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