Dirichlet Problems with an Indefinite and Unbounded Potential and Concave-Convex Nonlinearities

Abstract

We consider a parametric semilinear Dirichlet problem with an unbounded and indefinite potential. In the reaction we have the competing effects of a sublinear (concave) term and of a superlinear (convex) term. Using variational methods coupled with suitable truncation techniques, we prove two multiplicity theorems for small values of the parameter. Both theorems produce five nontrivial smooth solutions, and in the second theorem we provide precise sign information for all the solutions.

1. Introduction

Let Ω∈ℝ^𝑁 be a bounded domain with a 𝐶²-boundary 𝜕Ω. In this paper we study the following parametric nonlinear Dirichlet problem: (𝑧)(𝑧)𝑢(𝑧)((𝑧((𝑧𝑢|−Δ𝑢+𝛽=𝜆𝑔𝑧,𝑢))+𝑓𝑧,𝑢))inΩ,_𝜕Ω( (𝑃)=0,𝜆>0.^𝜆) Here 𝛽∈𝐿^𝑠(Ω) with 𝑠>𝑁/2(𝑁⩾2) is a potential function which may change sign (indefinite potential). Also 𝜆>0 is a parameter, and 𝑔,𝑓∶Ω×ℝ→ℝ are Carathéodory functions (i.e., for all 𝜁∈ℝ, functions 𝑧↦𝑔(𝑧,𝜁) and 𝑧↦𝑓(𝑧,𝜁) are measurable and for almost all 𝑧∈Ω, functions 𝜁↦𝑔(𝑧,𝜁) and 𝜁↦𝑓(𝑧,𝜁) are continuous). We assume that for almost all 𝑧∈Ω, the function 𝜁↦𝑔(𝑧,𝜁) is strictly sublinear near ±∞, while the function 𝜁↦𝑓(𝑧,𝜁) is superlinear near ±∞. So, problem ((𝑃)_𝜆) exhibits competing nonlinearities of the concave-convex type. This situation was first studied with 𝛽≡0 and the right hand side nonlinearity being 𝜆𝜁^𝑞−2𝜁+𝜁^𝑟−2𝜁, (1.1) with 1<𝑞<2<𝑟<2^∗=2𝑁𝑁−2if𝑁>2,+∞if𝑁=2, (1.2) by Ambrosetti et al. [1]. In [1], the authors focus on positive solutions and proved certain bifurcation-type phenomena as 𝜆>0 varies. Further results in this direction can be found in the works of I'lyasov [2], Li et al. [3], Lubyshev [4], and Rădulescu and Repovš [5]. In all the aforementioned works 𝛽≡0. We should also mention the recent work of Motreanu et al. [6], where the authors consider equations driven by the 𝑝-Laplacian, with concave term of the form |𝜁|^𝑞−2𝜁 (where 1<𝑞<𝑝) and a perturbation exhibiting an asymmetric behaviour at +∞ and at −∞ ((𝑝−1)-superlinear near +∞ and (𝑝−1)-sublinear near −∞). They prove multiplicity results producing four nontrivial solutions with sign information.

In this work, we prove two multiplicity results for problem ((𝑃)_𝜆) when the parameter 𝜆>0 is small. In both results we produce five nontrivial smooth solutions, and in the second we provide precise sign information for all the solutions. For the superlinear (“convex”) nonlinearity 𝑓(𝑧,⋅), we do not employ the usual in such cases Ambrosetti-Rabinowitz condition. Instead, we use a more general condition, which incorporates in our framework also superlinear perturbations with “slow” growth near ±∞, which do not satisfy the Ambrosetti-Rabinowitz condition. We should point out that none of the works mentioned earlier provide sign information for all the solutions (in particular, none of them produced a nodal (sign changing) solution) and all use the Ambrosetti-Rabinowitz condition to express the superlinearity of the “convex” contribution in the reaction.

Our approach is variational based on the critical point theory which is combined with suitable truncation techniques. In the next section we recall the main mathematical tools we will use in the analysis of problem ((𝑃)_𝜆). We also introduce the hypotheses on the terms 𝑔 and 𝑓.

2. Mathematical Background and Hypotheses

Let 𝑋 be a Banach space and let 𝑋^∗ be its topological dual. By ⟨⋅,⋅⟩ we denote the duality brackets for the pair (𝑋^∗,𝑋). Let 𝜑∈𝐶¹(𝑋). We say that 𝜑 satisfies the Cerami condition, if the following is true.

Every sequence {𝑥_𝑛}_𝑛⩾1⊆𝑋, such that {𝜑(𝑥_𝑛)}_𝑛⩾1⊆ℝ is bounded and

1+𝑥_𝑛𝜑^′𝑥_𝑛⟶0in𝑋^∗, (2.1)

admits a strongly convergent subsequence.

This compactness-type condition is in general weaker than the usual Palais-Smale condition. However, the Cerami condition suffices to prove a deformation theorem and from it we derive the minimax theory for certain critical values of 𝜑∈𝐶¹(𝑋) (see Gasiński and Papageorgiou [7] and Motreanu and Rădulescu [8]). In particular, we can have the following theorem, known in the literature as the mountain pass theorem.

Theorem 2.1. If 𝑋 is a Banach space, 𝜑∈𝐶¹(𝑋) satisfies the Cerami condition, 𝑥₀,𝑥₁∈𝑋,  𝜚>0‖𝑥₁−𝑥₀‖>𝜚: max𝜑𝑥₀,𝜑𝑥₁<inf𝜑(𝑥)∶𝑥−𝑥₀=𝜚=𝜂_𝜚,𝑐=inf_𝛾∈Γmax_[]𝑡∈0,1𝜑𝛾(𝑡), (2.2) where Γ=𝛾∈𝐶([0,1];𝑋)∶𝛾(0)=𝑥₀,𝛾(1)=𝑥₁. (2.3) Then 𝑐⩾𝜂_𝜚 and 𝑐 are a critical value of 𝜑.

In the analysis of problem ((𝑃)_𝜆), in addition to the Sobolev space 𝐻¹₀(Ω), we will also use the Banach space 𝐶¹₀Ω=𝑢∈𝐶¹Ω∶𝑢|_𝜕Ω=0. (2.4) This is an ordered Banach space with positive cone: 𝐶₊=𝑢∈𝐶¹₀Ω∶𝑢(𝑧)⩾0∀𝑧∈Ω. (2.5) This cone has a nonempty interior given by int𝐶₊=𝑢∈𝐶₊∶𝑢(𝑧)>0∀𝑧∈Ω,𝜕𝑢(𝑧)𝜕𝑛<0∀𝑧∈𝜕Ω, (2.6) where 𝑛(⋅) is the outward unit normal on 𝜕Ω.

In the proof of the second multiplicity theorem and in order to produce a nodal (sign changing) solution, we will also use critical groups. So, let us recall their definition. Let 𝜑∈𝐶¹(𝑋) and let 𝑐∈ℝ. We introduce the following sets: 𝜑^𝑐=(𝑥),𝐾𝑥∈𝑋∶𝜑⩽𝑐_𝜑=𝑥∈𝑋∶𝜑^′(𝑥)=0. (2.7) Also, if (𝑌₁,𝑌₂) is a topological pair with 𝑌₂⊆𝑌₁⊆𝑋 and 𝑘⩾0 is an integer, by 𝐻_𝑘(𝑌₁,𝑌₂) we denote the 𝑘th relative singular homology group for the pair (𝑌₂,𝑌₁) with integer coefficients. The critical groups of 𝜑 at an isolated critical point 𝑥₀∈𝑋 of 𝜑 with 𝜑(𝑥₀)=𝑐 are defined by 𝐶_𝑘𝜑,𝑥₀=𝐻_𝑘𝜑^𝑐∩𝑈,𝜑^𝑐∩𝑈⧵𝑥₀∀𝑘⩾0, (2.8) where 𝑈 is a neighbourhood of 𝑥₀, such that 𝐾_𝜑∩𝜑^𝑐∩𝑈={𝑥₀}. The excision property of singular homology implies that this definition is independent of the particular choice of the neighbourhood 𝑈.

Using the spectral theorem for compact self-adjoint operators, we can show that the differential operator𝐻¹₀(Ω)∋𝑢⟼−Δ𝑢+𝛽𝑢 (2.9) has a sequence of distinct eigenvalues {𝜆_𝑘}_𝑘⩾1, such that 𝜆_𝑘⟶+∞as𝑘⟶+∞. (2.10) The first eigenvalue is simple and admits the following variational characterization: 𝜆₁𝜎(𝑢)=inf‖𝑢‖²₂∶𝑢∈𝐻¹₀(Ω),𝑢≠0, (2.11) where 𝜎(𝑢)=‖∇𝑢‖²₂+_Ω𝛽(𝑧)𝑢(𝑧)²𝑑𝑧∀𝑢∈𝐻¹₀(Ω). (2.12) Moreover, the corresponding eigenfunction 𝑢∈𝐶¹₀(Ω) does not change sign, and in fact we can take 𝑢(𝑧)>0 for all 𝑧∈Ω (see Gasiński and Papageorgiou [9]). Using (2.11) and this property of the principal eigenfunction, we can have the following lemma (see Gasiński and Papageorgiou [9, Lemma 2.1]).

Lemma 2.2. If 𝜂∈𝐿^𝑠(Ω), 𝜂(𝑧)⩽𝜆₁ for almost all 𝑧∈Ω, 𝜂≠𝜆₁, then there exists 𝑐̂>0, such that 𝜎(𝑢)−_Ω𝜂𝑢²𝑑𝑧⩾𝑐‖̂𝑢‖²∀𝑢∈𝐻¹₀(Ω). (2.13)

Also, from Gasiński and Papageorgiou [9] we know that there exist 𝑐₀̂,𝑐₁̂>0, such that‖𝑢‖²⩽𝑐₀̂𝜎(𝑢)+𝑐₁̂‖𝑢‖²₂∀𝑢∈𝐻¹₀(Ω), (2.14) with ‖𝑢‖=‖∇𝑢‖₂, the norm of the Sobolev space 𝐻¹₀(Ω).

Next we state the hypotheses on the two components 𝑔 and 𝑓 of the reaction in problem ((𝑃)_𝜆).

Let𝐺(𝑧,𝜁)=^𝜁₀𝑔(𝑧,𝑠)𝑑𝑠,𝐹(𝑧,𝜁)=^𝜁₀𝑓(𝑧,𝑠)𝑑𝑠. (2.15)𝐻_𝑔:𝑔∶Ω×ℝ→ℝ is a Carathéodory function, such that 𝑔(𝑧,0)=0 for almost all 𝑧∈Ω.(i)For every 𝜚>0, there exists a function 𝑎_𝜚∈𝐿^∞(Ω)₊, such that 𝑔(𝑧,𝜁)⩽𝑎_𝜚(𝑧)foralmostall𝑧∈Ω,all𝜁⩽𝜚. (2.16)(ii)We have lim_𝜁→±∞𝑔(𝑧,𝜁)𝜁=0uniformlyforalmostall𝑧∈Ω. (2.17)(iii)There exist constants 𝑐₁,𝑐₂>0, 1<𝑞<𝜇<2 and 𝛿₀>0, such that 𝑐₁𝜁^𝑞(),()()⩽𝑔𝑧,𝜁𝜁foralmostall𝑧∈Ω,all𝜁∈ℝ𝜇𝐺𝑧,𝜁−𝑔𝑧,𝜁𝜁⩾𝑐₂𝜁^𝑞foralmostall𝑧∈Ω,all𝜁⩽𝛿₀. (2.18)(iv)For every 𝜚>0, we can find 𝛾_𝜚>0, such that for almost all 𝑧∈Ωthemap−𝜚,𝜚∋𝜁⟼𝑔(𝑧,𝜁)+𝛾_𝜚𝜁^𝑞−2𝜁isnondecreasing. (2.19)

Remark 2.3. Hypothesis 𝐻_𝑔(ii) implies that for almost all 𝑧∈Ω, the function 𝑔(𝑧,⋅) is strictly sublinear near ±∞. Hence 𝑔(𝑧,⋅) is the “concave” component in the reaction of ((𝑃)_𝜆) (the terminology “concave” and “convex” nonlinearities is due to Ambrosetti [1]). Note that hypothesis 𝐻_𝑔(iii) implies that for almost all 𝑧∈Ω, the function 𝑔(𝑧,⋅) has a similar growth near 0 that is, we have a concave term near zero. Hypothesis 𝐻_𝑔(iv) is weaker than assuming the monotonicity of 𝑔(𝑧,⋅) for almost all 𝑧∈Ω.

𝐻_𝑓:𝑓∶Ω×ℝ→ℝ is a Carathéodory function, such that 𝑓(𝑧,0)=0 for almost all 𝑧∈Ω.(i)There exist 𝑎∈𝐿^∞(Ω)₊, 𝑐>0 and 𝑟∈(2,2^∗), such that 𝑓(𝑧,𝜁)⩽𝑎(𝑧)+𝑐𝜁^𝑟−1foralmostall𝑧∈Ω,all𝜁∈ℝ. (2.20)(ii)We have lim_𝜁→±∞𝐹(𝑧,𝜁)𝜁²=+∞uniformlyforalmostall𝑧∈Ω. (2.21)(iii)There exist functions 𝜂₀,𝜂₀∈𝐿^∞(Ω)₊, such that 𝜂₀(𝑧)⩽𝜆₁ for almost all 𝑧∈Ω, 𝜂₀≠𝜆₁ and 𝜂₀(𝑧)⩽liminf_𝜁→0𝑓(𝑧,𝜁)𝜁⩽limsup_𝜁→0𝑓(𝑧,𝜁)𝜁⩽𝜂₀(𝑧) (2.22) uniformly for almost all 𝑧∈Ω.(iv)For every 𝜚>0, we can find 𝛾_𝜚>0, such that for almost all 𝑧∈Ω, we have

themap−𝜚,𝜚∋𝜁⟼𝑓(𝑧,𝜁)+𝛾_𝜚𝜁isnondecreasing. (2.23)

Remark 2.4. Hypothesis 𝐻_𝑓(ii) implies that for almost all 𝑧∈Ω, the function 𝐹(𝑧,⋅) is superquadratic near ±∞. Evidently, this is satisfied if the function 𝑓(𝑧,⋅) is superlinear near ±∞, that is, when lim_𝜁→±∞𝑓(𝑧,𝜁)𝜁=+∞uniformlyforalmostall𝑧∈Ω. (2.24) So, 𝑓(𝑧,𝜁) is the “convex” component of the reaction which “competes” with the “concave” component 𝑔(𝑧,𝜁).

Note that in 𝐻_𝑓, we did not include the Ambrosetti-Rabinowitz condition to characterize the superlinearity of 𝑓(𝑧,⋅). We recall that the Ambrosetti-Rabinowitz condition says that there exist 𝜏>2 and 𝑀>0, such that0<𝜏𝐹(𝑧,𝜁)⩽𝑓(𝑧,𝜁)𝜁uniformlyforalmostall𝑧∈Ω,all𝜁⩾𝑀,essinf_Ω𝐹(⋅,𝑀)>0. (2.25)(2.26) Integrating (2.25) and using (2.26), we obtain the weaker condition 𝑐₃𝜁^𝜏⩽𝐹(𝑧,𝜁)uniformlyforalmostall𝑧∈Ω,all𝜁⩾𝑀. (2.27) Therefore, for almost all 𝑧∈Ω, the function 𝜁⟼𝜆𝐺(𝑧,𝜁)+𝐹(𝑧,𝜁) (2.28) is superquadratic with at least 𝜏-growth near ±∞. Hence the Ambrosetti-Rabinowitz condition excludes superlinear perturbations with “slower” growth near ±∞. For this reason, here we employ a weaker condition. So, let 𝜉_𝜆(𝑧,𝜁)=𝜆𝑔(𝑧,𝜁)+𝑓(𝑧,𝜁)𝜁−2(𝜆𝐺(𝑧,𝜁)+𝐹(𝑧,𝜁)). (2.29)

We employ the following hypothesis.𝐻₀∶ For every 𝜆>0, there exists a function 𝜗^∗_𝜆∈𝐿¹(Ω)₊, such that 𝜉_𝜆(𝑧,𝜁)⩽𝜉_𝜆𝑧,𝑦+𝜗^∗_𝜆(𝑧)𝜉foralmostall𝑧∈Ω,all0⩽𝜁⩽𝑦,_𝜆(𝑧,𝜁)⩽𝜉_𝜆𝑧,𝑦+𝜗^∗_𝜆(𝑧)foralmostall𝑧∈Ω,all𝑦⩽𝜁⩽0. (2.30)

Remark 2.5. Hypothesis 𝐻₀ is a generalized version of a condition first introduced by Li and Yang [10], where the reader can find other possible extensions of the Ambrosetti-Rabinowitz condition and comparisons between them. Hypothesis 𝐻₀ is a quasimonotonicity condition on 𝜉_𝜆(𝑧,⋅), and it is satisfied if there exists 𝑀>0, such that for almost all 𝑧∈Ω, the function ()()𝜁⟼𝜆𝑔𝑧,𝜁+𝑓𝑧,𝜁𝜁 (2.31) is increasing on [𝑀,+∞] and is decreasing on [−∞,−𝑀] (see Li and Yang [10]).

Example 2.6. The following pairs of functions satisfy hypotheses 𝐻_𝑔, 𝐻_𝑓, and 𝐻₀ (for the sake of simplicity we drop the 𝑧-dependence): 𝑔₁(𝜁)=𝜁^𝑞−2𝜁,𝑓₁(𝜁)=𝜁^𝑝−2𝜁+𝜂₀𝑔𝜁,₂(𝜁)=𝜁^𝑞−2𝜁,𝑓₂(𝜁)=𝜁^𝑝−2𝜁ln1+𝜁+𝜂₀𝑔𝜁,₃(𝜁)=𝜁^𝑞−2𝜁−𝜁^𝜏−2𝜁,𝑓₃(𝜁)=𝜂₀𝜁−𝜁^𝑟−2𝜁𝜁𝜁if⩽1,_𝑝−2𝜁ln𝜁if𝜁>1, (2.32) with 1<𝑞<𝜏<2<𝑝<2^∗, 2<𝑟, 𝜂₀<𝜆₁.
Note that 𝑓₂ and 𝑓₃ do not satisfy the Ambrosetti-Rabinowitz condition (see (2.25)-(2.26)).

For every 𝑢∈𝐻¹₀(Ω), we set ‖𝑢‖=‖∇𝑢‖₂ (2.33) (by virtue of the Poincaré inequality). We mention that the notation ‖⋅‖ will be also used to denote the ℝ^𝑁-norm. It will always be clear from the context which norm is used. For 𝜁∈ℝ, let 𝜁^±=max{±𝜁,0}. Then for 𝑢∈𝑊₀^1,𝑝(Ω), we set 𝑢^±(⋅)=𝑢(⋅)^±. We have 𝑢^±∈𝐻¹₀(Ω), |𝑢|=𝑢⁺+𝑢⁻, and 𝑢=𝑢⁺−𝑢⁻. For a given measurable function ℎ∶Ω×ℝ→ℝ (e.g., a Carathéodory function), we set𝑁_ℎ(𝑢)(⋅)=ℎ(⋅,𝑢(⋅))∀𝑢∈𝑊₀^1,𝑝(Ω). (2.34) Finally, let 𝐴∈ℒ(𝐻¹₀(Ω),𝐻⁻¹(Ω)) be the operator, defined by 𝐴(𝑢),𝑦=_Ω∇𝑢,∇𝑦_ℝ𝑁𝑑𝑧∀𝑢,𝑦∈𝐻¹₀(Ω). (2.35) For the properties of the operator 𝐴 we refer to Gasiński and Papageorgiou [11, Proposition 3.1, page 852].

3. Solutions of Constant Sign

In this section, for 𝜆>0 small, we generate four nontrivial smooth solutions of constant sign (two positive and two negative). To this end, we introduce the following modifications of the nonlinearities 𝑔(𝑧,⋅) and 𝑓(𝑧,⋅):𝑔_±(𝑧,𝜁)=𝑔𝑧,±𝜁^±,𝑓_±(𝑧,𝜁)=𝑓𝑧,±𝜁^±+𝑐₁̂±𝜁^±, (3.1) with 𝑐₁̂>0 as in (2.14). These modifications are Carathéodory functions. We set𝐺_±(𝑧,𝜁)=^𝜁₀𝑔_±(𝑧,𝑠)𝑑𝑠,𝐹_±(𝑧,𝜁)=^𝜁₀𝑓_±(𝑧,𝑠)𝑑𝑠 (3.2) and consider the 𝐶¹-functionals 𝜑^±_𝜆∶𝐻¹₀(Ω)→ℝ, defined by𝜑^±_𝜆(𝑢)=12𝜎(𝑢)+𝑐₁̂2‖𝑢‖²₂−𝜆_Ω𝐺_±(𝑧,𝑢(𝑧))−_Ω𝐹_±(𝑧,𝑢(𝑧))𝑑𝑧∀𝑢∈𝐻¹₀(Ω). (3.3)

Proposition 3.1. If hypotheses 𝐻_𝑔, 𝐻_𝑓, and 𝐻₀ hold and 𝜆>0, then the functionals 𝜑^±_𝜆 satisfy the Cerami condition.

Proof. We do the proof for 𝜑⁺_𝜆, the proof for 𝜑⁻_𝜆 being similar.
So, let {𝑢_𝑛}_𝑛⩾1⊆𝐻¹₀(Ω) be a sequence, such that 𝜑⁺_𝜆𝑢_𝑛⩽𝑀₁∀𝑛⩾1, (3.4) for some 𝑀₁>0 and 1+𝑢_𝑛𝜑⁺_𝜆^′𝑢_𝑛⟶0in𝐻⁻¹(Ω). (3.5) From (3.5), we have 𝐴𝑢_𝑛,ℎ+_Ω𝛽+𝑐₁̂𝑢_𝑛ℎ𝑑𝑧−𝜆_Ω𝑔₊𝑧,𝑢_𝑛ℎ𝑑𝑧−_Ω𝑓₊𝑧,𝑢_𝑛⩽𝜀ℎ𝑑𝑧_𝑛‖ℎ‖1+𝑢_𝑛∀ℎ∈𝐻¹₀(Ω), (3.6) with 𝜀_𝑛↘0. In (3.6) we choose ℎ=−𝑢⁻_𝑛∈𝐻¹₀(Ω). Then 𝜎𝑢⁻_𝑛+𝑐₁̂𝑢⁻_𝑛²₂⩽𝜀_𝑛∀𝑛⩾1, (3.7) so 1𝑐₀̂𝑢⁻_𝑛²⩽𝜀_𝑛∀𝑛⩾1 (3.8) (see (2.14)), and hence 𝑢⁻_𝑛⟶0in𝐻¹₀(Ω). (3.9) From (3.4) and (3.9), we have 𝜎𝑢⁺_𝑛−𝜆_Ω2𝐺₊𝑧,𝑢⁺_𝑛𝑑𝑧−_Ω2𝐹𝑧,𝑢⁺_𝑛𝑑𝑧⩽𝑀₂∀𝑛⩾1, (3.10) for some 𝑀₂>0. Also, if in (3.6) we choose ℎ=𝑢⁺_𝑛∈𝐻¹₀(Ω), then −𝜎𝑢⁺_𝑛+𝜆_Ω𝑔𝑧,𝑢⁺_𝑛𝑢⁺_𝑛𝑑𝑧+_Ω𝑓𝑧,𝑢⁺_𝑛𝑢⁺_𝑛𝑑𝑧⩽𝜀_𝑛∀𝑛⩾1. (3.11) Adding (3.10) and (3.11), we obtain _Ω𝜉_𝜆𝑧,𝑢⁺_𝑛𝑑𝑧≤𝑀₃∀𝑛⩾1, (3.12) for some 𝑀₃>0 (see (2.29) for the definition of 𝜉_𝜆).
Claim 1. The sequence {𝑢⁺_𝑛}_𝑛⩾1⊆𝐻¹₀(Ω) is bounded.
Arguing by contradiction, suppose that the claim is not true. Then by passing to a subsequence if necessary, we may assume that 𝑢⁺_𝑛⟶+∞. (3.13) Let 𝑦_𝑛=𝑢⁺_𝑛𝑢⁺_𝑛∀𝑛⩾1. (3.14) Then 𝑦_𝑛=1∀𝑛⩾1. (3.15) And so, passing to a subsequence if necessary, we may assume that 𝑦_𝑛⟶𝑦weaklyin𝐻¹₀(Ω),𝑦_𝑛⟶𝑦in𝐿^𝑟(Ω). (3.16)(3.17) If 𝑦≠0, then 𝑢⁺_𝑛(𝑧)⟶+∞foralmostall𝑧∈Ω₊=𝑦>0 (3.18) (recall that 𝑦⩾0). So, by virtue of hypotheses 𝐻_𝑔(ii) and 𝐻_𝑓(ii), for almost all 𝑧∈Ω₊, we have lim_𝑛→+∞𝐺𝑧,𝑢⁺_𝑛(𝑧)𝑢⁺_𝑛²=lim_𝑛→+∞𝐺𝑧,𝑢⁺_𝑛(𝑧)𝑢⁺_𝑛(𝑧)²𝑦_𝑛(𝑧)²=0,lim_𝑛→+∞𝐹𝑧,𝑢⁺_𝑛(𝑧)𝑢⁺_𝑛²=lim_𝑛→+∞𝐹𝑧,𝑢⁺_𝑛(𝑧)𝑢⁺_𝑛(𝑧)²𝑦_𝑛(𝑧)²=+∞. (3.19) Then Fatou's lemma implies that lim_𝑛→+∞𝜆_Ω𝐺𝑧,𝑢⁺_𝑛𝑢⁺_𝑛²𝑑𝑧+_Ω𝐹𝑧,𝑢⁺_𝑛𝑢⁺_𝑛²𝑑𝑧=+∞. (3.20) But from (3.4) and (3.9), we have 𝜆_Ω𝐺𝑧,𝑢⁺_𝑛𝑑𝑧+_Ω𝐹𝑧,𝑢⁺_𝑛𝑑𝑧⩽𝑀₄+𝜎𝑢⁺_𝑛∀𝑛⩾1, (3.21) for some 𝑀₄>0, so 𝜆_Ω𝐺𝑧,𝑢⁺_𝑛𝑢⁺_𝑛²𝑑𝑧+_Ω𝐹𝑧,𝑢⁺_𝑛𝑢⁺_𝑛²𝑑𝑧⩽𝑀₅∀𝑛⩾1, (3.22) for some 𝑀₅>0 (since the sequence {𝜎(𝑦_𝑛)}_𝑛⩾1 is bounded in ℝ). Comparing (3.20) and (3.22), we reach a contradiction.
So, we have 𝑦=0. We fix 𝜇>0 and set 𝑣_𝑛=2𝜇^1/2𝑦_𝑛∀𝑛⩾1. (3.23) Evidently 𝑣_𝑛⟶0in𝐿^𝑟(Ω) (3.24) (see (3.17)). Hence by Krasnoselskii's theorem (see Gasiński and Papageorgiou [12, Proposition 1.4.14, page 87] and hypotheses 𝐻_𝑔(i) and 𝐻_𝑓(i)), we have _Ω𝐺𝑧,𝑣_𝑛(𝑧)𝑑𝑧⟶0,_Ω𝐹𝑧,𝑣_𝑛(𝑧)𝑑𝑧⟶0. (3.25) Since ‖𝑢⁺_𝑛‖→+∞, we can find 𝑛₀⩾1, such that 0<2𝜇^1/21𝑢⁺_𝑛<1∀𝑛⩾𝑛₀. (3.26) Let 𝑡_𝑛∈[0,1] be such that 𝜑⁺_𝜆𝑡_𝑛𝑢⁺_𝑛=max_0⩽𝑡⩽1𝜑⁺_𝜆𝑡𝑢⁺_𝑛. (3.27) By virtue of (3.26), we have 𝜑⁺_𝜆𝑡_𝑛𝑢⁺_𝑛⩾𝜑⁺_𝜆𝑣_𝑛=2𝜇𝜎𝑦_𝑛−𝜆_Ω𝐺𝑧,𝑣_𝑛𝑑𝑧−_Ω𝐹𝑧,𝑣_𝑛𝑑𝑧=2𝜇+2𝜇_Ω𝛽𝑦²_𝑛𝑑𝑧−𝜆_Ω𝐺𝑧,𝑣_𝑛𝑑𝑧−_Ω𝐹𝑧,𝑣_𝑛𝑑𝑧. (3.28) Note that _Ω𝛽𝑦²_𝑛𝑑𝑧⟶0. (3.29) This fact together with (3.25) and (3.28) implies that 𝜑⁺_𝜆𝑡_𝑛𝑢⁺_𝑛⩾𝜇∀𝑛⩾𝑛₁, (3.30) for some 𝑛₁⩾𝑛₀. Since 𝜇>0 is arbitrary, we infer that 𝜑⁺_𝜆𝑡_𝑛𝑢⁺_𝑛⟶+∞. (3.31) Note that 𝜑⁺_𝜆(0)=0,𝜑⁺_𝜆𝑢⁺_𝑛⩽𝑀₆∀𝑛⩾1, (3.32) for some 𝑀₆>0 (see (3.4) and (3.9)).
Hence (3.31) implies that there exists 𝑛₂⩾𝑛₁, such that 𝑡_𝑛∈(0,1)∀𝑛⩾𝑛₂. (3.33) And so from the choice of 𝑡_𝑛, we have 𝑑𝜑𝑑𝑡^+𝜆(𝑡𝑢⁺_𝑛)_{𝑡=𝑡𝑛}=0∀𝑛⩾𝑛₂, (3.34) so 𝜑⁺_𝜆^′𝑡_𝑛𝑢_𝑛,𝑢_𝑛=0∀𝑛⩾𝑛₂, (3.35) thus 𝜑⁺_𝜆^′𝑡_𝑛𝑢_𝑛,𝑡_𝑛𝑢_𝑛=0∀𝑛⩾𝑛₂, (3.36) and hence 𝜎𝑡_𝑛𝑢⁺_𝑛=𝜆_Ω𝑔𝑧,𝑡_𝑛𝑢⁺_𝑛𝑡_𝑛𝑢⁺_𝑛𝑑𝑧+_Ω𝑓𝑧,𝑡_𝑛𝑢⁺_𝑛𝑡_𝑛𝑢⁺_𝑛𝑑𝑧. (3.37) Hypothesis 𝐻₀ implies that _Ω𝜉_𝜆𝑧,𝑡_𝑛𝑢⁺_𝑛𝑑𝑧⩽_Ω𝜉_𝜆𝑧,𝑢⁺_𝑛𝑑𝑧+𝜗^∗_𝜆1∀𝑛⩾1, (3.38) so 2𝜑⁺_𝜆𝑡_𝑛𝑢⁺_𝑛⩽_Ω𝜉_𝜆𝑧,𝑢⁺_𝑛𝑑𝑧+𝜗^∗_𝜆1∀𝑛⩾1 (3.39) (see (3.37)), and thus 2𝜑⁺_𝜆𝑡_𝑛𝑢⁺_𝑛⩽𝑀₃+𝜗^∗_𝜆1∀𝑛⩾1 (3.40) (see (3.12)).
Comparing (3.31) and (3.40), we reach a contradiction. This proves the claim.
By virtue of the claim and (3.9), we have that the sequence {𝑢_𝑛}_𝑛⩾1⊆𝐻¹₀(Ω) is bounded. So, we may assume that 𝑢_𝑛⟶𝑢weaklyin𝐻¹₀(Ω),𝑢_𝑛⟶𝑢in𝐿^𝑟(Ω). (3.41)(3.42) In (3.6) we choose ℎ=𝑢_𝑛−𝑢, pass to the limit as 𝑛→+∞, and use (3.42). Then lim_𝑛→+∞𝐴𝑢_𝑛,𝑢_𝑛−𝑢=0, (3.43) so 𝑢_𝑛=∇𝑢_𝑛2⟶‖∇𝑢‖₂=‖𝑢‖ (3.44) (see Gasiński and Papageorgiou [11, Proposition 3.1, page 852]), and thus 𝑢_𝑛⟶𝑢in𝐻¹₀(Ω) (3.45) (by the Kadec-Klee property of Hilbert spaces). This proves that 𝜑⁺_𝜆 satisfies the Cerami condition.
Similarly we show that 𝜑⁻_𝜆 also satisfies the Cerami condition.

Our aim is to apply Theorem 2.1 (the mountain pass theorem) to two functionals 𝜑⁺_𝜆 and 𝜑⁻_𝜆. We have checked that both functionals satisfy the Cerami condition. So, it remains to show that they satisfy the mountain pass geometry as it is described in Theorem 2.1.

The next proposition is a crucial step in satisfying the mountain pass geometry for the two functionals 𝜑⁺_𝜆 and 𝜑⁻_𝜆.

Proposition 3.2. If hypotheses 𝐻_𝑔 and 𝐻_𝑓 hold, then there exist 𝜆^∗_±>0, such that for every 𝜆∈(0,𝜆^∗_±), we can find 𝜚^±_𝜆>0, such that inf𝜑^±_𝜆(𝑢)∶‖𝑢‖=𝜚^±_𝜆=𝜂^±_𝜆>0. (3.46)

Proof. Hypotheses 𝐻_𝑔(i) and (ii) imply that for a given 𝜀>0, we can find 𝑐₄=𝑐₄(𝜀)>0, such that 𝑔(𝑧,𝜁)⩽𝜀𝜁+𝑐₄𝜁^𝑞−1foralmostall𝑧∈Ω,all𝜁⩾0, (3.47) so 𝐺()⩽𝜀𝑧,𝜁2𝜁²+𝑐₄𝑞𝜁^𝑞foralmostall𝑧∈Ω,all𝜁⩾0. (3.48) Similarly hypotheses 𝐻_𝑓(i) and (iii) imply that for a given 𝜀>0, we can find 𝑐₅=𝑐₅(𝜀)>0, such that 𝑓(𝑧,𝜁)⩽𝜂₀(𝑧)+𝜀𝜁+𝑐₅𝜁^𝑟−1foralmostall𝑧∈Ω,all𝜁⩾0, (3.49) so 𝐹()⩽1𝑧,𝜁2𝜂₀(𝑧)+𝜀𝜁²+𝑐₅𝑟𝜁^𝑟foralmostall𝑧∈Ω,all𝜁⩾0. (3.50) Then, for 𝑢∈𝐻¹₀(Ω), we have 𝜑⁺_𝜆(𝑢)=12𝜎(𝑢)+𝑐₁̂2‖𝑢‖²₂−𝜆_Ω𝐺₊(𝑧,𝑢)𝑑𝑧−_Ω𝐹₊()⩾1𝑧,𝑢𝑑𝑧2𝜎𝑢⁺−12_Ω𝜂₀𝑢⁺²𝜀𝑑𝑧−2𝑢⁺²₂−𝑐₅𝑟𝑢^+𝑟_𝑟−𝜆𝜀2𝑢⁺²₂−𝜆𝑐₄𝑞𝑢^+𝑞_𝑞+12𝜎(𝑢⁻)+𝑐₁̂2‖𝑢⁻‖²₂⩾12𝜀()𝑐̂−𝜆+1𝜆₁𝑢⁺²+12𝑐₀̂‖𝑢⁻‖²−𝑐₆‖𝑢‖^𝑟+𝜆‖𝑢‖^𝑞, (3.51) for some 𝑐₆>0 (see (3.48), (3.50), (2.11), (2.14), and Lemma 2.2).
Choosing 𝜀∈(0,𝜆₁𝑐/̂(𝜆+1)), we have 𝜑⁺_𝜆(𝑢)⩾𝑐₇‖𝑢‖²−𝑐₆‖𝑢‖^𝑟+𝜆‖𝑢‖^𝑞=𝑐₇−𝑐₆‖𝑢‖^𝑟−2+𝜆‖𝑢‖^𝑞−2‖𝑢‖², (3.52) for some 𝑐₇>0. Let 𝜇_𝜆(𝑡)=𝑡^𝑟−2+𝜆𝑡^𝑞−2∀𝑡>0. (3.53) Since 𝑞<2<𝑟, we have lim_𝑡→0+𝜇_𝜆(𝑡)=lim_𝑟→+∞𝜇_𝜆(𝑡)=+∞. (3.54) Also 𝜇_𝜆 is continuous in (0,+∞). Therefore, we can find 𝑡₀∈(0,+∞), such that 𝜇_𝜆𝑡₀=inf_𝑡>0𝜇_𝜆(𝑡), (3.55) so 𝜇^′_𝜆𝑡₀=0, (3.56) and thus 𝑡₀=𝑡₀(𝜆)=𝜆2−𝑞𝑟−2^{1/(𝑟−𝑞)}. (3.57) Evidently 𝜇_𝜆𝑡₀⟶0as𝜆⟶0⁺. (3.58) Hence we can find 𝜆^∗₊>0, such that 𝜇_𝜆𝑡₀<𝑐₇𝑐₆∀𝜆∈0,𝜆^∗₊, (3.59) so inf𝜑⁺_𝜆(𝑢)∶‖𝑢‖=𝜚⁺_𝜆=𝑡₀(𝜆)=𝜂⁺_𝜆>0 (3.60) (see (3.52)).
Similarly, for 𝜑⁻_𝜆, we can find 𝜆^∗₋>0, such that for all 𝜆∈(0,𝜆^∗₋) there exists 𝜚⁻_𝜆>0, such that inf𝜑⁻_𝜆(𝑢)∶‖𝑢‖=𝜚⁻_𝜆=𝜂⁻_𝜆>0. (3.61)

With the next proposition we complete the mountain pass geometry for problem ((𝑃)_𝜆).

Proposition 3.3. If hypotheses 𝐻_𝑔 and 𝐻_𝑓 hold, 𝜆>0 and 𝑢∈int𝐶₊ with ‖𝑢‖₂=1, then 𝜑^±_𝜆(𝑡𝑢)⟶−∞as𝑡⟶±∞. (3.62)

Proof. By virtue of hypotheses 𝐻_𝑔(i) and (ii), for a given 𝜀>0, we can find 𝑐₈=𝑐₈(𝜀)>0, such that 𝐺()𝜀𝑧,𝜁⩾−2𝜁²−𝑐₈,foralmostall𝑧∈Ω,all𝜁∈ℝ. (3.63) Similarly, hypotheses 𝐻_𝑓(i) and (ii) imply that for any given 𝜉>0, we can find 𝑐₉=𝑐₉(𝜉)>0, such that 𝐹()⩾𝜉𝑧,𝜁2𝜁²−𝑐₉,foralmostall𝑧∈Ω,all𝜁∈ℝ. (3.64) Then, we have 𝜑⁺_𝜆()=𝑡𝑡𝑢²2𝜎(𝑢)+𝜆𝜀−𝜉2𝑡²+𝑐₁₀⩽𝑡²2‖𝑢‖²+𝛽_𝑠‖𝑢‖_2𝑠′+𝜆𝜀−𝜉+𝑐₁₀ (3.65) for some 𝑐₁₀>0 (see (3.63), (3.64) and recall that 𝑢∈int𝐶₊, ‖𝑢‖₂=1 and 1/𝑠+1/𝑠^′=1).
Since 𝜉>0 is arbitrary, choosing 𝜉>𝜆𝜀+‖𝑢‖²+𝛽_𝑠‖𝑢‖_2𝑠′, (3.66) from (3.65), we infer that 𝜑⁺_𝜆(𝑡𝑢)⟶−∞as𝑡⟶±∞. (3.67)

Now we are ready to produce the first two nontrivial smooth solutions of constant sign. In what follows, we set𝜆^∗=min𝜆^∗₋,𝜆^∗₊. (3.68)

Proposition 3.4. If hypotheses 𝐻_𝑔, 𝐻_𝑓, and 𝐻₀ hold and 𝜆∈(0,𝜆^∗], then problem ((𝑃)_𝜆) has at least two nontrivial smooth solutions of constant sign: 𝑢₀,𝑣₀∈𝐶¹₀Ω,with𝑣₀(𝑧)<0<𝑢₀(𝑧)∀𝑧∈Ω. (3.69)

Proof. Propositions 3.1, 3.2, and 3.3 permit the application of the mountain pass theorem (see Theorem 2.1) for the functional 𝜑⁺_𝜆, and so we obtain 𝑢₀∈𝐻¹₀(Ω), such that 𝜑⁺_𝜆(0)=0<𝜂⁺_𝜆⩽𝜑⁺_𝜆𝑢₀,𝜑⁺_𝜆^′𝑢₀=0. (3.70)(3.71)
From (3.70), we see that 𝑢₀≠0. From (3.71), we have 𝐴𝑢₀+𝛽𝑢₀+𝑐₁̂𝑢₀=𝜆𝑁_𝑔+𝑢₀𝑓+𝑁₊𝑢₀. (3.72) On (3.72) we act with −𝑢⁻₀∈𝐻¹₀(Ω) and obtain 𝜎𝑢⁻₀+𝑐₁̂𝑢⁻₀²₂=0, (3.73) so 1𝑐₀̂𝑢⁻₀²⩽0 (3.74) (see (2.14)); hence 𝑢₀⩾0, 𝑢₀≠0.
Therefore (3.72) becomes 𝐴𝑢₀+𝛽𝑢₀=𝜆𝑁_𝑔𝑢₀+𝑁_𝑓𝑢₀, (3.75) so −Δ𝑢₀(𝑧)+𝛽(𝑧)𝑢₀(𝑧)=𝜆𝑔𝑧,𝑢₀(𝑧)+𝑓𝑧,𝑢₀(𝑧)𝑢inΩ,_0𝜕Ω=0. (3.76) From the regularity theory for Dirichlet problems (see Struwe [13, pp. 217–219]), we have that 𝑢₀∈𝐶¹₀(Ω). Moreover, invoking the weak Harnack inequality of Pucci and Serrin [14, page 154], we have that 𝑢₀(𝑧)>0 for all 𝑧∈Ω.
Similarly working with 𝜑⁻_𝜆, this time we obtain a nontrivial smooth negative solution 𝑣₀∈𝐶¹₀(Ω) with 𝑣₀(𝑧)<0 for all 𝑧∈Ω.

We can improve the conclusion of this proposition by strengthening the condition on the potential 𝛽:𝐻_𝛽∶𝛽∈𝐿^𝑠(Ω) with 𝑠>𝑁/2 and 𝛽⁺∈𝐿^∞(Ω)₊.

Remark 3.5. So, the potential function is bounded from above but in general can be unbounded from below.

Proposition 3.6. If hypotheses 𝐻_𝑔, 𝐻_𝑓, 𝐻₀, and 𝐻_𝛽 hold and 𝜆∈(0,𝜆^∗), then problem ((𝑃)_𝜆) has at least two nontrivial smooth solutions of constant sign: 𝑢₀∈int𝐶₊,𝑣₀∈−int𝐶₊. (3.77)

Proof. From Proposition 3.4, we already have two solutions: 𝑢₀,𝑣₀∈𝐶¹₀Ω,with𝑣₀(𝑧)<0<𝑢₀(𝑧)∀𝑧∈Ω. (3.78) We have −Δ𝑢₀(𝑧)+𝛽(𝑧)𝑢₀(𝑧)=𝜆𝑔𝑧,𝑢₀(𝑧)+𝑓𝑧,𝑢₀(𝑧)⩾𝑓𝑧,𝑢₀(𝑧)foralmostall𝑧∈Ω (3.79) (see hypothesis 𝐻_𝑔(iii)). Let 𝜚=‖𝑢₀‖_∞ and let 𝛾_𝜚>0 be as postulated by hypothesis 𝐻_𝑓(iv). Then from (3.79), we have −Δ𝑢₀(𝑧)+𝛽(𝑧)+𝛾_𝜚𝑢₀(𝑧)⩾𝑓𝑧,𝑢₀(𝑧)+𝛾_𝜚𝑢₀(𝑧),foralmostall𝑧∈Ω, (3.80) so Δ𝑢₀(𝑧)⩽𝛽⁺_∞+𝛾_𝜚𝑢₀(𝑧)foralmostall𝑧∈Ω. (3.81) and thus 𝑢₀∈int𝐶₊ (see Vázquez [15] and Pucci and Serrin [14, page 120]).
Similarly for the negative solution 𝑣₀.

To continue and produce additional nontrivial smooth solutions of constant sign, we need to keep hypotheses 𝐻_𝛽.

Proposition 3.7. If hypotheses 𝐻_𝑔, 𝐻_𝑓, 𝐻₀ and 𝐻_𝛽 hold and 𝜆∈(0,𝜆^∗), then problem ((𝑃)_𝜆) has at least four nontrivial smooth solutions of constant sign: 𝑢₀,𝑢∈int𝐶₊,𝑢−𝑢₀∈int𝐶₊,𝑣₀,𝑣∈−int𝐶₊,𝑣₀−𝑣∈int𝐶₊. (3.82)

Proof. From Proposition 3.6, we already have two solutions: 𝑢₀∈int𝐶₊,𝑣₀∈−int𝐶₊. (3.83) We introduce the following truncation perturbation of the reaction of the problem ((𝑃)_𝜆): ℎ⁺_𝜆(𝑧,𝜁)=𝜆𝑔𝑧,𝑢₀(𝑧)+𝑓𝑧,𝑢₀(𝑧)+𝑐₁̂𝑢₀(𝑧)if𝜁⩽𝑢₀(𝑧),𝜆𝑔(𝑧,𝜁)+𝑓(𝑧,𝜁)+𝑐₁̂𝜁if𝑢₀(𝑧)<𝜁. (3.84) This is a Carathéodory function. We set 𝐻⁺_𝜆(𝑧,𝜁)=^𝜁₀ℎ⁺_𝜆(𝑧,𝑠)𝑑𝑠 (3.85) and consider the 𝐶¹-functional 𝜓⁺_𝜆∶𝐻¹₀(Ω)→ℝ, defined by 𝜓⁺_𝜆(𝑢)=12𝜎(𝑢)+𝑐₁̂2‖𝑢‖²₂−_Ω𝐻⁺_𝜆(𝑧,𝑢(𝑧))𝑑𝑧∀𝑢∈𝐻¹₀(Ω). (3.86)Claim 2. We have 𝐾_𝜓⁺_𝜆⊆[𝑢₀), where 𝑢₀=𝑢∈𝐻¹₀(Ω)∶𝑢₀(𝑧)⩽𝑢(𝑧)foralmostall𝑧∈Ω. (3.87)
Let 𝑢∈𝐾_𝜓⁺_𝜆. Then𝐴(𝑢)+𝛽+𝑐₁̂𝑢=𝑁_ℎ⁺_𝜆(𝑢). (3.88) On (3.88) we act with (𝑢₀−𝑢)⁺∈𝐻¹₀(Ω). Then 𝐴(𝑢),𝑢₀−𝑢⁺+_Ω𝛽+𝑐₁̂𝑢𝑢₀−𝑢⁺𝑑𝑧=_Ωℎ⁺_𝜆(𝑧,𝑢)𝑢₀−𝑢⁺=𝑑𝑧_Ω𝜆𝑔𝑧,𝑢₀+𝑓𝑧,𝑢₀+𝑐₁̂𝑢₀𝑢₀−𝑢⁺𝑢𝑑𝑧=⟨𝐴₀,𝑢₀−𝑢⁺⟩+_Ω𝛽+𝑐₁̂𝑢₀𝑢₀−𝑢⁺𝑑𝑧 (3.89) (see (3.84)), so 𝐴𝑢₀−𝑢,𝑢₀−𝑢⁺+_Ω𝛽𝑢₀−𝑢𝑢₀−𝑢⁺𝑑𝑧+c₁̂(𝑢₀−𝑢)⁺²₂=0, (3.90) thus 𝜎𝑢₀−𝑢⁺+𝑐₁̂𝑢₀−𝑢⁺²₂=0, (3.91) hence 1𝑐₀̂𝑢₀−𝑢⁺²⩽0 (3.92) (see (2.14)), and so finally 𝑢₀⩽𝑢. This proves Claim 2.Claim 3. We may assume that 𝑢₀ is a local minimizer of 𝜓⁺_𝜆.
Let 𝜇∈(𝜆,𝜆^∗), and consider problem (𝑃)_𝜇. As we did in the proof of Proposition 3.4, via the mountain pass theorem, we obtain a nontrivial smooth positive solution 𝑢_𝜇, and by virtue of the strong maximum principle, we have 𝑢_𝜇∈int𝐶₊ (see the proof of Proposition 3.7). Then −Δ𝑢_𝜇(𝑧)+𝛽(𝑧)𝑢_𝜇(𝑧)=𝜇𝑔𝑧,𝑢_𝜇(𝑧)+𝑓𝑧,𝑢_𝜇(𝑧)⩾𝜆𝑔𝑧,𝑢_𝜇(𝑧)+𝑓𝑧,𝑢_𝜇(𝑧)foralmostall𝑧∈Ω (3.93) (see 𝐻_𝑔(iii) and recall that 𝜆<𝜇).
We consider the following truncation perturbation of the reaction of problem ((𝑃)_𝜆): 𝛾⁺_𝜆(𝑧,𝜁)=()()0if𝜁<0,𝜆𝑔𝑧,𝜁+𝑓𝑧,𝜁+𝑐₁̂𝜁if0⩽𝜁⩽𝑢_𝜇(𝑧),𝜆𝑔𝑧,𝑢_𝜇(𝑧)+𝑓𝑧,𝑢_𝜇(𝑧)+𝑐₁̂𝑢_𝜇(𝑧)if𝑢_𝜇(𝑧)<𝜁. (3.94) This is a Carathéodory function. We set Γ⁺_𝜆(𝑧,𝜁)=^𝜁₀𝛾⁺_𝜆(𝑧,𝑠)𝑑𝑠 (3.95) and consider the 𝐶¹-functional 𝜓⁺_𝜆∶𝐻¹₀(Ω)→ℝ, defined by 𝜓⁺_𝜆(𝑢)=12𝜎(𝑢)+𝑐₁̂2‖𝑢‖²₂−_ΩΓ⁺_𝜆(𝑧,𝑢(𝑧))𝑑𝑧∀𝑢∈𝐻¹₀(Ω). (3.96) From (3.94), it is clear that 𝜓⁺_𝜆 is coercive. Also, using the Sobolev embedding theorem, we check that 𝜓⁺_𝜆 is sequentially weakly lower semicontinuous. So, by the Weierstrass theorem, we can find 𝑢_𝜆∈𝐻¹₀(Ω), such that 𝜓⁺_𝜆𝑢_𝜆=inf_{𝑢∈𝐻10(Ω)}𝜓⁺_𝜆(𝑢). (3.97) By virtue of hypothesis 𝐻_𝑓(iii), we can find 𝑐₁₁>0 and 𝛿₀>0, such that 𝐹()𝑐𝑧,𝜁⩾−¹¹2𝜁²foralmostall𝑧∈Ω,all𝜁⩽𝛿₀. (3.98) Let 𝑢∈int𝐶₊ and let 𝑡∈(0,1) be small, such that 𝑡𝑢⩽𝑢_𝜇,𝑡𝑢(𝑧)⩽𝛿₀∀𝑧∈Ω (3.99) (recall that 𝑢_𝜇∈int𝐶₊). Then, we have 𝜓⁺_𝜆()=𝑡𝑡𝑢²2𝜎()+𝑡𝑢²𝑐₁̂2‖𝑢⁻‖²₂−𝜆_Ω𝐺(𝑧,𝑡𝑢)𝑑𝑧−_Ω𝐹()⩽𝑡𝑧,𝑡𝑢𝑑𝑧₂2𝜎(𝑢)+𝑐₁̂+𝑐₁₁‖𝑢‖²₂−𝜆𝑡^𝑞𝑐₁𝑞‖𝑢‖^𝑞_𝑞 (3.100) (see (3.94), hypothesis 𝐻_𝑔(iii), and (3.98)). Since 𝑞<2, by choosing 𝑡∈(0,1) even smaller if necessary, we have 𝜓⁺_𝜆(𝑡𝑢)<0, (3.101) so 𝜓⁺_𝜆𝑢_𝜆<0=𝜓⁺_𝜆(0) (3.102) (see (3.97)); hence 𝑢_𝜆≠0.
From (3.97), we have 𝜓⁺_𝜆^′𝑢_𝜆=0, (3.103) so 𝐴𝑢_𝜆+𝛽+𝑐₁̂𝑢_𝜆=𝑁_𝛾⁺_𝜆𝑢_𝜆. (3.104) Acting on (3.104) with −𝑢⁻_𝜆∈𝐻¹₀(Ω), we obtain that 𝑢_𝜆⩾0, 𝑢_𝜆≠0. Also, acting on (3.104) with (𝑢_𝜆−𝑢_𝜇)⁺∈𝐻¹₀(Ω), we have ⟨𝐴𝑢_𝜆,𝑢_𝜆−𝑢_𝜇⁺⟩+_Ω𝛽+𝑐₁̂𝑢_𝜆𝑢_𝜆−𝑢_𝜇⁺𝑑𝑧=_Ω𝛾⁺_𝜆𝑧,𝑢_𝜆𝑢_𝜆−𝑢_𝜇⁺=𝑑𝑧_Ω𝜆𝑔𝑧,𝑢_𝜇+𝑓𝑧,𝑢_𝜇+𝑐₁̂𝑢_𝜇𝑢_𝜆−𝑢_𝜇⁺⩽𝐴𝑢𝑑𝑧_𝜇,𝑢_𝜆−𝑢_𝜇⁺+_Ω𝛽+𝑐₁̂𝑢_𝜇𝑢_𝜆−𝑢_𝜇⁺𝑑𝑧 (3.105) (see (3.94) and (3.93)), so ⟨𝐴𝑢_𝜆−𝑢_𝜇,𝑢_𝜆−𝑢_𝜇⁺⟩+_Ω𝛽+𝑐₁̂𝑢_𝜆−𝑢_𝜇(𝑢_𝜆−𝑢_𝜇)⁺𝑑𝑧⩽0, (3.106) thus 𝜎𝑢_𝜆−𝑢_𝜇⁺+𝑐₁̂𝑢_𝜆−𝑢_𝜇⁺²₂⩽0, (3.107) hence 1𝑐₀̂𝑢_𝜆−𝑢_𝜇⁺²⩽0 (3.108) (see (2.14)) and so finally 𝑢_𝜆⩽𝑢_𝜇. (3.109) So, we have proved that 𝑢_𝜆∈0,𝑢_𝜇=𝑢∈𝐻¹₀(Ω)∶0⩽𝑢(𝑧)⩽𝑢_𝜇(𝑧)foralmostall𝑧∈Ω. (3.110) Hence (3.104) becomes 𝐴𝑢_𝜆+𝛽𝑢_𝜆=𝜆𝑁_𝑔𝑢_𝜆+𝑁_𝑓𝑢_𝜆 (3.111) (see (3.94)), so −Δ𝑢_𝜆(𝑧)+𝛽(𝑧)𝑢_𝜆(𝑧)=𝜆𝑔𝑧,𝑢_𝜆(𝑧)+𝑓𝑧,𝑢_𝜆(𝑧)𝑢inΩ,_𝜆𝜕Ω=0, (3.112) thus 𝑢_𝜆∈int𝐶₊ (as in the proof of Proposition 3.6) and it is a solution of ((𝑃)_𝜆).
If 𝑢_𝜆≠𝑢₀, then this is the desired second nontrivial positive smooth solution of ((𝑃)_𝜆).
So, we may assume that 𝑢_𝜆=𝑢₀ and that there is no other solution of ((𝑃)_𝜆) in the order interval 𝑢₀,𝑢_𝜆=𝑢∈𝐻¹₀(Ω)∶𝑢₀(𝑧)⩽𝑢(𝑧)⩽𝑢_𝜇(𝑧)foralmostall𝑧∈Ω. (3.113) We introduce the following truncation of 𝛾⁺_𝜆(𝑧,⋅): 𝛾⁺_𝜆()=𝛾𝑧,𝜁^+𝜆𝑧,𝑢₀(𝑧)if𝜁<𝑢₀(𝑧),𝛾⁺_𝜆(𝑧,𝜁)if𝑢₀(𝑧)⩽𝜁. (3.114) This is a Carathéodory function. We set Γ⁺_𝜆(𝑧,𝜁)=^𝜁₀𝛾⁺_𝜆(𝑧,𝑠)𝑑𝑠 (3.115) and consider the 𝐶¹-functional 𝜓⁺_𝜆∶𝐻¹₀(Ω)→ℝ, defined by 𝜓⁺_𝜆(𝑢)=12𝜎(𝑢)+𝑐₁̂2‖𝑢‖²₂−_ΩΓ⁺_𝜆(𝑧,𝑢(𝑧))𝑑𝑧∀𝑢∈𝐻¹₀(Ω). (3.116) From (3.94) and (3.114), it follows that 𝜓⁺_𝜆 is coercive. Also, it is sequentially weakly lower semicontinuous. Hence, we can find 𝑢_𝜆∈𝐻¹₀(Ω), such that 𝜓⁺_𝜆𝑢_𝜆=inf_{𝑢∈𝐻10(Ω)}𝜓⁺_𝜆(𝑢), (3.117) so 𝜓⁺_𝜆^′𝑢_𝜆=0, (3.118) and thus 𝐴𝑢_𝜆+𝛽+𝑐₁̂𝑢_𝜆=𝑁𝛾⁺_𝜆𝑢_𝜆. (3.119) As before, acting on (3.119) with (𝑢₀−𝑢_𝜆)⁺∈𝐻¹₀(Ω) and with (𝑢_𝜆−𝑢_𝜇)⁺∈𝐻¹₀(Ω), we show that 𝑢_𝜆∈[𝑢₀,𝑢_𝜇]. Then, from (3.94) and (3.114), it follows that 𝐴𝑢_𝜆+𝛽𝑢_𝜆=𝜆𝑁_𝑔𝑢_𝜆+𝑁_𝑓𝑢_𝜆, (3.120) so 𝑢_𝜆∈int𝐶₊ is a solution of ((𝑃)_𝜆) in [𝑢₀,𝑢_𝜇], hence 𝑢_𝜆=𝑢₀.
Let 𝜚=‖𝑢₀‖_∞ and let 𝛾_𝜚>0 and 𝛾_𝜚>0 be as postulated by hypotheses 𝐻_𝑔(iv) and 𝐻_𝑓(iv), respectively. We have −Δ𝑢₀(𝑧)+𝛽(𝑧)+𝛾_𝜚𝑢₀(𝑧)+𝜆𝛾_𝜚𝑢₀(𝑧)^𝑞−1=𝜆𝑔𝑧,𝑢₀(𝑧)+𝜆𝛾_𝜚𝑢₀(𝑧)^𝑞−1+𝑓𝑧,𝑢₀(𝑧)+𝛾_𝜚𝑢₀(𝑧)⩽𝜆𝑔𝑧,𝑢_𝜇(𝑧)+𝜆𝛾_𝜚𝑢_𝜇(𝑧)^𝑞−1+𝑓𝑧,𝑢_𝜇(𝑧)+𝛾_𝜚𝑢_𝜇(𝑧)⩽𝜇𝑔𝑧,𝑢_𝜇(𝑧)+𝜇𝛾_𝜚𝑢_𝜇(𝑧)^𝑞−1+𝑓𝑧,𝑢_𝜇(𝑧)+𝛾_𝜚𝑢_𝜇(𝑧)=−Δ𝑢_𝜇(𝑧)+𝛽(𝑧)+𝛾_𝜚𝑢_𝜇(𝑧)+𝜇𝛾_𝜚𝑢_𝜇(𝑧)^𝑞−1foralmostall𝑧∈Ω (3.121) (see 𝐻_𝑔(iv) and (iii), 𝐻_𝑓(iv) and recall that 𝑢₀⩽𝑢_𝜇 and 𝜆<𝜇), so there exists 𝑐₁₂>0, such that Δ𝑢_𝜇−𝑢₀(𝑧)⩽𝛽⁺_∞+𝛾_𝜚𝑢_𝜇−𝑢₀(𝑧)+𝜇𝛾_𝜚𝑐₁₂𝑢_𝜇−𝑢₀(𝑧), (3.122) for almost all 𝑧∈Ω (recall that the function 𝜁↦𝜁^𝑞−1 is locally Lipschitz); thus 𝑢_𝜇−𝑢₀∈int𝐶₊ (3.123) (see Struwe [13] and Pucci and Serrin [14, page 120]). So, we have that 𝑢_𝜆=𝑢₀∈int_𝐶10(Ω)0,𝑢_𝜇. (3.124) Note that 𝜓⁺_𝜆|_[0,𝑢𝜇]=𝜓⁺_𝜆|_[0,𝑢𝜇] (3.125) (see (3.94) and (3.114)), so 𝑢₀isalocalC¹₀Ω-minimizerof𝜓⁺_𝜆 (3.126) (see (3.124)), and thus 𝑢₀isalocal𝐻¹₀(Ω)-minimizerof𝜓⁺_𝜆 (3.127) (Brézis and Nirenberg [16]). This proves Claim 3.
By virtue of Claim 3, as in Gasiński and Papageorgiou [17, proof of Theorem 3.4], we can find 𝜚_𝜆∈(0,1) small, such that 𝜓⁺_𝜆𝑢₀<inf𝜓⁺_𝜆(𝑢)∶𝑢−𝑢₀=𝜚_𝜆=𝜂⁺_𝜆. (3.128) As in Proposition 3.4, for 𝑢∈int𝐶₊ with ‖𝑢‖₂=1, we have 𝜓⁺_𝜆(𝑡𝑢)⟶−∞as𝑡⟶+∞. (3.129) Note that 𝜑⁺_𝜆=𝜓⁺_𝜆−𝜉⁺_𝜆 with 𝜉⁺_𝜆∈ℝ. Hence by virtue of Proposition 3.1, 𝜓⁺_𝜆 satisfies the Cerami condition. This fact together with (3.128) and (3.129) permits the use of the mountain pass theorem (see Theorem 2.1). So, we can find 𝑢∈𝐻¹₀(Ω), such that 𝜓⁺_𝜆𝑢₀<𝜂⁺_𝜆⩽𝜓⁺_𝜆()𝜓𝑢,_+𝜆′(𝑢)=0. (3.130)(3.131) From (3.130) we have 𝑢≠𝑢₀. From (3.131) and Claim 2, we have that 𝑢₀⩽𝑢. (3.132) Hence A(𝑢)+𝛽𝑢=𝜆𝑁_𝑔(𝑢)+𝑁_𝑓(𝑢) (3.133) (see (3.84)), and so 𝑢∈int𝐶₊ solves problem ((𝑃)_𝜆).
Moreover, as before, using the strong maximum principle (see Vázquez [15] and Pucci and Serrin [14, page 120]), we have𝑢−𝑢₀∈int𝐶₊. (3.134)
In a similar way, using 𝑣₀∈−int𝐶₊, we defineℎ⁻_𝜆()=()()𝑧,𝜁𝜆𝑔𝑧,𝜁+𝑓𝑧,𝜁+𝑐₁̂𝜁if𝜁<𝑣₀(𝑧),𝜆𝑔𝑧,𝑣₀(𝑧)+𝑓𝑧,𝑣₀(𝑧)+𝑐₁̂𝑣₀(𝑧)if𝑣₀(𝑧)⩽𝜁. (3.135) This is a Carathéodory function. We set 𝐻⁻_𝜆(𝑧,𝜁)=^𝜁₀ℎ⁻_𝜆(𝑧,𝑠)𝑑𝑠 (3.136) and consider the 𝐶¹-functional 𝜓⁻_𝜆∶𝐻¹₀(Ω)→ℝ, defined by 𝜓⁻_𝜆(𝑢)=12𝜎(𝑢)+𝑐₁̂2‖𝑢‖²₂−_Ω𝐻⁻_𝜆(𝑧,𝑢(𝑧))𝑑𝑧∀𝑢∈𝐻¹₀(Ω). (3.137) Reasoning as above, using this time 𝜓⁻_𝜆, we obtain a second negative smooth solution 𝑣∈−int𝐶₊ of problem ((𝑃)_𝜆), such that 𝑣₀−𝑣∈int𝐶₊. (3.138)

4. Five Solutions

In this section, we prove two multiplicity theorems, establishing five nontrivial smooth solutions when 𝜆∈(0,𝜆^∗). In the second multiplicity theorem, we provide sign information for all the solutions (i.e., we show that the fifth solution is actually nodal).

Theorem 4.1. If hypotheses 𝐻_𝑔,𝐻_𝑓,𝐻₀, and 𝐻_𝛽 hold and 𝜆∈(0,𝜆^∗), then problem ((𝑃)_𝜆) has at least five nontrivial smooth solutions: 𝑢₀,𝑢∈int𝐶₊,𝑢−𝑢₀∈int𝐶₊,𝑣₀,𝑣∈−int𝐶₊,𝑣₀−𝑣∈int𝐶₊𝑦₀∈𝐶¹₀Ω,𝑢₀−𝑦₀∈int𝐶₊,𝑦₀−𝑣₀∈int𝐶₊. (4.1)

Proof. From Proposition 3.7, we already have four nontrivial smooth solutions of constant sign: 𝑢₀,𝑢∈int𝐶₊,𝑢−𝑢₀∈int𝐶₊,𝑣₀,𝑣∈−int𝐶₊,𝑣₀−𝑣∈int𝐶₊. (4.2) We consider the following truncation perturbation of the reaction of problem ((𝑃)_𝜆): ℎ^∗_𝜆(𝑧,𝜁)=𝜆𝑔𝑧,𝑣₀(𝑧)+𝑓𝑧,𝑣₀(𝑧)+𝑐₁̂𝑣₀(𝑧)if𝜁<𝑣₀(𝑧),𝜆𝑔(𝑧,𝜁)+𝑓(𝑧,𝜁)if𝑣₀(𝑧)⩽𝜁⩽𝑢₀(𝑧),𝜆𝑔𝑧,𝑢₀(𝑧)+𝑓𝑧,𝑢₀(𝑧)+𝑐₁̂𝑢₀(𝑧)if𝑢₀(𝑧)<𝜁. (4.3) This is a Carathéodory function. We set 𝐻^∗_𝜆(𝑧,𝜁)=^𝜁₀ℎ^∗_𝜆(𝑧,𝑠)𝑑𝑠 (4.4) and consider the 𝐶¹-functional 𝜓^∗_𝜆∶𝐻¹₀(Ω)→ℝ, defined by 𝜓^∗_𝜆(𝑢)=12𝜎(𝑢)+𝑐₁̂2‖𝑢‖²₂−_Ω𝐻^∗_𝜆(𝑧,𝑢(𝑧))𝑑𝑧∀𝑢∈𝐻¹₀(Ω). (4.5) From (4.3), it follows that 𝜓^∗_𝜆 is coercive. Also, it is sequentially weakly lower semicontinuous. So, we can find 𝑦₀∈𝐻¹₀(Ω), such that 𝜓^∗_𝜆𝑦₀=inf_{𝑢∈𝐻10(Ω)}𝜓^∗_𝜆(𝑢). (4.6) As in the proof of Proposition 3.7, using (3.98), we show that 𝜓^∗_𝜆𝑦₀<0=𝜓^∗_𝜆(0), (4.7) hence 𝑦₀≠0. From (4.6), we have 𝜓^∗_𝜆^′𝑦₀=0, (4.8) so 𝐴𝑦₀+𝛽+𝑐₁̂𝑦₀=𝑁_ℎ^∗_𝜆𝑦₀. (4.9) On (4.9) we act with (𝑦₀−𝑢₀)⁺∈𝐻¹₀(Ω). Then 𝐴𝑦₀,𝑦₀−𝑢₀⁺+_Ω𝛽+𝑐₁̂𝑦₀𝑦₀−𝑢₀⁺𝑑𝑧=_Ω𝜆𝑔𝑧,𝑢₀+𝑓𝑧,𝑢₀+𝑐₁̂𝑢₀𝑦₀−𝑢₀⁺=𝐴𝑢𝑑𝑧₀,𝑦₀−𝑢₀⁺+_Ω𝛽+𝑐̂𝑢₀𝑦₀−𝑢₀⁺𝑑𝑧 (4.10) (see (4.3)), so 𝐴𝑦₀−𝑢₀,𝑦₀−𝑢₀⁺+_Ω𝛽+𝑐₁̂𝑦₀−𝑢₀𝑦₀−𝑢₀⁺𝑑𝑧=0, (4.11) thus 𝜎𝑦₀−𝑢₀⁺+𝑐₁̂𝑦₀−𝑢₀⁺²₂=0, (4.12) and hence 1𝑐₀̂𝑦₀−𝑢₀⁺²⩽0 (4.13) (see (2.14)); hence 𝑦₀⩽𝑢₀.
Similarly, acting on (4.9) with (𝑣₀−𝑦₀)⁺∈𝐻¹₀(Ω), we show that 𝑣₀⩽𝑦₀. Therefore, 𝑦₀∈𝑣₀,𝑢₀=𝑢∈𝐻¹₀(Ω)∶𝑣₀(𝑧)⩽𝑢(𝑧)⩽𝑢₀(𝑧)foralmostall𝑧∈Ω, (4.14) and so (4.9) becomes 𝐴𝑦₀+𝛽𝑦₀=𝜆𝑁_𝑔𝑦₀+𝑁_𝑓𝑦₀ (4.15) (see (4.3)); thus 𝑦₀∈𝐶¹₀Ω (4.16) (regularity theory; see Struwe [13]), and it solves problem ((𝑃)_𝜆).
Moreover, as in the proof of Proposition 3.7, using hypotheses 𝐻_𝑔(iv) and 𝐻_𝑓(iv), we also show that 𝑢₀−𝑦₀∈int𝐶₊,𝑦₀−𝑣₀∈int𝐶₊. (4.17)

Next we will improve the conclusion of Theorem 4.1 and show that the fifth solution 𝑦₀ is nodal (sign changing). To do this we need to strengthen a little bit the hypotheses on 𝑓(𝑧,⋅).

The new hypotheses of 𝑓 are the following:𝐻^′_𝑓:𝑓∶Ω×ℝ→ℝ is a Carathéodory function, such that 𝑓(𝑧,0)=0 for almost all 𝑧∈Ω,(i), (iii), (iv) are the same as the corresponding hypotheses 𝐻_𝑓(i), (iii), (iv),(ii) we have lim_𝜁→±∞𝑓(𝑧,𝜁)𝜁=+∞uniformlyforalmostall𝑧∈Ω. (4.18)

Remark 4.2. Hypothesis 𝐻^′_𝑓 is a slight restricted version of hypothesis 𝐻_𝑓. Note that if 𝜉̂(𝑧,𝜁)=𝑓(𝑧,𝜁)𝜁−2𝐹(𝑧,𝜁) (4.19) and there exists a function 𝜗^∗∈𝐿¹(Ω)₊, such that 𝜉̂(𝑧,𝜁)⩽𝜉̂𝑧,𝑦+𝜗^∗(𝑧)foralmostall𝑧∈Ω,all0⩽𝜁⩽𝑦or𝑦⩽𝜁⩽0, (4.20) then (4.20) and 𝐻_𝑓 (ii) imply 𝐻^′_𝑓(ii) (see Li and Yang [10]). Also, note that hypotheses 𝐻_𝑓(i), (ii), and (iii) imply that there exists 𝑐_∗>0, such that 𝑓(𝑧,𝜁)𝜁+𝑐_∗𝜁²⩾0foralmostall𝑧∈Ω,all𝜁∈ℝ. (4.21)

We consider the following auxiliary Dirichlet problem: −Δ𝑢(𝑧)+𝛽(𝑧)𝑢(𝑧)=𝜆𝑐₁|𝑢(𝑧)|^𝑞−2𝑢(𝑧)−𝑐_∗𝑢(𝑧)𝑢|inΩ,_𝜕Ω( (𝑄)=0.^𝜆) Here 𝑐₁>0 and 𝑞∈(1,2) are as in hypothesis 𝐻_𝑔(iii) and 𝑐_∗>0 is as in (4.21).

Proposition 4.3. For every 𝜆>0, problem ((𝑄)_𝜆) has a unique nontrivial positive solution 𝑢_𝜆∈int𝐶₊, and by oddness, we have that −𝑢_𝜆=𝑣_𝜆∈−int𝐶₊ is the unique negative solution of ((𝑄)_𝜆).

Proof. Let 𝑘⁺_𝜆∶Ω×ℝ→ℝ be the Carathéodory function, defined by 𝑘⁺_𝜆(𝑧,𝜁)=0if𝜁⩽0,𝜆𝑐₁𝜁^𝑞−1−𝑐₁̂−𝑐_∗𝜁if𝜁>0. (4.22) Clearly, we can always assume that 𝑐_∗⩾𝑐₁̂ (see (4.21)). Let 𝐾⁺_𝜆(𝑧,𝜁)=^𝜁₀𝑘⁺_𝜆(𝑧,𝑠)𝑑𝑠, (4.23) and consider the 𝐶¹-functional 𝜃⁺_𝜆∶𝐻¹₀(Ω)→ℝ, defined by 𝜃⁺_𝜆(𝑢)=12𝜎(𝑢)+𝑐₁̂2‖𝑢‖²₂−_Ω𝐾⁺_𝜆(𝑧,𝑢(𝑧))𝑑𝑧∀𝑢∈𝐻¹₀(Ω). (4.24) Using (2.14) and (4.22), we have 𝜃⁺_𝜆(𝑢)⩾12𝑐₀̂‖𝑢‖²+𝑐_∗−𝑐₁̂2𝑢⁺²₂−𝜆𝑐₁𝑞𝑢^+𝑞_𝑞. (4.25) Since 𝑞<2, from (4.25), we infer that 𝜃⁺_𝜆 is coercive. Also, it is sequentially weakly lower semicontinuous (recall that 𝑐_∗⩾𝑐₁̂). So, by the Weierstrass theorem, we can find 𝑢_𝜆∈𝐻¹₀(Ω), such that 𝜃⁺_𝜆𝑢_𝜆=inf_{𝑢∈𝐻10(Ω)}𝜃⁺_𝜆(𝑢). (4.26) As before (see the proof of Proposition 3.7), since 𝑞<2, we have 𝜃⁺_𝜆𝑢_𝜆<0=𝜃⁺_𝜆(0), (4.27) hence 𝑢_𝜆≠0. From (4.26), we have 𝜃⁺_𝜆^′𝑢_𝜆=0, (4.28) so 𝐴𝑢_𝜆+𝛽+𝑐₁̂𝑢_𝜆=𝑁_𝑘⁺_𝜆𝑢_𝜆. (4.29) Acting on (4.29) with −𝑢⁻_𝜆∈𝐻¹₀(Ω), we show that 𝑢_𝜆⩾0,𝑢_𝜆≠0 (see (2.14)). Then (4.29) becomes 𝐴𝑢_𝜆+𝛽+𝑐_∗𝑢_𝜆=𝜆𝑐₁𝑢_𝜆^𝑞−1, (4.30) so −Δ𝑢_𝜆(𝑧)+𝛽(𝑧)+𝑐_∗𝑢_𝜆(𝑧)⩾0foralmostall𝑧∈Ω, (4.31) thus Δ𝑢_𝜆(𝑧)⩽𝛽⁺_∞+𝑐_∗𝑢_𝜆(𝑧)foralmostall𝑧∈Ω (4.32) (see hypothesis 𝐻_𝛽), and hence 𝑢_𝜆∈int𝐶₊ (4.33) (see Vázquez [15] and Pucci and Serrin [14, page 120]).
We claim that this solution is the unique nontrivial positive solution of ((𝑄)_𝜆). To this end, let 𝑢,𝑣∈int𝐶₊ be two positive solutions of ((𝑄)_𝜆). We have _Ω𝜆𝑐₁𝑢^𝑞−1−𝑐_∗𝑢𝑢𝑢²−𝑣²=𝑑𝑧_Ω𝜆𝑐₁𝑢^𝑞−1−𝑐_∗𝑢𝑣𝑢−²𝑢=𝑑𝑧_Ω()𝑣−Δ𝑢𝑢−²𝑢𝑑𝑧+_Ω𝛽𝑢²−𝑣²=𝑑𝑧_Ω𝑣∇𝑢,∇𝑢−∇²𝑢_ℝ𝑁𝑑𝑧+_Ω𝛽𝑢²−𝑣²=‖‖𝑑𝑧∇𝑢²₂−_Ω∇𝑢,2𝑣𝑢𝑣∇𝑣−²𝑢²∇𝑢_ℝ𝑁𝑑𝑧+_Ω𝛽𝑢²−𝑣²=‖‖𝑑𝑧∇𝑢²₂−_Ω2𝑣𝑢(∇𝑢,∇𝑣)_ℝ𝑁+𝑑𝑧_Ω𝑣²𝑢²‖∇𝑢‖²𝑑𝑧+_Ω𝛽𝑢²−𝑣²𝑑𝑧. (4.34) Interchanging the roles of 𝑢 and 𝑣 in the above argument, we also have _Ω𝜆𝑐₁𝑣^𝑞−1−𝑐_∗𝑣𝑣𝑣²−𝑢²=‖‖𝑑𝑧∇𝑣²₂−_Ω2𝑢𝑣(∇𝑣,∇𝑢)_ℝ𝑁𝑑𝑧+_Ω𝑢²𝑣²‖∇𝑣‖²𝑑𝑧+_Ω𝛽𝑣²−𝑢²𝑑𝑧. (4.35) Adding (4.34) and (4.35), we obtain _Ω𝜆𝑐₁𝑢^𝑞−1−𝑐_∗𝑢𝑢−𝜆𝑐₁𝑣^𝑞−1−𝑐_∗𝑣𝑣𝑢²−𝑣²𝑢𝑑𝑧=∇𝑢−𝑣∇𝑣²₂+𝑣∇𝑣−𝑢∇𝑢²₂⩾0. (4.36) Since 𝑞<2, the function 𝜁↦(𝜆𝑐₁𝜁^𝑞−1−𝑐_∗𝜁)/𝜁 is strictly decreasing on (0,+∞). Hence, from (4.36), we infer that 𝑢=𝑣. This proves the uniqueness of the nontrivial positive solution 𝑢_𝜆∈int𝐶₊ of problem ((𝑄)_𝜆).
The oddness of problem ((𝑄)_𝜆) implies that 𝑣_𝜆=−𝑢_𝜆∈−int𝐶₊ is the unique nontrivial negative solution of ((𝑄)_𝜆).

Using Proposition 4.3, we can show that problem ((𝑃)_𝜆) (for 𝜆∈(0,𝜆^∗)) has extremal constant sign solutions; that is, it has a smallest nontrivial positive solution and a biggest nontrivial negative solution.

Proposition 4.4. If hypotheses 𝐻_𝑔, 𝐻^′_𝑓, 𝐻₀, and 𝐻_𝛽 hold and 𝜆∈(0,𝜆^∗), then problem ((𝑃)_𝜆) has a smallest nontrivial positive solution 𝑢^𝜆₊∈int𝐶₊ and a biggest nontrivial negative solution 𝑣^𝜆₋∈int−𝐶₊.

Proof. Let 𝑢 be a nontrivial positive solution of ((𝑃)_𝜆). From the proof of Proposition 3.6, we know that 𝑢∈int𝐶₊. We have −Δ𝑢(𝑧)+𝛽(𝑧)𝑢(𝑧)=𝜆𝑔(𝑧,𝑢(𝑧))+𝑓(𝑧,𝑢(𝑧))⩾𝜆𝑐₁𝑢(𝑧)^𝑞−1−𝑐_∗𝑢(𝑧), (4.37) for almost all 𝑧∈Ω (see hypothesis 𝐻_𝑔(iii) and (4.21)).
We consider the following Carathéodory function: 𝑘⁺_𝜆(𝑧,𝜁)=0if𝜁⩽0,𝜆𝑐₁𝜁^𝑞−1+𝑐₁̂−𝑐_∗𝜁if0⩽𝜁⩽𝑢,𝜆𝑐₁𝑢(𝑧)^𝑞−1+𝑐₁̂−𝑐_∗𝑢(𝑧)if𝑢<𝜁. (4.38) Let 𝐾⁺_𝜆(𝑧,𝜁)=^𝜁₀𝑘⁺_𝜆(𝑧,𝑠)𝑑𝑠, (4.39) and consider the 𝐶¹-functional 𝜉⁺_𝜆∶𝐻¹₀(Ω)→ℝ, defined by 𝜉⁺_𝜆(𝑢)=12𝜎(𝑢)+𝑐₁̂2‖𝑢‖²₂−_Ω𝐾⁺_𝜆(𝑧,𝑢(𝑧))𝑑𝑧∀𝑢∈𝐻¹₀(Ω). (4.40) From (4.38) and (2.14), it is clear that 𝜉⁺_𝜆 is coercive. Also 𝜉⁺_𝜆 is sequentially weakly lower semicontinuous. Thus we can find 𝑤₀∈𝐻¹₀(Ω), such that 𝜉⁺_𝜆𝑤₀=inf_{𝑢∈𝐻10(Ω)}𝜉⁺_𝜆(𝑢). (4.41) As before, the presence of the “concave” term 𝜆𝑐₁𝜁^𝑞−1 implies that 𝜉⁺_𝜆𝑤₀<0=𝜉⁺_𝜆(0), (4.42) that is, 𝑤₀≠0. From (4.41), we have 𝜉⁺_𝜆^′𝑤₀=0, (4.43) so 𝐴𝑤₀+𝛽+𝑐₁̂𝑤₀𝑘=𝑁_+𝜆𝑤₀. (4.44) On (4.44) we act with −𝑤⁻₀∈𝐻¹₀(Ω) and obtain 𝜎𝑤⁻₀+𝑐₁̂𝑤⁻₀²₂=0 (4.45) (see (4.38)), so 1𝑐₀̂𝑤⁻₀²⩽0 (4.46) (see (2.14)), and hence 𝑤₀⩾0, 𝑤₀≠0.
Also, on (4.44) we act with (𝑤₀−𝑢)⁺∈𝐻¹₀(Ω). Then⟨𝐴𝑤₀,𝑤₀−𝑢⁺⟩+_Ω𝛽+𝑐₁̂𝑤₀𝑤₀−𝑢⁺𝑑𝑧=_Ω𝑘⁺_𝜆𝑧,𝑤₀𝑤₀−𝑢⁺=𝑑𝑧_Ω𝜆𝑐₁𝑢^𝑞−1−𝑐_∗𝑢𝑤₀−𝑢⁺(𝑑𝑧⩽⟨𝐴𝑢),𝑤₀−𝑢⁺⟩+_Ω𝛽𝑤₀−𝑢⁺𝑑𝑧+𝑐₁̂𝑤₀−𝑢⁺²₂ (4.47) (see (4.38) and (4.37)), so 𝜎𝑤₀−𝑢⁺+𝑐₁̂𝑤₀−𝑢⁺²₂⩽0, (4.48) thus 1𝑐₀̂𝑤₀−𝑢⁺²⩽0 (4.49) (see (2.14)), and hence 𝑤₀⩽𝑢. (4.50) So, we have proved that 𝑤₀∈[0,𝑢]=𝑢∈𝐻¹₀(Ω)∶0⩽𝑢(𝑧)⩽𝑢(𝑧)foralmostall𝑧∈Ω. (4.51) This means that (4.44) becomes 𝐴𝑤₀+𝛽𝑤₀=𝜆𝑐₁𝑤₀^𝑞−1−𝑐_∗𝑤₀, (4.52) so 𝑤₀∈int𝐶₊ (regularity theory of Struwe [13] and strong maximum principle due to Vázquez [15] and Pucci Serrin [14, page 120]) and it solves problem ((𝑄)_𝜆). Thus 𝑤₀=𝑢_𝜆 (4.53) (see Proposition 4.3), and 𝑢_𝜆⩽𝑢. (4.54) This shows that every nontrivial positive solution 𝑢 of ((𝑃)_𝜆) satisfies 𝑢_𝜆⩽𝑢. (4.55) Similarly, we show that every nontrivial negative solution 𝑣 of problem ((𝑃)_𝜆) satisfies 𝑣⩽𝑣_𝜆=−𝑢_𝜆. (4.56)
Let 𝑆₊(𝜆) (resp., 𝑆₋(𝜆)) be the set of nontrivial positive (resp., negative) solutions of problem ((𝑃)_𝜆). Let 𝐶⊆𝑆₊(𝜆) be a chain (i.e., a totally ordered subset of 𝑆₊(𝜆)). From Dunford and Schwartz [18, page 336], we can find a sequence {𝑢_𝑛}_𝑛⩾1⊆𝐶, such thatinf𝐶=inf_𝑛⩾1𝑢_𝑛. (4.57) Lemma 1.5 of Heikkilä and Lakshmikantham [19, page 15] implies that we can have the sequence {𝑢_𝑛}_𝑛⩾1⊆𝐶 to be decreasing. Then we have 𝐴𝑢_𝑛+𝛽𝑢_𝑛=𝜆𝑁_𝑔𝑢_𝑛+𝑁_𝑓𝑢_𝑛,𝑢_𝜆⩽𝑢_𝑛⩽𝑢₁∀𝑛⩾1, (4.58) so thesequnece𝑢_𝑛𝑛⩾1⊆𝐻¹₀(Ω)isbounded. (4.59) Hence by passing to a suitable subsequence if necessary, we may assume that 𝑢_𝑛⟶𝑢_∗weaklyin𝐻¹₀(Ω),𝑢_𝑛⟶𝑢_∗in𝐿^2𝑠′(Ω)andin𝐿^𝑟(Ω). (4.60)(4.61) So, passing to the limit as 𝑛→+∞ in (4.58) and using (4.61), we obtain 𝐴𝑢_∗+𝛽𝑢_∗=𝜆𝑁_𝑔𝑢_∗+𝑁_𝑓𝑢_∗,𝑢_𝜆⩽𝑢_∗, (4.62) so inf𝐶=𝑢_∗∈𝑆₊(𝜆). (4.63) Since 𝐶 was an arbitrary chain, invoking the Kuratowski-Zorn lemma, we infer that 𝑆₊(𝜆) has a minimal element 𝑢^𝜆₊∈int𝐶₊. From Gasiński and Papageorgiou [17, Lemma 4.2, page 5763], we know that 𝑆₊(𝜆) is downward directed (i.e., if 𝑢₁,𝑢₂∈𝑆₊(𝜆); then we can find 𝑢∈𝑆₊(𝜆), such that 𝑢⩽𝑢₁ and 𝑢⩽𝑢₂). So, it follows that 𝑢^𝜆₊ is the smallest nontrivial positive solution of problem ((𝑃)_𝜆).
Similarly, we introduce the biggest nontrivial negative solution 𝑣^𝜆₋∈−int𝐶₊ of problem ((𝑃)_𝜆). Note that 𝑆₋(𝜆) is upward directed (i.e., if 𝑣₁,𝑣₂∈𝑆₋(𝜆), then we can find 𝑣∈𝑆₋(𝜆), such that 𝑣₁⩽𝑣 and 𝑣₂⩽𝑣; see Gasiński and Papageorgiou [17, Lemma 4.3, page 5764]).

Now that we have these extremal constant sign solutions, we can produce a nodal solution of problem ((𝑃)_𝜆) (with 𝜆∈(0,𝜆^∗)).

Theorem 4.5. If hypotheses 𝐻_𝑔, 𝐻^′_𝑓, 𝐻₀, and 𝐻_𝛽 hold and 𝜆∈(0,𝜆^∗), then problem ((𝑃)_𝜆) has at least five nontrivial smooth solutions: 𝑢₀,𝑢∈int𝐶₊,𝑢−𝑢₀∈int𝐶₊,𝑣₀,𝑣∈−int𝐶₊,𝑣₀−𝑣∈int𝐶₊𝑦₀∈𝐶¹₀Ωnodalwith𝑢₀−𝑦₀∈int𝐶₊,𝑦₀−𝑣₀∈int𝐶₊. (4.64) Moreover, problem ((𝑃)_𝜆) has a smallest nontrivial positive solution and a biggest negative solution.

Proof. The existence of extremal nontrivial constant sing solutions is guaranteed by Proposition 4.4. Let 𝑢^𝜆₊∈int𝐶₊ and 𝑣^𝜆₋∈−int𝐶₊ be these two extremal solutions.
We introduce the following truncation perturbation of the reaction of problem ((𝑃)_𝜆): 𝛾_𝜆(𝑧,𝜁)=𝜆𝑔𝑧,𝑣^𝜆₋(𝑧)+𝑓𝑧,𝑣^𝜆₋(𝑧)+𝑐₁̂𝑣^𝜆₋(𝑧)if𝜁<𝑣^𝜆₋(𝑧),𝜆𝑔(𝑧,𝜁)+𝑓(𝑧,𝜁)+𝑐₁̂𝜁if𝑣^𝜆₋(𝑧)⩽𝜁⩽𝑢^𝜆₊(𝑧),𝜆𝑔𝑧,𝑢^𝜆₊(𝑧)+𝑓𝑧,𝑢^𝜆₊(𝑧)+𝑐₁̂𝑢^𝜆₊(𝑧)if𝑢^𝜆₊(𝑧)<𝜁. (4.65) This is a Carathéodory function. We set Γ_𝜆(𝑧,𝜁)=^𝜁₀𝛾_𝜆(𝑧,𝑠)𝑑𝑠 (4.66) and consider the 𝐶¹-functional 𝜒_𝜆∶𝐻¹₀(Ω)→ℝ, defined by 𝜒_𝜆(𝑢)=12𝜎(𝑢)+𝑐₁̂2‖𝑢‖²₂−_ΩΓ_𝜆(𝑧,𝑢(𝑧))𝑑𝑧∀𝑢∈𝐻¹₀(Ω). (4.67) Also, we introduce 𝛾^±_𝜆(𝑧,𝜁)=𝛾_𝜆𝑧,±𝜁^±,Γ^±_𝜆(𝑧,𝜁)=^𝜁₀𝛾^±_𝜆(𝑧,𝑠)𝑑𝑠 (4.68) and consider the 𝐶¹-functionals 𝜒^±_𝜆∶𝐻¹₀(Ω)→ℝ, defined by 𝜒^±_𝜆(𝑢)=12𝜎(𝑢)+𝑐₁̂2‖𝑢‖²₂−_ΩΓ^±_𝜆(𝑧,𝑢(𝑧))𝑑𝑧∀𝑢∈𝐻¹₀(Ω). (4.69) As in the proof of Theorem 4.1, we show that 𝐾_𝜒𝜆⊆𝑣^𝜆₋,𝑢^𝜆₊,𝐾_𝜒⁺_𝜆⊆0,𝑢^𝜆₊,𝐾_𝜒⁻_𝜆⊆𝑣^𝜆₋,0 (4.70) (see (4.65)). The extremality of the solutions 𝑣^𝜆₋ and 𝑢^𝜆₊ implies that 𝐾_𝜒𝜆⊆𝑣^𝜆₋,𝑢^𝜆₊,𝐾_𝜒⁺_𝜆=0,𝑢^𝜆₊,𝐾_𝜒⁻_𝜆=𝑣^𝜆₋,0. (4.71)
Claim 4. Solutions 𝑢^𝜆₊ and 𝑣^𝜆₋ are both local minimizers of 𝜒_𝜆.
Evidently 𝜒⁺_𝜆 is coercive (see (4.65)) and sequentially weakly lower semicontinuous. So, we can find 𝑢₊∈𝐻¹₀(Ω), such that 𝜒⁺_𝜆𝑢₊=inf_{𝑢∈𝐻10(Ω)}𝜒⁺_𝜆(𝑢). (4.72) As before (see the proof of Proposition 3.7), the presence of the “concave” term implies that 𝜒⁺_𝜆𝑢₊<0=𝜒⁺_𝜆(0), (4.73) hence 𝑢₊≠0, and so 𝑢₊=𝑢^𝜆₊ (see (4.71)). Since 𝜒_𝜆𝐶+=𝜒⁺_𝜆𝐶+, (4.74) it follows that 𝑢₊=𝑢^𝜆₊∈int𝐶₊ is local 𝐶¹₀(Ω)-minimizers of 𝜒_𝜆; hence by Brézis and Nirenberg [16], it is also local 𝐻¹₀(Ω)-minimizers of 𝜒_𝜆.
Similarly this is for 𝑣^𝜆₋ using this time the functional 𝜒⁻_𝜆. This proves the claim.
Without any loss of generality, we may assume that 𝜒_𝜆𝑣^𝜆₋⩽𝜒_𝜆𝑢^𝜆₊. (4.75) The analysis is similar, if the opposite inequality holds. Because of the claim, we can find 𝜚∈(0,1) small, such that 𝜒_𝜆𝑣^𝜆₋⩽𝜒_𝜆𝑢^𝜆₊<inf𝜒_𝜆(𝑢)∶𝑢−𝑢^𝜆₊=𝜚=𝜂_𝜆 (4.76) (see Gasiński and Papageorgiou [17, proof of Theorem 3.4]).
Since 𝜒_𝜆 is coercive (see (4.65)), it satisfies the Cerami condition. This fact and (4.76) permit the use of the mountain pass theorem (see Theorem 2.1). So, we can find 𝑦₀∈𝐾_𝜒𝜆⊆[𝑣^𝜆₋,𝑢^𝜆₊] (see (4.71)), such that 𝜂_𝜆⩽𝜒_𝜆𝑦₀, (4.77) so 𝑦₀∉𝑣^𝜆₋,𝑢^𝜆₊ (4.78) (see (4.76)).
Since 𝑦₀ is a critical point of 𝜒_𝜆 of mountain pass type, we have 𝐶₁𝜒_𝜆,𝑦₀≠0 (4.79) (see e.g., Chang [20]). On the other hand, hypothesis 𝐻_𝑓(iii) implies that we can find 𝜉̂₁>0, such that 𝑓(𝑧,𝜁)𝜁−𝜇𝐹(𝑧,𝜁)⩽𝜉̂₁𝜁²foralmostall𝑧∈Ω,all𝜁⩽𝛿₁, (4.80) for some 𝛿₁⩽𝛿₀. This combined with hypothesis 𝐻_𝑔(iii) implies that ()()()()𝜁𝜁𝜇𝜆𝐺𝑧,𝜁+𝜇𝐹𝑧,𝜁⩾𝜆𝑔𝑧,𝜁𝜁+𝑓𝑧,𝜁>0foralmostall𝑧∈Ω,all⩽𝛿₂, (4.81) for some 𝛿₂⩽𝛿₁ and esssup_Ω𝜆𝐺⋅,𝛿₂+𝐹⋅,𝛿₂>0. (4.82) Hence invoking Proposition 2.1 of Jiu and Su [21], we infer that 𝐶_𝑘𝜒_𝜆,0=0∀𝑘⩾0. (4.83) Combining (4.79) and (4.83), we have that 𝑦₀≠0. Since 𝑦₀∈[𝑣^𝜆₋,𝑢^𝜆₊], the extremality of 𝑣^𝜆₋ and 𝑢^𝜆₊ implies that 𝑦₀ must be a nodal solution of problem ((𝑃)_𝜆), and the regularity theory (see Struwe [13]) implies that 𝑦₀∈𝐶¹₀(Ω).

Acknowledgments

This research has been partially supported by the Ministry of Science and Higher Education of Poland under Grants no. N201 542438 and N201 604640.