Sign-Changing Solutions for Nonlinear Elliptic Problems Depending on Parameters

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 𝐶²-boundary 𝜕Ω, and let 𝑉=𝑊^1,𝑝(Ω) and 𝑉₀=𝑊₀^1,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∈𝑉₀⧵{0} and parameters 𝑎∈ℝ, 𝑏∈ℝ such that−Δ_𝑝𝑢∈𝜕𝑗(⋅,𝑢,𝑎,𝑏)in𝑉^∗₀, (1.1) where Δ_𝑝𝑢=div(|∇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_𝑝𝑢∈𝜕𝑗(⋅,𝑢,𝜆)in𝑉^∗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)=^𝑠₀𝑓(𝑥,𝑡,𝜆)𝑑𝑡, (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∈𝑉₀⧵{0}∶−Δ_𝑝𝑢=𝑓(⋅,𝑢,𝜆)in𝑉^∗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:𝑎𝑗(𝑥,𝑠,𝑎,𝑏)=𝑝(𝑠⁺)^𝑝+𝑏𝑝(𝑠⁻)^𝑝+𝐺(𝑥,𝑠), (1.5) where 𝑠⁺=max{𝑠,0} and 𝑠⁻=max{−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^∞_loc(Ω×ℝ) and 𝐺 is given by 𝐺(𝑥,𝑠)∶=^𝑠₀𝑔(𝑥,𝑡)𝑑𝑡. (1.6) Problem (1.1) reduces then to the following parameter-dependent multi-valued elliptic problem:𝑢∈𝑉₀⧵{0}∶−Δ_𝑝𝑢∈𝑎(𝑢⁺)^𝑝−1−𝑏(𝑢⁻)^𝑝−1−𝜕𝐺(𝑥,𝑢)in𝑉^∗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∈𝑉₀⧵{0} and constants 𝑎∈ℝ, 𝑏∈ℝ such that−Δ_𝑝𝑢=𝑎(𝑢⁺)^𝑝−1−𝑏(𝑢⁻)^𝑝−1−𝑔(𝑥,𝑢)in𝑉^∗₀. (1.8) Multiple solution results for (1.8) were obtained by the authors in [4]. Furthermore, note that |𝑢|^𝑝−2𝑢=|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∈𝑉₀⧵{0} and a constant 𝜆∈ℝ such that−Δ_𝑝𝑢=𝜆|𝑢|^𝑝−2𝑢−𝑔(𝑥,𝑢)in𝑉^∗₀. (1.10) 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∈𝑉₀⧵{0} and a constant 𝜆∈ℝ such that−Δ_𝑝𝑢∈𝜆|𝑢|^𝑝−2𝑢−𝜕𝐺(𝑥,𝑢)in𝑉^∗₀. (1.11) 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,𝜆₀), with 𝜆₀>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)≥0fora.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|^𝑟fora.a.𝑥∈Ω∀(s,𝜆)∈ℝ×(0,𝜆). (2.2)(iii)For all 𝜆∈(0,𝜆) there exist constants 𝜇₀=𝜇₀(𝜆)>𝜆₂, 𝜈₀=𝜈₀(𝜆)>𝜇₀ and a set Ω_𝜆⊂Ω with Ω⧵Ω_𝜆 of Lebesgue measure zero such that 𝜇₀<liminf_𝑠→0𝑓(𝑥,𝑠,𝜆)|𝑠|^𝑝−2𝑠≤limsup_𝑠→0𝑓(𝑥,𝑠,𝜆)|𝑠|^𝑝−2𝑠≤𝜈₀ (2.3) uniformly with respect to 𝑥∈Ω_𝜆.

In H(𝑓)(iii), 𝜆₂ denotes the second eigenvalue of (−Δ_𝑝,𝑉₀). 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)=𝜆arctan(𝜆+𝜆₂𝜆|𝑠|^𝑝−2𝑠)+𝑐|𝑠|^𝑟−1𝑠∀(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: 𝑓(𝑠,𝜆)=𝜆arctan(𝑎₁𝜆+𝜆₂𝜆|𝑠|^𝑝−2𝑠)+𝑐₁|𝑠|^𝑟1−1𝑠if𝑠≤0,𝜆arctan(𝑎₂𝜆+𝜆₂𝜆𝑠^𝑝−1)+𝑐₂𝑠^𝑟2if𝑠>0, (2.5) with 𝜆>0, 𝑎₁≥1, 𝑎₂≥1, 𝑐₁>0, 𝑐₂>0, 𝑟₁>𝑝−1, 𝑟₂>𝑝−1.

2.2. Constant-Sign Solutions

The operator −Δ_𝑝∶𝑉₀→𝑉^∗₀ is maximal monotone and coercive; therefore there exists a unique solution 𝑒∈𝑉₀ of the Dirichlet problem𝑒∈𝑉₀∶−Δ_𝑝𝑒=1in𝑉^∗₀. (2.6) With 𝑠⁻=max{−𝑠,0} for 𝑠∈ℝ, and using −𝑒⁻∈𝑉₀ as a test function, we see that‖∇𝑒⁻‖^𝑝_𝑝=⟨−Δ_𝑝𝑒,−𝑒⁻⟩=−_Ω𝑒⁻(𝑥)𝑑𝑥≤0, (2.7) which implies that 𝑒≥0. From the nonlinear regularity theory (cf., e.g., [27, Theorem 1.5.6]) we have 𝑒∈𝐶¹₀(Ω). Then from the nonlinear strong maximum principle (see [28]) we infer that 𝑒∈int(𝐶¹₀(Ω)₊). Here int(𝐶¹₀(Ω)₊) denotes the interior of the positive cone 𝐶¹₀(Ω)₊={𝑢∈𝐶¹₀(Ω)∶𝑢(𝑥)≥0,∀𝑥∈Ω} in the Banach space 𝐶¹₀(Ω)={𝑢∈𝐶¹(Ω)∶𝑢(𝑥)=0,∀𝑥∈𝜕Ω}, given byint(𝐶¹₀(Ω)₊)={𝑢∈𝐶¹₀(Ω)∶𝑢(𝑥)>0,∀𝑥∈Ω,and𝜕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 𝜆₀∈(0,𝜆) with the property that if 𝜆∈(0,𝜆₀), one can choose 𝜉₀=𝜉₀(𝜆)∈(0,𝜃) such that 𝑐(𝜉₀‖𝑒‖_∞)^𝑟+‖𝑎(⋅,𝜆)‖_∞<𝜉₀^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). (2.10) 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 𝜆₁ the first eigenvalue of (−Δ_𝑝,𝑉₀) and by 𝜑₁ the eigenfunction of (−Δ_𝑝,𝑉₀) corresponding to 𝜆₁ satisfying 𝜑₁∈int(𝐶¹₀(Ω)₊),‖𝜑₁‖_𝑝=1. (2.11)

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 𝜇₀=𝜇₀(𝜆)>𝜆₁ and Ω_𝜆⊂Ω with Ω⧵Ω_𝜆 of Lebesgue measure zero such that 𝜇₀<liminf_𝑠→0𝑓(𝑥,𝑠,𝜆)|𝑠|^𝑝−2𝑠 (2.12) uniformly with respect to 𝑥∈Ω_𝜆.
Fix a constant 𝜃>0 and consider the corresponding number 𝜆₀∈(0,𝜆) obtained in Lemma 2.2. Then for any 𝜆∈(0,𝜆₀) the function 𝑢=𝜉₀𝑒∈int(𝐶¹₀(Ω)₊), with 𝜉₀∈(0,𝜃) given by Lemma 2.2, is a supersolution for problem (1.4), and the function 𝑢=𝜀𝜑₁∈int(𝐶¹₀(Ω)₊) is a subsolution of problem (1.4) provided that the number 𝜀>0 is sufficiently small.

Proof. For a fixed 𝜆∈(0,𝜆₀), from (2.9) and H(𝑓)(ii) we derive −Δ_𝑝𝑢=𝜉₀^𝑝−1>‖𝑎(⋅,𝜆)‖_∞+𝑐‖𝑢‖^𝑟_∞≥𝑓(⋅,𝑢(⋅),𝜆), (2.13) which says that 𝑢=𝜉₀𝑒 is a supersolution for problem (1.4).
On the other hand, by hypothesis we can find 𝜇=𝜇(𝜆)>𝜆₁ and 𝛿=𝛿(𝜆)>0 such that𝜇<𝑓(𝑥,𝑠,𝜆)|𝑠|^𝑝−2𝑠fora.a.𝑥∈Ω∀0<|𝑠|≤𝛿. (2.14) Choose 𝜀∈(0,𝛿/‖𝜑₁‖_∞). Then by (2.14) we have −Δ_𝑝(𝜀𝜑₁)=𝜆₁𝜀^𝑝−1𝜑₁^𝑝−1<𝜇𝜀^𝑝−1𝜑₁^𝑝−1<𝑓(𝑥,𝜀𝜑₁(𝑥),𝜆)fora.a.𝑥∈Ω, (2.15) which ensures that 𝑢=𝜀𝜑₁ 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 𝜇₀=𝜇₀(𝜆)>𝜆₁, 𝜈₀=𝜈₀(𝜆)>𝜇₀ and a set Ω_𝜆⊂Ω with Ω⧵Ω_𝜆 of Lebesgue measure zero such that 𝜇₀<liminf_𝑠→0𝑓(𝑥,𝑠,𝜆)|𝑠|^𝑝−2𝑠≤limsup_𝑠→0𝑓(𝑥,𝑠,𝜆)|𝑠|^𝑝−2𝑠≤𝜈₀ (2.16) uniformly with respect to 𝑥∈Ω_𝜆. Then for all 𝑏>0 there exists a number 𝜆₀∈(0,𝜆) with the property that if 𝜆∈(0,𝜆₀), then there is a constant 𝜉₀=𝜉₀(𝜆)∈(0,𝑏/‖𝑒‖_∞) such that problem (1.4) has a least positive solution 𝑢₊=𝑢₊(𝜆)∈int(𝐶¹₀(Ω)₊) in the order interval [0,𝜉₀𝑒] and a greatest negative solution 𝑢₋=𝑢₋(𝜆)∈−int(𝐶¹₀(Ω)₊) in the order interval [−𝜉₀𝑒,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 𝜆₀∈(0,𝜆) as therein. Fix 𝜆∈(0,𝜆₀). Lemma 2.3 ensures that 𝑢=𝜉₀𝑒∈int(𝐶¹₀(Ω)₊) is a supersolution for problem (1.4), with 𝜉₀∈(0,𝑏/‖𝑒‖_∞) given by Lemma 2.2, and 𝑢=𝜀𝜑₁∈int(𝐶¹₀(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 𝜀𝜑₁≤𝜉₀𝑒. Then by the method of sub-supersolution we know that in the order interval [𝜀𝜑₁,𝜉₀𝑒] there is a least (i.e., smallest) solution 𝑢_𝜀=𝑢_𝜀(𝜆)∈int(𝐶¹₀(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∈int(𝐶¹₀(Ω)₊) of problem (1.4) in the order interval [(1/𝑛)𝜑₁,𝜉₀𝑒]. Clearly, we have𝑢_𝑛↓𝑢₊pointwise, (2.17) with some function 𝑢₊∶Ω→ℝ satisfying 0≤𝑢₊≤𝜉₀𝑒. First we claim that 𝑢₊isasolutionofproblem(1.4). (2.18) Taking into account that 𝑢_𝑛 solves (1.4), and the fact that 𝑢_𝑛 belongs to the order interval [0,𝜉₀𝑒], from H(𝑓)(ii) we see that ‖∇𝑢_𝑛‖^𝑝_𝑝=_Ω𝑓(𝑥,𝑢_𝑛(𝑥),𝜆)𝑢_𝑛(𝑥)𝑑𝑥≤_Ω(𝑎(𝑥,𝜆)+𝑐𝜉^𝑟₀𝑒(𝑥)^𝑟)𝜉₀𝑒(𝑥)𝑑𝑥, (2.19) which implies the boundedness of the sequence (𝑢_𝑛) in 𝑉₀. Then due to (2.17) we have that 𝑢₊∈𝑉₀ as well as 𝑢_𝑛⇀𝑢₊in𝑉₀,𝑢_𝑛→𝑢₊in𝐿^𝑝(Ω)anda.e.inΩ. (2.20) Since 𝑢_𝑛 solves problem (1.4), one has ⟨−Δ_𝑝𝑢_𝑛,𝜑⟩=_Ω𝑓(𝑥,𝑢_𝑛(𝑥),𝜆)𝜑(𝑥)𝑑𝑥,∀𝜑∈𝑉₀. (2.21) Setting 𝜑=𝑢_𝑛−𝑢₊ in (2.21) gives ⟨−Δ_𝑝𝑢_𝑛,𝑢_𝑛−𝑢₊⟩=_Ω𝑓(𝑥,𝑢_𝑛(𝑥),𝜆)(𝑢_𝑛(𝑥)−𝑢₊(𝑥))𝑑𝑥. (2.22) As already noticed that the sequence (𝑓(⋅,𝑢_𝑛(⋅),𝜆) is uniformly bounded on Ω, so (2.20) and (2.22) yield lim_𝑛→∞⟨−Δ_𝑝𝑢_𝑛,𝑢_𝑛−𝑢₊⟩=0. (2.23) The 𝑆₊-property of −Δ_𝑝 on 𝑉₀ implies 𝑢_𝑛→𝑢₊in𝑉₀as𝑛→∞. (2.24) 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₊∈𝐶¹₀(Ω). The choice of 𝜉₀ guarantees that0≤𝑢₊(𝑥)≤𝜉₀𝑒(𝑥)≤𝑏fora.e.𝑥∈Ω. (2.25) Thus, from (2.18), assumptions H(𝑓)(ii) and (iii), and the boundedness of 𝑢₊, we get −Δ_𝑝𝑢₊(𝑥)=𝑓(𝑥,𝑢₊(𝑥),𝜆)≥−𝑐𝑢̂₊(𝑥)^𝑝−1fora.a.𝑥∈Ω, (2.26) with a constant 𝑐̂>0. Applying the nonlinear strong maximum principle (cf. [28]) we conclude that either 𝑢₊=0 or 𝑢₊∈int(𝐶¹₀(Ω)₊).
We claim that𝑢₊∈int(𝐶¹₀(Ω)₊). (2.27) Assume on the contrary that 𝑢₊=0. Then (2.17) becomes 𝑢_𝑛(𝑥)↓0∀𝑥∈Ω. (2.28) Since 𝑢_𝑛≥(1/𝑛)𝜑₁, we may consider 𝑢_𝑛=𝑢_𝑛‖∇𝑢_𝑛‖_𝑝∀𝑛. (2.29) Along a relabelled subsequence we may suppose 𝑢_𝑛⇀𝑢in𝑉₀,𝑢_𝑛→𝑢in𝐿^𝑝(Ω)anda.e.inΩ (2.30) for some 𝑢∈𝑉₀. 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_𝑝𝑢_𝑛,𝜑⟩=_Ω𝑓(𝑥,𝑢_𝑛(𝑥),𝜆)‖∇𝑢_𝑛‖_𝑝^𝑝−1𝜑𝑑𝑥,∀𝜑∈𝑉₀. (2.31) Setting 𝜑=𝑢_𝑛−𝑢 leads to ⟨−Δ_𝑝𝑢_𝑛,𝑢_𝑛−𝑢⟩=_Ω𝑓(𝑥,𝑢_𝑛(𝑥),𝜆)‖∇𝑢_𝑛‖_𝑝^𝑝−1(𝑢_𝑛−𝑢)𝑑𝑥. (2.32) By H(𝑓)(iii) we know that there exist constants 𝑐₀=𝑐₀(𝜆)>𝜆₁ and 𝛼=𝛼(𝜆)>0 such that |𝑓(𝑥,𝑠,𝜆)|≤𝑐₀|𝑠|^𝑝−1fora.a.𝑥∈Ω,∀|s|<𝛼, (2.33) while H(𝑓)(ii) entails |𝑓(𝑥,𝑠,𝜆)|≤𝑎(𝑥,𝜆)+𝑐|𝑠|^𝑟≤(‖𝑎(⋅,𝜆)‖_∞𝛼^𝑟+𝑐)|𝑠|^𝑟 (2.34) for a.a. 𝑥∈Ω and for all |𝑠|≥𝛼. Combining the two estimates gives |𝑓(𝑥,𝑠,𝜆)|≤𝑐₀|𝑠|^𝑝−1+𝑐₁|𝑠|^𝑟fora.a.𝑥∈Ω,∀𝑠∈ℝ (2.35) with a constant 𝑐₁=𝑐₁(𝜆)>0. Since 𝑢_𝑛∈[(1/𝑛)𝜑₁,𝜉₀𝑒], 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≤𝐶fora.a.𝑥∈Ω,∀𝑛. (2.36) We see from (2.36) that |𝑓(𝑥,𝑢_𝑛(𝑥),𝜆)|‖∇𝑢_𝑛‖_𝑝^𝑝−1|𝑢_𝑛(𝑥)−𝑢(𝑥)|≤𝐶𝑤(𝑥)^𝑝−1(𝑤(𝑥)+|𝑢(𝑥)|)fora.a.𝑥∈Ω. (2.37) Then, because the right-hand side of the above inequality is in 𝐿¹(Ω), by means of (2.30) and (2.36) we can apply Lebesgue's dominated convergence theorem to get lim_𝑛→∞Ω𝑓(𝑥,𝑢_𝑛(𝑥),𝜆)‖∇𝑢_𝑛‖_𝑝^𝑝−1(𝑢_𝑛−𝑢)𝑑𝑥=0. (2.38) Consequently, from (2.32) we obtain lim_𝑛→∞⟨−Δ_𝑝𝑢_𝑛,𝑢_𝑛−𝑢⟩=0. (2.39) The 𝑆₊-property of −Δ_𝑝 on 𝑉₀ implies 𝑢_𝑛→𝑢in𝑉₀as𝑛→∞. (2.40) On the basis of (2.31) and (2.40) it follows ⟨−Δ_𝑝𝑢,𝜑⟩=lim_𝑛→∞Ω𝑓(𝑥,𝑢_𝑛(𝑥),𝜆)‖∇𝑢_𝑛‖_𝑝^𝑝−1𝜑𝑑𝑥,∀𝜑∈𝑉₀. (2.41) Notice from (2.36) that |𝑓(𝑥,𝑢_𝑛(𝑥),𝜆)|‖∇𝑢_𝑛‖_𝑝^𝑝−1|𝜑(𝑥)|≤𝐶𝑤(𝑥)^𝑝−1|𝜑(𝑥)| (2.42) for a.a. 𝑥∈Ω and for all 𝜑∈𝑉₀. We are thus allowed to apply Fatou's lemma which in conjunction with (2.28), (2.30), and (2.16) ensures lim_𝑛→∞Ω𝑓(𝑥,𝑢_𝑛(𝑥),𝜆)‖∇𝑢_𝑛‖_𝑝^𝑝−1𝜑(𝑥)𝑑𝑥=lim_𝑛→∞Ω𝑓(𝑥,𝑢_𝑛(𝑥),𝜆)𝑢_𝑛(𝑥)^𝑝−1𝑢_𝑛(𝑥)^𝑝−1𝜑(𝑥)𝑑𝑥≥_Ωliminf_𝑛→∞(𝑓(𝑥,𝑢_𝑛(𝑥),𝜆)𝑢_𝑛(𝑥)^𝑝−1𝑢_𝑛(𝑥)^𝑝−1𝜑(𝑥))𝑑𝑥≥𝜇_0Ω𝑢(𝑥)^𝑝−1𝜑(𝑥)𝑑𝑥 (2.43) for all 𝜑∈𝑉_0,+∶=𝑉₀∩𝐿^𝑝(Ω)₊. Thanks to (2.41) we obtain ⟨−Δ_𝑝𝑢,𝜑⟩≥𝜇_0Ω𝑢(𝑥)^𝑝−1𝜑(𝑥)𝑑𝑥,∀𝜑∈𝑉_0,+. (2.44) 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 lim_𝑛→∞Ω𝑓(𝑥,𝑢_𝑛(𝑥),𝜆)‖∇𝑢_𝑛‖_𝑝^𝑝−1𝜑(𝑥)𝑑𝑥=lim_𝑛→∞Ω𝑓(𝑥,𝑢_𝑛(𝑥),𝜆)𝑢_𝑛(𝑥)^𝑝−1𝑢_𝑛(𝑥)^𝑝−1𝜑(𝑥)𝑑𝑥≤_Ωlimsup_𝑛→∞(𝑓(𝑥,𝑢_𝑛(𝑥),𝜆)𝑢_𝑛(𝑥)^𝑝−1𝑢_𝑛(𝑥)^𝑝−1𝜑(𝑥))𝑑𝑥≤𝜈_0Ω𝑢(𝑥)^𝑝−1𝜑(𝑥)𝑑𝑥 (2.45) for all 𝜑∈𝑉_0,+. Then (2.41) ensures ⟨−Δ_𝑝𝑢,𝜑⟩≤𝜈_0Ω𝑢(𝑥)^𝑝−1𝜑(𝑥)𝑑𝑥,∀𝜑∈𝑉_0,+. (2.46) Combining (2.44) and (2.46) results in 𝜇₀𝑢^𝑝−1≤−Δ_𝑝𝑢≤𝜈₀𝑢^𝑝−1a.e.inΩ, (2.47) which guarantees to have 𝑢∈𝐿^∞(Ω) (see [27, Theorem 1.5.5]). Since by (2.47) we know that Δ_𝑝𝑢∈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∈int(𝐶¹₀(Ω)₊). Consequently, there exist constants 𝑘₀>0 and 𝑘₁>0 such that 𝑘₀𝜑₁≤𝑢<𝑘₁𝜑₁a.e.inΩ. (2.48) Following [30] let us denote 𝐼(𝑢,𝑣)=⟨−Δ_𝑝𝑢𝑢,^𝑝−𝑣^𝑝𝑢^𝑝−1⟩−⟨−Δ_𝑝𝑢𝑣,^𝑝−𝑣^𝑝𝑣^𝑝−1⟩ (2.49) whenever (𝑢,𝑣)∈𝐷_𝐼, where 𝐷_𝐼={(𝑤₁,𝑤₂)∈(𝑉₀)²∶𝑤_𝑖𝑤≥0,^𝑖𝑤_𝑗∈𝐿^∞(Ω)for𝑖,𝑗∈{1,2}}. (2.50) Relation (2.48) justifies that (𝑘₁𝜑₁,𝑢)∈𝐷_𝐼. Then Proposition 1 of Anane [30] implies 𝐼(𝑘₁𝜑₁,𝑢)≥0. On the other hand a direct computation based on (2.48) and (2.47) shows 𝐼(𝑘₁𝜑₁,𝑢)=⟨−Δ_𝑝(𝑘₁𝜑₁),(𝑘₁𝜑₁)^𝑝−𝑢^𝑝(𝑘₁𝜑₁)^𝑝−1⟩−⟨−Δ_𝑝𝑢,(𝑘₁𝜑₁)^𝑝−𝑢^𝑝𝑢^𝑝−1⟩≤(𝜆₁−𝜇₀)_Ω((𝑘₁𝜑₁)^𝑝−𝑢^𝑝)𝑑𝑥<0. (2.51) 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∈𝑉₀ 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_𝑝𝑢∈𝐿^∞(Ω). Using Theorem 1.5.6 of [27] leads to 𝑢∈𝐶¹₀(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∈int(𝐶¹₀(Ω)₊), hence 𝑢∈[(1/𝑛)𝜑₁,𝑢] for 𝑛 sufficiently large. The fact that 𝑢_𝑛 is the least solution of (1.4) in [(1/𝑛)𝜑₁,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 𝜆₀∈(0,𝜆) with the property that if 𝜆∈(0,𝜆₀), then problem (1.4) has a (positive) solution 𝑢₊=𝑢₊(𝜆)∈int(𝐶¹₀(Ω)₊), a (negative) solution 𝑢₋=𝑢₋(𝜆)∈−int(𝐶¹₀(Ω)₊), and a nontrivial sign-changing solution 𝑢₀=𝑢₀(𝜆)∈𝐶¹₀(Ω) satisfying ‖𝑢₊‖_∞<𝑏, ‖𝑢₋‖_∞<𝑏, ‖𝑢₀‖_∞<𝑏.

Proof. Let 𝑏>0. Consider the positive number 𝜆₀ given by Theorem 2.4 and fix 𝜆∈(0,𝜆₀). Let 𝑢₊∈int(𝐶¹₀(Ω)₊) and 𝑢₋∈−int(𝐶¹₀(Ω)₊) be the two extremal solutions determined in Theorem 2.4. We introduce on Ω×ℝ the truncation functions 𝜏₊(𝑥,𝑠)=0if𝑠≤0,𝑠if0<𝑠<𝑢₊𝑢(𝑥),₊(𝑥)if𝑠≥𝑢₊𝜏(𝑥),₋𝑢(𝑥,𝑠)=⁻(𝑥)if𝑠≤𝑢₋(𝑥),𝑠if𝑢₋𝜏(𝑥)<𝑠<0,0if𝑠≥0,₀𝑢(𝑥,𝑠)=⁻(𝑥)if𝑠≤𝑢₋(𝑥),𝑠if𝑢₋(𝑥)<𝑠<𝑢₊𝑢(𝑥),₊(𝑥)if𝑠≥𝑢₊(𝑥) (2.52) and then define the following associated functionals: 𝐸₊1(𝑢)=𝑝‖∇𝑢‖^𝑝_𝑝−_Ω0^𝑢(𝑥)𝑓(𝑥,𝜏₊(𝑥,𝑠),𝜆)𝑑𝑠𝑑𝑥,∀𝑢∈𝑉₀,𝐸₋1(𝑢)=𝑝‖∇𝑢‖^𝑝_𝑝−_Ω0^𝑢(𝑥)𝑓(𝑥,𝜏₋(𝑥,𝑠),𝜆)𝑑𝑠𝑑𝑥,∀𝑢∈𝑉₀,𝐸₀1(𝑢)=𝑝‖∇𝑢‖^𝑝_𝑝−_Ω0^𝑢(𝑥)𝑓(𝑥,𝜏₀(𝑥,𝑠),𝜆)𝑑𝑠𝑑𝑥,∀𝑢∈𝑉₀. (2.53) It is clear that 𝐸₊,𝐸₋,𝐸₀∈𝐶¹(𝑉₀).
We observe that if 𝑣 is a critical point of 𝐸₊, then⟨−Δ_𝑝𝑣+Δ_𝑝𝑢₊,(𝑣−𝑢₊)⁺⟩=_Ω(𝑓(𝑥,𝜏₊(𝑥,𝑣(𝑥)),𝜆)−𝑓(𝑥,𝑢₊(𝑥),𝜆))(𝑣−𝑢₊)⁺𝑑𝑥=0 (2.54) which implies 𝑣≤𝑢₊. Similarly, it follows that 𝑣≥0. This leads to 𝑣isacriticalpointof𝐸₊⇒0≤𝑣(𝑥)≤𝑢₊(𝑥)fora.a.𝑥∈Ω. (2.55)
Since the function 𝐸₊ is coercive and weakly lower semicontinuous, there exists a global minimizer 𝑧₊∈𝑉₀ of it. Using (2.14), it is seen that𝐸₊(𝑧₊)=inf_𝑉0𝐸₊<0, (2.56) 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₊∈int(𝐶¹₀(Ω)₊), there exists a neighborhood 𝒰 of 𝑢₊ in the space 𝐶¹₀(Ω) such that 𝒰⊂𝐶¹₀(Ω)₊. Therefore 𝐸₀=𝐸₊ on 𝒰, which guarantees that 𝑢₊ is a local minimizer of 𝐸₀ on 𝐶¹₀(Ω). It results that 𝑢₊ is also a local minimizer of 𝐸₀ on the space 𝑉₀ (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 𝐸₀ on 𝑉₀.
As in the case of (2.55), we verify that every critical point of 𝐸₀ belongs to the set {𝑢∈𝑉₀∶𝑢₋(𝑥)≤𝑢(𝑥)≤𝑢₊(𝑥)a.e.𝑥∈Ω}, which implies that every critical point of 𝐸₀ is a solution to problem (1.4). The functional 𝐸₀ is coercive, weakly lower semicontinuous, with inf_𝑉0𝐸₀<0. Thus 𝐸₀ has a global minimizer 𝑦₀∈𝑉₀ with 𝑦₀≠0. The above properties ensure that 𝑦₀ is a nontrivial solution of problem (1.4) belonging to the order interval [𝑢₋,𝑢₊]. Assume 𝑦₀≠𝑢₊ and 𝑦₀≠𝑢₋. We claim that 𝑦₀ changes sign. Indeed, if not, 𝑦₀ would have constant sign, for instance 𝑦₀≥0 a.e. on Ω. Using assumptions H(𝑓)(ii) and (iii) and the boundedness of 𝑦₀, we may apply the strong maximum principle which leads to 𝑦₀>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 𝑢₀=𝑦₀.
Thus, the proof reduces to consider the cases 𝑦₀=𝑢₊ or 𝑦₀=𝑢₋. To make a choice, suppose 𝑦₀=𝑢₊. We may also admit that 𝑢₋ is a strict local minimizer of 𝐸₀. This is true since on the contrary we would find (infinitely many) critical points 𝑥₀ of 𝐸₀ belonging to the order interval [𝑢₋,𝑢₊] which are different from 0, 𝑢₋, 𝑢₊, and if 𝑥₀ 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 that𝐸₀(𝑢₊)≤𝐸₀(𝑢₋)<inf{𝐸₀(𝑢)∶𝑢∈𝜕𝐵_𝜌(𝑢₋)}, (2.57) where 𝜕𝐵_𝜌(𝑢₋)={𝑢∈𝑉₀∶‖𝑢−𝑢₋‖=𝜌}. Relation (2.57) in conjunction with the Palais-Smale condition (which holds for 𝐸₀ due to its coercivity) enables us to apply the mountain pass theorem to the functional 𝐸₀ (see, e.g., [31]). In this way we get 𝑢₀∈𝑉₀ satisfying 𝐸′₀(𝑢₀)=0 and inf{𝐸₀(𝑢)∶𝑢∈𝜕𝐵_𝜌(𝑢₋)}≤𝐸₀(𝑢₀)=inf_𝛾∈Γmax_{𝑡∈[−1,1]}𝐸₀(𝛾(𝑡)), (2.58) where Γ={𝛾∈𝐶([−1,1],𝑉₀)∶𝛾(−1)=𝑢₋,𝛾(1)=𝑢₊}. (2.59) We infer from (2.57) and (2.58) that 𝑢₀≠𝑢₋and 𝑢₀≠𝑢₊.
The next step in the proof is to show that𝐸₀(𝑢₀)<0. (2.60) By the equality in (2.58), it suffices to produce a path 𝛾∈Γ such that 𝐸₀(𝛾(𝑡))<0∀𝑡∈[−1,1]. (2.61) Let 𝑆=𝑉₀∩𝜕𝐵^𝐿𝑝₁^(Ω), where 𝜕𝐵^𝐿𝑝₁^(Ω)={𝑢∈𝐿^𝑝(Ω)∶‖𝑢‖_𝑝=1}, and 𝑆_𝐶=𝑆∩𝐶¹₀(Ω) be endowed with the topologies induced by 𝑉₀ and 𝐶¹₀(Ω), respectively. We set Γ_0,𝐶={𝛾∈𝐶([−1,1],𝑆_𝐶)∶𝛾(−1)=−𝜑₁,𝛾(1)=𝜑₁}. (2.62) Making use of the first inequality in assumption H(𝑓) (iii), we fix numbers 𝜇>𝜆₂ and 𝛿>0 such that (2.14) holds, and then let 𝜌₀∈(0,𝜇−𝜆₂). We recall the following variational expression for 𝜆₂ given by Cuesta et al. [32]: 𝜆₂=inf_𝛾∈Γ0max_{𝑢∈𝛾([−1,1])}‖∇𝑢‖^𝑝_𝑝, (2.63) where Γ₀={𝛾∈𝐶([−1,1],𝑆)∶𝛾(−1)=−𝜑₁,𝛾(1)=𝜑₁}. (2.64) By (2.63) there exists 𝛾∈Γ₀ such that max_{𝑡∈[−1,1]}‖∇𝛾(𝑡)‖^𝑝_𝑝<𝜆₂+𝜌₀2. (2.65) Choose some number 𝑟 with 0<𝑟≤(𝜆₂+𝜌₀)^1/𝑝−(𝜆₂+𝜌₀/2)^1/𝑝. The density of 𝑆_𝐶 in 𝑆 implies that Γ_0,𝐶 is dense in Γ₀; so there is 𝛾₀∈Γ_0,𝐶 satisfying max_{𝑡∈[−1,1]}‖∇𝛾(𝑡)−∇𝛾₀(𝑡)‖_𝑝<𝑟. (2.66) Then the choice of 𝑟 establishes max_{𝑡∈[−1,1]}‖∇𝛾₀(𝑡)‖^𝑝_𝑝<𝜆₂+𝜌₀. (2.67) The boundedness of the set 𝛾₀([−1,1])(Ω) in ℝensures the existence of some 𝜀₁>0 such that 𝜀₁|𝑢(𝑥)|≤𝛿∀𝑥∈Ω∀𝑢∈𝛾₀([−1,1]). (2.68) Since 𝑢₊,−𝑢₋∈int(𝐶¹₀(Ω)₊) (see Theorem 2.4), for every 𝑢∈𝛾₀([−1,1]) and any bounded neighborhood 𝑉_𝑢 of 𝑢 in 𝐶¹₀(Ω) there exist positive numbers ℎ_𝑢 and 𝑗_𝑢 such that 𝑢₊−1ℎ𝑣∈int(𝐶¹₀(Ω)₊),−𝑢₋+1𝑗𝑣∈int(𝐶¹₀(Ω)₊), (2.69) whenever ℎ≥ℎ_𝑢, 𝑗≥𝑗_𝑢, and 𝑣∈𝑉_𝑢. This fact and the compactness of 𝛾₀([−1,1]) in 𝐶¹₀(Ω) allow to determine a number 𝜀₀>0 for which one has 𝑢₋(𝑥)≤𝜀𝑢(𝑥)≤𝑢₊(𝑥)∀𝑥∈Ω,𝑢∈𝛾₀([−1,1]),𝜀∈(0,𝜀₀). (2.70) We now focus on the continuous path 𝜀𝛾₀ in 𝐶¹₀(Ω) joining −𝜀𝜑₁and 𝜀𝜑₁ with a fixed constant 𝜀 satisfying 0<𝜀<𝑚𝑖𝑛{𝜀₀,𝜀₁}. By (2.70), (2.67), (2.68), (2.14) with 𝜇>𝜆₂, and taking into account the choice of 𝜌₀ as well as 𝛾₀([−1,1])⊂𝜕𝐵^𝐿𝑝₁^(Ω) we obtain 𝐸₀(𝜀𝛾₀𝜀(𝑡))=^𝑝𝑝‖∇𝛾₀(𝑡)‖^𝑝_𝑝−_Ω^𝜀𝛾0₀^(𝑡)(𝑥)𝑓(𝑥,𝜏₀=𝜀(𝑥,𝑠),𝜆)𝑑𝑠𝑑𝑥_𝑝𝑝‖∇𝛾₀(𝑡)‖^𝑝_𝑝−_Ω^𝜀𝛾0₀^(𝑡)(𝑥)≤𝜀𝑓(𝑥,𝑠,𝜆)𝑑𝑠𝑑𝑥_𝑝𝑝(𝜆₂+𝜌₀−𝜇)<0∀𝑡∈[−1,1]. (2.71)
At this point we apply the second deformation lemma (see, e.g., [27, page 366]) to the 𝐶¹ functional 𝐸₊∶𝑉₀→ℝ. Towards this let us denote𝑐₊=𝑐₊(𝜆)=𝐸₊(𝜀𝜑₁),𝑚₊=𝑚₊(𝜆)=𝐸₊(𝑢₊𝐸),_𝑐++={𝑢∈𝑉₀∶𝐸₊(𝑢)≤𝑐₊}. (2.72) 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]→𝑉₀ by 𝛾₊(𝑡)∶=(𝜂(𝑡,𝜀𝜑₁))⁺∶=max{𝜂(𝑡,𝜀𝜑₁),0} (2.73) for all 𝑡∈[0,1], it is seen that 𝛾₊ is a continuous path in 𝑉₀ joining 𝜀𝜑₁ and 𝑢₊. (Note the mapping 𝑤↦𝑤⁺ is continuous from 𝑉₀ into itself.) The properties of the deformation 𝜂 imply 𝐸₀(𝛾₊(𝑡))=𝐸₊(𝛾₊(𝑡))≤𝐸₊(𝜂(𝑡,𝜀𝜑₁))=𝐸₊(𝜀𝜑₁)=𝐸₀(𝜀𝜑₁)<0 (2.74) for all 𝑡∈[0,1]. Similarly, applying the second deformation lemma to the functional 𝐸₋, we construct a continuous path 𝛾₋∶[0,1]→𝑉₀ joining 𝑢₋ and −𝜀𝜑₁ such that 𝐸₀(𝛾₋(𝑡))<0∀𝑡∈[0,1]. (2.75) The union of the curves 𝛾₋, 𝜀𝛾₀, 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 𝑢₀≠0. Recalling that the critical points of 𝐸₀ are in the order interval {𝑢∈𝑉₀∶𝑢₋(𝑥)≤𝑢(𝑥)≤𝑢₊(𝑥)a.e.𝑥∈Ω} we derive that 𝑢₀ is a nontrivial solution of (1.4) distinct from 𝑢₋ and 𝑢₊, with 𝑢₋≤𝑢₀≤𝑢₊. By the nonlinear regularity theory we have that 𝑢₀∈𝐶¹₀(Ω). The extremality properties of the constant sign solutions 𝑢₋ and 𝑢₊ as described in Theorem 2.4 force 𝑢₀ 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|^𝑟fora.a.𝑥∈Ω∀(s,𝜆)∈ℝ×(0,𝜆). (2.76)(iii)For all 𝜆∈(0,𝜆) there exist constants 𝜇₀=𝜇₀(𝜆)>𝜆₂, 𝜈₀=𝜈₀(𝜆)>𝜇₀ and a set Ω_𝜆⊂Ω with Ω⧵Ω_𝜆 of Lebesgue measure zero such that 𝜇₀<liminf_𝑠→0𝑓(𝑥,𝑠,𝜆)|𝑠|^𝑝−2𝑠≤limsup_𝑠→0𝑓(𝑥,𝑠,𝜆)|𝑠|^𝑝−2𝑠≤𝜈₀ (2.77) 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₊,𝜆)∀𝑥∈Ω,𝑓(𝑥,𝑠,𝜆)<0∀𝑥∈Ω,all𝑠∈(b₋,0),𝑓(𝑥,𝑠,𝜆)>0∀𝑥∈Ω,all𝑠∈(0,b₊). (2.78)(v)For every 𝜆∈(0,𝜆), there exist 𝑀=𝑀(𝜆)>0 and 𝜇=𝜇(𝜆)>𝑝 such that0<𝜇𝐹(𝑥,𝑠,𝜆)≤𝑓(𝑥,𝑠,𝜆)𝑠∀𝑥∈Ω,all|s|≥𝑀. (2.79)

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 𝜆₀∈(0,𝜆) with the property that if 𝜆∈(0,𝜆₀), then problem (1.4) has a minimal (positive) solution 𝑢₊=𝑢₊(𝜆)∈int(𝐶¹₀(Ω)₊), a maximal (negative) solution 𝑢₋=𝑢₋(𝜆)∈−int(𝐶¹₀(Ω)₊), and two nontrivial sign-changing solutions 𝑢₀=𝑢₀(𝜆),𝑤₀=𝑤₀(𝜆)∈𝐶¹₀(Ω) satisfying ‖𝑢₊‖_∞<𝑏, ‖𝑢₋‖_∞<𝑏, 𝑢₋≤𝑢₀≤𝑢₊ a.e. in Ω (so ‖𝑢₀‖_∞<𝑏) and ‖𝑤₀‖_∞≥𝑏, where 𝑏∶=min{𝑏₊,|𝑏₋|}.

Proof. Since hypotheses H'(𝑓) are stronger than H(𝑓), we can apply Theorem 2.5 with 𝑏=min{𝑏₊,|𝑏₋|}, which ensures the existence of a number 𝜆₀∈(0,𝜆) such that for every 𝜆∈(0,𝜆₀), problem (1.4) possesses a positive solution 𝑢₊=𝑢₊(𝜆)∈int(𝐶¹₀(Ω)₊), a negative solution 𝑢₋=𝑢₋(𝜆)∈−int(𝐶¹₀(Ω)₊), and a sign-changing solution 𝑢₀=𝑢₀(𝜆)∈𝐶¹₀(Ω) with −𝑏<𝑢₋≤𝑢₀≤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 𝑤₀=𝑤₀(𝜆)∈𝐶¹₀(Ω) (by the nonlinear regularity theory) with max_Ω𝑤₀≥𝑏₊ and min_Ω𝑤₀≤𝑏₋. Therefore, we have ‖𝑤₀‖_∞≥𝑏, which shows that the sign-changing solutions 𝑢₀ and 𝑤₀ are different. This completes the proof.

Remark 2.7. In fact, under hypotheses H'(𝑓), for 𝜆∈(0,𝜆₀), 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 (𝜇₁,𝜇₂)∈ℝ² for which the problem𝑢∈𝑉₀∶−Δ_𝑝𝑢=𝜇₁(𝑢⁺)^𝑝−1−𝜇₂(𝑢⁻)^𝑝−1in𝑉^∗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 𝜆₁×ℝ and ℝ×𝜆₁ with 𝜆₁ 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