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 parameter λ > 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 ⊆ RN be a bounded domain with a C2-boundary ∂Ω. In this paper we study the following nonlinear nonhomogeneous Neumann problem −div a(∇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 map a : RN → RN 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 the p-Laplacian (1 < p < +∞) and the (p, q)-Laplacian (1 < q < p < +∞), that is, the sum of a p-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 term u → [ξ(z) + λ]up−1 with the potential function ξ ∈ 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 Carathéodory 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 function x→ f(z, x) is (p− 1)-superlinear near +∞. However, the superlinearity of f(z, ·) is not expressed via the usual for such 2010 Mathematics 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 2 N. S. PAPAGEORGIOU, C. VETRO, F. VETRO EJDE-2020/12 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 that f(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 parameter λ > 0 of the set of positive solu- tions of problem (1.1). More precisely, we show that there exists a critical parameter value λ∗ > 0 such that • for all λ > λ∗ problem (1.1) has at least two positive solutions; • for λ = λ∗ problem (1.1) has at least a positive solution; • for all λ ∈ (0, λ∗) problem (1.1) has no positive solution. In addition we show that for every λ ∈ L = [λ∗,+∞), problem (1.1) has a small- est positive solution uλ and we examine the monotonicity and continuity properties of the map λ→ uλ. With the conditions valid on the negative semiaxis R− = (−∞, 0], we can have analogous results for the negative solutions. In particular we can produce a biggest negative solution vλ for problem (1.1). When the conditions are bilateral (that is, valid for all x ∈ R and not only on the semiaxes), then using the two extremal con- stant sign solutions uλ and vλ, 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-Rǎdulescu [11]. In [8] the differential operator is also nonhomogeneous but the conditions on the map a(·) are more restrictive excluding, for example, the important case of the (p, q)-Laplacian. Also ξ ≡ 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 the p-Laplacian and ξ ≡ 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 space W 1,p(Ω) and the Banach space C1(Ω). By ‖ · ‖ we denote the norm of W 1,p(Ω) defined by ‖u‖ = [‖u‖pp + ‖∇u‖pp]1/p for all u ∈W 1,p(Ω). The Banach space C1(Ω) is ordered with positive (order) cone C+ = {u ∈ C1(Ω) : u(z) ≥ 0 for all z ∈ Ω}. This cone has a nonempty interior given by D+ = {u ∈ C1(Ω) : u(z) > 0 for all z ∈ Ω}. We will also consider another order cone for C1(Ω), namely the cone Ĉ+ = { u ∈ C1(Ω) : u(z) ≥ 0 for all z ∈ Ω, ∂u ∂n ∣∣ ∂Ω∩u−1(0) ≤ 0 } . This cone too has a nonempty interior int Ĉ+ = { u ∈ Ĉ+ : u(z) > 0 for all z ∈ Ω, ∂u ∂n ∣∣ ∂Ω∩u−1(0) < 0 } . EJDE-2020/12 POSITIVE AND NODAL SOLUTIONS 3 Given x ∈ R, we set x± = max{±x, 0}. For any measurable function u : Ω → RN , we define u±(z) = u(z)± for all z ∈ Ω. If u ∈W 1,p(Ω), then u± ∈W 1,p(Ω). If u, v ∈W 1,p(Ω) and v ≤ 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 intC1(Ω)[v, u] we denote the interior in C1(Ω) of [v, u] ∩ C1(Ω). Let X be a Banach space, ϕ ∈ C1(X,R) and c ∈ R. We define Kϕ = {x ∈ X : ϕ′(x) = 0} (the critical set of ϕ), ϕc = {x ∈ X : ϕ(x) ≤ c} (the sublevel set of ϕ at c). Let (A,B) be a topological pair such that B ⊆ A ⊆ X. By Hk(A,B), k ∈ N0, we denote the kth-relative singular homology group for the pair (A,B) with integer coefficients. If u ∈ Kϕ is isolated and ϕ(u) = c, then the critical groups of ϕ at u are defined by Ck(ϕ, u) = Hk(ϕc ∩ U,ϕc ∩ U \ {u}) for all k ∈ N0, with U being a neighborhood of u such that Kϕ ∩ ϕc ∩ 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 〈·, ·〉 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: un w−→ u in X and lim sup n→+∞ 〈A(un), un − u〉 ≤ 0 ⇒ un → u in X. Also, we say that ϕ ∈ C1(X,R) satisfies the “C-condition”, if the following property holds: Every sequence {un}n≥1 ⊆ X such that {ϕ(un)}n≥1 ⊆ R is bounded and (1 + ‖un‖X)ϕ′(un)→ 0 in X∗ as n → +∞, admits a strongly convergent subsequence. If h1, h2 ∈ L∞(Ω), then we write h1 � h2 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, by Nf (·) we denote the Nemytskii operator corresponding to f , that is, Nf (u)(·) = f(·, u(·)) for every u : Ω→ R measurable, and by | · |N we denote the Lebesgue measure on RN . Let ϑ ∈ C1(0,+∞) with ϑ(t) > 0 for all t > 0. We assume that 0 < ĉ ≤ ϑ′(t)t ϑ(t) ≤ c0 and c1t p−1 ≤ ϑ(t) ≤ c2[ts−1 + tp−1] for all t > 0, with 1 ≤ s < p < +∞, c1, c2 > 0. The hypotheses on the map a(·) are as follows: (H1) a(y) = a0(|y|)y for all y ∈ RN with a0(t) > 0 for all t > 0, and (i) a0 ∈ C1(0,+∞), t → a0(t)t is strictly increasing on (0,+∞), a0(t)t→ 0+ as t→ 0+ and limt→0+ a′0(t)t a0(t) > −1; 4 N. S. PAPAGEORGIOU, C. VETRO, F. VETRO EJDE-2020/12 (ii) there exists c3 > 0 such that |∇a(y)| ≤ c3 ϑ(|y|) |y| for all y ∈ RN \ {0}; (iii) (∇a(y)ξ, ξ)RN ≥ ϑ(|y|) |y| |ξ| 2 for all y ∈ RN \ {0}, ξ ∈ RN ; (iv) If G0(t) = ∫ t 0 a0(s)sds, then there exist q ∈ (1, p) and c∗, c4 > 0 such that lim sup t→0+ qG0(t) tq ≤ c∗, t→ G0(t1/q) is convex, c4t p ≤ a0(t)t2 − qG0(t) for all t > 0, 0 ≤ pG0(t)− a0(t)t2 for all t > 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 primitive G0(·) is strictly convex and strictly increasing. We set G(y) = G0(|y|) for all y ∈ RN . Then G(·) is convex and ∇G(y) = G′0(|y|) y |y| = a0(|y|)y = a(y) for all y ∈ RN \ {0}. So, G(·) is the primitive of a(·). This fact and the convexity of G(·) imply that G(y) ≤ (a(y), y)RN for all y ∈ RN . (2.1) Hypotheses (H1) lead to the following lemma summarizing the main properties of the map y → a(y) (see Papageorgiou-Rǎdulescu [9]). Lemma 2.2. If hypotheses (H1)(i)(ii)(iii) hold, then (a) a(·) is continuous, strictly monotone, hence maximal monotone; (b) there exists c5 > 0 such that |a(y)| ≤ c5[|y|s−1 + |y|p−1] for all y ∈ RN ; (c) (a(y), y)RN ≥ c1 p−1 |y| p for all y ∈ RN . 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 exists c6 > 0 such that c1 p(p−1) |y| p ≤ G(y) ≤ c6[1 + |y|p] for all y ∈ RN . 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−2y with 1 < p < +∞. This map corresponds to the p-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 a p-Laplacian and of a q-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−2 2 y with 1 < p < +∞. This map corresponds to the extended capillary differential operator. EJDE-2020/12 POSITIVE AND NODAL SOLUTIONS 5 (d) a(y) = [1 + 1 1+|y|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]). Let A : W 1,p(Ω)→W 1,p(Ω)∗ be the nonlinear operator defined by 〈A(u), h〉 = ∫ Ω (a(∇u),∇h)RNdz for all u, h ∈W 1,p(Ω). From Gasiński-Papageorgiou [4] (Problem 2.192, p. 279), we have the following result. Proposition 2.5. If hypotheses (H1) hold, then the map A(·) is continuous, mono- tone (hence maximal monotone too) and of type (S)+. The following strong comparison principle by Papageorgiou-Rǎdulescu-Repovš [13], will be useful in our analysis of problem (1.1). Proposition 2.6. If hypotheses (H1) hold, ξ̂ ∈ L∞(Ω) with ξ̂(z) ≥ 0 for a.a. z ∈ Ω, h1, h2 ∈ L∞(Ω) with 0 < η ≤ h2(z)− h1(z) for a.a. z ∈ Ω and u, v ∈ C1,α(Ω) with α ∈ (0, 1], v ≤ u and − div a(∇v(z)) + ξ̂(z)|v(z)|p−2v(z) = h1(z) for a.a. z ∈ Ω, − div a(∇u(z)) + ξ̂(z)|u(z)|p−2u(z) = h2(z) for a.a. z ∈ Ω, then u− v ∈ int Ĉ+. Next we introduce hypotheses on the potential function ξ(z) and on the reaction term f(z, x). (H2) ξ ∈ L∞(Ω). (H3) f : Ω × R → R is a Carathéodory function such that f(z, 0) = 0 for a.a. z ∈ Ω and (i) η(z)xp−1 ≤ f(z, x) ≤ α(z)[1 + xr−1] for a.a. z ∈ Ω, all x ≥ 0, with η, α ∈ L∞(Ω), ξ+ � η and p < r < p∗ = { Np N−p if N > p +∞ if p ≥ N ; (ii) if F (z, x) = ∫ x 0 f(z, s)ds, then limx→+∞ F (z,x) xp = +∞ uniformly for a.a. z ∈ Ω; (iii) if d(z, x) = f(z, x)x− pF (z, x), then there exists e ∈ L1(Ω) such that d(z, x) ≤ d(z, y) + e(z) for a.a. z ∈ Ω, all 0 ≤ x ≤ y and d(z, x)→ +∞ for a.a. z ∈ Ω as x→ +∞; (iv) with q ∈ (1, p) as in hypothesis (H1)(iv), we have limx→0+ f(z,x) xq−1 = +∞ uniformly for a.a. z ∈ Ω; (v) for each ρ > 0 there exists ξ̂ρ > 0 such that for a.a. z ∈ Ω the function x→ f(z, x) + ξ̂ρx 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, all x ≤ 0. (2.2) Hypotheses (H3)(ii)(iii) imply that lim x→+∞ f(z, x) xp−1 = +∞ uniformly for a.a. z ∈ Ω. 6 N. S. PAPAGEORGIOU, C. VETRO, F. VETRO EJDE-2020/12 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 exist ϑ > p and M > 0 such that 0 < ϑF (z, x) ≤ f(z, x)x for a.a. z ∈ Ω, all x ≥M, (2.3) 0 < ess infΩ F (·,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, all x ≥M , some c7 > 0, ⇒ c7x ϑ−1 ≤ f(z, x) for a.a. z ∈ Ω, all x ≥M , (see (2.3)). (2.5) From (2.5) we see that the AR-condition implies that f(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 exists M > 0 such that for a.a. z ∈ Ω the function x→ 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 if x ≤ 1, xp−1 lnx+ η(z)xτ−1 if 1 < x, with η ∈ 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 follows γ : W 1,p(Ω)→ R is the C1-functional defined by γ(u) = ∫ Ω pG(∇u)dz + ∫ Ω ξ(z)|u|pdz for all u ∈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, then L 6= ∅ and, for every λ ∈ L, ∅ 6= Sλ ⊆ D+. Proof. Let µ > ‖ξ‖∞ (see hypothesis (H2)) and consider the following auxiliary Neumann problem −div a(∇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 that µ > ‖ξ‖∞, we see that the left hand side of (3.1) is continuous, strictly monotone and coercive. Therefore EJDE-2020/12 POSITIVE AND NODAL SOLUTIONS 7 problem (3.1) admits a unique solution u ∈ W 1,p(Ω), u 6= 0. Moreover, the non- linear regularity theory (see [7]) and the nonlinear maximum principle (see [15]), imply that u ∈ D+. We set M0 = ‖Nf (u)‖∞ (see hypothesis (H3) (i)), m0 = min Ω u > 0 (recall that u ∈ D+), λ = µ+ M0 mp−1 0 > 0. We have − div a(∇u(z)) + [ξ(z) + λ]u(z)p−1 = −div a(∇u(z)) + [ξ(z) + µ]u(z)p−1 +M0 (u(z) m0 )p−1 ≥ 1 +M0 (see (3.1) and recall that u(z) ≥ m0 for all z ∈ Ω) > f(z, u(z)) for a.a. z ∈ Ω. (3.2) We introduce the Carathéodory function f̂(z, x) = { f(z, x+) if x ≤ u(z), f(z, u(z)) if u(z) < x. (3.3) (see (2.2)). We set F̂ (z, x) = ∫ x 0 f̂(z, s)ds and consider the C1-functional ϕ̂ : W 1,p(Ω) → R defined by ϕ̂(u) = 1 p γ(u) + λ p ‖u‖pp − ∫ Ω F̂ (z, u)dz for all u ∈W 1,p(Ω). From (3.3) and since λ > µ > ‖ξ‖∞, we see that ϕ̂(·) is coercive. Also using the Sobolev embedding theorem, we show that ϕ̂(·) is sequentially weakly lower semicontinuous. So, by the Weierstrass-Tonelli theorem, we can find u0 ∈W 1,p(Ω) such that ϕ̂(u0) = inf[ϕ̂(u) : 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 q xq for a.a. z ∈ Ω, all x ∈ [0, δ]. (3.5) Given u ∈ D+, we choose t ∈ (0, 1) small such that t|∇u(z)| ≤ δ and tu(z) ≤ δ for all z ∈ Ω. (3.6) Using (3.5) and (3.6), we have ϕ̂(tu) ≤ c∗0 q tq‖∇u‖qq + tp p [‖ξ‖∞ + λ]‖u‖pp − η q tq‖u‖qq = tq[c8 − ηc9] + tpc10 for some c8, c9, c10 > 0. Since η > c∗0 is arbitrary, by choosing η > c8 c9 we obtain ϕ̂(tu) ≤ c10t p − c11t q for all t > 0, and some c11 > 0. 8 N. S. PAPAGEORGIOU, C. VETRO, F. VETRO EJDE-2020/12 Recall that q < p. So, by choosing t ∈ (0, 1) even smaller if necessary, we have ϕ̂(tu) < 0 implies ϕ̂(u0) < 0 = ϕ̂(0) (see (3.4)) which in turn implies u0 6= 0. From (3.4) we have that ϕ̂′(u0) = 0 implies 〈A(u0), h〉+ ∫ Ω [ξ(z) + λ]|u0|p−2u0hdz = ∫ Ω f̂(z, u0)hdz ∀h ∈W 1,p(Ω). (3.7) In (3.7) first we choose h = −u−0 ∈W 1,p(Ω). Then c1 p− 1 ‖∇u−0 ‖pp + ∫ Ω [ξ(z) + λ](u−0 )pdz ≤ 0 (see Lemma 2.2 and (3.3)), ⇒ c12‖u−0 ‖p ≤ 0 for some c12 > 0 (recall λ > µ > ‖ξ‖∞), ⇒ u0 ≥ 0, u0 6= 0. Next in (3.7) we choose h = (u0 − u)+ ∈W 1,p(Ω). Then we have 〈A(u0), (u0 − u)+〉+ ∫ Ω [ξ(z) + λ]up−1 0 (u0 − u)+dz = ∫ Ω f(z, u)(u0 − u)+dz (see (3.3)), ≤ 〈A(u), (u0 − u)+〉+ ∫ Ω [ξ(z) + λ]up−1(u0 − u)+dz (see (3.2)), which implies u0 ≤ u (see Proposition 2.5 and recall that λ > µ > ‖ξ‖∞). So, we have proved that u0 ∈ [0, u], u0 6= 0. (3.8) From (3.3), (3.7) and (3.8) it follows that − div a(∇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 div a(∇u0(z)) ≤ [‖η‖∞ + ‖ξ‖∞ + 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 set L, namely we show that L is an upper half line. In addition we establish a kind of monotonicity property for the solution multifunction λ→ Sλ. Proposition 3.2. If hypotheses (H1)–(H3) hold, λ ∈ L, uλ ∈ Sλ ⊆ D+ and η > λ, then η ∈ L and there exists uη ∈ Sη ⊆ D+ such that uλ − uη ∈ int Ĉ+. Proof. We have − div a(∇uλ(z)) + [ξ(z) + λ]uλ(z)p−1 = f(z, uλ(z)) < −div a(∇uλ(z)) + [ξ(z) + η]uλ(z)p−1 for a.a. z ∈ Ω (since η > λ). EJDE-2020/12 POSITIVE AND NODAL SOLUTIONS 9 We introduce the Carathéodory function k(z, x) = { f(z, x+) + µ̂(x+)p−1 if x ≤ uλ(z), f(z, uλ(z)) + µ̂uλ(z)p−1 if uλ(z) < x, (3.11) where µ̂ ≥ ‖ξ‖∞. We set K(z, x) = ∫ x 0 k(z, s)ds and consider the C1-functional ψ̂η : W 1,p(Ω)→ R defined by ψ̂η(u) = 1 p γ(u) + η + µ̂ p ‖u‖pp − ∫ Ω K(z, u)dz for all u ∈W 1,p(Ω). From (3.11) and since µ̂ ≥ ‖ξ‖∞ and η > λ > 0, we see that ψ̂η(·) is coercive. Also it is sequentially weakly lower semicontinuous. So, we can find uη ∈ W 1,p(Ω) such that ψ̂η(uη) = inf[ψ̂η(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 ψ̂η(uη) < 0 = ψ̂η(0) ⇒ uη 6= 0. From (3.12) we have that ψ̂′η(uη) = 0 implies 〈A(uη), h〉+ ∫ Ω [ξ(z)+η+µ]|uη|p−2uηhdz = ∫ Ω k(z, uη)hdz ∀h ∈W 1,p(Ω). (3.13) In (3.13) we choose h = −u−η ∈ W 1,p(Ω) and h = (uη − uλ)+ ∈ W 1,p(Ω) and as in the proof of Proposition 3.1, we show that uη ∈ [0, uλ], uη 6= 0. (3.14) From (3.11), (3.13) and (3.14) we infer that η ∈ L and uη ∈ Sη ⊆ D+ (see Proposition 3.1), uη ≤ uλ. Let ρ = ‖uλ‖∞ and let ξ̂ρ > 0 be as postulated by hypothesis (H3)(v). We have − div a(∇uλ) + [ξ(z) + η + ξ̂ρ]u p−1 λ = f(z, uλ) + ξ̂ρu p−1 λ + (η − λ)up−1 λ (since uλ ∈ Sλ) ≥ f(z, uη) + ξ̂ρu p−1 η + (η − λ)up−1 η (see (H3)(v) and recall uη ≤ uλ) > −div a(∇uη) + [ξ(z) + η + ξ̂ρ]u p−1 η for a.a. z ∈ Ω (since uη ∈ Sη). (3.15) Let mη = minΩ uη > 0 (recall that uη ∈ D+). We have (η − λ)up−1 η ≥ (η − λ)mp−1 η > 0 (since η > λ). Then from (3.15) and Proposition 2.6, it follows that uλ − uη ∈ int Ĉ+. � Let λ∗ = inf L. Proposition 3.3. If hypotheses (H1)–(H3) hold, then λ∗ > 0. Proof. Let ϕλ : W 1,p(Ω) → R be the energy (Euler) functional for problem (1.1) defined by ϕλ(u) = 1 p γ(u) + λ p ‖u‖pp − ∫ Ω F (z, u)dz for all u ∈W 1,p(Ω). 10 N. S. PAPAGEORGIOU, C. VETRO, F. VETRO EJDE-2020/12 Arguing by contradiction, suppose that λ∗ = 0. Let {λn}n≥1 ⊆ L such that λn ↓ 0. We fix λ > λ1. For every n ∈ N and ûn ∈ Sλn ⊆ D+, on account of Proposition 3.2 and its proof we can find unλ ∈ Sλ ⊆ D+ such that ϕλ(unλ) < 0, unλ ≤ ûn. We have − div a(∇un+1 λ ) + [ξ(z) + λn]un+1 λ ≤ f(z, un+1 λ ) for a.a. z ∈ Ω, (3.16) −div a(∇ûn+1) + [ξ(z) + λn]ûp−1 n+1 ≥ f(z, ûn+1) for a.a. z ∈ Ω. (3.17) With µ̂ ≥ ‖ξ‖∞ we introduce the Carathéodory function kn(z, x) =  f(z, un+1 λ (z)) + µ̂un+1 λ (z)p−1 if x < un+1 λ (z), f(z, x) + µ̂xp−1 if un+1 λ (z) ≤ x ≤ ûn+1(z), f(z, ûn+1(z)) + µ̂ûn+1(z)p−1 if ûn+1 < x. (3.18) We set Kn(z, x) = ∫ x 0 kn(z, s)ds and consider the C1-functional ϕ̃λn : W 1,p(Ω)→ R defined by ϕ̃λn(u) = 1 p γ(u) + λn + µ̂ p ‖u‖pp − ∫ Ω Kn(z, u)dz for all u ∈W 1,p(Ω), with µ̂ ≥ ‖ξ‖∞. Evidently ϕ̃λn(·) is coercive (see (3.18)) and sequentially weakly lower semicontinuous and so we can find un ∈W 1,p(Ω) such that ϕ̃λn(un) = inf[ϕ̃λn(u) : u ∈W 1,p(Ω)], ⇒ ϕ̃′λn(un) = 0, ⇒ 〈A(un), h〉+ ∫ Ω [ξ(z) + λn + µ̂]|un|p−2unhdz = ∫ Ω kn(z, un)hdz (3.19) for all h ∈W 1,p(Ω). Choosing h = (unλ − un)+ ∈W 1,p(Ω) and h = (un − ûn+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 ∈ [unλ, ûn+1] ∩D+ (by the nonlinear regularity theory). We have ϕ̃λn(unλ) ≤ 1 p γ(unλ) + λn p ‖unλ‖pp − ∫ Ω f(z, unλ)unλdz (see (3.18)) ≤ 1 p γ(unλ) + λ p ‖unλ‖pp − ∫ Ω pF (z, unλ)dz + ‖e‖1 (see (H3)(iii)) ≤ 1 p γ(unλ) + λ p ‖unλ‖pp − ∫ Ω F (z, unλ)dz + ‖e‖1 (since F ≥ 0) = ϕλ(unλ) + ‖e‖1 < ‖e‖1, which implies ϕ̃λn(un) < ‖e‖1 for all n ∈ N (see (3.19)). This in turn implies ϕλn(un) ≤ c13 for some c13 > 0 and all n ∈ N (see (3.18)). Therefore we have produced a sequence {un}n≥1 ⊆W 1,p(Ω) such that un ∈ Sλn ⊆ D+ and ϕλn(un) ≤ c13 for all n ∈ N. (3.20) EJDE-2020/12 POSITIVE AND NODAL SOLUTIONS 11 From (3.20) we have 〈A(un), h〉+ ∫ Ω [ξ(z) + λn]up−1 n hdz = ∫ Ω f(z, un)hdz for all h ∈W 1,p(Ω), and all n ∈ N, (3.21) γ(un) + λn‖un‖pp − ∫ Ω pF (z, un)dz ≤ pc13 for all n ∈ N. (3.22) In (3.21) we choose h = un ∈W 1,p(Ω). Then − γ(un)− λn‖un‖pp + ∫ Ω f(z, un)undz = 0 for all n ∈ N. (3.23) We add (3.22) and (3.23) to obtain∫ Ω [f(z, un)un − pF (z, un)]dz = ∫ Ω d(z, un)dz ≤ pc13 for all n ∈ 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 ‖un‖ → +∞. (3.25) We set yn = un/‖un‖ for n ∈ N. Then ‖yn‖ = 1, yn ≥ 0 for all n ∈ N. So, we may assume that yn w−→ y in W 1,p(Ω) and yn → y in Lr(Ω), y ≥ 0. First, we assume that y 6= 0. Let Ω+ = {z ∈ Ω : y(z) > 0}. Then |Ω+|N > 0 (recall that y ≥ 0). From (3.25) it follows that un(z) → +∞ for all z ∈ Ω+. So, we have d(z, un(z))→ +∞ for a.a. z ∈ Ω (see hypothesis (H3)(iii)). This implies∫ 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, all x ≥ 0. (3.27) Then we have∫ Ω d(z, un)dz = ∫ Ω+ d(z, un)dz + ∫ Ω\Ω+ d(z, un)dz ≥ ∫ Ω+ d(z, un)dz − ‖e‖1 for all n ∈ N (see (3.27)), which implies ∫ Ω d(z, un)dz → +∞ as n → +∞ (see (3.26)). This contradicts (3.24). Now we assume that y = 0. Let τ > 0 and set vn = (pτ)1/pyn ∈W 1,p(Ω) for all n ∈ N. Let γ : W 1,p(Ω)→ R be the C1-functional defined by γ(u) = c1 p− 1 ‖∇u‖pp + ∫ Ω ξ(z)|u|pdz for all u ∈W 1,p(Ω). We introduce the C1-functionals ϕλn : W 1,p(Ω)→ R, n ∈ N, defined by ϕλn(u) = 1 p γ(u) + λn p ‖u‖pp − ∫ Ω F (z, u)dz for all u ∈W 1,p(Ω). 12 N. S. PAPAGEORGIOU, C. VETRO, F. VETRO EJDE-2020/12 Let tn ∈ [0, 1] be such that ϕλn(tnun) = max[ϕλn(tun) : 0 ≤ t ≤ 1] for all n ∈ N. (3.28) On account of (3.25), we see that we can find n0 ∈ N such that (pτ)1/p 1 ‖un‖ ≤ 1 for all n ≥ n0. (3.29) From (3.28) and (3.29) it follows that ϕλn(tnun) ≥ ϕλn(vn) = τ [γ(yn) + [λn + µ̂]‖yn‖pp]− ∫ Ω [F (z, vn) + µ̂ p vpn]dz for all n ≥ n0, with µ̂ ≥ ‖ξ‖∞ ≥ τc14 − ∫ Ω [F (z, vn) + µ̂ p vpn]dz for some c14 > 0, all n ≥ n0 (since µ̂ ≥ ‖ξ‖∞). (3.30) Evidently ∫ Ω [F (z, vn) + µ̂ p v p n]dz → 0 as n→ +∞ (recall y = 0). Hence from (3.30) it follows that ϕλn(tnun) ≥ τ 2 c14 for all n ≥ n1 ≥ n0. Since τ > 0 is arbitrary, we infer that ϕλn(tnun)→ +∞ as n→ +∞. (3.31) We have ϕλn(0) = 0 and ϕλn(un) ≤ ϕλn(un) ≤ c13 for all n ∈ N (see (3.20)). Then on account of (3.31), we have tn ∈ (0, 1) for all n ≥ n2. (3.32) From (3.28) and (3.32) it follows that d dt ϕλn(tun) ∣∣ t=tn = 0 for all n ≥ n2, ⇒ 〈ϕ′λn(tnun), tnun〉 = 0 for all n ≥ n2 (by the chain rule), ⇒ c1 p− 1 ‖∇(tnun)‖pp + ∫ Ω [ξ(z) + λn](tnun)pdz = ∫ Ω f(z, tnun)(tnun)dz for all n ≥ n2, ⇒ pϕλn(tnun) ≤ ∫ Ω d(z, tnun)dz ≤ ∫ Ω d(z, un)dz + ‖e‖1 for all n ≥ n2 (see hypothesis (H3) (iii) and (3.32)), ⇒ pϕλn(tnun) ≤ pc13 + ‖e‖1 for all n ≥ 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 un w−→ u∗ in W 1,p(Ω) and un → u∗ in Lr(Ω). (3.33) In (3.21) we choose h = un − u∗ ∈W 1,p(Ω), pass to the limit as n→ +∞ and use (3.33). Then we obtain lim n→+∞ 〈A(un), un − u∗〉 = 0, EJDE-2020/12 POSITIVE AND NODAL SOLUTIONS 13 which implies un → u∗ in W 1,p(Ω) (see Proposition 2.5). (3.34) So, if in (3.21) we pass to the limit as n → +∞ and use (3.34) and the fact that λn ↓ 0 (recall we have assumed that λ∗ = 0), we obtain 〈A(u∗), h〉+ ∫ Ω ξ(z)up−1 ∗ hdz = ∫ Ω f(z, u∗)hdz for all h ∈W 1,p(Ω). (3.35) In (3.35) we choose h ≡ 1. Then∫ Ω ξ(z)up−1 ∗ dz = ∫ Ω f(z, u∗)dz ≥ ∫ Ω η(z)up−1 ∗ dz (see (H3)(i)), which implies ∫ Ω [η(z)− ξ(z)]up−1 ∗ dz ≤ 0. (3.36) Note that hypotheses (H3)(i),(iv) imply that we can find c15 > 0 such that f(z, x) ≥ xq−1 − c15x r−1 for a.a. z ∈ Ω, all x ≥ 0. Evidently we can always assume that c15 > ‖ξ‖∞. We consider the following auxiliary Neumann problem −div a(∇u(z)) + ξ(z)u(z)p−1 = u(z)q−1 − c15u(z)r−1 in Ω, ∂u ∂n = 0, u > 0. From [9, Proposition 3.5] we know that this problem has a unique positive solution ũ ∈ D+. Let λ ∈ L and u ∈ Sλ ⊆ D+. We introduce the Carathéodory function β(z, x) = { (x+)q−1 − c15(x+)r−1 + µ̂(x+)p−1 if x ≤ u(z), u(z)q−1 − c15u(z)r−1 + µ̂u(z)p−1 if u(z) < x, (3.37) with µ̂ ≥ ‖ξ‖∞. We set B(z, x) = ∫ x 0 β(z, s)ds and consider the C1-functional σ̃λ : W 1,p(Ω)→ R defined by σ̃λ(u) = 1 p γ(u) + λ+ µ̂ p ‖u‖pp − ∫ Ω B(z, u)dz for u ∈W 1,p(Ω). The direct method of calculus of variations gives ũ0 ∈W 1,p(Ω) such that σ̃λ(ũ0) = inf[σ̃λ(u) : u ∈W 1,p(Ω)] < 0 = σ̃λ(0) (since q < p). So, ũ0 6= 0 and ũ0 ∈ Kσ̃λ ⊆ [0, u] ∩ C+ (see (3.37) and use the nonlinear regularity theory). Hence from (3.37) we infer that ũ0 = ũ ∈ D+ and so ũ ≤ u for all u ∈ Sλ, all λ ∈ L. It follows that ũ ≤ u∗ ⇒ ∫ Ω [η(z)− ξ(z)]up−1 ∗ dz > 0 (since ξ ≺ η) which contradicts (3.36). So, we conclude that λ∗ > 0. � Proposition 3.4. If hypotheses (H1)–(H3) hold and λ ∈ (λ∗,+∞), then problem (1.1) admits at least two positive solutions u0, û ∈ Sλ ⊆ D+. 14 N. S. PAPAGEORGIOU, C. VETRO, F. VETRO EJDE-2020/12 Proof. Let λ∗ < θ < λ < η. By Proposition 3.2, we can find uθ ∈ Sθ ⊆ D+, u0 ∈ Sλ ⊆ D+ and uη ∈ Sη ⊆ D+ such that uθ − u0 ∈ int Ĉ+ and u0 − uη ∈ int Ĉ+, ⇒ u0 ∈ intC1(Ω)[uη, uθ]. (3.38) We introduce the Carathéodory function j(z, x) = { f(z, uη(z)) + µ̂uη(z)p−1 if x ≤ uη(z), f(z, x) + µ̂xp−1 if uη(z) < x, (3.39) with µ̂ ≥ ‖ξ‖∞. We set J(z, x) = ∫ x 0 j(z, s)ds and consider the C1-functional ψ̂λ : W 1,p(Ω)→ R defined by ψ̂λ(u) = 1 p γ(u) + λ+ µ̂ p ‖u‖pp − ∫ Ω J(z, u)dz for all u ∈W 1,p(Ω). In addition, we introduce the following truncation of j(z, ·), j̃(z, x) = { j(z, x) if x ≤ uθ(z), j(z, uθ(z)) if uθ(z) < x. (3.40) This is a Carathéodory function. We set J̃(z, x) = ∫ x 0 j̃(z, s)ds and consider the C1-functional ψ̃λ : W 1,p(Ω)→ R defined by ψ̃λ(u) = 1 p γ(u) + λ+ µ̂ p ‖u‖pp − ∫ Ω J̃(z, u)dz for all u ∈W 1,p(Ω). From (3.39), (3.40) and the nonlinear regularity theory of Lieberman [7], we have Kψ̂λ ⊆ [uη) ∩D+ and Kψ̃λ ⊆ [uη, uθ] ∩D+. (3.41) From (3.39), (3.40), (3.41), we see that we may assume that Kψ̃λ = {u0}. (3.42) Otherwise we already have a second positive solution of (1.1), distinct from u0 and the proof is complete. Clearly ψ̃λ(·) is coercive (see (3.40)) and sequentially weakly lower semicontinu- ous. So, we can find ũ0 ∈W 1,p(Ω) such that ψ̃λ(ũ0) = inf[ψ̃λ(u) : u ∈W 1,p(Ω)] ⇒ ũ0 ∈ Kψ̃λ ⇒ ũ0 = u0 (see (3.42)). From (3.39) and (3.40) we see that ψ̂λ ∣∣ [uη,uθ] = ψ̃λ ∣∣ [uη,uθ] . Then from (3.38) it follows that u0 ∈ D+ is a local C1(Ω)-minimizer of ψ̂λ, ⇒ u0 ∈ D+ is a local W 1,p(Ω)-minimizer of ψ̂λ (3.43) (see Papageorgiou-Rǎdulescu [10]). From (3.41) we can assume that Kψ̂λ is finite. (3.44) EJDE-2020/12 POSITIVE AND NODAL SOLUTIONS 15 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 ψ̂λ(u0) < inf[ψ̂λ(u) : ‖u− u0‖ = ρ] = m̂λ. (3.45) By (H3)(ii), for u ∈ D+ we have ψ̂λ(tu)→ −∞ as t→ +∞. (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 ψ̂λ(·) satisfies the C-condition. (3.47) Then (3.45), (3.46), (3.47) permit the use of the mountain pass theorem. So, we can find û ∈W 1,p(Ω) such that û ∈ Kψ̂λ ⊆ [uη) ∩D+ (see (3.41)) and m̂λ ≤ ψ̂λ(û) (see (3.45)), ⇒ û 6= u0 (see (3.45)) and û ∈ Sλ ⊆ D+ (see (3.39)). � Proposition 3.5. If hypotheses (H1)–(H3) hold, then λ∗ ∈ L. Proof. Let {λn}n≥1 ⊆ L such that λn ↓ λ∗. From the proof of Proposition 3.3, we know that we can find un ∈ Sλn ⊆ D+, n ∈ N, such that ũ ≤ un and ϕλn(un) ≤ c16 for some c16 > 0, all n ∈ N. As in the proof of Proposition 3.3, we can show that un → u∗ in W 1,p(Ω). Then in the limit as n→ +∞ we have ũ ≤ u∗ and 〈A(u∗), h〉+ ∫ Ω [ξ(z) + λ∗]u p−1 ∗ hdz = ∫ Ω f(z, u∗)hdz, for all h ∈W 1,p(Ω), which implies u∗ ∈ Sλ∗ ⊆ D+, and so λ∗ ∈ L. � Note that Proposition 3.5 implies that L = [λ∗,+∞). Summarizing our results on the dependence of the set of positive solutions of (1.1) on the parameter λ > 0, we can state the following bifurcation-type result for big values of λ > 0. Theorem 3.6. If hypotheses (H1)–(H3) hold, then there exists a critical parameter value λ∗ > 0 such that (a) for all λ > λ∗ problem (1.1) has at least two positive solutions u0, û ∈ D+, u0 6= û; (b) for λ = λ∗ problem (1.1) has at least one positive solution u∗ ∈ D+; (c) for all λ ∈ (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 ûλ ∈ D+. Proof. From Papageorgiou-Rǎdulescu-Repovš [12] (see the proof of Proposition 7), we know that the solution set Sλ is downward directed (that is, if u1, u2 ∈ Sλ, then we can find u ∈ Sλ 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 inf Sλ = inf n≥1 un and 0 ≤ un ≤ u1 for all n ∈ N. (3.48) 16 N. S. PAPAGEORGIOU, C. VETRO, F. VETRO EJDE-2020/12 We have 〈A(un), h〉+ ∫ Ω [ξ(z) + λ]up−1 n hdz = ∫ Ω f(z, un)hdz for all h ∈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-Rǎdulescu [10] implies that we can find c16 > 0 such that un ∈ L∞(Ω) and ‖un‖∞ ≤ c16 for all n ∈ N. Then the nonlinear regularity theory of Lieberman [7] implies that there exist α ∈ (0, 1) and c17 > 0 such that un ∈ C1,α(Ω), ‖un‖C1,α(Ω) ≤ c17 for all n ∈ N. (3.50) From (3.50), the compact embedding of C1,α(Ω) into C1(Ω), and the monotonicity of {un}n≥1, we have un → ûλ in C1(Ω). (3.51) From the proof of Proposition 3.3, we know that ũ ≤ un for all n ∈ N, ⇒ ũ ≤ ûλ (see (3.51)), hence ûλ 6= 0. If in (3.49) we pass to the limit as n→ +∞ and use (3.51), we obtain 〈A(ûλ), h〉+ ∫ Ω [ξ(z) + λ]ûp−1 λ hdz = ∫ Ω f(z, ûλ)hdz for all h ∈W 1,p(Ω), ⇒ ûλ ∈ Sλ ⊆ D+ and ûλ = inf Sλ. � Next we examine the properties of the map L 3 λ→ ûλ ∈ C1(Ω). Proposition 3.8. If hypotheses (H1)–(H3) hold, then the map σ : L → C1(Ω) defined by σ(λ) = ûλ has the following properties: (a) σ(·) is strictly decreasing in the sense that λ∗ ≤ λ < η implies ûλ − ûη ∈ int Ĉ+; (b) σ(·) is right continuous. Proof. (a) Let ûλ ∈ D+ be the minimal positive solution of (1.1) (λ ∈ L). Accord- ing to Proposition 3.2, we can find uη ∈ Sη ⊆ D+ such that ûλ − uη ∈ int Ĉ+, ⇒ ûλ − ûη ∈ int Ĉ+ (since ûη ≤ uη) ⇒ σ(·) is strictly decreasing. (b) Let λn ↓ λ ∈ L. As in the proof of Proposition 3.3, we can find un ∈W 1,p(Ω) such that un ∈ Sλn ⊆ D+ and ϕλn(un) ≤ c18 for some c18 > 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 ≤ ûλn ≤ un for all n ∈ N, ⇒ {ûλn}n≥1 ⊆W 1,p(Ω). EJDE-2020/12 POSITIVE AND NODAL SOLUTIONS 17 From this and the nonlinear regularity theory of Lieberman [7] (see the proof of Proposition 11), we obtain (at least for a subsequence) that ûλn → ũλ in C1(Ω). (3.52) If ũλ 6= ûλ, then we can find z0 ∈ Ω such that ûλ(z0) < ũλ(z0) implies ûλ(z0) < ûλn(z0) for all n ≥ n0 (see (3.52)). This contradicts (a). Therefore by Urysohn’s criterion, for the original sequence we have ûλn → ûλ in C1(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 reaction f(z, x) are as follows (H3’) f : Ω × R → R is a Carathéodory 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), ξ � η and p < r < p∗; (ii) if F (z, x) = ∫ x 0 f(z, s)ds, then limx→−∞ F (z,x) |x|p = +∞ uniformly for a.a. z ∈ Ω; (iii) if d(z, x) = f(z, x)x− pF (z, x), then there exists e ∈ L1(Ω) such that d(z, x) ≤ d(z, y) + e(z) for a.a. z ∈ Ω, all y < x ≤ 0 and d(z, x)→ +∞ for a.a. z ∈ Ω as x→ −∞; (iv) with q ∈ (1, p) as in hypothesis (H1) (iv), we have limx→0− f(z,x) |x|q−2x = +∞ uniformly for a.a. z ∈ Ω; (v) for every ρ > 0 there exists ξ̂ρ > 0 such that for a.a. z ∈ Ω the function x→ f(z, x) + ξ̂ρ|x|p−2x is nondecreasing on [−ρ, 0]. We introduce the two sets: L′ = {λ > 0 : problem (1.1) has a negative solution}, S′λ = 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- rameter λ > 0. Theorem 3.9. If hypotheses (H1), (H2), (H3′) hold, then there exists a critical parameter value λ′∗ > 0 such that (a) for all λ > λ′∗ problem (1.1) has at least two negative solutions v0, v̂ ∈ −D+, v0 6= v̂; (b) for λ = λ′∗ problem (1.1) has at least one negative solution v∗ ∈ −D+; (c) for all λ ∈ (0, λ′∗) problem (1.1) has no negative solutions. We can generate extremal negative solutions. In this case S′λ is upward directed, that is, if v1, v2 ∈ S′λ ⊆ −D+, we can find v ∈ S′λ such that v1 ≤ 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 ∈ L′ = [λ′∗,+∞), then problem (1.1) has a biggest negative solution v̂λ ∈ −D+ and the map σ′ : L′ → C1(Ω) defined by σ′(λ) = v̂λ is strictly increasing in the sense that λ′∗ ≤ λ < η implies v̂η − v̂λ ∈ int Ĉ+ and σ′(·) is also right continuous. 18 N. S. PAPAGEORGIOU, C. VETRO, F. VETRO EJDE-2020/12 4. Nodal solutions Let λ̂∗ = max{λ∗, λ′∗}. Suppose that the conditions of f(z, ·) are bilateral (that is, valid on all of R). Then Proposition 3.8 and 3.10 guarantee that for all λ ≥ λ̂∗ problem (1.1) has a smallest positive solution ûλ ∈ D+ and a biggest negative solution v̂λ ∈ −D+. Using these two extremal constant sign solutions of (1.1), we can produce a nodal (sign-changing) solution. Now the hypotheses on the reaction f(z, x) are as follows: (H3”) f : Ω × R → R is a Carathéodory 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), ξ � η and p < r < p∗; (ii) if F (z, x) = ∫ x 0 f(z, s)ds, then limx→±∞ F (z,x) |x|p = +∞ uniformly for a.a. z ∈ Ω; (iii) if d(z, x) = f(z, x)x− pF (z, x), then there exists e ∈ L∞(Ω) such that d(z, x) ≤ d(z, y) + e(z) for a.a. z ∈ Ω, all 0 ≤ x ≤ y or y ≤ x ≤ 0 and d(z, x)→ +∞ for a.a. z ∈ Ω as x→ ±∞; (iv) with q < p as in hypothesis (H1) (iv), there exists τ ∈ (1, q) and δ0 > 0 such that ĉ0|x|τ ≤ f(z, x)x ≤ τF (z, x) for a.a. z ∈ Ω, all |x| ≤ δ0, some ĉ0 > 0; (v) for every ρ > 0, there exists ξ̂ρ > 0 such that for a.a. z ∈ Ω, the function x→ f(z, x) + ξ̂ρ|x|p−2x is nondecreasing on [−ρ, ρ]. Note that now our condition on f(z, ·) near zero is stronger than before. Proposition 4.1. If hypotheses (H1), (H2), (H3”) hold and λ ≥ λ̂∗, then problem (1.1) admits a nodal solution yλ ∈ C1(Ω). Proof. Using the extremal constant sign solutions ûλ ∈ D+ and v̂λ ∈ −D+, we introduce the Carathéodory function γ̂(z, x) =  f(z, v̂λ(z)) + µ̂|v̂λ(z)|p−2v̂λ(z) if x < v̂λ(z), f(z, x) + µ̂|x|p−2x if v̂λ(z) ≤ x ≤ ûλ(z), f(z, ûλ(z)) + µ̂ûλ(z)p−1 if ûλ(z) < x, (4.1) with µ̂ ≥ ‖ξ‖∞. Also we consider the positive and negative truncations of γ̂(z, ·), namely the Carathéodory functions γ̂±(z, x) = γ̂(z,±x±). (4.2) We set Γ̂(z, x) = ∫ x 0 γ̂(z, s)ds and Γ̂±(z, x) = ∫ x 0 γ̂±(z, s)ds and consider the C1- functionals σ̂, σ̂± : W 1,p(Ω)→ R defined by σ̂(u) = 1 p γ(u) + λ+ µ̂ p ‖u‖pp − ∫ Ω Γ̂(z, u)dz σ̂±(u) = 1 p γ(u) + λ+ µ̂ p ‖u‖pp − ∫ Ω Γ̂±(z, u)dz for all u ∈W 1,p(Ω). Using (4.1) and (4.2), we can easily show that Kσ̂ ⊆ [v̂λ, ûλ] ∩ C1(Ω), Kσ̂+ ⊆ [0, ûλ] ∩ C+,Kσ̂− ⊆ [v̂λ, 0] ∩ (−C+). The extremality of ûλ and v̂λ implies that Kσ̂ ⊆ [v̂λ, ûλ] ∩ C1(Ω), Kσ̂+ = {0, ûλ}, Kσ̂− = {0, v̂λ}. (4.3) EJDE-2020/12 POSITIVE AND NODAL SOLUTIONS 19 From (4.1) and (4.2) it is clear that σ̂+(·) is coercive. Also, it is sequentially weakly lower semicontinuous. So, we can find ũλ ∈W 1,p(Ω) such that σ̂+(ũλ) = inf[σ̂+(u) : u ∈W 1,p(Ω)]. (4.4) By hypothesis (H3”)(iv), we have σ̂+(ũλ) < 0 = σ̂+(0), ⇒ ũλ 6= 0, ⇒ ũλ = ûλ ∈ D+ (see (4.3), (4.4)). Since σ̂+ ∣∣ C+ = σ̂ ∣∣ C+ it follows that ûλ ∈ D+ is a local C1(Ω)-minimizer of σ̂(·), ⇒ ûλ ∈ D+ is a local W 1,p(Ω)-minimizer of σ̂(·) (4.5) (see Papageorgiou-Rǎdulescu [10]). Similarly, using this time σ̂−(·), we show that v̂λ ∈ −D+ is a local W 1,p(Ω)-minimizer of σ̂(·). (4.6) We can assume that σ̂(v̂λ) ≤ σ̂(ûλ). The reasoning is the same if the opposite inequality holds, using (4.6) instead of (4.5). By (4.3) and the extremality of ûλ and v̂λ, we see that we can assume that Kσ̂ ⊆ C1(Ω) is finite. (4.7) Otherwise we already have an infinity of smooth nodal solutions and so we are done. By (4.5), (4.7) and [14, Theorem 5.7.6, p. 367,], we can find ρ ∈ (0, 1) small such that σ̂(v̂λ) ≤ σ̂(ûλ) < inf[σ̂(u) : ‖u− ûλ‖ = ρ] = m̂λ, ‖v̂λ − ûλ‖ > ρ. (4.8) From (4.1) and since µ̂ ≥ ‖ξ‖∞, we see that σ̂(·) is coercive. Hence σ̂(·) satisfies the C-condition. (4.9) Then (4.8) and (4.9) permit the use of the mountain pass theorem. So, we can find yλ ∈W 1,p(Ω) such that yλ ∈ Kσ̂ and m̂λ ≤ σ̂(yλ), ⇒ yλ ∈ C1(Ω) and yλ 6∈ {ûλ, v̂λ}(see (4.7), (4.8)). (4.10) Also. from Papageorgiou-Rǎdulescu-Repovš [14, Theorem 6.5.8, p. 431], we have C1(σ̂, yλ) 6= 0. (4.11) On the other hand by (H3”)(iv) and Papageorgiou-Rǎdulescu [9, Proposition 3.7], we have Ck(σ̂, 0) = 0 for all k ∈ N0. (4.12) Comparing (4.11) and (4.12), we infer that yλ 6= 0. Therefore from (4.3) and (4.10), we have that yλ ∈ C1(Ω) is a nodal solution of (1.1). � Now we can state a multiplicity theorem for problem (1.1). Theorem 4.2. 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USSR Izv., 29 (1987), no. 1, 33–66. Nikolaos S. Papageorgiou National Technical University, Department of Mathematics, Zografou campus, 15780, Athens, Greece Email address: npapg@math.ntua.gr Calogero Vetro University of Palermo, Department of Mathematics and Computer Science, Via Archi- rafi 34, 90123, Palermo, Italy Email address: calogero.vetro@unipa.it Francesca Vetro (corresponding author) Nonlinear Analysis Research Group, Faculty of Mathematics and Statistics, Ton Duc Thang University, Ho Chi Minh City, Vietnam Email address: francescavetro@tdtu.edu.vn 1. Introduction 2. Mathematical Background - Hypotheses 3. Positive Solutions 4. Nodal solutions References