Electronic Journal of Differential Equations, Vol. 2022 (2022), No. 77, pp. 1–15. ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu KIRCHHOFF SYSTEMS INVOLVING FRACTIONAL p-LAPLACIAN AND SINGULAR NONLINEARITY MOUNA KRATOU Abstract. In this work we consider the fractional Kirchhoff equations with singular nonlinearity, M (∫ R2N |u(x)− u(y)|p |x− y|N+sp dxdy ) (−∆)spu = λa(x)|u|q−2u+ 1− α 2− α− β c(x)|u|−α|v|1−β , in Ω, M (∫ R2N |v(x)− v(y)|p |x− y|N+sp dxdy ) (−∆)spv = µb(x)|v|q−2v + 1− β 2− α− β c(x)|u|1−α|v|−β , in Ω, u = v = 0, in RN \ Ω, where Ω is a bounded domain in RN with smooth boundary, N > ps, s ∈ (0, 1), 0 < α < 1, 0 < β < 1, 2 − α − β < p ≤ pθ < q < p∗s , p∗s = Np N−sp is the fractional Sobolev exponent, λ, µ are two parameters, a, b, c ∈ C(Ω) are non- negative weight functions, M(t) = k + ltθ−1 with k > 0, l, θ ≥ 1, and (−∆)sp is the fractional p-laplacian operator. We prove the existence of multiple non- negative solutions by studying the nature of the Nehari manifold with respect to the parameters λ and µ. 1. Introduction Let Ω be a bounded domain in RN with smooth boundary ∂Ω, N > ps, s ∈ (0, 1), and p∗s = Np N−ps . The purpose of this work is to study the existence of multiple solutions for the following Kirchhoff equations with fractional p-Laplacian operator and singular nonlinearity, M (∫ R2N |u(x)− u(y)|p |x− y|N+sp dxdy ) (−∆)spu = λa(x)|u|q−2u+ 1− α 2− α− β c(x)|u|−α|v|1−β , in Ω, 2020 Mathematics Subject Classification. 34B15, 37C25, 35R20. Key words and phrases. Kirchhoff-type equations; fractional p-Laplace operator; Nehari manifold; singular elliptic system; multiple positive solutions. ©2022. This work is licensed under a CC BY 4.0 license. Submitted September 11, 2022. Published November 21, 2022. 1 2 M. KRATOU EJDE-2022/77 M (∫ R2N |v(x)− v(y)|p |x− y|N+sp dxdy ) (−∆)spv = µb(x)|v|q−2v + 1− β 2− α− β c(x)|u|1−α|v|−β , in Ω, (1.1) u = v = 0, in RN \ Ω, where 0 < α < 1, 0 < β < 1, 2 − α − β < p ≤ pθ < q < p∗s, p ∗ s = N N−ps is the fractional Sobolev exponent, λ, µ are two parameters, a, b, c ∈ C(Ω) are non-negative weight functions with compact support in Ω, M(t) = k + ltθ−1 with k > 0, l, θ ≥ 1, and (−∆)sp is the fractional p-Laplacian operator defined as (−∆)spu(x) = 2 lim ε↘0 ∫ RN\Bε |u(x)− u(y)|p−2(u(x)− u(y)) |x− y|N+sp dy, x ∈ RN . Problems of this type describe diffusion processes in heterogeneous or com- plex medium (anomalous diffusion) caused by random displacements executed by jumpers that able to walk to neighboring and nearby sites and also excursions to remote sites by way of Lévy flights. They also can be used in modeling turbulence, chaotic dynamics, plasma physics and financial dynamics for more details see [1, 5] and references therein. Recently, a great deal of attention has been focused on studying this kind of non- local problems. We refer the readers to [17, 18, 19, 20, 21] for Kirchhoff problems involving the laplace operator and a singular term. For fractional Kirchhoff prob- lem involving a singular term of type u−γ has been studied in [10], by combining variational methods with an appropriate truncation argument. For further details on the fractional system, we refer the interested readers to [22, 32]. Problem (1.1) without a Kirchhoff coefficient has been studied extensively in recent years. In the case of the problem involving the fractional p-laplace exis- tence results via Morse theory has been treated in Iannizzotto-Liu-Perera-Squassina [16]. The critical case is treated in Perera-Squassina-Yang [23] with additional new abstract result based on a pseudo-index related to the Z2-cohomological index. These restrictions are used to prove the existence of a range of the validity of the Palais-Smale condition. Note that, in this work, the bifurcation and multiplicity results is obtained for some restrictions on the parameter λ. Moreover, by Nehari manifold and fibering maps the multiplicity of solutions has been investigated in [3, 7, 11, 12, 28, 30]. In particular, in [3], the authors considered the problem (−∆)spu = λ|u|q−2u+ 2α α+ β |u|α−2u|v|β , in Ω, (−∆)spv = µ|v|q−2v + 2β α+ β |u|α|v|β−2v, in Ω, u = v = 0, in RN \ Ω, where Ω is a bounded domain in Rn with smooth boundary ∂Ω, N > sp, s ∈ (0, 1), p < α + β < p∗, p∗s = Np N−sp is the fractional Sobolev exponent, λ, µ are two parameters. The authors considered the associated Nehari manifold using the fibering maps and showed the existence of solutions when the pair of parameters (λ, µ) satisfies certain conditions. In the local setting (s = 1), problem (1.1) without a Kirchhoff coefficient has an extensive literature. We refer the reader to the monographs of Ghergu-Radulescu EJDE-2022/77 NEHARI MANIFOLD APPROACH FOR SINGULAR PROBLEMS 3 [13] for a more general presentation of these results and to the survey article of Crandall-Rabinowitz-Tartar [6]. After this, many authors have considered the prob- lem above for laplacian, p-Laplacian, N -Laplacian operators, using the technique used in [6] or a combination of this approach with the Nehari’s and Perron’s meth- ods. Among the references we like to mention [4, 8, 14, 15, 26, 29, 24]. Motivated by above results, we show the existence and multiplicity of nontrivial, non-negative solutions of the singular fractional p-Kirchhoff system (1.1). To state our result, we introduce some notation. Let [u]s,p = (∫ R2N |u(x)− u(y)|p |x− y|N+ps dxdy )1/p be the Gagliardo seminorm of a measurable function u : R2N → R. Let W s,p(RN ) := {u ∈ Lp(RN ) : [u]s,p <∞} be the usual fractional Sobolev space endowed with the norm ‖u‖s,p := ( ‖u‖pLp(Ω) + [u]ps,p )1/p . We denote Q = R2N \ ( (RN \ Ω)× (RN \ Ω) ) and define the space X := { u : RN → R Lebesgue measurable : u\Ω ∈ Lp(Ω) and∫ Q |u(x)− u(y)|p |x− y|N+sp dx dy <∞ } with the norm ‖u‖X = ( ‖u‖Lp(Ω) + ∫ Q |u(x)− u(y)|p |x− y|N+ps dxdy )1/p . Through this paper we consider the space X0 to be the completion of the space C∞0 (Ω) in X, which is can be defined with the norm ‖u‖X0 = (∫ Q |u(x)− u(y)|p |x− y|N+sp dxdy )1/p . It is readily seen that (X0, ‖.‖) is a uniformly convex Banach space and that the embedding X0 ↪→ Lq(Ω) is continuous for all 1 ≤ q ≤ p∗s, and compact for all 1 ≤ q < p∗s. We define the best constant S of the embedding as S = inf{‖u‖pX0 : u ∈ X0, |u|pp∗s = 1}. (1.2) The dual space of (X0, ‖ · ‖) is denoted by (X∗, ‖.‖∗), and 〈·, ·〉 denotes the usual duality between X0 and X∗. When r+r′ ∈ (p, p∗), then, for any u ∈ X0, we obtain ‖u‖Lr+r′ (Ω) ≤ S‖u‖X0 . (1.3) Let E = X0 × X0 be the Cartesian product of two Hilbert spaces, which is a reflexive Banach space endowed with the norm ‖(u, v)‖ = (∫ Q |u(x)− u(y)|p |x− y|N+ps dxdy + ∫ Q |v(x)− v(y)|p |x− y|N+ps dxdy )1/p . (1.4) Definition 1.1. We say that (u, v) ∈ E is a weak solution of problem (1.1) if u, v > 0 in Ω, one has M(‖u‖X0 ) ∫ Q |u(x)− u(y)|p−2(u(x)− u(y))(φ(x)− φ(y)) |x− y|N+sp dx dy 4 M. KRATOU EJDE-2022/77 +M(‖v‖X0 ) ∫ Q |v(x)− v(y)|p−2(v(x)− v(y))(ψ(x)− ψ(y)) |x− y|N+sp dx dy = ∫ Ω ( λa(x)|u|q−2uφ+ µb(x)|v|q−2vφ ) dx+ 1− α 2− α− β ∫ Ω c(x)u−αv1−βψ dx + 1− β 2− α− β ∫ Ω c(x)u1−αv−βψ dx. for all (φ, ψ) ∈ E. We give below the precise statements of results that we will prove. Theorem 1.2. Let s ∈ (0, 1), N > sp and Ω be a bounded domain in Rn. If 0 < α < 1, 0 < β < 1, 2− α− β < p ≤ pθ < q < p∗s, then, there exists a number Λ0 = (q + α+ β − 2 ‖c‖∞k(q − p) ) p p+α+β−2 ( 2− α− β − q k(2− α− β − p) |Ω| p∗s−q p∗s )− p p−q S 2−α−β p+α+β−2 , such that for 0 < (λ‖a‖∞) p p−q + (µ‖b‖∞) p p−q < Λ0, problem (1.1) has at least two nontrivial positive solutions. The rest of this article is organized as follows. Section 2 is devoted to proof some lemmas in preparation for the proof of our main result. While, existence of two solutions,Theorem 1.2, will be presented in Sections 3 and 4. 2. Nehari manifold and fibering map analysis In this section, we collect some basic results on a Nehari manifold and we give the analysis of the fibering maps. Associated to problem (1.1) we define the functional Eλ,µ : E → R given by Eλ,µ(u, v) = k p ‖(u, v)‖p + l pθ ‖(u, v)‖pθ − 1 q ∫ Ω ( λa(x)|u|q + µb(x)|v|q ) dx − 1 2− α− β ∫ Ω c(x)(u+)1−α(v+)1−β dx. As usual, r+ = max{r, 0} and r− = max{−r, 0} for r ∈ R. Notice that Eλ,µ is not a C1 functional in E, and hence classical variational methods are not applicable. Notice that (u, v) is a weak solution of problem (1.1), then u, v > 0 in Ω and satisfy the equation k‖(u, v)‖p + l‖(u, v)‖pθ − λ ∫ Ω a(x)|u|q dx − µ ∫ Ω b(x)|v|q dx− ∫ Ω c(x)|u|1−α|v|1−β dx = 0. (2.1) One can easily verify that the energy functional Eλ,µ(u, v) is not bounded below on the space E. But we will show that Eλ,µ(u, v) is bounded below on the Nehari manifold defined below, and we will extract solutions by minimizing the functional on suitable subsets. The Nehari manifold is defined as Nλ,µ = { (u, v) ∈ E \ {(0, 0)}; k p ‖(u, v)‖p + l pθ ‖(u, v)‖pθ − λ ∫ Ω a(x)|u|q dx − µ ∫ Ω b(x)|v|q dx− ∫ Ω c(x)|u|1−α|v|1−β dx = 0 } . EJDE-2022/77 NEHARI MANIFOLD APPROACH FOR SINGULAR PROBLEMS 5 We note that Nλ,µ contains every solution of problem (1.1). Now as we know that the Nehari manifold is closely related to the behavior of the functions Φu,v : t 7→ Eλ,µ(tu, tv) for t > 0 defined by Φu,v(t) = ktp p ‖(u, v)‖p + ltpθ pθ ‖(u, v)‖pθ − tq q ∫ Ω ( λa(x)|u|q + µb(x)|v|q ) dx − t2−α−β 2− α− β ∫ Ω c(x)|u|1−α|v|1−β dx, which gives Φ′u,v(t) = ktp−1‖(u, v)‖p + ltpθ−1‖(u, v)‖pθ − tq−1 ∫ Ω ( λa(x)|u|q + µb(x)|v|q ) dx − t1−α−β ∫ Ω c(x)|u|1−α|v|1−β dx, (2.2) and Φ′′u,v(t) = (p− 1)ktp−2‖(u, v)‖p + l(pθ − 1)tpθ−2‖(u, v)‖pθ − (q − 1)tq−2 ∫ Ω ( λa(x)|u|q + µb(x)|v|q ) dx − (1− α− β)t−α−β ∫ Ω c(x)|u|1−α|v|1−β dx. (2.3) Such maps are called fibering maps and were introduced by Drábek and Pohozaev in [9]. By Hölder’s inequality and Sobolev inequalities, one has∫ Ω (λa(x)|u|q + µb(x)|v|q) dx ≤ |Ω| p∗s−q p∗s ( λ‖a‖∞‖u‖qp∗s + µ‖b‖∞‖v‖qp∗s ) ≤ |Ω| p∗s−q p∗s S− q p (λ‖a‖∞‖u‖q + µ‖b‖∞‖v‖q) ≤ |Ω| p∗s−q p∗s S− q p ( (λ‖a‖∞) p p−q + µ‖b‖∞) p p−q ) p−q p (‖u‖q + ‖v‖q) ≤ C|Ω| p∗s−q p∗s S− q p ( (λ‖a‖∞) p p−q + (µ‖b‖∞) p p−q ) p−q p ‖(u, v)‖q (2.4) and using Young’s and Sobolev inequalities, we obtain∫ Ω c(x)|u|1−α|v|1−β dx ≤ ‖c‖∞ ( 1− α 2− α− β ∫ Ω |u|2−α−β dx+ 1− β 2− α− β ∫ Ω |v|2−α−β dx ) ≤ ‖c‖∞S− 2−α−β p ‖(u, v)‖2−α−β . (2.5) Lemma 2.1. Let (u, v) ∈ E \{(0, 0)}. Then (tu, tv) ∈ Nλ,µ if and only if Φ′u,v(t) = 0. Proof. The result is a consequence of the fact that Φ′u,v(t) = 〈E′λ,µ(u, v), (u, v)〉 = ktp−1‖(u, v)‖p + ltpθ−1‖(u, v)‖pθ 6 M. KRATOU EJDE-2022/77 − tq−1 (∫ Ω λa(x)|u|q dx− ∫ Ω µb(x)|v|q dx ) − t1−α−β ∫ Ω c(x)|u|1−α|v|1−β dx = 0 if and only if (tu, tv) ∈ Nλ,µ. � From lemma 2.1, we have that the elements in Nλ,µ correspond to station- ary points of the maps Φu,v(tu, tv) and in particular, (u, v) ∈ Nλ,µ if and only if Φ′u,v(1) = 0. Hence, it is natural to split Nλ,µ into three parts corresponding to local minima, local maxima and points of inflection Φu,v(t) defined as follows: N+ λ,µ = {(u, v) ∈ Nλ,µ : Φ′′u,v(1) > 0} = {(tu, tv) ∈ E \ {0, 0} : Φ′u,v(t) = 0,Φ′′u,v(t) > 0}, N−λ,µ = {(u, v) ∈ Nλ,µ : Φ′′u,v(1) < 0} = {(tu, tv) ∈ E \ {0, 0} : Φ′u,v(t) = 0,Φ′′u(t) < 0}, N 0 λ,µ = {(u, v) ∈ Nλ,µ : Φ′′u,v(1) = 0} = {(tu, tv) ∈ E \ {0, 0} : Φ′u,v(t) = 0,Φ′′u,v(t) = 0}. Our first result is as follows. Lemma 2.2. If (u, v) is a minimizer of Eλ,µ on Nλ,µ such that (u, v) 6∈ N 0 λ,µ. Then, (u, v) is a critical point for Eλ,µ. For a proof of the above lemma see [31]. Lemma 2.3. Eλ,µ is coercive and bounded below on Nλ,µ. Proof. Since (u, v) ∈ Nλ,µ, then using (2.1) and the embedding of X0 in L2−α−β(Ω), we obtain Eλ,µ(u, v) = k (1 p − 1 q ) ‖(u, v)‖p + l ( 1 pθ − 1 q ) ‖(u, v)‖pθ − ( 1 2− α− β − 1 q ) ∫ Ω c(x)|u|1−α|v|1−β dx. Then by (2.5), we obtain Eλ,µ(u, v) ≥ k (1 p − 1 q ) ‖(u, v)‖p + l ( 1 pθ − 1 q ) ‖(u, v)‖pθ − ( 1 2− α− β − 1 q ) ‖c‖∞S− 2−α−β 2 ‖(u, v)‖2−α−β . Since 2 − α − β < p ≤ pθ, it follows that Eλ,µ is coercive and bounded below on Nλ,µ. This completes the proof. � Lemma 2.4. For each (u, v) ∈ N−λ,µ (respectively N+ λ,µ) with u, v ≥ 0, and all (φ, ψ) ∈ Nλ,µ with (φ, ψ) ≥ 0, there exist ε > 0 and a continuous function h = h(r) > 0 such that for all r ∈ R with |r| < ε we have h(0) = 1 and h(r)(u+ rφ, v+ rψ) ∈ N−λ,µ (respectively N+ λ,µ). Proof. We introduce the function f : R× R −→ R defined by f(t, r) = ktp+α+β−2‖(u+ rφ, v + rψ)‖p + ltpθ+α+β−2‖(u+ rφ, v + rψ)‖pθ EJDE-2022/77 NEHARI MANIFOLD APPROACH FOR SINGULAR PROBLEMS 7 − (q + α+ β − 2)tq+α+β−3 ∫ Ω (λa(x)(u+ rφ)q + µb(x)(v + rψ)q) dx − ∫ Ω c(x)(u+ rφ)1−α(v + rψ)1−βdx. Hence, ft(t, r) = k(p+ α+ β − 2)tp+α+β−3‖(u+ rφ, v + rψ)‖p + l(pθ + α+ β − 2)tpθ+α+β−3‖(u+ rφ, v + rψ)‖pθ − tq+α+β−2 ∫ Ω (λa(x)(u+ rφ)q + µb(x)(v + rψ)q) dx. Then, ft is continuous on R × R. Now, since (u, v) ∈ N−λ,µ ⊂ Nλ,µ, we have f(1, 0) = 0, and ft(1, 0) = k(p+ α+ β − 2)‖(u, v)‖p + l(pθ + α+ β − 2)‖(u, v)‖pθ − (q + α+ β − 2) ∫ Ω (λa(x)uq + µb(x)vq) dx < 0. Therefore, applying the implicit function theorem to the function f at the point (1, 0) we obtain a δ > 0 and a positive continuous function h = h(r) > 0, r ∈ R, |r| < δ satisfying h(0) = 1 and h(r)(u + rφ, v + rψ) ∈ Nλ,µ, for all r ∈ R, |r| < δ. Hence, taking ε > 0 possibly smaller enough (ε < δ), we obtain h(r)(u+ rφ, v + rψ) ∈ N−λ,µ, ∀r ∈ R, |r| < ε. The case (u, v) ∈ N+ λ,µ may be obtained in the same way. This completes the proof. � Lemma 2.5. There exists Λ0 = (q + α+ β − 2 ‖c‖∞k(q − p) ) p p+α+β−2 ( 2− α− β − q k(2− α− β − p) |Ω| 2∗s−q 2∗s )− p p−q S 2−α−β p+α+β−2 , such that for 0 < (λ‖a‖∞) p p−q + (µ‖b‖∞) p p−q < Λ0 the following holds: (1) If ∫ Ω ( λa(x)|u|q + µb(x)|v|q ) dx > 0, then, there exist a unique Tl > 0 and unique t0 < Tl < t1 such that Φu,v(t0) = Φu,v(t1), Φ′u,v(t0) < 0 < Φ′u,v(t1); that is, (t0u, t0v) ∈ N+ λ,µ, (t1u, t1v) ∈ N−λ,µ and Eλ,µ(t0u, t0v) = min 0≤t≤t1 Eλ,µ(tu, tv), Eλ,µ(t1u, t1v) = max t≥Tl Eλ,µ(tu, tv). (2) If ∫ Ω ( λa(x)|u|q + µb(x)|v|q ) dx < 0, then there exist a unique Tl > 0 such that (Tlu, Tlv) ∈ N−λ,µ and Eλ,µ(Tlu, Tlv) = maxt≥0Eλ,µ(tu, tv). Proof. (1) ∫ Ω ( λa(x)|u|q + µb(x)|v|q ) dx > 0, We introduce the function ψu,v : R+ −→ R define by ψu,v(t) = ktp−q‖(u, v)‖p + ltpθ−q‖(u, v)‖pθ − t2−α−β−q ∫ Ω c(x)|u|1−α|v|1−β dx. 8 M. KRATOU EJDE-2022/77 Note that (tu, tv) ∈ Nλ,µ if and only if ψu,v(t) = ∫ Ω ( λa(x)|u|q + µb(x)|v|q ) dx. Now, the first derivative of the function ψ is ψ′u,v(t) = k(p− q)tp−q−1‖(u, v)‖p + (pθ − q)ltpθ−q−1‖(u, v)‖pθ − (2− α− β − q)t1−α−β−q ∫ Ω c(x)|u|1−α|v|1−β dx = t−q−1 ( k(p− q)tp‖(u, v)‖p + (pθ − q)ltpθ‖(u, v)‖pθ − (2− α− β − q)t−α−β+2 ∫ Ω c(x)|u|1−α|v|1−β dx ) (2.6) It is clear that ψu,v(t)→ −∞ as t→∞. Moreover, using (2.6), it is simple to see that limt→0+ ψ′u,v(t) > 0 and limt→∞ ψ′u,v(t) < 0. That is there exists Tl > 0 such that ψu,v(t) is increasing on (0, Tl), decreasing on (Tl,∞) and ψ′u,v(Tl) = 0. So, ψu,v(Tl) = kT p−ql ‖(u, v)‖p + lT pθ−ql ‖(u, v)‖pθ − T 2−α−β−q l ∫ Ω c(x)|u|1−α|v|1−β dx. where Tl is the solution of k(p− q)tp‖(u, v)‖p + (pθ − q)ltpθ‖(u, v)‖pθ − (2− α− β − q)t−α−β+2 ∫ Ω c(x)|u|1−α|v|1−β dx = 0. (2.7) Then, using (2.7), we obtain T0 := ( (2− α− β − q) ∫ Ω c(x)|u|1−α|v|1−β dx k(p− q)‖(u, v)‖p ) 1 p+β+α−2 ≤ Tl. (2.8) From inequality (2.8), we can find a constant C = C(p, q, α, β) > 0 such that ψu,v(Tl) ≥ ψu,v(T0) ≥ kT p−q0 ‖(u, v)‖p − T 2−α−β−q 0 ∫ Ω c(x)|u|1−α|v|1−β dx ≥ k ( α+ β q + α+ β − 2 )(q + α+ β − 2 k(q − 2) ) 2−q β+α ‖(u, v)‖2 q+α+β−2 β+α( ∫ Ω c(x)|u|1−α|v|1−β dx ) q−2 β+α − |Ω| 2∗s−q 2∗s S− q 2 ( (λ‖a‖∞) 2 2−q + (µ‖b‖∞) 2 2−q ) 2−q 2 ‖(u, v)‖q > 0 if and only if (λ‖a‖∞) 2 2−q + (µ‖b‖∞) 2 2−q < ( k(q − 2) ‖c‖∞(q + α+ β − 2) )− 2 α+β (q + α+ β − 2 k(α+ β) |Ω| 2∗s−q 2∗s )− 2 2−q S α+β−2 α+β + q 2−q = Λ0. Then, there exists exactly two points t0 < Tl and t1 > Tl with ψ′u,v(t0) = ∫ Ω ( λa(x)|u|q + µb(x)|v|q ) dx = ψ′u,v(t1). EJDE-2022/77 NEHARI MANIFOLD APPROACH FOR SINGULAR PROBLEMS 9 Also, ψ′u,v(t0) > 0 and ψ′u,v(t1) < 0. That is, (t0u, t0v) ∈ N+ λ,µ and (t1u, t1v) ∈ N−λ,µ. Since Φ′u,v(t) = tq ( ψu,v(t)− ∫ Ω ( λa(x)|u|q + µb(x)|v|q ) dx ) . Thus, Φ′u,v(t) < 0 for all t ∈ [0, t0) and Φ′u,v(t) > 0 for all t ∈ (t0, t1). Hence Eλ,µ(t0u, t0v) = min0≤t≤t1 Eλ,µ(tu, tv). In the same way, Φ′u,v(t) > 0 for all t ∈ (t0, t1), Φ′u,v(t) = 0 and Φ′u,v(t) < 0 for all t ∈ (t1,∞) that is Eλ,µ(t1u, t1v) = max t≥Tl Eλ,µ(tu, tv). (2) ∫ Ω ( λa(x)|u|q + µb(x)|v|q ) dx < 0. So ψu,v(t) → −∞ as t → ∞. Therefore, for all (λ, µ) there exists Tl > 0 such that (Tlu, Tlv) ∈ N−λ,µ and Eλ,µ(Tlu, Tlv) = maxt≥0Eλ,µ(tu, tv). � As a consequence of Lemma 2.5, we have the following result. Lemma 2.6. There exists Λ0 = (q + α+ β − 2 ‖c‖∞k(q − p) ) p p+α+β−2 ( 2− α− β − q k(2− α− β − p) |Ω| p∗s−q p∗s )− p p−q S 2−α−β p+α+β−2 , such that for 0 < (λ‖a‖∞) p p−q + (µ‖b‖∞) p p−q < Λ0, we have N±λ,µ 6= ∅ and N 0 λ,µ = {0}. Proof. Firstly, using Lemma 2.4, we conclude that N±λ,µ are non-empty for all (λ, µ) with 0 < (λ‖a‖∞) p p−q + (µ‖b‖∞) p p−q < Λ0. Now, we proceed by contradiction to prove that N 0 λ,µ = {0} for all (λ, µ) with 0 < (λ‖a‖∞) p p−q + (µ‖b‖∞) p p−q < Λ0. Let (u, v) ∈ N 0 λ,µ. Then, we have two cases. Case 1 (u, v) ∈ Nλ,µ and ∫ Ω ( λa(x)|u|q + µb(x)|v|q ) dx = 0. Using (2.2) and (2.3) with t = 1, it follows that (p− 1)k‖(u, v)‖p + l(pθ − 1)‖(u, v)‖pθ − (1− α− β) ∫ Ω c(x)|u|1−α|v|1−β dx = (p+ α+ β − 2)k‖(u, v)‖p + l(pθ + α+ β − 2)‖(u, v)‖pθ > 0 which is a contradiction. Case 2 Let (u, v) ∈ Nλ,µ and ∫ Ω ( λa(x)|u|q + µb(x)|v|q ) dx = 0. Using (2.2) and (2.3) with t = 1, it follows that (p− q)k‖(u, v)‖p + l(pθ − q)‖(u, v)‖pθ (2.9) = −(q + α+ β) ∫ Ω c(x)|u|1−α|v|1−β dx, (2.10) (2− α− β − p)k‖(u, v)‖p + l(2− α− β − pθ)‖(u, v)‖pθ = (2− α− β − q) ∫ Ω ( λa(x)|u|q + µb(x)|v|q ) dx. (2.11) Now, we define Jλ,µ : Nλ,µ → R by Jλ,µ(u, v) = 2− α− β − p 2− α− β − q k‖(u, v)‖p + 2− α− β − pθ 2− α− β − q l‖(u, v)‖pθ − ∫ Ω ( λa(x)|u|q + µb(x)|v|q ) dx. 10 M. KRATOU EJDE-2022/77 Hence, from (2.11), Jλ,µ(u, v) = 0 for all (u, v) ∈ N 0 λ,µ. Moreover, Jλ,µ(u, v) ≥ 2− α− β − p 2− α− β − q k‖(u, v)‖p − ∫ Ω ( λa(x)|u|q + µb(x)|v|q ) dx ≥ 2− α− β − p 2− α− β − q k‖(u, v)‖p − C|Ω| p∗s−q p∗s S− q p ( (λ‖a‖∞) p p−q + (µ‖b‖∞) p p−q ) p−q p ‖(u, v)‖q ≥ ‖(u, v)‖q (2− α− β − p 2− α− β − q k‖(u, v)‖p−q − C|Ω| p∗s−q p∗s S− q p ( (λ‖a‖∞) p p−q + (µ‖b‖∞) p p−q ) p−q p ) . Then, using (2.5) and (2.9), we obtain ‖(u, v)‖ ≥ 1 ‖c‖∞ S− 2−α−β p(p+α+β−2) ( k(p− q) 2− α− β − q )− 1 p+α+β−2 . (2.12) By (2.12) we obtain Jλ,µ(u, v) ≥ ‖(u, v)‖q (2− α− β − p 2− α− β − q k ( k(p− q)‖c‖∞S 2−α−β p(p+α+β−2) )( k(p− q) 2− α− β − q ) q−p p+α+β−2 − C|Ω| p∗s−q p∗s S− q p ( (λ‖a‖∞) p p−q + (µ‖b‖∞) p p−q ) p−q p ) . This implies that for 0 < (λ‖a‖∞) p p−q + (µ‖b‖∞) p p−q < Λ0, we have Jλ,µ(u, v) > 0, for all (u, v) ∈ N 0 λ,µ, which is a contradiction. The proof is complete. � By Lemmas 2.3 and 2.4, for 0 < (λ‖a‖∞) p p−q + (µ‖b‖∞) p p−q < Λ0, we can write Nλ,µ = N+ λ,µ ∪N − λ,µ and define c+λ,µ = inf (u,v)∈N+ λ,µ Eλ,µ(u, v), c−λ,µ = inf (u,v)∈N−λ,µ Eλ,µ(u, v). 3. Existence of a minimizer on N+ λ,µ In this section, we will show that the minimum of Eλ,µ is achieved in N+ λ,µ. Also, we show that this minimizer is also a solution of problem (1.1). Lemma 3.1. If 0 < (λ‖a‖∞) p p−q + (µ‖b‖∞) p p−q < Λ0, then for all (u, v) ∈ N+ λ,µ, c+λ,µ < 0. Proof. Let (u+ 0 , v + 0 ) ∈ N+ λ,µ, then we have Φ′′ (u+ 0 ,v + 0 ) (1) > 0 which from (2.1) gives∫ Ω c(x)|u|1−α|v|1−β dx < k(p− q) 2− α− β − q ‖(u, v)‖p + l(pθ − q) 2− α− β − q ‖(u, v)‖pθ. (3.1) EJDE-2022/77 NEHARI MANIFOLD APPROACH FOR SINGULAR PROBLEMS 11 Hence, using (2.1) with (3.1), we have Eλ,µ(u, v) ≤ k( 1 p − 1 q )‖(u, v)‖p + l( 1 pθ − 1 q )‖(u, v)‖pθ − ( 1 2− α− β − 1 q ) ∫ Ω c(x)|u|1−α|v|1−β dx ≤ [ k( 1 p − 1 q )− ( 1 2− α− β − 1 q ) k(p− q) 2− α− β − q ] ‖(u, v)‖p + [ l( 1 pθ − 1 q )− ( 1 2− α− β − 1 q ) l(pθ − q) 2− α− β − q ] ‖(u, v)‖pθ. (3.2) Thus, by (3.2), we obtain Eλ,µ(u, v) < − (k(q − p)(p+ α+ β − 2) pq(2− α− β) ‖(u, v)‖p + l(q − p)(p+ α+ β − 2) pq(2− α− β) ‖(u, v)‖pθ ) < 0. Therefore, c+λ,µ < 0 follows from the definition c+λ,µ. This completes the proof. � Theorem 3.2. If 0 < (λ‖a‖∞) p p−q + (µ‖b‖∞) p p−q < Λ0, then there exists (u+ 0 , v + 0 ) in N+ λ,µ satisfying Eλ,µ(u+ 0 , v + 0 ) = inf(u,v)∈N+ λ,µ Eλ,µ(u, v). Proof. Since Eλ,µ is bounded below on Nλ,µ and so is on N+ λ,µ. Then, there exists {(u+ n , v + n )} ⊂ N+ λ,µ a sequence such that Eλ,µ(u+ n , v + n )→ inf (u,v)∈N+ λ,µ Eλ,µ(u, v) as n→∞. Since Eλ,µ is coercive, {un, vn} is bounded in E. Then there exists a subsequence, still denoted by (u+ n , v + n ) and (u+ 0 , v + 0 ) ∈ E such that, as n→∞, u+ n ⇀ u+ 0 , v + n ⇀ v+ 0 weakly in X0, u+ n → u+ 0 , v + n → v+ 0 strongly in Lr(Ω) for 1 ≤ r < p∗s, u+ n → u+ 0 , v + n → v+ 0 a.e. in Ω. By Vitali’s theorem (see [25, pp. 133]), we claim that lim n→∞ ∫ Ω a(x)|u+ n |1−αdx = ∫ Ω a(x)|u+ 0 |1−αdx. (3.3) Indeed, we only need to prove that { ∫ Ω a(x)|u+ n |1−αdx, n ∈ N} is equi-absolutely- continuous. Note that {un} is bounded, by the Sobolev embedding theorem, so there exists a constant C > 0 such that |un|p∗s ≤ C < ∞. Moreover, by Hölder inequality we have∫ Ω a(x)u1−αdx ≤ ‖a‖∞ ∫ Ω |u|1−αdx ≤ ‖a‖∞|Ω| p∗s p∗s+α−1 |u|1−αp∗s . (3.4) From (3.4), for every ε > 0, setting δ = ( ε ‖a‖∞C1−α ) p∗s p∗s+γ−1 , 12 M. KRATOU EJDE-2022/77 when A ⊂ Ω with meas(A) < δ, we have∫ A a(x)|u+ n |1−αdx ≤ ‖a‖∞‖u‖1−αp∗s ( measA ) p∗s+α−1 p∗s ≤ ‖a‖∞C1−αδ p∗s+α−1 p∗s < ε. Thus, our claim is true. Similarly, lim n→∞ ∫ Ω b(x)|v+ n |1−βdx = ∫ Ω b(x)|v+ 0 |1−βdx. (3.5) On the other hand, by [2] there exists l ∈ Lr(RN ) such that |u+ n (x)| ≤ l(x), |v+ n (x)| ≤ l(x), as k →∞ for 1 ≤ r < p∗s. Therefore by the Dominated convergence Theorem,∫ Ω ( λ|u+ n |q + µ|v+ n |q ) dx→ ∫ Ω ( λ|u+ 0 |q + µ|v+ 0 |q ) dx. Moreover, by Lemma 2.5, there exists t0 such that (t0u + 0 , t0v + 0 ) ∈ N+ λ,µ. Now, we shall prove that u+ n → u+ 0 strongly in X0, v+ n → v+ 0 strongly in X0. Suppose otherwise, then ‖(u+ 0 , v + 0 )‖E ≤ lim inf n→∞ ‖(u+ n , v + n )‖E . On the other hand, since (u+ n , v + n ) ∈ N+ λ,µ, one has lim n→∞ Φ′ u+ n ,v + n (t0) = lim n→∞ ( ktp−1 0 ‖(u+ n , v + n )‖p + ltpθ−1 0 ‖(u+ n , v + n )‖pθ − tq−1 0 ∫ Ω ( λa(x)|u+ n |q + µb(x)|v+ n )|q ) dx− t1−α−β0 ∫ Ω c(x)|u+ n |1−α|v+ n |1−β dx ) > ktp−1 0 ‖(u+ 0 , v + 0 )‖p + ltpθ−1 0 ‖(u+ 0 , v + 0 )‖pθ − tq−1 0 ∫ Ω ( λa(x)|u+ 0 |q + µb(x)|v+ 0 )|q ) dx − t1−α−β0 ∫ Ω c(x)|u+ 0 |1−α|v + 0 |1−β dx = Φ′ u+ 0 ,v + 0 (t0) = 0. So, Φ′ u+ n ,v + n (t0) > 0 for n large enough. Moreover, (u+ n , v + n ) ∈ N+ λ,µ, and we can see for all n that Φ′ u+ n ,v + n (t) < 0 for t ∈ (0, t) and Φ′ u+ n ,v + n (1) = 0. Thus we must have t0 > 1. Moreover Φu+ n ,v + n (1) is decreasing for t ∈ (0, t0) and that is Eλ,µ(t0u + 0 , t0v + 0 ) < Eλ,µ(u+ 0 , v + 0 ) = lim n→∞ Eλ,µ(u+ n , v + n ) = inf (u,v)∈N+ λ,µ Eλ,µ(u, v) which gives a contradiction. Thus, u+ n → u+ 0 strongly in X0, v+ n → v+ 0 strongly in X0 and Eλ,µ(u+ 0 , v + 0 ) = inf(u,v)∈N+ λ,µ Eλ,µ(u, v). The proof of is complete. � 4. Existence of a minimizer on N−λ,µ In this section, we shall show the existence of a solution to problem (1.1) by proving the existence of minimizer of Eλ,µ on N−λ,µ. Lemma 4.1. If 0 < (λ‖a‖∞) p p−q + (µ‖b‖∞) p p−q < Λ0, then for all (u, v) ∈ N+ λ,µ, c−λ,µ > d0 for some d0 = d0(α, β, p, q, a, b, λ, µ, |Ω|) > 0. EJDE-2022/77 NEHARI MANIFOLD APPROACH FOR SINGULAR PROBLEMS 13 Proof. Let (u−0 , v − 0 ) ∈ N−λ,µ, then we have Φ′′ u−0 ,v − 0 (1) < 0 which from (2.1) gives∫ Ω c(x)|u|1−α|v|1−β dx > k(p− q) 2− α− β − q ‖(u, v)‖p + l(pθ − q) 2− α− β − q ‖(u, v)‖pθ. (4.1) Therefore using (2.5), we obtain ‖(u, v)‖ > 1 ‖c‖∞ S− 2−α−β p(p+α+β−2) ( k(p− q) 2− α− β − q )− 1 p+α+β−2 . (4.2) Hence, using (2.4) and (4.2), one has Eλ,µ(u, v) ≥ k (1 p − 1 2− α− β ) ‖(u, v)‖p − (1 q − 1 2− α− β ) |Ω| p∗s−q p∗s × S− q p ( (λ‖a‖∞) p p−q + (µ‖b‖∞) p p−q ) p−q p ‖(u, v)‖q = ‖(u, v)‖q [ k (1 p − 1 2− α− β ) ‖(u, v)‖p−q − (1 q − 1 2− α− β ) |Ω| p∗s−q p∗s × S− q p ( (λ‖a‖∞) p p−q + (µ‖b‖∞) p p−q ) p−q p ] > ‖(u, v)‖q [ k (1 p − 1 2− α− β ) S (p−q) p ( p− q 2− α− β − q ) q−p p+α+β−2 − (1 q − 1 2− α− β ) |Ω| p∗s−q p∗s S− q p ( (λ‖a‖∞) p p−q + (µ‖b‖∞) p p−q ) p−q p ] . Thus, if 0 < (λ‖a‖∞) p p−q + (µ‖b‖∞) p p−q < Λ0, then Eλ,µ(u, v) > d0 for all (u, v) ∈ N−λ,µ for some d0 = d0(α, β, p, q, a, b, λ, µ, |Ω|) > 0. Therefore c−λ,µ > d0 follows from the definition c−λ,µ. This completes the proof. � Theorem 4.2. If 0 < (λ‖a‖∞) p p−q + (µ‖b‖∞) p p−q < Λ0, then there exists (u−0 , v − 0 ) in N−λ,µ satisfying Eλ,µ(u−0 , v − 0 ) = inf(u,v)∈N−λ,µ Eλ,µ(u, v). Proof. Since Eλ,µ is bounded below on Nλ,µ and so on N−λ,µ. Then, there exists {(u−n , v−n )} ⊂ N−λ,µ, a sequence such that Eλ,µ(u−n , v − n )→ inf (u,v)∈N−λ,µ Eλ,µ(u, v) as n→∞. Since Eλ,µ is coercive, {(un, vn)} is bounded in E. Then there exists a subsequence, still denoted by (u−n , v − n ) and (u−0 , v − 0 ) ∈ E such that, as n→∞, u+ n ⇀ u−0 , v − n ⇀ v−0 weakly in X0, u−n → u−0 , v − n → v−0 strongly in Lr(Ω) for 1 ≤ r < p∗s, u−n → u−0 , v − n → v−0 a.e. in Ω. Moreover, as in Lemma 3.2, we have lim n→∞ ∫ Ω |u−n |1−αdx = ∫ Ω |u−0 |1−αdx, lim n→∞ ∫ Ω |v−n |1−βdx = ∫ Ω |v−0 |1−βdx,∫ Ω ( λa(x)|u+ n |q + µb(x)|v+ n |q ) dx→ ∫ Ω ( λa(x)|u+ 0 |q + µb(x)|v+ 0 |q ) dx. 14 M. KRATOU EJDE-2022/77 Moreover, by Lemma 2.5, there exists t1 such that (t1u − 0 , t1v − 0 ) ∈ N−λ,µ. Now, we prove that u−n → u−0 strongly in X0, v−n → v−0 strongly in X0. Suppose otherwise, then ‖(u−0 , v − 0 )‖E ≤ lim inf n→∞ ‖(u−n , v−n )‖E . Thus, since (u−n , v − n ) ∈ N−λ,µ, Eλ,µ(tu−0 , tv − 0 ) ≤ Eλ,µ(u−0 , v − 0 ), for all t ≥ 0 we have Eλ,µ(t1u − 0 , t1v − 0 ) < lim n→∞ Eλ,µ(t1u − n , t1v − n ) ≤ lim n→∞ Eλ,µ(u−n , v − n ) = c−λ,µ. which gives a contradiction. Thus, u−n → u−0 strongly in X0, v−n → v−0 strongly in X0 and Eλ,µ(u−0 , v − 0 ) = inf(u,v)∈N−λ,µ Eλ,µ(u, v). The proof is complete. � Proof of Theorem 1.2. Let us start by proving the existence of non-negative solu- tions. First, by Theorems 3.2, 4.2, there exist (u+ 0 , v + 0 ) ∈ N+ λ,µ, (u−0 , v − 0 ) ∈ N−λ,µ satisfying Eλ,µ(u+ 0 , v + 0 ) = inf (u,v)∈N+ λ,µ Eλ,µ(u, v), Eλ,µ(u−0 , v − 0 ) = inf (u,v)∈N−λ,µ Eλ,µ(u, v). Moreover, since Eλ,µ(u+ 0 , v + 0 ) = Eλ,µ(|u+ 0 |, |v + 0 |) and (|u+ 0 |, |v + 0 |) ∈ N + λ,µ. Similarly we have Eλ(u−0 , v − 0 ) = Eλ,µ(|u−0 |, |v − 0 |) and (|u−0 |, |v − 0 |) ∈ N − λ,µ, so we may assume (u±0 , v ± 0 ) ≥ 0. By Lemma 2.2, (u±0 , v ± 0 ) are the nontrivial non-negatives solutions of problem (1.1). Finally, it remain to show that the solutions found in Theorems 3.2, 4.2, are distinct. Since N−λ,µ ∩N + λ,µ = ∅, then (u±0 , v ± 0 ) are distinct. The proof of complete. � Acknowledgments. The author would like to thank the anonymous referees for their carefully reading this paper and their useful comments. References [1] D. Applebaum; Lévy Processes and Stochastic Calculus, second ed., Camb. Stud. Adv. Math., 116, Cambridge University Press, Cambridge, 2009. [2] H. Brezis; Analyse fonctionelle in: Théorie et Applications, Masson, Paris, 1983. [3] W. Chen and S. Deng; The Nehari manifold for a fractional p-Laplacian system involving concave-convex nonlinearities, Nonlinear Analysis: Real World Applications, 27 (2016), 80- 92. [4] M. M. Coclite, G. Palmieri; On a singular nonlinear Dirichlet problem, Comm. Partial Dif- ferential Equations, 14 (1989), 1315-1327. [5] A. Cotsiolis, N. Tavoularis; Best constants for Sobolev inequalities for higher order fractional derivatives, J. Math. Anal. Appl., 295 (2004), 225-236. [6] M. G. Crandall, P. H. Rabinowitz, L. Tartar; On a Dirichlet problem with a singular nonlin- earity, Comm. Partial Differential Equations, 2 (1977), 193–222. [7] A. Daoues, A. Hammami, K. Saoudi; Multiple positive solutions for a nonlocal PDE with crit- ical Sobolev-Hardy and singular nonlinearities via perturbation method, Fractional Calculus and Applied Analysis, 23(3) (2020), 837-860. [8] R. Dhanya, J. Giacomoni, S. Prashanth, K. Saoudi; Global bifurcation and local multiplicity results for elliptic equations with singular nonlinearity of super exponential growth in R2, Advances in Differential Equations, 17 (3-4) (2012), 369-400. [9] P. Drabek, S. I. Pohozaev; Positive solutions for the p-Laplacian: application of the fibering method, Proc. Royal Soc. Edinburgh Sect A, 127 (1997), 703-726. [10] A. Fiscella; A fractional Kirchhoff problem involving a singular term and a critical nonlin- earity, arXiv preprint arXiv:1703.07861. EJDE-2022/77 NEHARI MANIFOLD APPROACH FOR SINGULAR PROBLEMS 15 [11] A. Ghanmi, K. Saoudi; The Nehari manifold for a singular elliptic equation involving the fractional Laplace operator, Fractional Differential Calculus, 6 (2) (2016), 201-217. [12] A. Ghanmi, K. Saoudi; A multiplicity results for a singular problem involving the fractional p-Laplacian operator, Complex variables and elliptic equations, 61 (9) (2016) 1199-1216. [13] M. Ghergu, V. Radulescu; Singular elliptic problems: bifurcation and asymptotic analysis, Oxford Lecture Series in Mathematics and its Applications, 37, The Clarendon Press, Oxford University Press, Oxford, 2008. [14] M. Ghergu, V. Radulescu; Singular elliptic problems with lack of compactness, Ann. Mat. Pura Appl., 185(1) (2006), 63-79. [15] J. Giacomoni, K. Saoudi; Multiplicity of positive solutions for a singular and critical problem, Nonlinear Anal., 71(9) (2009), 4060-4077. [16] A. Iannizzotto, S. Liu, K. Perera, M. Squassina; Existence results for fractional p-Laplacian problems via Morse theory, Adv. Calc. Var., 9(2) (2016), 101-125. [17] C. Y. Lei, J. F. Liao, C. L. Tang; Multiple positive solutions for Kirchhoff type of problems with singularity and critical exponents, J. Math. Anal. Appl., 421 (2015), 521-538. [18] J. F. Liao, X. F. Ke, C. Y. Lei, C. L. Tang; A uniqueness result for Kirchhoff type problems with singularity, Appl. Math. Lett., 59 (2016), 24-30. [19] J. F. Liao, P. Zhang, J. Liu, C. L. Tang; Existence and multiplicity of positive solutions for a class of Kirchhoff type problems with singularity, J. Math. Anal. Appl., 430 (2015), 1124-1148. [20] X. Liu, Y. Sun; Multiple positive solutions for Kirchhoff type of problems with singularity, Commun. Pure Appl. Anal., 12 (2013), 721-733. [21] R. Q. Liu, C. L. Tang, J. F. Liao, X. P. Wu; Positive solutions of Kirchhoff type problem with singular and critical nonlinearities in dimension four, Commun. Pure Appl. Anal., 15 (2016), 1841-1856. [22] G. Molica Bisci, V. Radulescu, R. Servadei; Variational Methods for Nonlocal Fractional Problems, Encyclopedia of Mathematics and its Applications, 162, Cambride University Press, Cambridge, (2016), xvi+383 pp. [23] K. Perera, M. Squassina, Y. Yang; Bifurcation and multiplicity results for critical fractional p-Laplacian problems, Math. Nachr., 289 (2-3) (2016), 332-342.. [24] V. Radulescu; Combined effects in nonlinear singular elliptic problems with convection, Rev. Roumaine Math. Pures Appl. 53 (5-6) (2008), 543-553. [25] W. Rudin; Real and complex analysis, McGraw-Hill, New York, London, (1966). [26] K. Saoudi; Existence and non-existence for a singular problem with variables potentials; Electronic Journal of Differential equations, 2017 (291) (2017), 1-9. [27] K. Saoudi; A fractional Kirchhof system with singular nonlinearities; Analysis and Mathe- matical Physics, 9 (2019), 1463-1480. [28] K. Saoudi; A critical fractional elliptic equation with singular nonlinearities; Fractional Cal- culus and Applied Analysis, 20 (6) (2017), 1507-1530. [29] K. Saoudi, M. Kratou; Existence of multiple solutions for a singular and quasilinear equation, Complex Var. Elliptic Equ., 60 (7) (2015), 893-925. [30] K. Saoudi, M. Kratou, E. Al Zahrani; Uniqueness and existence of solutions for a singular system with nonlocal operator via perturbation method, Journal of Applied Analysis and Computation, 10 (4) (2020), 1311-1325. [31] G. Tarantello; On nonhomogenous elliptic involving critical Sobolev exponent, Ann. Inst. H. Poincare Anal. Non Lineare, 9 (1992), 281-304. [32] J. Yang, H. Chen, Z. Feng; Multiple positive solutions to the fractional Kirchhoff problem with critical indefinite nonlinearities, Electron. J. Differential Equations, 2020 (101) (2020), 1-21. Mouna Kratou College of sciences at Dammam, University of Imam Abdulrahman Bin Faisal, 31441 Dammam, Saudi Arabia. Basic and Applied Scientific Research Center, Imam Abdulrahman Bin Faisal University, P.O. Box 1982, 31441, Dammam, Saudi Arabia Email address: mmkratou@iau.edu.sa 1. Introduction 2. Nehari manifold and fibering map analysis 3. Existence of a minimizer on N,+ 4. Existence of a minimizer on N-, Acknowledgments References