Singular $p$-biharmonic problems involving the Hardy-Sobolev exponent

This paper is concerned with existence results for the singular $p$-biharmonic problem involving the Hardy potential and the critical Hardy-Sobolev exponent. More precisely, by using variational methods combined with the Mountain pass theorem and the Ekeland variational principle, we establish the existence and multiplicity of solutions. To illustrate the usefulness of our results, an illustrative example is also presented.


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 where Ω ⊂ R n is a regular bounded domain, µ and λ are positive parameters, min(q, r) ≥ p, q ≤ p * (α), and r ≤ p * .
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 nonlinearity, and the Hardy potential.Moreover, it appears in many applications, such as non-Newtonian fluids, viscous fluids, traveling waves in suspension bridges, and 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.

Preliminaries
We begin by recalling some necessary facts related to the Hardy-Sobolev exponent 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 for every ϕ ∈ W 2,p (R N ), where W 2,p (R N ) denotes the Sobolev space which is defined by 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 (R N ) can be endowed with the following norm The space W 2,p (R N ) is continuously embeddable into L σ (R N ) for every p ≤ σ ≤ p * , and compactly embeddable into L σ loc (R N ), for every p ≤ σ < p * .Moreover, for every ϕ ∈ W 2,p (R N ), one has where |ϕ| p denotes the usual L p (R N )-norm and S σ is defined by Hereafter, for simplicity, we shall denote E := W 2,p (R N ).We define the weighted Lebesgue space L r (R N , f ) by and endow it with the norm ) is a uniformly convex Banach space.Dhifli and Alsaedi [12] proved that under hypothesis (H3), the embedding E → L r (R N , f ) is continuous and compact.Moreover, one has the estimate Now, let us introduce the notion of weak solutions. where We define the energy functional J µ : E → R, by 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 J µ .
Definition 2.2.We say that a function Φ ∈ C 1 (F, R), where F is a Banach space, satisfies the Palais-Smale condition, if every sequence To prove Theorem 1.1, we need the following result which is proved in Ambrosetti and Rabinowitz [2, Theorem 2.4].

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 J µ (ϕ) ≥ η > 0.
Proof.Let ϕ be a positive function in C ∞ c (E).Then for every s > 0 we have Since p < p * (α), it follows that J µ (sϕ) → −∞, as s → ∞.Therefore there exists s 0 > ρ ϕ large enough, such that J µ (s 0 ϕ) < 0. If we now set e = s 0 ϕ, then e > ρ and J µ (e) < 0. This completes the proof.Proof.Let {ϕ n } be a Palais-Smale sequence, which means that J µ (ϕ n ) is bounded and J µ (ϕ n ) → 0, as n → ∞.Therefore there exist m 1 > 0 and m 2 > 0 such that J µ (ϕ n ) ≤ m 1 and |J µ (ϕ n )| ≤ m 2 .Letting θ := min(r, p * (α)), we obtain by hypothesis (H1) that and since θ = min(r, p * (α)) > p, it follows that the sequence {ϕ n } is bounded in E. Therefore (up to a subsequence) there exists ϕ ∈ E such that ϕ n → ϕ a.e. in R n , so, by (H1), (H2) and the Dominated convergence theorem, One can now show by a standard argument that the weak limit u of {ϕ n } is a critical point of J µ and thus J µ (ϕ) = 0. Let w n := ϕ n − ϕ.Then w n converges weakly to zero.Moreover, by Brezis and Lieb [5, Lemma 3], we obtain and from (3.1) we have hence for n large enough, (3.3) On the other hand, one has so by letting n tend to infinity, we obtain )l, and using the last equation and (3.3) we obtain (3.4)However, we have J µ (ϕ), ϕ = 0, for every ϕ ∈ E. So, from (H2) we obtain and since r ∈ (p, p * ) and p < p * (α), it follows that J µ (ϕ) ≥ 0. This is in contradiction with (3.4).Since l = 0, we see by (3.2) that {ϕ n } converges strongly to ϕ in E. This completes the proof.

Proof of Theorem 1.2
The proof is divided into several lemmas.
Proof.Let ϕ ∈ E. Invoking hypotheses (H2), (H3), equations (2.3), (2.5), and the Hölder inequality, we obtain where It is not difficult to prove that h attains its global maximum at Set Then for every µ ∈ (0, µ 0 ), we have and since h is continuous, we can find ρ > 0 such that thus for every ϕ ∈ E with ϕ = ρ, we have This completes the proof.
Since the proof of the above lemma is very similar to that of Lemma 3.2, we omit it.Proof.Let {ϕ n } be a Palais-Smale sequence.By the argument from the previous section, it follows that there exist m 1 > 0 and m 2 > 0, such that J µ (ϕ n ) ≤ m 1 and Let us prove that {ϕ n } is bounded.If not, then up to a subsequence we can assume that ϕ n → ∞, as n → ∞.By hypotheses (H2) and (H3), we obtain Let ε > 0 be such that If we consider the functional J µ : B(0, ρ) → R, then by the Ekeland variational principle there exists ψ ε ∈ B(0, ρ), such that so by (4.4), we have which implies that ψ ε ∈ B(0, ρ).

An Application
As an application of our results, we shall consider the problem 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.

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].
In the second part of the paper, the Mountain pass theorem is combined with the Ekeland variational principle to prove the existence of two nontrivial solutions.Our second main result of this paper generalizes the work of Perrera and Zou [19].
We note that the manipulation of the critical Hardy nonlinearity is more complicated and the improvement method used here is an application of the Brezis-Lieb lemma.As the foundation for further improvements, we aim to obtain even stronger results for problems with discontinuous nonlinearities.
Since r < p, a contradiction is obtained by letting n in the last inequality tend to infinity, therefore {ϕ n } 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, µ 0 ), where µ 0 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 ψ µ as a critical point for J µ .Moreover, as in the proof of (3.5),