Multiplicity of solutions for a singular system with sign-changing potential

: This paper focuses on a singular system with a sign-changing potential in Γ, a bounded domain with a Lipschitz boundary in ℝ d . By imposing appropriate conditions on the weight potential, which is allowed to change sign, we establish the existence of multiple solutions using the shape optimization approach. This study representsoneoftheearliestendeavorstoexploreandanalyzetheoccurrenceofmultiplesolutionsinfractional singular systems involving sign-changing potentials. By explicitly addressing this particular aspect, our paper contributes significantly to the limited body of literature that exists in this specific field.


Introduction
Let Γ ⊂ ℝ d be a bounded domain with Lipschitz boundary, where d > ps and 0 < s < 1.This paper is concerned with the following fractional order singular systems with sign-changing potentials: where η is a positive parameter, ps > t > 0, 0 < β < 1 < p < p * s (t) with p * s (t) = p(d−t) d−ps and ν(x) is a changing sign potential that will be specified later.We define the following fractional operator by |x − y| ps+d dy for all x ∈ ℝ d .
Problem (1.1) has received a lot of attention recently.Applications in fields such as fluid dynamics, astronomy, optimization, electromagnetic, water waves, probability theory, phase transitions, and others provide as inspiration for the research of this type of topic.Please refer to [6,9,10,17,19,22,24,25,27,28] and references therein for further information on applications.
Our topic is mathematically interesting since there is a lack of regularizing impact for problem (1.1) owing to the presence of changing sign data, which effects the existence and multiplicity of solutions.It is worth noticing that if ν = 0 and Γ is a smooth cone, problem (1.1) does not permit a solution using the shape optimization approach presented in [30].
We cite [1-3, 5, 7, 11, 18, 21, 26] and the references therein as evidence that the topic has been well explored for different types of PDEs.They demonstrated the existence of solutions for ν under reasonable conditions.However, in our knowledge there is no results on a singular system with sign-changing potentials.For example, the following one-dimensional Schrödinger equations with a quasi-periodic potential has been study in [12]: (1.2) Fei [14] also obtained The existence of homoclinic orbits for Hamiltonian systems with the potentials changing sign.Meanwhile, Tiihonen [32] proved a multiplicity result for free boundary problems on the basis of shape optimization methods.In [15,29,33] using the method mentioned above, the authors study the singular system governed by different types of partial differential equations, which is similar to problem (1.2).In this context, the current problem (1.1) is novel, not only due to a singularity, but also due to the sign-changing weight potentials.Borrowing ideas from [30] and [31] we will prove multiplicity results of problem (1.1) when ν changes sign.By establishing a global integrability result that will be useful for the following and using shape optimization approach, we prove that problem (1.1) has at least one non-trivial mild solution.Taking into account the fact that problem (1.1) has a gradient-like structure of the pullback attractor, we prove the existence of multiple solutions by finding an expression for optimal control in terms of dual variable.
Before stating our result, we say that ν satisfies condition (A1) if there exists a sufficiently small positive number ϵ such that (see [20]) where G is the Green function associated with the operator (−Δ p ) s .
The article is organized as follows: we present some notations and basic lemmas in section 2. Section 3 is devoted to prove the existence of a solution that is a local minimizer in H 0 of Ω η,ν with respect to (1.1).The approximated problem is studied in Section 4. Section 5 is devoted to establish the multiplicity of solutions.

Basic lemmas
We recall some definitions of function spaces and basic lemmas that will be used in later sections.Let Γ ⊂ ℝ d and . We define the space (H, ‖ ⋅ ‖ H ) by with the standard norm .
Here ‖f‖ p refers to the L p -norm of f .The space H 0 is further defined by which is endowed with the norm .
We define the best Sobolev constant by . (2.1) where μ ∈ H 0 , then we say that f ∈ H 0 is a mild solution to (1.1).

Existence of a mild solution to problem (1.1)
As well as showing that a mild solution exists, we also demonstrate that it is a constrained minimizer of the semi-ratio dependent generalized Ω η,ν .Our first basic lemma is as follows.
Since ‖f‖ is sufficiently small, we infer that inf which together with the definition of the infimum (3.1) yields that a minimizing sequence {f i } exists.Moreover, it follows from the reflexivity of H 0 that there exists f η and a subsequence f i such that By the Concentration-Compactness Lemma [13], we have ) By letting i → ∞, it follows from Hölder inequality that Using similar arguments as above, we obtain Combining (3.3), (3.4), and (3.5), we infer that In addition, it follows from (3.3)-(3.4)and sufficiently large i that Combining (3.2), (3.6) and (3.7), one has Since F ς 0 is closed and convex, we infer that f η ∈ F ς 0 .Then, it follows from (3.1) that Ω η,ν (f η ) = c 5 < 0 and one has f η ̸ ≡ 0, that is a minimizer of Ω η,ν on H 0 .We shall prove that f η > 0 is a mild solution to problem (1.1).Let 0 ≤ μ ∈ H 0 .There exists a sufficiently small positive number r such that Passing to the limit as r → 0 + in the above inequality, one has that lim inf where ξ ∈ (0, 1), and a.e. in Γ.It follows from the Riemann-Lebesgue Lemma that Combining (3.8) and (3.9), one has that where μ ≥ 0 a.e. in ℝ d .It follows from Lemma 3.1 and the fact Ω η,ν < 0 that f η ∈ F ς 0 .Then there exists ρ ∈ (0, 1) such that (1 + r)f η ∈ B ς 0 , where |r| ≤ ρ.If we define the functional P η,ν by then it achieves a golbal minimum at r = 0, because f η is a constrained minimizer of P η,ν in B ς 0 .Moreover, We define Ψ ∈ H 0 by Ψ := (f + η + ϵμ) + , replace μ with it in (3.10), which together with (3.11), one has that Since the measure of Γ ϵ and Γ ϵ tend to zero as ϵ → 0 + , it follows that |x − y| ps+d dx dy → 0 as ϵ → 0 + .Then passing to the limit as ϵ → 0 + , one has If we change μ by −μ, then the above equality holds, because μ is an arbitrary test function, which yields that f η is a mild solution to problem (1.1).At last, settting μ = f − η in (2.1), one has that f η ≥ 0.Moreover, since I η = C < 0, then f η ̸ ≡ 0. So it follows from the maximum principle that f η is a mild solution to problem (1.1).We complete the proof.

Existence of a solution of the fractional singular system
In this section, we firstly introduce the following fractional singular system We define the functional energy Ω i,η,ν : H 0 → ℝ with respect to (4.1) by It follows from the definition of Ω i,η,ν that Ω i,η,ν is Fréchet differentiable, where μ ∈ H 0 , and Obviously, the solutions of the fractional singular system (4.1) are critical points of Ω i,η,ν .
If we set f ∈ H 0 such that f + ̸ ≡ 0 and r > 0, then it follows from the fact 1 So we prove the existence of g η ∈ H 0 such that ‖g η ‖ > ς 0 and Ω i,η,ν (g η ) < σ 0 , and we also complete the proof.
Lemma 4.2.Let 0 < β < 1. Suppose that ν satisfies condition (A1).Then Ω i,η,ν satisfies the structural condition at any controlled level c 6 ∈ ℝ with the property that c 6 < (sp−t) p(d−t) A d−t sp−t − ψ η for any η > 0, where Proof.If we consider that {f j } ⊂ H 0 is a structural minimizing Palais-Smale sequence for the functional energy Ω i,η,ν at a controlled level c 6 ∈ ℝ, where c 6 satisfies Ω i,η,ν (f j ) → c 6 and Ω  i,η,ν (f j ) → 0 as j → ∞, ( it follows from the shape optimization approach and the Cauchy-Schwarz inequality that there exists a positive constant c 7 and ϵ > 0 such that Moreover, one has which together with (4.4) yields that ‖f − j ‖ → 0 as j → ∞.Hence, choosing j large enough, we have which yields that {f j } is a sequence of positive continuous functions.Since the sequence {f j } is bounded, it follows that there exists {f j } ⊂ H 0 , g η ∈ H 0 and two positive numbers ϖ and ν such that f j ⇀ g η weakly in H 0 , and So we assume that ν > 0. In a similar way, from the Faedo-Galerkin approximations, one has        f j − g η Then we can apply the Lebesgue Dominated Convergence Theorem and get that which yields that So f j → g η strongly as j → ∞ in H 0 .Since Ω  i,η,ν (f j ) → 0 as j → ∞, one has that It follows from the Brezis-Wainger inequality [13] that It follows from the Hardy-Littlewood-Sobolev inequality that where It is now clear that it converges to zero, once we apply the Hölder inequality and the cooperative character of the system.Indeed, ).
Therefore, the operator P τ is continuous.So Al p ≤ ϖ p * s (t) , which yields that l = 0. Hence f j → g η as j → ∞ in H 0 .We complete the proof.Otherwise, we suppose that It follows from (4.6), (4.7), the Young inequality and the Hölder inequality that which is a contradiction.So ϖ = 0 and f j → g η .We complete the proof of Lemma 4.2.
So we can infer that there exist four positive constants c 10 , c 11 , c 12 and c 13 such that Thus, by (4.11), we infer that there exists a positive constant η 1 such that, for any η ∈ (0, η 1 ) Finally, we complete the proof.
Lemma 4.4.Let 0 < β < 1. Suppose that ν satisfies condition (A1).Then there exists a mild solution g i ∈ H 0 to the problem (4.1) such that where σ 0 and ψ η are defined in Lemmas So it follows from Lemmas 4.1-4.3 that Ω i,η,ν satisfies the structural condition at the level c i,η,ν .Now there exists a non-smooth critical point g i for I i,η,ν at the Gevrey level c i,η,ν .In addition, Ω i,η,ν (g i ) = c i,η,ν > σ 0 > 0. We infer that g i is a non-regular point of Ω i,η,ν and it is also a solution to (4.1).Then, if we apply (4.4) and replace μ with g − i in (4.2), then ‖g i ‖ = 0.So g i is positive.Finally, by the shape optimization approach, we know that g i is a mild solution to the problem (4.1).We complete the proof of Lemma 4.4.

Multiplicity of solutions to problem (1.1)
Firstly, we consider that {g i } i is a family of mild function defined in Lemma 4.4.It follows from Lemma 4.4 and the Cauchy-Schwarz inequality that So we get that {g i } is bounded in H 0 because of β ∈ (0, 1).The reflexivity of H 0 allows one to use Banach-Alaoglu theorem and obtain a subsequence, we denote it by {g i }.Then there exists a function g η such that g i ⇀ g η weakly in H 0 , and Next we shall show that g i → g η strongly in H 0 , which yields that ‖g i − g η ‖ → 0 as i → ∞.
If ν = 0, then it follows from (5.1) that ‖g i ‖ → 0 as i → ∞.Now, we suppose that ν > 0. Since 0 ≤ g i a.e. in Γ, it follows from the Hölder inequality and (5.1) that  (5.5) Since {g i } i is bounded in H 0 , it follows from the shape optimization approach that there exist Γ ⊂ Γ and a positive number c 14 such that g i ≤ c 14 a.e. in Γ, (5.6) for any integer i.If we define μ ∈ C ∞ 0 (Γ) such that supp(δ) = Γ ⊂ Γ, it follows from (5.6) that 0 ≤ | δ If we replace f by g i in (4.2), pass to the limit in g i as i → ∞, and apply change of variables in (5.1), then one has that ( which contradict with the fact (5.10).So Ω η,ν (g η ) = lim i→∞ Ω i,η,ν (g i ), which yields that g η is a solution of (1.1).

c γ 14 a
.e. in Γ, which together with the Dominated Convergence Theorem and (5.1) yield that lim i→∞