Similar results hold for the negative solutions and in this case we have a biggest negative solution

Electronic Journal of Differential Equations, Vol. 2020 (2020), No. 12, pp. 1–20.

ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu

POSITIVE AND NODAL SOLUTIONS FOR NONLINEAR NONHOMOGENEOUS PARAMETRIC NEUMANN PROBLEMS

NIKOLAOS S. PAPAGEORGIOU, CALOGERO VETRO, FRANCESCA VETRO

Abstract. We consider a parametric Neumann problem driven by a nonlinear nonhomogeneous differential operator plus an indefinite potential term. The reaction term is superlinear but does not satisfy the Ambrosetti-Rabinowitz condition. First we prove a bifurcation-type result describing in a precise way the dependence of the set of positive solutions on the parameterλ >0. We also show the existence of a smallest positive solution. Similar results hold for the negative solutions and in this case we have a biggest negative solution.

Finally using the extremal constant sign solutions we produce a smooth nodal solution.

1. Introduction

Let Ω ⊆R^N be a bounded domain with a C²-boundary ∂Ω. In this paper we study the following nonlinear nonhomogeneous Neumann problem

−diva(∇u(z)) + [ξ(z) +λ]u(z)^p−1=f(z, u(z)) in Ω,

∂u

∂n= 0 on∂Ω, u >0, λ >0, 1< p <+∞. (1.1) In this problem the mapa:R^N →R^N involved in the definition of the differential operator is strictly monotone and continuous, thus maximal monotone too. Also it satisfies certain other regularity and growth conditions listed in hypotheses (H1) (see Section 2). These conditions are not restrictive and incorporate in our frame- work many differential operators of interest such as thep-Laplacian (1< p <+∞) and the (p, q)-Laplacian (1< q < p <+∞), that is, the sum of ap-Laplacian and of a q-Laplacian. The differential operator of (1.1) is not homogeneous and this is a source of difficulties in the analysis of problem (1.1). There is also a para- metric potential termu→ [ξ(z) +λ]u^p−1 with the potential functionξ∈ L^∞(Ω) being indefinite (that is, sign-changing). Hence the left hand side of (1.1) is not in general coercive and this is another feature of problem (1.1) that complicates our arguments. The reaction term f(z, x) is a Carath´eodory function (that is, for all x∈R, z → f(z, x) is measurable and for a.a. z ∈Ω, x→f(z, x) is continuous).

We assume that for a.a.z∈Ω the functionx→f(z, x) is (p−1)-superlinear near +∞. However, the superlinearity off(z,·) is not expressed via the usual for such

2010Mathematics Subject Classification. 35J20, 35J60, 58E05.

Key words and phrases. Nonlinear nonhomogeneous differential operator;

nonlinear regularity theory; nonlinear maximum principle; strong comparison;

bifurcation-type theorem; nodal solution; critical group.

c

2020 Texas State University.

Submitted February 11, 2019. Published January 24, 2020.

1

problems Ambrosetti-Rabinowitz condition (the AR-condition for short). Instead we employ an alternative less restrictive condition which permits the consideration of (p−1)-superlinear nonlinearities with “slower” growth near +∞. Near 0⁺ we assume thatf(z,·) is (q−1)-superlinear with 1< q < p.

Using variational tools from the critical point theory together with suitable trun- cation, perturbation and comparison techniques, we prove a bifurcation-type result which describes the dependence on the parameterλ >0 of the set of positive solu- tions of problem (1.1). More precisely, we show that there exists a critical parameter valueλ_∗>0 such that

• for allλ > λ∗ problem (1.1) has at least two positive solutions;

• forλ=λ_∗ problem (1.1) has at least a positive solution;

• for allλ∈(0, λ_∗) problem (1.1) has no positive solution.

In addition we show that for everyλ∈ L= [λ_∗,+∞), problem (1.1) has a small- est positive solutionuλand we examine the monotonicity and continuity properties of the mapλ→uλ.

With the conditions valid on the negative semiaxisR−= (−∞,0], we can have analogous results for the negative solutions. In particular we can produce a biggest negative solutionvλ for problem (1.1). When the conditions are bilateral (that is, valid for allx∈Rand not only on the semiaxes), then using the two extremal con- stant sign solutionsu_λandv_λ, we produce a nodal (sign-changing) solution for prob- lem (1.1). Our work here continues and extends the ones by Motreanu-Motreanu- Papageorgiou [8], Averna-Papageorgiou-Tornatore [1] and Papageorgiou-Rˇadulescu [11]. In [8] the differential operator is also nonhomogeneous but the conditions on the mapa(·) are more restrictive excluding, for example, the important case of the (p, q)-Laplacian. Alsoξ≡0 and the authors do not prove the precise dependence on λ >0 of the set of positive solutions (bifurcation-type result). In [1] the differential operator is thep-Laplacian andξ≡0. The authors do not prove the existence of nodal solutions. Finally in [11] the equation is semilinear driven by the Laplacian, but the boundary condition is Robin. It is an interesting open problem whether we can extend our work here to Robin boundary value problems.

2. Mathematical Background - Hypotheses

In the analysis of problem (1.1) we will use the Sobolev spaceW^1,p(Ω) and the Banach spaceC¹(Ω). Byk · kwe denote the norm ofW^1,p(Ω) defined by

kuk= [kuk^p_p+k∇uk^p_p]^1/p for allu∈W^1,p(Ω).

The Banach spaceC¹(Ω) is ordered with positive (order) cone C+={u∈C¹(Ω) :u(z)≥0 for allz∈Ω}.

This cone has a nonempty interior given by

D+={u∈C¹(Ω) :u(z)>0 for allz∈Ω}.

We will also consider another order cone forC¹(Ω), namely the cone Cb+=

u∈C¹(Ω) :u(z)≥0 for allz∈Ω, ∂u

∂n

∂Ω∩u⁻¹(0)≤0 . This cone too has a nonempty interior

intCb₊=

u∈Cb₊:u(z)>0 for allz∈Ω, ∂u

∂n

∂Ω∩u⁻¹(0)<0 .

Given x∈R, we setx^± = max{±x,0}. For any measurable function u: Ω→ R^N, we defineu^±(z) =u(z)^± for allz∈Ω. Ifu∈W^1,p(Ω), thenu^±∈W^1,p(Ω). If u, v∈W^1,p(Ω) andv≤u, we define

[v, u] ={h∈W^1,p(Ω) :v(z)≤h(z)≤u(z) for a.a. z∈Ω}, [v) =

h∈W^1,p(Ω) :v(z)≤h(z) for a.a. z∈Ω . By int_C1(Ω)[v, u] we denote the interior inC¹(Ω) of [v, u]∩C¹(Ω).

LetX be a Banach space,ϕ∈C¹(X,R) andc∈R. We define Kϕ={x∈X :ϕ⁰(x) = 0} (the critical set ofϕ), ϕ^c ={x∈X:ϕ(x)≤c} (the sublevel set of ϕatc).

Let (A, B) be a topological pair such that B ⊆A⊆X. ByHk(A, B),k∈N0, we denote thek^th-relative singular homology group for the pair (A, B) with integer coefficients. If u∈K_ϕ is isolated andϕ(u) =c, then the critical groups ofϕatu are defined by

Ck(ϕ, u) =Hk(ϕ^c∩U, ϕ^c∩U\ {u}) for allk∈N⁰,

with U being a neighborhood of usuch that Kϕ∩ϕ^c∩U = {u}. The excision property of singular homology, implies that this definition is independent of the isolating neighborhood.

Let X^∗ be the topological dual of X and denote by h·,·i the duality brackets of the pair (X^∗, X). A map A: X →X^∗ is said to be of type (S)+ if it has the following property:

u_n −^w→uinX and lim sup

n→+∞

hA(u_n), u_n−ui ≤0 ⇒ u_n→uinX.

Also, we say thatϕ∈C¹(X,R) satisfies the “C-condition”, if the following property holds:

Every sequence{un}n≥1⊆Xsuch that{ϕ(un)}n≥1⊆Ris bounded and (1 +kunkX)ϕ⁰(u_n)→0 inX^∗ asn→+∞,admits a strongly convergent subsequence.

Ifh1, h2∈L^∞(Ω), then we writeh1h2 when we have h₁(z)≤h₂(z) for a.a. z∈Ω and the above inequality is strict on a set of positive measure.

Finally for any measurable function f : Ω×R → R, byNf(·) we denote the Nemytskii operator corresponding tof, that is,

Nf(u)(·) =f(·, u(·)) for everyu: Ω→Rmeasurable, and by| · |N we denote the Lebesgue measure onR^N.

Letϑ∈C¹(0,+∞) withϑ(t)>0 for allt >0. We assume that 0<bc≤ϑ⁰(t)t

ϑ(t) ≤c₀ and c₁t^p−1≤ϑ(t)≤c₂[t^s−1+t^p−1] for allt >0, with 1≤s < p <+∞,c1, c2>0.

The hypotheses on the mapa(·) are as follows:

(H1) a(y) =a₀(|y|)y for ally∈R^N witha₀(t)>0 for allt >0, and

(i) a0 ∈ C¹(0,+∞), t → a0(t)t is strictly increasing on (0,+∞), a0(t)t→0⁺ as t→0⁺ and lim_t→0+

a⁰₀(t)t a0(t) >−1;

(ii) there existsc3>0 such that|∇a(y)| ≤c3ϑ(|y|)

|y| for ally∈R^N \ {0};

(iii) (∇a(y)ξ, ξ)_RN ≥ ^ϑ(|y|)_|y| |ξ|² for ally∈R^N \ {0},ξ∈R^N; (iv) If G₀(t) =Rt

0a₀(s)sds, then there exist q∈(1, p) and c^∗, c₄ >0 such that

lim sup

t→0⁺

qG0(t)

t^q ≤c^∗, t→G₀(t^1/q) is convex, c4t^p≤a0(t)t²−qG0(t) for allt >0,

0≤pG0(t)−a0(t)t² for allt >0.

Remark 2.1. Hypotheses (H1)(i)(ii)(iii) are dictated by the nonlinear regularity theory of Lieberman [7] and the nonlinear maximum principle of Pucci-Serrin [15].

Hypothesis (H1)(iv) is motivated by the particular needs of our problem. However, as the examples below illustrate, it is not restrictive and it is satisfied in all cases of interest.

From the above hypotheses we see that the primitiveG₀(·) is strictly convex and strictly increasing. We setG(y) =G₀(|y|) for ally∈R^N. ThenG(·) is convex and

∇G(y) =G⁰₀(|y|)y

|y| =a₀(|y|)y=a(y) for ally∈R^N \ {0}.

So,G(·) is the primitive ofa(·). This fact and the convexity ofG(·) imply that G(y)≤(a(y), y)_RN for ally∈R^N. (2.1) Hypotheses (H1) lead to the following lemma summarizing the main properties of the mapy→a(y) (see Papageorgiou-Rˇadulescu [9]).

Lemma 2.2. If hypotheses(H1)(i)(ii)(iii) hold, then

(a) a(·)is continuous, strictly monotone, hence maximal monotone;

(b) there existsc5>0 such that |a(y)| ≤c5[|y|^s−1+|y|^p−1] for ally∈R^N; (c) (a(y), y)_RN ≥_p−1^c1 |y|^p for ally∈R^N.

This lemma and (2.1) lead to the following growth estimates for the primitive G(·).

Corollary 2.3. If hypotheses(H1)(i)(ii)(iii)hold, then there existsc₆>0such that

c₁

p(p−1)|y|^p≤G(y)≤c₆[1 +|y|^p]for all y∈R^N.

The following examples show that the framework provided by hypotheses (H1) is broad.

Example 2.4. The following maps satisfy hypotheses (H1) (see [9]):

(a) a(y) =|y|^p−2ywith 1< p <+∞. This map corresponds to thep-Laplacian differential operator.

(b) a(y) =|y|^p−2y+|y|^q−2y with 1< q < p <+∞. This map corresponds to the (p, q)-Laplacian differential operator, that is, the sum of ap-Laplacian and of aq-Laplacian. Such operators arise in many problems of mathemat- ical physics and correspond to the so-called double phase equations. In this direction we mention the works of Cherfils-Il’yasov [2] (reaction-diffusion systems) and of Zhikov [16] (problems in elasticity theory).

(c) a(y) = [1 +|y|²]^p−22 y with 1 < p < +∞. This map corresponds to the extended capillary differential operator.

(d) a(y) = [1 +_1+|y|¹ p]|y|^p−2y with 1 < p < +∞. This map corresponds to a differential operator which arises in problems of plasticity theory (see Fuchs-Li [3]).

LetA:W^1,p(Ω)→W^1,p(Ω)^∗ be the nonlinear operator defined by hA(u), hi=

Z

Ω

(a(∇u),∇h)_RNdz for allu, h∈W^1,p(Ω).

From Gasi´nski-Papageorgiou [4] (Problem 2.192, p. 279), we have the following result.

Proposition 2.5. If hypotheses(H1)hold, then the mapA(·)is continuous, mono- tone(hence maximal monotone too)and of type(S)₊.

The following strong comparison principle by Papageorgiou-Rˇadulescu-Repovˇs [13], will be useful in our analysis of problem (1.1).

Proposition 2.6. If hypotheses(H1)hold,ξb∈L^∞(Ω)withξ(z)b ≥0for a.a. z∈Ω, h1, h2∈L^∞(Ω) with 0< η≤h2(z)−h1(z)for a.a. z∈Ωandu, v∈C^1,α(Ω) with α∈(0,1],v≤uand

−diva(∇v(z)) +ξ(z)|v(z)|b ^p−2v(z) =h1(z) for a.a. z∈Ω,

−diva(∇u(z)) +ξ(z)|u(z)|b ^p−2u(z) =h2(z) for a.a. z∈Ω, thenu−v∈intCb₊.

Next we introduce hypotheses on the potential functionξ(z) and on the reaction termf(z, x).

(H2) ξ∈L^∞(Ω).

(H3) f : Ω×R→ Ris a Carath´eodory function such that f(z,0) = 0 for a.a.

z∈Ω and

(i) η(z)x^p−1 ≤ f(z, x)≤ α(z)[1 +x^r−1] for a.a. z ∈ Ω, all x≥ 0, with η, α∈L^∞(Ω),ξ⁺η andp < r < p^∗ =

( _{N p}

N−p ifN > p +∞ ifp≥N; (ii) if F(z, x) = Rx

0 f(z, s)ds, then lim_x→+∞^F(z,x)_xp = +∞ uniformly for a.a.z∈Ω;

(iii) ifd(z, x) =f(z, x)x−pF(z, x), then there existse∈L¹(Ω) such that d(z, x)≤d(z, y) +e(z) for a.a.z∈Ω, all 0≤x≤yandd(z, x)→+∞

for a.a.z∈Ω asx→+∞;

(iv) withq∈(1, p) as in hypothesis (H1)(iv), we have lim_x→0+ f(z,x) x^q−1 = +∞

uniformly for a.a.z∈Ω;

(v) for eachρ >0 there existsξbρ>0 such that for a.a.z∈Ω the function x→f(z, x) +ξbρx^p−1 is nondecreasing on [0, ρ].

Remark 2.7. Since initially (Section 3) our aim is to produce positive solutions for problem (1.1) and all the above conditions of f(z,·) concern the positive semiaxis R+= [0,+∞), without any loss of generality we may assume that

f(z, x) = 0 for a.a. z∈Ω, allx≤0. (2.2) Hypotheses (H3)(ii)(iii) imply that

x→+∞lim f(z, x)

x^p−1 = +∞ uniformly for a.a. z∈Ω.

Therefore the reaction f(z,·) is (p−1)-superlinear near +∞. Usually such problems are treated using the AR-condition which leads to an easy verification of the C-condition for the energy (Euler) functional of the problem. We recall that the AR-condition (unilateral version due to (2.2)), says that there existϑ > pand M >0 such that

0< ϑF(z, x)≤f(z, x)x for a.a.z∈Ω, allx≥M , (2.3)

0<ess infΩF(·, M). (2.4)

Integrating (2.3) and using (2.4), we obtain the following weaker condition c₇x^ϑ ≤F(z, x) for a.a.z∈Ω, allx≥M, somec₇>0,

⇒ c7x^ϑ−1≤f(z, x) for a.a.z∈Ω, allx≥M, (see (2.3)). (2.5) From (2.5) we see that the AR-condition implies thatf(z,·) has at least (ϑ−1)- polynomial growth. In this work, we replace the AR-condition by the quasimono- tonicity condition on d(z,·) stated in hypothesis (H3)(iii). This hypothesis is a slight generalization of a condition used by Li-Yang [6]. This condition is satisfied if there existsM >0 such that for a.a.z∈Ω the functionx→ ^f(z,x)_xp−1 is nondecreas- ing on [M,+∞). Hence from (2.5) we infer that the quasimonotonicity condition on d(z,·) is more general than the AR-condition. It permits the consideration of superlinear nonlinearities with “slower” growth near +∞. To see this, consider the following function

f(z, x) =

(η(z)(x⁺)^q−1 ifx≤1, x^p−1lnx+η(z)x^τ−1 if 1< x,

withη ∈L^∞(Ω),ξ⁺η and 1< τ, q < p. This function satisfies hypotheses (H3) but fails to satisfy the AR-condition (see (2.3), (2.4)).

In what followsγ:W^1,p(Ω)→Ris theC¹-functional defined by γ(u) =

Z

Ω

pG(∇u)dz+ Z

Ω

ξ(z)|u|^pdz for allu∈W^1,p(Ω).

3. Positive Solutions

In this section we study the dependence on the parameter λ > 0 of the set of positive solutions. So, we introduce the following two sets:

L={λ >0 : problem (1.1) has a positive solution}, Sλ= set of positive solutions of (1.1).

We start with the following result about these two sets.

Proposition 3.1. If hypotheses (H1)–(H3)hold, thenL 6=∅ and, for everyλ∈ L,

∅ 6=Sλ⊆D+.

Proof. Let µ > kξk_∞ (see hypothesis (H2)) and consider the following auxiliary Neumann problem

−diva(∇u(z)) + [ξ(z) +µ]u(z)^p−1= 1 in Ω,

∂u

∂n= 0 on∂Ω.

(3.1) Using Lemma 2.2, Proposition 2.5 and the fact thatµ >kξk∞, we see that the left hand side of (3.1) is continuous, strictly monotone and coercive. Therefore

problem (3.1) admits a unique solution u∈ W^1,p(Ω), u6= 0. Moreover, the non- linear regularity theory (see [7]) and the nonlinear maximum principle (see [15]), imply thatu∈D+. We set

M₀=kNf(u)k∞ (see hypothesis (H3) (i)), m0= min

Ω

u >0 (recall that u∈D+), λ=µ+ M0

m^p−1₀ >0.

We have

−diva(∇u(z)) + [ξ(z) +λ]u(z)^p−1

=−diva(∇u(z)) + [ξ(z) +µ]u(z)^p−1+M₀ u(z) m0

p−1

≥1 +M0 (see (3.1) and recall thatu(z)≥m0 for allz∈Ω)

> f(z, u(z)) for a.a. z∈Ω.

(3.2)

We introduce the Carath´eodory function fb(z, x) =

(f(z, x⁺) ifx≤u(z),

f(z, u(z)) ifu(z)< x. (3.3) (see (2.2)).

We set F(z, x) =b Rx

0 fb(z, s)dsand consider theC¹-functional ϕb:W^1,p(Ω) →R defined by

ϕ(u) =b 1

pγ(u) +λ pkuk^p_p−

Z

Ω

Fb(z, u)dz for allu∈W^1,p(Ω).

From (3.3) and since λ > µ > kξk∞, we see that ϕ(·) is coercive. Also usingb the Sobolev embedding theorem, we show that ϕ(·) is sequentially weakly lowerb semicontinuous. So, by the Weierstrass-Tonelli theorem, we can findu0∈W^1,p(Ω) such that

ϕ(ub ₀) = inf[ϕ(u) :b u∈W^1,p(Ω)]. (3.4) Hypotheses (H1)(iv) and (H3)(iv) imply that given η > c^∗₀ > c^∗, we can find δ∈(0, m0] such that

G(y)≤ c^∗₀

q|y|^q for all|y| ≤δ, F(z, x)≥ η

qx^q for a.a. z∈Ω, allx∈[0, δ].

(3.5)

Givenu∈D₊, we chooset∈(0,1) small such that

t|∇u(z)| ≤δ and tu(z)≤δ for allz∈Ω. (3.6) Using (3.5) and (3.6), we have

ϕ(tu)b ≤c^∗₀

q t^qk∇uk^q_q+t^p

p[kξk∞+λ]kuk^p_p−η qt^qkuk^q_q

=t^q[c8−ηc9] +t^pc10 for somec8, c9, c10>0.

Sinceη > c^∗₀ is arbitrary, by choosingη > ^c_c⁸

9 we obtain

ϕ(tu)b ≤c₁₀t^p−c₁₁t^q for allt >0, and somec₁₁>0.

Recall that q < p. So, by choosing t ∈ (0,1) even smaller if necessary, we have ϕ(tu)b <0 impliesϕ(ub 0)<0 =ϕ(0) (see (3.4)) which in turn impliesb u06= 0.

From (3.4) we have thatϕb⁰(u0) = 0 implies hA(u0), hi+

Z

Ω

[ξ(z) +λ]|u0|^p−2u₀hdz= Z

Ω

fb(z, u₀)hdz ∀h∈W^1,p(Ω). (3.7) In (3.7) first we chooseh=−u⁻₀ ∈W^1,p(Ω). Then

c1

p−1k∇u⁻₀k^p_p+ Z

Ω

[ξ(z) +λ](u⁻₀)^pdz≤0 (see Lemma 2.2 and (3.3)),

⇒ c12ku⁻₀k^p≤0 for somec12>0 (recallλ > µ >kξk∞),

⇒ u₀≥0, u₀6= 0.

Next in (3.7) we chooseh= (u₀−u)⁺∈W^1,p(Ω). Then we have hA(u0),(u0−u)⁺i+

Z

Ω

[ξ(z) +λ]u^p−1₀ (u0−u)⁺dz

= Z

Ω

f(z, u)(u0−u)⁺dz (see (3.3)),

≤ hA(u),(u0−u)⁺i+ Z

Ω

[ξ(z) +λ]u^p−1(u0−u)⁺dz (see (3.2)),

which impliesu0≤u(see Proposition 2.5 and recall thatλ > µ >kξk∞). So, we have proved that

u0∈[0, u], u06= 0. (3.8)

From (3.3), (3.7) and (3.8) it follows that

−diva(∇u0(z)) + [ξ(z) +λ]u0(z)^p−1=f(z, u0(z)) for a.a. z∈Ω,

∂u0

∂n = 0 on∂Ω.

(3.9) From (3.9) and [10, Proposition 2.10], we have u₀ ∈L^∞(Ω). Then the nonlinear regularity theory of Lieberman [7] implies that u₀ ∈ C₊ \ {0}. From (3.9) and hypothesis (H3)(i), we have that

diva(∇u0(z))≤[kηk_∞+kξk_∞+λ]u0(z)^p−1 for a.a. z∈Ω. (3.10) The nonlinear maximum principle of Pucci-Serrin [15, pp. 111, 120] and (3.10) imply that u0 ∈ D+. Therefore we conclude that λ ∈ L 6= ∅ and for all λ ∈ L,

∅ 6=Sλ⊆D+.

In the next proposition, we prove a structural property of the setL, namely we show thatLis an upper half line. In addition we establish a kind of monotonicity property for the solution multifunctionλ→Sλ.

Proposition 3.2. If hypotheses(H1)–(H3)hold,λ∈ L,u_λ∈S_λ⊆D₊andη > λ, thenη∈ L and there existsu_η∈S_η⊆D₊ such that u_λ−u_η ∈intCb₊.

Proof. We have

−diva(∇uλ(z)) + [ξ(z) +λ]uλ(z)^p−1

=f(z, u_λ(z))

<−diva(∇u_λ(z)) + [ξ(z) +η]u_λ(z)^p−1 for a.a. z∈Ω (since η > λ).

We introduce the Carath´eodory function k(z, x) =

(f(z, x⁺) +µ(xb ⁺)^p−1 ifx≤u_λ(z),

f(z, uλ(z)) +µub λ(z)^p−1 ifuλ(z)< x, (3.11) where µb ≥ kξk_∞. We set K(z, x) = Rx

0 k(z, s)ds and consider theC¹-functional ψbη:W^1,p(Ω)→Rdefined by

ψbη(u) =1

pγ(u) +η+µb p kuk^p_p−

Z

Ω

K(z, u)dz for allu∈W^1,p(Ω).

From (3.11) and sinceµb≥ kξk_∞andη > λ >0, we see thatψbη(·) is coercive. Also it is sequentially weakly lower semicontinuous. So, we can finduη ∈W^1,p(Ω) such that

ψbη(uη) = inf[ψbη(u) :u∈W^1,p(Ω)]. (3.12) As before (see the proof of Proposition 3.1) on account of hypotheses (H1)(iv) and (H3)(iv) we have

ψb_η(u_η)<0 =ψb_η(0) ⇒ u_η 6= 0.

From (3.12) we have thatψb_η⁰(uη) = 0 implies hA(u_η), hi+

Z

Ω

[ξ(z)+η+µ]|u_η|^p−2u_ηhdz= Z

Ω

k(z, u_η)hdz ∀h∈W^1,p(Ω). (3.13) In (3.13) we chooseh=−u⁻_η ∈W^1,p(Ω) andh= (uη−uλ)⁺ ∈W^1,p(Ω) and as in the proof of Proposition 3.1, we show that

u_η∈[0, u_λ], u_η 6= 0. (3.14)

From (3.11), (3.13) and (3.14) we infer that

η∈ Landuη∈Sη⊆D+ (see Proposition 3.1), uη ≤uλ.

Letρ=kuλk_∞and letξbρ>0 be as postulated by hypothesis (H3)(v). We have

−diva(∇uλ) + [ξ(z) +η+ξbρ]u^p−1_λ

=f(z, uλ) +ξbρu^p−1_λ + (η−λ)u^p−1_λ (since uλ∈Sλ)

≥f(z, uη) +ξbρu^p−1_η + (η−λ)u^p−1_η (see (H3)(v) and recalluη ≤uλ)

>−diva(∇uη) + [ξ(z) +η+ξb_ρ]u^p−1_η for a.a.z∈Ω (sinceu_η∈S_η).

(3.15)

Letmη= min_Ωuη >0 (recall thatuη∈D+). We have

(η−λ)u^p−1_η ≥(η−λ)m^p−1_η >0 (since η > λ).

Then from (3.15) and Proposition 2.6, it follows thatu_λ−u_η∈intCb₊. Letλ_∗= infL.

Proposition 3.3. If hypotheses(H1)–(H3) hold, thenλ_∗>0.

Proof. Letϕλ :W^1,p(Ω) →R be the energy (Euler) functional for problem (1.1) defined by

ϕλ(u) =1

pγ(u) +λ pkuk^p_p−

Z

Ω

F(z, u)dz for allu∈W^1,p(Ω).

Arguing by contradiction, suppose thatλ∗= 0. Let{λn}n≥1⊆ Lsuch thatλn ↓0.

We fixλ > λ1. For everyn∈Nandubn∈Sλ_n ⊆D+, on account of Proposition 3.2 and its proof we can finduⁿ_λ∈Sλ⊆D+such thatϕλ(uⁿ_λ)<0,uⁿ_λ ≤bun. We have

−diva(∇uⁿ⁺¹_λ ) + [ξ(z) +λn]uⁿ⁺¹_λ ≤f(z, uⁿ⁺¹_λ ) for a.a. z∈Ω, (3.16)

−diva(∇ubn+1) + [ξ(z) +λn]bu^p−1_n+1≥f(z,bun+1) for a.a.z∈Ω. (3.17) Withbµ≥ kξk∞we introduce the Carath´eodory function

kn(z, x) =







f(z, uⁿ⁺¹_λ (z)) +µub ⁿ⁺¹_λ (z)^p−1 ifx < uⁿ⁺¹_λ (z),

f(z, x) +bµx^p−1 ifuⁿ⁺¹_λ (z)≤x≤ubn+1(z), f(z,bun+1(z)) +µbubn+1(z)^p−1 ifubn+1< x.

(3.18)

We setKn(z, x) =Rx

0 kn(z, s)dsand consider theC¹-functionalϕeλ_n :W^1,p(Ω)→R defined by

ϕe_λn(u) =1

pγ(u) +λ_n+bµ p kuk^p_p−

Z

Ω

K_n(z, u)dz for allu∈W^1,p(Ω), with bµ≥ kξk_∞. Evidentlyϕeλ_n(·) is coercive (see (3.18)) and sequentially weakly lower semicontinuous and so we can findun ∈W^1,p(Ω) such that

ϕeλ_n(un) = inf[ϕeλ_n(u) :u∈W^1,p(Ω)],

⇒ ϕe⁰_λ

n(u_n) = 0,

⇒ hA(un), hi+ Z

Ω

[ξ(z) +λn+µ]|ub n|^p−2unhdz= Z

Ω

kn(z, un)hdz

(3.19)

for allh∈W^1,p(Ω). Choosingh= (uⁿ_λ−un)⁺∈W^1,p(Ω) andh= (un−ubn+1)⁺∈ W^1,p(Ω) and using (3.16), (3.17) and (3.18), we show (see also the proof of Propo- sition 3.1) that

un ∈[uⁿ_λ,ubn+1]∩D+ (by the nonlinear regularity theory).

We have

ϕe_λn(uⁿ_λ)≤1

pγ(uⁿ_λ) +λn

p kuⁿ_λk^p_p− Z

Ω

f(z, uⁿ_λ)uⁿ_λdz (see (3.18))

≤1

pγ(uⁿ_λ) +λ

pkuⁿ_λk^p_p− Z

Ω

pF(z, uⁿ_λ)dz+kek₁ (see (H3)(iii))

≤1

pγ(uⁿ_λ) +λ

pkuⁿ_λk^p_p− Z

Ω

F(z, uⁿ_λ)dz+kek1 (since F≥0)

=ϕ_λ(uⁿ_λ) +kek₁

<kek1,

which implies ϕe_λn(u_n) < kek1 for all n ∈ N (see (3.19)). This in turn implies ϕ_λn(u_n)≤c₁₃for some c₁₃>0 and alln∈N(see (3.18)).

Therefore we have produced a sequence{un}n≥1⊆W^1,p(Ω) such that

u_n ∈S_λn⊆D₊ and ϕ_λn(u_n)≤c₁₃ for alln∈N. (3.20)

From (3.20) we have hA(un), hi+

Z

Ω

[ξ(z) +λn]u^p−1_n hdz

= Z

Ω

f(z, un)hdz for allh∈W^1,p(Ω), and alln∈N,

(3.21)

γ(un) +λnkunk^p_p− Z

Ω

pF(z, un)dz≤pc13 for alln∈N. (3.22) In (3.21) we chooseh=un ∈W^1,p(Ω). Then

−γ(un)−λnkunk^p_p+ Z

Ω

f(z, un)undz= 0 for alln∈N. (3.23) We add (3.22) and (3.23) to obtain

Z

Ω

[f(z, u_n)u_n−pF(z, u_n)]dz= Z

Ω

d(z, u_n)dz≤pc₁₃ for alln∈N. (3.24) We will show that {un}n≥1 ⊆ W^1,p(Ω) is bounded. Arguing indirectly, suppose that at least for a subsequence we have

ku_nk →+∞. (3.25)

We setyn =un/kunkforn∈N. Thenkynk= 1,yn≥0 for alln∈N. So, we may assume that

yn

−w→y in W^1,p(Ω) and yn→y in L^r(Ω), y≥0.

First, we assume thaty6= 0. Let Ω+={z∈Ω :y(z)>0}. Then|Ω+|N >0 (recall that y≥0). From (3.25) it follows thatun(z)→+∞for allz∈Ω+. So, we have d(z, u_n(z))→+∞for a.a.z∈Ω (see hypothesis (H3)(iii)). This implies

Z

Ω+

d(z, un)dz→+∞ (by Fatou’s lemma). (3.26) From hypothesis (H3)(iii) we have

d(z, x)≥ −e(z) for a.a. z∈Ω, allx≥0. (3.27) Then we have

Z

Ω

d(z, u_n)dz= Z

Ω₊

d(z, u_n)dz+ Z

Ω\Ω₊

d(z, u_n)dz

≥ Z

Ω+

d(z, un)dz− kek1 for alln∈N(see (3.27)), which implies R

Ωd(z, un)dz → +∞ as n → +∞ (see (3.26)). This contradicts (3.24).

Now we assume thaty= 0. Letτ >0 and setv_n= (pτ)^1/py_n∈W^1,p(Ω) for all n∈N. Letγ:W^1,p(Ω)→Rbe theC¹-functional defined by

γ(u) = c₁

p−1k∇uk^p_p+ Z

Ω

ξ(z)|u|^pdz for allu∈W^1,p(Ω).

We introduce theC¹-functionalsϕ_λn :W^1,p(Ω)→R,n∈N, defined by ϕ_λn(u) = 1

pγ(u) +λn

p kuk^p_p− Z

Ω

F(z, u)dz for allu∈W^1,p(Ω).

Lettn∈[0,1] be such that ϕ_λ

n(t_nu_n) = max[ϕ_λn(tu_n) : 0≤t≤1] for alln∈N. (3.28) On account of (3.25), we see that we can findn0∈Nsuch that

(pτ)^1/p 1

kunk ≤1 for alln≥n0. (3.29) From (3.28) and (3.29) it follows that

ϕ_λ

n(t_nu_n)≥ϕ_λ

n(v_n)

=τ[γ(y_n) + [λ_n+µ]kyb _nk^p_p]− Z

Ω

[F(z, v_n) +µb pv_n^p]dz for alln≥n0, withµb≥ kξk_∞

≥τ c14− Z

Ω

[F(z, vn) +µb pv_n^p]dz

for somec14>0, all n≥n0 (sinceµb≥ kξk_∞).

(3.30)

EvidentlyR

Ω[F(z, v_n) +^µb_pv_n^p]dz→0 asn→+∞(recally= 0). Hence from (3.30) it follows that

ϕ_λn(tnun)≥τ

2c14 for alln≥n1≥n0. Sinceτ >0 is arbitrary, we infer that

ϕ_λ

n(t_nu_n)→+∞ asn→+∞. (3.31)

We have

ϕ_λn(0) = 0 andϕ_λn(un)≤ϕλn(un)≤c13 for alln∈N(see (3.20)).

Then on account of (3.31), we have

tn ∈(0,1) for alln≥n2. (3.32) From (3.28) and (3.32) it follows that

d

dtϕ_λn(tun) _t=t

n = 0 for alln≥n2,

⇒ hϕ⁰_λ

n(t_nu_n), t_nu_ni= 0 for alln≥n₂ (by the chain rule),

⇒ c₁

p−1k∇(tnun)k^p_p+ Z

Ω

[ξ(z) +λn](tnun)^pdz= Z

Ω

f(z, tnun)(tnun)dz for alln≥n2,

⇒pϕ_λn(tnun)≤ Z

Ω

d(z, tnun)dz≤ Z

Ω

d(z, un)dz+kek1

for alln≥n2 (see hypothesis (H3) (iii) and (3.32)),

⇒ pϕ_λn(tnun)≤pc13+kek1 for alln≥n2 (see (3.24)), which contradicts (3.31).

So, we have that{un}n≥1⊆W^1,p(Ω) is bounded. We may assume that

u_n−^w→u_∗ in W^1,p(Ω) and u_n→u_∗ inL^r(Ω). (3.33) In (3.21) we chooseh=un−u_∗∈W^1,p(Ω), pass to the limit asn→+∞and use (3.33). Then we obtain

n→+∞lim hA(un), un−u∗i= 0,

which implies

u_n→u_∗ inW^1,p(Ω) (see Proposition 2.5). (3.34) So, if in (3.21) we pass to the limit asn→+∞ and use (3.34) and the fact that λ_n↓0 (recall we have assumed thatλ_∗= 0), we obtain

hA(u_∗), hi+ Z

Ω

ξ(z)u^p−1_∗ hdz= Z

Ω

f(z, u_∗)hdz for allh∈W^1,p(Ω). (3.35) In (3.35) we chooseh≡1. Then

Z

Ω

ξ(z)u^p−1_∗ dz= Z

Ω

f(z, u_∗)dz≥ Z

Ω

η(z)u^p−1_∗ dz (see (H3)(i)), which implies

Z

Ω

[η(z)−ξ(z)]u^p−1_∗ dz≤0. (3.36) Note that hypotheses (H3)(i),(iv) imply that we can findc₁₅>0 such that

f(z, x)≥x^q−1−c₁₅x^r−1 for a.a. z∈Ω, allx≥0.

Evidently we can always assume that c₁₅ > kξk∞. We consider the following auxiliary Neumann problem

−diva(∇u(z)) +ξ(z)u(z)^p−1=u(z)^q−1−c₁₅u(z)^r−1in Ω,

∂u

∂n = 0, u >0.

From [9, Proposition 3.5] we know that this problem has a unique positive solution ue∈D+. Letλ∈ Landu∈Sλ⊆D+. We introduce the Carath´eodory function

β(z, x) =

((x⁺)^q−1−c15(x⁺)^r−1+µ(xb ⁺)^p−1 ifx≤u(z),

u(z)^q−1−c₁₅u(z)^r−1+µu(z)b ^p−1 ifu(z)< x, (3.37) withµb≥ kξk∞.

We setB(z, x) =Rx

0 β(z, s)dsand consider theC¹-functionaleσλ:W^1,p(Ω)→R defined by

eσλ(u) = 1

pγ(u) +λ+µb p kuk^p_p−

Z

Ω

B(z, u)dz foru∈W^1,p(Ω).

The direct method of calculus of variations givesue₀∈W^1,p(Ω) such that σe_λ(eu₀) = inf[eσ_λ(u) :u∈W^1,p(Ω)]<0 =σe_λ(0) (sinceq < p).

So,eu₀6= 0 and eu₀∈K

eσ_λ ⊆[0, u]∩C₊ (see (3.37) and use the nonlinear regularity theory). Hence from (3.37) we infer thatue₀=ue∈D₊ and soue≤ufor allu∈S_λ, allλ∈ L. It follows that

eu≤u_∗ ⇒ Z

Ω

[η(z)−ξ(z)]u^p−1_∗ dz >0 (sinceξ≺η)

which contradicts (3.36). So, we conclude thatλ_∗>0.

Proposition 3.4. If hypotheses (H1)–(H3) hold and λ∈(λ_∗,+∞), then problem (1.1)admits at least two positive solutions u₀,bu∈S_λ⊆D₊.

Proof. Let λ∗ < θ < λ < η. By Proposition 3.2, we can find uθ ∈ Sθ ⊆ D+, u0∈Sλ⊆D+ anduη ∈Sη⊆D+ such that

uθ−u0∈intCb+ andu0−uη∈intCb+,

⇒ u0∈int_C1(Ω)[uη, uθ]. (3.38) We introduce the Carath´eodory function

j(z, x) =

(f(z, u_η(z)) +bµu_η(z)^p−1 ifx≤u_η(z),

f(z, x) +µxb ^p−1 ifuη(z)< x, (3.39) with µb ≥ kξk_∞. We set J(z, x) = Rx

0 j(z, s)ds and consider the C¹-functional ψbλ:W^1,p(Ω)→Rdefined by

ψb_λ(u) =1

pγ(u) +λ+µb p kuk^p_p−

Z

Ω

J(z, u)dz for allu∈W^1,p(Ω).

In addition, we introduce the following truncation ofj(z,·), ej(z, x) =

(j(z, x) ifx≤u_θ(z),

j(z, uθ(z)) ifuθ(z)< x. (3.40) This is a Carath´eodory function. We setJe(z, x) = Rx

0 ej(z, s)ds and consider the C¹-functional ψe_λ:W^1,p(Ω)→Rdefined by

ψeλ(u) =1

pγ(u) +λ+µb p kuk^p_p−

Z

Ω

Je(z, u)dz for allu∈W^1,p(Ω).

From (3.39), (3.40) and the nonlinear regularity theory of Lieberman [7], we have K

ψbλ ⊆[u_η)∩D₊ andK

ψeλ ⊆[u_η, u_θ]∩D₊. (3.41) From (3.39), (3.40), (3.41), we see that we may assume that

K

ψeλ ={u0}. (3.42)

Otherwise we already have a second positive solution of (1.1), distinct fromu0and the proof is complete.

Clearlyψeλ(·) is coercive (see (3.40)) and sequentially weakly lower semicontinu- ous. So, we can findeu0∈W^1,p(Ω) such that

ψe_λ(eu₀) = inf[ψe_λ(u) :u∈W^1,p(Ω)]

⇒ eu₀∈K

ψeλ

⇒ eu0=u0 (see (3.42)).

From (3.39) and (3.40) we see that ψbλ

_[u

η,uθ]=ψeλ

_[u

η,uθ]. Then from (3.38) it follows that

u₀∈D₊ is a localC¹(Ω)-minimizer of ψb_λ,

⇒ u₀∈D₊ is a localW^1,p(Ω)-minimizer ofψb_λ

(3.43) (see Papageorgiou-Rˇadulescu [10]).

From (3.41) we can assume that

Kψb_λ is finite. (3.44)

Using (3.43), (3.44) and [14, Theorem 5.7.6, p. 367,], we see that we can find ρ∈(0,1) small such that

ψbλ(u0)<inf[ψbλ(u) :ku−u0k=ρ] =mbλ. (3.45) By (H3)(ii), foru∈D+ we have

ψb_λ(tu)→ −∞ ast→+∞. (3.46)

Moreover, reasoning as in the proof of Proposition 3.3 (see the part of the proof from (3.20) up to (3.33)), we can show that

ψbλ(·) satisfies theC-condition. (3.47) Then (3.45), (3.46), (3.47) permit the use of the mountain pass theorem. So, we can findub∈W^1,p(Ω) such that

bu∈K

ψb_λ ⊆[uη)∩D+ (see (3.41)) andmbλ≤ψbλ(bu) (see (3.45)),

⇒ ub6=u₀ (see (3.45)) andbu∈S_λ⊆D₊ (see (3.39)).

Proposition 3.5. If hypotheses(H1)–(H3) hold, thenλ∗∈ L.

Proof. Let{λn}_n≥1⊆ L such thatλn↓λ_∗. From the proof of Proposition 3.3, we know that we can findun ∈Sλ_n⊆D+,n∈N, such that

ue≤un and ϕλ_n(un)≤c16 for somec16>0, all n∈N.

As in the proof of Proposition 3.3, we can show thatu_n →u_∗ in W^1,p(Ω). Then in the limit asn→+∞we have

eu≤u∗ andhA(u∗), hi+ Z

Ω

[ξ(z) +λ∗]u^p−1_∗ hdz= Z

Ω

f(z, u∗)hdz,

for allh∈W^1,p(Ω), which impliesu_∗∈S_λ∗⊆D₊, and soλ_∗∈ L.

Note that Proposition 3.5 implies thatL= [λ_∗,+∞). Summarizing our results on the dependence of the set of positive solutions of (1.1) on the parameterλ >0, we can state the following bifurcation-type result for big values ofλ >0.

Theorem 3.6. If hypotheses(H1)–(H3)hold, then there exists a critical parameter value λ_∗>0 such that

(a) for allλ > λ_∗ problem (1.1)has at least two positive solutionsu₀,ub∈D₊, u₀6=u;b

(b) forλ=λ_∗ problem (1.1)has at least one positive solutionu_∗∈D₊; (c) for allλ∈(0, λ_∗)problem (1.1)has no positive solutions.

Next we show that for every λ∈ L = [λ∗,+∞), problem (1.1) has a smallest positive solution.

Proposition 3.7. If hypotheses (H1)–(H3) hold and λ ∈ L = [λ_∗,+∞), then problem (1.1)has a smallest positive solution buλ∈D+.

Proof. From Papageorgiou-Rˇadulescu-Repovˇs [12] (see the proof of Proposition 7), we know that the solution setS_λis downward directed (that is, ifu₁, u₂∈S_λ, then we can findu∈S_λ such that u≤u₁,u≤u₂). Then invoking [5, Lemma 3.10, p.

178], we can find a decreasing sequence{un}_n≥1⊆Sλ such that infS_λ= inf

n≥1u_n and 0≤u_n≤u₁for alln∈N. (3.48)

We have hA(un), hi+

Z

Ω

[ξ(z) +λ]u^p−1_n hdz= Z

Ω

f(z, un)hdz for allh∈W^1,p(Ω). (3.49) Choosing h=un ∈W^1,p(Ω) in (3.49) and using (3.48), we infer that {un}_n≥1 ⊆ W^1,p(Ω) is bounded. Proposition 7 in Papageorgiou-Rˇadulescu [10] implies that we can findc16>0 such that

u_n ∈L^∞(Ω) andku_nk_∞≤c₁₆for alln∈N.

Then the nonlinear regularity theory of Lieberman [7] implies that there existα∈ (0,1) andc17>0 such that

un ∈C^1,α(Ω), kunk_C1,α(Ω)≤c17 for alln∈N. (3.50) From (3.50), the compact embedding ofC^1,α(Ω) intoC¹(Ω), and the monotonicity of{un}_n≥1, we have

un→ubλin C¹(Ω). (3.51)

From the proof of Proposition 3.3, we know that ue≤un for alln∈N,

⇒ eu≤ub_λ (see (3.51)), hencebu_λ6= 0.

If in (3.49) we pass to the limit asn→+∞and use (3.51), we obtain hA(buλ), hi+

Z

Ω

[ξ(z) +λ]bu^p−1_λ hdz= Z

Ω

f(z,ubλ)hdz for allh∈W^1,p(Ω),

⇒ ubλ∈Sλ⊆D+ andubλ= infSλ.

Next we examine the properties of the mapL 3λ→buλ∈C¹(Ω).

Proposition 3.8. If hypotheses (H1)–(H3) hold, then the map σ : L → C¹(Ω) defined byσ(λ) =ub_λ has the following properties:

(a) σ(·) is strictly decreasing in the sense that λ∗ ≤λ < η implies buλ−ubη ∈ intCb+;

(b) σ(·)is right continuous.

Proof. (a) Letubλ∈D+ be the minimal positive solution of (1.1) (λ∈ L). Accord- ing to Proposition 3.2, we can finduη∈Sη ⊆D+ such that

ubλ−uη ∈intCb+,

⇒ ub_λ−ub_η∈intCb₊ (since bu_η ≤u_η)

⇒ σ(·) is strictly decreasing.

(b) Letλn ↓λ∈ L. As in the proof of Proposition 3.3, we can findun∈W^1,p(Ω) such that

u_n∈S_λn ⊆D₊ andϕ_λn(u_n)≤c₁₈ for somec₁₈>0, all n∈N.

From this it follows that {un}_n≥1⊆W^1,p(Ω) is bounded (see the proof of Propo- sition 3.3). We have

0≤ub_λn≤u_n for alln∈N,

⇒ {bu_λn}_n≥1⊆W^1,p(Ω).

From this and the nonlinear regularity theory of Lieberman [7] (see the proof of Proposition 11), we obtain (at least for a subsequence) that

bu_λn→eu_λ inC¹(Ω). (3.52) If eu_λ 6=bu_λ, then we can find z₀ ∈Ω such that ub_λ(z₀)<ue_λ(z₀) implies ub_λ(z₀)<

ubλ_n(z0) for all n≥n0 (see (3.52)). This contradicts (a). Therefore by Urysohn’s criterion, for the original sequence we have

ub_λn→ub_λ inC¹(Ω) ⇒ σ(·) is right continuous.

If we impose on f(z,·) similar conditions valid on the negative semiaxis R− = (−∞,0], we can have analogous results for the negative solutions.

Now the hypotheses on the reactionf(z, x) are as follows

(H3’) f : Ω×R→ Ris a Carath´eodory function such that f(z,0) = 0 for a.a.

z∈Ω and

(i) η(z)|x|^p ≤ f(z, x)x ≤ α(z)[1 +|x|^r] for a.a. z ∈ Ω, all x ≤ 0, with η, α∈L^∞(Ω),ξηand p < r < p^∗;

(ii) if F(z, x) = Rx

0 f(z, s)ds, then lim_x→−∞^F(z,x)_|x|p = +∞ uniformly for a.a.z∈Ω;

(iii) ifd(z, x) =f(z, x)x−pF(z, x), then there existse∈L¹(Ω) such that d(z, x)≤d(z, y) +e(z) for a.a.z∈Ω, ally < x≤0 andd(z, x)→+∞

for a.a.z∈Ω asx→ −∞;

(iv) with q∈(1, p) as in hypothesis (H1) (iv), we have lim_x→0⁻ _|x|^f(z,x)q−2x = +∞uniformly for a.a.z∈Ω;

(v) for everyρ >0 there existsξb_ρ>0 such that for a.a.z∈Ω the function x→f(z, x) +ξbρ|x|^p−2xis nondecreasing on [−ρ,0].

We introduce the two sets:

L⁰={λ >0 : problem (1.1) has a negative solution}, S_λ⁰ = set of negative solutions of (1.1).

Reasoning as we did above for positive solutions, we have the following bifurcation- type result describing the dependence of the set of negative solutions on the pa- rameterλ >0.

Theorem 3.9. If hypotheses (H1), (H2), (H3⁰) hold, then there exists a critical parameter value λ⁰_∗>0 such that

(a) for allλ > λ⁰_∗problem(1.1)has at least two negative solutionsv0,vb∈ −D+, v06=bv;

(b) forλ=λ⁰_∗ problem (1.1)has at least one negative solutionv∗∈ −D+; (c) for allλ∈(0, λ⁰_∗)problem (1.1)has no negative solutions.

We can generate extremal negative solutions. In this caseS⁰_λis upward directed, that is, if v₁, v₂ ∈S⁰_λ ⊆ −D+, we can find v ∈ S_λ⁰ such thatv₁ ≤v, v₂ ≤v (see [12]). So, in this case we produce the biggest negative solution for problem (1.1).

Proposition 3.10. If hypotheses (H1), (H2), (H3’)hold and λ∈ L⁰ = [λ⁰_∗,+∞), then problem (1.1)has a biggest negative solutionbv_λ∈ −D₊and the mapσ⁰:L⁰→ C¹(Ω) defined by σ⁰(λ) = vbλ is strictly increasing in the sense that λ⁰_∗ ≤ λ < η impliesbv_η−bv_λ∈intCb₊ andσ⁰(·)is also right continuous.

4. Nodal solutions

Letbλ∗= max{λ∗, λ⁰_∗}. Suppose that the conditions off(z,·) are bilateral (that is, valid on all ofR). Then Proposition 3.8 and 3.10 guarantee that for allλ≥λb∗

problem (1.1) has a smallest positive solution bu_λ ∈ D₊ and a biggest negative solutionbv_λ∈ −D+. Using these two extremal constant sign solutions of (1.1), we can produce a nodal (sign-changing) solution.

Now the hypotheses on the reactionf(z, x) are as follows:

(H3”) f : Ω×R→ Ris a Carath´eodory function such that f(z,0) = 0 for a.a.

z∈Ω and

(i) η(z)|x|^p ≤ f(z, x)x ≤ α(z)[1 +|x|^r] for a.a. z ∈ Ω, all x ∈ R, with η, α∈L^∞(Ω),ξηand p < r < p^∗;

(ii) if F(z, x) = Rx

0 f(z, s)ds, then limx→±∞F(z,x)

|x|^p = +∞ uniformly for a.a.z∈Ω;

(iii) ifd(z, x) =f(z, x)x−pF(z, x), then there existse∈L^∞(Ω) such that d(z, x)≤d(z, y) +e(z) for a.a. z∈Ω, all 0≤x≤y ory≤x≤0 and d(z, x)→+∞for a.a.z∈Ω as x→ ±∞;

(iv) withq < pas in hypothesis (H1) (iv), there existsτ∈(1, q) andδ0>0 such thatbc0|x|^τ ≤f(z, x)x≤τ F(z, x) for a.a.z∈Ω, all|x| ≤δ0, some bc0>0;

(v) for every ρ > 0, there exists ξbρ > 0 such that for a.a. z ∈ Ω, the functionx→f(z, x) +ξb_ρ|x|^p−2xis nondecreasing on [−ρ, ρ].

Note that now our condition onf(z,·) near zero is stronger than before.

Proposition 4.1. If hypotheses(H1), (H2), (H3”) hold andλ≥bλ∗, then problem (1.1)admits a nodal solution y_λ∈C¹(Ω).

Proof. Using the extremal constant sign solutions bu_λ ∈ D₊ and bv_λ ∈ −D₊, we introduce the Carath´eodory function

bγ(z, x) =







f(z,bv_λ(z)) +µ|bbv_λ(z)|^p−2bv_λ(z) ifx <bv_λ(z),

f(z, x) +µ|x|b ^p−2x ifbvλ(z)≤x≤ubλ(z), f(z,bu_λ(z)) +µbub_λ(z)^p−1 ifbu_λ(z)< x,

(4.1) with bµ≥ kξk_∞. Also we consider the positive and negative truncations ofbγ(z,·), namely the Carath´eodory functions

γb±(z, x) =bγ(z,±x^±). (4.2) We set Γ(z, x) =b Rx

0 bγ(z, s)ds and Γb_±(z, x) = Rx

0 bγ_±(z, s)ds and consider the C¹- functionalsbσ,bσ_± :W^1,p(Ω)→Rdefined by

bσ(u) = 1

pγ(u) +λ+bµ p kuk^p_p−

Z

Ω

Γ(z, u)dzb bσ_±(u) =1

pγ(u) +λ+µb p kuk^p_p−

Z

Ω

bΓ_±(z, u)dz for allu∈W^1,p(Ω).

Using (4.1) and (4.2), we can easily show that K

bσ⊆[bv_λ,ub_λ]∩C¹(Ω), K

σb+⊆[0,bu_λ]∩C₊, K

bσ−⊆[bv_λ,0]∩(−C₊).

The extremality ofubλand bvλ implies that K

σb⊆[bv_λ,ub_λ]∩C¹(Ω), K

bσ₊={0,bu_λ}, K

σb− ={0,vb_λ}. (4.3)