# Onpz–Laplacian System Involving Critical Nonlinearities

Authors: Ahmed Aberqi; Jaouad Bennouna; Omar Benslimane; Maria Alessandra Ragusa
Journal: Journal of Function Spaces
 (2022)

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## Abstract

In this paper, we deal with the existence of at least two nonnegative nontrivial solutions to apz–Laplacian system involving critical nonlinearity in the context of Sobolev spaces with variable exponents on complete manifolds. We have established our main results by exploring both Nehari’s method and doing a refined analysis on the associated fiber map and some variational techniques.


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## Body

## 1. Introduction

In the present work, we investigate the existence of nonnegative nontrivial solutions to the following system:(1)S−Δg,pzw=αwqz−2w+2azaz+bzwaz−2wybzinM,−Δg,pzy=βyqz−2y+2bzaz+bzwazybz−2yinM,w∂M=y∂M=0.Here,M,g is a complete compact Riemannian N-manifold, α,β∈ℝ∗+ to be specified later, and p,q,a,b∈CM¯ satisfying the assumptions (23) and (24) in Section 3. −Δg,pz is the Laplacian operator on M,g.In recent years, several researchers have been interested in equations or systems involving thepz–Laplacian, not only for their application in several scientific fields, such as fluid filtration in porous media, constrained heating, elastoplasticity, and optimal control, but also for their mathematical importance in the theory of function spaces with variable exponents. For example, in [1], Zhang proved the existence of positive solutions under some conditions of the following class of pz–Laplacian systems:
(2)S1−Δpzw=αfz,yinQ,−Δqzw=αgz,winQ,w∂Q=y∂Q=0,in bounded open set Q⊂ℝN, without assuming the symmetric radial conditions. And by using the subsuper solution technique, Boulaaras et al. in [2] have studied the asymptotic behavior of the system S1. In [3], Aberqi et al. established the existence of a renormalized solution for a class of nonlinear parabolic systems using the Gagliardo-Nirenberg theorem, with the source term being less regular. In addition, we refer to the work of Marino and Winkert [4] who studied this kind of system with nongrowth conditions, governed by a double-phase operator. For the systems with singular source data, we refer to Saoudi [5] and Papageorgiou et al. [6]. For more results, we refer to [7–11], as well as to [12], and the references therein.Before explaining the novelty of this paper, we give an overview of the literature on this kind of system inW1,pQ. Adriouch and El Hamidi in [13] proved the existence and multiplicity of solutions to the following system:
(3)S2−Δpw=αwp1−2w+a+1wwa−1yb+1inQ,−Δqy=βyq−2y+b+1wa+1yb−1yinQ,by using the variational techniques. Chen and Wu in [14] examined the semilinear version of S2 with more general parametric functions fα, gβ, and convex-concave critical nonlinearity. Mercuri and Willem in [15] proved a representation theorem for Palais-Smale sequences involving the p–Laplacian and critical nonlinearities. For a deeper comprehension, see [16–18].Next, we will mention some papers that deal with the same problem with the fractionalp–Laplacian. We refer to Chen and Squassina [19], Pawan and Sreenadh [20], and Biswas and Tiwari [21] for fractional pz–Laplacian. Readers may refer to the references given therein for more background.Our goal in the present contribution is to study this kind of system with nonstandard convex-concave nonlinearity, in the Sobolev spaces on the complete manifold. We prove the existence of nonnegative nontrivial solutions using the Nehari manifold technique. However, we address the challenges due to the fact thatΔpz is not homogeneous, also due to the non-Euclidean framework of the system. Moreover, we do not have enough background on this space, such as embedding results, Hölder inequality, and the relation between the wpz and ρqzDw, including the pertinent result proven in ([22], Proposition 2.5). This is the first existing result in this field to the best of our knowledge.The theorem below contains our main result.Theorem 1.
LetM,g satisfy the Bvolα,y property. Then, there exists a constant K such that if 0<α+β<K, then the system S admits at least two nonnegative weak solutions.The organization of this contribution is given as follows. We start in Section2 by presenting some definitions and properties of Lebesgue spaces with variable exponents on a bounded set of Q⊂ℝN and on a complete manifold M. After that, in Section 3, we give some properties of the Nehari manifold and set up the variational framework of the system S. Then, we establish the existence of two nonnegative nontrivial solutions to the system S.


## 2. Notations and Basic Properties

This section is devoted to recalling some definitions and properties which will be used in the next sections (see [22–27]).Consider an open-bounded setQ of ℝN with N≥2. We define the Lebesgue space with variable exponent Lν·Q as the set of all measurable function w:Q↦ℝ such that
(4)ρν·w=∫Qwzνzdz<∞,endowed with the Luxembourg norm. And the associated Sobolev space is given by
(5)W1,νzQ=w∈LνzQ:Dw∈LνzQ,with the norm
(6)wW1,νzQ=wLνzQ+DwLνzQ∀w∈W1,νzQ.AndW01,νzQ=C0∞Q¯W1,νzQ.Lemma 2 (see [12]).
Letν1z∈L∞Q such that ν1≥0,ν1≠0. Let ν2:Q⟶ℝ be a measurable function such that ν1zν2z≥1 a.e. in Q. Then, for every w∈Lν1zν2zQ,(7)wLν2zQν1·≤wLν1zν2zQν1−+wLν1zν2zQν1+.


### 2.1. Sobolev Spaces on Manifolds

Definition 3 (see [22]).
Letw∈CcM and Qi,φii∈I be an atlas of M,(8)∫Mwzdvgz=∑k∈J∫φkQkdetgij1/2ηkuoφk−1zdz,where dvg=detgij1/2dz, gij are the components of the Riemannian metric g in the chart and dz is the Lebesgue measure of ℝN.Definition 4 (see [22]).
The Sobolev spaceLkq·M is the completion of Ckq·M with respect to the norm wLkq·, where
(9)Ckq·M=w∈C∞MsuchthatDjw∈Lq·M;j=0,⋯,k,wLkq·=∑j=0kDjwLq·,with Dkw2=∏1≤α≤kgiαjα. Dkwi1⋯ikDkwj1⋯jk is the norm of the k-th covariant derivative of w. If Q is a subset of M, then Lk,0q·Q is the completion of Ckq·M∩C0Q with respect to.Lkq·, where C0Q denotes the vector space of continuous functions whose support is a compact subset of Q.Definition 5 (see [22]).
Letξ:a,b⟶M a curve of class C1. The length of ξ is
(10)ℓξ=∫abgdξds,dξds1/2ds.
Letu,v∈M2, we define the distance between u and v by
(11)dgu,v=infℓξξ:a,b⟶M:ξa=uandξb=v.Definition 6 (see [22]).
Log-Hölder continuity: lett:M⟶ℝ; we say that t is log-Hölder continuous, if there exists c>0 such that
(12)tu−tv≤c.loge+dgu,v−1−1,∀u,v∈M.The set of Log-Hölder continuous functions onM will be denoted by PlogM, which is linked to PlogℝN by the proposition below.Proposition 7 (see [24, 25]).
LetQ,ϕ be a chart of M,q∈PlogM, such that
(13)12δij≤gij≤2δij,δijisthedeltaKroneckersymbol,like bilinear forms. Then, q∘ϕ−1∈PlogϕQ.Definition 8 (see [22]).
We say thatM,g has property Bvolα,y, if the Ricci tensor of g noted by Rcg verifies Rcg≥αN−1g for some α, and for all u∈M, there exists some y>0 such that B1zg≥y where B1z are the balls of radius 1 centered at some point z in terms of the volume of smaller concentric balls.
To compare the functionals·q· and ρq··, one has the relation
(14)minρq·w1/q−,ρq·w1/q+≤wLq·≤maxρq·w1/q−,ρq·w1/q+.Proposition 9 (see [23]).
Hölder’s inequality: for allw∈Lq·M and y∈Lq′·M, we have
(15)∫Mwzyzdvgz≤rqwLq·M.yLq′·M,
whererq is a positive constant depending on q− and q+.Definition 10 (see [22, 26]).
We define the Sobolev space onM,g by
(16)W1,qzM=w∈LqzM:Dkw∈LqzMk=1,2,⋯,n,endowed by the norm
(17)wW1,qzM=wLqzM+∑k=1nDkwLqzM,and we define W01,qzM as the closure of C∞M in W1,qzM.Theorem 11 (see [22, 23]).
Letaz,bz,qz,pz∈CM¯∩L∞M, with M as a compact Riemannian manifold.
(i)
If(18)pz<q∗z=NqzN−qzforz∈M¯,qz<N.
Then, we have(19)W1,qzM↪↪LpzM.(ii)
If(20)az+bz<q∗zforz∈M¯,qz<N.
Then, we have(21)W1,qzM↪↪Laz+bzM.Proposition 12 (see [24]).
We haveW1,qxM=W01,qxM, if M,g is complete.


## 2.1. Sobolev Spaces on Manifolds

Definition 3 (see [22]).
Letw∈CcM and Qi,φii∈I be an atlas of M,(8)∫Mwzdvgz=∑k∈J∫φkQkdetgij1/2ηkuoφk−1zdz,where dvg=detgij1/2dz, gij are the components of the Riemannian metric g in the chart and dz is the Lebesgue measure of ℝN.Definition 4 (see [22]).
The Sobolev spaceLkq·M is the completion of Ckq·M with respect to the norm wLkq·, where
(9)Ckq·M=w∈C∞MsuchthatDjw∈Lq·M;j=0,⋯,k,wLkq·=∑j=0kDjwLq·,with Dkw2=∏1≤α≤kgiαjα. Dkwi1⋯ikDkwj1⋯jk is the norm of the k-th covariant derivative of w. If Q is a subset of M, then Lk,0q·Q is the completion of Ckq·M∩C0Q with respect to.Lkq·, where C0Q denotes the vector space of continuous functions whose support is a compact subset of Q.Definition 5 (see [22]).
Letξ:a,b⟶M a curve of class C1. The length of ξ is
(10)ℓξ=∫abgdξds,dξds1/2ds.
Letu,v∈M2, we define the distance between u and v by
(11)dgu,v=infℓξξ:a,b⟶M:ξa=uandξb=v.Definition 6 (see [22]).
Log-Hölder continuity: lett:M⟶ℝ; we say that t is log-Hölder continuous, if there exists c>0 such that
(12)tu−tv≤c.loge+dgu,v−1−1,∀u,v∈M.The set of Log-Hölder continuous functions onM will be denoted by PlogM, which is linked to PlogℝN by the proposition below.Proposition 7 (see [24, 25]).
LetQ,ϕ be a chart of M,q∈PlogM, such that
(13)12δij≤gij≤2δij,δijisthedeltaKroneckersymbol,like bilinear forms. Then, q∘ϕ−1∈PlogϕQ.Definition 8 (see [22]).
We say thatM,g has property Bvolα,y, if the Ricci tensor of g noted by Rcg verifies Rcg≥αN−1g for some α, and for all u∈M, there exists some y>0 such that B1zg≥y where B1z are the balls of radius 1 centered at some point z in terms of the volume of smaller concentric balls.
To compare the functionals·q· and ρq··, one has the relation
(14)minρq·w1/q−,ρq·w1/q+≤wLq·≤maxρq·w1/q−,ρq·w1/q+.Proposition 9 (see [23]).
Hölder’s inequality: for allw∈Lq·M and y∈Lq′·M, we have
(15)∫Mwzyzdvgz≤rqwLq·M.yLq′·M,
whererq is a positive constant depending on q− and q+.Definition 10 (see [22, 26]).
We define the Sobolev space onM,g by
(16)W1,qzM=w∈LqzM:Dkw∈LqzMk=1,2,⋯,n,endowed by the norm
(17)wW1,qzM=wLqzM+∑k=1nDkwLqzM,and we define W01,qzM as the closure of C∞M in W1,qzM.Theorem 11 (see [22, 23]).
Letaz,bz,qz,pz∈CM¯∩L∞M, with M as a compact Riemannian manifold.
(i)
If(18)pz<q∗z=NqzN−qzforz∈M¯,qz<N.
Then, we have(19)W1,qzM↪↪LpzM.(ii)
If(20)az+bz<q∗zforz∈M¯,qz<N.
Then, we have(21)W1,qzM↪↪Laz+bzM.Proposition 12 (see [24]).
We haveW1,qxM=W01,qxM, if M,g is complete.


## 3. Proof of the Main Results

In this section, we prove our main result, and we note thatJ=W01,qzM×W01,qzM, endowed with norm w,ypz=Dwzpz+Dyzpz. In what follows, DM is the space of C∞ functions with compact support in M.


### 3.1. Nehari Manifold Analysis forS

First, we define the weak solution of systemS as follows.Definition 13.
We say thatw,y∈J is a weak solution of the system S, if w,y∈J one has
(22)∫MDwzpz−2gDwz,Dϕzdvgz+∫MDyzpz−2gDyz,Dψzdvgz=∫Mαwzqz−2wzϕz+βyzqz−2yzψzdvgz+∫M2azaz+bzwzaz−2wzyzbzϕzdvgz+∫M2bzaz+bzwzazyzbz−2yzψzdvgz,for all ϕ,ψ∈DM×DM.The functionsp,q,a,b∈CM¯ are assumed to satisfy the following assumption:
(23)1<q−≤q+<p−≤p+<a−+b−≤a++b+<∞,and the following condition holds
(24)p−a++b+<p−−q+a++b+−q+a−+b−−q+p+−q−.To prove our main result, we will use Nehari manifold and fibering maps. The fact thatw,y is a weak solution is equivalent to being a critical point of the following functional Eα,β:J⟶ℝ defined as
(25)Eα,βw,y=∫M1pzDwzpz+Dyzpzdvgz−∫M1qzαwzqz+βyzqzdvgz−∫M2az+bzwzazyzbzdvgz.By a direct calculation, we haveEα,β∈C1J,ℝ and
(26)E′α,βw,y,ϕ,ψ=∫Mαwpz−2wϕdvgz+∫Mαypz−2yψdvgz−∫Mαwqz−2wϕ+βyqz−2yψdvgz−∫M2azaz+bzwaz−2wybzϕdvgz−∫M2bzaz+bzwazybz−2yψdvgz,for any ϕ,ψ∈DM×DM.Consider the Nehari manifold(27)Nα,β=w,y∈J\0,0:<E′α,βw,y,w,y≥0.Then,w,y∈Nα,β if and only if
(28)∫MDwpz+Dypzdvgz−∫Mαwqz+βyqzdvgz−∫M2azaz+bzwazybzdvgz−∫M2bzaz+bzwazybzdvgz=0,which implies that
(29)∫MDwpz+Dypzdvgz−∫Mαwqz+βyqzdvgz−2∫Mwazybzdvgz=0.The Nehari manifoldNα,β is closely linked to the behavior of the function of the form ζw,y:s↦Eα,βsw,sy for s>0 defined by
(30)ζw,ys=E′α,βsw,sy=∫Mspzpzawzpz+ayzpzdvgz−∫Msqzqzαwzqz+βyzqzdvgz−∫M2saz+bzaz+bzwzazyzbzdvgz.Lemma 14.
Letw,y∈J\0,0, then sw,sy∈Nα,β if and only if ζ′w,ys=0.Proof.
The result is a consequence of the fact that(31)ζ′w,ys=E′α,βsw,sy,w,y=0.From this lemma, we have that the elements inNα,β correspond to stationary points of the maps ζw,y.Hence, we note that(32)ζw,y′s=∫Mspz−1Dwzpz+Dyzpzdvgz−∫Msqz−1αwzqz+βyzqzdvgz−2∫Msaz+bz−1wzazyzbzdvgz,(33)ζ′′w,ys=∫Mpz−1spz−2Dwzpz+Dyzpzdvgz−∫Mqz−1sqz−2αwzqz+βyzqzdvgz−2∫Maz+bz−1saz+bz−2wzazyzbzdvgz.By Lemma14, w,y∈Nα,β if and only if ζ′w,y1=0. Hence, according to (32), we have for w,y∈Nα,β that
(34)ζ′′w,y1=∫Mpz−1Dwzpz+Dyzpzdvgz−∫Mqz−1αwzqz+βyzqzdvgz−2∫Maz+bz−1wzazyzbzdvgz=2∫Mpz−az+bzwzazyzbzdvgz+∫Mpz−qzαwzqz+βyzqzdvgz=∫Mpz−qzDwzpz+Dyzpzdvgz−2∫Maz+bz−qzwzazyzbzdvgz=∫Mpz−az+bzDwzpz+Dyzpzdvgz+∫Maz+bz−qzαwzqz+βyzqzdvgz.Thus, it is natural to splitNα,β into three corresponding to local minima, local maxima, and points of inflexion of ζw,y, i.e.,
(35)Nα,β+=w,y∈Nα,β:ζ′′w,y1>0,Nα,β−=w,y∈Nα,β:ζ′′w,y1<0,Nα,β0=w,y∈Nα,β:ζ′′w,y1=0.Lemma 15.
Letw,y∈J, then we have
(i)(36)∫Mαwzqz+βyzqzdvgz≤c1α+β×maxmaxwW01,qzMq−,yW01,qzMq−,maxwW01,qzMq+,yW01,qzMq+.
(ii)(37)∫Mwzazyxbzdvgz≤c2maxmaxwW01,qzMa−+b−,yW01,qzMa−+b−,maxwW01,qzMa++b+,yW01,qzMa++b+,for some constants c1,c2>0.Proof.
i Using Theorem 11 (i), and Lemma 2, we get
(38)∫Mαwzqz+βyzqzdvgz≤2αwLaz+bz/qzMq·+2βyLaz+bz/qzMq·≤2αwLaz+bzMq−+wLaz+bzMq++2βyLaz+bzMq−+yLaz+bzMq+≤2maxcn,p−,q−,a−+a−q−,cn,p−,q−,a−+b−q+×αwW01,qzMq−+wW01,qzMq++βyW01,qzMq−+yW01,qzMq+.
Hence,(39)∫Mαwzqz+βyzqzdvgz≤c1α+β×maxmaxwW01,qzMq−,yW01,qzMq−,maxwW01,qzMq+,yW01,qzMq+.(ii) By Young’s inequality, Lemma 2, and Theorem 11 (ii), we have that
(40)∫Mwzazyzbzdvgz≤∫Mazaz+bzwzaz+bz+bzaz+bzyzaz+bzdvgz≤wLaz+bzMa++b++wLaz+bzMa−+b−+yLaz+bzMa++b++yLaz+bzMa−+b−≤2maxcn,a−+b−,q−q−,cn,a−+b−,q−q+×wW01,qzMa++b++wW01,qzMa−+b−+yW01,qzMa++b++yW01,qzMa−+b−.
Hence,(41)∫Mwzazyzbzdvgz≤c2maxmaxwW01,qzMa−+b−,yW01,qzMa−+b−maxwW01,qzMa++b+,yW01,qzMa++b+.Lemma 16.
For eachα,β∈ℝ2\0,0, there exists a constant K1>0 such that for any 0<α+β<K1, we have Nα,β0=∅.Proof.
Suppose otherwise, thatNα,β0≠∅ for all α,β∈ℝ2\0,0. Let w,y∈Nα,β0, such that w,y>1. Then, by Lemma 15, (34), and the definition of Nα,β0, we have
(42)0=ζ′′w,y1≤p+−a−+b−w,yp−+c1α,βa++b+−q−w,yq−,that is,
(43)a−+b−−p+w,yp−≤a++b+−q−c1α,βw,yq−.
Then,(44)w,y≤a++b+−q−a−+b−−p+c1α+β1/p−−q−.
Analogously,(45)0=ζ′′w,y1≥p−−q+w,yp−−2a++b+−q−c2w,ya++b+,2a++b+−q−c2w,ya++b+≥p−−q+w,yp−.
Then,(46)w,ya++b+−p−≥p−−q+2a++b+−q−c2,thus,
(47)w,y≥p−−q+2c2a++b+−q−1/a++b+−p−.
According to (44) and (47), we deduce that
(48)α+β>1c1a−+b−−p+a++b+−q−p−−q+2c2a++b+−q−p−−q−/a++b+−p−=K1,which is a contradiction. Hence, we can conclude that for any 0<α+β<K1, we have Nα,β=∅∀α,β∈ℝ2\0,0.Lemma 17.
Ifw,y is a minimizing of Eα,β on Nα,β such that w,y∉Nα,β0. Then, w,y is a critical point of Eα,β.Proof.
Letw,y be a local minimizing of Eα,β in any subset of Nα,β. Then, in any case, w,y is a minimizer of Eα,β, under the constraint
(49)Lα,β=E′α,βw,y,w,y=0.
Sincew,y∉Nα,β0, the constraint is nondegenerate in w,y, then by the theory of Lagrange multipliers, there exists σ∈ℝ such that
(50)Eα,βw,y=σLα,βw,y.
Thus,(51)E′α,βw,y,w,y=σL′α,βw,y,w,y=σζ′′w,y1.
Sinceζ′′w,y1≠0 and w,y∉Nα,β0, we obtain that σ=0, which completes the proof.Lemma 18.
For everyα,β∈ℝ2\0,0 such that 0<α+β<K1. The functional Eα,β is bounded and coercive on Nα,β.Proof.
For anyw,y∈Nα,β, according to (23), (24), (29), and Lemma 15, we get
(52)Eα,βw,y=∫M1pzDwzpz+Dyzpzdvgz−∫M1qzαwzqz+βyzqzdvgz−∫M1az+bzDwzpz+Dyzpzdvgz+∫M1az+bzαwzqz+βyzqzdvgz≥1p+−1a−+b−∫MDwzpz+Dyzpzdvgz+1a++b+−1q−∫Mαwzqz+βyzqzdvgz≥1p+−1a−+b−w,yp−+1a++b+−1q−c1α+βw,yq+.
Asp−>q+, then Eα,βw,y⟶∞ as w,y⟶∞. It follows that Eα,β is coercive and bounded below on Nα,β for 0<α+β<K1.Lemma 19.
(i)
Ifw,y∈Nα,β+, then ∫Mαwzqz+βyzqzdvgz>0.(ii)
Ifw,y∈Nα,β−, then ∫Mwzazyzbzdvgz>0.Proof.
i Since w,y∈Nα,β+, we have ζ′′w,y1>0. Then, using (23) and (34), we get
(53)0<ζ′′w,y1<p+−a−+b−∫MDwzpz+Dyzpzdvgz+a++b+−q−∫Mαwzqz+βyzqzdvgz,then,
(54)a−+b−−p+w,yp+<a++b+−q−∫Mαwqz+βyqzdvgz.
Hence,(55)∫Mαwzqz+βyzqzdvgz>a−+b−−p+a++b+−q−w,yp+>0.(ii) Since w,y∈Nα,β−, we have ζ′′w,y1<0. Thus, according to (23) and (34), we obtain that
(56)0>ζ′′w,y1≥p−−q+∫MDwzpz+Dyzpzdvgz−2a++b+−q−∫Mwzazyzbzdvgz,then,
(57)2a++b+−q−∫Mwazybzdvgz>p−−q+∫MDwpz+Dypzdvgz.
Hence,(58)∫Mwzazyzbzdvgz>p−−q+2a++b+−q−w,yp−>0.Remark 20.
As a consequence of Lemmas16–18, we have for every α,β∈ℝ+2 with α+β<K1,Nα,β=Nα,β−∪Nα,β+, and Eα,β is coercive and bounded below on Nα,β− and Nα,β+. We define
(59)σα,β=infw,y∈Nα,βEα,βw,y,σα,β+=infw,y∈Nα,β+Eα,βw,y,andσα,β−=infw,y∈Nα,β−Eα,βw,y.Lemma 21.
The following facts hold:(i)
Ifα+β<K1, then σα,β≤σα,β+<0(ii)
Ifα+β<K2, then we have σα,β−>c0 for some c0a,b,p,q,α,β>0Proof.
i Let w,y∈Nα,β+; by (34), we have
(60)0<p+−q−∫MDwpz+Dypzdvgz−2a−+b−−q+∫Mwazybzdvgz,then,
(61)∫Mwazybzdvgz<p+−q−2a−+b−−q+∫MDwpz+Dypzdvgz.
Hence, by (61) and (29), we have
(62)Eα,βw,y≤1p−∫MDwzpz+Dyzpzdvgz−1q+∫Mαwzqz+βyzqzdvgz−2a++b+∫Mwzazyzbzdvgz=1p−−1q+∫MDwzpz+Dyzpzdvgz+21q+−1a++b+∫Mwzazyzazdvgz≤1p−−1q+∫MDwzpz+Dyzpzdvgz+21q+−1a++b+.p+−q−2a−+b−−q+×∫MDwzpz+Dyzpzdvgz=1p−−1q++1q+−1a++b+.p+−q−a−+b−−q+×∫MDwzpz+Dyzpzdvgz=q+−p−a++b++p−a++b+−q+×p+−q−a−+b−−q+×p−q+a++b+−1w,yp+.
According to (24) and (23), we get Eα,βw,y<0. Therefore, from the definition of σα,β and σα,β+, it follows thatσα,β≤σα,β+<0.(73)(ii) Let w,y∈Nα,β−; by (47), we have
(63)p−−q+2a++b+−q−w,yp−<∫Mwzazyzbzdvgz,and by Lemma 15 (ii), we get
(64)p−−q+2a++b+−q−w,yp−<c2w,ya++b+,then,
(65)w,ya++b+−p−>p−−q+2c2a++b+−q−.
Hence,(66)w,y>p−−q+2c2a++b+−q−1/a++b+−p−.
According to (29), (66), and Lemma 15 (i), we deduce that
(67)Eα,βw,y≥1p+∫MDwzpz+Dyzpzdvgz−1q−∫Mαwzqz+βyzqzdvgz−2a−+b−∫Mwzazyzbzdvgz=1p+−1a−+b−∫MDwzpz+Dyzpzdvgz+1a−+b−−1q−∫Mαwzqz+βyzqzdvgz≥1p+−1a−+b−w,yp++c1α+β1a−+b+−1q−w,yq−≥1p+−1a−+b−p−−q+2c2a++b+−q−p+a++b+−p−+1a−+b+−1q−c1α+βp−−q+2c2a++b+−q−q−a++b+−p−=c0.
Thus, if we choose(68)α+β<1c1.q−a−+b−−p+p+a−+b−−q−.p−−q+2c2a++b+−q−p+−q−/a++b+−p−=k2,we deduce that Eα,βw,y>c0 for some positive constant c0 depending on a,b,p,q,α, and β.Lemma 22.
For eachw,y∈J\0,0, there exists a constant K3>0 such that for all α+β<K3, we have the following:
(i)
If∫Mαwzqz+βyzqzdvgz=0, then there exists a unique s−w,s−y>0 such that s−w,s−y∈Nα,β− and Eα,βs−w,s−y=sups>0Eα,βsw,sy.(ii)
If∫Mαwzqz+βyzqzdvgz>0, then there exist smax>0 and unique numbers 0<s+<smax<s−, such as s+w,s+y∈Nα,β+,s−w,s−y∈Nα,β− and(69)Eα,βs+w,s+y=info<s<smaxEα,βsw,sy;Eα,βs−w,s−y=sups≥0Eα,βsw,sy.Proof.
Before tackling our proof, we definesmax as follows:
(70)smax=∫Mpz−qzDwzpz+Dyzpzdvgz2∫Maz+bz−qzwzazyzbzdvgz,for every w,y∈Nα,β−.
Hence, we have thatζw,ys is increasing for s∈0,smax and decreasing for s∈smax,+∞ and achieves its maximum. We set Rs=∫Mspz−qzDwzpz+Dyzpzdvgz−2∫Msaz+bz−qzwzazyzbzdvgz; by Lemma 19, we have that
(71)R0=0andRs⟶−∞whens⟶∞,with
(72)R′s=∫Mpz−qzspz−qz−1Dwzpz+Dyzpzdvgz−2∫Maz+bz−qzsaz+bz−qz−1wzazyzbzdvgz.(i)
For0<s<1 which is sufficiently small, we have(73)ζw,ys>sp+p+∫MDwzpz+Dyzpzdvgz−2ta++b+a++b+∫Mwzazyzbzdvgz>0,and for s>1 which is sufficiently large, we get
(74)ζw,ys<tp+p+∫MDwzpz+Dyzpzdvgz−2sa++b+a++b+∫Mwzazyzbzdvgz<0.
Sinceζw,ys achieves its maximum, then by Lemma 14, ζ′w,ys=E′α,βs−w,s−y,w,y=0. On the other hand, if ∫Mαwzqz+βyzqzdvgz≤0, then there is a unique s−>smax such that Rs−=∫Mαwzqz+βyzqzdvgz and since
(75)ζ′′s−w,s−y,s−w,s−y=∫Mpz−qzs−pzDwzpz+Dyzpzdvgz−2∫Maz+bz−qzs−az+bzwzazyzbzdvgz≤s−1+q−R′smax,E′α,βs−w,s−y,s−w,s−y≤s−p−−1w,yp−−s−q+−1∫Mαwzqz+βyzqzdvgz−2s−a−+b−−q+∫Mwzazyzbzdvgz≤s−q+−1Rs−−∫Mαwzqz+βyzqzdvgz=0.
We obtain thats−w,s−y∈Nα,β−.
Fors>1, we get by (74) and (34) that
(76)E′α,βss−w,ss−y,s−w,s−y≤sp−−1s−p−−1w,yp−−4sa++b+−1s−a++b+∫Mwzazyzbzdvgz<0,and for 0<s<1, we deduce again by (73) and (34) that
(77)E′α,βss−w,ss−y,s−w,s−y≥sp+−1s−p+−1w,yp+−4sa−+b−−1s−a−+b−∫Mwzazyzbzdvgz>0.
Thus,s− is unique, which achieves the proof.
(ii)
If∫Mαwzqz+βyzqzdvgz>0, we have(78)0<∫Mαwzqz+βyzqzdvgz≤c1α,βw,yq−≤Rsmaxforα+β<K3.
Therefore, there are uniques+ and s− such that 0<s+<smax<s−,(79)Rs+=∫Mαwzqz+βyzqzdvgz=Rs−,R′s+>0>R′s−,thus, by (i), we have s+w,s+y∈Nα,β+,s−w,s−y∈Nα,β−,Eα,βs−w,s−y≥Eα,βsw,sy≥Eα,βs+w,s+yforeachs∈s+,s− and Eα,βs+w,s+y≤Eα,βsw,syforeachs∈0,s+. Hence,
(80)Eα,βs+w,s+y=inf0<s<smaxEα,βsw,sy;Eα,βs−w,s−y=sups≥0Eα,βsw,sy.


### 3.2. Existence of Nonnegative Solutions

This section is devoted to proving the existence of minimizers inNα,β+, Nα,β−, also to show the existence of two nonnegative solutions of system S.Lemma 23.
Forα+β<K=minK1,k2, the functional Eα,β has a minimizer w0,y0 in Nα,β+, which satisfies the following assumptions:
(i)
Eα,βw0+,y0+=σα,β+<0(ii)
w0+,y0+ is a solution of SProof.
i Thanks to Lemma 18, Eα,β is bounded below on Nα,β, which in particular is bounded below in Nα,β+. Then, there exists a minimizing sequence wn+,yn+⊂Nα,β+ such that
(81)limn⟶+∞Eα,βwn+,yn+=infw,y∈Nα,β+Eα,βw,y=σα,β+<0.
Since,Eα,β is coercive, wn+,yn+ is bounded on J. Hence, we suppose that, without loss generality, wn+,yn+⟶w0+,y0+ on J, and by the compact embedding (Theorem 11), we have
(82)wn+⟶w0+stronglyinLpzMandLaz+bzMasn⟶+∞.yn+⟶y0+stronglyinLpzMandLaz+bzMasn⟶+∞.wn+z⟶w0+andyn+z⟶y0+za.e.inMasn⟶+∞.
Now, we shall demonstrate thatwn+⟶w0+ and yn+⟶y0+ in W01,qzM as n⟶+∞. Otherwise, let wn+⟶w0+ or yn+⟶y0+ in W01,qzM as n⟶+∞. Then, we have
(83)ρq·w0+<limn⟶+∞infρq·wn+,ρq·y0+<limn⟶+∞infρq·yn+,using (82), we obtain that
(84)∫Mw0+pzdvgz=limn⟶+∞inf∫Mwnpzdvgz,∫My0+pzdvgz=limn⟶+∞inf∫Mynpzdvgz,since, E′α,βwn+,yn+,wn+,yn+=0, we get
(85)Eα,βwn+,yn+≥1p+−1a−+b−wn+,yn+p++1a−+b−−1q−wn+,yn+q−.
That is,(86)limn⟶+∞Eα,βwn+,yn+≥1p+−1a−+b−limn⟶+∞wn+,yn+p++1a−+b−−1q−limn⟶+∞wn+,yn+q−.
By (82) and (83), we have
(87)σα,β+>1p+−1a−+b−w0+,y0+p++1a−+b−−1q−w0+,y0+q−.
Since,p+>q− for w,y>1, we deduce that
(88)σα,β+=infw,y∈Nα,β+Eα,βw,y>0,which is a contradiction with Lemma 21. Hence,
(89)wn+⟶w0+inW01,qzM,yn+⟶y0+inW01,qzM,limn⟶+∞Eα,βwn+,yn+=Eα,βw0+,y0+=σα,β+.
Consequently,w0+,y0+ is a minimizer of Eα,β on Nα,β+.
(ii) According to Lemma17, we deduce that w0+,y0+ is a solution of S.Lemma 24.
Letwn− and yn− be any two bounded sequences in W01,qzM. Then,
(90)limn⟶+∞∫Mwn−zazyn−xbzdvgz=∫Mw0−zazy0−zbzdvgz.Proof.
Similar to the proof ([21], Theorem 5.2), we will omit it.Lemma 25.
Ifα+β<K=minK1,K2, then Eα,β has a minimizer w0−,y0− in Nα,β− such that
(i)
Eα,βw0−,y0−=σα,β−>0(ii)
w0−,y0− is a solution of SProof.
i As Eα,β is bounded below on Nα,β and so on Nα,β−. Then, there exists a minimizing sequence wn−,yn−⊂Nα,β− such that
(91)limn⟶+∞Eα,βwn−,yn−=infw,y∈Nα,β−Eα,βw,y=σα,β−>0.
AsEα,β is coercive, wn−,yn− is bounded in J, and thus, there exists w0−,y0−∈J such that up to a subsequence wn−,yn−⊂w0−,y0− and according to Theorem 11, we obtain
(92)wn−⟶w0−stronglyinLpzMandLaz+bzMasn⟶+∞,yn−⟶y0−stronglyinLpzMandLaz+bzMasn⟶+∞,wn−x⟶w0−andyn−x⟶y0−xa.e.inMasn⟶+∞.
According to (92) and Lemma 24, we deduce that
(93)∫Mαwn−∣qz+βyn−qzdvgz⟶∫Mαw0−∣qz+βy0−qzdvgz,∫Mwn−zazyn−zbzdvgz⟶∫Mw0−zazy0−zbzdvgz.
On the other hand, ifw0−,y0−∈Nα,β−, then there exists a constant s>0 such that sw0−,sy0−∈Nα,β−, and according to (92) and (93), we have
(94)limn⟶+∞Eα,βswn−,syn−=limn⟶+∞inf∫M1pzDswn−zpz+Dsyn−zpzdvgz−∫M1qzαswn−zqz+βsyn−zqzdvgz−∫M2az+bzswn−zazsyn−zbzdvgz≥limn⟶+∞inf∫M1pzDswn−zpz+Dsyn−zpzdvgz−limn⟶+∞inf∫M1qzαswn−zqz+βsyn−zqzdvgz−limn⟶+∞inf∫M2az+bzswn−zazsyn−zbzdvgz>∫M1pzDsw0−zpz+Dsy0−zdvgz−∫M1qzαsw0−zqz+βsy0−zqzdvgz−∫M2az+bzsw0−zazsy0−zbzdvgz=Eα,βsw0−,sy0−.
Considering (32) and (94), we get
(95)limn⟶+∞ζ′wn−,yn−s=limn⟶+∞inf∫Mspz−1Dwn−zpz+Dyn−zpzdvgz−∫Msqz−1αwn−zqz+βyn−zqzdvgz−2∫Msaz+bz−1wn−zazyn−zbzdvgz≥sp−−1limn⟶+∞inf∫MDwn−zpz+Dyn−zpzdvgz−sq+−1limn⟶+∞inf∫Mαwn−zqz+βyn−zqzdvgz−2sa++b+−1limn⟶+∞inf∫Mwn−zazyn−zbzdvgz>sp−−1∫MDw0−zpz+Dy0−zpzdvgz−sq+−1∫Mαw0−zqz+βy0−zqzdvgz−2sa++b+−1∫Mw0−zazy0−zbzdvgz=ζ′w0−,y0−s=0.
Forn large enough, ζ′wn−,yn−s>0. Since wn−,yn−∈Nα,β− for all n∈ℕ, we have ζ′wn−,yn−1=0 and ζ′wn−,yn−s<0 for every n∈ℕ. By Lemma 22, we get ζ′wn−,yn−s<0 for s>0, then from (95), we must have s<1. Since w0−,y0−∈Nα,β− and by Lemma 22, we conclude that 1 is the global maximum for ζwn−,yn−s. Therefore, from Lemma 23, it follows that
(96)Eα,βw0−,y0−<limn⟶+∞Eα,βwn−,yn−≤infw,y∈Nα,β−Eα,βw,y.
It contradicts thatw0−,y0−∈Nα,β−. Hence, wn−,yn−⟶w0−,y0− strongly in J as n⟶+∞ and w0,y0∈Nα,β. Using the fact that
(97)Eα,βw0−,y0−=infw,y∈Nα,βEα,βw,y>0,and Lemma 15, we conclude that w0−,y0−∈Nα,β−.
(ii) From Lemma17, w0−,y0− is a solution of S.Proof of Theorem 1.
From Lemma23 and Lemma 25, there are w+,y+∈Nα,β+ and w−,y−∈Nα,β− such that
(98)Eα,βw+,y+=infw,y∈Nα,β+Eα,βw,yandEα,βw−,y−=infw,y∈Nα,β−Eα,βw,y.
Moreover,Eα,βw±,y±=Eα,βw±,y±; hence, we can assume w±≥0,y±≥0. From Lemma 17, w±,y± are two critical points of Eα,βand, thus, are nonnegative nontrivial solutions of system S.


## 3.1. Nehari Manifold Analysis forS

First, we define the weak solution of systemS as follows.Definition 13.
We say thatw,y∈J is a weak solution of the system S, if w,y∈J one has
(22)∫MDwzpz−2gDwz,Dϕzdvgz+∫MDyzpz−2gDyz,Dψzdvgz=∫Mαwzqz−2wzϕz+βyzqz−2yzψzdvgz+∫M2azaz+bzwzaz−2wzyzbzϕzdvgz+∫M2bzaz+bzwzazyzbz−2yzψzdvgz,for all ϕ,ψ∈DM×DM.The functionsp,q,a,b∈CM¯ are assumed to satisfy the following assumption:
(23)1<q−≤q+<p−≤p+<a−+b−≤a++b+<∞,and the following condition holds
(24)p−a++b+<p−−q+a++b+−q+a−+b−−q+p+−q−.To prove our main result, we will use Nehari manifold and fibering maps. The fact thatw,y is a weak solution is equivalent to being a critical point of the following functional Eα,β:J⟶ℝ defined as
(25)Eα,βw,y=∫M1pzDwzpz+Dyzpzdvgz−∫M1qzαwzqz+βyzqzdvgz−∫M2az+bzwzazyzbzdvgz.By a direct calculation, we haveEα,β∈C1J,ℝ and
(26)E′α,βw,y,ϕ,ψ=∫Mαwpz−2wϕdvgz+∫Mαypz−2yψdvgz−∫Mαwqz−2wϕ+βyqz−2yψdvgz−∫M2azaz+bzwaz−2wybzϕdvgz−∫M2bzaz+bzwazybz−2yψdvgz,for any ϕ,ψ∈DM×DM.Consider the Nehari manifold(27)Nα,β=w,y∈J\0,0:<E′α,βw,y,w,y≥0.Then,w,y∈Nα,β if and only if
(28)∫MDwpz+Dypzdvgz−∫Mαwqz+βyqzdvgz−∫M2azaz+bzwazybzdvgz−∫M2bzaz+bzwazybzdvgz=0,which implies that
(29)∫MDwpz+Dypzdvgz−∫Mαwqz+βyqzdvgz−2∫Mwazybzdvgz=0.The Nehari manifoldNα,β is closely linked to the behavior of the function of the form ζw,y:s↦Eα,βsw,sy for s>0 defined by
(30)ζw,ys=E′α,βsw,sy=∫Mspzpzawzpz+ayzpzdvgz−∫Msqzqzαwzqz+βyzqzdvgz−∫M2saz+bzaz+bzwzazyzbzdvgz.Lemma 14.
Letw,y∈J\0,0, then sw,sy∈Nα,β if and only if ζ′w,ys=0.Proof.
The result is a consequence of the fact that(31)ζ′w,ys=E′α,βsw,sy,w,y=0.From this lemma, we have that the elements inNα,β correspond to stationary points of the maps ζw,y.Hence, we note that(32)ζw,y′s=∫Mspz−1Dwzpz+Dyzpzdvgz−∫Msqz−1αwzqz+βyzqzdvgz−2∫Msaz+bz−1wzazyzbzdvgz,(33)ζ′′w,ys=∫Mpz−1spz−2Dwzpz+Dyzpzdvgz−∫Mqz−1sqz−2αwzqz+βyzqzdvgz−2∫Maz+bz−1saz+bz−2wzazyzbzdvgz.By Lemma14, w,y∈Nα,β if and only if ζ′w,y1=0. Hence, according to (32), we have for w,y∈Nα,β that
(34)ζ′′w,y1=∫Mpz−1Dwzpz+Dyzpzdvgz−∫Mqz−1αwzqz+βyzqzdvgz−2∫Maz+bz−1wzazyzbzdvgz=2∫Mpz−az+bzwzazyzbzdvgz+∫Mpz−qzαwzqz+βyzqzdvgz=∫Mpz−qzDwzpz+Dyzpzdvgz−2∫Maz+bz−qzwzazyzbzdvgz=∫Mpz−az+bzDwzpz+Dyzpzdvgz+∫Maz+bz−qzαwzqz+βyzqzdvgz.Thus, it is natural to splitNα,β into three corresponding to local minima, local maxima, and points of inflexion of ζw,y, i.e.,
(35)Nα,β+=w,y∈Nα,β:ζ′′w,y1>0,Nα,β−=w,y∈Nα,β:ζ′′w,y1<0,Nα,β0=w,y∈Nα,β:ζ′′w,y1=0.Lemma 15.
Letw,y∈J, then we have
(i)(36)∫Mαwzqz+βyzqzdvgz≤c1α+β×maxmaxwW01,qzMq−,yW01,qzMq−,maxwW01,qzMq+,yW01,qzMq+.
(ii)(37)∫Mwzazyxbzdvgz≤c2maxmaxwW01,qzMa−+b−,yW01,qzMa−+b−,maxwW01,qzMa++b+,yW01,qzMa++b+,for some constants c1,c2>0.Proof.
i Using Theorem 11 (i), and Lemma 2, we get
(38)∫Mαwzqz+βyzqzdvgz≤2αwLaz+bz/qzMq·+2βyLaz+bz/qzMq·≤2αwLaz+bzMq−+wLaz+bzMq++2βyLaz+bzMq−+yLaz+bzMq+≤2maxcn,p−,q−,a−+a−q−,cn,p−,q−,a−+b−q+×αwW01,qzMq−+wW01,qzMq++βyW01,qzMq−+yW01,qzMq+.
Hence,(39)∫Mαwzqz+βyzqzdvgz≤c1α+β×maxmaxwW01,qzMq−,yW01,qzMq−,maxwW01,qzMq+,yW01,qzMq+.(ii) By Young’s inequality, Lemma 2, and Theorem 11 (ii), we have that
(40)∫Mwzazyzbzdvgz≤∫Mazaz+bzwzaz+bz+bzaz+bzyzaz+bzdvgz≤wLaz+bzMa++b++wLaz+bzMa−+b−+yLaz+bzMa++b++yLaz+bzMa−+b−≤2maxcn,a−+b−,q−q−,cn,a−+b−,q−q+×wW01,qzMa++b++wW01,qzMa−+b−+yW01,qzMa++b++yW01,qzMa−+b−.
Hence,(41)∫Mwzazyzbzdvgz≤c2maxmaxwW01,qzMa−+b−,yW01,qzMa−+b−maxwW01,qzMa++b+,yW01,qzMa++b+.Lemma 16.
For eachα,β∈ℝ2\0,0, there exists a constant K1>0 such that for any 0<α+β<K1, we have Nα,β0=∅.Proof.
Suppose otherwise, thatNα,β0≠∅ for all α,β∈ℝ2\0,0. Let w,y∈Nα,β0, such that w,y>1. Then, by Lemma 15, (34), and the definition of Nα,β0, we have
(42)0=ζ′′w,y1≤p+−a−+b−w,yp−+c1α,βa++b+−q−w,yq−,that is,
(43)a−+b−−p+w,yp−≤a++b+−q−c1α,βw,yq−.
Then,(44)w,y≤a++b+−q−a−+b−−p+c1α+β1/p−−q−.
Analogously,(45)0=ζ′′w,y1≥p−−q+w,yp−−2a++b+−q−c2w,ya++b+,2a++b+−q−c2w,ya++b+≥p−−q+w,yp−.
Then,(46)w,ya++b+−p−≥p−−q+2a++b+−q−c2,thus,
(47)w,y≥p−−q+2c2a++b+−q−1/a++b+−p−.
According to (44) and (47), we deduce that
(48)α+β>1c1a−+b−−p+a++b+−q−p−−q+2c2a++b+−q−p−−q−/a++b+−p−=K1,which is a contradiction. Hence, we can conclude that for any 0<α+β<K1, we have Nα,β=∅∀α,β∈ℝ2\0,0.Lemma 17.
Ifw,y is a minimizing of Eα,β on Nα,β such that w,y∉Nα,β0. Then, w,y is a critical point of Eα,β.Proof.
Letw,y be a local minimizing of Eα,β in any subset of Nα,β. Then, in any case, w,y is a minimizer of Eα,β, under the constraint
(49)Lα,β=E′α,βw,y,w,y=0.
Sincew,y∉Nα,β0, the constraint is nondegenerate in w,y, then by the theory of Lagrange multipliers, there exists σ∈ℝ such that
(50)Eα,βw,y=σLα,βw,y.
Thus,(51)E′α,βw,y,w,y=σL′α,βw,y,w,y=σζ′′w,y1.
Sinceζ′′w,y1≠0 and w,y∉Nα,β0, we obtain that σ=0, which completes the proof.Lemma 18.
For everyα,β∈ℝ2\0,0 such that 0<α+β<K1. The functional Eα,β is bounded and coercive on Nα,β.Proof.
For anyw,y∈Nα,β, according to (23), (24), (29), and Lemma 15, we get
(52)Eα,βw,y=∫M1pzDwzpz+Dyzpzdvgz−∫M1qzαwzqz+βyzqzdvgz−∫M1az+bzDwzpz+Dyzpzdvgz+∫M1az+bzαwzqz+βyzqzdvgz≥1p+−1a−+b−∫MDwzpz+Dyzpzdvgz+1a++b+−1q−∫Mαwzqz+βyzqzdvgz≥1p+−1a−+b−w,yp−+1a++b+−1q−c1α+βw,yq+.
Asp−>q+, then Eα,βw,y⟶∞ as w,y⟶∞. It follows that Eα,β is coercive and bounded below on Nα,β for 0<α+β<K1.Lemma 19.
(i)
Ifw,y∈Nα,β+, then ∫Mαwzqz+βyzqzdvgz>0.(ii)
Ifw,y∈Nα,β−, then ∫Mwzazyzbzdvgz>0.Proof.
i Since w,y∈Nα,β+, we have ζ′′w,y1>0. Then, using (23) and (34), we get
(53)0<ζ′′w,y1<p+−a−+b−∫MDwzpz+Dyzpzdvgz+a++b+−q−∫Mαwzqz+βyzqzdvgz,then,
(54)a−+b−−p+w,yp+<a++b+−q−∫Mαwqz+βyqzdvgz.
Hence,(55)∫Mαwzqz+βyzqzdvgz>a−+b−−p+a++b+−q−w,yp+>0.(ii) Since w,y∈Nα,β−, we have ζ′′w,y1<0. Thus, according to (23) and (34), we obtain that
(56)0>ζ′′w,y1≥p−−q+∫MDwzpz+Dyzpzdvgz−2a++b+−q−∫Mwzazyzbzdvgz,then,
(57)2a++b+−q−∫Mwazybzdvgz>p−−q+∫MDwpz+Dypzdvgz.
Hence,(58)∫Mwzazyzbzdvgz>p−−q+2a++b+−q−w,yp−>0.Remark 20.
As a consequence of Lemmas16–18, we have for every α,β∈ℝ+2 with α+β<K1,Nα,β=Nα,β−∪Nα,β+, and Eα,β is coercive and bounded below on Nα,β− and Nα,β+. We define
(59)σα,β=infw,y∈Nα,βEα,βw,y,σα,β+=infw,y∈Nα,β+Eα,βw,y,andσα,β−=infw,y∈Nα,β−Eα,βw,y.Lemma 21.
The following facts hold:(i)
Ifα+β<K1, then σα,β≤σα,β+<0(ii)
Ifα+β<K2, then we have σα,β−>c0 for some c0a,b,p,q,α,β>0Proof.
i Let w,y∈Nα,β+; by (34), we have
(60)0<p+−q−∫MDwpz+Dypzdvgz−2a−+b−−q+∫Mwazybzdvgz,then,
(61)∫Mwazybzdvgz<p+−q−2a−+b−−q+∫MDwpz+Dypzdvgz.
Hence, by (61) and (29), we have
(62)Eα,βw,y≤1p−∫MDwzpz+Dyzpzdvgz−1q+∫Mαwzqz+βyzqzdvgz−2a++b+∫Mwzazyzbzdvgz=1p−−1q+∫MDwzpz+Dyzpzdvgz+21q+−1a++b+∫Mwzazyzazdvgz≤1p−−1q+∫MDwzpz+Dyzpzdvgz+21q+−1a++b+.p+−q−2a−+b−−q+×∫MDwzpz+Dyzpzdvgz=1p−−1q++1q+−1a++b+.p+−q−a−+b−−q+×∫MDwzpz+Dyzpzdvgz=q+−p−a++b++p−a++b+−q+×p+−q−a−+b−−q+×p−q+a++b+−1w,yp+.
According to (24) and (23), we get Eα,βw,y<0. Therefore, from the definition of σα,β and σα,β+, it follows thatσα,β≤σα,β+<0.(73)(ii) Let w,y∈Nα,β−; by (47), we have
(63)p−−q+2a++b+−q−w,yp−<∫Mwzazyzbzdvgz,and by Lemma 15 (ii), we get
(64)p−−q+2a++b+−q−w,yp−<c2w,ya++b+,then,
(65)w,ya++b+−p−>p−−q+2c2a++b+−q−.
Hence,(66)w,y>p−−q+2c2a++b+−q−1/a++b+−p−.
According to (29), (66), and Lemma 15 (i), we deduce that
(67)Eα,βw,y≥1p+∫MDwzpz+Dyzpzdvgz−1q−∫Mαwzqz+βyzqzdvgz−2a−+b−∫Mwzazyzbzdvgz=1p+−1a−+b−∫MDwzpz+Dyzpzdvgz+1a−+b−−1q−∫Mαwzqz+βyzqzdvgz≥1p+−1a−+b−w,yp++c1α+β1a−+b+−1q−w,yq−≥1p+−1a−+b−p−−q+2c2a++b+−q−p+a++b+−p−+1a−+b+−1q−c1α+βp−−q+2c2a++b+−q−q−a++b+−p−=c0.
Thus, if we choose(68)α+β<1c1.q−a−+b−−p+p+a−+b−−q−.p−−q+2c2a++b+−q−p+−q−/a++b+−p−=k2,we deduce that Eα,βw,y>c0 for some positive constant c0 depending on a,b,p,q,α, and β.Lemma 22.
For eachw,y∈J\0,0, there exists a constant K3>0 such that for all α+β<K3, we have the following:
(i)
If∫Mαwzqz+βyzqzdvgz=0, then there exists a unique s−w,s−y>0 such that s−w,s−y∈Nα,β− and Eα,βs−w,s−y=sups>0Eα,βsw,sy.(ii)
If∫Mαwzqz+βyzqzdvgz>0, then there exist smax>0 and unique numbers 0<s+<smax<s−, such as s+w,s+y∈Nα,β+,s−w,s−y∈Nα,β− and(69)Eα,βs+w,s+y=info<s<smaxEα,βsw,sy;Eα,βs−w,s−y=sups≥0Eα,βsw,sy.Proof.
Before tackling our proof, we definesmax as follows:
(70)smax=∫Mpz−qzDwzpz+Dyzpzdvgz2∫Maz+bz−qzwzazyzbzdvgz,for every w,y∈Nα,β−.
Hence, we have thatζw,ys is increasing for s∈0,smax and decreasing for s∈smax,+∞ and achieves its maximum. We set Rs=∫Mspz−qzDwzpz+Dyzpzdvgz−2∫Msaz+bz−qzwzazyzbzdvgz; by Lemma 19, we have that
(71)R0=0andRs⟶−∞whens⟶∞,with
(72)R′s=∫Mpz−qzspz−qz−1Dwzpz+Dyzpzdvgz−2∫Maz+bz−qzsaz+bz−qz−1wzazyzbzdvgz.(i)
For0<s<1 which is sufficiently small, we have(73)ζw,ys>sp+p+∫MDwzpz+Dyzpzdvgz−2ta++b+a++b+∫Mwzazyzbzdvgz>0,and for s>1 which is sufficiently large, we get
(74)ζw,ys<tp+p+∫MDwzpz+Dyzpzdvgz−2sa++b+a++b+∫Mwzazyzbzdvgz<0.
Sinceζw,ys achieves its maximum, then by Lemma 14, ζ′w,ys=E′α,βs−w,s−y,w,y=0. On the other hand, if ∫Mαwzqz+βyzqzdvgz≤0, then there is a unique s−>smax such that Rs−=∫Mαwzqz+βyzqzdvgz and since
(75)ζ′′s−w,s−y,s−w,s−y=∫Mpz−qzs−pzDwzpz+Dyzpzdvgz−2∫Maz+bz−qzs−az+bzwzazyzbzdvgz≤s−1+q−R′smax,E′α,βs−w,s−y,s−w,s−y≤s−p−−1w,yp−−s−q+−1∫Mαwzqz+βyzqzdvgz−2s−a−+b−−q+∫Mwzazyzbzdvgz≤s−q+−1Rs−−∫Mαwzqz+βyzqzdvgz=0.
We obtain thats−w,s−y∈Nα,β−.
Fors>1, we get by (74) and (34) that
(76)E′α,βss−w,ss−y,s−w,s−y≤sp−−1s−p−−1w,yp−−4sa++b+−1s−a++b+∫Mwzazyzbzdvgz<0,and for 0<s<1, we deduce again by (73) and (34) that
(77)E′α,βss−w,ss−y,s−w,s−y≥sp+−1s−p+−1w,yp+−4sa−+b−−1s−a−+b−∫Mwzazyzbzdvgz>0.
Thus,s− is unique, which achieves the proof.
(ii)
If∫Mαwzqz+βyzqzdvgz>0, we have(78)0<∫Mαwzqz+βyzqzdvgz≤c1α,βw,yq−≤Rsmaxforα+β<K3.
Therefore, there are uniques+ and s− such that 0<s+<smax<s−,(79)Rs+=∫Mαwzqz+βyzqzdvgz=Rs−,R′s+>0>R′s−,thus, by (i), we have s+w,s+y∈Nα,β+,s−w,s−y∈Nα,β−,Eα,βs−w,s−y≥Eα,βsw,sy≥Eα,βs+w,s+yforeachs∈s+,s− and Eα,βs+w,s+y≤Eα,βsw,syforeachs∈0,s+. Hence,
(80)Eα,βs+w,s+y=inf0<s<smaxEα,βsw,sy;Eα,βs−w,s−y=sups≥0Eα,βsw,sy.


## 3.2. Existence of Nonnegative Solutions

This section is devoted to proving the existence of minimizers inNα,β+, Nα,β−, also to show the existence of two nonnegative solutions of system S.Lemma 23.
Forα+β<K=minK1,k2, the functional Eα,β has a minimizer w0,y0 in Nα,β+, which satisfies the following assumptions:
(i)
Eα,βw0+,y0+=σα,β+<0(ii)
w0+,y0+ is a solution of SProof.
i Thanks to Lemma 18, Eα,β is bounded below on Nα,β, which in particular is bounded below in Nα,β+. Then, there exists a minimizing sequence wn+,yn+⊂Nα,β+ such that
(81)limn⟶+∞Eα,βwn+,yn+=infw,y∈Nα,β+Eα,βw,y=σα,β+<0.
Since,Eα,β is coercive, wn+,yn+ is bounded on J. Hence, we suppose that, without loss generality, wn+,yn+⟶w0+,y0+ on J, and by the compact embedding (Theorem 11), we have
(82)wn+⟶w0+stronglyinLpzMandLaz+bzMasn⟶+∞.yn+⟶y0+stronglyinLpzMandLaz+bzMasn⟶+∞.wn+z⟶w0+andyn+z⟶y0+za.e.inMasn⟶+∞.
Now, we shall demonstrate thatwn+⟶w0+ and yn+⟶y0+ in W01,qzM as n⟶+∞. Otherwise, let wn+⟶w0+ or yn+⟶y0+ in W01,qzM as n⟶+∞. Then, we have
(83)ρq·w0+<limn⟶+∞infρq·wn+,ρq·y0+<limn⟶+∞infρq·yn+,using (82), we obtain that
(84)∫Mw0+pzdvgz=limn⟶+∞inf∫Mwnpzdvgz,∫My0+pzdvgz=limn⟶+∞inf∫Mynpzdvgz,since, E′α,βwn+,yn+,wn+,yn+=0, we get
(85)Eα,βwn+,yn+≥1p+−1a−+b−wn+,yn+p++1a−+b−−1q−wn+,yn+q−.
That is,(86)limn⟶+∞Eα,βwn+,yn+≥1p+−1a−+b−limn⟶+∞wn+,yn+p++1a−+b−−1q−limn⟶+∞wn+,yn+q−.
By (82) and (83), we have
(87)σα,β+>1p+−1a−+b−w0+,y0+p++1a−+b−−1q−w0+,y0+q−.
Since,p+>q− for w,y>1, we deduce that
(88)σα,β+=infw,y∈Nα,β+Eα,βw,y>0,which is a contradiction with Lemma 21. Hence,
(89)wn+⟶w0+inW01,qzM,yn+⟶y0+inW01,qzM,limn⟶+∞Eα,βwn+,yn+=Eα,βw0+,y0+=σα,β+.
Consequently,w0+,y0+ is a minimizer of Eα,β on Nα,β+.
(ii) According to Lemma17, we deduce that w0+,y0+ is a solution of S.Lemma 24.
Letwn− and yn− be any two bounded sequences in W01,qzM. Then,
(90)limn⟶+∞∫Mwn−zazyn−xbzdvgz=∫Mw0−zazy0−zbzdvgz.Proof.
Similar to the proof ([21], Theorem 5.2), we will omit it.Lemma 25.
Ifα+β<K=minK1,K2, then Eα,β has a minimizer w0−,y0− in Nα,β− such that
(i)
Eα,βw0−,y0−=σα,β−>0(ii)
w0−,y0− is a solution of SProof.
i As Eα,β is bounded below on Nα,β and so on Nα,β−. Then, there exists a minimizing sequence wn−,yn−⊂Nα,β− such that
(91)limn⟶+∞Eα,βwn−,yn−=infw,y∈Nα,β−Eα,βw,y=σα,β−>0.
AsEα,β is coercive, wn−,yn− is bounded in J, and thus, there exists w0−,y0−∈J such that up to a subsequence wn−,yn−⊂w0−,y0− and according to Theorem 11, we obtain
(92)wn−⟶w0−stronglyinLpzMandLaz+bzMasn⟶+∞,yn−⟶y0−stronglyinLpzMandLaz+bzMasn⟶+∞,wn−x⟶w0−andyn−x⟶y0−xa.e.inMasn⟶+∞.
According to (92) and Lemma 24, we deduce that
(93)∫Mαwn−∣qz+βyn−qzdvgz⟶∫Mαw0−∣qz+βy0−qzdvgz,∫Mwn−zazyn−zbzdvgz⟶∫Mw0−zazy0−zbzdvgz.
On the other hand, ifw0−,y0−∈Nα,β−, then there exists a constant s>0 such that sw0−,sy0−∈Nα,β−, and according to (92) and (93), we have
(94)limn⟶+∞Eα,βswn−,syn−=limn⟶+∞inf∫M1pzDswn−zpz+Dsyn−zpzdvgz−∫M1qzαswn−zqz+βsyn−zqzdvgz−∫M2az+bzswn−zazsyn−zbzdvgz≥limn⟶+∞inf∫M1pzDswn−zpz+Dsyn−zpzdvgz−limn⟶+∞inf∫M1qzαswn−zqz+βsyn−zqzdvgz−limn⟶+∞inf∫M2az+bzswn−zazsyn−zbzdvgz>∫M1pzDsw0−zpz+Dsy0−zdvgz−∫M1qzαsw0−zqz+βsy0−zqzdvgz−∫M2az+bzsw0−zazsy0−zbzdvgz=Eα,βsw0−,sy0−.
Considering (32) and (94), we get
(95)limn⟶+∞ζ′wn−,yn−s=limn⟶+∞inf∫Mspz−1Dwn−zpz+Dyn−zpzdvgz−∫Msqz−1αwn−zqz+βyn−zqzdvgz−2∫Msaz+bz−1wn−zazyn−zbzdvgz≥sp−−1limn⟶+∞inf∫MDwn−zpz+Dyn−zpzdvgz−sq+−1limn⟶+∞inf∫Mαwn−zqz+βyn−zqzdvgz−2sa++b+−1limn⟶+∞inf∫Mwn−zazyn−zbzdvgz>sp−−1∫MDw0−zpz+Dy0−zpzdvgz−sq+−1∫Mαw0−zqz+βy0−zqzdvgz−2sa++b+−1∫Mw0−zazy0−zbzdvgz=ζ′w0−,y0−s=0.
Forn large enough, ζ′wn−,yn−s>0. Since wn−,yn−∈Nα,β− for all n∈ℕ, we have ζ′wn−,yn−1=0 and ζ′wn−,yn−s<0 for every n∈ℕ. By Lemma 22, we get ζ′wn−,yn−s<0 for s>0, then from (95), we must have s<1. Since w0−,y0−∈Nα,β− and by Lemma 22, we conclude that 1 is the global maximum for ζwn−,yn−s. Therefore, from Lemma 23, it follows that
(96)Eα,βw0−,y0−<limn⟶+∞Eα,βwn−,yn−≤infw,y∈Nα,β−Eα,βw,y.
It contradicts thatw0−,y0−∈Nα,β−. Hence, wn−,yn−⟶w0−,y0− strongly in J as n⟶+∞ and w0,y0∈Nα,β. Using the fact that
(97)Eα,βw0−,y0−=infw,y∈Nα,βEα,βw,y>0,and Lemma 15, we conclude that w0−,y0−∈Nα,β−.
(ii) From Lemma17, w0−,y0− is a solution of S.Proof of Theorem 1.
From Lemma23 and Lemma 25, there are w+,y+∈Nα,β+ and w−,y−∈Nα,β− such that
(98)Eα,βw+,y+=infw,y∈Nα,β+Eα,βw,yandEα,βw−,y−=infw,y∈Nα,β−Eα,βw,y.
Moreover,Eα,βw±,y±=Eα,βw±,y±; hence, we can assume w±≥0,y±≥0. From Lemma 17, w±,y± are two critical points of Eα,βand, thus, are nonnegative nontrivial solutions of system S.



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