Nonhomogeneous Nonlinear Dirichlet Problems with a 𝑝-Superlinear Reaction

Abstract

We consider a nonlinear Dirichlet elliptic equation driven by a nonhomogeneous differential operator and with a Carathéodory reaction 𝑓(𝑧,𝜁), whose primitive 𝑓(𝑧,𝜁) is 𝑝-superlinear near ±∞, but need not satisfy the usual in such cases, the Ambrosetti-Rabinowitz condition. Using a combination of variational methods with the Morse theory (critical groups), we show that the problem has at least three nontrivial smooth solutions. Our result unifies the study of “superlinear” equations monitored by some differential operators of interest like the 𝑝-Laplacian, the (𝑝,𝑞)-Laplacian, and the 𝑝-generalized mean curvature operator.

1. Introduction

The motivation for this paper comes from the work of Wang [1] on superlinear Dirichlet equations. More precisely, let Ω⊆ℝ^𝑁 be a domain with a 𝐶²-boundary 𝜕Ω. Wang [1] studied the following Dirichlet problem:−Δ𝑢(𝑧)=𝑓(𝑢(𝑧))inΩ,𝑢|_𝜕Ω=0. (1.1) He assumes that 𝑓∈𝐶¹(ℝ), 𝑓(0)=𝑓′(0)=0, |𝑓′(𝜁)|≤𝑐(1+|𝜁|^𝑟−2), with 1<𝑟<2^∗, where2^∗=2𝑁𝑁−2if𝑁≥3,+∞if𝑁=1,2 (1.2) and that there exist 𝜇>2 and 𝑀>0, such that0<𝜇𝐹(𝜁)≤𝑓(𝜁)𝜁∀𝜁≥𝑀, (1.3) where𝐹(𝜁)=^𝜁₀𝑓(𝑠)𝑑𝑠 (1.4) (this is the so-called Ambrosetti-Rabinowitz condition). Under these hypotheses, Wang [1] proved that problem (1.1) has at least three nontrivial solutions.

The aim of this work is to establish the result of Wang [1] for a larger class of nonlinear Dirichlet problems driven by a nonhomogeneous nonlinear differential operator. In fact, our formulation unifies the treatment of “superlinear’’ equations driven by the 𝑝-Laplacian, the (𝑝,𝑞)-Laplacian, and the 𝑝-generalized mean curvature operators. In addition, our reaction term 𝑓(𝑧,𝜁) is 𝑧 dependent, need not be 𝐶¹ in the 𝜁-variable, and in general does not satisfy the Ambrosetti-Rabinowitz condition. Instead, we employ a weaker “superlinear’’ condition, which incorporates in our framework functions with “slower’’ growth near ±∞. An earlier extension of the result of Wang [1] to equations driven by the 𝑝-Laplacian was obtained by Jiang [2, Theorem  12, p.1236] with a continuous “superlinear’’ reaction 𝑓(𝑧,𝜁) satisfying the Ambrosetti-Rabinowitz condition.

So, let Ω⊆ℝ^𝑁 be as above. The problem under consideration is the following:−div𝑎(∇𝑢(𝑧))=𝑓(𝑧,𝑢(𝑧))a.e.inΩ,𝑢|_𝜕Ω=0. (1.5) Here 𝑎∶ℝ^𝑁→ℝ^𝑁 is a map which is strictly monotone and satisfies certain other regularity conditions. The precise conditions on 𝑎 are formulated in hypotheses 𝐻(𝑎). These hypotheses are rather general, and as we already mentioned, they unify the treatment of various differential operators of interest. The reaction 𝑓(𝑧,𝜁) is a Carathéodory function (i.e., for all 𝜁∈ℝ, the function 𝑧↦𝑓(𝑧,𝜁) is measurable and for almost all 𝑧∈Ω, the function 𝜁↦𝑓(𝑧,𝜁) is continuous). We assume that the primitive𝐹(𝑧,𝜁)=^𝜁₀𝑓(𝑧,𝑠)𝑑𝑠 (1.6) exhibits 𝑝-superlinear growth near ±∞. However, we do not employ the usual in such cases, the Ambrosetti-Rabinowitz condition. Instead we use a weaker condition (see hypotheses 𝐻(𝑓)), which permits the consideration of a broader class of reaction terms.

Our approach is variational based on the critical point theory combined with Morse theory (critical groups). In the next section for easy reference, we present the main mathematical tools that we will use in the paper. We also state the precise hypotheses on the maps 𝑎 and 𝑓 and explore some useful consequences of them.

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 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. Nevertheless, the Cerami condition suffices to have a deformation theorem, and from it the minimax theory of certain critical values of 𝜑 is derive (see, e.g., Gasiński and Papageorgiou [3]). In particular, we can state the following theorem, known in the literature as the “mountain pass theorem.’’

Theorem 2.1. If 𝜑∈𝐶¹(𝑋) satisfies the Cerami condition, 𝑥₀,𝑥₁∈𝑋 are such that ‖𝑥₁−𝑥₀‖>𝜚>0, and max𝜑𝑥₀,𝜑𝑥₁<inf𝜑(𝑥)∶𝑥−𝑥₀=𝜚=𝜂_𝜚,𝑐=inf_𝛾∈Γmax_0≤𝑡≤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 is a critical value of 𝜑.

In the analysis of problem (1.5) in addition to the Sobolev space 𝑊₀^1,𝑝(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(⋅) denotes the outward unit normal on 𝜕Ω.

In what follows, by 𝜆₁ we denote the first eigenvalue of (−Δ_𝑝,𝑊₀^1,𝑝(Ω)), where Δ_𝑝 denotes the 𝑝-Laplace operator, defined by Δ_𝑝𝑢=div‖∇𝑢‖^𝑝−2∇𝑢∀𝑢∈𝑊₀^1,𝑝(Ω). (2.7) We know (see, e.g., Gasiński and Papageorgiou [3]) that 𝜆₁>0 is isolated and simple (i.e., the corresponding eigenspace is one-dimensional) and 𝜆₁‖‖=inf∇𝑢^𝑝_𝑝‖𝑢‖^𝑝_𝑝∶𝑢∈𝑊₀^1,𝑝(Ω),𝑢≠0. (2.8) In this variational characterization of 𝜆₁, the infimum is realized on the corresponding one-dimensional eigenspace. From (2.8), we see that the elements of the eigenspace do not change sign. In what follows, by 𝑢₁ we denote the 𝐿^𝑝-normalized (i.e., ‖𝑢₁‖_𝑝=1) positive eigenfunction corresponding to 𝜆₁>0. The nonlinear regularity theory for the 𝑝-Laplacian equations (see, e.g., Gasiński and Papageorgiou [3, p. 737]) and the nonlinear maximum principle of Vázquez [4] imply that 𝑢₁∈int𝐶₊.

Now, let 𝜑∈𝐶¹(𝑋) and let 𝑐∈ℝ. We introduce the following notation: 𝜑^𝑐=(𝑥),𝐾𝑥∈𝑋∶𝜑≤𝑐_𝜑=𝑥∈𝑋∶𝜑^′(𝑥),𝐾=0_𝑐𝜑=𝑥∈𝐾_𝜑∶𝜑(𝑥)=𝑐. (2.9)

Let (𝑌₁,𝑌₂) be a topological pair with 𝑌₂⊆𝑌₁⊆𝑋. For every integer 𝑘≥0, by 𝐻_𝑘(𝑌₁,𝑌₂) we denote the 𝑘th relative singular homology group for the pair (𝑌₁,𝑌₂) with integer coefficients. The critical groups of 𝜑 at an isolated point 𝑥₀∈𝐾_𝜑 with 𝜑(𝑥₀)=𝑐 (i.e., 𝑥₀∈𝐾^𝑐_𝜑) are defined by 𝐶_𝑘𝜑,𝑥₀=𝐻_𝑘𝜑^𝑐∩𝑈,𝜑^𝑐∩𝑈⧵𝑥₀∀𝑘≥0, (2.10) 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.

Suppose that 𝜑∈𝐶¹(𝑋) satisfies the Cerami condition and 𝜑(𝐾_𝜑)>−∞. Let 𝑐<𝑖𝑛𝑓𝜑(𝐾_𝜑). The critical groups of 𝜑 at infinity are defined by 𝐶_𝑘𝜑,∞=𝐻_𝑘𝑋,𝜑^𝑐∀𝑘≥0. (2.11) The second deformation theorem (see, e.g., Gasiński and Papageorgiou [3, p. 628]) guarantees that this definition is independent of the particular choice of the level 𝑐<𝑖𝑛𝑓𝜑(𝐾_𝜑).

Suppose that 𝐾_𝜑 is finite. We set 𝑀(𝑡,𝑥)=_𝑘≥0rank𝐶_𝑘𝜑,𝑥𝑡^𝑘∀𝑡∈ℝ,𝑥∈𝐾_𝜑,𝑃(𝑡,∞)=_𝑘≥0rank𝐶_𝑘𝜑,∞𝑡^𝑘∀𝑡∈ℝ. (2.12) The Morse relation says that _{𝑥∈𝐾𝜑}𝑀(𝑡,𝑥)=𝑃(𝑡,∞)+(1+𝑡)𝑄(𝑡), (2.13) where 𝑄(𝑡)=_𝑘≥0𝛽_𝑘𝑡^𝑘 (2.14) is a formal series in 𝑡∈ℝ with nonnegative integer coefficients 𝛽_𝑘∈ℕ.

Now we will introduce the hypotheses on the maps 𝑎(𝑦) and 𝑓(𝑧,𝜁). So, let ℎ∈𝐶¹(0,+∞) be such that 0<𝑡ℎ^′(𝑡)ℎ(𝑡)≤𝑐₀∀𝑡>0, (2.15) for some 𝑐₀>0 and 𝑐₁𝑡^𝑝−1≤ℎ(𝑡)≤𝑐₂1+|𝑡|^𝑝−1∀𝑡>0, (2.16) for some 𝑐₁,𝑐₂>0.

The hypotheses on the map 𝑎(𝑦) are the following:

𝐻(𝑎)∶𝑎(𝑦)=𝑎₀(‖𝑦‖)𝑦, where 𝑎₀(𝑡)>0 for all 𝑡>0 and(i)𝑎∈𝐶(ℝ^𝑁;ℝ^𝑁)∩𝐶¹(ℝ^𝑁⧵{0};ℝ^𝑁),(ii)there exists 𝑐₃>0, such that∇𝑎𝑦≤𝑐₃ℎ𝑦𝑦∀𝑦∈ℝ^𝑁⧵{0}, (2.17)(iii)we have∇𝑎𝑦𝜉,𝜉_ℝ𝑁≥ℎ𝑦𝑦𝜉²∀𝑦∈ℝ^𝑁⧵{0},𝜉∈ℝ^𝑁, (2.18)(iv)if 𝐺∶ℝ^𝑁→ℝ^𝑁 is a function, such that ∇𝐺(𝑦)=𝑎(𝑦) for 𝑦∈ℝ^𝑁 and 𝐺(0)=0, then there exists 𝑐₄>0, such that𝑝𝐺𝑦−𝑎𝑦,𝑦_ℝ𝑁≥−𝑐₄∀𝑦∈ℝ^𝑁. (2.19)

Remark 2.2. Let 𝐺₀(𝑡)=^𝑡₀𝑎₀(𝑠)𝑠𝑑𝑠∀𝑡≥0. (2.20) Evidently 𝐺₀ is strictly convex and strictly increasing on ℝ₊=[0,+∞). We set 𝐺𝑦=𝐺₀𝑦∀𝑦∈ℝ^𝑁. (2.21) Then 𝐺 is convex, 𝐺(0)=0, and ∇𝐺𝑦=𝐺^′₀𝑦𝑦𝑦=𝑎₀𝑦𝑦=𝑎𝑦∀𝑦∈ℝ^𝑁⧵{0}. (2.22) Hence the primitive function 𝐺(𝑦) used in hypothesis 𝐻(𝑎)(𝑖𝑣) is uniquely defined. Note that the convexity of 𝐺 implies that 𝐺𝑦≤𝑎𝑦,𝑦_ℝ𝑁∀𝑦∈ℝ^𝑁. (2.23)
Hypotheses 𝐻(𝑎) and (2.23) lead easily to the following lemma summarizing the main properties of 𝑎.

Lemma 2.3. If hypotheses 𝐻(𝑎) hold, then(a)the map 𝑦↦𝑎(𝑦) is maximal monotone and strictly monotone,(b)there exists 𝑐₅>0, such that 𝑎𝑦≤𝑐₅1+𝑦^𝑝−1∀𝑦∈ℝ^𝑁, (2.24)(c)we have𝑎𝑦,𝑦_ℝ𝑁≥𝑐₁𝑦𝑝−1^𝑝∀𝑦∈ℝ^𝑁 (2.25) (where 𝑐₁>0 is as in (2.16)).

From the above lemma and the integral form of the mean value theorem, we have the following result.

Corollary 2.4. If hypotheses 𝐻(𝑎) hold, then there exists 𝑐₆>0, such that 𝑐₁𝑝𝑦𝑝−1^𝑝≤𝐺𝑦≤𝑐₆1+𝑦^𝑝∀𝑦∈ℝ^𝑁. (2.26)

Example 2.5. The following maps satisfy hypotheses 𝐻(𝑎):(a)𝑎(𝑦)=‖𝑦‖^𝑝−2𝑦 with 1<𝑝<+∞.This map corresponds to the 𝑝-Laplace differential operator Δ_𝑝𝑢=div‖∇𝑢‖^𝑝−2∇𝑢∀𝑢∈𝑊₀^1,𝑝(Ω). (2.27)(b)𝑎(𝑦)=‖𝑦‖^𝑝−2𝑦+𝜇‖𝑦‖^𝑝−2𝑦 with 2≤𝑞<𝑝<+∞,𝜇≥0.This map corresponds to the (𝑝,𝑞)-Laplace differential operator Δ_𝑝𝑢+𝜇Δ_𝑞𝑢∀𝑢∈𝑊₀^1,𝑝(Ω). (2.28) This is an important operator occurring in quantum physics (see Benci et al. [5]). Recently there have been some papers dealing with the existence and multiplicity of solutions for equations driven by such operators. We mention the works of Cingolani and Degiovanni [6], Figueiredo [7], and Sun [8]:(c)𝑎(𝑦)=(1+‖𝑦‖²)^(𝑝−2)/2𝑦, with 2≤𝑝<+∞.This map corresponds to the 𝑝-generalized mean curvature operator div1+‖∇𝑢‖^{2(𝑝−2)/2}∇𝑢∀𝑢∈𝑊₀^1,𝑝(Ω). (2.29) Such equations were investigated by Chen-Shen [9]:(d)𝑎(𝑦)=‖𝑦‖^𝑝−2𝑦+‖𝑦‖^𝑝−2𝑦/(1+‖𝑦‖^𝑝), with 1<𝑝<+∞,(e)𝑎(𝑦)=‖𝑦‖^𝑝−2𝑦+ln(1+‖𝑦‖^𝑝−2)𝑦, with 2≤𝑝<+∞.
Let 𝐴∶𝑊₀^1,𝑝(Ω)→𝑊^−1,𝑝′(Ω)=𝑊₀^1,𝑝(Ω)^∗ (with 1/𝑝+1/𝑝′=1) be the nonlinear map, defined by 𝐴(𝑢),𝑦=_Ω𝑎(∇𝑢(𝑧)),∇𝑦(𝑧)_ℝ𝑁𝑑𝑧∀𝑢,𝑦∈𝑊₀^1,𝑝(Ω). (2.30) From Gasiński and Papageorgiou [10, Proposition  3.1, p. 852], we have the following result for this map.

Proposition 2.6. If hypotheses 𝐻(𝑎) hold, then the map 𝐴∶𝑊₀^1,𝑝(Ω)→𝑊^−1,𝑝′(Ω) defined by (2.30) is bounded, continuous, strictly monotone, hence maximal monotone too, and of type (𝑆)₊; that is, if 𝑢_𝑛→𝑢 weakly in 𝑊₀^1,𝑝(Ω) and limsup_𝑛→+∞𝐴𝑢_𝑛,𝑢_𝑛−𝑢≤0, (2.31) then 𝑢_𝑛→𝑢 in 𝑊₀^1,𝑝(Ω).

The next lemma is an easy consequence of (2.8) and of the fact that 𝑢₁∈int𝐶₊ (see, e.g., Papageorgiou and Kyritsi-Yiallourou [11, p. 356]).

Proposition 2.7. If 𝜗∈𝐿^∞(Ω)₊, 𝜗(𝑧)≤(𝑐₁/𝑝(𝑝−1))𝜆₁ for almost all 𝑧∈Ω, 𝜗≠(𝑐₁/𝑝(𝑝−1))𝜆₁, then there exists 𝜉₀>0, such that 𝑐₁‖‖𝑝−1∇𝑢^𝑝_𝑝−_Ω𝜗(𝑧)|𝑢(𝑧)|^𝑝𝑑𝑧≥𝜉₀‖∇𝑢‖^𝑝_𝑝∀𝑢∈𝑊₀^1,𝑝(Ω). (2.32)
The hypotheses on the reaction 𝑓(𝑧,𝜁) are the following:
𝐻(𝑓)∶𝑓∶Ω×ℝ→ℝ is a Carathéodory function, such that 𝑓(𝑧,0)=0 for almost all 𝑧∈Ω and(i)there exist 𝑎∈𝐿^∞(Ω)₊, 𝑐>0 and 𝑟∈(𝑝,𝑝^∗), with 𝑝^∗=𝑁𝑝𝑁−𝑝if𝑝<𝑁,+∞if𝑝≥𝑁, (2.33) such that 𝑓(𝑧,𝜁)≤𝑎(𝑧)+𝑐𝜁^𝑟−1foralmostall𝑧∈Ω,all𝜁∈ℝ; (2.34)(ii)if 𝐹(𝑧,𝜁)=^𝜁₀𝑓(𝑧,𝑠)𝑑𝑠, (2.35) then lim_𝜁→±∞𝐹(𝑧,𝜁)𝜁^𝑝=+∞uniformlyforalmostall𝑧∈Ω, (2.36)(iii)there exist 𝜏∈((𝑟−𝑝)max{1,𝑁/𝑝},𝑝^∗) and 𝛽₀>0, such that liminf_𝜁→±∞𝑓(𝑧,𝜁)𝜁−𝑝𝐹(𝑧,𝜁)𝜁^𝜏≥𝛽₀uniformlyforalmostall𝑧∈Ω, (2.37)(iv)there exists 𝜗∈𝐿^∞(Ω)₊, such that 𝜗(𝑧)≤(𝑐₁/𝑝(𝑝−1))𝜆₁ for almost all 𝑧∈Ω, 𝜗≠(𝑐₁/𝑝(𝑝−1))𝜆₁, and limsup_𝜁→0𝑝𝐹(𝑧,𝜁)𝜁^𝑝≤𝜗(𝑧)uniformlyforalmostall𝑧∈Ω; (2.38)(v)for every 𝜚>0, there exists 𝜉_𝜚>0, such that 𝑓(𝑧,𝜁)𝜁+𝜉_𝜚𝜁^𝑝≥0foralmostall𝑧∈Ω,all𝜁≤𝜚. (2.39)

Remark 2.8. Hypothesis 𝐻(𝑓)(ii) implies that for almost all 𝑧∈Ω, the primitive 𝐹(𝑧,⋅) is 𝑝-superlinear. Evidently, this condition is satisfied if lim_𝜁→±∞𝑓(𝑧,𝜁)𝜁^𝑝−2𝜁=+∞uniformlyforalmostall𝑧∈Ω, (2.40) that is, for almost all 𝑧∈Ω, the reaction 𝑓(𝑧,⋅) is (𝑝−1)-superlinear. However, we do not use the usual for “superlinear’’ problems, the Ambrosetti-Rabinowitz condition. We recall that this condition says that there exist 𝜇>0 and 𝑀>0, such that 0<𝜇𝐹(𝑧,𝜁)≤𝑓(𝑧,𝜁)𝜁foralmostall𝑧∈Ω,all𝜁≥𝑀,esssup_Ω𝐹(⋅,𝑀)>0. (2.41) Integrating (2.41), we obtain the weaker condition 𝑐₇𝜁^𝜇≤𝐹(𝑧,𝜁)foralmostall𝑧∈Ω,all𝜁≥𝑀, (2.42) with 𝑐₇>0. Evidently from (2.42) we have the much weaker condition lim_𝜁→±∞𝐹(𝑧,𝜁)𝜁^𝑝=+∞uniformlyforalmostall𝑧∈Ω. (2.43) Here, the 𝑝-superlinearity condition (2.43) is coupled with hypothesis 𝐻(𝑓)(iii), which is weaker than the Ambrosetti-Rabinowitz condition (2.41). Indeed, suppose that the Ambrosetti-Rabinowitz condition is satisfied. We may assume that 𝜇>(𝑟−𝑝)max{1,𝑁/𝑝}. Then 𝑓(𝑧,𝜁)𝜁−𝑝𝐹(𝑧,𝜁)𝜁^𝜇=𝑓(𝑧,𝜁)𝜁−𝜇𝐹(𝑧,𝜁)𝜁^𝜇+𝜇−𝑝𝐹(𝑧,𝜁)𝜁^𝜇≥𝜇−𝑝𝑐₇ (2.44) (see (2.41) and (2.42)). Therefore, hypothesis 𝐻(𝑓)(iii) is satisfied.

Example 2.9. The following function satisfies hypotheses 𝐻(𝑓), but not the Ambrosetti-Rabinowitz condition. For the sake of simplicity we drop the 𝑧-dependence: 𝑓(𝜁)=𝜁^𝑞−2𝜁if𝜁𝑝𝜁≤1,_𝑝−2𝜁𝜁+1ln𝑝if𝜁>1, (2.45) with 1<𝑝<𝑞<+∞.
In this work, for every 𝑢∈𝑊₀^1,𝑝(Ω), we set ‖𝑢‖=∇_𝑢𝑝 (2.46) (by virtue of the Poincaré inequality). We mention that the notation ‖⋅‖ will also be used to denote the ℝ^𝑁-norm. However, no confusion is possible, since it is always clear from the context, whose norm is used. For every 𝜁∈ℝ, we set 𝜁^±=max±𝜁,0, (2.47) and for 𝑢∈𝑊₀^1,𝑝(Ω), we define 𝑢^±(⋅)=𝑢(⋅)^±. (2.48) Then 𝑢^±∈𝑊₀^1,𝑝(Ω), and we have 𝑢=𝑢⁺−𝑢⁻,|𝑢|=𝑢⁺+𝑢⁻. (2.49) By |⋅|_𝑁 we denote the Lebesgue measure on ℝ^𝑁. Finally, for a given measurable function ℎ∶Ω×ℝ→ℝ (e.g., a Carathéodory function), we define 𝑁_ℎ(𝑢)(⋅)=ℎ(⋅,𝑢(⋅))∀𝑢∈𝑊₀^1,𝑝(Ω) (2.50) (the Nemytskii map corresponding to ℎ(⋅,⋅)).

3. Three-Solution Theorem

In this section, we prove a multiplicity theorem for problem (1.5), producing three nontrivial smooth solutions, two of which have constant sign (one positive, the other negative).

First we produce two constant sign solutions of (1.5). For this purpose, we introduce the positive and negative truncations of 𝑓(𝑧,⋅), namely:𝑓_±(𝑧,𝜁)=𝑓𝑧,±𝜁^±∀(𝑧,𝜁)∈Ω×ℝ. (3.1) Both are Carathéodory functions. We set𝐹_±(𝑧,𝜁)=^𝜁₀𝑓_±(𝑧,𝑠)𝑑𝑠 (3.2) and consider the 𝐶¹-functionals 𝜑_±∶𝑊₀^1,𝑝(Ω)→ℝ, defined by𝜑_±(𝑢)=_Ω𝐺(∇𝑢(𝑧))𝑑𝑧−_Ω𝐹_±(𝑧,𝑢(𝑧))𝑑𝑧∀𝑢∈𝑊₀^1,𝑝(Ω). (3.3) Also, let 𝜑∶𝑊₀^1,𝑝(Ω)→ℝ be the 𝐶¹-energy functional for problem (1.5), defined by𝜑(𝑢)=_Ω𝐺(∇𝑢(𝑧))𝑑𝑧−_Ω𝐹(𝑧,𝑢(𝑧))𝑑𝑧∀𝑢∈𝑊₀^1,𝑝(Ω). (3.4)

Proposition 3.1. If hypotheses 𝐻(𝑎) and 𝐻(𝑓) hold, then the functionals 𝜑_± satisfy the Cerami condition.

Proof. We do the proof for 𝜑₊, the proof for 𝜑₋ being similar.
So, let {𝑢_𝑛}_𝑛≥1⊆𝑊₀^1,𝑝(Ω) be a sequence, such that 𝜑₊𝑢_𝑛≤𝑀₁∀𝑛≥1, (3.5) for some 𝑀₁>0, and 1+𝑢_𝑛𝜑′₊𝑢_𝑛⟶0in𝑊^−1,𝑝′(Ω). (3.6) From (3.6), we have 𝜑′₊𝑢_𝑛≤𝜀,𝑣^𝑛‖𝑣‖1+𝑢_𝑛∀𝑣∈𝑊₀^1,𝑝(Ω), (3.7) with 𝜀_𝑛↘0, so 𝐴𝑢_𝑛,𝑣−_Ω𝑓₊𝑧,𝑢_𝑛≤𝜀𝑣𝑑𝑧^𝑛‖𝑣‖1+𝑢_𝑛∀𝑛≥1. (3.8) In (3.8), first we choose 𝑣=−𝑢⁻_𝑛∈𝑊₀^1,𝑝(Ω). Then using Lemma 2.3(c), we have 𝑐₁𝑝−1∇𝑢⁻_𝑛^𝑝_𝑝≤𝜀_𝑛∀𝑛≥1, (3.9) so 𝑢⁻_𝑛⟶0in𝑊₀^1,𝑝(Ω). (3.10) Next, in (3.8) we choose 𝑣=𝑢⁺_𝑛∈𝑊₀^1,𝑝(Ω). We obtain −_Ω𝑎∇𝑢⁺_𝑛,∇𝑢⁺_𝑛ℝ𝑁+_Ω𝑓𝑧,𝑢⁺_𝑛𝑢⁺_𝑛𝑑𝑧≤𝜀_𝑛∀𝑛≥1. (3.11) From (3.5) and (3.10), we have _Ω𝑝𝐺∇𝑢⁺_𝑛𝑑𝑧−_Ω𝑝𝐹𝑧,𝑢⁺_𝑛𝑑𝑧≤𝑀₂∀𝑛≥1 (3.12) for some 𝑀₂>0. Adding (3.11) and (3.12), we obtain _Ω𝑝𝐺∇𝑢⁺_𝑛−𝑎∇𝑢⁺_𝑛,∇𝑢⁺_𝑛ℝ𝑁𝑑𝑧+_Ω𝑓𝑧,𝑢⁺_𝑛𝑢⁺_𝑛−𝑝𝐹𝑧,𝑢⁺_𝑛𝑑𝑧≤𝑀₃∀𝑛≥1, (3.13) for some 𝑀₃>0, so _Ω𝑓𝑧,𝑢⁺_𝑛𝑢⁺_𝑛−𝑝𝐹𝑧,𝑢⁺_𝑛𝑑𝑧≤𝑀₄=𝑀₃+𝑐₄|Ω|_𝑁∀𝑛≥1 (3.14) (see hypothesis 𝐻(𝑎)(iv)).
Hypotheses 𝐻(𝑓)(i) and (iii) imply that we can find 𝛽₁∈(0,𝛽₀) and 𝑐₈>0, such that 𝛽₁𝜁^𝜏−𝑐₈≤𝑓(𝑧,𝜁)𝜁−𝑝𝐹(𝑧,𝜁)foralmostall𝑧∈Ω,all𝜁∈ℝ. (3.15) Using (3.15) and (3.14), we obtain 𝛽₁𝑢⁺_𝑛^𝜏_𝜏≤𝑀₅∀𝑛≥1, (3.16) with 𝑀₅=𝑀₄+𝑐₈|Ω|_𝑁>0 and so thesequence𝑢⁺_𝑛𝑛≥1⊆𝐿^𝜏(Ω)isbounded. (3.17) First suppose that 𝑁≠𝑝. From hypothesis 𝐻(𝑓)(iii), it is clear that we can always assume that 𝜏≤𝑟<𝑝^∗. So, we can find 𝑡∈[0,1), such that 1𝑟=1−𝑡𝜏+𝑡𝑝^∗. (3.18) Invoking the interpolation inequality (see, e.g., Gasiński and Papageorgiou [3, p. 905]), we have 𝑢⁺_𝑛𝑟≤𝑢⁺_𝑛𝜏^1−𝑡𝑢⁺_𝑛^𝑡_𝑝∗, (3.19) so 𝑢⁺_𝑛^𝑟_𝑟≤𝑀₆𝑢⁺_𝑛^𝑡𝑟∀𝑛≥1, (3.20) for some 𝑀₆>0 (see (3.17) and use the Sobolev embedding theorem).
Recall that _Ω𝑎∇𝑢⁺_𝑛,∇𝑢⁺_𝑛ℝ𝑁𝑑𝑧−_Ω𝑓𝑧,𝑢⁺_𝑛𝑢⁺_𝑛𝑑𝑧≤𝜀_𝑛∀𝑛≥1. (3.21) From hypothesis 𝐻(𝑓)(i), we have 𝑓(𝑧,𝜁)𝜁≤𝑎(𝑧)+𝑐̂𝜁^𝑟foralmostall𝑧∈Ω,all𝜁∈ℝ, (3.22) with 𝑎∈𝐿^∞(Ω)₊, 𝑐̂>0. Therefore, from (3.21) and Lemma 2.3(c), we have 𝑐₁𝑝−1∇𝑢⁺_𝑛^𝑝_𝑝≤𝑐₉1+𝑢⁺_𝑛^𝑟_𝑟∀𝑛≥1, (3.23) for some 𝑐₉>0 and so 𝑢⁺_𝑛^𝑝≤𝑐₁₀1+𝑢⁺_𝑛^𝑡𝑟∀𝑛≥1, (3.24) for some 𝑐₁₀>0 (see (3.20)). The hypothesis on 𝜏 (see 𝐻(𝑓)(iii)) implies that 𝑡𝑟<𝑝, and so thesequence𝑢⁺_𝑛𝑛≥1⊆𝑊₀^1,𝑝(Ω)isbounded, (3.25) and thus thesequence𝑢_𝑛𝑛≥1⊆𝑊₀^1,𝑝(Ω)isbounded (3.26) (see (3.26)).
Now, suppose that 𝑁=𝑝. In this case, we have 𝑝^∗=+∞, while from the Sobolev embedding theorem, we have that 𝑊₀^1,𝑝(Ω)⊆𝐿^𝑞(Ω) for all 𝑞∈[1,+∞). So, we need to modify the previous argument. Let 𝜗>𝑟≥𝜏. Then we choose 𝑡∈[0,1), such that 1𝑟=1−𝑡𝜏+𝑡𝜗, (3.27) so 𝜗()𝑡𝑟=𝑟−𝜏.𝜗−𝜏 (3.28) Note that 𝜗(𝑟−𝜏)𝜗−𝜏⟶𝑟−𝜏as𝜗⟶+∞=𝑝^∗. (3.29) Since 𝑁=𝑝, we have 𝑟−𝜏<𝑝 (see 𝐻(𝑓)(𝑖𝑖𝑖)). Therefore, for large 𝜗>𝑟, we have that 𝑡𝑟<𝑝 (see (3.28)). Hence, if in the previous argument, we replace 𝑝^∗ with such a large 𝜗>𝑟, again we reach (3.26).
Because of (3.26), we may assume that 𝑢_𝑛⟶𝑢weaklyin𝑊₀^1,𝑝(Ω),𝑢_𝑛⟶𝑢in𝐿^𝑝(Ω). (3.30)(3.31) In (3.8), we choose 𝑣=𝑢_𝑛−𝑢∈𝑊₀^1,𝑝(Ω), pass to the limit as 𝑛→+∞, and use (3.30). Then liminf_𝑛→+∞𝐴𝑢_𝑛,𝑢_𝑛−𝑢=0, (3.32) so 𝑢_𝑛⟶𝑢in𝑊₀^1,𝑝(Ω) (3.33) (see Proposition 2.6). This proves that 𝜑₊ satisfies the Cerami condition.
Similarly we show that 𝜑₋ satisfies the Cerami condition.

With some obvious minor modifications in the above proof, we can also have the following result.

Proposition 3.2. If hypotheses 𝐻(𝑎) and 𝐻(𝑓) hold, then the functional 𝜑 satisfies the Cerami condition.

Next we determine the structure of the trivial critical point 𝑢=0 for the functionals 𝜑_± and 𝜑.

Proposition 3.3. If hypotheses 𝐻(𝑎) and 𝐻(𝑓) hold, then 𝑢=0 is a local minimizer for the functionals 𝜑_± and 𝜑.

Proof. By virtue of hypotheses 𝐻(𝑓)(𝑖) and (𝑖𝑣), for a given 𝜀>0 we can find 𝑐_𝜀>0, such that 𝐹()≤1𝑧,𝜁𝑝(𝜗(𝑧)+𝜀)𝜁^𝑝+𝑐_𝜀𝜁^𝑟foralmostall𝑧∈Ω,all𝜁∈ℝ. (3.34) Then for all 𝑢∈𝑊₀^1,𝑝(Ω), we have 𝜑₊(𝑢)=_Ω𝐺(∇𝑢)𝑑𝑧−_Ω𝐹𝑧,𝑢⁺≥𝑐𝑑𝑧₁𝑝‖‖𝑝−1∇𝑢^𝑝_𝑝−1𝑝_Ω𝜗𝑢^+𝑝𝜀𝑑𝑧−𝑝𝜆₁‖𝑢‖^𝑝−𝑐₁₁‖𝑢‖^𝑟≥1𝑝𝜉₀−𝜀𝜆₁‖𝑢‖^𝑝−𝑐₁₁‖𝑢‖^𝑟, (3.35) for some 𝑐₁₁>0 (see Corollary 2.4, (2.8), (3.34), and Proposition 2.7). Choosing 𝜀∈(0,𝜆₁𝜉₀), we have 𝜑₊(𝑢)≥𝑐₁₂‖𝑢‖^𝑝−𝑐₁₁‖𝑢‖^𝑟∀𝑢∈𝑊₀^1,𝑝(Ω), (3.36) for some 𝑐₁₂>0. Since 𝑟>𝑝, from (3.36), it follows that we can find small 𝜚∈(0,1), such that 𝜑₊(𝑢)>0∀𝑢,with0<‖𝑢‖≤𝜚, (3.37) so 𝑢=0isalocalminimizerof𝜑₊. (3.38) Similarly, we show that 𝑢=0 is a local minimizer for the functionals 𝜑₋ and 𝜑.

We may assume that 𝑢=0 is an isolated critical point of 𝜑₊ (resp., 𝜑₋). Otherwise, we already have a sequence of distinct positive (resp., negative) solutions of (1.5) and so we are done. Moreover, as in Gasiński and Papageorgiou [12, proof of Theorem  3.4,] we can find small 𝜚_±∈(0,1), such thatinf𝜑_±(𝑢)∶‖𝑢‖=𝜚_±=𝜂_±>0. (3.39) By virtue of hypothesis 𝐻(𝑓)(ii) (the 𝑝-superlinear condition), we have the next result, which completes the mountain pass geometry for problem (1.1).

Proposition 3.4. If hypotheses 𝐻(𝑎) and 𝐻(𝑓) hold and 𝑢∈int𝐶₊, then 𝜑_±(𝑡𝑢)→−∞ as 𝑡→±∞.

Proof. By virtue of hypotheses 𝐻(𝑓)(i) and (ii), for a given 𝜉>0, we can find 𝑐₁₃=𝑐₁₃(𝜉)>0, such that 𝜉𝜁^𝑝−𝑐₁₃≤𝐹(𝑧,𝜁)foralmostall𝑧∈Ω,all𝜁∈ℝ. (3.40) Then for 𝑢∈int𝐶₊ and 𝑡>0, we have 𝜑₊(𝑡𝑢)=_Ω𝐺(𝑡∇𝑢)𝑑𝑧−_Ω𝐹(𝑧,𝑡𝑢)𝑑𝑧≤𝑐₁₄1+𝑡^𝑝‖𝑢‖^𝑝−𝜉𝑡^𝑝‖𝑢‖^𝑝_𝑝+𝑐₁₃|Ω|_𝑁=𝑡^𝑝𝑐₁₄‖𝑢‖^𝑝−𝜉‖𝑢‖^𝑝_𝑝+𝑐₁₅, (3.41) for some 𝑐₁₄>0 and with 𝑐₁₅=𝑐₁₄+𝑐₁₃|Ω|_𝑁>0 (see Corollary 2.4 and (3.40)).
Choosing 𝜉>𝑐₁₄(‖𝑢‖^𝑝/‖𝑢‖^𝑝_𝑝), from (3.41), it follows that 𝜑₊(𝑡𝑢)⟶−∞as𝑡⟶+∞. (3.42) Similarly, we show that 𝜑₋(𝑡𝑢)⟶−∞as𝑡⟶−∞. (3.43)

Now we are ready to produce two constant sign smooth solutions of (1.5).

Proposition 3.5. If hypotheses 𝐻(𝑎) and 𝐻(𝑓) hold, then problem (1.5) has at least two nontrivial constant sign smooth solutions 𝑢₀∈int𝐶₊,𝑣₀∈−int𝐶₊. (3.44)

Proof. From (3.39), we have 𝜑₊(0)=0<inf𝜑₊(𝑢)∶‖𝑢‖=𝜚₊=𝜂₊. (3.45) Moreover, according to Proposition 3.4, for 𝑢∈int𝐶₊, we can find large 𝑡>0, such that 𝜑₊(𝑡𝑢)≤𝜑₊(0)=0<𝜂₊,‖𝑡𝑢‖>𝜚₊. (3.46) Then because of (3.45), (3.46), and Proposition 3.1, we can apply the mountain pass theorem (see Theorem 2.1) and find 𝑢₀∈𝑊₀^1,𝑝(Ω), such that 𝜑₊(0)=0<𝜂₊≤𝜑₊𝑢₀,𝜑^′₊𝑢₀=0. (3.47)(3.48) From (3.47) we see that 𝑢₀≠0. From (3.48), we have 𝐴𝑢₀=𝑁_𝑓+𝑢₀. (3.49) On (3.49) we act with −𝑢⁻₀∈𝑊₀^1,𝑝(Ω) and obtain 𝑐₁𝑝−1∇𝑢⁻₀^𝑝_𝑝≤0 (3.50) (see Lemma 2.3(c)), so 𝑢₀≥0,𝑢₀≠0. (3.51) Then, from (3.49), we have −div𝑎∇𝑢₀(𝑧)=𝑓𝑧,𝑢₀(𝑧)a.e.in𝑢Ω,₀|_𝜕Ω=0. (3.52) Theorem  7.1 of Ladyzhenskaya and Uraltseva [13, p. 286] implies that 𝑢₀∈𝐿^∞(Ω). Then from Lieberman [14, p. 320], we have that 𝑢₀∈𝐶₀^1,𝛼(Ω) for some 𝛼∈(0,1). Let 𝜚=‖𝑢₀‖_∞, and let 𝜉_𝜚>0 be as postulated by hypothesis 𝐻(𝑓)(𝑣). Then −div𝑎∇𝑢₀(𝑧)+𝜉_𝜚𝑢₀(𝑧)^𝑝−1≥0foralmostall𝑧∈Ω (3.53) (see (3.52) and hypothesis 𝐻(𝑓)(𝑖𝑣)), so div𝑎∇𝑢₀(𝑧)≤𝜉_𝜚𝑢₀(𝑧)^𝑝−1foralmostall𝑧∈Ω. (3.54) Then, from Theorem  5.5.1 of Pucci and Serrin [15, p. 120], we have that 𝑢₀∈int𝐶₊.
Similarly, working with 𝜑₋, we obtain another constant sign smooth solution 𝑣₀∈−int𝐶₊.

Next, using the Morse theory (critical groups), we will produce a third nontrivial smooth solution. To this end, first we compute the critical groups of 𝜑_± at infinity (see also Wang [1] and Jiang [2]).

Proposition 3.6. If hypotheses 𝐻(𝑎) and 𝐻(𝑓) hold, then 𝐶_𝑘𝜑_±,∞=0∀𝑘≥0. (3.55)

Proof. We do the proof for 𝜑₊, the proof for 𝜑₋ being similar.
By virtue of hypotheses 𝐻(𝑓)(i) and (ii), for a given 𝜉>0, we can find 𝑐₁₆=𝑐₁₆(𝜉)>0, such that 𝐹₊(𝑧,𝜁)≥𝜉𝜁^+𝑝−𝑐₁₆foralmostall𝑧∈Ω,all𝜁∈ℝ. (3.56) Let 𝐸₊=𝑢∈𝜕𝐵₁∶𝑢⁺≠0, (3.57) where 𝜕𝐵₁=𝑢∈𝑊₀^1,𝑝(Ω)∶‖𝑢‖=1. (3.58) For 𝑢∈𝐸₊ and 𝑡>0, we have 𝜑₊(𝑡𝑢)=_Ω𝐺(𝑡∇𝑢)𝑑𝑧−_Ω𝐹₊(𝑧,𝑡𝑢)𝑑𝑧≤𝑐₁₇1+𝑡^𝑝−𝜉𝑡^𝑝𝑢^+𝑝_𝑝−𝑐₁₆|Ω|_𝑁=𝑡^𝑝𝑐₁₇−𝜉𝑢^+𝑝_𝑝+𝑐₁₇+𝑐₁₆|Ω|_𝑁 (3.59) for some 𝑐₁₇>0 (see Corollary 2.4, (3.56) and recall that ‖𝑢‖=‖∇𝑢‖_𝑝=1).
Choosing 𝜉>𝑐₁₇/‖𝑢⁺‖^𝑝_𝑝, we see that 𝜑₊(𝑡𝑢)⟶−∞as𝑡⟶+∞. (3.60) Hypothesis 𝐻(𝑓)(iii) implies that we can find 𝛽₁∈(0,𝛽₀) and 𝑀₇>0, such that 𝑓₊(𝑧,𝜁)𝜁−𝑝𝐹(𝑧,𝜁)≥𝛽₁𝜁^𝜏foralmostall𝑧∈Ω,all𝜁≥𝑀₇. (3.61) Then for all 𝑦∈𝑊₀^1,𝑝(Ω), we have _Ω𝑝𝐹₊𝑧,𝑦−𝑓₊𝑧,𝑦𝑦𝑑𝑧≤−_{{𝑦≥𝑀7}}𝛽₁𝑦^𝜏𝑑𝑧+𝑐₁₈, (3.62) for some 𝑐₁₈>0 (see (3.61)). Let 𝑐₁₉=𝑐₁₈+𝑐₄|Ω|_𝑁>0 (see hypothesis 𝐻(𝑎)(iv)) and choose 𝛾<−𝑐₁₉. Because of (3.60), for 𝑢∈𝐸₊ and for large 𝑡>0, we have 𝜑₊(𝑡𝑢)=_Ω𝐺(𝑡∇𝑢)𝑑𝑧−_Ω𝐹₊(𝑧,𝑡𝑢)𝑑𝑧≤𝛾. (3.63) Since 𝜑₊(0)=0, we can find 𝑡^∗>0, such that 𝜑₊𝑡^∗𝑢=𝛾. (3.64) Also, for large 𝑡>0, we have 𝑑𝜑𝑑𝑡⁺(𝑡𝑢)=⟨𝜑^′₊()=1𝑡𝑢,𝑢⟩𝑡𝜑^′₊()=1𝑡𝑢,𝑡𝑢𝑡_Ω(𝑎(𝑡∇𝑢),𝑡∇𝑢)_ℝ𝑁𝑑𝑧−_Ω𝑓₊()≤1𝑧,𝑡𝑢𝑡𝑢𝑑𝑧𝑡_Ω𝑝𝐺(𝑡∇𝑢)𝑑𝑧+𝑐₄|Ω|_𝑁−_Ω𝑝𝐹₊(𝑧,𝑡𝑢)𝑑𝑧+𝑐₁₈≤1𝑡𝛾+𝑐₁₈+𝑐₄|Ω|_𝑛<0 (3.65) (see hypothesis 𝐻(𝑎)(iv), (3.62), (3.63), and recall that 𝑝𝜑₊(𝑡𝑢)≤𝜑₊(𝑡𝑢)≤𝛾<0). Hence, by the implicit function theorem, 𝑡^∗ is unique and in fact there is a unique function 𝜇₊∈𝐶(𝐸₊), such that 𝜑₊𝜇₊(𝑢)𝑢=𝛾∀𝑢∈𝐸₊. (3.66) Let 𝐷₊=𝑢∈𝑊₀^1,𝑝(Ω)∶𝑢⁺≠0. (3.67) We set 𝜇₊(𝑢)=1‖𝑢‖𝜇₊𝑢‖𝑢‖∀𝑢∈𝐷₊. (3.68) Then 𝜇₊∈𝐶(𝐷₊) and 𝜑₊𝜇₊(𝑢)𝑢=𝛾∀𝑢∈𝐷₊. (3.69) Moreover, if 𝜑₊(𝑢)=𝛾, then 𝜇₊(𝑢)=1. We set 𝜇₊(𝑢)=1if𝜑₊(𝑢)≤𝛾,𝜇₊(𝑢)if𝜑₊(𝑢)>𝛾. (3.70) Evidently 𝜇₊∈𝐶(𝐸₊). Let ℎ₊∶[0,1]×𝐷₊→𝐷₊ be defined by ℎ₊(𝑡,𝑢)=(1−𝑡)𝑢+𝑡𝜇₊(𝑢)𝑢. (3.71) Clearly ℎ₊ is continuous and ℎ₊(0,𝑢)=𝑢,ℎ₊(1,𝑢)=𝜇₊(𝑢)𝑢∈𝜑^𝛾₊,ℎ₊(𝑡,𝑢)=𝑢∀𝑡∈[0,1],𝑢∈𝜑^𝛾₊ (3.72) (see (3.70)) and so 𝜑^𝛾₊isastrictdeformationretractof𝐷₊. (3.73) It is easy to see that 𝐷₊ is contractible in itself. Hence 𝐻_𝑘𝑊₀^1,𝑝(Ω),𝐷₊=0∀𝑘≥0 (3.74) (see Granas and Dugundji [16, p. 389]), so 𝐻_𝑘𝑊₀^1,𝑝(Ω),𝜑^𝛾₊=0∀𝑘≥0 (3.75) (see (3.73)) and thus 𝐶_𝑘𝜑₊,∞=0∀𝑘≥0 (3.76) (choosing 𝛾<0 negative enough).
The same applies for 𝜑₋, using this time the sets 𝐸₋=𝑢∈𝜕𝐵₁∶𝑢⁻,𝐷≠0₋=𝑢∈𝑊₀^1,𝑝(Ω)∶𝑢⁻≠0. (3.77)

With suitable changes in the above proof, we can have the following result.

Proposition 3.7. If hypotheses 𝐻(𝑎) and 𝐻(𝑓) hold, then 𝐶_𝑘𝜑,∞=0∀𝑘≥0. (3.78)

Proof. As before, hypotheses 𝐻(𝑓)(i) and (ii) imply that for a given 𝜉>0, we can find 𝑐₂₀=𝑐₂₀(𝜉)>0, such that 𝐹(𝑧,𝜁)≥𝜉𝜁^𝑝−𝑐₂₀foralmostall𝑧∈Ω,all𝜁∈ℝ. (3.79) Let 𝑢∈𝜕𝐵₁={𝑢∈𝑊₀^1,𝑝(Ω)∶‖𝑢‖=1} and 𝑡>0. Then 𝜑(𝑡𝑢)=_Ω𝐺(𝑡∇𝑢)𝑑𝑧−_Ω𝐹(𝑧,𝑡𝑢)𝑑𝑧≤𝑐₂₁1+𝑡^𝑝−𝜉𝑡^𝑝‖𝑢‖^𝑝_𝑝+𝑐₂₀|Ω|_𝑁≤𝑡^𝑝𝑐₂₁−𝜉‖𝑢‖^𝑝_𝑝+𝑐₂₁+𝑐₂₀|Ω|_𝑁 (3.80) for some 𝑐₂₁>0 (see Corollary 2.4, (3.79) and recall that ‖𝑢‖=1). Choosing 𝜉>𝑐₂₁/‖𝑢‖^𝑝_𝑝, we see that 𝜑(𝑡𝑢)⟶−∞as𝑡⟶+∞. (3.81) Hypothesis 𝐻(𝑓)(iii) implies that we can find 𝛽₁∈(0,𝛽₀) and 𝑀₈>0, such that 𝑓(𝑧,𝜁)𝜁−𝑝𝐹(𝑧,𝜁)≥𝛽₁𝜁^𝜏foralmostall𝑧∈Ω,all𝜁≥𝑀₈. (3.82) Then, for any 𝑦∈𝑊₀^1,𝑝(Ω), we have 𝑝𝐹𝑧,𝑦−𝑓𝑧,𝑦𝑦_Ω𝑑=_{{|𝑦|<𝑀8}}𝑝𝐹𝑧,𝑦−𝑓𝑧,𝑦𝑦𝑑𝑧+_{{|𝑦|≥𝑀8}}𝑝𝐹𝑧,𝑦−𝑓𝑧,𝑦𝑦𝑑𝑧≤−_{{|𝑦|≥𝑀8}}𝛽𝑦^𝜏𝑑𝑧+𝑐₂₂, (3.83) for some 𝑐₂₂>0 (see hypothesis 𝐻(𝑓)(i)).
Let 𝑐₂₃=𝑐₂₂+𝑐₄|Ω|_𝑁 (see hypothesis 𝐻(𝑎)(iv)) and choose 𝛾<−𝑐₂₃. Because of (3.81), for a given 𝑢∈𝜕𝐵₁ and for large 𝑡>0, we have 𝜑(𝑡𝑢)=_Ω𝐺(𝑡∇𝑢)𝑑𝑧−_Ω𝐹(𝑧,𝑡𝑢)𝑑𝑧≤𝛾. (3.84) We also have 𝑑𝜑()=𝜑𝑑𝑡𝑡𝑢^′()=1𝑡𝑢,𝑢𝑡𝜑^′()=1𝑡𝑢,𝑡𝑢𝑡_Ω(𝑎(𝑡∇𝑢),𝑡∇𝑢)_ℝ𝑁−_Ω𝑓()≤1𝑧,𝑡𝑢𝑡𝑢𝑑𝑧𝑡_Ω𝑝𝐺(𝑡∇𝑢)𝑑𝑧+𝑐₄|Ω|_𝑁−_Ω𝑝𝐹(𝑧,𝑡𝑢)𝑑𝑧+𝑐₂₂≤1𝑡𝛾+𝑐₂₂+𝑐₄|Ω|_𝑁<0 (3.85) (see hypothesis 𝐻(𝑓)(iv), (3.83), (3.84) and note that 𝑝𝜑(𝑡𝑢)≤𝜑(𝑡𝑢)≤𝛾<0).
The implicit function theorem implies that there exists unique 𝜇∈𝐶(𝜕𝐵₁), such that 𝜑𝜇(𝑢)𝑢=𝛾∀𝑢∈𝜕𝐵₁. (3.86) We define (𝑢)=1𝜇‖𝑢‖𝜇𝑢‖𝑢‖∀𝑢≠0. (3.87) Then 𝜇∈𝐶(𝑊₀^1,𝑝(Ω)⧵{0}) and 𝜑𝜇(𝑢)𝑢=𝛾∀𝑢∈𝑊₀^1,𝑝(Ω)⧵{0}. (3.88) Moreover, if 𝜑(𝑢)=𝛾, then 𝜇(𝑢)=1. We introduce (𝑢)=1𝜇if𝜑(𝑢)(𝑢)≤𝛾,𝜇if𝜑(𝑢)>𝛾. (3.89) Evidently 𝑢∈𝐶(𝑊₀^1,𝑝(Ω)⧵{0}). Let ℎ∶[0,1]×(𝑊₀^1,𝑝(Ω)⧵{0})→𝑊₀^1,𝑝(Ω)⧵{0} be defined by ℎ(𝑡,𝑢)=(1−𝑡)𝑢+𝑡𝜇(𝑢)𝑢. (3.90) Then ℎ(0,𝑢)=𝑢,ℎ(1,𝑢)=𝜇(𝑢)𝑢∈𝜑^𝛾,ℎ(𝑡,𝑢)=𝑢∀𝑡∈[0,1],𝑢∈𝜑^𝛾, (3.91) so 𝜑^𝛾isastrongdeformationretractof𝑊₀^1,𝑝(Ω)⧵{0}. (3.92) Via the radial retraction, we see that 𝜕𝐵₁ is a retract of 𝑊₀^1,𝑝(Ω)⧵{0} and 𝑊₀^1,𝑝(Ω)⧵{0} is deformable into 𝜕𝐵₁. Invoking Theorem  6.5 of Dugundji [17, p. 325], we have that 𝜕𝐵₁isadeformationretractof𝑊₀^1,𝑝(Ω)⧵{0}, (3.93) so 𝜕𝐵₁and𝜑^𝛾arehomotopyequivalent. (3.94) Thus 𝐻_𝑘𝑊₀^1,𝑝(Ω),𝜕𝐵₁=𝐻_𝑘𝑊₀^1,𝑝(Ω),𝜑^𝛾∀𝑘≥0, (3.95) and finally 𝐶_𝑘𝜑,∞=𝐻_𝑘𝑊₀^1,𝑝(Ω),𝜕𝐵₁∀𝑘≥0. (3.96) But 𝜕𝐵₁ is an absolute retract of 𝑊₀^1,𝑝(Ω) (see, e.g., Gasiński and Papageorgiou [3, p. 691]), hence contractible in itself. Therefore 𝐻_𝑘𝑊₀^1,𝑝(Ω),𝜕𝐵₁=0∀𝑘≥0, (3.97) so 𝐶_𝑘𝜑,∞=0∀𝑘≥0. (3.98)

Now we are ready to produce the third nontrivial smooth solution for problem (1.5).

Theorem 3.8. If hypotheses 𝐻(𝑎) and 𝐻(𝑓) hold, then problem (1.5) has at least three nontrivial smooth solutions 𝑢₀∈int𝐶₊,𝑣₀∈−int𝐶₊,𝑦₀∈𝐶¹₀Ω⧵{0}. (3.99)

Proof. From Proposition 3.5, we already have two nontrivial constant sign and smooth solutions 𝑢₀∈int𝐶₊,𝑣₀∈−int𝐶₊. (3.100) We assume that 𝐾_𝜑=0,𝑢₀,𝑣₀. (3.101) Otherwise we already have a third nontrivial solution 𝑦₀, and by the nonlinear regularity theory, 𝑦₀∈𝐶¹₀(Ω), so we completed the proof.
Claim 1. 𝐶_𝑘(𝜑₊,𝑢₀)=𝐶_𝑘(𝜑₋,𝑣₀)=𝛿_𝑘,1ℤ for all 𝑘≥0.
We do the proof for the pair (𝜑₊,𝑢₀), the proof for the pair (𝜑₋,𝑣₀) being similar.
We start by noting that 𝐾_𝜑+={0,𝑢₀}. Indeed, suppose that 𝑢∈𝐾_𝜑+⧵{0}. Then 𝐴(𝑢)=𝑁_𝑓+(𝑢). (3.102) Acting with −𝑢⁻∈𝑊₀^1,𝑝(Ω), we obtain 𝑐₁‖𝑝−1∇𝑢⁻‖^𝑝_𝑝≤0 (3.103) (see Lemma 2.3(c)) and so 𝑢₀≥0, 𝑢₀≠0. Moreover, by nonlinear regularity (see Ladyzhenskaya and Uraltseva [13] and Lieberman [14]), we have that 𝑢∈𝐶₊⧵{0}. Since 𝜑′|_𝐶+=𝜑′₊|_𝐶+, we infer that 𝑢∈𝐾_𝜑={0,𝑢₀,𝑣₀}, and hence 𝑢=𝑢₀.
Choose 𝛾,𝜗∈ℝ, such that 𝛾<0=𝜑₊(0)<𝜗<𝜂₊≤𝜑₊𝑢₀ (3.104) (see (3.47)), and consider the following set: 𝜑^𝛾₊⊆𝜑^𝜗₊⊆𝑊₀^1,𝑝(Ω)=𝑊. (3.105) We consider the long exact sequence of singular homology groups corresponding to the above triple. We have ⋯⟶𝐻_𝑘𝑊,𝜑−→^𝛾₊^𝑖∗𝐻_𝑘𝑊,𝜑−→^𝜗₊^𝜕∗𝐻_𝑘−1𝜑^𝜗₊,𝜑^𝛾₊⟶⋯∀𝑘≥0, (3.106) where 𝑖_∗ is the group homomorphism induced by the embedding 𝑖∶𝜑^𝛾₊→𝜑^𝜗₊ and 𝜕_∗ is the boundary homomorphism. Note that 𝐻_𝑘𝑊,𝜑^𝛾₊=𝐶_𝑘𝜑₊,∞∀𝑘≥0 (3.107) (since 𝛾<0, 𝐾_𝜑+={0,𝑢₀} and 0=𝜑₊(0)<𝜂₊≤𝜑₊(𝑢₀); see (3.47)), so 𝐻_𝑘𝑊,𝜑^𝛾₊=0∀𝑘≥0 (3.108) (see Proposition 3.6).
From the choice of 𝜗>0, the only critical value of 𝜑₊ in the interval (𝛾,𝜗) is 0. Hence 𝐻_𝑘−1𝜑^𝜗₊,𝜑^𝛾₊=𝐶_𝑘−1𝜑₊,0=𝛿_𝑘−1,0ℤ=𝛿_𝑘,1ℤ∀𝑘≥0 (3.109) (see Proposition 3.3).
Finally, for the same reason, we have 𝐻_𝑘𝑊,𝜑^𝜗₊=𝐶_𝑘𝜑₊,𝑢₀∀𝑘≥0. (3.110)
From (3.108), (3.109), and (3.110), it follows that in (3.106) only the tail (i.e., 𝑘=1) is nontrivial. The rank theorem implies that rank𝐻₁𝑊,𝜑^𝜗₊=rankim𝜕_∗+rankker𝜕_∗=rankim𝜕_∗+rankim𝑖_∗=1+0=1 (3.111) (by virtue of the exactness of (3.106)), so rank𝐶₁𝜑₊,𝑢₀≤1 (3.112) (see (3.110)).
But 𝑢₀ is a critical point of mountain pass type for 𝜑₊. Hence rank𝐶₁𝜑₊,𝑢₀≥1. (3.113) From (3.112) and (3.113) and since 𝐶_𝑘𝜑₊,𝑢₀=𝐻_𝑘𝑊,𝜑^𝜗₊=0∀𝑘≠1, (3.114) we infer that 𝐶_𝑘𝜑₊,𝑢₀=𝛿_𝑘,1ℤ∀𝑘≥0. (3.115)
Similarly, we show that 𝐶_𝑘𝜑₋,𝑣₀=𝛿_𝑘,1ℤ∀𝑘≥0. (3.116)
This proves Claim 1.
Claim 2. 𝐶_𝑘(𝜑,𝑢₀)=𝐶_𝑘(𝜑₊,𝑢₀),𝐶_𝑘(𝜑,𝑣₀)=𝐶_𝑘(𝜑₋,𝑣₀) for all 𝑘≥0.
We consider the homotopy ℎ₁(𝑡,𝑢)=(1−𝑡)𝜑₊(𝑢)+𝑡𝜑(𝑢)∀(𝑡,𝑢)∈[0,1]×𝑊₀^1,𝑝(Ω). (3.117) Clearly 𝑢₀∈𝐾_{ℎ(𝑡,⋅)} for all 𝑡∈[0,1]. We will show that there exists 𝜚>0, such that 𝐵_𝜚𝑢₀∩𝐾_{ℎ1(𝑡,⋅)}=𝑢₀∀𝑡∈[0,1], (3.118) where 𝐵_𝜚𝑢₀=𝑢∈𝑊₀^1,𝑝(Ω)∶𝑢−𝑢₀<𝜚. (3.119) Arguing by contradiction, suppose that (3.118) is not true for any 𝜚>0. Then we can find two sequences {𝑡_𝑛}_𝑛≥1⊆[0,1] and {𝑢_𝑛}_𝑛≥1⊆𝑊₀^1,𝑝(Ω)⧵{𝑢₀}, such that 𝑡_𝑛⟶𝑡in[],𝑢0,1_𝑛⟶𝑢₀in𝑊₀^1,𝑝(Ω),ℎ_𝑡𝑛^′𝑢_𝑛=0∀𝑛≥1. (3.120)
For every 𝑛≥1, we have 𝐴𝑢_𝑛=1−𝑡_𝑛𝑁_𝑓+𝑢_𝑛+𝑡_𝑛𝑁_𝑓𝑢_𝑛, (3.121) so −div𝑎∇𝑢_𝑛(𝑧)=1−𝑡_𝑛𝑓𝑧,𝑢⁺_𝑛(𝑧)+𝑡_𝑛𝑓𝑧,𝑢_𝑛(𝑧)a.e.in𝑢Ω,_𝑛|_𝜕Ω=0. (3.122) From Ladyzhenskaya and Uraltseva [13, p. 286], we know that we can find 𝑀₉>0, such that 𝑢_𝑛∞≤𝑀₉∀𝑛≥1. (3.123) Then from Lieberman [14, p. 320], we infer that there exist 𝛼∈(0,1) and 𝑀₁₀>0, such that 𝑢_𝑛∈𝐶₀^1,𝛼Ω,𝑢_{𝑛𝐶01,𝛼(Ω)}≤𝑀₁₀∀𝑛≥1. (3.124) The compactness of the embedding 𝐶₀^1,𝛼(Ω) into 𝐶¹₀(Ω) and (3.120) imply that 𝑢_𝑛⟶𝑢₀in𝐶¹₀Ω, (3.125) so, there exists 𝑛₀≥1, such that 𝑢_𝑛∈int𝐶₊∀𝑛≥𝑛₀ (3.126) (recall that 𝑢₀∈int𝐶₊; see Proposition 3.5) and thus {𝑢_𝑛}_𝑛≥𝑛0⊆𝐾_𝜑+ are distinct solutions of (1.5), a contradiction.
Therefore (3.118) holds for some 𝜚>0. Invoking the homotopy invariance property of critical groups, we have 𝐶_𝑘𝜑,𝑢₀=𝐶_𝑘𝜑₊,𝑢₀∀𝑘≥0. (3.127) Similarly, we show that 𝐶_𝑘𝜑,𝑣₀=𝐶_𝑘𝜑₋,𝑣₀∀𝑘≥0. (3.128) This proves Claim 2.
From Claims 1 and 2, we have that 𝐶_𝑘𝜑,𝑢₀=𝐶_𝑘𝜑,𝑣₀=𝛿_𝑘,1ℤ∀𝑘≥0. (3.129) Also, we have 𝐶_𝑘𝜑,0=𝛿_𝑘,0ℤ∀𝑘≥0 (3.130) (see Proposition 3.3) and 𝐶_𝑘𝜑,∞=0∀𝑘≥0 (3.131) (see Proposition 3.7).
Since 𝐾_𝜑={0,𝑢₀,𝑣₀}, from (3.129), (3.130), (3.131), and the Morse relation (2.13) with 𝑡=−1, we have 2(−1)¹+(−1)⁰=0, (3.132) a contradiction.
Therefore, there exists 𝑦₀∈𝐾_𝜑, 𝑦₀∉{0,𝑢₀,𝑣₀}. So, 𝑦₀ solves (1.5), and by the nonlinear regularity theory, 𝑦₀∈𝐶¹₀(Ω).

Remark 3.9. Even in the Hilbert space case (i.e., 𝑝=2), our result is more general than that of Wang [1], since we go beyond the Laplace differential operator and our hypotheses on the reaction 𝑓 are considerably more general.

Acknowledgments

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