Electronic Journal of Differential Equations, Vol. 2023 (2023), No. 61, pp. 1–12. ISSN: 1072-6691. URL: https://ejde.math.txstate.edu, https://ejde.math.unt.edu DOI: 10.58997/ejde.2023.61 SINGULAR p-BIHARMONIC PROBLEMS INVOLVING THE HARDY-SOBOLEV EXPONENT AMOR DRISSI, ABDELJABBAR GHANMI, DUŠAN D. REPOVŠ Abstract. This article concerns the existence and multiplicity of solutions for the singular p-biharmonic problem involving the Hardy potential and the critical Hardy-Sobolev exponent. To this end we use variational methods com- bined with the Mountain pass theorem and the Ekeland variational principle. We illustrate the usefulness of our results with and example. 1. Introduction Recently, a lot of attention has been paid to the study of problems involving the p-Laplacian operator and the p-biharmonic operator. We refer the reader to Alsaedi et al. [1], Cung et al. [10], Huang and Liu [15], and Sun and Wu [24, 25]. The reason for studying these problems is their applications in fields such as quantum mechanics, flame propagation, and traveling waves in suspension bridges; for more applications see Bucur and Valdinoci [6], and Lazer and McKenna [16]. Problems involving Hardy terms have been extensively investigated by several authors, see, e.g., Bhakta et al. [3, 4], Ghoussoub and Yuan [13], and Guan et al. [14]. Various problems involving the critical Hardy-Sobolev exponent have been widely studied, see, e.g., Chaharlang and Razani [7], Chen et al. [9], Perera and Zou [19], Pérez- Llanos and Primo [20], Wang [26], and Wang and Zhao [27]. In particular, Ghoussoub and Yuan [13] used variational methods to study the existence of solutions of the problem −∆pϕ = λ|ϕ|r−2ϕ+ µ |ϕ|q−2ϕ |x|α in Ω, ϕ = 0 on ∂Ω, where Ω ⊂ Rn is a regular bounded domain, µ and λ are positive parameters, min(q, r) ≥ p, q ≤ p∗(α), and r ≤ p∗. 2020 Mathematics Subject Classification. 31B30, 35J35, 49J35. Key words and phrases. p-Laplacian operator; p-Biharmonic equation; Variational method; Existence of solutions; Hardy potential; Critical Hardy-Sobolev exponent; Ekeland variational principle; Mountain pass geometry. ©2023. This work is licensed under a CC BY 4.0 license. Submitted March 8, 2023. Published September 18, 2023. 1 2 A. DRISSI, A. GHANMI, D. D. REPOVŠ EJDE-2023/61 Perrera and Zou [19] investigated the critical Hardy p-Laplacian problem −∆pϕ = λ|ϕ|p−2ϕ+ |ϕ|p∗(α)−2ϕ |x|α in Ω, ϕ = 0 on ∂Ω. (1.1) More precisely, they used variational methods to establish the multiplicity of solu- tions of problem (1.1). Recently, Wang [26] considered the problem ∆2 pϕ = h(x, ϕ) + µ |ϕ|r−2ϕ |x|s in Ω, ϕ = ∆ϕ = 0 on ∂Ω. (1.2) He used the Mountain pass theorem to establish the existence of solutions of prob- lem (1.2). Moreover, the existence of multiple solutions was established by applying the Fountain Theorem. Motivated by the above mentioned results, we study in the existence and mul- tiplicity of solutions of the following singular p-biharmonic problem involving the Hardy potential and the critical Hardy-Sobolev exponent, ∆2 pϕ− λ |ϕ|p−2ϕ |x|2p + ∆pϕ = µf(x)h(ϕ) + |ϕ|p∗(α)−2ϕ |x|α in RN , (1.3) where 0 ≤ α < 2p, 1 < p < N 2 , λ > 0, µ > 0, and p∗(α) := p(N−α) N−2p . Here, ∆p and ∆2 p are the p-Laplacian and the p-biharmonic operator, respectively, defined by ∆pϕ := div(|∇ϕ|p−2∇ϕ), ∆2 pϕ := ∆(|∆ϕ|p−2∆ϕ), respectively, f is a positive function, h is a continuous function. We use the following hypotheses: (H1) There exists r ∈ (p, p∗) such that f ∈ L∞(RN ) and |h(ϕ)| ≤ c1|ϕ|r−1 for every ϕ ∈ E and some positive constant c1, where p∗ := pN N−2p and the space E := W 2,p(RN ) is defined in Section 2. (H2) There exists σ > 0 such that for every y ∈ RN , we have 0 < rH(ϕ) ≤ h(ϕ)ϕ, |ϕ| ≥ σ > 0, where H(t) := ∫ t 0 h(s)ds. (H3) There exist c1 > 0, 1 < r < p, and s ∈ ( p∗ p∗−r , p p−r ) such that 0 < f ∈ L p∗ p∗−r (RN ) ∩ Lsloc(RN ), and |h(ϕ)| ≤ c1|ϕ|r−1 for for every ϕ ∈ E. The following are the main results of this article. Theorem 1.1. Suppose that (H1), (H2) hold. Then for every µ > 0, the singular p-biharmonic problem (1.3) has at least one nontrivial weak solution, provided that λ > 0 is small enough. Theorem 1.2. Suppose that (H2), (H3) hold. Then there exists µ0 > 0 such that for every µ ∈ (0, µ0), the singular p-biharmonic problem (1.3) has at least two nontrivial weak solutions, provided that λ > 0 is small enough. Note that the singular p-biharmonic problem (1.3) is very important since it contains the p-biharmonic operator, the p-Laplacian operator, the singular nonlin- earity, and the Hardy potential. Moreover, it appears in many applications, such as non-Newtonian fluids, viscous fluids, traveling waves in suspension bridges, and EJDE-2023/61 SINGULAR p-BIHARMONIC PROBLEMS 3 various other physical phenomena, see, e.g., Chen et al. [8], Lazer and McKenna [16], and Ružička [22]. The article is organized as follows: In Section 2, we present some variational framework related to problem (1.3). In Section 3, we prove Theorem 1.1. In Section 4 we combine the Mountain pass theorem with the Ekeland variational principle to prove the multiplicity of solutions of problem (1.3) (Theorem 1.2). In Section 5 we present an example that illustrate our main results. Finally, in Section 6 we summarize the main contributions of this article. 2. Preliminaries We begin by recalling some necessary facts related to the Hardy-Sobolev expo- nent nonlinearity. We finish this section by presenting the variational framework related to problem (1.3). For other necessary background material we refer to the comprehensive monograph by Papageorgiou et al. [18]. It is well-known that the Hardy-Sobolev exponent is closely related to the Rellich inequality (see Davies and Hinz [11, p. 520])∫ RN |ϕ(x)|p |x|2p dx ≤ ( p2 N(p− 1)(N − 2p) )p ∫ RN |∆ϕ(x)|p dx, (2.1) for every ϕ ∈ W 2,p(RN ), where W 2,p(RN ) denotes the Sobolev space which is defined by W 2,p(RN ) := { ϕ ∈ Lp(RN ) : ∆ϕ, |∇ϕ| ∈ Lp(RN ) } . For more details about this space, see Davies and Hinz [11], Mitidieri [17], and Rellich [21]. According to the Rellich inequality (2.1), W 2,p(RN ) can be endowed with the following norm ‖ϕ‖ := (∫ RN |∆ϕ(x)|p − λ |ϕ(x)|p |x|2p + |∇ϕ(x)|p dx )1/p , provided that 0 < λ < (N(p− 1)(N − 2p) p2 )p . (2.2) The space W 2,p(RN ) is continuously embeddable into Lσ(RN ) for every p ≤ σ ≤ p∗, and compactly embeddable into Lσloc(RN ), for every p ≤ σ < p∗. Moreover, for every ϕ ∈W 2,p(RN ), one has |ϕ|σ ≤ S−1/p σ ‖ϕ‖, (2.3) where |ϕ|p denotes the usual Lp(RN )-norm and Sσ is defined by Sσ := inf ϕ∈W 2,p(RN ),ϕ6=0 ∫ RN |∆ϕ(x)|p − λ |ϕ(x)|p |x|2p + |∇ϕ(x)|p dx(∫ RN |x−α||ϕ(x)|σ dx )p/σ . (2.4) Hereafter, for simplicity, we shall denote E := W 2,p(RN ). We define the weighted Lebesgue space Lr(RN, f) by Lr(RN, f) := { ϕ : RN → R : ϕ is measurable and ∫ RN f(x)|ϕ(x)|r dx <∞ } , and endow it with the norm ‖ϕ‖r,f := (∫ RN f(x)|ϕ(x)|r dx )1/r . 4 A. DRISSI, A. GHANMI, D. D. REPOVŠ EJDE-2023/61 Then Lr(RN, f) is a uniformly convex Banach space. Dhifli and Alsaedi [12] proved that under hypothesis (H3), the embedding E ↪→ Lr(RN , f) is continuous and compact. Moreover, one has the estimate ‖ϕ‖rr,f ≤ S −r/p p∗ |f | p∗ p∗−r ‖ϕ‖r, for every ϕ ∈ E. (2.5) Now, let us introduce the notion of weak solutions. Definition 2.1. A function ϕ ∈ E is said to be a weak solution of problem (1.3), provided that Λ(ϕ,ψ) = µ ∫ RN f(x)h(ϕ)ψ dx+ ∫ RN |x|−αϕp ∗(α)−2ϕψ dx, for every ψ ∈ E, where Λ(ϕ,ψ) := ∫ RN |∆ϕ|p−2∆ϕ∆ψ − λ |ϕ| p−2ϕψ |x|2p + |∇ϕ|p−2∇ϕ∇ψ dx. We define the energy functional Jµ : E → R, by Jµ(ϕ) := 1 p ‖ϕ‖p − µ ∫ RN f(x)h(ϕ)ϕdx− 1 p∗(α) ∫ RN |x|−αϕp ∗(α) dx. Note that a function ϕ ∈ E is a weak solution of (1.3), if it satisfies J ′µ(ϕ) = 0, i.e., ϕ is a critical value for Jµ. Definition 2.2. We say that a function Φ ∈ C1(F,R), where F is a Banach space, satisfies the Palais-Smale condition, if every sequence {ϕn} ⊂ F , such that Φ(ϕn) is bounded and Φ′(ϕn)→ 0 in F ∗, as n→∞, contains a convergent subsequence. To prove Theorem 1.1, we need the following result which is proved in Ambrosetti and Rabinowitz [2, Theorem 2.4]. Theorem 2.3 (Mountain pass theorem). Let Φ ∈ C1(F,R), where F is a Ba- nach space, and suppose that ϕ ∈ F is such that ||ϕ|| > r, for some r > 0, and inf ||ψ||=r Φ(ψ) > Φ(0) > Φ(ϕ). If in addition, Φ satisfies the Palais-Smale condi- tion at level c, then c is a critical value of Φ, where c := infγ∈Γ maxs∈[0,1] Φ(γ(s)) and Γ = {γ ∈ C([0, 1], F ) : (γ(0), γ(1)) = (0, ϕ)}. 3. Proof of Theorem 1.1 In this section, we shall prove the first main result of this paper. More precisely, under suitable conditions, we shall prove that the functional energy associated with problem (1.3) satisfies the Mountain pass geometry. First, we shall prove several lemmas. Lemma 3.1. Under hypotheses (H1) and (H2), there exist ρ > 0 and η > 0 such that ‖ϕ‖ = ρ implies Jµ(ϕ) ≥ η > 0. Proof. Let ϕ ∈ E. From (H1), (H2) and (2.3), we obtain Jµ(ϕ) = 1 p ‖ϕ‖p − µ ∫ RN f(x)H(ϕ) dx− 1 p∗(α) ∫ RN |x|−αup ∗(α) dx ≥ 1 p ‖ϕ‖p − µ r ∫ RN f(x)h(ϕ) dx− 1 p∗(α) S − p ∗(α) p p∗(α) ‖ϕ‖ p∗(α) ≥ 1 p ‖ϕ‖p − µ r c1‖f‖∞|u|rr − 1 p∗(α) S − p ∗(α) p p∗(α) ‖ϕ‖ p∗(α) EJDE-2023/61 SINGULAR p-BIHARMONIC PROBLEMS 5 ≥ 1 p ‖ϕ‖p − µ r c1‖f‖∞S−r/pr ‖ϕ‖r − 1 p∗(α) S − p ∗(α) p p∗(α) ‖ϕ‖ p∗(α) ≥ ‖ϕ‖p {1 p − µ r c1‖f‖∞S−r/pr ‖ϕ‖r−p − 1 p∗(α) S − p ∗(α) p p∗(α) ‖ϕ‖ p∗(α)−p } , and since min(r, p∗(α)) > p, we obtain lim ‖ϕ‖→0 {1 p − µ r c1‖f‖∞S−r/pr ‖ϕ‖r−p − 1 p∗(α) S − p ∗(α) p p∗(α) ‖ϕ‖ p∗(α)−p } = 1 p > 0, therefore, for ρ > 0 small enough, if ‖ϕ‖ = ρ, we obtain η := ρp (1 p − µ r c1‖f‖∞S−r/pr ρr−p − 1 p∗(α) S − p ∗(α) p p∗(α) ρp ∗(α)−p ) > 0, thus ‖ϕ‖ = ρ =⇒ Jµ(ϕ) ≥ η > 0. This completes the proof. � Lemma 3.2. Under the hypotheses of Lemma 3.1, there exists e ∈ E such that ‖e‖ > ρ and Jµ(e) < 0. Proof. Let ϕ be a positive function in C∞c (E). Then for every s > 0 we have Jµ(sϕ) = sp p ‖ϕ‖p − µ ∫ RN f(x)H(sϕ) dx− sp ∗(α) p∗(α) ∫ RN |x|−αϕp ∗(α) dx ≤ sp p ‖ϕ‖p − sp ∗(α) p∗(α) ∫ RN |x|−αϕp ∗(α) dx. Since p < p∗(α), it follows that Jµ(sϕ) → −∞, as s → ∞. Therefore there exists s0 > ρ ‖ϕ‖ large enough, such that Jµ(s0ϕ) < 0. If we now set e = s0ϕ, then ‖e‖ > ρ and Jµ(e) < 0. This completes the proof. � Lemma 3.3. Under the hypotheses of Lemma 3.1, Jµ satisfies the Palais-Smale condition. Proof. Let {ϕn} be a Palais-Smale sequence, which means that Jµ(ϕn) is bounded and J ′µ(ϕn) → 0, as n → ∞. Therefore there exist m1 > 0 and m2 > 0 such that Jµ(ϕn) ≤ m1 and |J ′µ(ϕn)| ≤ m2. Letting θ := min(r, p∗(α)), we obtain by hypothesis (H1) that θm1 +m2 ≥ θJµ(ϕn)− 〈J ′µ(ϕn), ϕn〉 ≥ θ p ‖ϕn‖p − µθ ∫ RN f(x)H(ϕn) dx− θ p∗(α) ∫ RN |x|−α|ϕn|p ∗(α) dx − ‖ϕn‖p + µ ∫ RN f(x)h(ϕn)ϕn dx+ ∫ RN |x|−αϕp ∗(α) n dx ≥ ( θ p − 1)‖ϕn‖p + µ(r − θ) ∫ RN f(x)H(ϕn) dx + (1− θ p∗(α) ) ∫ RN |x|−α|ϕn|p ∗(α) dx ≥ ( θ p − 1)‖ϕn‖p, and since θ = min(r, p∗(α)) > p, it follows that the sequence {ϕn} is bounded in E. Therefore (up to a subsequence) there exists ϕ ∈ E such that ϕn ⇀ ϕ weakly in E, 6 A. DRISSI, A. GHANMI, D. D. REPOVŠ EJDE-2023/61 ϕn → ϕ strongly in Lr(RN ), ϕn → ϕ a.e. in Rn, so, by (H1), (H2) and the Dominated convergence theorem, lim n→∞ ∫ RN f(x)H(ϕn) dx = ∫ RN f(x)H(ϕ) dx. (3.1) One can now show by a standard argument that the weak limit u of {ϕn} is a critical point of Jµ and thus J ′µ(ϕ) = 0. Let wn := ϕn − ϕ. Then wn converges weakly to zero. Moreover, by Brezis and Lieb [5, Lemma 3], we obtain |wn|p ∗(α) p∗(α) = |ϕn|p ∗(α) p∗(α) − |ϕ| p∗(α) p∗(α) + o(1), therefore, lim n→∞ ∫ RN |x|−α|ϕn|p ∗(α) − |x|−α|wn|p ∗(α) dx = ∫ RN |x|−α|ϕ|p ∗(α) dx, and from (3.1) we have 〈J ′µ(ϕn), ϕn〉 − 〈J ′µ(ϕ), ϕ〉 = ‖wn‖p − ∫ RN |x|−α|wn|p ∗(α) dx+ o(1), hence for n large enough, ‖wn‖p = ∫ RN |x|−α|wn|p ∗(α) dx+ o(1), thus lim n→∞ ‖wn‖p = lim n→∞ ∫ RN |x|−α|wn|p ∗(α) = l ≥ 0. (3.2) If l > 0, then by combining equation (2.4) with (3.2) we obtain l ≥ S p p∗(α)−p p∗(α) . (3.3) On the other hand, one has Jµ(ϕn)− Jµ(ϕ) = 1 p ‖wn‖p − 1 p∗(α) ∫ RN |x|−α|wn|p ∗(α) dx+ o(1), so by letting n tend to infinity, we obtain c− Jµ(ϕ) = ( 1 p − 1 p∗(α) )l, and using the last equation and (3.3) we obtain Jµ(ϕ) + ( 1 p − 1 p∗(α) )l = c < ( 1 p − 1 p∗(α) )S p p∗(α)−p λ , which implies that Jµ(ϕ) < 0. (3.4) However, we have 〈J ′µ(ϕ), ϕ〉 = 0, for every ϕ ∈ E. So, from (H2) we obtain ‖ϕ‖p = µ ∫ RN f(x)h(ϕ)ϕdx+ ∫ RN |x|−α|ϕ|p ∗(α) dx ≥ rµ ∫ RN f(x)H(ϕ) dx+ ∫ RN |x|−α|ϕ|p ∗(α) dx; EJDE-2023/61 SINGULAR p-BIHARMONIC PROBLEMS 7 therefore, Jµ(ϕ) = 1 p ‖ϕ‖p − µ ∫ RN f(x)H(ϕ) dx− 1 p∗(α) ∫ RN |x|−α|u|p ∗(α) dx ≥ 1 p ( rµ ∫ RN f(x)H(ϕ) dx+ ∫ RN |x|−α|u|p ∗(α) dx ) − µ ∫ RN f(x)H(ϕ) dx− 1 p∗(α) ∫ RN |x|−α|u|p ∗(α) dx ≥ µ( r p − 1) ∫ RN f(x)H(ϕ) dx+ ( 1 p − 1 p∗(α) ) ∫ RN |x|−α|u|p ∗(α) dx, and since r ∈ (p, p∗) and p < p∗(α), it follows that Jµ(ϕ) ≥ 0. This is in contradic- tion with (3.4). Since l = 0, we see by (3.2) that {ϕn} converges strongly to ϕ in E. This completes the proof. � Proof of Theorem 1.1. By Lemma 3.1, there exist ρ ∈ (0,∞) and η ∈ (0,∞) such that inf‖ϕ‖=ρ Jµ(ϕ) ≥ η > 0. On the other hand, by Lemma 3.2, there exists e ∈ E such that ρ ≤ ‖e‖ and Jµ(e) < 0 < inf ‖ϕ‖=ρ Jµ(ϕ), hence, combining Lemma 3.3 and Theorem 2.3, we can establish the existence of a critical point ϕµ. Moreover, ϕµ is characterized by Jµ(ϕµ) = inf γ∈Γ max t∈[0,1] Jµ(γ(t)), where Γ := {γ ∈ C([0, 1], X) : (γ(0), γ(1)) = (0, e)}, so if we take γ(s) = se, then there exists s0 ∈ [0, 1] such that ‖s0e‖ = ρ, hence invoking Lemma 3.2, we obtain Jµ(ϕµ) ≥ η > 0. (3.5) This completes the proof. � 4. Proof of Theorem 1.2 The proof is divided into several lemmas. Lemma 4.1. Under hypotheses (H2) and (H3), there exist positive constants µ0, ρ, and η such that for every µ ∈ (0, µ0), ‖ϕ‖ = ρ implies Jµ(ϕ) ≥ η > 0. 8 A. DRISSI, A. GHANMI, D. D. REPOVŠ EJDE-2023/61 Proof. Let ϕ ∈ E. Invoking hypotheses (H2), (H3), equations (2.3), (2.5), and the Hölder inequality, we obtain Jµ(ϕ) = 1 p ‖ϕ‖p − µ ∫ RN f(x)H(ϕ) dx− 1 p∗(α) ∫ RN |x|−αϕp ∗(α) dx ≥ 1 p ‖ϕ‖p − µ r ∫ RN f(x)h(ϕ) dx− 1 p∗(α) S − p ∗(α) p p∗ ‖ϕ‖p ∗(α) ≥ 1 p ‖ϕ‖p − µ r c1‖ϕ‖rr,f − 1 p∗(α) S − p ∗(α) p p∗ ‖ϕ‖p ∗(α) ≥ 1 p ‖ϕ‖p − µ r c1‖f‖ p∗ p∗−r S −r/p p∗ ‖ϕ‖r − 1 p∗(α) S − p ∗(α) p p∗ ‖ϕ‖p ∗(α) ≥ ‖ϕ‖r (1 p ‖ϕ‖p−r − µ r c1‖f‖ p∗ p∗−r S −r/p p∗ − 1 p∗(α) S − p ∗(α) p p∗ ‖ϕ‖p ∗(α)−r ) ≥ ‖ϕ‖r ( h ( ‖ϕ‖ ) − µ r c1‖f‖ p∗ p∗−r S −r/p p∗ ) , (4.1) where h(s) := 1 p sp−r − 1 p∗(α) S − p ∗(α) p p∗ sp ∗(α)−r. It is not difficult to prove that h attains its global maximum at s0 := (p∗(α)(p− r)S p∗(α) p p∗ p ( p∗(α)− r ) ) 1 p∗−p . Set µ0 := rf(s0) c1‖f‖ p∗ p∗−r S −r/p p∗ . (4.2) Then for every µ ∈ (0, µ0), we have h(s0)− µ r c1‖f‖ p∗ p∗−r S −r/p p∗ > 0, and since h is continuous, we can find ρ > 0 such that h(ρ)− µ r c1‖f‖ p∗ p∗−r S −r/p p∗ > 0, thus for every ϕ ∈ E with ‖ϕ‖ = ρ, we have Jµ(ϕ) ≥ η := ρr ( h(ρ)− µ r c1‖f‖ p∗ p∗−r S −r/p p∗ ) > 0. This completes the proof. � Lemma 4.2. There exists e ∈ E such that ‖e‖ > ρ and Jµ(e) < 0. Since the proof of the above lemma is very similar to that of Lemma 3.2, we omit it. Lemma 4.3. Under hypotheses (H2) and (H3), the functional Jµ satisfies the Palais-Smale condition. Proof. Let {ϕn} be a Palais-Smale sequence. By the argument from the previous section, it follows that there exist m1 > 0 and m2 > 0, such that Jµ(ϕn) ≤ m1 and |J ′µ(ϕn)| ≤ m2. EJDE-2023/61 SINGULAR p-BIHARMONIC PROBLEMS 9 Let us prove that {ϕn} is bounded. If not, then up to a subsequence we can assume that ‖ϕn‖ → ∞, as n→∞. By hypotheses (H2) and (H3), we obtain p∗(α)m1 +m2 ≥ p∗(α)Jµ(ϕn)− 〈J ′µ(ϕn), ϕn〉 ≥ p∗(α) p ‖ϕn‖p − µp∗(α) ∫ RN f(x)H(ϕn) dx − p∗(α) p∗(α) ∫ RN |x|−αϕp ∗(α) n dx− ‖ϕn‖p + µ ∫ RN f(x)h(ϕn)ϕn dx+ ∫ RN |x|−αϕp ∗(α) n dx ≥ (p∗(α) p − 1 ) ‖ϕn‖p + µ(r − p∗(α)) ∫ RN f(x)H(ϕn) dx ≥ (p∗(α) p − 1 ) ‖ϕn‖p − µ(p∗(α)− r)c1‖f‖ p∗ p∗−r S −r/p p∗ ‖ϕn‖r. Since r < p, a contradiction is obtained by letting n in the last inequality tend to infinity, therefore {ϕn} is indeed bounded. The rest of the proof is analogous to the proof of Lemma 3.3. This completes the proof. � Proof of Theorem 1.2. Let µ ∈ (0, µ0), where µ0 is defined in (4.2). Combining Lemmas 4.1, 4.2, and 4.3 with Theorem 2.3, we can deduce that problem (1.3) has a weak solution ψµ as a critical point for Jµ. Moreover, as in the proof of (3.5), one has Jµ(ψµ) ≥ η > 0. (4.3) Now, by Lemma 4.1, we can see that infψ∈∂B(0,ρ) Jµ(ψ) > 0. Moreover, by Lemma 4.2, and equation (4.1), we obtain −∞ < c := inf ψ∈B(0,ρ) (Jµ(ψ)) < 0. Let ε > 0 be such that 0 < ε < inf ψ∈∂B(0,ρ) Jµ(ψ)− inf ψ∈B(0,ρ) Jµ(ψ). (4.4) If we consider the functional Jµ : B(0, ρ) → R, then by the Ekeland variational principle there exists ψε ∈ B(0, ρ), such that c ≤ Jµ(ψε) ≤ c+ ε Jµ(ψε) < Jµ(ψ) + ε||ψ − ψε||, ψ 6= ψε, (4.5) so by (4.4), we have Jµ(ψε) ≤ inf ψ∈B(0,ρ) Jµ(ψ) + ε ≤ inf ψ∈B(0,ρ) Jµ(ψ) + ε < inf ψ∈∂B(0,ρ) Jµ(ψ), (4.6) which implies that ψε ∈ B(0, ρ). On the other hand, if we define the functional Φµ : B(0, ρ) → R by Φµ(ψ) := Jµ(ψ) + ε‖ψ − ψε‖, then ψε is a global minimum of Φµ. Therefore, for s ∈ (0, 1) small enough, we have Φµ(ψε + sψ)− Φµ(ψε) s ≥ 0, for every ψ ∈ B(0, 1), 10 A. DRISSI, A. GHANMI, D. D. REPOVŠ EJDE-2023/61 i.e., Jµ(ψε + sψ)− Jµ(ψε) s + ε‖ψ‖ ≥ 0. By letting s tend to zero, we obtain 〈J ′µ(ψε), ψ〉 + ε‖ψ‖ ≥ 0. This implies that ‖J ′µ(ψε)‖ ≤ ε. If we put wn := ψ 1 n , we obtain {wn} ⊂ B(0, ρ). Moreover, Jµ(wn)→ c < 0, and J ′µ(wn) → 0, as n → ∞. Since {wn} ⊂ B(0, ρ), it follows that {wn} is bounded in E. So, up to a subsequence still denoted by wn, there exists ψµ ∈ E, such that {wn} converges weakly to ψµ ∈ E. Invoking Lemma 4.3, we see that wn → ψµ strongly in E. Now, from the fact that Jµ ∈ C1(E,R) implies t J ′µ(wn)→ Jµ(ψµ), as n→∞, we have J ′µ(ψµ) = 0 and Jµ(ψµ) < 0, (4.7) hence ψµ is a nontrivial weak solution of (1.3). Moreover, by combining (4.3) with (4.7), we obtain that Jµ(ϕµ) < 0 < Jµ(ψµ), i.e., uµ and ψµ are distinct. This completes the proof. � 5. An Application As an application of our results, we shall consider the problem ∆2 pϕ− λ |ϕ|p−2ϕ |x|2p + ∆pϕ = µf(x)|ϕ|r−2ϕ+ |ϕ|p∗(p)−2ϕ |x|p in RN , (5.1) where 1 < p < N 2 and λ > 0. We note that problems of type (5.1) describe the deformations of an elastic beam. Also, they give a model for considering traveling waves in suspension bridges. It is not difficult to see that 1 < α = p < 2p and h(ϕ) = |ϕ|r−2ϕ satisfies the second inequality of hypotheses (H1) and (H3), with c1 = 1 > 0. Moreover, a simple calculation shows that H(ϕ) = 1 r |ϕ| r which satisfies rH(ϕ) = h(ϕ)ϕ, so hypothesis (H2) is also satisfied for every σ > 0. Hence if r ∈ (p, p∗) and f ∈ L∞(RN ), then Theorem 1.1 implies that for every µ > 0, there exists λ0 > 0 such that for every λ ∈ (0, λ0), problem (5.1) has a nontrivial solution. Moreover, if 1 < r < p and 0 < f ∈ L p∗ p∗−r (RN ) ∩ Lsloc(RN ), for some s ∈ ( p∗ p∗ − r , p p− r ) , then Theorem 1.2 implies the existence of λ0 > 0 and µ0 > 0 such that for every λ ∈ (0, λ0) and µ ∈ (0, µ0), problem (5.1) has at least two nontrivial solutions. 6. Conclusion The variational method has a long and rich history, and it has given rise to the functional energy. The Mountain pass theorem is used in the first part of this paper to prove the existence of a nontrivial solution for a p-biharmonic problem involving the Hardy-Sobolev exponent. Our first main result generalizes the paper of Ghoussoub and Yuan [13]. 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Zhao; Nonuniformly nonlinear elliptic equations of p-biharmonic type, J. Math. Anal. Appl. 348 (2008), 730–738. Amor Drissi Department of Mathematics, Faculty of Sciences, University of Tunis El Manar, 2092 Tunis, Tunisia Email address: amor.drissi@ipeiem.utm.tn Abdeljabbar Ghanmi Department of Mathematics, Faculty of Sciences, University of Tunis El Manar, 2092 Tunis, Tunisia Email address: abdeljabbar.ghanmi@lamsin.rnu.tn Dušan D. Repovš Faculty of Education and Faculty of Mathematics and Physics, University of Ljubljana, and Institute of Mathematics, Physics and Mechanics, SI-1000, Slovenia Email address: dusan.repovs@guest.arnes.si 1. Introduction 2. Preliminaries 3. Proof of Theorem ?? 4. Proof of Theorem ?? 5. An Application 6. Conclusion Acknowledgments References