functions of −∆ with homogeneous Dirichlet boundary conditions and with asso- ciated eigenvalues {λk}k, given u ∈ V0(Ω), we can write u =
∞
X
k=1
bkek(x). Hence, by Proposition 2.1, we get that v =h-ext(u) ∈ H0,L1 (C) is given by
v(x, y) =
∞
X
k=1
bkek(x)e−λ
1 2 ky.
Finally, let us define the square root of the Laplacian (or half-Laplacian) in a bounded domain Ω of RN. For any continuous function u in ¯Ω, there exists a unique function v such that
∆Cv = 0 in C
v(x, 0) = u(x) on Ω × {0} ' Ω,
v = 0 on ∂LC,
(6)
where ∆C denotes the Laplacian operator in C, so that v is the generalized har- monic extension of u in C which is 0 on ∂LC.
Now, consider the operator T defined as T u(x) := −∂v
∂y(x, 0). (7)
It is readily seen that the system
∆Cw = 0 in C
w(x, 0) = −∂v∂y(x, 0) = T u(x) on Ω × {0} ' Ω,
w = 0 on ∂LC,
admits the unique solution w(x, y) = −∂v∂y(x, y), and thus by (6) we immediately have
T (T u)(x) = −∂w
∂y(x, 0) = ∂2v
∂y2(x, 0) = (−∆v)(x, 0).
In conclusion, T2= ∆, i.e the operator T mapping the Dirichlet datum u to the Neumann datum −∂v∂y(·, 0) is a square root of the Laplacian −∆ in Ω.
In light of the previous “Dirichlet to Neumann” procedure, by Remark 2.1 we have that
√−∆u(x) = ∂v
∂ν
Ω×{y=0}
= −∂v
∂y
Ω×{y=0}
=
∞
X
k=1
bkλ1/2k ek(x),
where ν = −y. Moreover, it is natural to give the following Definition 2.1. We say that u ∈ H01/2(Ω) is a weak solution of
(√−∆u = g(x, u) in Ω
u = 0 su ∂Ω, (*)
whenever the harmonic extension v ∈ H0,L1 (C) of u is a weak solution of
∆v = 0 in C
v = 0 on ∂LC
∂v
∂ν = g(x, u) on Ω × {0} ,
(**)
that is if Z
C
Dv · Dξ dxdy = Z
Ω×{0}
g(x, v(x, 0))ξ(x, 0) dx ∀ξ ∈ H0,L1 (C).
An important regularity result for√
−∆ is available:
Theorem 2.1 (Theorem 1.9, [9])). Let Ω be a bounded domain of RN of class C2,α, α ∈ (0, 1) . If g ∈ V0(Ω)∗ and u is a weak solution of
(√−∆u = g(x) in Ω
u = 0 on ∂Ω, (P)
then:
1. If g ∈ L2(Ω), then u ∈ H01(Ω).
2. If g ∈ H01(Ω), then u ∈ H2(Ω) ∩ H01(Ω).
3. If g ∈ L∞(Ω), then u ∈ Cα(Ω).
4. If g ∈ Cα(Ω) and g|∂Ω≡ 0, then u ∈ C1,α(Ω).
5. If g ∈ C1,α(Ω) and g|∂Ω≡ 0, then u ∈ C2,α(Ω).
Of course, applying Theorem 2.1 to weak solutions of (√−∆u = f (x, u) in Ω
u = 0 on ∂Ω, (Q)
we can obtained related results for nonlinear problems.
3 The bifurcation parameter
We consider the problem
(√−∆u = λ(βu − g(x, u)) in Ω
u = 0 on ∂Ω, (Pλβ)
where N ≥ 2, λ ∈ R, Ω ⊂ RN is a bounded domain of class C2,α for some 0 < α ≤ 1, g : Ω × R −→ R is a Carath´eodory perturbation (i.e., for all s ∈ R the map x 7→ g(x, s) is measurable and for a.e. x ∈ Ω the map x 7→ g(x, s) is continuous). Concerning the function β, we assume that it is a sign changing measurable weight (for a first presentation of such classes of weights, we refer to [24]).
In view of the considerations of the previous section, studying problem (Pλβ) is equivalent to studying the extension problem in the cylinder
−∆v = 0 in C
v = 0 on ∂L(C)
∂v
∂ν = λ(βv − g(x, v)) in Ω × {0} .
(Qβλ)
Of course, (Qβλ) has a variational structure, so that v solves (Qβλ) if and only if it is a critical point for the functional I : H0,L1 (C) → R defined as
I(u) =1 2
Z
C
|Du|2dxdy − λ Z
Ω×{0}
F (x, u) dx, (8)
where
F (x, u) = Z u
0
[βs − g(x, s)]ds.
From now on, all integrals of the form Z
Ω×{0}
will be simply denoted by Z
Ω
.
Now, we set λ∗= inf
Z
C
|Du|2 dxdy : u ∈ H0,L1 (C) with Z
Ω
βu2= 1
. (9)
In order to ensure that λ∗ is well defined, we impose the following condition on the weight function β:
H(β): β ∈ Lq(Ω) with q > N and β+= max{β, 0} 6= 0.
Remark 3.1. The assumption β+= max{β, 0} 6= 0 implies that the set
u ∈ H0,L1 (C) with Z
Ω
βu2= 1
is not empty, so that λ∗ is well defined.
Proposition 3.1. If hypothesis H(β) holds, then λ∗> 0.
Proof. Let u ∈ H0,L1 (C) be such that R
Ωβu2 = 1. Since u ∈ H0,L1 (C), then u ∈ L2](Ω), and we note that
1 N + 1
N N −1
= 1
N +N − 1 N = 1.
Then, using H¨older’s inequality and Lemma 2.1, we get 1 =
Z
Ω
βu2dx ≤ β+
LN(Ω)kuk2L2](Ω)≤ |Ω|q−NqN β+
Lq(Ω)kuk22](Ω)
≤ C1
β+
Lq(Ω)kDuk2L2(C)
for some universal constant C1> 0.
Hence, inf
u∈ H01(C)
Z
C
|Du|2 dxdy
= λ∗≥ (C1 β+
Lq(Ω))−1> 0.
We introduce the following general maximum principles.
Proposition 3.2. Let Ω be a smooth bounded domain of RN and let u ∈ C1(Ω) be a solution of
√−∆u + B(x, u, Du) ≥ 0 in Ω
u ≥ 0 in Ω
u = 0 on ∂Ω,
with B = B(x, s, ξ) of class C1in ξ, B ∈ Liploc in s and such that B(x, 0, 0) = 0 for a.e. x ∈ Ω. Then u > 0 in Ω or u ≡ 0 in Ω.
Proof. Let v =h-ext(u) be the harmonic extension of u, so that v satisfies
−∆v = 0 in C,
v ≥ 0 in C,
v = 0 on ∂L(C),
∂v
∂ν ≥ −B(x, v(x, 0), Dv(x, 0)) in Ω.
Suppose by contradiction that u is not identically zero and that there exists x0∈ Ω such that u(x0) = 0, so that v(x0, 0) = 0. By [22, Theorem 2.7.1] we find
∂νv(x0, 0) < 0. (10)
By (10), the assumptions on B and the equivalence between problem (P ) and (P0), we find
0 >√
−∆u(x0) ≥ −B(x0, u(x0), Du(x0)) = 0, and a contradiction arises.
Remark 3.2. In the case of weak solution, the previous result still holds true by [22, Theorem 5.4.1].
Next, let us consider the problem
(√−∆u = λ∗βu in Ω
u = 0 on ∂Ω. (Pλ∗)
We start with the following regularity result (see [16] for a related result in RN).
Proposition 3.3. Let Ω be a smooth bounded domain of RN and let u be a weak solution of (Pλ). If H(β) holds, then u ∈ Cα(Ω).
Proof. Set v =h-ext(u), so that v ∈ H0,L1 (C) solves
−∆v = 0 in C v = 0 on ∂L(C)
∂v
∂ν = λβv in Ω × {0} .
Now, we will use the Moser iteration technique. If M > 0 we define vM = min {v+, M } and thus vM ∈ H0,L1 (C) ∩ L∞(C). For any nonnegative number k we set φ = vM2k+1, so that Dφ = (2k + 1)v2kMDvM.
By definition of weak solution with φ = v2k+1M as test function, we get (2k + 1)
Z
C
vM2kDv · DvMdxdy = λ Z
Ω
βvvM2k+1dx, that is
(2k + 1) Z
C
vM2k|DvM|2dxdy = λ Z
Ω
βv2k+2M dx. (11)
We note that (2k + 1)
Z
C
vM2k|DvM|2dxdy =(2k + 1)
(k + 1)2 < DvMk+1, Dvk+1M >
=(2k + 1) (k + 1)2
Z
C
|DvMk+1|2dxdy.
(12)
In addition, by applying the H¨older inequality to the right-hand side of (11), we obtain
λ Z
Ω
βv2k+2M dx ≤ λ β+
Lq(Ω)
(v+)k+1
2
L2q0(Ω). (13) By applying (12) and (13), from (11) one gets
(2k + 1) (k + 1)2
Z
C
|DvMk+1|2dxdy ≤ λ β+
q
(v+)k+1
2
2q0. (14)
Since lim
M →∞vM(x, y) = v+(x, y) for a.e. (x, y) ∈ C, applying Fatou’s Lemma, we get
(v+)k+1 H1
0,L(C)≤ k + 1
√2k + 1
√ λ
β+
1/2 q
(v+)k+1
2q0. (15) By Lemma 2.1, we get the existence of S > 0 such that
(v+)k+1
2] ≤ k + 1
√2k + 1S√ λ
β+
1/2 q
(v+)k+1
2q0, (16) where, with some abuse of notation, we have set kvkLq(Ω) = kv(·, 0)kLq(Ω×{0})
for any v ∈ H0,L1 (C) and all q ≥ 1.
Since q > N , then 2q0< 2]. Now, define k0= 0 and by recurrence 2(kn+ 1)q0= (kn−1+ 1)2].
It is easy see that ki > ki−1 and that kn → ∞ as n → ∞. Moreover, (16) implies that
if (v+) ∈ L2(ki−1+1)q0(Ω) then (v+) ∈ L(ki−1+1)2](Ω) = L2(ki+1)q0(Ω), and hence (v+) ∈ Lr(Ω) for every r ≥ 1. Moreover, by (15) we obtain that
Z
C
|D(v+)ki+1|2≤ Ci for every i ≥ 0, or equivalently (v+)ki+1∈ H0,L1 (C) for every i ≥ 0.
Analogously, choosing vM := min{M, v−}, we can prove that v− ∈ Lr(Ω) for every r ≥ 1.
Now, since v ∈ Lr(Ω) for all r < ∞, then βv ∈ Lq(Ω) with q > q > N . Now, following the proof of Proposition 3.1 (iii) in [9], for (x, y) ∈ C we introduce the function w(x, y) = Ry
0 v(x, t)dt, which turns out to solve the homogeneous Dirichlet problem
(−∆w(x, y) = λβ(x)v(x, 0) in C,
w = 0 on ∂C.
Then, performing the odd reflection woddof w in Ω×R, by the Calder´on-Zygmund inequality, we obtain that wodd ∈ W2,q(Ω × (−R, R)) for every R > 0. In particular, w ∈ C1,α( ¯Ω), and so, by the very definition of v = wy, we get that v ∈ Cα(C), and so
u ∈ Cα(Ω).
Now, let us consider
C+=u ∈ Cα(Ω) : u(x) ≥ 0 ∀x ∈ Ω , whose interior set is
int C+=u ∈ Cα(Ω) : u(x) > 0 ∀x ∈ Ω . Let e1 be the first eigenfunction of √
−∆ with associated first eigenvalue λ∗ and L2 weight β, i.e. a solution of problem (Pλ∗). We first prove that e1 does exist. In fact, we have
Proposition 3.4. If H(β) holds, then there exists e1∈ H01/2(Ω) ∩ int C+ which is the first eigenfunction with associated eigenvalue λ∗ of problem (Pλ∗).
Proof. Let us consider the functional J (u) = 1
2 Z
C
|Du|2dxdy constrained on the set
M =
u ∈ H0,L1 (C) Z
Ω
βu2dx = 1
,
and note that it is sequentially weakly lower semicontinuous and coercive; so by the Weierstrass Theorem we can find e1∈ H0,L1 (C) such that
J (e1) = infn
J (u)| u ∈ H0,L1 (C)o . Thus, there exists λ∗∈ R such that J0(e1)v = λ∗R
Ωβe1vdx for every v ∈ H0,L1 (C), that is e1 is the first eigenfunction for the problem
(∆e1= 0 in C,
∂e1
∂ν = λ∗βe1 in Ω.
Moreover, it is clear that we can assume e1≥ 0 in Ω, and by Proposition 3.3 we get that e1∈ C+.
Finally, by Remark 3.2, e1∈ int C+.
In general no other properties can be deduced for e1if no other hypotheses on β are given. However, we can recover the classical features of the first eigenvalue with additional assumptions. In particular, from Theorem 2.1.4-5 we have the following regularity result:
Proposition 3.5. If β ∈ C1,α(Ω), then e1∈ C2,α(Ω).
4 The bifurcation theorem
On the absorption term g we assume that
H(g): g : Ω×R −→ R is a Carath´eodory function such that g(x, s) = 0 for a.e. x ∈ Ω and for all s ≤ 0 and g(x, s) ≥ 0 for a.e. x ∈ Ω and for all s > 0. Moreover:
1. there exists g0∈ L∞(Ω) such that:
lim inf
s→∞
g(x, s)
s ≥ g0(x) uniformly for a.e. x ∈ Ω (17) and
ess inf
Ω (g0− β) = µ > 0. (18)
2.
lim
s→0+
g(x, s)
s = 0 uniformly for a.e. x ∈ Ω (19) 3. for every u ∈ L∞(Ω) there exists ρ = ρ(u) ≥ 0 such that
|g(x, u(x))| ≤ ρ for a.e. x ∈ Ω. (20) 4.
For a.e. x ∈ Ω the function u 7→ g(x, u)
u is strictly increasing in (0, ∞).
(21) Remark 4.1. Hypothesis (17) does not exclude the classical case, that is
lim inf
s→∞
g(x, s) s = ∞, as it happens, for example, if g(x, s) = |s|p−2s, p > 2.
Remark 4.2. Hypothesis (20) is very general and does exclude any a priori bound on the growth of the function at infinity, as it is usual in variational problems of this type.
We shall prove the following result:
Proposition 4.1. If hypotheses H(β), H(g), (17), (18), (19) and (20) hold, then for all λ > λ∗ there exists a solution u ∈ Cα( ¯Ω) of problem (Pλβ) such that u(x) > 0 for all x ∈ Ω.
Proof. By virtue of hypothesis (17), it follows that for all ∈ (0, µ) there exists M = M () > 0 such that for all s > M
g(x, s) ≥ (g0(x) − )s for a.e. x ∈ Ω. (22) Now we fix ψ > M and we consider the following truncation for the reaction term:
h(x, s) =
0 if s ≤ 0
β(x)s − g(x, s) if 0 ≤ s ≤ ψ β(x)ψ − g(x, ψ) if s ≥ ψ.
(23)
Of course, h is still a Carath´eodory function. Setting H(x, s) =Rs
0 h(x, s)ds, by H(g) the functional Iψ: H0,L1 (C) −→ R defined by
Iψ(v) = 1
2kDvk2L2(C)− λ Z
Ω
H(x, v(x)) dx
is of class C1 and sequentially weakly lower semicontinuous. Moreover, by the very definition of the truncation h(x, s), we have that Iψ is coercive. Therefore, by the Weierstrass Theorem we can find v ∈ H0,L1 (C) such that
Iψ(v) = infIψ(v) | v ∈ H0,L1 (C) = m, (24) that is m is a critical value for Iψ.
Now we must check that v 6= 0. By virtue of hypothesis (19), in correspon- dence of the previous there exists δ = δ() > 0 with δ < min{M, ψ} such that
g(x, s) ≤ s ∀s ∈ [0, δ] and a.e. x ∈ Ω, so that
G(x, s) = Z s
0
g(x, s) ds ≤ s2
2 ∀s ∈ [0, δ] and a.e. x ∈ Ω. (25) By Proposition 3.4, we can find t ∈ (0, 1) such that te1∈ [0, δ] for all x ∈ Ω.
Then, by (25)
Iψ(te1) =t2
2||De1||22− λ Z
Ω
H(x, te1)dx
=t2
2 kDe1k22− λt2 2
Z
Ω
βe21dx + λ Z
Ω
G(x, te1) dx
≤t2
2 kDe1k22− λt2 2
Z
Ω
βe21dx +λ
2t2ke1k22
=t2
2(λ∗− λ) Z
Ω
βe21dx +λ
2t2ke1k22. Recalling thatR
Ωβe21dx = 1 and λ∗< λ, choosing > 0 sufficiently small, then we have Iψ(te1) < 0 and by (24) we find that
m = Iψ(v) < Iψ(0) = 0, (26) that is v 6= 0, as claimed.
Since v is a critical point for Iψ then Iψ0(¯v) = 0, that is
A(v) = λh(x, v), (27)
where A : H0,L1 (C) −→ (H0,L1 (C))0 is the linear map defined by
(A(u), v) = Z
C
Du · Dv dxdy for all u, v ∈ H01(C).
Now, we act on (27) with −(v)− ∈ H0,L1 (C), obtaining (A(v), −(v)−) =
Z
C
D(v)−
2 dxdy = 0,
so that v−= 0, that is v ≥ 0.
Next we act on (27) with (¯v − ψ)+∈ H0,L1 (C) and by definition of h we find (A(v), (v − ψ)+) = λ
Z
Ω
h(x, ¯v)(v − ψ)+dx =
= Z
Ω
(β ¯v − g(x, ¯v))(v − ψ)+dx ≤
≤ Z
Ω
(βψ − g(x, ψ))(v − ψ)+dx.
(28)
First, let us prove that Z
Ω
(βψ − g(x, ψ))(v − ψ)+dx ≤ 0. (29) Indeed, recalling that ψ > M and < µ, using (22), by (18) we have
βψ − g(x, ψ) ≤ βψ − g0(x)ψ + ψ ≤ ( − µ)ψ < 0. (30) Then, by (28),
Z
{x∈C : ¯v>ψ}
|D¯v|2dxdy ≤ 0, so that, from (28), (29) and (30), we find that
(βψ − g(x, ψ))(v − ψ)+= 0 in Ω.
Using again (30), this implies that (v − ψ)+ = 0 in Ω, that is ¯v ≤ ψ in Ω.
Therefore, v is not only a solution of the truncation problem, but also of problem (Qλ); then ¯u = trΩv is a solution of problem (P ). Finally, (20) and elliptic¯ regularity for Neumann problems in Lipschitz domains (for instance, see [18]) imply that v ∈ Cα( ¯C), and so u ∈ Cα( ¯Ω).
Proposition 4.2. If hypotheses H(β) and (21) hold, then the nontrivial nonneg- ative solution of problem (Pλβ) is unique.
Proof. We consider two nontrivial nonnegative solutions u, v ∈ Cα(Ω) of problem Pλβ and we test with u2u+−v2 and v2v+−u2, which clearly belong to H0,L1 (C). Then
Z
C
Du · Du2− v2
u + dxdy = λ Z
Ω
βuu2− v2 u + dx − λ
Z
Ω
g(x, u)u2− v2
u + dx (31) Z
C
Dv · Dv2− u2
v + dxdy = λ Z
Ω
βvv2− u2 v + dx − λ
Z
Ω
g(x, v)v2− u2
v + dx. (32) Now
Du2− v2
u + = u2+ 2u
(u + )2Du −2vDv
u + + v2
(u + )2Du, (33)
and an analogous form for Dv2u+−u2. Adding (31) and (32), we find
Z
C
Du · Du2− v2 u + dxdy +
Z
C
Dv · Dv2− u2 u + dxdy
= λ Z
Ω
β
v3
v + − vu2
v + + u3
u + − v2u u +
dx
− λ Z
Ω
g(x, u)u2− v2 u + dx − λ
Z
Ω
g(x, v)v2− u2 v + dx.
By (33) this reads as 0 =
Z
C
Du · u2+ 2u
(u + )2Du − 2vDv
u + + v2 (u + )2Du
dxdy
+ Z
C
Dv v2+ 2v
(v + )2Dv −2uDu
v + + u2 (v + )2Dv
dxdy + λ
Z
Ω
g(x, u)u2− v2 u + dx + λ
Z
Ω
g(x, v)v2− u2 v + dx
− λ Z
Ω
β
v3
v + − vu2
v + + u3
u + − v2u u +
dx
= Z
C
|D log(u + )|2+ |D log(v + )|2− 2 hD log(u + ), D log(v + )i (u2+ v2)dxdy
+ λ Z
Ω
g(x, u)u2− v2 u + dx + λ
Z
Ω
g(x, v)v2− u2 v + dx
− λ Z
Ω
β
v3
v + − vu2
v + + u3
u + − v2u u +
dx + 2
Z
C
h|D log(u + )|2u + |D log(v + )|2v − (u + v)hD log(v + ), D log(u + )ii dxdy, i.e.
Z
C
|D log(u + ) − D log(v + )|2(u2+ v2)dxdy + λ
Z
Ω
g(x, u)
u + −g(x, v) v +
(u2− v2)dx + 2
Z
C
h|D log(u + )|2u + |D log(v + )|2v − (u + v)hD log(v + ), D log(u + )ii dxdy
− λ Z
Ω
β
v3
v + − vu2 v + + u3
u + − v2u u +
dx = 0.
(34) Now, by (20) we have
g(x, u)
u + −g(x, v) v +
(u2− v2)
≤ ρ(kuk∞)
kuk∞ +ρ(kvk∞) kvk∞
|u2− v2| ∈ L1(Ω), and by the Lebesgue Theorem we immediately find that
Z
Ω
g(x, u)
u + −g(x, v) v +
(u2− v2)dx −→
Z
Ω
g(x, u)
u −g(x, v) v
(u2− v2)dx
as → 0.
Second, 2
Z
C
h|D log(u + )|2u + |D log(v + )|2v − (u + v)hD log(v + ), D log(u + )ii dxdy
≤ 2
|Du|2
(u + )2 + |Dv|2
(v + )2 + |Du||Dv|
(u + )(v + )
≤|Du|2
2 +|Dv|2
2 + |Du||Dv| ∈ L1(C),
and again the Lebesgue Theorem implies that the integral under consideration tends to 0 as → 0.
Moreover, it is easy to see that, similarly, Z
Ω
β
v3
v + − vu2
v + + u3
u + − v2u u +
dx → 0 as → 0.
In conclusion, passing to the lim inf in (34), Fatou’s Lemma implies that 0 ≥
Z
C
|D log(u) − D log(v)|2(u2+ v2)dxdy + λ Z
Ω
g(x, u)
u −g(x, v) v
(u2− v2)dx.
Since both integrals are nonnegative by (21), we immediately get that u = v, as desired.
Next we prove that:
Proposition 4.3. If hypotheses H(β) and H(g) hold, problem (Pλβ) has no pos- itive solutions when λ ∈ (0, λ∗] .
Proof. Suppose that there exists a positive solution v06= 0 for (Qβλ), that is Z
C
Dv0· Dv dxdy = λ Z
Ω
[βv0− g(x, v0)]v dx ∀ v ∈ H0,L1 (C).
In particular Z
C
|Dv0|2dxdy = λ Z
Ω
[βv02− g(x, v0)v0]dx < λ Z
Ω
βv02dx Comparing with the definition of λ∗, we have
λ∗ Z
Ω
βv02≤ Z
C
|Dv0|2 dxdy < λ Z
Ω
βv02dx, and we reach a contradiction, since λ ≤ λ∗.
In conclusion, we can state the following theorem:
Theorem 4.1. If hypotheses H(β), H(g), (17), (18), (19), (20), (21) hold, then there exists λ∗> 0 such that
