Multiplicity Results for a Perturbed Elliptic Neumann Problem

Abstract

The existence of three solutions for elliptic Neumann problems with a perturbed nonlinear term depending on two real parameters is investigated. Our approach is based on variational methods.

1. Introduction

Here and in the sequel, Ω⊂ℝ^𝑛 is a bounded open set, with a boundary of class 𝐶¹, 𝑞∈𝐿^∞(Ω) with 𝑒𝑠𝑠inf_Ω𝑞>0, 𝑝>𝑛; 𝑓∶Ω×ℝ→ℝ and 𝑔∶Ω×ℝ→ℝ are 𝐿¹-Carathéodory functions.

The aim of this paper is to study the following perturbed boundary value problem with Neumann conditions:−Δ_𝑝𝑢+𝑞(𝑥)|𝑢|^𝑝−2𝑢=𝜆𝑓(𝑥,𝑢)+𝜇𝑔(𝑥,𝑢),inΩ,𝜕𝑢𝜕𝜈=0,on𝜕Ω,( (𝑃_𝜆,𝜇)) where Δ_𝑝=div(|∇𝑢|^𝑝−2∇𝑢) is the 𝑝-Laplacian, 𝜈 is the outer unit normal to 𝜕Ω, 𝜆 and 𝜇 are positive real parameters.

Nonlinear boundary value problems involving the 𝑝-Laplacian operator Δ_𝑝 (with 𝑝≠2) arise from a variety of physical problems. They are used in non-Newtonian fluids, reaction-diffusion problems, flow through porous media, and petroleum extraction (see, e.g., [1, 2]).

In the last years, several researchers have studied nonlinear problems of this type through different approaches. In [1], the authors have obtained results on the existence of a solution for the problem −Δ_𝑝𝑢+𝑔(𝑥,𝑢)=𝑓(𝑢),inΩ,𝜕𝑢𝜕𝜈=0,on𝜕Ω, (1.1) by using the perturbation result on sums of ranges of nonlinear accretive operators. Subsequently, Wei and Agarwal, in [2], have studied the same problem by developing some new techniques in the wake of [1]. Problem (𝑃_𝜆,𝜇), when 𝑞=0, 𝜆=1 and 𝑔 does not depend on 𝑢, has been studied in [3]. In this paper, the authors have obtained the existence of at least three solutions for small 𝜇, by using Implicit Function Theorem and Morse Theory. By using variational methods and in particular critical point results given by Ricceri in [4], Faraci, in her nice paper [5], has dealt with a Neumann Problem involving the 𝑝-Laplacian (for any 𝑝) of type −Δ_𝑝𝑢+𝑞(𝑥)|𝑢|^𝑝−2𝑢=𝑓(𝑥,𝑢)+𝜇𝑔(𝑥,𝑢),inΩ,𝜕𝑢𝜕𝜈=0,on𝜕Ω. (1.2) In particular, [5, Theorems 8, 9] assure the existence of three solutions for the problem given above.

In the present paper, we establish some results (Theorems 3.1, 3.2), which assure the existence of at least three weak solutions for the problem (𝑃_𝜆,𝜇). In particular the following result is a consequence of Theorem 3.2.

Theorem 1.1. Let 𝑓∶ℝ→ℝ be a nonnegative continuous function such that lim_𝑡→0+𝑓(𝑡)𝑡=0,⁶₀𝑓(𝑡)𝑑𝑡<6¹₀𝑓(𝑡)𝑑𝑡. (1.3) Then, for every 𝜆∈]3/4∫¹₀𝑓(𝑡)𝑑𝑡,9/2∫⁶₀𝑓(𝑡)𝑑𝑡[ and for every positive continuous function 𝑔∶[0,1]×ℝ→ℝ there exists 𝛿^⋆>0 such that, for each 𝜇∈]0,𝛿^∗[, the problem −𝑢^′′𝑢+𝑢=𝜆𝑓(𝑢)+𝜇𝑔(𝑥,𝑢)in]0,1[,_′(0)=𝑢^′(1) (1.4) has at least three nonzero classical solutions.

With respect to [3, 5], we stress that our results hold under different assumptions (see Remarks 3.4 and 3.5). In particular, in Theorem 1.1, no asymptotic condition at infinity is required on the nonlinear term. We also point out that in Theorems 3.1 and 3.2, precise estimates of parameters 𝜆 and 𝜇 are given.

2. Preliminaries and Basic Notations

Our main tools are three critical point theorems that we recall here in a convenient form. The first has been obtained in [6], and it is a more precise version of Theorem 3.2 of [7]. The second has been established in [7].

Theorem 2.1 (see [6, Theorem 2.6]). Let 𝑋 be a reflexive real Banach space, Φ∶𝑋→ℝ a coercive, continuously Gâteaux differentiable and sequentially weakly lower semicontinuous functional whose Gâteaux derivative admits a continuous inverse on 𝑋^∗, Ψ∶𝑋→ℝ a continuously Gâteaux differentiable functional whose Gâteaux derivative is compact such that Φ(0)=Ψ(0)=0. (2.1) Assume that there exist 𝑟>0 and 𝑥∈𝑋, with 𝑟<Φ(𝑥), such that: (𝑎₁)sup_{Φ(𝑥)≤𝑟}Ψ(𝑥)/𝑟<Ψ(𝑥)/Φ(𝑥); (𝑎₂)for each 𝜆∈Λ_𝑟∶=]Φ(𝑥)/Ψ(𝑥),𝑟/sup_{Φ(𝑥)≤𝑟}Ψ(𝑥)[ the functional Φ−𝜆Ψ is coercive. Then, for each 𝜆∈Λ_𝑟, the functional Φ−𝜆Ψ has at least three distinct critical points in 𝑋.

Theorem 2.2 (see [7, Corollary 3.1]). Let 𝑋 be a reflexive real Banach space, Φ∶𝑋→ℝ a convex, coercive and continuously Gâteaux differentiable functional whose Gâteaux derivative admits a continuous inverse on 𝑋^∗, Ψ∶𝑋→ℝ a continuously Gâteaux differentiable functional whose Gâteaux derivative is compact such that inf_𝑋Φ=Φ(0)=Ψ(0)=0. (2.2) Assume that there exist two positive constants 𝑟₁,𝑟₂, and 𝑥∈𝑋, with 2𝑟₁<Φ(𝑥)<𝑟₂/2, such that (𝑏₁)sup_𝑥∈Φ⁻¹_{(]−∞,𝑟1[)}Ψ(𝑥)/𝑟₁<(2/3)(Ψ(𝑥)/Φ(𝑥)); (𝑏₂)sup_𝑥∈Φ⁻¹_{(]−∞,𝑟2[)}Ψ(𝑥)/𝑟₂<(1/3)(Ψ(𝑥)/Φ(𝑥)); (𝑏₃)for each 𝜆∈Λ^′_𝑟1,𝑟2∶=](3/2)(Φ(𝑥)/Ψ(𝑥)), min{𝑟₁/sup_𝑥∈Φ⁻¹_{(]−∞,𝑟1[)}Ψ(𝑥), 𝑟₂/2sup_𝑥∈Φ⁻¹_{(]−∞,𝑟2[)}Ψ(𝑥)}[ and for every 𝑥₁,𝑥₂∈𝑋, which are local minima for the functional Φ−𝜆Ψ, and such that Ψ(𝑥₁)≥0 and Ψ(𝑥₂)≥0 one has inf_𝑡∈[0,1]Ψ(𝑡𝑥₁+(1−𝑡)𝑥₂)≥0. Then, for each 𝜆∈Λ^′_𝑟1,𝑟2 the functional Φ−𝜆Ψ admits three critical points which lie in Φ⁻¹(]−∞,𝑟₂[).

Now we recall some basic definitions and notations.

A function ℎ∶Ω×ℝ→ℝ is called an 𝐿¹-Carathéodory function if 𝑥→ℎ(𝑥,𝑡) is measurable for all 𝑡∈ℝ, 𝑡→ℎ(𝑥,𝑡) is continuous for almost every 𝑥∈Ω, for all 𝑀>0 one has sup_{|𝑡|≤𝑀}|ℎ(𝑥,𝑡)|∈𝐿¹(Ω). Clearly, if ℎ is continuous in Ω×ℝ, then it is 𝐿¹-Carathéodory.

We also recall that a weak solution of the problem (𝑃_𝜆,𝜇) is any 𝑢∈𝑊^1,𝑝(Ω), such that _Ω(|∇𝑢(𝑥)|^𝑝−2∇𝑢(𝑥)∇𝑣(𝑥))𝑑𝑥+_Ω(𝑞(𝑥)|𝑢(𝑥)|^𝑝−2−𝑢(𝑥)𝑣(𝑥))𝑑𝑥_Ω[𝜆𝑓(𝑥,𝑢(𝑥))+𝜇𝑔(𝑥,𝑢(𝑥))]𝑣(𝑥)𝑑𝑥=0,∀𝑣∈𝑊^1,𝑝(Ω). (2.3)

Put𝑘=sup_{𝑢∈𝑊1,𝑝(Ω)⧵{0}}sup_𝑥∈Ω|𝑢(𝑥)|(∫_Ω|∇𝑢(𝑥)|^𝑝𝑑𝑥+∫_Ω𝑞(𝑥)|𝑢(𝑥)|^𝑝𝑑𝑥)^1/𝑝. (2.4)

If Ω is convex, an explicit upper bound for the constant 𝑘 is𝑘≤2^{(𝑝−1)/𝑝}1max{(∫_Ω)𝑞(𝑥)𝑑𝑥^1/𝑝,diam(Ω)𝑛^1/𝑝(𝑝−1𝑝−𝑛𝑚(Ω))^{(𝑝−1)/𝑝}‖𝑞‖_∞∫_Ω}𝑞(𝑥)𝑑𝑥 (2.5) (see, e.g., [8, Remark 1]).

Put 𝐹(𝑥,𝜉)=∫^𝜉₀𝑓(𝑥,𝑡)𝑑𝑡 for all (𝑥,𝜉)∈Ω×ℝ and 𝐺(𝑥,𝜉)=∫^𝜉₀𝑔(𝑥,𝑡)𝑑𝑡 for all (𝑥,𝜉)∈Ω×ℝ.

Moreover, set 𝐺^𝑐∶=∫_Ωmax_{|𝜉|≤𝑐}𝐺(𝑥,𝜉)𝑑𝑥 for all 𝑐>0 and 𝐺_𝑑∶=inf_Ω×[0,𝑑]𝐺 for all 𝑑>0. Clearly, 𝐺^𝑐≥0 and 𝐺_𝑑≤0.

3. Main Results

In this section, we present our main results on the existence of at least three weak solutions for the problem (𝑃_𝜆,𝜇).

In order to introduce our first result, fixing 𝑐,𝑑>0 such that ‖𝑞‖₁𝑑^𝑝/∫_Ω𝐹(𝑥,𝑑)𝑑𝑥<𝑐^𝑝/𝑘^𝑝∫_Ωmax_{|𝜉|≤𝑐}𝐹(𝑥,𝜉)𝑑𝑥 and picking 𝜆∈Λ∶=]‖𝑞‖₁𝑑^𝑝/𝑝∫_Ω 𝐹(𝑥,𝑑)𝑑𝑥,  𝑐^𝑝/𝑝𝑘^𝑝∫_Ω max_{|𝜉|≤𝑐}𝐹(𝑥,𝜉)𝑑𝑥[, put𝑐𝛿∶=min{^𝑝−𝜆𝑝𝑘^𝑝∫_Ωmax_{|𝜉|≤𝑐}𝐹(𝑥,𝜉)𝑑𝑥𝑝𝑘^𝑝𝐺^𝑐,𝑑^𝑝‖𝑞‖₁−𝑝𝜆∫_Ω𝐹(𝑥,𝑑)𝑑𝑥|Ω|𝑝𝐺_𝑑},1𝛿∶=min{𝛿,max{0,𝑝|Ω|𝑘^𝑝limsup_{|𝜉|→+∞}[(sup_𝑥∈Ω𝐺(𝑥,𝜉))/𝜉^𝑝]}}, (3.1)(3.2) where we read 𝑟/0=+∞ so that, for instance, 𝛿=+∞ when limsup_{|𝜉|→+∞} (sup_𝑥∈Ω𝐺(𝑥,𝜉)/𝜉^𝑝)≤0 and 𝐺_𝑑=𝐺^𝑐=0.

Theorem 3.1. Let 𝑓∶Ω×ℝ→ℝ be an 𝐿¹-Carathéodory function, and put 𝐹(𝑥,𝜉)=∫^𝜉₀𝑓(𝑥,𝑡)𝑑𝑡 for all (𝑥,𝜉)∈Ω×ℝ. Assume that there exist two positive constants 𝑐,𝑑, with 𝑐<(𝑘‖𝑞‖₁^1/𝑝)𝑑, such that (i)∫_Ωmax_{|𝜉|≤𝑐}𝐹(𝑥,𝜉)𝑑𝑥/𝑐^𝑝<1𝑘^𝑝‖𝑞‖₁∫_Ω𝐹(𝑥,𝑑)𝑑𝑥𝑑^𝑝, where 𝑘 is given by (2.4); (ii)limsup_{|𝜉|→+∞}[(sup_𝑥∈Ω𝐹(𝑥,𝜉))/𝜉^𝑝]≤0. Then, for every 𝜆∈Λ∶=]‖𝑞‖₁𝑑^𝑝/𝑝∫_Ω𝐹(𝑥,𝑑)𝑑𝑥,𝑐^𝑝/𝑝𝑘^𝑝∫_Ωmax_|𝜉|<𝑐𝐹(𝑥,𝜉)𝑑𝑥[ and for every 𝐿¹(Ω)-Carathédory function 𝑔∶Ω×ℝ→ℝ such that (iii)limsup_{|𝜉|→+∞}[(sup_𝑥∈Ω𝐺(𝑥,𝜉))/𝜉^𝑝]<+∞, where 𝐺(𝑥,𝜉)=∫^𝜉₀𝑔(𝑥,𝑡)𝑑𝑡 for all (𝑥,𝜉)∈Ω×ℝ, there exists 𝛿>0 given by (3.2) such that, for each 𝜇∈[0,𝛿[, Problem (𝑃_𝜆,𝜇) has at least three weak solutions.

Proof. Fix 𝜆, 𝑔, and 𝜇 as in the conclusion. Take 𝑋=𝑊^1,𝑝(Ω) endowed with the norm ‖𝑢‖=(_Ω|∇𝑢(𝑥)|^𝑝𝑑𝑥+_Ω𝑞(𝑥)|𝑢(𝑥)|^𝑝𝑑𝑥)^1/𝑝. (3.3) On the space 𝐶⁰(Ω), we consider the norm ‖𝑢‖_∞∶=sup_𝑥∈Ω|𝑢(𝑥)|. Since 𝑝>𝑛, 𝑋 is compactly embedded in 𝐶⁰(Ω), we have ‖𝑢‖_∞≤𝑘‖𝑢‖. (3.4) Put, for each 𝑢∈𝑋, 1Φ(𝑢)=𝑝‖𝑢‖^𝑝,Ψ(𝑢)=_Ω𝜇[𝐹(𝑥,𝑢(𝑥))+𝜆𝐺(𝑥,𝑢(𝑥))]𝑑𝑥. (3.5) Since the critical points of the functional Φ−𝜆Ψ on 𝑋 are weak solutions of problem (𝑃_𝜆,𝜇), our aim is to apply Theorem 2.1 to Φ and Ψ. To this end, taking into account that the regularity assumptions of Theorem 2.1 on Φ and Ψ are satisfied, we will verify (𝑎₁) and (𝑎₂).
Put 𝑟=1/𝑝(𝑐/𝑘)^𝑝 taking into account (3.4), one hassup_{Φ(𝑢)≤𝑟}Ψ(𝑢)=sup_{Φ(𝑢)≤𝑟Ω}𝜇[𝐹(𝑥,𝑢(𝑥))+𝜆𝐺(𝑥,𝑢(𝑥))]𝑑𝑥≤_Ωmax_{|𝜉|≤𝑐}𝜇𝐹(𝑥,𝜉)𝑑𝑥+𝜆𝐺^𝑐. (3.6) Now, fix 𝑢=𝑑. Clearly 𝑢∈𝑋 and moreover Φ(𝑢)>𝑟. One has Ψ(𝑢)=_Ω𝜇𝐹(𝑥,𝑑)𝑑𝑥+𝜆_Ω𝐺(𝑥,𝑑)𝑑𝑥,Φ(1𝑢)=𝑝‖𝑢‖^𝑝=1𝑝𝑑^𝑝‖𝑞‖₁. (3.7) Hence, Ψ(𝑢)Φ(≥𝑝𝑢)‖𝑞‖₁∫_Ω𝐹(𝑥,𝑑)𝑑𝑥𝑑^𝑝+𝑝|Ω|‖𝑞‖₁𝜇𝜆𝐺_𝑑𝑑^𝑝,sup_{Φ(𝑢)≤𝑟}Ψ(𝑢)𝑟≤𝑝𝑘^𝑝∫_Ωmax_{|𝜉|≤𝑐}𝐹(𝑥,𝜉)𝑑𝑥𝑐^𝑝+𝑝𝑘^𝑝𝜇𝜆𝐺^𝑐𝑐^𝑝. (3.8) Since 𝜇<𝛿, one has 𝑐𝜇<^𝑝−𝜆𝑝𝑘^𝑝∫_Ωmax_|𝜉|<𝑐𝐹(𝑥,𝜉)𝑑𝑥𝑝𝑘^𝑝𝐺^𝑐,thismeans𝑝𝑘^𝑝∫_Ωmax_{|𝜉|≤𝑐}𝐹(𝑥,𝜉)𝑑𝑥𝑐^𝑝+𝑝𝑘^𝑝𝜇𝜆𝐺^𝑐𝑐^𝑝<1𝜆. (3.9) Furthermore, 𝜇<(𝑑^𝑝‖𝑞‖₁−𝑝𝜆∫_Ω𝐹(𝑥,𝑑)𝑑𝑥)/𝑝|Ω|𝐺_𝑑; this means that (𝑝/‖𝑞‖₁)(∫_Ω𝐹(𝑥,𝑑)𝑑𝑥/𝑑^𝑝)+(𝑝|Ω|/‖𝑞‖₁)(𝜇/𝜆)(𝐺_𝑑/𝑑^𝑝)>1/𝜆.
Then,𝑝𝑘^𝑝∫_Ωmax_{|𝜉|≤𝑐}𝐹(𝑥,𝜉)𝑑𝑥𝑐^𝑝+𝑝𝑘^𝑝𝜇𝜆𝐺^𝑐𝑐^𝑝<1𝜆<𝑝‖𝑞‖₁∫_Ω𝐹(𝑥,𝑑)𝑑𝑥𝑑^𝑝+|Ω|𝑝‖𝑞‖₁𝜇𝜆𝐺_𝑑𝑑^𝑝. (3.10) Hence, from (3.8) and (3.10), condition (𝑎₁) of Theorem 2.1 is verified.
Finally, since 𝜇<𝛿, we can fix 𝑙>0 such that limsup_{|𝜉|→+∞}[(sup_𝑥∈Ω𝐺(𝑥,𝜉))/𝜉^𝑝]<𝑙 and 𝜇𝑙<1/|Ω|𝑝𝑘^𝑝. Therefore, there exists a function ℎ∈𝐿¹(Ω) such that𝐺(𝑥,𝜉)≤𝑙𝜉^𝑝+ℎ(𝑥) (3.11) for each (𝑥,𝜉)∈Ω×ℝ.
Now, fix 0<𝜀<(1/𝑝|Ω|𝑘^𝑝𝜆)−(𝜇𝑙/𝜆); from (ii) there is a function ℎ_𝜀∈𝐿¹(Ω) such that𝐹(𝑥,𝜉)≤𝜀𝜉^𝑝+ℎ_𝜀(𝑥) (3.12) for each (𝑥,𝜉)∈Ω×ℝ. It follows that, for each 𝑢∈𝑋, 1Φ(𝑢)−𝜆Ψ(𝑢)≥(𝑝−𝜆|Ω|𝑘^𝑝𝜀−𝜇|Ω|𝑘^𝑝𝑙)‖𝑢‖^𝑝−𝜆‖ℎ_𝜀‖₁−𝜇‖ℎ‖₁. (3.13) This leads to the coercivity of Φ−𝜆Ψ, and condition (𝑎₂) of Theorem 2.1 is verified. Since, from (3.8) and (3.10), 𝜆∈]Φ(𝑢)Ψ(,𝑟𝑢)sup_{Φ(𝑢)≤𝑟}Ψ(𝑢)[, (3.14) Theorem 2.1 assures the existence of three critical points for the functional Φ−𝜆Ψ, and the proof is complete.

Now, we state a variant of Theorem 3.1. Here no asymptotic condition on 𝑔 is requested; on the other hand, the functions 𝑓, 𝑔 are supposed to be nonnegative.

Fix 𝑐₁,𝑑,𝑐₂>0 such that 3‖𝑞‖₁𝑑^𝑝/2∫_Ω𝐹(𝑥,𝑑)𝑑𝑥<1/𝑘^𝑝min{𝑐^𝑝₁/∫_Ω𝐹(𝑥,𝑐₁)𝑑𝑥,  𝑐^𝑝₂/2∫_Ω𝐹(𝑥,𝑐₂)𝑑𝑥} and picking 𝜆∈Λ∶=]3‖𝑞‖₁𝑑^𝑝2𝑝∫_Ω,1𝐹(𝑥,𝑑)𝑑𝑥𝑝𝑘^𝑝𝑐min{^𝑝1∫_Ω𝐹(𝑥,𝑐₁,𝑐)𝑑𝑥2∫_Ω𝐹(𝑥^𝑝2,𝑐₂)𝑑𝑥}[, (3.15) put 𝛿^∗𝑐∶=min{^𝑝1−𝜆𝑝𝑘^𝑝∫_Ω𝐹(𝑥,𝑐₁)𝑑𝑥𝑝𝑘^𝑝𝐺^𝑐1,𝑐^𝑝₂−2𝜆𝑝𝑘^𝑝∫_Ω𝐹(𝑥,𝑐₂)𝑑𝑥2𝑝𝑘^𝑝𝐺^𝑐2}. (3.16)

Theorem 3.2. Assume that there exist three positive constants 𝑐₁,𝑐₂,𝑑, with 2^1/𝑝𝑐₁<𝑘‖𝑞‖₁^1/𝑝𝑑<2^−1/𝑝𝑐₂, such that (j)𝑓(𝑥,𝜉)≥0 for each (𝑥,𝜉)∈Ω×[0,𝑐₂]; (jj)∫_Ω𝐹(𝑥,𝑐₁)𝑑𝑥/𝑐^𝑝₁<(2/3𝑘^𝑝‖𝑞‖₁)(∫_Ω𝐹(𝑥,𝑑)𝑑𝑥/𝑑^𝑝);(jjj)∫_Ω𝐹(𝑥,𝑐₂)𝑑𝑥/𝑐^𝑝₂<(1/3𝑘^𝑝‖𝑞‖₁)(∫_Ω𝐹(𝑥,𝑑)𝑑𝑥/𝑑^𝑝). Then, for every 𝜆∈Λ∶=]3‖𝑞‖₁𝑑^𝑝2𝑝∫_Ω,1𝐹(𝑥,𝑑)𝑑𝑥𝑝𝑘^𝑝𝑐min{^𝑝1∫_Ω𝐹(𝑥,𝑐₁,𝑐)𝑑𝑥2∫_Ω𝐹(𝑥^𝑝2,𝑐₂)𝑑𝑥}[ (3.17) and for every nonnegative 𝐿¹-Carathédory function 𝑔∶Ω×ℝ→ℝ, there exists 𝛿^∗>0 given by (3.16) such that, for each 𝜇∈]0,𝛿^∗[, the problem (𝑃_𝜆,𝜇) has at least three weak solutions 𝑢_𝑖, 𝑖=1,2,3, such that 0≤𝑢_𝑖(𝑥)<𝑐₂,∀𝑥∈Ω,𝑖=1,2,3. (3.18)

Proof. Without loss of generality, we can assume 𝑓(𝑥,𝑡)≥0 for all (𝑥,𝑡)∈Ω×ℝ. Fix 𝜆, 𝑔, and 𝜇 as in the conclusion and take 𝑋, Φ and Ψ as in the proof of Theorem 3.1.
We observe that the regularity assumptions of Theorem 2.2 on Φ and Ψ are satisfied.
Then, our aim is to verify (𝑏₁) and (𝑏₂). To this end, put 𝑢 as in Theorem 3.1, 𝑟₁=(1/𝑝)(𝑐₁/𝑘)^𝑝, and 𝑟₂=(1/𝑝)(𝑐₂/𝑘)^𝑝. Therefore, one has 2𝑟₁<Φ(𝑢)<𝑟₂/2 and, since 𝜇<𝛿^∗ and 𝐺_𝑑=0, one hassup_{Φ(𝑢)<𝑟1}Ψ(𝑢)𝑟₁≤𝑝𝑘^𝑝∫_Ω𝐹(𝑥,𝑐₁)𝑑𝑥𝑐^𝑝₁+𝑝𝑘^𝑝𝜇𝜆𝐺^𝑐1𝑐^𝑝₁<1𝜆<2𝑝3‖𝑞‖₁∫_Ω𝐹(𝑥,𝑑)𝑑𝑥𝑑^𝑝+𝜇𝜆2|Ω|𝑝3‖𝑞‖₁𝐺_𝑑𝑑^𝑝≤23Ψ(𝑢)Φ(,𝑢)2sup_{Φ(𝑢)<𝑟2}Ψ(𝑢)𝑟₂≤2𝑝𝑘^𝑝∫_Ω𝐹(𝑥,𝑐₂)𝑑𝑥𝑐^𝑝₂+2𝑝𝑘^𝑝𝜇𝜆𝐺^𝑐2𝑐^𝑝₂<1𝜆<2𝑝3‖𝑞‖₁∫_Ω𝐹(𝑥,𝑑)𝑑𝑥𝑑^𝑝+𝜇𝜆2|Ω|𝑝3‖𝑞‖₁𝐺_𝑑𝑑^𝑝≤23Ψ(𝑢)Φ(.𝑢) (3.19) Therefore, (𝑏₁) and (𝑏₂) of Theorem 2.2 are verified.
Finally, we verify that Φ−𝜆Ψ satisfies assumption (𝑏₃) of Theorem 2.2.
Let 𝑢₁ and 𝑢₂ be two local minima for Φ−𝜆Ψ. Then 𝑢₁ and 𝑢₂ are critical points for Φ−𝜆Ψ, and so, they are weak solutions for the problem (𝑃_𝜆,𝜇). For every positive parameter 𝜆,𝜇, and for every (𝑥,𝑡)∈Ω×[0,+∞[ one has (𝜆𝑓+𝜇𝑔)(𝑥,𝑡)≥0, hence, owing to the Weak Maximum Principle (see for instance [9]) we obtain 𝑢₁(𝑥)≥0,𝑢₂(𝑥)≥0,forall𝑥∈Ω. Then, it follows that 𝑠𝑢₁+(1−𝑠)𝑢₂≥0,forall𝑠∈[0,1], and that (𝜆𝑓+𝜇𝑔)(𝑥,𝑠𝑢₁+(1−𝑠)𝑢₂)≥0, and, hence, Ψ(𝑠𝑢₁+(1−𝑠)𝑢₂)≥0,forall𝑠∈[0,1].
From Theorem 2.2, the functional Φ−𝜆Ψ has at least three distinct critical points which are weak solutions of (𝑃_𝜆,𝜇) and the conclusion is achieved.

Example 3.3. Consider the following problem −Δ₃𝑢+|𝑢|𝑢=320|𝑥|²𝑢³1+𝑢⁸+𝜇(|𝑥|²+1)|𝑢|³,inΩ,𝜕𝑢𝜕𝜈=0,on𝜕Ω, (3.20) where Ω={𝑥∈ℝ²∶|𝑥|<1}. Then, owing to Theorem 3.2, for each 𝜇∈]0,10⁻³[, the problem given above has at least three weak solutions. It is enough to choose, for instance, 𝑐₁=10⁻⁶,𝑐₂=10² and 𝑑=1.

Remark 3.4. We observe that [5, Theorem 8] cannot be applied to the problem of Example 3.3 since the assumption(𝑁^′₆)thereexist𝑠>1,𝛼>0,𝛽∈𝐿¹(Ω)with𝑠<𝑝such that|𝐺(𝑥,𝑢)|≤𝛼|𝑢|^𝑠+𝛽(𝑥) does not hold. Moreover, also [5, Theorem 9] cannot be applied since condition (𝑁₇^′′)limsup_𝑢→0+(inf_𝑥∈Ω∫^𝑢₀𝑔(𝑥,𝑡)𝑑𝑡/|𝑢|^𝑝)=+∞ is not verified.
Finally, we observe that [5, Theorem 7] and [5, Theorem 9] ensure only two solutions and two nonzero solutions, respectively.
Furthermore, contrary to theorems in [5], owing to our results, we have precise values of 𝜇 for which the problem admits solutions.
On the other hand, we observe that in [5], the case 𝑝≤𝑛 is investigated too.

Remark 3.5. It is easy to verify that our results and theorems in [3] are mutually independent. In particular, we remark that none of theorems in [3] can be applied to the problem in Example 3.3. We also observe that in [3] no estimate of small 𝜇 is given.

Proof of Theorem 1.1. Our aim is to apply Theorem 3.2 by choosing 𝑛=1,𝑝=2,and𝑞=1. Put 𝑑=1 and 𝑐₂=6. Therefore, taking into account that 𝑘=√2, one has 3‖𝑞‖₁𝑑^𝑝=32𝑝𝐹(𝑑)4∫¹₀,1𝑓(𝑡)𝑑𝑡𝑝𝑘^𝑝𝑐^𝑝₂2𝐹(𝑐₂)=92∫⁶₀.𝑓(𝑡)𝑑𝑡 (3.21) Moreover, since lim_𝑡→0⁺(𝐹(𝑡)/𝑡²)=0, there is a positive constant 𝑐₁<1, such that (𝐹(𝑐₁)/𝑐²₁)<(1/3)∫¹₀𝑓(𝑡)𝑑𝑡 and 𝑐²₁/𝐹(𝑐₁)>18/∫⁶₀𝑓(𝑡)𝑑𝑡. Hence, a simple computation shows that all assumptions of Theorem 3.2 are satisfied, and the conclusion follows.

Remark 3.6. We explicitly observe that in Theorem 3.2, as well as in Theorem 1.1, no asymptotic condition at infinity on the nonlinear term is requested.

Example 3.7. The function 𝑓∶ℝ→ℝ1𝑓(𝑥)=0,𝑡≤2,12𝑡−1,2<𝑡≤1,33𝑡−32𝑡,1<𝑡≤263,12568,263𝑒256<𝑡≤6,_𝑡−𝑒⁶+18,𝑡>6, (3.22) satisfies all assumptions of Theorem 1.1.

Remark 3.8. We explicitly observe that the very nice Theorem 1 of [10] cannot be applied to the function of Example 3.7 since lim_𝑡→+∞(𝑓(𝑡)/𝑡)=+∞.