Abstract
In this paper, we will focus on following nonlocal quasilinear elliptic system with singular nonlinearities:
where \(\Omega \subset {I\!\!R}^N\) be a smooth bounded domain, \(s_1,\,s_2\in (0,1)\), \(\alpha _1\), \(\alpha _2\), \(\beta _1\), \(\beta _2\) are suitable positive constants, \((-\Delta )_{p}^{s_1}\) and \((-\Delta )_{q}^{s_2}\) are the fractional \(p-\text {Laplacian}\) and \(q-\text {Laplacian}\) 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.
Similar content being viewed by others
1 Introduction
In this work, we consider the existence of positive solution of the following nonlocal quasilinear system:
where \(\Omega \) be a smooth bounded domain of \({I\!\!R}^N\), \(s_1, s_2\in (0,1)\) with \(s_1\ne s_1\), \(\alpha _1\), \(\alpha _2\), \(\beta _1\) and \(\beta _2\) are positive constants. Here, \((-\Delta )_{p}^{s_2}\)( resp. \((-\Delta )_{q}^{s_2}\)) is the fractional \(p-\text {Laplacian}\) (resp. \(q-\text {Laplacian}\)), defined by
where \((t_1,g_1)=(s_1,p)\), \((t_2,g_2)=(s_2,q)\) with \(p,q>1\) and \(\text {P.V}\) is the principal value .
In the local case, \(t_i=1\) for \(i=1,2\), the operator defined in (1.2) is reduced to \(\Delta _{g_i}u=div(|\nabla u|^{g_i-2}\nabla u)\) that the well-known \(g_i-\text {Laplacian}\) operator with \(g_i>1\) and \(g_i\in \{p,q\}\).
Before giving our main results, let us briefly recall literature.
\(\bullet \) Equation: Notice that System (1.1) can be seen as a version of the singular scalar equations
where \(s\in (0,1)\), \(\alpha ,\beta >0\), \(p>1\) and \(\lambda \) is real positive parameter. Several works are devoted to classes of problems (1.3).
For \(s=1\), \(1<p<N\) and \(\lambda =0\), 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 \(\lambda \ne 0\), problem (1.3) has been treated in [18], where the author has used the variational method to show that for \(0<\lambda<\Lambda <\infty \), problem (1.3) has two solutions. This paper was generalized for \(p-\text {Laplacian}\) operator in [16] where the authors have showed the existence of two solutions using the variational method for \(0<\alpha <1\) and \(p-1<\beta <\frac{PN}{N-p}-1\) (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 \(u^{-\alpha }\) replaced by \((u_n+\frac{1}{n})^{-\alpha }\) (see also [12, 27, 30]) for more general context. The cases, when \(0<s<1\) and \(p\ne 2\), 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.
\(\bullet \) System : The case of systems with \(p,q-\text {Laplacians}\) 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 \((\alpha _i,\beta _i)\in (0,\theta _i)\) 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 \(\Omega \) be a bounded regular domain in \({I\!\!R}^N\), \(s_1,s_2\in (0,1)\), \(p\in [1, \frac{N}{s_1})\), \(q\in [1, \frac{N}{s_2})\) \(p'\in [1,p^{*}_{s_1}]\), \(q'\in [1,q^{*}_{s_2}]\) where \(p'\) and \(q'\) are conjugate exponents of p and q, respectively. Assume that \(\alpha _i,\beta _i \in (0,\gamma _i)\) for \(i=1,2\), such that
Then, System (1.1) possesses a nontrivial solution in \(W^{s_1,p}_0(\Omega )\times W^{s_2,q}_0(\Omega )\).
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.
2 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 \(\Omega \subset {I\!\!R}^N\) be an arbitrary open-bounded set . For \(p>1\) and \(s\in (0,1)\), we denoted by
the fractional order Sobolev space endowed with the norm
We set
Then, \(W_0^{s,p}(\Omega )\) endowed with the norm
where
is Banach space; we refer to [10, 23] for more details and properties of the fractional Sobolev spaces.
Theorem 2.1
(Fractional Sobolev inequality [10]) Assume that \(0<s<1, p>1\) satisfy \(ps<N\). Then, there exists a positive constant \(S\equiv S(N,s,p)\), such that for all \(v\in C_{0}^{\infty }({I\!\!R}^N)\)
where \(p^{*}_{s}= \dfrac{pN}{N-ps}\) is critical Sobolev exponent.
Let us consider now the following quasilinear problem:
where \(\Omega \subset {I\!\!R}^N\) be open-bounded domain, \(1<p<\infty \), \(0<s<1\), \(f\in W^{-s,p'}(\Omega )\) (we shall denoted \(W^{-s,p'}(\Omega )\) is the dual of the reflexive Banach space of \(W_0^{s,p}(\Omega )\) and \(p'=\frac{p}{p-1}\).
Definition 2.2
For \(f\in W^{-s,p'}(\Omega )\), we say that \(u\in W_0^{s,p}(\Omega )\) is a weak solution to (2.1) if
for all \(v\in W^{s,p}_0(\Omega )\).
Proposition 2.3
[33]. Let \(s\in (0,1)\) and \(1<p<\infty \). Then, for every \(f\in W^{-s,p'}(\Omega )\), the Dirichlet problem (2.1) has a unique weak solution \(u\in W_0^{s,p}(\Omega )\). Moreover
Proposition 2.4
[33]. Let \(s\in (0,1)\) and \(1<p<\infty \). Then, we have that
-
(1)
\((-\Delta )^s_p : W_0^{s,p}(\Omega ) \rightarrow W^{-s,p'}(\Omega )\) is strictly monotone, continuous, coercive, and bounded.
-
(2)
\(((-\Delta )^s_p)^{-1} : W^{-s,p'}(\Omega )\rightarrow W_0^{s,p}(\Omega )\) is locally Lipschitz continuous if \(p\in (1,2)\) and is Lipschitz continuous if \(p\ge 2\).
-
(3)
The composed operator \(W^{-s,p'}(\Omega )\hookrightarrow W_0^{s,p}(\Omega )\hookrightarrow L^{q}(\Omega )\) is compact if \(1\le q<\frac{pN}{N-ps}\).
Lemma 2.5
(Strong maximum principle [24]). Let \(u\in W_0^{s,p}(\Omega )\) satisfy
Then, u has lower semi-continuous representative in \(\Omega \), which is either identically 0 or positive.
Theorem 2.6
(Hardy inequality [19]). Let \(0<s<1\) and \(1<p<\infty \) be such that \(sp<N\). Assume that \(\Omega \subset {I\!\!R}^N\) is a (bounded) uniform domain with a (locally) \((s, p)-\text {uniformly fat boundary}\). Then, \(\Omega \) admits an \((s, p)-\text {Hardy inequality}\), 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.
Theorem 2.7
Let E be a Banach space and \(T: \mathbb {R}^{+}\times E\rightarrow E\) a continuous and compact operator, such that \(T(0,u)=0\) for all \(u\in E\). Then, the equation
possesses an unbounded continuum \(F\subset \mathbb {R}^{+}\times E\) of solutions with \((0,0)\in F\).
3 Proof of the main result
In this section, we focus to prove the existence of nontrivial solution to (1.1) under some hypothesis on \(\alpha _1\), \(\alpha _2\), \(\beta _1\), \(\beta _2\), \(s_1\), and \(s_2\)
Before proving Theorem 1.1, we begin with the following auxiliary system:
First, we begin by the following Lemma.
Lemma 3.1
Under the hypothesis of Theorem 1.1. Then, for \(\delta ,\sigma >0\) fixed and for all \(\lambda >0\), System (3.1) possesses a nontrivial solution in \(W^{s_1,p}_0(\Omega )\times W^{s_2,q}_0(\Omega )\).
Proof
We consider now the following approximating system:
where \((\phi \,\,, \psi )\in L^{p'}(\Omega )\times L^{q'}(\Omega )\) are be fixed.
First of all, we observe that:
For \(\psi \in L^{q'}(\Omega )\) and \(\sigma >0\), we have that \(\frac{1}{|\psi |^{\alpha _1}+\sigma }\in L^{p'}(\Omega )\).
On the other hand, by hypothesis, \(p'\beta _1<q'\) , then we get,
Hence, \(|\psi |^{\beta _1}\in L^{p'}(\Omega )\).
By same way as before, we obtain that \(\frac{1}{|\phi |^{\alpha _2}+\delta }\in L^{q'}(\Omega ) \) and \(|\phi |^{\beta _2}\in L^{q'}(\Omega )\) for \(\beta _2q'<p'\).
Now, using Proposition 2.3 for each \((\lambda ,\phi ,\psi )\in {I\!\!R}^{+}\times L^{p'}(\Omega )\times L^{q'}(\Omega )\), system (3.2) possesses a unique weak solution (u, v) in \(W^{s_1,p}_0(\Omega )\times W^{s_2,q}_0(\Omega )\), that is
for all \(\xi \in W^{s_1,p}_0(\Omega )\) and
for all \(\zeta \in W^{s_2,q}_0(\Omega )\).
Hence, the following operator:
is well defined. Let us show that
is compact.
In fact, let \(\{(\lambda _n,\phi _n,\psi _n)_n\}\) be a bounded sequence in \(\mathbb {R}^{+}\times L^{p'}(\Omega )\times L^{q'}(\Omega )\), such that
Therefore, it follows that \(\{u_n\}\) and \(\{v_n\}\) are bounded, respectively, in \(W^{s_1,p}_0(\Omega )\) and \(W^{s_2,q}_0(\Omega )\).
Thus, we get the existence a subsequence \((u_n, v_n)_n\) and \((u,v)\in W^{s_1,p}_0(\Omega )\times W^{s_2,q}_0(\Omega )\), such that
-
1.
\(u_n\rightharpoonup u\) weakly in \(W^{s_1,p}_0(\Omega )\),
-
2.
\(u_n \rightarrow u\) in \(L^{\tau }\) strongly for every \(\tau \in [1, p^{*}_{s_1})\),
-
3.
\(v_n \rightharpoonup v\) weakly in \(W^{s_2,q}_0(\Omega )\),
-
4.
\(v_n \rightarrow v\) in \(L^{\theta }(\Omega )\) strongly for every \(\theta \in [1, q^{*}_{s_2})\).
Now, using \(\xi =u_n-u\) as test function in first equation of system (3.3), we get
On the other hand, if \(p\ge 2\), we obtain that
Thus
Since \(W^{s_1,p}_0(\Omega )\hookrightarrow L^p(\Omega )\) is compact, hence, it follows that \(u_n\rightarrow u\) strongly in \(W^{s_1,p}_0(\Omega )\). As before, by similar reasoning , we get \(v_n\rightarrow v\) strongly in \(W^{s_2,q}_0(\Omega )\). 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 \((0,0,0)\in F\) 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 \((u,v)=S(\lambda ,u,v)\), which means that, if \((\lambda ,u,v)\in F\), then, \((\lambda ,u,v)\) solve (3.1). It remains only to show that \((1,u,v)\in F\) which corresponds to our required solution.
It clear that for every \(\lambda >0\) by strong maximum principle \(u,v>0\) in \(\Omega \).
We argue by contradiction. Assume that there exist \(\lambda ^{*}\), such that for all \(\lambda \le \lambda ^{*}\), we have \((\lambda ,\, u,\,v)\in F\).
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
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 \(p,q>1\), \(\beta _1<p-1\) and \(\beta _2<q-1\), then from the last inequality, we get \(||u||^{p}_{W^{s_1,p}_0(\Omega )}\) and \(||v||^{q}_{W^{s_2,q}_0(\Omega )}\) are bounded and this provides the contradiction, and consequently, F must be unbounded with respect to \(\lambda \) and in particular for \(\lambda =1\), on have \((1,\,u,\,v)\in F\) which gives a solution to (3.1). \(\square \)
Now, we are able to prove our main result.
Proof of Theorem 1.1. Using Lemma 3.1, we deduce that the system
has a solution \((u_n,v_n)\in W^{s_1,p}_0(\Omega )\times W^{s_2,q}_0(\Omega )\).
Now, we claim
In fact, let \((w_1, w_2)\in W^{s_1,p}_0(\Omega )\times W^{s_2,q}_0(\Omega ) \) are the nontrivial solutions to
and
where \(0<m_1=\min _{t\ge 0}\Big \{\frac{1}{t^{\alpha _1+1}}+t^{\beta _1} \Big \} \) and \(0<m_2=\min _{t\ge 0}\Big \{\frac{1}{t^{\alpha _2}+1}+t^{\beta _2}\Big \}\).
Since \(p,q>1\), then using the comparison principle (see [20] ) and Lemma 2.5, we get
as desired. Let us show that the sequences \(\{u_n\}_n\) and \(\{v_n\}_n\) are bounded in \(W^{s_1,p}_0(\Omega )\) and \( W^{s_2,q}_0(\Omega )\), respectively.
First, we take \(u_n\) as test function in first equation of (3.4), and we get
Since \(\beta _1 p'<q'\) and \(q'\in [1, q^*_{s_2}]\), 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, \(w_2(x)\ge Cd^{s_2}(x)\); therefore, \(v_n\ge w_2(x)\ge Cd^{s_2}(x)\) in \(\Omega \).
Therefore, we get
Since \(s_1>\alpha _1 s_2\), then, using Hölder and Hardy inequalities, we reach that
Therefore, we conclude that
By the same computation as in above, if \(s_2>\alpha _2 s_1\), \(\beta _2 q'<p'\) and \(p'\in [1, p^{*}_{s_1}]\) are satisfies, we can show that
Since \(p,q> 1\) and \(\beta _1,\beta _2 <\min \{p-1, q-1\}\) and from (3.8) and (3.9), we reach that \(\{u_n\}_n\) and \(\{v_n\}_n\) are bounded in \(W^{s_1,p}_0(\Omega )\) and in \( W^{s_2,q}_0(\Omega )\), respectively. Therefore, there exist two measurable functions \(u\in W^{s_1,p}_0(\Omega ) \) and \(v\in W^{s_2,q}_0(\Omega )\), such that
-
(1)
\(u_n\rightharpoonup u\) weakly in \(W^{s_1,p}_0(\Omega )\),
-
(2)
\(u_n \rightarrow u\) in \(L^{\tau }\) strongly for every \(\tau \in [1, p^{*}_{s_1})\),
-
(3)
\(v_n \rightharpoonup v\) weakly in \(W^{s_2,q}_0(\Omega )\),
-
(4)
\(v_n \rightarrow v\) in \(L^{\theta }(\Omega )\) strongly for every \(\theta \in [1, q^{*}_{s_2})\).
-
(5)
\(u_n (x) \rightarrow u(x)\) a.e in \(\Omega \),
-
(6)
\(v_n (x)\rightarrow v (x)\) a.e in \(\Omega \).
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.
Corollary 3.2
Let \(\Omega \) be a bounded regular domain of \({I\!\!R}^N\), \(s\in (0,1)\), \(p\in [1, \frac{N}{s})\), \(q\in [1, \frac{N}{s})\), \(p'\in [1,p^{*}_{s}]\), \(q'\in [1,q^{*}_{s}]\) where \(p'\) and \(q'\) are conjugate exponents of p and q, respectively. Assume that \(\alpha _i,\beta _i \in (0,\gamma _i)\) for \(i=1,2\), such that
Then, System (1.1) possesses a nontrivial solution in \(W^{s_1,p}_0(\Omega )\times W^{s_2,q}_0(\Omega )\).
Remark 1
Notice that, if we take, \(s_1=s_2=1\) in Theorem 1.1, we get the result obtained in [1].
Data availability
Not applicable.
References
Alves, C.O.: Correa On the existence of positive solution for a class of singular systems involving quasilinear operator. Appl. Math. Comput. 185(1), 727–736 (2007)
Alves, C.O.; Corrêa, F.J.S.A.; Gonçalves, J.V.A.: Existence of solutions for some classes of singular Hamiltonian systems. Adv. Nonlinear Stud. 5, 265–278 (2005)
Arcoya, D.; Moreno-Mérida, L.: Multiplicity of solutions for a Dirichlet problem with a strongly singular nonlinearity. Nonlinear Anal. 95, 281–291 (2014)
Bal, K.; Garain, P.: Multiplicity of solution for a quasilinear equation with singular nonlinearity. Mediterr. J. Math. 17, 91 (2020). https://doi.org/10.1007/s00009-020-01515-5
Barrios, B.; DeBonis, I.; Medina, M.; Peral, I.: Semilinear problems for the fractional Laplacian with a singular nonlinearity. J. Open. Math. 13, 91–107 (2015)
Berdan, N.E.; Diáz, J.I.; Rakotoson, J.M.: The uniform Hopf inequality for discontinuous coefficients and optimal regularity in BMO for singular problems. J. Math. Anal. Appl. 437, 350–379 (2016)
Boccardo, L.; Orsina, L.: Semilinear elliptic equations with singular nonlinearities. Calculus Var. Partial Differ. Equ. 37(3/4), 363–380 (2010)
Boukarabila, Y.O.: Anisotropic system with singular and regular nonlinearities. Complex Variables Elliptic Equ (2019). https://doi.org/10.1080/17476933.2019.1606802
Crandall, M.G.; Rabinowitz, P.H.; Tartar, L.: On a Dirichlet problem with a singular nonlinearity. Commun. Partial Differ. Equ. 2(2), 193–222 (1977)
Di Nezza, E.; Palatucci, G.; Valdinoci, E.: Hitchhiker’s guide to the fractional Sobolev spaces. Bull. Sci. Math. 136(5), 521–573 (2012)
Diaz, J.I.; Morel, J.M.; Oswald, L.: An elliptic equation with singular nonlinearity. Commun. Partial Differ. Equ. 12(12), 1333–1344 (1987)
Ghanmi, A.; Saoudi, K.: The Nehari manifold for a singular elliptic equation involving the fractional Laplace operator. J. Fract. Differ. Calc. 6(2), 201–217 (2016)
Ghanmi, A.; Saoudi, K.: A multiplicity results for a singular problem involving the fractional p-Laplacian operator. J. Complex Variables Elliptic Equ. 61(9), 1199–1216 (2016)
Giacomoni, J.; Saoudi, K.: Multiplicity of positive solutions for a singular and critical problem. Nonlinear Anal. 71(9), 4060–4077 (2009)
Giacomoni, J.; Sreenadh, K.: Multiplicity results for a singular and quasilinear equation. Discrete Continuous Dyn. Syst. 2007(Special), 429–435 (2007)
Giacomoni, J.; Schindler, I.; Takác, P.: Sobolev versus Hölder local minimizers and existence of multiple solutions for a singular quasilinear equation. Annali della Scuola Normale Superiore di Pisa Tome 6(1), 117–158 (2007)
Giacomoni, J.; Hernandez, J.; Sauvy, P.: Quasilinear and singular elliptic systems. Adv. Nonlinear Anal. 2, 1–41 (2013). https://doi.org/10.1515/anona-2012-0019
Haitao, Y.: Multiplicity and asymptotic behavior of positive solutions for a singular semilinear elliptic problem. J. Differ. Equ. 189(2), 487–512 (2003)
Ihnatsyeva, L.; Lehrbäck, J.; Tuominen, H.; Väahäkangas, A.V.: Fractional Hardy inequalities and visibility of the boundary. arXiv:1305.4616
Jarohs, S.: Strong comparison principle for the fractional p-Laplacian and applications to starshaped rings. Adv. Nonlinear Stud. 18, 691–704 (2018)
Lazer, A.C.; McKenna, P.J.: On a singular nonlinear elliptic boundary-value problem. Proc. Am. Math. Soc. 111(3), 721–730 (1991)
Manouni, E.; Perera, K.; Shivaji, R.: On singular quasimonotone (p, q)-Laplacian systems. Proc. R. Soc. Edinb. Sect. A 142, 585–594 (2012)
Molica Bisci, G.; Radulescu, V.D.; Servadei, R.: Variational Methods for Nonlocal Fractional Problems. Cambridge University Press, Cambridge (2016)
Mosconi, S.; Squassina, M.: Nonlocal problems at nearly critical growth. Nonlinear Anal. 136, 84–101 (2016)
Motreanu, D.; Moussaoui, A.: A quasilinear singular elliptic system without cooperative. Acta Math. Sci. 34(3), 905–916 (2014)
Mukherjee, T.; Sreenadh, K.: On Dirichlet problem for fractional \(p-\text{ Laplacian }\) with singular nonlinearity. Adv. Nonlinear Anal. (2016). https://doi.org/10.1515/anona-2016-0113
Panda, A.; Choudhuri, D.; Kumar, Giri R.: Existence of positive solutions for a singular elliptic problem with critical exponent and measure data. Rocky Mt. J. Math. 51(3), 973–988 (2021). https://doi.org/10.1216/rmj.2021.51.973
Rabinowitz, P.H.: Some global results for nonlinear eigenvalue problems. J. Funct. Anal. 7(3), 487–513 (1971)
Rosen, G.: Minimum value for \(c\) in the sobolev inequality \(|\varphi ^{3}|\le c |\varphi |^{3}\). SIAM J. Appl. Math. 21(1), 30–32 (1971)
Saoudi, K.: A critical fractional elliptic equation with singular nonlinearities. J. Fract. Calc. Appl. Anal. 20(6), 1507–1530 (2017)
Saoudi, K.; Ghosh, S.; Choudhuri, D.: Multiplicity and Hölder regularity of solutions for a nonlocal elliptic PDE involving singularity. J. Math. Phys. 60, 101509 (2019). https://doi.org/10.1063/1.5107517
Sun, Y.; Shaoping, W.; Long, Y.: Combined effects of singular and superlinear nonlinearities in some singular boundary value problems. J. Differ. Equ. 176(2), 511–531 (2001)
Warma, M.: Local Lipschitz continuity of the inverse of the fractional \(p-\text{ Laplacian }\), Hölder type continuity and continuous dependence of solutions to associated parabolic equations on bounded domains. Nonlinear Anal. Theory Methods Appl. 135, 129–157 (2016)
Acknowledgements
This project is partially supported by project C00L03UN380120220001, DGRSDT, Algeria.
Funding
Not applicable.
Author information
Authors and Affiliations
Corresponding author
Ethics declarations
Conflict of interest
The author declares that he has no conflict of interest.
Informed consent
Not applicable.
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/.
About this article
Cite this article
Biroud, K. Nonlocal fractional system involving the fractional p, q-Laplacians and singular potentials. Arab. J. Math. 11, 497–505 (2022). https://doi.org/10.1007/s40065-022-00382-0
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s40065-022-00382-0