International Journal of Differential Equations
Volume 2010 (2010), Article ID 536236, 33 pages
doi:10.1155/2010/536236
Research Article

Sign-Changing Solutions for Nonlinear Elliptic Problems Depending on Parameters

Siegfried Carl1 and Dumitru Motreanu2

1Institut für Mathematik, Martin-Luther-Universität Halle-Wittenberg, 06099 Halle, Germany
2Département de Mathématiques, Université de Perpignan, 66860 Perpignan, France

Received 18 September 2009; Accepted 23 November 2009

Academic Editor: Thomas Bartsch

Copyright © 2010 Siegfried Carl and Dumitru Motreanu. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

The study of multiple solutions for quasilinear elliptic problems under Dirichlet or nonlinear Neumann type boundary conditions has received much attention over the last decades. The main goal of this paper is to present multiple solutions results for elliptic inclusions of Clarke's gradient type under Dirichlet boundary condition involving the 𝑝 -Laplacian which, in general, depend on two parameters. Assuming different structure and smoothness assumptions on the nonlinearities generating the multivalued term, we prove the existence of multiple constant-sign and sign-changing (nodal) solutions for parameters specified in terms of the Fučik spectrum of the 𝑝 -Laplacian. Our approach will be based on truncation techniques and comparison principles (sub-supersolution method) for elliptic inclusions combined with variational and topological arguments for, in general, nonsmooth functionals, such as, critical point theory, Mountain Pass Theorem, Second Deformation Lemma, and the variational characterization of the “beginning”of the Fu\v cik spectrum of the 𝑝 -Laplacian. In particular, the existence of extremal constant-sign solutions and their variational characterization as global (resp., local) minima of the associated energy functional will play a key-role in the proof of sign-changing solutions.

1. Introduction

Let Ω ⊂ ℝ 𝑁 be a bounded domain with a 𝐶 2 -boundary 𝜕 Ω , and let 𝑉 = 𝑊 1 , 𝑝 ( Ω ) and 𝑉 0 = 𝑊 0 1 , 𝑝 ( Ω ) , 1 < 𝑝 < + ∞ , denote the usual Sobolev spaces with their dual spaces 𝑉 ∗ and 𝑉 ∗ 0 , respectively. We consider the following nonlinear multi-valued elliptic boundary value problem under Dirichlet boundary condition: find 𝑢 ∈ 𝑉 0 ⧵ { 0 } and parameters 𝑎 ∈ ℝ , 𝑏 ∈ ℝ such that − Δ 𝑝 𝑢 ∈ 𝜕 𝑗 ( ⋅ , 𝑢 , 𝑎 , 𝑏 ) i n 𝑉 ∗ 0 , ( 1 . 1 ) where Δ 𝑝 𝑢 = d i v ( | ∇ 𝑢 | 𝑝 − 2 ∇ 𝑢 ) is the 𝑝 -Laplacian, and 𝑠 ↦ 𝜕 𝑗 ( 𝑥 , 𝑠 , 𝑎 , 𝑏 ) denotes Clarke's generalized gradient of some locally Lipschitz function 𝑠 ↦ 𝑗 ( 𝑥 , 𝑠 , 𝑎 , 𝑏 ) which depends on 𝑥 ∈ Ω and the parameters 𝑎 , 𝑏 . For 𝑎 = 𝑏 = ∶ 𝜆 problem (1.1) reduces to − Δ 𝑝 𝑢 ∈ 𝜕 𝑗 ( ⋅ , 𝑢 , 𝜆 ) i n 𝑉 ∗ 0 , ( 1 . 2 ) which may be considered as a nonlinear and nonsmooth eigenvalue problem. We are going to study the existence of multiple solutions of (1.1) for two different classes of 𝑗 which are in some sense complementary. Our presentation is based on and extends the authors' recent results obtained in [1–3]. For the first class of 𝑗 we let 𝑎 = 𝑏 = 𝜆 and assume the following structure of 𝑗 : 𝑗 ( 𝑥 , 𝑠 , 𝜆 ) = 𝑠 0 𝑓 ( 𝑥 , 𝑡 , 𝜆 ) 𝑑 𝑡 , ( 1 . 3 ) where 𝑓 ∶ Ω × ℝ × ( 0 , 𝜆 ) → ℝ is such that 𝑓 ( ⋅ , ⋅ , 𝜆 ) ∶ Ω × ℝ → ℝ is a Carathéodory function. Problem (1.1) reduces then to the following nonlinear eigenvalue problem: 𝑢 ∈ 𝑉 0 ⧵ { 0 } ∶ − Δ 𝑝 𝑢 = 𝑓 ( ⋅ , 𝑢 , 𝜆 ) i n 𝑉 ∗ 0 , ( 1 . 4 ) which will be considered in Section 2 when the parameter 𝜆 is small enough.

The second class of 𝑗 has the following structure: 𝑎 𝑗 ( 𝑥 , 𝑠 , 𝑎 , 𝑏 ) = 𝑝 𝑠 + 𝑝 + 𝑏 𝑝 ( 𝑠 − ) 𝑝 + 𝐺 ( 𝑥 , 𝑠 ) , ( 1 . 5 ) where 𝑠 + = m a x { 𝑠 , 0 } and 𝑠 − = m a x { − 𝑠 , 0 } is the positive and negative part of 𝑠 , respectively, and 𝐺 ∶ Ω × ℝ → ℝ is assumed to be the primitive of a measurable function 𝑔 ∶ Ω × ℝ → ℝ that is merely bounded on bounded sets; that is, 𝑔 ∈ 𝐿 ∞ l o c ( Ω × ℝ ) and 𝐺 is given by 𝐺 ( 𝑥 , 𝑠 ) ∶ = 𝑠 0 𝑔 ( 𝑥 , 𝑡 ) 𝑑 𝑡 . ( 1 . 6 ) Problem (1.1) reduces then to the following parameter-dependent multi-valued elliptic problem: 𝑢 ∈ 𝑉 0 ⧵ { 0 } ∶ − Δ 𝑝 𝑢 𝑢 ∈ 𝑎 + 𝑝 − 1 − 𝑏 ( 𝑢 − ) 𝑝 − 1 − 𝜕 𝐺 ( 𝑥 , 𝑢 ) i n 𝑉 ∗ 0 , ( 1 . 7 ) which will be studied in Section 3 for parameters 𝑎 and 𝑏 large enough. Note that 𝑠 ↦ 𝜕 𝐺 ( 𝑥 , 𝑠 ) stands for the generalized Clarke's gradient of the locally Lipschitz function 𝑠 ↦ 𝐺 ( 𝑥 , 𝑠 ) . Obviously, if 𝑔 ∶ Ω × ℝ → ℝ is a Carathéodory function, that is, 𝑥 ↦ 𝑔 ( 𝑥 , 𝑠 ) is measurable in Ω for all 𝑠 ∈ ℝ and 𝑠 ↦ 𝑔 ( 𝑥 , 𝑠 ) is continuous in ℝ for a.a. 𝑥 ∈ Ω , then 𝜕 𝐺 ( 𝑥 , 𝑠 ) = { 𝑔 ( 𝑥 , 𝑠 ) } is single-valued, and thus problem (1.7) reduces to the following nonlinear elliptic problem depending on parameters 𝑎 and 𝑏 : find 𝑢 ∈ 𝑉 0 ⧵ { 0 } and constants 𝑎 ∈ ℝ , 𝑏 ∈ ℝ such that − Δ 𝑝 𝑢 𝑢 = 𝑎 + 𝑝 − 1 − 𝑏 ( 𝑢 − ) 𝑝 − 1 − 𝑔 ( 𝑥 , 𝑢 ) i n 𝑉 ∗ 0 . ( 1 . 8 ) Multiple solution results for (1.8) were obtained by the authors in [4]. Furthermore, note that | 𝑢 | 𝑝 − 2 𝑢 = | 𝑢 | 𝑝 − 2 𝑢 + − 𝑢 − = 𝑢 + 𝑝 − 1 − ( 𝑢 − ) 𝑝 − 1 . ( 1 . 9 ) Therefore, if one assumes, in addition, 𝑎 = 𝑏 = ∶ 𝜆 , then (1.8) reduces to the nonlinear elliptic eigenvalue problem: find 𝑢 ∈ 𝑉 0 ⧵ { 0 } and a constant 𝜆 ∈ ℝ such that − Δ 𝑝 𝑢 = 𝜆 | 𝑢 | 𝑝 − 2 𝑢 − 𝑔 ( 𝑥 , 𝑢 ) i n 𝑉 ∗ 0 . ( 1 . 1 0 ) In a recent paper (see [5]) the authors considered the eigenvalue problem (1.10) for a Carathéodory function 𝑔 . Combining the method of sub-supersolution with variational techniques and assuming certain growth conditions of 𝑠 ↦ 𝑔 ( 𝑥 , 𝑠 ) at infinity and at zero the authors were able to prove the existence of at least three nontrivial solutions including one that changes sign. The results in [5] improve among others recent results obtained in [6]. For 𝑎 = 𝑏 = ∶ 𝜆 , (1.7) reduces to the corresponding multivalued eigenvalue problem: find 𝑢 ∈ 𝑉 0 ⧵ { 0 } and a constant 𝜆 ∈ ℝ such that − Δ 𝑝 𝑢 ∈ 𝜆 | 𝑢 | 𝑝 − 2 𝑢 − 𝜕 𝐺 ( 𝑥 , 𝑢 ) i n 𝑉 ∗ 0 . ( 1 . 1 1 ) The existence of multiple solutions for (1.11) has been shown recently in [7] where techniques for single-valued problems developed in [5] and hemivariational methods applied in [8] have been used. Multiplicity results for (1.11) have been obtained also in [9].

The existence of multiple solutions for semilinear and quasilinear elliptic problems has been studied by a number of authors, for example, [10–24]. All these papers deal with nonlinearities ( 𝑥 , 𝑠 ) ↦ 𝑔 ( 𝑥 , 𝑠 ) that are sufficiently smooth.

2. Problem (1.4) for 𝜆 being Small

The aim of this section is to provide an existence result of multiple solutions for all values of the parameter 𝜆 in an interval ( 0 , 𝜆 0 ) , with 𝜆 0 > 0 , guaranteeing that for any such 𝜆 there exist at least three nontrivial solutions of problem (1.4), two of them having opposite constant sign and the third one being sign-changing (or nodal). More precisely, we demonstrate that under suitable assumptions there exist a smallest positive solution, a greatest negative solution, and a sign-changing solution between them, whereas the notions smallest and greatest refer to the underlying natural partial ordering of functions. This continues the works of Jin [25] (where 𝑝 = 2 and 𝑓 ( 𝑥 , 𝑠 , 𝜆 ) is Hölder continuous with respect to ( 𝑥 , 𝑠 ) ∈ Ω × ℝ for every fixed 𝜆 ) and of Motreanu-Motreanu-Papageogiou [26]. In these cited works one obtains three nontrivial solutions, two of which being of opposite constant sign, but without knowing that the third one changes sign. Here we derive the new information of having, in addition, a sign-changing solution by strengthening the unilateral condition for the right-hand side of the equation in (1.4) at zero. Furthermore, under additional hypotheses, we demonstrate that one can obtain two sign-changing solutions.

2.1. Hypotheses and Example

Let 𝐿 𝑞 ( Ω ) + , 1 ≤ 𝑞 ≤ + ∞ , denote the positive cone of 𝐿 𝑞 ( Ω ) given by 𝐿 𝑞 ( Ω ) + = { 𝑣 ∈ 𝐿 𝑞 ( Ω ) ∶ 𝑣 ( 𝑥 ) ≥ 0 f o r a . a . 𝑥 ∈ Ω } . ( 2 . 1 ) We impose the following hypotheses on the nonlinearity 𝑓 ( 𝑥 , 𝑠 , 𝜆 ) in problem (1.4). H ( 𝑓 ) 𝑓 ∶ Ω × ℝ × ( 0 , 𝜆 ) → ℝ , with 𝜆 > 0 , is a function such that 𝑓 ( 𝑥 , 0 , 𝜆 ) = 0 for a.a. 𝑥 ∈ Ω , whenever 𝜆 ∈ ( 0 , 𝜆 ) , and one has the following. (i)For all 𝜆 ∈ ( 0 , 𝜆 ) , 𝑓 ( ⋅ , ⋅ , 𝜆 ) is Carathéodory (i.e., 𝑓 ( ⋅ , 𝑠 , 𝜆 ) is measurable for all 𝑠 ∈ ℝ and 𝑓 ( 𝑥 , ⋅ , 𝜆 ) is continuous for almost all 𝑥 ∈ Ω ). (ii)There are constants 𝑐 > 0 , 𝑟 > 𝑝 − 1 , and functions 𝑎 ( ⋅ , 𝜆 ) ∈ 𝐿 ∞ ( Ω ) + ( 𝜆 ∈ ( 0 , 𝜆 ) ) with ‖ 𝑎 ( ⋅ , 𝜆 ) ‖ ∞ → 0 as 𝜆 ↓ 0 such that | | | | 𝑓 ( 𝑥 , 𝑠 , 𝜆 ) ≤ 𝑎 ( 𝑥 , 𝜆 ) + 𝑐 | 𝑠 | 𝑟 f o r a . a . 𝑥 ∈ Ω ∀ ( s , 𝜆 ) ∈ ℝ × 0 , 𝜆 . ( 2 . 2 ) (iii)For all 𝜆 ∈ ( 0 , 𝜆 ) there exist constants 𝜇 0 = 𝜇 0 ( 𝜆 ) > 𝜆 2 , 𝜈 0 = 𝜈 0 ( 𝜆 ) > 𝜇 0 and a set Ω 𝜆 ⊂ Ω with Ω ⧵ Ω 𝜆 of Lebesgue measure zero such that 𝜇 0 < l i m i n f 𝑠 → 0 𝑓 ( 𝑥 , 𝑠 , 𝜆 ) | 𝑠 | 𝑝 − 2 𝑠 ≤ l i m s u p 𝑠 → 0 𝑓 ( 𝑥 , 𝑠 , 𝜆 ) | 𝑠 | 𝑝 − 2 𝑠 ≤ 𝜈 0 ( 2 . 3 ) uniformly with respect to 𝑥 ∈ Ω 𝜆 .

In H ( 𝑓 ) (iii), 𝜆 2 denotes the second eigenvalue of ( − Δ 𝑝 , 𝑉 0 ) . As mentioned in the Introduction, the strengthening with respect to [26] (see also [25]) of the unilateral condition for the right-hand side 𝑓 in (1.4), which enables us to obtain, in addition, sign-changing solutions, consists in adding the part involving the limit superior in H ( 𝑓 ) (iii).

Let us provide an example where all the assumptions formulated in H ( 𝑓 ) are fulfilled.

Example 2.1. For the sake of simplicity we drop the 𝑥 dependence for the function 𝑓 in the right-hand side of (1.4). The function 𝑓 ∶ ℝ × ( 0 , + ∞ ) → ℝ given by 𝑓 ( 𝑠 , 𝜆 ) = 𝜆 a r c t a n 𝜆 + 𝜆 2 𝜆 | 𝑠 | 𝑝 − 2 𝑠 + 𝑐 | 𝑠 | 𝑟 − 1 𝑠 ∀ ( s , 𝜆 ) ∈ ℝ × ( 0 , + ∞ ) , ( 2 . 4 ) with 𝑐 > 0 and 𝑟 > 𝑝 − 1 , satisfies hypotheses H ( 𝑓 ) . Next we give an example of function 𝑓 ∶ ℝ × ( 0 , + ∞ ) → ℝ verifying assumptions H ( 𝑓 ) which is generally not odd with respect to 𝑠 : ⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩ 𝑎 𝑓 ( 𝑠 , 𝜆 ) = 𝜆 a r c t a n 1 𝜆 + 𝜆 2 𝜆 | 𝑠 | 𝑝 − 2 𝑠 + 𝑐 1 | 𝑠 | 𝑟 1 − 1 𝑎 𝑠 i f 𝑠 ≤ 0 , 𝜆 a r c t a n 2 𝜆 + 𝜆 2 𝜆 𝑠 𝑝 − 1 + 𝑐 2 𝑠 𝑟 2 i f 𝑠 > 0 , ( 2 . 5 ) with 𝜆 > 0 , 𝑎 1 ≥ 1 , 𝑎 2 ≥ 1 , 𝑐 1 > 0 , 𝑐 2 > 0 , 𝑟 1 > 𝑝 − 1 , 𝑟 2 > 𝑝 − 1 .

2.2. Constant-Sign Solutions

The operator − Δ 𝑝 ∶ 𝑉 0 → 𝑉 ∗ 0 is maximal monotone and coercive; therefore there exists a unique solution 𝑒 ∈ 𝑉 0 of the Dirichlet problem 𝑒 ∈ 𝑉 0 ∶ − Δ 𝑝 𝑒 = 1 i n 𝑉 ∗ 0 . ( 2 . 6 ) With 𝑠 − = m a x { − 𝑠 , 0 } for 𝑠 ∈ ℝ , and using − 𝑒 − ∈ 𝑉 0 as a test function, we see that ‖ ∇ 𝑒 − ‖ 𝑝 𝑝 = − Δ 𝑝 𝑒 , − 𝑒 − = − Ω 𝑒 − ( 𝑥 ) 𝑑 𝑥 ≤ 0 , ( 2 . 7 ) which implies that 𝑒 ≥ 0 . From the nonlinear regularity theory (cf., e.g., [27, Theorem 1.5.6]) we have 𝑒 ∈ 𝐶 1 0 ( Ω ) . Then from the nonlinear strong maximum principle (see [28]) we infer that 𝑒 ∈ i n t ( 𝐶 1 0 ( Ω ) + ) . Here i n t ( 𝐶 1 0 ( Ω ) + ) denotes the interior of the positive cone 𝐶 1 0 ( Ω ) + = { 𝑢 ∈ 𝐶 1 0 ( Ω ) ∶ 𝑢 ( 𝑥 ) ≥ 0 , ∀ 𝑥 ∈ Ω } in the Banach space 𝐶 1 0 ( Ω ) = { 𝑢 ∈ 𝐶 1 ( Ω ) ∶ 𝑢 ( 𝑥 ) = 0 , ∀ 𝑥 ∈ 𝜕 Ω } , given by 𝐶 i n t 1 0 Ω + = 𝑢 ∈ 𝐶 1 0 Ω ∶ 𝑢 ( 𝑥 ) > 0 , ∀ 𝑥 ∈ Ω , a n d 𝜕 𝑢 , 𝜕 𝑛 ( 𝑥 ) < 0 , ∀ 𝑥 ∈ 𝜕 Ω ( 2 . 8 ) where 𝑛 = 𝑛 ( 𝑥 ) is the outer unit normal at 𝑥 ∈ 𝜕 Ω .

Lemma 2.2. Let the data 𝑐 , 𝑟 , and 𝑎 ( ⋅ , 𝜆 ) be as in H ( 𝑓 ) (ii). Then for every constant 𝜃 > 0 there is 𝜆 0 ∈ ( 0 , 𝜆 ) with the property that if 𝜆 ∈ ( 0 , 𝜆 0 ) , one can choose 𝜉 0 = 𝜉 0 ( 𝜆 ) ∈ ( 0 , 𝜃 ) such that 𝑐 𝜉 0 ‖ 𝑒 ‖ ∞ 𝑟 + ‖ 𝑎 ( ⋅ , 𝜆 ) ‖ ∞ < 𝜉 0 𝑝 − 1 . ( 2 . 9 )

Proof. On the contrary there would exist a constant 𝜃 > 0 and a sequence 𝜆 𝑛 ↓ 0 as 𝑛 → ∞ such that 𝑐 𝜉 ‖ 𝑒 ‖ ∞ 𝑟 + ‖ ‖ 𝑎 ⋅ , 𝜆 𝑛 ‖ ‖ ∞ ≥ 𝜉 𝑝 − 1 ∀ 𝑛 ∈ ℕ , 𝜉 ∈ ( 0 , 𝜃 ) . ( 2 . 1 0 ) Letting 𝑛 → ∞ we get 𝑐 ‖ 𝑒 ‖ 𝑟 ∞ 𝜉 𝑟 − 𝑝 + 1 ≥ 1 for all 𝜉 ∈ ( 0 , 𝜃 ) because we have ‖ 𝑎 ( ⋅ , 𝜆 ) ‖ ∞ → 0 as 𝜆 ↓ 0 . Since 𝑟 > 𝑝 − 1 , a contradiction is achieved as 𝜉 ↓ 0 . Therefore (2.9) holds true.

We denote by 𝜆 1 the first eigenvalue of ( − Δ 𝑝 , 𝑉 0 ) and by 𝜑 1 the eigenfunction of ( − Δ 𝑝 , 𝑉 0 ) corresponding to 𝜆 1 satisfying 𝜑 1 𝐶 ∈ i n t 1 0 Ω + , ‖ ‖ 𝜑 1 ‖ ‖ 𝑝 = 1 . ( 2 . 1 1 )

Lemma 2.3. Assume H ( 𝑓 ) (i) and (ii) and the following weaker form of hypothesis H ( 𝑓 ) (iii): for all 𝜆 ∈ ( 0 , 𝜆 ) there exist 𝜇 0 = 𝜇 0 ( 𝜆 ) > 𝜆 1 and Ω 𝜆 ⊂ Ω with Ω ⧵ Ω 𝜆 of Lebesgue measure zero such that 𝜇 0 < l i m i n f 𝑠 → 0 𝑓 ( 𝑥 , 𝑠 , 𝜆 ) | 𝑠 | 𝑝 − 2 𝑠 ( 2 . 1 2 ) uniformly with respect to 𝑥 ∈ Ω 𝜆 .
Fix a constant 𝜃 > 0 and consider the corresponding number 𝜆 0 ∈ ( 0 , 𝜆 ) obtained in Lemma 2.2. Then for any 𝜆 ∈ ( 0 , 𝜆 0 ) the function 𝑢 = 𝜉 0 𝑒 ∈ i n t ( 𝐶 1 0 ( Ω ) + ) , with 𝜉 0 ∈ ( 0 , 𝜃 ) given by Lemma 2.2, is a supersolution for problem (1.4), and the function 𝑢 = 𝜀 𝜑 1 ∈ i n t ( 𝐶 1 0 ( Ω ) + ) is a subsolution of problem (1.4) provided that the number 𝜀 > 0 is sufficiently small.

Proof. For a fixed 𝜆 ∈ ( 0 , 𝜆 0 ) , from (2.9) and H ( 𝑓 ) (ii) we derive − Δ 𝑝 𝑢 = 𝜉 0 𝑝 − 1 > ‖ 𝑎 ( ⋅ , 𝜆 ) ‖ ∞ ‖ ‖ + 𝑐 𝑢 ‖ ‖ 𝑟 ∞ ≥ 𝑓 ⋅ , , 𝑢 ( ⋅ ) , 𝜆 ( 2 . 1 3 ) which says that 𝑢 = 𝜉 0 𝑒 is a supersolution for problem (1.4).
On the other hand, by hypothesis we can find 𝜇 = 𝜇 ( 𝜆 ) > 𝜆 1 and 𝛿 = 𝛿 ( 𝜆 ) > 0 such that 𝜇 < 𝑓 ( 𝑥 , 𝑠 , 𝜆 ) | 𝑠 | 𝑝 − 2 𝑠 f o r a . a . 𝑥 ∈ Ω ∀ 0 < | 𝑠 | ≤ 𝛿 . ( 2 . 1 4 ) Choose 𝜀 ∈ ( 0 , 𝛿 / ‖ 𝜑 1 ‖ ∞ ) . Then by (2.14) we have − Δ 𝑝 𝜀 𝜑 1 = 𝜆 1 𝜀 𝑝 − 1 𝜑 1 𝑝 − 1 < 𝜇 𝜀 𝑝 − 1 𝜑 1 𝑝 − 1 < 𝑓 𝑥 , 𝜀 𝜑 1 ( 𝑥 ) , 𝜆 f o r a . a . 𝑥 ∈ Ω , ( 2 . 1 5 ) which ensures that 𝑢 = 𝜀 𝜑 1 is a subsolution of problem (1.4).

The following result which asserts the existence of two solutions of problem (1.4) having opposite constant sign and being extremal plays an important role in the proof of the existence of sign-changing solutions.

Theorem 2.4. Assume H ( 𝑓 ) (i) and (ii) and the following weaker form of H ( 𝑓 ) (iii): for all 𝜆 ∈ ( 0 , 𝜆 ) there exist constants 𝜇 0 = 𝜇 0 ( 𝜆 ) > 𝜆 1 , 𝜈 0 = 𝜈 0 ( 𝜆 ) > 𝜇 0 and a set Ω 𝜆 ⊂ Ω with Ω ⧵ Ω 𝜆 of Lebesgue measure zero such that 𝜇 0 < l i m i n f 𝑠 → 0 𝑓 ( 𝑥 , 𝑠 , 𝜆 ) | 𝑠 | 𝑝 − 2 𝑠 ≤ l i m s u p 𝑠 → 0 𝑓 ( 𝑥 , 𝑠 , 𝜆 ) | 𝑠 | 𝑝 − 2 𝑠 ≤ 𝜈 0 ( 2 . 1 6 ) uniformly with respect to 𝑥 ∈ Ω 𝜆 . Then for all 𝑏 > 0 there exists a number 𝜆 0 ∈ ( 0 , 𝜆 ) with the property that if 𝜆 ∈ ( 0 , 𝜆 0 ) , then there is a constant 𝜉 0 = 𝜉 0 ( 𝜆 ) ∈ ( 0 , 𝑏 / ‖ 𝑒 ‖ ∞ ) such that problem (1.4) has a least positive solution 𝑢 + = 𝑢 + ( 𝜆 ) ∈ i n t ( 𝐶 1 0 ( Ω ) + ) in the order interval [ 0 , 𝜉 0 𝑒 ] and a greatest negative solution 𝑢 − = 𝑢 − ( 𝜆 ) ∈ − i n t ( 𝐶 1 0 ( Ω ) + ) in the order interval [ − 𝜉 0 𝑒 , 0 ] .

Proof. Since the proof of the existence of the greatest negative solution follows the same lines, we only provide the arguments for the existence of the least positive solution.
Applying Lemma 2.3 for 𝜃 = 𝑏 / ‖ 𝑒 ‖ ∞ we find 𝜆 0 ∈ ( 0 , 𝜆 ) as therein. Fix 𝜆 ∈ ( 0 , 𝜆 0 ) . Lemma 2.3 ensures that 𝑢 = 𝜉 0 𝑒 ∈ i n t ( 𝐶 1 0 ( Ω ) + ) is a supersolution for problem (1.4), with 𝜉 0 ∈ ( 0 , 𝑏 / ‖ 𝑒 ‖ ∞ ) given by Lemma 2.2, and 𝑢 = 𝜀 𝜑 1 ∈ i n t ( 𝐶 1 0 ( Ω ) + ) is a subsolution for problem (1.4) if 𝜀 > 0 is small enough. Passing eventually to a smaller 𝜀 > 0 , we may assume that 𝜀 𝜑 1 ≤ 𝜉 0 𝑒 . Then by the method of sub-supersolution we know that in the order interval [ 𝜀 𝜑 1 , 𝜉 0 𝑒 ] there is a least (i.e., smallest) solution 𝑢 𝜀 = 𝑢 𝜀 ( 𝜆 ) ∈ i n t ( 𝐶 1 0 ( Ω ) + ) of problem (1.4) (see [29]).
We thus obtain that for every positive integer 𝑛 sufficiently large there is a least solution 𝑢 𝑛 ∈ i n t ( 𝐶 1 0 ( Ω ) + ) of problem (1.4) in the order interval [ ( 1 / 𝑛 ) 𝜑 1 , 𝜉 0 𝑒 ] . Clearly, we have 𝑢 𝑛 ↓ 𝑢 + p o i n t w i s e , ( 2 . 1 7 ) with some function 𝑢 + ∶ Ω → ℝ satisfying 0 ≤ 𝑢 + ≤ 𝜉 0 𝑒 . First we claim that 𝑢 + i s a s o l u t i o n o f p r o b l e m ( 1 . 4 ) . ( 2 . 1 8 ) Taking into account that 𝑢 𝑛 solves (1.4), and the fact that 𝑢 𝑛 belongs to the order interval [ 0 , 𝜉 0 𝑒 ] , from H ( 𝑓 ) (ii) we see that ‖ ‖ ∇ 𝑢 𝑛 ‖ ‖ 𝑝 𝑝 = Ω 𝑓 𝑥 , 𝑢 𝑛 ( 𝑢 𝑥 ) , 𝜆 𝑛 ( 𝑥 ) 𝑑 𝑥 ≤ Ω 𝑎 ( 𝑥 , 𝜆 ) + 𝑐 𝜉 𝑟 0 𝑒 ( 𝑥 ) 𝑟 𝜉 0 𝑒 ( 𝑥 ) 𝑑 𝑥 , ( 2 . 1 9 ) which implies the boundedness of the sequence ( 𝑢 𝑛 ) in 𝑉 0 . Then due to (2.17) we have that 𝑢 + ∈ 𝑉 0 as well as 𝑢 𝑛 ⇀ 𝑢 + i n 𝑉 0 , 𝑢 𝑛 ⟶ 𝑢 + i n 𝐿 𝑝 ( Ω ) a n d a . e . i n Ω . ( 2 . 2 0 ) Since 𝑢 𝑛 solves problem (1.4), one has − Δ 𝑝 𝑢 𝑛 = , 𝜑 Ω 𝑓 𝑥 , 𝑢 𝑛 ( 𝑥 ) , 𝜆 𝜑 ( 𝑥 ) 𝑑 𝑥 , ∀ 𝜑 ∈ 𝑉 0 . ( 2 . 2 1 ) Setting 𝜑 = 𝑢 𝑛 − 𝑢 + in (2.21) gives − Δ 𝑝 𝑢 𝑛 , 𝑢 𝑛 − 𝑢 + = Ω 𝑓 𝑥 , 𝑢 𝑛 ( 𝑢 𝑥 ) , 𝜆 𝑛 ( 𝑥 ) − 𝑢 + ( 𝑥 ) 𝑑 𝑥 . ( 2 . 2 2 ) As already noticed that the sequence ( 𝑓 ( ⋅ , 𝑢 𝑛 ( ⋅ ) , 𝜆 ) is uniformly bounded on Ω , so (2.20) and (2.22) yield l i m 𝑛 → ∞ − Δ 𝑝 𝑢 𝑛 , 𝑢 𝑛 − 𝑢 + = 0 . ( 2 . 2 3 ) The 𝑆 + -property of − Δ 𝑝 on 𝑉 0 implies 𝑢 𝑛 ⟶ 𝑢 + i n 𝑉 0 a s 𝑛 ⟶ ∞ . ( 2 . 2 4 ) The strong convergence in (2.24) and Lebesgue's dominated convergence theorem permit to pass to the limit in (2.21) that results in (2.18).
By (2.18) and the nonlinear regularity theory (cf., e.g., Theorem 1.5.6 in [27]) it turns out 𝑢 + ∈ 𝐶 1 0 ( Ω ) . The choice of 𝜉 0 guarantees that 0 ≤ 𝑢 + ( 𝑥 ) ≤ 𝜉 0 𝑒 ( 𝑥 ) ≤ 𝑏 f o r a . e . 𝑥 ∈ Ω . ( 2 . 2 5 ) Thus, from (2.18), assumptions H ( 𝑓 ) (ii) and (iii), and the boundedness of 𝑢 + , we get − Δ 𝑝 𝑢 + ( 𝑥 ) = 𝑓 𝑥 , 𝑢 + ( 𝑥 ) , 𝜆 ≥ − ̂ 𝑐 𝑢 + ( 𝑥 ) 𝑝 − 1 f o r a . a . 𝑥 ∈ Ω , ( 2 . 2 6 ) with a constant ̂ 𝑐 > 0 . Applying the nonlinear strong maximum principle (cf. [28]) we conclude that either 𝑢 + = 0 or 𝑢 + ∈ i n t ( 𝐶 1 0 ( Ω ) + ) .
We claim that 𝑢 + 𝐶 ∈ i n t 1 0 Ω + . ( 2 . 2 7 ) Assume on the contrary that 𝑢 + = 0 . Then (2.17) becomes 𝑢 𝑛 ( 𝑥 ) ↓ 0 ∀ 𝑥 ∈ Ω . ( 2 . 2 8 ) Since 𝑢 𝑛 ≥ ( 1 / 𝑛 ) 𝜑 1 , we may consider ̃ 𝑢 𝑛 = 𝑢 𝑛 ‖ ‖ ∇ 𝑢 𝑛 ‖ ‖ 𝑝 ∀ 𝑛 . ( 2 . 2 9 ) Along a relabelled subsequence we may suppose ̃ 𝑢 𝑛 ⇀ ̃ 𝑢 i n 𝑉 0 , ̃ 𝑢 𝑛 ⟶ ̃ 𝑢 i n 𝐿 𝑝 ( Ω ) a n d a . e . i n Ω ( 2 . 3 0 ) for some ̃ 𝑢 ∈ 𝑉 0 . Moreover, one can find a function 𝑤 ∈ 𝐿 𝑝 ( Ω ) + such that | ̃ 𝑢 𝑛 ( 𝑥 ) | ≤ 𝑤 ( 𝑥 ) for almost all 𝑥 ∈ Ω . Relation (2.21) reads − Δ 𝑝 ̃ 𝑢 𝑛 = , 𝜑 Ω 𝑓 𝑥 , 𝑢 𝑛 ( 𝑥 ) , 𝜆 ‖ ‖ ∇ 𝑢 𝑛 ‖ ‖ 𝑝 𝑝 − 1 𝜑 𝑑 𝑥 , ∀ 𝜑 ∈ 𝑉 0 . ( 2 . 3 1 ) Setting 𝜑 = ̃ 𝑢 𝑛 − ̃ 𝑢 leads to − Δ 𝑝 ̃ 𝑢 𝑛 , ̃ 𝑢 𝑛 = − ̃ 𝑢 Ω 𝑓 𝑥 , 𝑢 𝑛 ( 𝑥 ) , 𝜆 ‖ ‖ ∇ 𝑢 𝑛 ‖ ‖ 𝑝 𝑝 − 1 ̃ 𝑢 𝑛 − ̃ 𝑢 𝑑 𝑥 . ( 2 . 3 2 ) By H ( 𝑓 ) (iii) we know that there exist constants 𝑐 0 = 𝑐 0 ( 𝜆 ) > 𝜆 1 and 𝛼 = 𝛼 ( 𝜆 ) > 0 such that | | | | 𝑓 ( 𝑥 , 𝑠 , 𝜆 ) ≤ 𝑐 0 | 𝑠 | 𝑝 − 1 f o r a . a . 𝑥 ∈ Ω , ∀ | s | < 𝛼 , ( 2 . 3 3 ) while H ( 𝑓 ) (ii) entails | | | | 𝑓 ( 𝑥 , 𝑠 , 𝜆 ) ≤ 𝑎 ( 𝑥 , 𝜆 ) + 𝑐 | 𝑠 | 𝑟 ≤ ‖ ‖ 𝑎 ( ⋅ , 𝜆 ) ∞ 𝛼 𝑟 + 𝑐 | 𝑠 | 𝑟 ( 2 . 3 4 ) for a.a. 𝑥 ∈ Ω and for all | 𝑠 | ≥ 𝛼 . Combining the two estimates gives | | | | 𝑓 ( 𝑥 , 𝑠 , 𝜆 ) ≤ 𝑐 0 | 𝑠 | 𝑝 − 1 + 𝑐 1 | 𝑠 | 𝑟 f o r a . a . 𝑥 ∈ Ω , ∀ 𝑠 ∈ ℝ ( 2 . 3 5 ) with a constant 𝑐 1 = 𝑐 1 ( 𝜆 ) > 0 . Since 𝑢 𝑛 ∈ [ ( 1 / 𝑛 ) 𝜑 1 , 𝜉 0 𝑒 ] , 𝑟 > 𝑝 − 1 and (2.35) holds, there exists a constant 𝐶 > 0 such that | | 𝑓 𝑥 , 𝑢 𝑛 | | ( 𝑥 ) , 𝜆 𝑢 𝑛 ( 𝑥 ) 𝑝 − 1 ≤ 𝐶 f o r a . a . 𝑥 ∈ Ω , ∀ 𝑛 . ( 2 . 3 6 ) We see from (2.36) that | | 𝑓 𝑥 , 𝑢 𝑛 | | ( 𝑥 ) , 𝜆 ‖ ‖ ∇ 𝑢 𝑛 ‖ ‖ 𝑝 𝑝 − 1 | | ̃ 𝑢 𝑛 | | ( 𝑥 ) − ̃ 𝑢 ( 𝑥 ) ≤ 𝐶 𝑤 ( 𝑥 ) 𝑝 − 1 | | | | 𝑤 ( 𝑥 ) + ̃ 𝑢 ( 𝑥 ) f o r a . a . 𝑥 ∈ Ω . ( 2 . 3 7 ) Then, because the right-hand side of the above inequality is in 𝐿 1 ( Ω ) , by means of (2.30) and (2.36) we can apply Lebesgue's dominated convergence theorem to get l i m 𝑛 → ∞ Ω 𝑓 𝑥 , 𝑢 𝑛 ( 𝑥 ) , 𝜆 ‖ ‖ ∇ 𝑢 𝑛 ‖ ‖ 𝑝 𝑝 − 1 ̃ 𝑢 𝑛 − ̃ 𝑢 𝑑 𝑥 = 0 . ( 2 . 3 8 ) Consequently, from (2.32) we obtain l i m 𝑛 → ∞ − Δ 𝑝 ̃ 𝑢 𝑛 , ̃ 𝑢 𝑛 − ̃ 𝑢 = 0 . ( 2 . 3 9 ) The 𝑆 + -property of − Δ 𝑝 on 𝑉 0 implies ̃ 𝑢 𝑛 ⟶ ̃ 𝑢 i n 𝑉 0 a s 𝑛 ⟶ ∞ . ( 2 . 4 0 ) On the basis of (2.31) and (2.40) it follows − Δ 𝑝 ̃ 𝑢 , 𝜑 = l i m 𝑛 → ∞ Ω 𝑓 𝑥 , 𝑢 𝑛 ( 𝑥 ) , 𝜆 ‖ ‖ ∇ 𝑢 𝑛 ‖ ‖ 𝑝 𝑝 − 1 𝜑 𝑑 𝑥 , ∀ 𝜑 ∈ 𝑉 0 . ( 2 . 4 1 ) Notice from (2.36) that | | 𝑓 𝑥 , 𝑢 𝑛 | | ( 𝑥 ) , 𝜆 ‖ ‖ ∇ 𝑢 𝑛 ‖ ‖ 𝑝 𝑝 − 1 | | | | 𝜑 ( 𝑥 ) ≤ 𝐶 𝑤 ( 𝑥 ) 𝑝 − 1 | | | | 𝜑 ( 𝑥 ) ( 2 . 4 2 ) for a.a. 𝑥 ∈ Ω and for all 𝜑 ∈ 𝑉 0 . We are thus allowed to apply Fatou's lemma which in conjunction with (2.28), (2.30), and (2.16) ensures l i m 𝑛 → ∞ Ω 𝑓 𝑥 , 𝑢 𝑛 ( 𝑥 ) , 𝜆 ‖ ‖ ∇ 𝑢 𝑛 ‖ ‖ 𝑝 𝑝 − 1 𝜑 ( 𝑥 ) 𝑑 𝑥 = l i m 𝑛 → ∞ Ω 𝑓 𝑥 , 𝑢 𝑛 ( 𝑥 ) , 𝜆 𝑢 𝑛 ( 𝑥 ) 𝑝 − 1 ̃ 𝑢 𝑛 ( 𝑥 ) 𝑝 − 1 ≥ 𝜑 ( 𝑥 ) 𝑑 𝑥 Ω l i m i n f 𝑛 → ∞ 𝑓 𝑥 , 𝑢 𝑛 ( 𝑥 ) , 𝜆 𝑢 𝑛 ( 𝑥 ) 𝑝 − 1 ̃ 𝑢 𝑛 ( 𝑥 ) 𝑝 − 1 𝜑 ( 𝑥 ) 𝑑 𝑥 ≥ 𝜇 0 Ω ̃ 𝑢 ( 𝑥 ) 𝑝 − 1 𝜑 ( 𝑥 ) 𝑑 𝑥 ( 2 . 4 3 ) for all 𝜑 ∈ 𝑉 0 , + ∶ = 𝑉 0 ∩ 𝐿 𝑝 ( Ω ) + . Thanks to (2.41) we obtain − Δ 𝑝 ̃ 𝑢 , 𝜑 ≥ 𝜇 0 Ω ̃ 𝑢 ( 𝑥 ) 𝑝 − 1 𝜑 ( 𝑥 ) 𝑑 𝑥 , ∀ 𝜑 ∈ 𝑉 0 , + . ( 2 . 4 4 ) Owing to (2.42) we may once again use Fatou's lemma; so according to (2.28), (2.30), and the last part of (2.16), we find l i m 𝑛 → ∞ Ω 𝑓 𝑥 , 𝑢 𝑛 ( 𝑥 ) , 𝜆 ‖ ‖ ∇ 𝑢 𝑛 ‖ ‖ 𝑝 𝑝 − 1 𝜑 ( 𝑥 ) 𝑑 𝑥 = l i m 𝑛 → ∞ Ω 𝑓 𝑥 , 𝑢 𝑛 ( 𝑥 ) , 𝜆 𝑢 𝑛 ( 𝑥 ) 𝑝 − 1 ̃ 𝑢 𝑛 ( 𝑥 ) 𝑝 − 1 ≤ 𝜑 ( 𝑥 ) 𝑑 𝑥 Ω l i m s u p 𝑛 → ∞ 𝑓 𝑥 , 𝑢 𝑛 ( 𝑥 ) , 𝜆 𝑢 𝑛 ( 𝑥 ) 𝑝 − 1 ̃ 𝑢 𝑛 ( 𝑥 ) 𝑝 − 1 𝜑 ( 𝑥 ) 𝑑 𝑥 ≤ 𝜈 0 Ω ̃ 𝑢 ( 𝑥 ) 𝑝 − 1 𝜑 ( 𝑥 ) 𝑑 𝑥 ( 2 . 4 5 ) for all 𝜑 ∈ 𝑉 0 , + . Then (2.41) ensures − Δ 𝑝 ̃ 𝑢 , 𝜑 ≤ 𝜈 0 Ω ̃ 𝑢 ( 𝑥 ) 𝑝 − 1 𝜑 ( 𝑥 ) 𝑑 𝑥 , ∀ 𝜑 ∈ 𝑉 0 , + . ( 2 . 4 6 ) Combining (2.44) and (2.46) results in 𝜇 0 ̃ 𝑢 𝑝 − 1 ≤ − Δ 𝑝 ̃ 𝑢 ≤ 𝜈 0 ̃ 𝑢 𝑝 − 1 a . e . i n Ω , ( 2 . 4 7 ) which guarantees to have ̃ 𝑢 ∈ 𝐿 ∞ ( Ω ) (see [27, Theorem 1.5.5]). Since by (2.47) we know that Δ 𝑝 ̃ 𝑢 ∈ 𝐿 ∞ ( Ω ) , we are in a position to address Theorem 1.5.6 in [27], which provides ̃ 𝑢 ∈ 𝐶 1 , 𝛽 ( Ω ) with some 𝛽 ∈ ( 0 , 1 ) . This regularity up to the boundary and the fact that ̃ 𝑢 ≥ 0 a.e. in Ω and (2.47) enable us to refer to the strong maximum principle (see Theorem 5 of Vázquez [28]). Recalling that ̃ 𝑢 does not vanish identically on Ω (because ‖ ∇ ̃ 𝑢 ‖ 𝑝 = 1 ) we deduce that ̃ 𝑢 ( 𝑥 ) > 0 for all 𝑥 ∈ Ω and ( 𝜕 ̃ 𝑢 / 𝜕 𝑛 ) ( 𝑥 ) < 0 for all 𝑥 ∈ 𝜕 Ω which amounts to saying ̃ 𝑢 ∈ i n t ( 𝐶 1 0 ( Ω ) + ) . Consequently, there exist constants 𝑘 0 > 0 and 𝑘 1 > 0 such that 𝑘 0 𝜑 1 ≤ ̃ 𝑢 < 𝑘 1 𝜑 1 a . e . i n Ω . ( 2 . 4 8 ) Following [30] let us denote 𝐼 ( 𝑢 , 𝑣 ) = − Δ 𝑝 𝑢 𝑢 , 𝑝 − 𝑣 𝑝 𝑢 𝑝 − 1 − − Δ 𝑝 𝑢 𝑣 , 𝑝 − 𝑣 𝑝 𝑣 𝑝 − 1 ( 2 . 4 9 ) whenever ( 𝑢 , 𝑣 ) ∈ 𝐷 𝐼 , where 𝐷 𝐼 = 𝑤 1 , 𝑤 2 ∈ 𝑉 0 2 ∶ 𝑤 𝑖 𝑤 ≥ 0 , 𝑖 𝑤 𝑗 ∈ 𝐿 ∞ . ( Ω ) f o r 𝑖 , 𝑗 ∈ { 1 , 2 } ( 2 . 5 0 ) Relation (2.48) justifies that ( 𝑘 1 𝜑 1 , ̃ 𝑢 ) ∈ 𝐷 𝐼 . Then Proposition 1 of Anane [30] implies 𝐼 ( 𝑘 1 𝜑 1 , ̃ 𝑢 ) ≥ 0 . On the other hand a direct computation based on (2.48) and (2.47) shows 𝐼 𝑘 1 𝜑 1 = , ̃ 𝑢 − Δ 𝑝 𝑘 1 𝜑 1 , 𝑘 1 𝜑 1 𝑝 − ̃ 𝑢 𝑝 𝑘 1 𝜑 1 𝑝 − 1 − − Δ 𝑝 𝑘 ̃ 𝑢 , 1 𝜑 1 𝑝 − ̃ 𝑢 𝑝 ̃ 𝑢 𝑝 − 1 ≤ 𝜆 1 − 𝜇 0 Ω 𝑘 1 𝜑 1 𝑝 − ̃ 𝑢 𝑝 𝑑 𝑥 < 0 . ( 2 . 5 1 ) This contradiction proves that the claim in (2.27) holds true.
In view of (2.18) it remains to establish that 𝑢 + is the smallest positive solution of problem (1.4) in the interval [ 0 , 𝑢 ] . Let 𝑢 ∈ 𝑉 0 be a positive solution to (1.4) in [ 0 , 𝑢 ] . Since 𝑢 ∈ 𝐿 ∞ ( Ω ) , then (1.4) and H ( 𝑓 ) (ii) allow to deduce that − Δ 𝑝 𝑢 ∈ 𝐿 ∞ ( Ω ) . Using Theorem 1.5.6 of [27] leads to 𝑢 ∈ 𝐶 1 0 ( Ω ) . Then, as 𝑢 is a solution to (1.4) and 𝑢 ∈ [ 0 , 𝑢 ] , with ‖ 𝑢 ‖ ∞ < 𝑏 , by means of hypotheses H ( 𝑓 ) (ii) and (iii), we are able to apply the strong maximum principle. So we get 𝑢 ∈ i n t ( 𝐶 1 0 ( Ω ) + ) , hence 𝑢 ∈ [ ( 1 / 𝑛 ) 𝜑 1 , 𝑢 ] for 𝑛 sufficiently large. The fact that 𝑢 𝑛 is the least solution of (1.4) in [ ( 1 / 𝑛 ) 𝜑 1 , 𝑢 ] ensures 𝑢 𝑛 ≤ 𝑢 . Taking into account (2.17), we obtain 𝑢 + ≤ 𝑢 . This completes the proof.

2.3. Sign-Changing Solution

The main result of this section is as follows.

Theorem 2.5. Under hypotheses 𝐻 ( 𝑓 ) , for all 𝑏 > 0 , there exists a number 𝜆 0 ∈ ( 0 , 𝜆 ) with the property that if 𝜆 ∈ ( 0 , 𝜆 0 ) , then problem (1.4) has a (positive) solution 𝑢 + = 𝑢 + ( 𝜆 ) ∈ i n t ( 𝐶 1 0 ( Ω ) + ) , a (negative) solution 𝑢 − = 𝑢 − ( 𝜆 ) ∈ − i n t ( 𝐶 1 0 ( Ω ) + ) , and a nontrivial sign-changing solution 𝑢 0 = 𝑢 0 ( 𝜆 ) ∈ 𝐶 1 0 ( Ω ) satisfying ‖ 𝑢 + ‖ ∞ < 𝑏 , ‖ 𝑢 − ‖ ∞ < 𝑏 , ‖ 𝑢 0 ‖ ∞ < 𝑏 .

Proof. Let 𝑏 > 0 . Consider the positive number 𝜆 0 given by Theorem 2.4 and fix 𝜆 ∈ ( 0 , 𝜆 0 ) . Let 𝑢 + ∈ i n t ( 𝐶 1 0 ( Ω ) + ) and 𝑢 − ∈ − i n t ( 𝐶 1 0 ( Ω ) + ) be the two extremal solutions determined in Theorem 2.4. We introduce on Ω × ℝ the truncation functions 𝜏 + ⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩ ( 𝑥 , 𝑠 ) = 0 i f 𝑠 ≤ 0 , 𝑠 i f 0 < 𝑠 < 𝑢 + ( 𝑢 𝑥 ) , + ( 𝑥 ) i f 𝑠 ≥ 𝑢 + ( 𝜏 𝑥 ) , − ⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩ 𝑢 ( 𝑥 , 𝑠 ) = − ( 𝑥 ) i f 𝑠 ≤ 𝑢 − ( 𝑥 ) , 𝑠 i f 𝑢 − 𝜏 ( 𝑥 ) < 𝑠 < 0 , 0 i f 𝑠 ≥ 0 , 0 ⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩ 𝑢 ( 𝑥 , 𝑠 ) = − ( 𝑥 ) i f 𝑠 ≤ 𝑢 − ( 𝑥 ) , 𝑠 i f 𝑢 − ( 𝑥 ) < 𝑠 < 𝑢 + 𝑢 ( 𝑥 ) , + ( 𝑥 ) i f 𝑠 ≥ 𝑢 + ( 𝑥 ) ( 2 . 5 2 ) and then define the following associated functionals: 𝐸 + 1 ( 𝑢 ) = 𝑝 ‖ ∇ 𝑢 ‖ 𝑝 𝑝 − Ω 0 𝑢 ( 𝑥 ) 𝑓 𝑥 , 𝜏 + ( 𝑥 , 𝑠 ) , 𝜆 𝑑 𝑠 𝑑 𝑥 , ∀ 𝑢 ∈ 𝑉 0 , 𝐸 − 1 ( 𝑢 ) = 𝑝 ‖ ∇ 𝑢 ‖ 𝑝 𝑝 − Ω 0 𝑢 ( 𝑥 ) 𝑓 𝑥 , 𝜏 − ( 𝑥 , 𝑠 ) , 𝜆 𝑑 𝑠 𝑑 𝑥 , ∀ 𝑢 ∈ 𝑉 0 , 𝐸 0 1 ( 𝑢 ) = 𝑝 ‖ ∇ 𝑢 ‖ 𝑝 𝑝 − Ω 0 𝑢 ( 𝑥 ) 𝑓 𝑥 , 𝜏 0 ( 𝑥 , 𝑠 ) , 𝜆 𝑑 𝑠 𝑑 𝑥 , ∀ 𝑢 ∈ 𝑉 0 . ( 2 . 5 3 ) It is clear that 𝐸 + , 𝐸 − , 𝐸 0 ∈ 𝐶 1 ( 𝑉 0 ) .
We observe that if 𝑣 is a critical point of 𝐸 + , then − Δ 𝑝 𝑣 + Δ 𝑝 𝑢 + , 𝑣 − 𝑢 + + = Ω 𝑓 𝑥 , 𝜏 + ( 𝑥 , 𝑣 ( 𝑥 ) ) , 𝜆 − 𝑓 𝑥 , 𝑢 + ( 𝑥 ) , 𝜆 𝑣 − 𝑢 + + 𝑑 𝑥 = 0 ( 2 . 5 4 ) which implies 𝑣 ≤ 𝑢 + . Similarly, it follows that 𝑣 ≥ 0 . This leads to 𝑣 i s a c r i t i c a l p o i n t o f 𝐸 + ⟹ 0 ≤ 𝑣 ( 𝑥 ) ≤ 𝑢 + ( 𝑥 ) f o r a . a . 𝑥 ∈ Ω . ( 2 . 5 5 )
Since the function 𝐸 + is coercive and weakly lower semicontinuous, there exists a global minimizer 𝑧 + ∈ 𝑉 0 of it. Using (2.14), it is seen that 𝐸 + 𝑧 + = i n f 𝑉 0 𝐸 + < 0 , ( 2 . 5 6 ) and so 𝑧 + ≠ 0 . Relation (2.55) shows that 𝑧 + is a nontrivial solution of problem (1.4) belonging to the order interval [ 0 , 𝑢 + ] . Via assumptions H ( 𝑓 ) (ii) and (iii) and the boundedness of 𝑧 + , we may apply the strong maximum principle which ensures 𝑧 + > 0 on Ω . In view of the minimality property of 𝑢 + as stated in Theorem 2.4, it follows that 𝑧 + = 𝑢 + . In fact, 𝑢 + is the unique global minimizer of 𝐸 + .
Since 𝑢 + ∈ i n t ( 𝐶 1 0 ( Ω ) + ) , there exists a neighborhood 𝒰 of 𝑢 + in the space 𝐶 1 0 ( Ω ) such that 𝒰 ⊂ 𝐶 1 0 ( Ω ) + . Therefore 𝐸 0 = 𝐸 + on 𝒰 , which guarantees that 𝑢 + is a local minimizer of 𝐸 0 on 𝐶 1 0 ( Ω ) . It results that 𝑢 + is also a local minimizer of 𝐸 0 on the space 𝑉 0 (see [27], pages 655-656 ). Employing the functional 𝐸 − and proceeding as in the case of 𝑢 + , we establish that 𝑢 − is a local minimizer of 𝐸 0 on 𝑉 0 .
As in the case of (2.55), we verify that every critical point of 𝐸 0 belongs to the set { 𝑢 ∈ 𝑉 0 ∶ 𝑢 − ( 𝑥 ) ≤ 𝑢 ( 𝑥 ) ≤ 𝑢 + ( 𝑥 ) a . e . 𝑥 ∈ Ω } , which implies that every critical point of 𝐸 0 is a solution to problem (1.4). The functional 𝐸 0 is coercive, weakly lower semicontinuous, with i n f 𝑉 0 𝐸 0 < 0 . Thus 𝐸 0 has a global minimizer 𝑦 0 ∈ 𝑉 0 with 𝑦 0 ≠ 0 . The above properties ensure that 𝑦 0 is a nontrivial solution of problem (1.4) belonging to the order interval [ 𝑢 − , 𝑢 + ] . Assume 𝑦 0 ≠ 𝑢 + and 𝑦 0 ≠ 𝑢 − . We claim that 𝑦 0 changes sign. Indeed, if not, 𝑦 0 would have constant sign, for instance 𝑦 0 ≥ 0 a.e. on Ω . Using assumptions H ( 𝑓 ) (ii) and (iii) and the boundedness of 𝑦 0 , we may apply the strong maximum principle which leads to 𝑦 0 > 0 on Ω . This is impossible because it contradicts the minimality property of the solution 𝑢 + as given by Theorem 2.4. According to the claim, we obtain the conclusion of the theorem setting 𝑢 0 = 𝑦 0 .
Thus, the proof reduces to consider the cases 𝑦 0 = 𝑢 + or 𝑦 0 = 𝑢 − . To make a choice, suppose 𝑦 0 = 𝑢 + . We may also admit that 𝑢 − is a strict local minimizer of 𝐸 0 . This is true since on the contrary we would find (infinitely many) critical points 𝑥 0 of 𝐸 0 belonging to the order interval [ 𝑢 − , 𝑢 + ] which are different from 0 , 𝑢 − , 𝑢 + , and if 𝑥 0 does not change sign, taking into account the strong maximum principle, the extremality properties of the solutions 𝑢 − , 𝑢 + given in Theorem 2.4 will be contradicted. A straightforward argument allows then to find 𝜌 ∈ ( 0 , ‖ 𝑢 + − 𝑢 − ‖ ) such that 𝐸 0 𝑢 + ≤ 𝐸 0 𝑢 − 𝐸 < i n f 0 ( 𝑢 ) ∶ 𝑢 ∈ 𝜕 𝐵 𝜌 𝑢 − , ( 2 . 5 7 ) where 𝜕 𝐵 𝜌 ( 𝑢 − ) = { 𝑢 ∈ 𝑉 0 ∶ ‖ 𝑢 − 𝑢 − ‖ = 𝜌 } . Relation (2.57) in conjunction with the Palais-Smale condition (which holds for 𝐸 0 due to its coercivity) enables us to apply the mountain pass theorem to the functional 𝐸 0 (see, e.g., [31]). In this way we get 𝑢 0 ∈ 𝑉 0 satisfying 𝐸 ′ 0 ( 𝑢 0 ) = 0 and 𝐸 i n f 0 ( 𝑢 ) ∶ 𝑢 ∈ 𝜕 𝐵 𝜌 𝑢 − ≤ 𝐸 0 𝑢 0 = i n f 𝛾 ∈ Γ m a x [ ] 𝑡 ∈ − 1 , 1 𝐸 0 ( 𝛾 ( 𝑡 ) ) , ( 2 . 5 8 ) where [ ] Γ = 𝛾 ∈ 𝐶 − 1 , 1 , 𝑉 0 ∶ 𝛾 ( − 1 ) = 𝑢 − , 𝛾 ( 1 ) = 𝑢 + . ( 2 . 5 9 ) We infer from (2.57) and (2.58) that 𝑢 0 ≠ 𝑢 − and 𝑢 0 ≠ 𝑢 + .
The next step in the proof is to show that 𝐸 0 𝑢 0 < 0 . ( 2 . 6 0 ) By the equality in (2.58), it suffices to produce a path ̂ 𝛾 ∈ Γ such that 𝐸 0 [ ] . ( ̂ 𝛾 ( 𝑡 ) ) < 0 ∀ 𝑡 ∈ − 1 , 1 ( 2 . 6 1 ) Let 𝑆 = 𝑉 0 ∩ 𝜕 𝐵 𝐿 𝑝 1 ( Ω ) , where 𝜕 𝐵 𝐿 𝑝 1 ( Ω ) = { 𝑢 ∈ 𝐿 𝑝 ( Ω ) ∶ ‖ 𝑢 ‖ 𝑝 = 1 } , and 𝑆 𝐶 = 𝑆 ∩ 𝐶 1 0 ( Ω ) be endowed with the topologies induced by 𝑉 0 and 𝐶 1 0 ( Ω ) , respectively. We set Γ 0 , 𝐶 = [ ] 𝛾 ∈ 𝐶 − 1 , 1 , 𝑆 𝐶 ∶ 𝛾 ( − 1 ) = − 𝜑 1 , 𝛾 ( 1 ) = 𝜑 1 . ( 2 . 6 2 ) Making use of the first inequality in assumption H ( 𝑓 ) (iii), we fix numbers 𝜇 > 𝜆 2 and 𝛿 > 0 such that (2.14) holds, and then let 𝜌 0 ∈ ( 0 , 𝜇 − 𝜆 2 ) . We recall the following variational expression for 𝜆 2 given by Cuesta et al. [32]: 𝜆 2 = i n f 𝛾 ∈ Γ 0 m a x [ ] ) 𝑢 ∈ 𝛾 ( − 1 , 1 ‖ ∇ 𝑢 ‖ 𝑝 𝑝 , ( 2 . 6 3 ) where Γ 0 = ( [ ] 𝛾 ∈ 𝐶 − 1 , 1 , 𝑆 ) ∶ 𝛾 ( − 1 ) = − 𝜑 1 , 𝛾 ( 1 ) = 𝜑 1 . ( 2 . 6 4 ) By (2.63) there exists 𝛾 ∈ Γ 0 such that m a x 𝑡 ∈ [ − 1 , 1 ] ‖ ∇ 𝛾 ( 𝑡 ) ‖ 𝑝 𝑝 < 𝜆 2 + 𝜌 0 2 . ( 2 . 6 5 ) Choose some number 𝑟 with 0 < 𝑟 ≤ ( 𝜆 2 + 𝜌 0 ) 1 / 𝑝 − ( 𝜆 2 + 𝜌 0 / 2 ) 1 / 𝑝 . The density of 𝑆 𝐶 in 𝑆 implies that Γ 0 , 𝐶 is dense in Γ 0 ; so there is 𝛾 0 ∈ Γ 0 , 𝐶 satisfying m a x 𝑡 ∈ [ − 1 , 1 ] ‖ ‖ ∇ 𝛾 ( 𝑡 ) − ∇ 𝛾 0 ‖ ‖ ( 𝑡 ) 𝑝 < 𝑟 . ( 2 . 6 6 ) Then the choice of 𝑟 establishes m a x 𝑡 ∈ [ − 1 , 1 ] ‖ ‖ ∇ 𝛾 0 ‖ ‖ ( 𝑡 ) 𝑝 𝑝 < 𝜆 2 + 𝜌 0 . ( 2 . 6 7 ) The boundedness of the set 𝛾 0 ( [ − 1 , 1 ] ) ( Ω ) in ℝ ensures the existence of some 𝜀 1 > 0 such that 𝜀 1 | | | | 𝑢 ( 𝑥 ) ≤ 𝛿 ∀ 𝑥 ∈ Ω ∀ 𝑢 ∈ 𝛾 0 ( [ ] − 1 , 1 ) . ( 2 . 6 8 ) Since 𝑢 + , − 𝑢 − ∈ i n t ( 𝐶 1 0 ( Ω ) + ) (see Theorem 2.4), for every 𝑢 ∈ 𝛾 0 ( [ − 1 , 1 ] ) and any bounded neighborhood 𝑉 𝑢 of 𝑢 in 𝐶 1 0 ( Ω ) there exist positive numbers ℎ 𝑢 and 𝑗 𝑢 such that 𝑢 + − 1 ℎ 𝐶 𝑣 ∈ i n t 1 0 Ω + , − 𝑢 − + 1 𝑗 𝐶 𝑣 ∈ i n t 1 0 Ω + , ( 2 . 6 9 ) whenever ℎ ≥ ℎ 𝑢 , 𝑗 ≥ 𝑗 𝑢 , and 𝑣 ∈ 𝑉 𝑢 . This fact and the compactness of 𝛾 0 ( [ − 1 , 1 ] ) in 𝐶 1 0 ( Ω ) allow to determine a number 𝜀 0 > 0 for which one has 𝑢 − ( 𝑥 ) ≤ 𝜀 𝑢 ( 𝑥 ) ≤ 𝑢 + ( 𝑥 ) ∀ 𝑥 ∈ Ω , 𝑢 ∈ 𝛾 0 ( [ ] − 1 , 1 ) , 𝜀 ∈ 0 , 𝜀 0 . ( 2 . 7 0 ) We now focus on the continuous path 𝜀 𝛾 0 in 𝐶 1 0 ( Ω ) joining − 𝜀 𝜑 1 and 𝜀 𝜑 1 with a fixed constant 𝜀 satisfying 0 < 𝜀 < 𝑚 𝑖 𝑛 { 𝜀 0 , 𝜀 1 } . By (2.70), (2.67), (2.68), (2.14) with 𝜇 > 𝜆 2 , and taking into account the choice of 𝜌 0 as well as 𝛾 0 ( [ − 1 , 1 ] ) ⊂ 𝜕 𝐵 𝐿 𝑝 1 ( Ω ) we obtain 𝐸 0 𝜀 𝛾 0 = 𝜀 ( 𝑡 ) 𝑝 𝑝 ‖ ‖ ∇ 𝛾 0 ‖ ‖ ( 𝑡 ) 𝑝 𝑝 − Ω 𝜀 𝛾 0 0 ( 𝑡 ) ( 𝑥 ) 𝑓 𝑥 , 𝜏 0 = 𝜀 ( 𝑥 , 𝑠 ) , 𝜆 𝑑 𝑠 𝑑 𝑥 𝑝 𝑝 ‖ ‖ ∇ 𝛾 0 ‖ ‖ ( 𝑡 ) 𝑝 𝑝 − Ω 𝜀 𝛾 0 0 ( 𝑡 ) ( 𝑥 ) 𝑓 ≤ 𝜀 ( 𝑥 , 𝑠 , 𝜆 ) 𝑑 𝑠 𝑑 𝑥 𝑝 𝑝 𝜆 2 + 𝜌 0 [ ] . − 𝜇 < 0 ∀ 𝑡 ∈ − 1 , 1 ( 2 . 7 1 )
At this point we apply the second deformation lemma (see, e.g., [27, page 366]) to the 𝐶 1 functional 𝐸 + ∶ 𝑉 0 → ℝ . Towards this let us denote 𝑐 + = 𝑐 + ( 𝜆 ) = 𝐸 + 𝜀 𝜑 1 , 𝑚 + = 𝑚 + ( 𝜆 ) = 𝐸 + 𝑢 + , 𝐸 𝑐 + + = 𝑢 ∈ 𝑉 0 ∶ 𝐸 + ( 𝑢 ) ≤ 𝑐 + . ( 2 . 7 2 ) It was already shown that 𝑢 + is the unique global minimizer of 𝐸 + , and so we have 𝑚 + < 𝑐 + . Taking into account (2.55), 𝐸 + has no critical values in the interval ( 𝑚 + , 𝑐 + ] (for, otherwise, the minimality of the positive solution 𝑢 + of (1.4) would be contradicted). Using also that the functional 𝐸 + satisfies the Palais-Smale condition (because it is coercive), the second deformation lemma can be applied to 𝐸 + yielding a continuous mapping 𝜂 ∈ 𝐶 ( [ 0 , 1 ] × 𝐸 𝑐 + + , 𝐸 𝑐 + + ) such that 𝜂 ( 0 , 𝑢 ) = 𝑢 and 𝜂 ( 1 , 𝑢 ) = 𝑢 + for all 𝑢 ∈ 𝐸 𝑐 + + , as well as 𝐸 + ( 𝜂 ( 𝑡 , 𝑢 ) ) ≤ 𝐸 + ( 𝑢 ) whenever 𝑡 ∈ [ 0 , 1 ] and 𝑢 ∈ 𝐸 𝑐 + + . Introducing 𝛾 + ∶ [ 0 , 1 ] → 𝑉 0 by 𝛾 + ( 𝜂 𝑡 ) ∶ = 𝑡 , 𝜀 𝜑 1 + 𝜂 ∶ = m a x 𝑡 , 𝜀 𝜑 1 , 0 ( 2 . 7 3 ) for all 𝑡 ∈ [ 0 , 1 ] , it is seen that 𝛾 + is a continuous path in 𝑉 0 joining 𝜀 𝜑 1 and 𝑢 + . (Note the mapping 𝑤 ↦ 𝑤 + is continuous from 𝑉 0 into itself.) The properties of the deformation 𝜂 imply 𝐸 0 𝛾 + ( 𝑡 ) = 𝐸 + 𝛾 + ( 𝑡 ) ≤ 𝐸 + 𝜂 𝑡 , 𝜀 𝜑 1 = 𝐸 + 𝜀 𝜑 1 = 𝐸 0 𝜀 𝜑 1 < 0 ( 2 . 7 4 ) for all 𝑡 ∈ [ 0 , 1 ] . Similarly, applying the second deformation lemma to the functional 𝐸 − , we construct a continuous path 𝛾 − ∶ [ 0 , 1 ] → 𝑉 0 joining 𝑢 − and − 𝜀 𝜑 1 such that 𝐸 0 𝛾 − [ ] ( 𝑡 ) < 0 ∀ 𝑡 ∈ 0 , 1 . ( 2 . 7 5 ) The union of the curves 𝛾 − , 𝜀 𝛾 0 , and 𝛾 + gives rise to a path ̂ 𝛾 ∈ Γ . We see from (2.75), (2.71), and (2.74) that (2.61) is satisfied. Hence (2.60) holds, and so 𝑢 0 ≠ 0 . Recalling that the critical points of 𝐸 0 are in the order interval { 𝑢 ∈ 𝑉 0 ∶ 𝑢 − ( 𝑥 ) ≤ 𝑢 ( 𝑥 ) ≤ 𝑢 + ( 𝑥 ) a . e . 𝑥 ∈ Ω } we derive that 𝑢 0 is a nontrivial solution of (1.4) distinct from 𝑢 − and 𝑢 + , with 𝑢 − ≤ 𝑢 0 ≤ 𝑢 + . By the nonlinear regularity theory we have that 𝑢 0 ∈ 𝐶 1 0 ( Ω ) . The extremality properties of the constant sign solutions 𝑢 − and 𝑢 + as described in Theorem 2.4 force 𝑢 0 to be sign-changing. This completes the proof.

2.4. Two Sign-Changing Solutions

The goal of this section is to show that under hypotheses stronger than those in Theorem 2.5, problem (1.4) possesses at least two sign-changing solutions.

The new hypotheses on the nonlinearity 𝑓 ( 𝑥 , 𝑠 , 𝜆 ) in problem (1.4) are the following. H ( 𝑓 ) 𝑓 ∶ Ω × ℝ × ( 0 , 𝜆 ) → ℝ , with 𝜆 > 0 , is a function such that 𝑓 ( 𝑥 , 0 , 𝜆 ) = 0 for a.a. 𝑥 ∈ Ω , whenever 𝜆 ∈ ( 0 , 𝜆 ) . (i)For all 𝜆 ∈ ( 0 , 𝜆 ) , 𝑓 ( ⋅ , ⋅ , 𝜆 ) ∈ 𝐶 ( Ω × ℝ ) .(ii)There are constants 𝑐 > 0 , 𝑟 ∈ ( 𝑝 − 1 , 𝑝 ∗ − 1 ) , and functions 𝑎 ( ⋅ , 𝜆 ) ∈ 𝐿 ∞ ( Ω ) + ( 𝜆 ∈ ( 0 , 𝜆 ) ) with ‖ 𝑎 ( ⋅ , 𝜆 ) ‖ ∞ → 0 as 𝜆 ↓ 0 such that | | | | 𝑓 ( 𝑥 , 𝑠 , 𝜆 ) ≤ 𝑎 ( 𝑥 , 𝜆 ) + 𝑐 | 𝑠 | 𝑟 f o r a . a . 𝑥 ∈ Ω ∀ ( s , 𝜆 ) ∈ ℝ × 0 , 𝜆 . ( 2 . 7 6 ) (iii)For all 𝜆 ∈ ( 0 , 𝜆 ) there exist constants 𝜇 0 = 𝜇 0 ( 𝜆 ) > 𝜆 2 , 𝜈 0 = 𝜈 0 ( 𝜆 ) > 𝜇 0 and a set Ω 𝜆 ⊂ Ω with Ω ⧵ Ω 𝜆 of Lebesgue measure zero such that 𝜇 0 < l i m i n f 𝑠 → 0 𝑓 ( 𝑥 , 𝑠 , 𝜆 ) | 𝑠 | 𝑝 − 2 𝑠 ≤ l i m s u p 𝑠 → 0 𝑓 ( 𝑥 , 𝑠 , 𝜆 ) | 𝑠 | 𝑝 − 2 𝑠 ≤ 𝜈 0 ( 2 . 7 7 ) uniformly with respect to 𝑥 ∈ Ω 𝜆 .(iv)There exist constants 𝑏 − < 0 < 𝑏 + such that for all 𝜆 ∈ ( 0 , 𝜆 ) we have 𝑓 𝑥 , 𝑏 − , 𝜆 = 0 = 𝑓 𝑥 , 𝑏 + b , 𝜆 ∀ 𝑥 ∈ Ω , 𝑓 ( 𝑥 , 𝑠 , 𝜆 ) < 0 ∀ 𝑥 ∈ Ω , a l l 𝑠 ∈ − , 𝑓 , 0 ( 𝑥 , 𝑠 , 𝜆 ) > 0 ∀ 𝑥 ∈ Ω , a l l 𝑠 ∈ 0 , b + . ( 2 . 7 8 ) (v)For every 𝜆 ∈ ( 0 , 𝜆 ) , there exist 𝑀 = 𝑀 ( 𝜆 ) > 0 and 𝜇 = 𝜇 ( 𝜆 ) > 𝑝 such that 0 < 𝜇 𝐹 ( 𝑥 , 𝑠 , 𝜆 ) ≤ 𝑓 ( 𝑥 , 𝑠 , 𝜆 ) 𝑠 ∀ 𝑥 ∈ Ω , a l l | s | ≥ 𝑀 . ( 2 . 7 9 )

We notice that hypotheses H ' ( 𝑓 ) are stronger than H ( 𝑓 ) . In particular, for every 𝜆 ∈ ( 0 , 𝜆 ) , we added the Ambrosetti-Rabinowitz condition for 𝑓 ( ⋅ , ⋅ , 𝜆 ) (see hypothesis H ' ( 𝑓 ) (v)).

We state now the main result of this section, which produces two sign-changing solutions for problem (1.4).

Theorem 2.6. Assume that hypotheses H ' ( 𝑓 ) are fulfilled. Then there exists a number 𝜆 0 ∈ ( 0 , 𝜆 ) with the property that if 𝜆 ∈ ( 0 , 𝜆 0 ) , then problem (1.4) has a minimal (positive) solution 𝑢 + = 𝑢 + ( 𝜆 ) ∈ i n t ( 𝐶 1 0 ( Ω ) + ) , a maximal (negative) solution 𝑢 − = 𝑢 − ( 𝜆 ) ∈ − i n t ( 𝐶 1 0 ( Ω ) + ) , and two nontrivial sign-changing solutions 𝑢 0 = 𝑢 0 ( 𝜆 ) , 𝑤 0 = 𝑤 0 ( 𝜆 ) ∈ 𝐶 1 0 ( Ω ) satisfying ‖ 𝑢 + ‖ ∞ < 𝑏 , ‖ 𝑢 − ‖ ∞ < 𝑏 , 𝑢 − ≤ 𝑢 0 ≤ 𝑢 + a.e. in Ω (so ‖ 𝑢 0 ‖ ∞ < 𝑏 ) and ‖ 𝑤 0 ‖ ∞ ≥ 𝑏 , where 𝑏 ∶ = m i n { 𝑏 + , | 𝑏 − | } .

Proof. Since hypotheses H ' ( 𝑓 ) are stronger than H ( 𝑓 ) , we can apply Theorem 2.5 with 𝑏 = m i n { 𝑏 + , | 𝑏 − | } , which ensures the existence of a number 𝜆 0 ∈ ( 0 , 𝜆 ) such that for every 𝜆 ∈ ( 0 , 𝜆 0 ) , problem (1.4) possesses a positive solution 𝑢 + = 𝑢 + ( 𝜆 ) ∈ i n t ( 𝐶 1 0 ( Ω ) + ) , a negative solution 𝑢 − = 𝑢 − ( 𝜆 ) ∈ − i n t ( 𝐶 1 0 ( Ω ) + ) , and a sign-changing solution 𝑢 0 = 𝑢 0 ( 𝜆 ) ∈ 𝐶 1 0 ( Ω ) with − 𝑏 < 𝑢 − ≤ 𝑢 0 ≤ 𝑢 + < 𝑏 . The proof of Theorem 2.5 shows that 𝑢 + and 𝑢 − can be chosen to be the minimal positive solution and the maximal negative solution, respectively.
On the other hand, hypotheses H ' ( 𝑓 ) enable us to apply Theorem 1.1 of Bartsch et al. [33]. It follows that there exists a sign-changing solution 𝑤 0 = 𝑤 0 ( 𝜆 ) ∈ 𝐶 1 0 ( Ω ) (by the nonlinear regularity theory) with m a x Ω 𝑤 0 ≥ 𝑏 + and m i n Ω 𝑤 0 ≤ 𝑏 − . Therefore, we have ‖ 𝑤 0 ‖ ∞ ≥ 𝑏 , which shows that the sign-changing solutions 𝑢 0 and 𝑤 0 are different. This completes the proof.

Remark 2.7. In fact, under hypotheses H ' ( 𝑓 ) , for 𝜆 ∈ ( 0 , 𝜆 0 ) , problem (1.4) admits at least six nontrivial solutions: two positive solutions, two negative solutions, and two sign-changing solutions, as seen in Theorem 5 in [34].

3. Problem (1.7) for Parameters 𝑎 and 𝑏 being Large

The main goal of this section is to provide a detailed multiplicity analysis of the nonsmooth elliptic problem (1.7) in dependence of the two parameters 𝑎 and 𝑏 . Conditions in terms of the Fučik spectrum are formulated that ensure the existence of sign-changing solutions. As for the precise formulation of this result we recall the Fučik spectrum, see, for example, [13].

The set Σ 𝑝 of those points ( 𝜇 1 , 𝜇 2 ) ∈ ℝ 2 for which the problem 𝑢 ∈ 𝑉 0 ∶ − Δ 𝑝 𝑢 = 𝜇 1 𝑢 + 𝑝 − 1 − 𝜇 2 ( 𝑢 − ) 𝑝 − 1 i n 𝑉 ∗ 0 ( 3 . 1 ) has a nontrivial solution is called the Fučik spectrum of the negative 𝑝 -Laplacian on Ω . Hence, Σ 𝑝 clearly contains the two lines 𝜆 1 × ℝ and ℝ × 𝜆 1 with 𝜆 1 being the first Dirichlet eigenvalue of − Δ 𝑝 . In addition, the spectrum 𝜎 ( − Δ 𝑝 ) of the negative 𝑝 -Laplacian has an unbounded sequence of variational eigenvalues ( 𝜆 𝑙 ) , 𝑙 ∈ ℕ , satisfying a standard min-max characterization, and Σ 𝑝 contains the corresponding sequence of points ( 𝜆 𝑙 , 𝜆 𝑙 ) , 𝑙 ∈ ℕ . A first nontrivial curve 𝒞 in Σ 𝑝 through ( 𝜆 2 , 𝜆 2 ) asymptotic to 𝜆 1 × ℝ and ℝ × 𝜆 1 at infinity was constructed and variationally characterized by a mountain-pass procedure by Cuesta et al. [32] (see Figure 1), which implies the existence of a continuous path in { 𝑢 ∈ 𝑉 0 ∶ 𝐼 ( 𝑎 , 𝑏 ) ( 𝑢 ) < 0 , ‖ 𝑢 ‖ 𝑝 = 1 } joining − 𝜑 1 and 𝜑 1 provided ( 𝑎 , 𝑏 ) is above the curve 𝒞 . Here the functional 𝐼 ( 𝑎 , 𝑏 ) on 𝑉 0 is given by 𝐼 ( 𝑎 , 𝑏 ) ( 𝑢 ) = Ω | | | | ∇ 𝑢 𝑝 𝑢 − 𝑎 + 𝑝 − 𝑏 ( 𝑢 − ) 𝑝 𝑑 𝑥 . ( 3 . 2 ) The hypothesis on the parameters 𝑎 and 𝑏 that will finally ensure the existence of sign-changing solutions is as follows. (H)Let ( 𝑎 , 𝑏 ) ∈ ℝ 2 + be above the curve 𝒞 of the Fučik spectrum constructed in [32]; see Figure 1.

536236.fig.001
Figure 1: Fučik Spectrum.
3.1. Hypotheses, Definitions, and Preliminaries

We impose the following hypotheses on the nonlinearity 𝑔 ∶ Ω × ℝ → ℝ whose primitive is 𝐺 of problem (1.7) ( g 1 ) ( 𝑥 , 𝑠 ) ↦ 𝑔 ( 𝑥 , 𝑠 ) is measurable in each variable separately. ( g 2 ) There exists 𝑐 > 0 , and 𝑞 ∈ [ 𝑝 , 𝑝 ∗ ) such that | | | | 𝑔 ( 𝑥 , 𝑠 ) ≤ 𝑐 1 + | 𝑠 | 𝑞 − 1 ( 3 . 3 ) for a.a. 𝑥 ∈ Ω and for all 𝑠 ∈ ℝ , where 𝑝 ∗ denotes the critical Sobolev exponent which is 𝑝 ∗ = 𝑁 𝑝 / ( 𝑁 − 𝑝 ) if 𝑝 < 𝑁 , and 𝑝 ∗ = + ∞ if 𝑝 ≥ 𝑁 . ( g 3 ) One has l i m 𝑠 → 0 𝑔 ( 𝑥 , 𝑠 ) | 𝑠 | 𝑝 − 2 𝑠 = 0 u n i f o r m l y w i t h r e s p e c t t o a . a . 𝑥 ∈ Ω . ( 3 . 4 ) ( g 4 ) One has l i m | 𝑠 | → ∞ 𝑔 ( 𝑥 , 𝑠 ) | 𝑠 | 𝑝 − 2 𝑠 = + ∞ u n i f o r m l y w i t h r e s p e c t t o a . a . 𝑥 ∈ Ω . ( 3 . 5 )

In view of assumptions (g1) and (g2) the function 𝑠 ↦ 𝐺 ( 𝑥 , 𝑠 ) is locally Lipschitz and the functional 𝒢 ∶ 𝐿 𝑞 ( Ω ) → ℝ defined by 𝒢 ( 𝑢 ) ∶ = Ω 𝐺 ( 𝑥 , 𝑢 ( 𝑥 ) ) 𝑑 𝑥 , 𝑢 ∈ 𝐿 𝑞 ( Ω ) ( 3 . 6 ) is well defined and locally Lipschitz continuous as well. The generalized gradients 𝜕 𝐺 ( 𝑥 , ⋅ ) and 𝜕 𝒢 can be characterized as follows: Define for every ( 𝑥 , 𝑡 ) ∈ Ω × ℝ , 𝑔 1 ( 𝑥 , 𝑡 ) ∶ = l i m 𝛿 → 0 + e s s i n f | 𝜏 − 𝑡 | < 𝛿 𝑔 ( 𝑥 , 𝜏 ) , 𝑔 2 ( 𝑥 , 𝑡 ) ∶ = l i m 𝛿 → 0 + e s s s u p | 𝜏 − 𝑡 | < 𝛿 𝑔 ( 𝑥 , 𝜏 ) . ( 3 . 7 ) Proposition 1.7 in [35] ensures that 𝑔 𝜕 𝐺 ( 𝑥 , 𝜉 ) = 1 ( 𝑥 , 𝜉 ) , 𝑔 2 ( 𝑥 , 𝜉 ) , ( 3 . 8 ) while Theorem 4.5.19 of [36] implies 𝜕 𝒢 ( 𝑢 ) ⊆ 𝑤 ∈ 𝐿 𝑞 ′ ( Ω ) ∶ 𝑔 1 ( 𝑥 , 𝑢 ( 𝑥 ) ) ≤ 𝑤 ( 𝑥 ) ≤ 𝑔 2 ( 𝑥 , 𝑢 ( 𝑥 ) ) a . e . i n Ω ( 3 . 9 ) with 𝑞 ′ ∶ = 𝑞 / ( 𝑞 − 1 ) . The next result is an immediate consequence of [37, Proposition 2.1.5].

Lemma 3.1. Suppose 𝑢 𝑛 → 𝑢 in 𝑉 0 , 𝑤 𝑛 ⇀ 𝑤 in 𝐿 𝑞 ( Ω ) , and 𝑤 𝑛 ∈ 𝜕 𝒢 ( 𝑢 𝑛 ) for all 𝑛 ∈ ℕ . Then 𝑤 ∈ 𝜕 𝒢 ( 𝑢 ) .

Definition 3.2. A function 𝑢 ∈ 𝑉 0 is called a solution of (1.7) if there is an 𝜂 ∈ 𝐿 𝑞 ( Ω ) such that 𝜂 ( 𝑥 ) ∈ 𝜕 𝐺 ( 𝑥 , 𝑢 ( 𝑥 ) ) f o r a . a . 𝑥 ∈ Ω , Ω | | | | ∇ 𝑢 𝑝 − 2 ∇ 𝑢 ∇ 𝜑 𝑑 𝑥 + Ω 𝑢 𝜂 − 𝑎 + 𝑝 − 1 + 𝑏 ( 𝑢 − ) 𝑝 − 1 𝜑 𝑑 𝑥 = 0 , ∀ 𝜑 ∈ 𝑉 0 . ( 3 . 1 0 )

Remark 3.3. Due to assumption (g3) we have 𝑔 1 ( 𝑥 , 0 ) ≤ 0 ≤ 𝑔 2 ( 𝑥 , 0 ) for almost all 𝑥 ∈ Ω . Hence, in view of (3.8), problem (1.7) always possesses the trivial solution.

Definition 3.4. A function 𝑢 ∈ 𝑉 ∶ = 𝑊 1 , 𝑝 ( Ω ) is called a subsolution of (1.7) if 𝑢 | 𝜕 Ω ≤ 0 , and if there is an 𝜂 ∈ 𝐿 𝑞 ( Ω ) such that 𝜂 ( 𝑥 ) ∈ 𝜕 𝐺 𝑥 , 𝑢 ( 𝑥 ) f o r a . a . 𝑥 ∈ Ω , Ω | | ∇ 𝑢 | | 𝑝 − 2 ∇ 𝑢 ∇ 𝜑 𝑑 𝑥 + Ω 𝜂 𝑢 − 𝑎 + 𝑝 − 1 𝑢 + 𝑏 − 𝑝 − 1 𝜑 𝑑 𝑥 ≤ 0 , ∀ 𝜑 ∈ 𝑉 0 ∩ 𝐿 𝑝 ( Ω ) + . ( 3 . 1 1 )

Similarly, we define a supersolution as follows.

Definition 3.5. A function 𝑢 ∈ 𝑉 is called a supersolution of (1.7) if 𝑢 | 𝜕 Ω ≥ 0 , and if there is an 𝜂 ∈ 𝐿 𝑞 ( Ω ) such that 𝜂 ( 𝑥 ) ∈ 𝜕 𝐺 𝑥 , 𝑢 ( 𝑥 ) f o r a . a . 𝑥 ∈ Ω , Ω | | ∇ 𝑢 | | 𝑝 − 2 ∇ 𝑢 ∇ 𝜑 𝑑 𝑥 + Ω 𝜂 − 𝑎 𝑢 + 𝑝 − 1 + 𝑏 𝑢 − 𝑝 − 1 𝜑 𝑑 𝑥 ≥ 0 , ∀ 𝜑 ∈ 𝑉 0 ∩ 𝐿 𝑝 ( Ω ) + . ( 3 . 1 2 )

Lemma 3.6. Let 𝑒 be the uniquely defined solution of (2.6). If 𝑎 > 𝜆 1 , then there exists a constant 𝛼 𝑎 > 0 such that for any 𝑏 ∈ ℝ the function 𝛼 𝑎 𝑒 is a positive supersolution of problem (1.7).

Proof. Let 𝑎 > 𝜆 1 . By (g4) there is 𝑠 𝑎 > 0 such that 𝑔 ( 𝑥 , 𝑠 ) 𝑠 𝑝 − 1 > 𝑎 f o r a . a . 𝑥 ∈ Ω ∀ 𝑠 > s a , ( 3 . 1 3 ) and by (g2) we get | | 𝑔 ( 𝑥 , 𝑠 ) − 𝑎 𝑠 𝑝 − 1 | | ≤ | | | | 𝑔 ( 𝑥 , 𝑠 ) + 𝑎 𝑠 𝑝 − 1 ≤ 𝑐 𝑎 , f o r a . a . 𝑥 ∈ Ω ∀ 𝑠 ∈ 0 , s a , ( 3 . 1 4 ) which implies 𝑔 ( 𝑥 , 𝑠 ) ≥ 𝑎 𝑠 𝑝 − 1 − 𝑐 𝑎 f o r a . a . 𝑥 ∈ Ω ∀ 𝑠 ≥ 0 , ( 3 . 1 5 ) and thus in view of the definition of 𝑔 2 we obtain 𝑔 2 ( 𝑥 , 𝑠 ) ≥ 𝑎 𝑠 𝑝 − 1 − 𝑐 𝑎 f o r a . a . 𝑥 ∈ Ω ∀ 𝑠 ≥ 0 , ( 3 . 1 6 ) Let 𝑢 = 𝛼 𝑎 𝑒 , where 𝛼 𝑎 is a positive constant to be specified. Then we get − Δ 𝑝 𝑢 − 𝑎 𝑢 + 𝑝 − 1 + 𝑏 𝑢 − 𝑝 − 1 + 𝑔 2 𝑥 , 𝑢 = 𝛼 𝑎 𝑝 − 1 − 𝑎 𝛼 𝑎 𝑝 − 1 𝑒 𝑝 − 1 + 𝑔 2 𝑥 , 𝛼 𝑎 𝑒 ≥ 𝛼 𝑎 𝑝 − 1 − 𝑐 𝑎 , ( 3 . 1 7 ) which shows that for 𝛼 𝑎 ∶ = 𝑐 𝑎 1 / ( 𝑝 − 1 ) the function 𝛼 𝑎 𝑒 is in fact a supersolution of (1.7) with 𝜂 ( 𝑥 ) = 𝑔 2 ( 𝑥 , 𝑢 ( 𝑥 ) ) .

In a similar way the following lemma on the existence of a negative subsolution can be proved.

Lemma 3.7. Let 𝑒 be the uniquely defined solution of (2.6). If 𝑏 > 𝜆 1 , then there exists a constant 𝛽 𝑏 > 0 such that for any 𝑎 ∈ ℝ the function − 𝛽 𝑏 𝑒 is a negative subsolution of problem (1.7).

In the next lemma we demonstrate that small constant multiples of 𝜑 1 may be sub- and supersolutions of (1.7). More precisely we have the following result.

Lemma 3.8. Let 𝜑 1 be the normalized positive eigenfunction corresponding to the first eigenvalue 𝜆 1 of ( − Δ 𝑝 , 𝑉 0 ) . If 𝑎 > 𝜆 1 , then for 𝜀 > 0 sufficiently small and any 𝑏 ∈ ℝ the function 𝜀 𝜑 1 is a positive subsolution of problem (1.7). If 𝑏 > 𝜆 1 , then for 𝜀 > 0 sufficiently small and any 𝑎 ∈ ℝ the function − 𝜀 𝜑 1 is a negative supersolution of problem (1.7).

Proof. By (g3) there is a constant 𝛿 𝑎 > 0 such that | | | | 𝑔 ( 𝑥 , 𝑠 ) | 𝑠 | 𝑝 − 1 < 𝑎 − 𝜆 1 f o r a . a . 𝑥 ∈ Ω ∀ 0 < | 𝑠 | ≤ 𝛿 𝑎 , ( 3 . 1 8 ) which implies | | | | ≤ 𝑔 ( 𝑥 , 𝑠 ) 𝑎 − 𝜆 1 𝑠 𝑝 − 1 f o r a . a . 𝑥 ∈ Ω , ∀ 𝑠 ∶ 0 ≤ 𝑠 ≤ 𝛿 𝑎 . ( 3 . 1 9 ) Define 𝑢 = 𝜀 𝜑 1 with 𝜀 > 0 . Applying (3.19) and the definition of 𝑔 1 we get − Δ 𝑝 𝑢 𝑢 − 𝑎 + 𝑝 − 1 𝑢 + 𝑏 − 𝑝 − 1 + 𝑔 1 𝑥 , 𝑢 = 𝜆 1 𝜀 𝜑 1 𝑝 − 1 − 𝑎 𝜀 𝜑 1 𝑝 − 1 + 𝑔 1 𝑥 , 𝜀 𝜑 1 ≤ 𝜆 1 − 𝑎 𝜀 𝜑 1 𝑝 − 1 + 𝑎 − 𝜆 1 𝜀 𝜑 1 𝑝 − 1 = 0 ( 3 . 2 0 ) provided 0 ≤ 𝜀 𝜑 1 ≤ 𝛿 𝑎 . The latter can be satisfied by choosing 𝜀 sufficiently small such that 𝜀 ∈ ( 0 , 𝛿 𝑎 / ‖ 𝜑 1 ‖ ∞ ) , where ‖ 𝜑 1 ‖ ∞ stands for the supremum-norm of 𝜑 1 . This proves that 𝜀 𝜑 1 is a subsolution if 𝜀 ∈ ( 0 , 𝛿 𝑎 / ‖ 𝜑 1 ‖ ∞ ) . In a similar way one can show that for 𝜀 sufficiently small the function − 𝜀 𝜑 1 is a negative supersolution.

Applying a recently obtained comparison result that holds for even more general elliptic inclusions (see [38, Theorem 4.1, Corollary 4.1] we immediately obtain the following theorem.

Theorem 3.9. Let hypotheses (g1)-(g2) be satisfied and assume the existence of a subsolution 𝑢 and supersolution 𝑢 of (1.7) such that 𝑢 ≤ 𝑢 . Then there exist extremal solutions of (1.7) within [ 𝑢 , 𝑢 ] .

3.2. Extremal Constant-Sign Solutions and Their Variational Characterization

Combining the results of Lemmas 3.6, 3.7, and 3.8 and Theorem 3.9 we immediately deduce the existence of nontrivial positive solutions of problem (1.7) provided the parameter 𝑎 satisfies 𝑎 > 𝜆 1 that and the existence of negative solutions of problem (1.7) provided that the parameter 𝑏 satisfies 𝑏 > 𝜆 1 . Our main goal of this section is to show that problem (1.7) has a smallest positive solution 𝑢 + ∈ i n t ( 𝐶 1 0 ( Ω ) + ) and a greatest negative solution 𝑢 − ∈ − i n t ( 𝐶 1 0 ( Ω ) + ) . More precisely the following result will be shown.

Theorem 3.10. Let hypotheses (g1)–(g4) be fulfilled. For every 𝑎 > 𝜆 1 and 𝑏 ∈ ℝ there exists a smallest positive solution 𝑢 + = 𝑢 + ( 𝑎 ) ∈ i n t ( 𝐶 1 0 ( Ω ) + ) of (1.7) within the order interval [ 0 , 𝛼 𝑎 𝑒 ] with the constant 𝛼 𝑎 > 0 as in Lemma 3.6. For every 𝑏 > 𝜆 1 and 𝑎 ∈ ℝ there exists a greatest negative solution 𝑢 − = 𝑢 − ( 𝑏 ) ∈ − i n t ( 𝐶 1 0 ( Ω ) + ) of (1.7) within the order interval [ − 𝛽 𝑏 𝑒 , 0 ] with the constant 𝛽 𝑏 > 0 as in Lemma 3.7.

Proof. Let 𝑎 > 𝜆 1 . Lemmas 3.6 and 3.8 ensure that 𝑢 = 𝛼 𝑎 𝑒 ∈ i n t ( 𝐶 1 0 ( Ω ) + ) is a supersolution of problem (1.7) and 𝑢 = 𝜀 𝜑 1 ∈ i n t ( 𝐶 1 0 ( Ω ) + ) is a subsolution of problem (1.7) provided that 𝜀 > 0 is sufficiently small. We may choose 𝜀 > 0 such that, in addition, 𝜀 𝜑 1 ≤ 𝛼 𝑎 𝑒 . Thus by Theorem 3.9 there exists a smallest and a greatest solution of (1.7) within the ordered interval [ 𝜀 𝜑 1 , 𝛼 𝑎 𝑒 ] . Let us denote the smallest solution by 𝑢 𝜀 . Moreover, the nonlinear regularity theory for the 𝑝 -Laplacian (cf., e.g., [27, Theorem 1.5.6]) and Vázquez's strong maximum principle [28] ensure that 𝑢 𝜀 ∈ i n t ( 𝐶 1 0 ( Ω ) + ) . Thus for every positive integer 𝑛 sufficiently large there is a smallest solution 𝑢 𝑛 ∈ i n t ( 𝐶 1 0 ( Ω ) + ) of problem (1.7) within [ ( 1 / 𝑛 ) 𝜑 1 , 𝛼 𝑎 𝑒 ] . In this way we inductively construct a sequence ( 𝑢 𝑛 ) of smallest solutions which is monotone decreasing; that is, we have 𝑢 𝑛 ↓ 𝑢 + p o i n t w i s e ( 3 . 2 1 ) with some function 𝑢 + ∶ Ω → ℝ satisfying 0 ≤ 𝑢 + ≤ 𝛼 𝑎 𝑒 .Claim 1. 𝑢 + is a solution of problem (1.7).As 𝑢 𝑛 ∈ i n t ( 𝐶 1 0 ( Ω ) + ) and 𝑢 𝑛 are solutions of (1.7) we have ‖ ‖ ∇ 𝑢 𝑛 ‖ ‖ 𝑝 𝑝 = Ω 𝑎 𝑢 𝑛 ( 𝑥 ) 𝑝 − 𝜂 𝑛 ( 𝑢 𝑥 ) 𝑛 ( 𝑥 ) 𝑑 𝑥 , ( 3 . 2 2 ) where 𝜂 𝑛 ∈ 𝐿 𝑞 ( Ω ) and 𝜂 𝑛 ( 𝑥 ) ∈ 𝜕 𝐺 ( 𝑥 , 𝑢 𝑛 ( 𝑥 ) ) for almost all 𝑥 ∈ Ω . Since 𝑢 𝑛 ∈ [ ( 1 / 𝑛 ) 𝜑 1 , 𝛼 𝑎 𝑒 ] , the last equation together with (g2) implies that the sequence ( 𝑢 𝑛 ) is bounded in 𝑉 0 . Taking into account (3.21) we obtain that 𝑢 + ∈ 𝑉 0 and 𝑢 𝑛 ⇀ 𝑢 + i n 𝑉 0 , 𝑢 𝑛 ⟶ 𝑢 + i n 𝐿 𝑝 ( Ω ) a n d a . e . i n Ω . ( 3 . 2 3 ) The solution 𝑢 𝑛 of (1.7) satisfies − Δ 𝑝 𝑢 𝑛 = , 𝜑 Ω 𝑎 𝑢 𝑛 𝑝 − 1 − 𝜂 𝑛 𝜑 𝑑 𝑥 , ∀ 𝜑 ∈ 𝑉 0 , ( 3 . 2 4 ) which yields with 𝜑 = 𝑢 𝑛 − 𝑢 + in (3.24) the equation − Δ 𝑝 𝑢 𝑛 , 𝑢 𝑛 − 𝑢 + = Ω 𝑎 𝑢 𝑛 𝑝 − 1 − 𝜂 𝑛 𝑢 𝑛 − 𝑢 + 𝑑 𝑥 . ( 3 . 2 5 ) Using the convergence properties (3.23) of ( 𝑢 𝑛 ) and (g2) as well as the uniform boundedness of the sequence ( 𝑢 𝑛 ) , we get by applying Lebesgue's dominated convergence theorem l i m 𝑛 → ∞ − Δ 𝑝 𝑢 𝑛 , 𝑢 𝑛 − 𝑢 + = 0 , ( 3 . 2 6 ) which by the 𝑆 + -property of − Δ 𝑝 on 𝑉 0 implies 𝑢 𝑛 ⟶ 𝑢 + i n 𝑉 0 a s 𝑛 ⟶ ∞ . ( 3 . 2 7 ) Since 𝑢 𝑛 are uniformly bounded, from (g2) we see that there exists a constant 𝑐 > 0 such that | | 𝜂 𝑛 | | ( 𝑥 ) ≤ 𝑐 a . e . i n Ω , ∀ 𝑛 ∈ ℕ , ( 3 . 2 8 ) and thus we get (for some subsequence if necessary) 𝜂 𝑛 ⇀ 𝜂 + in 𝐿 𝑞 ( Ω ) . By the strong convergence (3.27), Lemma 3.1 can be applied to show that 𝜂 + ( 𝑥 ) ∈ 𝜕 𝐺 ( 𝑥 , 𝑢 + ( 𝑥 ) ) for almost every 𝑥 ∈ Ω . Passing to the limit in (3.24) (for some subsequence if necessary) proves Claim 1.
As 𝑢 + belongs, in particular, to 𝐿 ∞ ( Ω ) , Claim 1 and Assumption (g2) implies Δ 𝑝 𝑢 + ∈ 𝐿 ∞ ( Ω ) . The nonlinear regularity theory (cf., e.g., Theorem 1.5.6 in [27]) ensures that 𝑢 + ∈ 𝐶 1 , 𝛾 ( Ω ) for some 𝛾 ∈ ( 0 , 1 ) , so 𝑢 + ∈ 𝐶 1 0 ( Ω ) . In view of (g2) (g3) a constant ̃ 𝑐 𝑎 > 0 can be found such that | | | | 𝑔 ( 𝑥 , 𝑠 ) ≤ ̃ 𝑐 𝑎 𝑠 𝑝 − 1 f o r a . a . 𝑥 ∈ Ω ∀ 0 ≤ s ≤ 𝛼 a ‖ e ‖ ∞ , ( 3 . 2 9 ) which yields in conjunction with Claim 1 that Δ 𝑝 𝑢 + ≤ ̃ 𝑐 𝑎 𝑢 + 𝑝 − 1 . ( 3 . 3 0 ) We now apply Vázquez's strong maximum principle [28] where in its statement the function 𝛽 is chosen as 𝛽 ( 𝑠 ) = ̃ 𝑐 𝑎 𝑠 𝑝 − 1 for all 𝑠 > 0 , which is possible because ∫ 0 + ( 1 / ( 𝑠 𝛽 ( 𝑠 ) ) 1 / 𝑝 ) 𝑑 𝑠 = + ∞ . This result guarantees that if 𝑢 + ≠ 0 , then 𝑢 + > 0 in Ω and 𝜕 𝑢 + / 𝜕 𝑛 < 0 on 𝜕 Ω which means that 𝑢 + ∈ i n t ( 𝐶 1 0 ( Ω ) + ) .Claim 2. 𝑢 + ∈ i n t ( 𝐶 1 0 ( Ω ) + ) .Suppose that Claim 2 does not hold. Then by Vázquez's strong maximum principle we must have 𝑢 + = 0 , and thus the sequence ( 𝑢 𝑛 ) satisfies 𝑢 𝑛 ( 𝑥 ) ↓ 0 ∀ 𝑥 ∈ Ω . ( 3 . 3 1 ) Setting ̃ 𝑢 𝑛 = 𝑢 𝑛 ‖ ‖ ∇ 𝑢 𝑛 ‖ ‖ 𝑝 ∀ 𝑛 , ( 3 . 3 2 ) we may suppose that along a relabelled subsequence one has ̃ 𝑢 𝑛 ⇀ ̃ 𝑢 i n 𝑉 0 , ̃ 𝑢 𝑛 ⟶ ̃ 𝑢 i n 𝐿 𝑝 ( Ω ) a n d a . e i n Ω ( 3 . 3 3 ) with some ̃ 𝑢 ∈ 𝑉 0 , and there is a function 𝑤 ∈ 𝐿 𝑝 ( Ω ) + such that | | ̃ 𝑢 𝑛 | | ( 𝑥 ) ≤ 𝑤 ( 𝑥 ) f o r a l m o s t a l l 𝑥 ∈ Ω . ( 3 . 3 4 ) Since 𝑢 𝑛 are positive solutions of (1.7), we get for ̃ 𝑢 𝑛 the following variational equation: − Δ 𝑝 ̃ 𝑢 𝑛 , 𝜑 = 𝑎 Ω ̃ 𝑢 𝑛 𝑝 − 1 𝜑 𝑑 𝑥 − Ω 𝜂 𝑛 𝑢 𝑛 𝑝 − 1 ̃ 𝑢 𝑛 𝑝 − 1 𝜑 𝑑 𝑥 , ∀ 𝜑 ∈ 𝑉 0 . ( 3 . 3 5 ) With the special test function 𝜑 = ̃ 𝑢 𝑛 − ̃ 𝑢 in (3.35) we obtain − Δ 𝑝 ̃ 𝑢 𝑛 , ̃ 𝑢 𝑛 − ̃ 𝑢 = 𝑎 Ω ̃ 𝑢 𝑛 𝑝 − 1 ̃ 𝑢 𝑛 − ̃ 𝑢 𝑑 𝑥 − Ω 𝜂 𝑛 𝑢 𝑛 𝑝 − 1 ̃ 𝑢 𝑛 𝑝 − 1 ̃ 𝑢 𝑛 − ̃ 𝑢 𝑑 𝑥 . ( 3 . 3 6 ) From (3.29) and (3.34) we get the estimate | | 𝜂 𝑛 | | ( 𝑥 ) 𝑢 𝑛 ( 𝑥 ) 𝑝 − 1 ̃ 𝑢 𝑛 𝑝 − 1 | | ( 𝑥 ) ̃ 𝑢 𝑛 | | ( 𝑥 ) − ̃ 𝑢 ( 𝑥 ) ≤ ̃ 𝑐 𝑎 𝑤 ( 𝑥 ) 𝑝 − 1 | | | | 𝑤 ( 𝑥 ) + ̃ 𝑢 ( 𝑥 ) f o r a . a . 𝑥 ∈ Ω . ( 3 . 3 7 ) As the right-hand side of the last inequality is in 𝐿 1 ( Ω ) , we may apply Lebesgue's dominated convergence theorem, which in conjunction with (3.33) yields l i m 𝑛 → ∞ Ω 𝜂 𝑛 𝑢 𝑛 𝑝 − 1 ̃ 𝑢 𝑛 𝑝 − 1 ̃ 𝑢 𝑛 − ̃ 𝑢 𝑑 𝑥 = 0 . ( 3 . 3 8 ) From (3.33) and (3.36) we conclude l i m 𝑛 → ∞ − Δ 𝑝 ̃ 𝑢 𝑛 , ̃ 𝑢 𝑛 − ̃ 𝑢 = 0 , ( 3 . 3 9 ) which in view of the 𝑆 + -property of − Δ 𝑝 on 𝑉 0 results in ̃ 𝑢 𝑛 ⟶ ̃ 𝑢 i n 𝑉 0 a s 𝑛 ⟶ ∞ , ( 3 . 4 0 ) and therefore, in particular, ‖ ∇ ̃ 𝑢 ‖ 𝑝 = 1 . Taking into account (g3), (3.31), and (3.40), we may pass to the limit in (3.35) which results in − Δ 𝑝 ̃ 𝑢 , 𝜑 = 𝑎 Ω ̃ 𝑢 𝑝 − 1 𝜑 𝑑 𝑥 , ∀ 𝜑 ∈ 𝑉 0 . ( 3 . 4 1 ) As ̃ 𝑢 ≠ 0 , relation (3.41) expresses the fact that ̃ 𝑢 ≥ 0 is an eigenfunction of ( − Δ 𝑝 , 𝑉 0 ) corresponding to the eigenvalue 𝑎 . As 𝑎 > 𝜆 1 , this is impossible according to Anane [30], because ̃ 𝑢 must change sign. This contradiction proves that Claim 2 holds true. Note that unlike in the proof of Theorem 2.4, here the contradiction is achieved by the sign-changing property of eigenfunctions belonging to eigenvalues bigger than 𝜆 1 .Claim 3. 𝑢 + ∈ i n t ( 𝐶 1 0 ( Ω ) + ) is the smallest positive solution of (1.7) in [ 0 , 𝛼 𝑎 𝑒 ] .We already know that 𝑢 + ∈ [ 0 , 𝛼 a 𝑒 ] . Assume that 𝑢 ∈ 𝑉 0 is any positive solution of (1.7) belonging to [ 0 , 𝛼 𝑎 𝑒 ] . Since 𝑢 ∈ 𝐿 ∞ ( Ω ) , then by (1.7) and (g3) we deduce Δ 𝑝 𝑢 ∈ 𝐿 ∞ ( Ω ) . Using [27, Theorem 1.56] we derive 𝑢 ∈ 𝐶 1 0 ( Ω ) , and applying Vázquez's strong maximum principle [28] we infer 𝑢 ∈ i n t ( 𝐶 1 0 ( Ω ) + ) , which yields 𝑢 ∈ [ ( 1 / 𝑛 ) 𝜑 1 , 𝛼 𝑎 𝑒 ] for 𝑛 sufficiently large. This in conjunction with the fact that 𝑢 𝑛 is the least solution of (1.7) in [ ( 1 / 𝑛 ) 𝜑 1 , 𝛼 𝑎 𝑒 ] ensures 𝑢 𝑛 ≤ 𝑢 if 𝑛 is large enough. In view of (3.21), we obtain 𝑢 + ≤ 𝑢 , which proves Claim 3.
The proof of the existence of the greatest negative solution 𝑢 − = 𝑢 − ( 𝑏 ) ∈ − i n t ( 𝐶 1 0 ( Ω ) + ) of (1.7) within the ordered interval [ − 𝛽 𝑏 𝑒 , 0 ] can be done in a similar way. This completes the proof of the theorem.

Under hypotheses (g1)–(g4), Theorem 3.10 ensures the existence of extremal positive and negative solutions of (1.7) for all 𝑎 > 𝜆 1 and 𝑏 > 𝜆 1 denoted by 𝑢 + = 𝑢 + ( 𝑎 ) ∈ i n t ( 𝐶 1 0 ( Ω ) + ) and 𝑢 − = 𝑢 − ( 𝑏 ) ∈ − i n t ( 𝐶 1 0 ( Ω ) + ) , respectively. In what follows we are going to characterize these extremal solutions as global (local) minimizers of certain locally Lipschitz functionals that are generated by truncation procedures. To this end let us introduce truncation functions related to the extremal solutions 𝑢 + and 𝑢 − as follows: 𝜏 + ⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩ ( 𝑥 , 𝑠 ) = 0 i f 𝑠 < 0 , 𝑠 i f 0 ≤ 𝑠 ≤ 𝑢 + ( 𝑢 𝑥 ) , + ( 𝑥 ) i f 𝑠 > 𝑢 + ( 𝜏 𝑥 ) , − ⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩ 𝑢 ( 𝑥 , 𝑠 ) = − ( 𝑥 ) i f 𝑠 < 𝑢 − ( 𝑥 ) , 𝑠 i f 𝑢 − 𝜏 ( 𝑥 ) ≤ 𝑠 ≤ 0 , 0 i f 𝑠 > 0 , 0 ⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩ 𝑢 ( 𝑥 , 𝑠 ) = − ( 𝑥 ) i f 𝑠 < 𝑢 − ( 𝑥 ) , 𝑠 i f 𝑢 − ( 𝑥 ) ≤ 𝑠 ≤ 𝑢 + 𝑢 ( 𝑥 ) , + ( 𝑥 ) i f 𝑠 > 𝑢 + ( 𝑥 ) . ( 3 . 4 2 ) The truncations 𝜏 + , 𝜏 − , 𝜏 0 ∶ Ω × ℝ → ℝ are continuous, uniformly bounded, and Lipschitzian with respect to 𝑠 . The extremal positive and negative solutions 𝑢 + and 𝑢 − of (1.7), respectively, ensured by Theorem 3.10 satisfy − Δ 𝑝 𝑢 + 𝑢 = 𝑎 + 𝑝 − 1 − 𝜂 + , − Δ 𝑝 𝑢 − 𝑢 = − 𝑏 − 𝑝 − 1 − 𝜂 − i n 𝑉 ∗ 0 , ( 3 . 4 3 ) where 𝜂 + , 𝜂 − ∈ 𝐿 𝑞 ( Ω ) and 𝜂 + ( 𝑥 ) ∈ 𝜕 𝐺 𝑥 , 𝑢 + ( 𝑥 ) , 𝜂 − ( 𝑥 ) ∈ 𝜕 𝐺 𝑥 , 𝑢 − ( 𝑥 ) ( 3 . 4 4 ) for a.a. 𝑥 ∈ Ω . By means of 𝜂 + , 𝜂 − we introduce the following truncations of the nonlinearity 𝑔 ∶ Ω × ℝ → ℝ : 𝑔 + ⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩ ( 𝑥 , 𝑠 ) ∶ = 0 i f 𝑠 < 0 , 𝑔 ( 𝑥 , 𝑠 ) f o r 0 ≤ 𝑠 ≤ 𝑢 + ( 𝜂 𝑥 ) , + ( 𝑥 ) w h e n 𝑠 > 𝑢 + ( 𝑔 𝑥 ) , − ⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩ 𝜂 ( 𝑥 , 𝑠 ) ∶ = − ( 𝑥 ) i f 𝑠 < 𝑢 − ( 𝑥 ) , 𝑔 ( 𝑥 , 𝑠 ) f o r 𝑢 − 𝑔 ( 𝑥 ) ≤ 𝑠 ≤ 0 , 0 w h e n 𝑠 > 0 , 0 ⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩ 𝜂 ( 𝑥 , 𝑠 ) ∶ = − ( 𝑥 ) i f 𝑠 < 𝑢 − ( 𝑥 ) , 𝑔 ( 𝑥 , 𝑠 ) f o r 𝑢 − ( 𝑥 ) ≤ 𝑠 ≤ 𝑢 + 𝜂 ( 𝑥 ) , + ( 𝑥 ) w h e n 𝑠 > 𝑢 + ( 𝑥 ) ( 3 . 4 5 ) and define functionals 𝐸 + , 𝐸 − , 𝐸 0 by 𝐸 + 1 ( 𝑢 ) = 𝑝 ‖ ∇ 𝑢 ‖ 𝑝 𝑝 − Ω 0 𝑢 ( 𝑥 ) 𝑎 𝜏 + ( 𝑥 , 𝑠 ) 𝑝 − 1 − 𝑔 + 𝐸 ( 𝑥 , 𝑠 ) 𝑑 𝑠 𝑑 𝑥 , − 1 ( 𝑢 ) = 𝑝 ‖ ∇ 𝑢 ‖ 𝑝 𝑝 + Ω 0 𝑢 ( 𝑥 ) 𝑏 | | 𝜏 − | | ( 𝑥 , 𝑠 ) 𝑝 − 1 + 𝑔 − 𝐸 ( 𝑥 , 𝑠 ) 𝑑 𝑠 𝑑 𝑥 , 0 1 ( 𝑢 ) = 𝑝 ‖ ∇ 𝑢 ‖ 𝑝 𝑝 − Ω 0 𝑢 ( 𝑥 ) 𝑎 𝜏 + ( 𝑥 , 𝑠 ) 𝑝 − 1 | | 𝜏 − 𝑏 − | | ( 𝑥 , 𝑠 ) 𝑝 − 1 − 𝑔 0 ( 𝑥 , 𝑠 ) 𝑑 𝑠 𝑑 𝑥 . ( 3 . 4 6 ) Due to (g2) the functionals 𝐸 + , 𝐸 − , 𝐸 0 ∶ 𝑉 0 → ℝ are locally Lipschitz continuous. Moreover, in view of the truncations involved these functionals are bounded below, coercive, and weakly lower semicontinuous such that their global minimizers exist. The following lemma provides a characterization of the critical points of these functionals.

Lemma 3.11. Let 𝑢 + and 𝑢 − be the extremal constant-sign solutions of (1.7). Then the following holds. (i)A critical point 𝑣 ∈ 𝑉 0 of 𝐸 + is a (nonnegative) solution of (1.7) satisfying 0 ≤ 𝑣 ≤ 𝑢 + .(ii)A critical point 𝑤 ∈ 𝑉 0 of 𝐸 − is a (nonpositive) solution of (1.7) satisfying 𝑢 − ≤ 𝑤 ≤ 0 .(iii)A critical point 𝑧 ∈ 𝑉 0 of 𝐸 0 is a solution of (1.7) satisfying 𝑢 − ≤ 𝑧 ≤ 𝑢 + .

Proof. To prove (i) let 𝑣 be a critical point of 𝐸 + , that is, 0 ∈ 𝜕 𝐸 + ( 𝑣 ) , which results in 𝑣 ∈ 𝑉 0 ∶ − Δ 𝑝 𝑣 = 𝑎 𝜏 + ( 𝑥 , 𝑣 ) 𝑝 − 1 − 𝑤 i n 𝑉 ∗ 0 ( 3 . 4 7 ) for some 𝑤 ∈ 𝐿 𝑞 ( Ω ) and such that 𝑤 ( 𝑥 ) ∈ 𝜕 𝐺 + ( 𝑥 , 𝑣 ( 𝑥 ) ) almost everywhere in Ω , with 𝐺 + ( 𝑥 , 𝜉 ) ∶ = 𝜉 0 𝑔 + ( 𝑥 , 𝑡 ) 𝑑 𝑡 , ( 𝑥 , 𝜉 ) ∈ Ω × ℝ . ( 3 . 4 8 ) Let us show that 𝑣 ≤ 𝑢 + holds. As 𝑢 + is a positive solution of (1.7), it satisfies the first equation in (3.43), and by subtracting that equation from (3.47) and applying the special test function 𝜑 = ( 𝑣 − 𝑢 + ) + we get Ω | | | | ∇ 𝑣 𝑝 − 2 | | ∇ 𝑣 − ∇ 𝑢 + | | 𝑝 − 2 ∇ 𝑢 + ∇ 𝑣 − 𝑢 + + = 𝑑 𝑥 Ω 𝑎 𝜏 + ( 𝑥 , 𝑣 ) 𝑝 − 1 − 𝑢 + 𝑝 − 1 − 𝑤 − 𝜂 + 𝑣 − 𝑢 + + 𝑑 𝑥 . ( 3 . 4 9 ) By the definition of the truncations introduced above we have 𝜏 + ( 𝑥 , 𝑣 ) 𝑝 − 1 ( 𝑥 ) − 𝑢 + 𝑝 − 1 ( 𝑥 ) = 0 and 𝑤 ( 𝑥 ) − 𝜂 + ( 𝑥 ) = 0 for a.a. 𝑥 ∈ { 𝑣 > 𝑢 + } , and thus the right-hand side of (3.49) is zero which leads to Ω | | | | ∇ 𝑣 𝑝 − 2 | | ∇ 𝑣 − ∇ 𝑢 + | | 𝑝 − 2 ∇ 𝑢 + ∇ 𝑣 − 𝑢 + + 𝑑 𝑥 = 0 , ( 3 . 5 0 ) and hence it follows ∇ ( 𝑣 − 𝑢 + ) + = 0 . Because ( 𝑣 − 𝑢 + ) + ∈ 𝑉 0 , this implies ( 𝑣 − 𝑢 + ) + = 0 which proves 𝑣 ≤ 𝑢 + . To prove 0 ≤ 𝑣 we test (3.47) with 𝜑 = 𝑣 − = m a x { − 𝑣 , 0 } ∈ 𝑉 0 and get Ω | | | | ∇ 𝑣 𝑝 − 2 ∇ 𝑣 ∇ 𝑣 − 𝑑 𝑥 = Ω 𝑎 𝜏 + ( 𝑥 , 𝑣 ) 𝑝 − 1 𝑣 − 𝑤 − 𝑑 𝑥 = 0 , ( 3 . 5 1 ) which results in ‖ ∇ 𝑣 − ‖ 𝑝 𝑝 = 0 , and thus 𝑣 − = 0 , that is, 0 ≤ 𝑣 . Thus the critical point 𝑣 of 𝐸 + which is a solution of the Dirichlet problem (3.47) satisfies 0 ≤ 𝑣 ≤ 𝑢 + , and therefore 𝜏 + ( 𝑥 , 𝑣 ) = 𝑣 . Because 𝜕 𝐺 + ( 𝑥 , 𝑣 ( 𝑥 ) ) ⊂ 𝜕 𝐺 ( 𝑥 , 𝑣 ( 𝑥 ) ) , it follows 𝑤 ∈ 𝜕 𝐺 ( 𝑥 , 𝑣 ( 𝑥 ) ) , and therefore 𝑣 must be a solution of (1.7). This proves (i). In just the same way one can prove also (ii) and (iii) which is omitted.

The following lemma provides a variational characterization of the extremal constant-sign solutions 𝑢 + and 𝑢 − .

Lemma 3.12. Let 𝑎 > 𝜆 1 and 𝑏 > 𝜆 1 . Then the extremal positive solution 𝑢 + of (1.7) is the unique global minimizer of the functional 𝐸 + , and the extremal negative solution 𝑢 − of (1.7) is the unique global minimizer of the functional 𝐸 − . Both 𝑢 + and 𝑢 − are local minimizers of 𝐸 0 .

Proof. The functional 𝐸 + ∶ 𝑉 0 → ℝ is bounded below, coercive, and weakly lower semicontinuous. Thus there exists a global minimizer 𝑣 + ∈ 𝑉 0 of 𝐸 + , that is, 𝐸 + 𝑣 + = i n f 𝑢 ∈ 𝑉 0 𝐸 + ( 𝑢 ) , ( 3 . 5 2 ) As 𝑣 + is a critical point of 𝐸 + , so by Lemma 3.11 it is a nonnegative solution of (1.7) satisfying 0 ≤ 𝑣 + ≤ 𝑢 + . Since 𝑎 − 𝜆 1 > 0 , there is a 𝜈 𝑎 > 0 such that 𝑎 − 𝜆 1 − 𝜈 𝑎 > 0 . By (g3) we infer the existence of a ̃ 𝛿 𝑎 > 0 such that | | | | ≤ 𝑔 ( 𝑥 , 𝑠 ) 𝑎 − 𝜆 1 − 𝜈 𝑎 𝑠 𝑝 − 1 ̃ 𝛿 , ∀ 𝑠 ∶ 0 < 𝑠 ≤ 𝑎 , ( 3 . 5 3 ) and thus for 𝜀 > 0 sufficiently small such that 𝜀 𝜑 1 ≤ 𝑢 + ‖ ‖ 𝜑 , 𝜀 1 ‖ ‖ 𝐿 ∞ ( Ω ) ≤ ̃ 𝛿 𝑎 , ( 3 . 5 4 ) we obtain (note ‖ 𝜑 1 ‖ 𝑝 = 1 ) 𝐸 + 𝑣 + ≤ 𝐸 + 𝜀 𝜑 1 = 𝜆 1 𝑝 𝜀 𝑝 + Ω 𝜀 𝜑 1 0 ( 𝑥 ) − 𝑎 𝑠 𝑝 − 1 ≤ 𝜆 + 𝑔 ( 𝑥 , 𝑠 ) 𝑑 𝑠 𝑑 𝑥 1 𝑝 𝜀 𝑝 + Ω 𝜀 𝜑 1 0 ( 𝑥 ) − 𝜆 1 − 𝜈 𝑎 𝑠 𝑝 − 1 𝑑 𝑠 𝑑 𝑥 < 0 . ( 3 . 5 5 ) Hence it follows that 𝐸 + ( 𝑣 + ) < 0 , and thus 𝑣 + ≠ 0 . Applying nonlinear regularity theory for the 𝑝 -Laplacian (cf., e.g., [27, Theorem 1.5.6]) and Vázquez's strong maximum principle, we see that 𝑣 + ∈ i n t ( 𝐶 1 0 ( Ω ) + ) . As 𝑢 + is the smallest positive solution of (1.7) in [ 0 , 𝛼 𝑎 𝑒 ] and 0 ≤ 𝑣 + ≤ 𝑢 + , it follows 𝑣 + = 𝑢 + , which shows that the global minimizer 𝑣 + must be unique and equal to 𝑢 + . By similar arguments one can show that the global minimizer 𝑣 − of 𝐸 − must be unique and 𝑣 − = 𝑢 − . It remains to prove that 𝑢 + and 𝑢 − are local minimizers of 𝐸 0 . Let us show this last assertion for 𝑢 + only. By definition we have 𝐸 + ( 𝑢 ) = 𝐸 0 ( 𝑢 ) ∀ 𝑢 ∈ 𝑉 0 w i t h 𝑢 ≥ 0 . ( 3 . 5 6 ) Since 𝑢 + is a global minimizer of 𝐸 + and 𝑢 + ∈ i n t ( 𝐶 1 0 ( Ω ) + ) , it follows that 𝑢 + is a local minimizer of 𝐸 0 with respect to the 𝐶 1 topology. Due to a result by Motreanu and Papageorgiou in [39, Proposition 4], we conclude that 𝑢 + is also a local minimizer of 𝐸 0 with respect to the 𝑉 0 topology. This completes the proof of the lemma.

Lemma 3.13. The functional 𝐸 0 ∶ 𝑉 0 → ℝ has a global minimizer 𝑣 0 which is a nontrivial solution of (1.7) satisfying 𝑢 − ≤ 𝑣 0 ≤ 𝑢 + .

Proof. One easily verifies that 𝐸 0 ∶ 𝑉 0 → ℝ is coercive and weakly lower semicontinuous, and thus a global minimizer 𝑣 0 exists which is a critical point of 𝐸 0 . Apply Lemma 3.11(iii) and note that, for example, 𝐸 0 ( 𝑢 + ) = 𝐸 + ( 𝑢 + ) < 0 , which shows that 𝑣 0 ≠ 0 .

3.3. Sign-Changing Solutions

Theorem 3.10 ensures the existence of a smallest positive solution 𝑢 + ∈ i n t ( 𝐶 1 0 ( Ω ) + ) in [ 0 , 𝛼 𝑎 𝑒 ] and a greatest negative solution 𝑢 − ∈ − i n t ( 𝐶 1 0 ( Ω ) + ) of (1.7) in [ − 𝛽 𝑏 𝑒 , 0 ] . The idea to show the existence of sign-changing solutions is to prove the existence of nontrivial solutions 𝑢 0 of (1.7) satisfying 𝑢 − ≤ 𝑢 0 ≤ 𝑢 + with 𝑢 0 ≠ 𝑢 − and 𝑢 0 ≠ 𝑢 + , which then must be sign-changing, because 𝑢 + and 𝑢 − are the extremal constant-sign solutions.

Theorem 3.14. Let hypotheses (g1)–(g4) and (H) be satisfied. Then problem (1.7) has a smallest positive solution 𝑢 + ∈ i n t ( 𝐶 1 0 ( Ω ) + ) in [ 0 , 𝛼 𝑎 𝑒 ] , a greatest negative solution 𝑢 − ∈ − i n t ( 𝐶 1 0 ( Ω ) + ) in [ − 𝛽 𝑏 𝑒 , 0 ] , and a nontrivial sign-changing solution 𝑢 0 ∈ 𝐶 1 0 ( Ω ) with 𝑢 − ≤ 𝑢 0 ≤ 𝑢 + .

Proof. Clearly the existence of the extremal positive and negative solution 𝑢 + and 𝑢 − follows from Theorem 3.10, because (H), in particular, implies that 𝑎 > 𝜆 1 and 𝑏 > 𝜆 1 . As for the existence of a sign-changing solution we first note that by Lemma 3.13 it follows that the global minimizer 𝑣 0 of 𝐸 0 is a nontrivial solution of (1.7) satisfying 𝑢 − ≤ 𝑣 0 ≤ 𝑢 + . Therefore, if 𝑣 0 ≠ 𝑢 + and 𝑣 0 ≠ 𝑢 − , then 𝑣 0 = 𝑢 0 must be a sign-changing solution as asserted, because 𝑢 − is the greatest negative and 𝑢 + is the smallest positive solution of (1.7). Thus, we still need to prove the existence of sign-changing solutions in case that either 𝑣 0 = 𝑢 − or 𝑣 0 = 𝑢 + .
Let us consider the case 𝑣 0 = 𝑢 + only, since the case 𝑣 0 = 𝑢 − can be treated quite similarly. By Lemma 3.12, 𝑢 − is a local minimizer of 𝐸 0 . Without loss of generality we may even assume that 𝑢 − is a strict local minimizer of 𝐸 0 , because on the contrary we would find (infinitely many) critical points 𝑧 of 𝐸 0 that are sign-changing solutions thanks to 𝑢 − ≤ 𝑧 ≤ 𝑢 + and the extremality of the solutions 𝑢 − , 𝑢 + obtained in Theorem 3.10 which proves the assertion.
Therefore, it remains to prove the existence of sign-changing solutions under the assumptions that the global minimizer 𝑣 0 of 𝐸 0 is equal to 𝑢 + , and 𝑢 − is a strict local minimizer of 𝐸 0 . This implies the existence of 𝜌 ∈ ( 0 , ‖ 𝑢 − − 𝑢 + ‖ ) such that 𝐸 0 𝑢 + ≤ 𝐸 0 𝑢 − 𝐸 < i n f 0 ( 𝑢 ) ∶ 𝑢 ∈ 𝜕 𝐵 𝜌 𝑢 − , ( 3 . 5 7 ) where 𝜕 𝐵 𝜌 ( 𝑢 − ) = { 𝑢 ∈ 𝑉 0 ∶ ‖ 𝑢 − 𝑢 − ‖ = 𝜌 } . The functional 𝐸 0 satisfies the Palais-Smale condition, because it is bounded below, locally Lipschitz continuous, and coercive; see, for example, [40, Corollary 2.4]. Thus in view of (3.57) we may apply the nonsmooth version of Ambrosetti-Rabinowitz's Mountain-Pass Theorem (see, e.g., [41, Theorem 3.4]) which ensures the existence of a critical point 𝑢 0 ∈ 𝑉 0 satisfying 0 ∈ 𝜕 𝐸 0 ( 𝑢 0 ) and 𝐸 i n f 0 ( 𝑢 ) ∶ 𝑢 ∈ 𝜕 𝐵 𝜌 𝑢 − ≤ 𝐸 0 𝑢 0 = i n f 𝛾 ∈ Γ m a x [ ] 𝑡 ∈ − 1 , 1 𝐸 0 ( 𝛾 ( 𝑡 ) ) , ( 3 . 5 8 ) where [ ] Γ = 𝛾 ∈ 𝐶 − 1 , 1 , 𝑉 0 ∶ 𝛾 ( − 1 ) = 𝑢 − , 𝛾 ( 1 ) = 𝑢 + . ( 3 . 5 9 ) It is clear from (3.57) and (3.58) that 𝑢 0 ≠ 𝑢 − and 𝑢 0 ≠ 𝑢 + , and thus 𝑢 0 is a sign-changing solution provided 𝑢 0 ≠ 0 . To prove the latter we claim 𝐸 0 𝑢 0 < 0 ( 3 . 6 0 ) for which it suffices to construct a path ̂ 𝛾 ∈ Γ such that 𝐸 0 [ ] . ( ̂ 𝛾 ( 𝑡 ) ) < 0 ∀ 𝑡 ∈ − 1 , 1 ( 3 . 6 1 ) The construction of such a path ̂ 𝛾 can be done by adopting an approach due to the authors in [3] and applying the Second Deformation Lemma for locally Lipschitz functionals as it can be found in [42, Theorem 2.10]. This completes the proof.

Remark 3.15. The multiplicity results and the existence of sign-changing solutions obtained in this section generalize very recent results due to the authors obtained in [3, 4, 7]. Moreover, if the function 𝑡 ↦ 𝑔 ( 𝑥 , 𝑡 ) is continuous on ℝ and 𝑎 = 𝑏 = 𝜆 , then 𝜕 𝐺 ( 𝑥 , 𝜉 ) = { 𝑔 ( x , 𝜉 ) } , and problem (1.7) reduces to 𝑢 ∈ 𝑉 0 ∶ − Δ 𝑝 𝑢 = 𝜆 | 𝑢 | 𝑝 − 2 𝑢 − 𝑔 ( 𝑥 , 𝑢 ) i n 𝑉 ∗ 0 . ( 3 . 6 2 ) However, even in this setting the results obtained here are more general than obtained in [6, Theorem 3.9], because we do not assume that 𝑔 ( 𝑥 , 𝑡 ) 𝑡 ≥ 0 for all 𝑡 ∈ ℝ .

Remark 3.16. Theorem 3.14 improves also Corollary 3.2 of [8]. In fact, let 𝑝 = 2 , let 𝑢 ∈ 𝑉 0 be a solution of (1.7) in case 𝑎 = 𝑏 = 𝜆 and 𝑔 ( 𝑥 , 𝑡 ) ≡ 𝑔 ( 𝑡 ) , ( 𝑥 , 𝑡 ) ∈ Ω × ℝ with 𝜂 ∈ 𝐿 𝑞 ( Ω ) satisfying 𝜂 ( 𝑥 ) ∈ 𝜕 𝐺 ( 𝑢 ( 𝑥 ) ) . By definition of Clarke's gradient we have, for any 𝜑 ∈ 𝑉 0 , 𝜂 ( 𝑥 ) 𝜑 ( 𝑥 ) ≤ 𝐺 0 ( 𝑢 ( 𝑥 ) ; 𝜑 ( 𝑥 ) ) a . e . i n Ω . ( 3 . 6 3 ) As 𝑢 is a solution, the following holds: 𝑢 ∈ 𝑉 0 and ( 𝑝 = 2 ), Ω ∇ 𝑢 ∇ 𝜑 𝑑 𝑥 = 𝜆 Ω 𝑢 𝜑 𝑑 𝑥 − Ω 𝜂 𝜑 𝑑 𝑥 , ( 3 . 6 4 ) which yields − Ω ∇ 𝑢 ∇ 𝜑 𝑑 𝑥 + 𝜆 Ω 𝑢 𝜑 𝑑 𝑥 = Ω 𝜂 𝜑 𝑑 𝑥 ≤ Ω 𝐺 0 ( 𝑢 ; 𝜑 ) 𝑑 𝑥 , ∀ 𝜑 ∈ 𝑉 0 , ( 3 . 6 5 ) That is, 𝑢 turns out to be a solution of the hemivariational inequality studied in [8]. Since the hypotheses of [8, Corollary 3.2] imply (g1)–(g4), the assertion follows.

Remark 3.17. Multiplicity results for a nonsmooth version of problem (1.4) in form of (1.2) can be established under structure conditions for Clarke's gradient 𝜕 𝑗 similar to H(f).
Multiplicity and sign-changing solutions results have been obtained recently in [43] for the following nonlinear Neumann-type boundary value problem: find 𝑢 ∈ 𝑉 ⧵ { 0 } and parameters 𝑎 , 𝑏 ∈ ℝ such that − Δ 𝑝 𝑢 = 𝑓 ( 𝑥 , 𝑢 ) − | 𝑢 | 𝑝 − 2 | | | | 𝑢 i n Ω , ∇ 𝑢 𝑝 − 2 𝜕 𝑢 𝑢 𝜕 𝜈 = 𝑎 + 𝑝 − 1 − 𝑏 ( 𝑢 − ) 𝑝 − 1 + 𝑔 ( 𝑥 , 𝑢 ) o n 𝜕 Ω . ( 3 . 6 6 ) For problem (3.66) conditions on the parameters have been given in terms of the “Steklov-Fučik” spectrum to ensure multiplicity results.

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