Solutions to a (p1,…,pn)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(p_1, \ldots ,p_n)$$\end{document}-Laplacian Problem with Hardy Potentials

We are concerned with the existence and multiplicity of weak solutions for a general form of a (p1,…,pn)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(p_1, \ldots ,p_n)$$\end{document}-Laplacian elliptic problem including singular terms. Our approaches are mainly based on critical points theory.


Introduction
During the note, from start to finish, Ω ⊂ ℝ N , (N ≥ 3) is a bounded domain with smooth boundary and n ∈ ℕ . We assume that and i s, i = 1, … , n , are positive real functions on Ω such that the following conditions hold: We consider the Dirichlet problem (1.1) 1 < p i < ∞ & 1 < q i < N i= 1, … , n, such that max 1≤i≤n q i =q < p = min 1≤i≤n p i (not necessarily unique) for which p i * =p.
A generalization of Laplace operator is the non-homogeneous differential operator Δ p + Δ q , called (p, q)-Laplacian. Usually solutions to (p, q)-Laplacian problems are the steady state solutions of the reaction diffusion systems-mathematical models which correspond to several physical phenomena. This system has a wide range of applications in physics and related sciences like chemical reaction design, biophysics, plasma physics, geology, and ecology. These equations also arise in the study of soliton-like solutions of the nonlinear Schrödinger equation as a model for elementary particles for example waves in a discrete electrical lattice (see [6-8, 11, 16, 34]). These problems have been intensively studied in the last decades; for instance, the existence of a nontrivial solution for the following (p, q)-Laplacian equation with p-critical exponent in ℝ N has been probed by Chaves [5] where 1 < q ≤ p < r + 1 < p * , p < N, > 0 is a parameter and g ∈ L p * p * −r−1 (ℝ N ) is positive in an open set. In 2019, Behboudi et al. [2] verified the existence of two weak solutions for the following problem where 2 ≤ q < p < N (one can see [9,10,12,13,17,[20][21][22][23][24][25]33] and references therein for the importance of study of these kinds of problems).
Here, we study a general form of the (p, q)-Laplacian problems and we are interested in the existence and multiplicity of weak solutions for the problem ( P ). We show that there exists at least one weak solution to the problem ( P).

Definition 1.1 (Weak solution)
We say that u ∈ W 1,p 0 (Ω) is a weak solution of the problem ( P ) if u satisfies the Dirichlet boundary conditions and the following integral equality is true We point out that our approaches also fit with slightly different versions of the problem ( P ), e.g., with p(x)-Laplacian operator or the Heisenberg p-Laplacian operator or even the weighted Heisenberg p-Laplacian operator on the left hand side. Interested reader can see more details in [18,19,[26][27][28][29][30][31][32] and the references therein.

Notations and Some Remarks
For 1 < q i < N and 1 < p i < ∞ , i = 1, ⋯ , n , we set and in the same way | ⋅ | p denotes the usual norm of Lebesgue space L p (Ω) , i.e., for p ∈ {p i , q i ∶ i = 1, … , n} and the Sobolev space W 1,p (Ω) is defined naturally, that is equipped with the norm where ∇u = ( u In the next theorem, we recall standard compact embeddings in the Sobolev spaces [1, Theorem 2.6.2].
The following is the classical Hardy's inequality proved in [14].
Subsequently, we will need the definition of (PS) [r] -condition.
and fixed some r ∈ [−∞, +∞] . We say that I satisfies the Palais-Smale condition cut off upper at r (in short the (PS) [r] -condition), if, for every sequence {u k } k∈ℕ in X the following states are satisfied: then, {u k } k∈ℕ admits a convergent subsequence. When r = +∞ , we write (PS)-condition instead of (PS) ∞ -condition. From now on, we set endowed with the norm As a special case of Theorem 2.1, one has the compact embedding for 1 ≤ r <p * ; additionally, for any u ∈ , there exists K r > 0 such that (i) if ‖u‖ ≥ 1 , then by applying the assumption (1.2) and Hardy and Hölder inequalities one has the following estimate.
Let f ∶ Ω × ℝ → ℝ be a Carathéodory function satisfying the growth condition ( F ) and define the function Therefore, the functional Ψ ∶ → ℝ, is well-defined and by (2.2), we gain the following estimate: Besides, Ψ is continuously Gâteaux differentiable functional, with the following compact derivative for every u, v ∈ (see [15]). Now, put Clearly, the critical points of I are the weak solutions of Problem ( P).

Remark 2.3 The functional I satisfies the (PS)-compactness condition.
Proof Let {u k } ⊂ be a Palais-Smale sequence, i.e.,

From Remark 2.2 and relation (2.4), we gain
Since I � (u k ) → 0 , for large enough k, one has because 1 ≤ <p , which means that {u k } is a bounded sequence in the reflexive Banach space . So, there exists û ∈ such that, up to the subsequences, • u k ⇀û in ; • u k →û in L s (Ω), 1 ≤ s <p * ; • u k (x) →û(x) a.e. in Ω; • there exist w s ∈ L s (Ω) such that |u k (x)| ≤ w s (x) a.e. in Ω and for all n ∈ ℕ where 1 ≤ s <p * .
On the other hand, u k ,û ∈ for k ∈ ℕ , from Minkowski inequality, we have Using the above matters and Lebesgue's dominated convergence theorem, we deduce is the measure of the unit ball in ℝ N . Clearly, 0 < Φ(ū ) < 1 . On the other hand, as we mentioned before, the functional I satisfies the (PS)-condition and Φ(u k ) < r for all n ∈ ℕ , so (PS) [r] -condition holds. Now, bearing in mind the next theorem, which is a particular case of Bonanno's theorem [3,Theorem 2.3], in the case r = 1.

Theorem 3.1 Let X be a real Banach space, Φ, Ψ ∶ X → ℝ be two continuously Gâteaux differentiable functionals such that
Assume that there exists r > 0 and x ∈ X , with 0 < Φ(x) < r , such that Then, for each ∈ Λ r , there is x 0, ∈ Φ −1 (]0, r[) such that I � (x 0, ) ≡ X * and According to the top contents, Theorem 3.1 guarantees the existence of local minimum point ū for I; indeed, the next theorem has been proved which is the first result of this paper.

Theorem 3.2 Assume that f is a Carathéodory function that satisfies the growth condition ( F ) and F is defined by (2.3) and for each 1 ≤ i ≤ n following limit equality is true:
Then, for every ∈]0, * [ with Problem ( P ) has at least one nontrivial weak solution.

Multiplicity of Weak Solutions
In this section, our goal is to present enough conditions such that our problem has multiple solutions.
One of the main tools of this section is the following abstract result which was proved in [3, Theorem 3.2].
Theorem 4.1 Let X be a real Banach space, Φ, Ψ ∶ X → ℝ be two continuously Gâteaux differentiable functionals such that Φ is bounded from below and Φ(0) = Ψ(0) = 0 . Fix r > 0 and assume that for each the functional I ∶= Φ − Ψ satisfies the (PS)-condition and it is unbounded from below. Then, for each the functional I admits two distinct critical points.
Let , Φ, Ψ be as before and there exist >p and r > 0 such that for each x ∈ Ω and |t| ≥ r . Integrating condition (4.1), there exist and such that for each (x, t) ∈ Ω × ℝ . Fixed û ∈ �{ } , for any t > 1 , one has Then, the assumption >p > q guarantees that I is unbounded from below. According to Remark 2.3, I fulfils (PS)-condition. Therefore, thanks to Theorem 4.1, for each ∈]0, * [ , the functional I admits two distinct critical points that are weak solutions of our problem. In reality, we prove the next theorem which is the second result of this paper. , be a continuously Gâteaux differentiable whose Gâteaux derivative is compact such that Assume that there exist r > 0 and x ∈ X , with r < Φ(x) , such that Then, for each ∈ Λ r , the functional Φ − Ψ has at least three distinct critical points in X.
Remark 4. 1 We point out that if f (x, 0) ≠ 0 , then, by applying Theorems 4.1 and 4.3 , we obtain the existence of at least two and three non-trivial weak solutions, respectively.