Nonlocal fractional system involving the fractional p, q-Laplacians and singular potentials

In this paper, we will focus on following nonlocal quasilinear elliptic system with singular nonlinearities: (S)(-Δ)ps1u=1vα1+vβ1inΩ,(-Δ)qs2u=1uα2+uβ2inΩ,u,v=0in(IRN\Ω),u,v>0inΩ,\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} (S) \left\{ \begin{array}{ll} (-\Delta )_{p}^{s_1} u =\dfrac{1}{v^{\alpha _1}}+v^{\beta _1}&{} \text { in }\Omega , \\ (-\Delta )_{q}^{s_2} u =\dfrac{1}{u^{\alpha _2}}+u^{\beta _2}&{} \text { in }\Omega , \\ u,v=0 &{} \text { in } ({I\!\!R}^N\setminus \Omega ) , \\ u,v>0 &{} \text { in } \Omega , \end{array} \right. \end{aligned}$$\end{document}where Ω⊂IRN\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Omega \subset {I\!\!R}^N$$\end{document} be a smooth bounded domain, s1,s2∈(0,1)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$s_1,\,s_2\in (0,1)$$\end{document}, α1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\alpha _1$$\end{document}, α2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\alpha _2$$\end{document}, β1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\beta _1$$\end{document}, β2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\beta _2$$\end{document} are suitable positive constants, (-Δ)ps1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(-\Delta )_{p}^{s_1}$$\end{document} and (-Δ)qs2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(-\Delta )_{q}^{s_2}$$\end{document} are the fractional p-Laplacian\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$p-\text {Laplacian}$$\end{document} and q-Laplacian\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$q-\text {Laplacian}$$\end{document} operators. Using approximating arguments, Rabinowitz bifurcation Theorem, and fractional Hardy inequality, we are able to show the existence of positive solution to the above system.

In the local case, t i = 1 for i = 1, 2, the operator defined in (1.2) is reduced to g i u = div(|∇u| g i −2 ∇u) that the well-known g i − Laplacian operator with g i > 1 and g i ∈ {p, q}.
Before giving our main results, let us briefly recall literature.
• Equation: Notice that System (1.1) can be seen as a version of the singular scalar equations where s ∈ (0, 1), α, β > 0, p > 1 and λ is real positive parameter. Several works are devoted to classes of problems (1.3). For s = 1, 1 < p < N and λ = 0, the existence of weak solution and regularity of solutions have been widely studied in [6,7,9,11,21,29] and the references therein. In the case s = 1, p = 2, and λ = 0, problem (1.3) has been treated in [18], where the author has used the variational method to show that for 0 < λ < < ∞, problem (1.3) has two solutions. This paper was generalized for p − Laplacian operator in [16] where the authors have showed the existence of two solutions using the variational method for 0 < α < 1 and p−1 < β < P N N − p −1 (see also [4]). Other related works can be found [3,14,15,32] and their corresponding references.
Recently, the study of fractional elliptic equations with singular nonlinearity attracted lot of interests by researchers in nonlinear analysis. In [5], for p = 2 and 0 < s < 1, the authors studied the existence of distributional solutions of problem (1.3) using the uniform estimates of {u n } which are solutions of the regularized problems with singular term u −α replaced by (u n + 1 n ) −α (see also [12,27,30]) for more general context. The cases, when 0 < s < 1 and p = 2, have been considered in [26] where the authors have showed the existence of multiple solutions to (1.3) using variational methods. Readers may refer to the work in [13,31] and the references therein.
Needless to say, the references mentioned above do not exhaust the rich literature on the subject.
• System : The case of systems with p, q − Laplacians and s 1 = s 2 = 1, System (1.1) with singular nonlinearities was treated in [1], the authors have showed using Rabinowitz bifurcation theorem and a Hardy-Sobolev inequality the existence of the weak solution, for every (α i , β i ) ∈ (0, θ i ) with i = 1, 2 and We refer the readers, [2,17,22,25] for more general context and the references therein.
Recently, System (1.1) has been treated by another type of operator, notably an anisotropic operator; see [8].
Our main interest in this work is to analyze System (1.1). We will consider principally nonlinearities with concave-convex structure. It is clear that one of the main difficulties to show some control of the singular term near the boundary of the domain. The existence of solutions will be proved using approximation technics, the classical Rabinowitz bifurcation Theorem, and Hopf's lemma. Our main existence result is stated in the following theorem.

Theorem 1.1 Let be a bounded regular domain in IR
where p and q are conjugate exponents of p and q, respectively. Assume that α i , β i ∈ (0, γ i ) for i = 1, 2, such that Then, System (1.1) possesses a nontrivial solution in W The paper is organized as follows. In the next section, we recall some basic notions and properties like fractional Sobolev spaces, notion of solution, and beside that some inequalities and useful lemmas are included, as well as strong maximum principle and Rabinowitz bifurcation Theorem that will be used along in this paper. In the last section, we prove the main existence results of this work.

The functional setting and tools
In this section, we collect some well-known results on Sobolev spaces and give some tools as they are needed to prove our main results.
Let ⊂ IR N be an arbitrary open-bounded set . For p > 1 and s ∈ (0, 1), we denoted by the fractional order Sobolev space endowed with the norm is Banach space; we refer to [10,23] for more details and properties of the fractional Sobolev spaces.
Theorem 2.6 (Hardy inequality [19]). Let 0 < s < 1 and 1 < p < ∞ be such that sp < N . Assume that ⊂ IR N is a (bounded) uniform domain with a (locally) (s, p) − uniformly fat boundary. Then, admits an (s, p) − Hardy inequality, that is, there is a constant C > 0, such that Finally, we recall the classical Rabinowitz result, see [28], that will be used systematically in this paper.

Proof of the main result
In this section, we focus to prove the existence of nontrivial solution to (1.1) under some hypothesis on α 1 , α 2 , β 1 , β 2 , s 1 , and s 2 Before proving Theorem 1.1, we begin with the following auxiliary system: First, we begin by the following Lemma.
We consider now the following approximating system: where (φ , ψ) ∈ L p ( ) × L q ( ) are be fixed. First of all, we observe that: For ψ ∈ L q ( ) and σ > 0, we have that 1 |ψ| α 1 +σ ∈ L p ( ). On the other hand, by hypothesis, p β 1 < q , then we get, Hence, |ψ| β 1 ∈ L p ( ). By same way as before, we obtain that 1 |φ| α 2 +δ ∈ L q ( ) and |φ| β 2 ∈ L q ( ) for β 2 q < p . Now, using Proposition 2.3 for each (λ, φ, ψ) ∈ IR + × L p ( ) × L q ( ), system (3.2) possesses a unique weak solution (u, v) in W for all ζ ∈ W s 2 ,q 0 ( ). Hence, the following operator: is well defined. Let us show that is compact. In fact, let {(λ n , φ n , ψ n ) n } be a bounded sequence in R + × L p ( ) × L q ( ), such that . Now, using ξ = u n − u as test function in first equation of system (3.3), we get On the other hand, if p ≥ 2, we obtain that Thus Since W s 1 , p 0 ( ) → L p ( ) is compact, hence, it follows that u n → u strongly in W s 1 , p 0 ( ). As before, by similar reasoning , we get v n → v strongly in W s 2 ,q 0 ( ). The case 1 < q < 2 and 1 < p < 2 is made using similar arguments, and we will omit its proof. Therefore Consequently, mapping S is compact. and claim follows. Hence, using the same computation, we get easily that S is continuous. On the other hand, we observe that (0, 0, 0) ∈ F and S(0, u, v) = (0, 0), then we are in the conditions of Theorem 2.7. Hence, we get an component F of solutions to (u, v) = S(λ, u, v), which means that, if (λ, u, v) ∈ F, then, (λ, u, v) solve (3.1). It remains only to show that (1, u, v) ∈ F which corresponds to our required solution.
It clear that for every λ > 0 by strong maximum principle u, v > 0 in . We argue by contradiction. Assume that there exist λ * , such that for all λ ≤ λ * , we have (λ, u, v) ∈ F. Using u as test function of the first equation of (3.1), we obtain that where in the last inequalities, we have used the fact W s 1 , p 0 ( ) → L p ( ) and W s 2 ,q 0 ( ) → L q ( ). Similarly, using v as test function in the second equation of (3.1) and by taking into consideration the following immersions: it follows that: Hence, combining the above estimate, we obtain that Since p, q > 1, β 1 < p − 1 and β 2 < q − 1, then from the last inequality, we get ||u|| Now, we are able to prove our main result. Proof of Theorem 1.1. Using Lemma 3.1, we deduce that the system In fact, let (w 1 , where 0 < m 1 = min t≥0 1 t α 1 +1 + t β 1 and 0 < m 2 = min t≥0 1 t α 2 +1 + t β 2 . Since p, q > 1, then using the comparison principle (see [20] ) and Lemma 2.5, we get u n ≥ w 1 > 0 and v n ≥ w 2 > 0 for every n ∈ IN as desired. Let us show that the sequences {u n } n and {v n } n are bounded in W s 1 , p 0 ( ) and W s 2 ,q 0 ( ), respectively. First, we take u n as test function in first equation of (3.4), and we get Since β 1 p < q and q ∈ [1, q * s 2 ], thus, using Hölder and Sobolev inequalities , it follows that: Now, we will estimate the first integral in the right-hand side of inequality (3.7). By Hopf's lemma (see [24]), we get that, dx.
Hence, using classical arguments, we get the desired result. A direct consequence of our result in the case where s 1 = s 2 = s is the following.