A sub-supersolution method for nonlinear elliptic singular systems with natural growth and some applications

A SUB-SUPERSOLUTION METHOD FOR NONLINEAR ELLIPTIC SINGULAR SYSTEMS WITH NATURAL GROWTH AND SOME APPLICATIONS JOS´ E CARMONA, PEDRO J. MART´ INEZ-APARICIO, AND ANTONIO SU´ AREZ Abstract. In this paper we give a sub-supersolution method for nonlinear elliptic singular systems with quadratic gradient whose model system is the following            −∆u+vβ|∇u|2 uα=f1(x, u, v) in Ω, −∆v+uµ|∇v|2 vγ=f2(x, u, v) in Ω, u=v= 0 on ∂Ω, where Ω is a smooth bounded domain of RN(N≥3), β, µ ≥0, 0 < α, γ < 1 and regular f1, f2functions. Moreover, we apply it to prove existence of solution for some systems, including the classical Lotka-Volterra models with gradient terms. Specifically, we study the competition and the symbiotic LoktaVolterra systems. 1. Introduction The aim of this paper is to provide a sub-supersolution method for the following nonlinear elliptic singular system with natural growth (1.1)              −∆u+g1(v)|∇u|2 uα=f1(x, u, v) in Ω, −∆v+g2(u)|∇v|2 vγ=f2(x, u, v) in Ω, u=v= 0 on ∂Ω, where Ω is a smooth bounded domain of RN(N≥3) the functions g1, g2∈ C([0,+∞)) and f1, f2∈C(Ω ×[0,+∞)×[0,+∞)) verifying some general conditions detailed below. Regarding the literature there are several papers about equations with quadratic gradient terms. The existence of solutions of the equation (1.2) −∆u+g(u)|∇u|2=a(x) in Ω, u= 0 on ∂Ω, for every function a(x) in a given Lebesgue space has been systematically studied in [4, 5, 6] and references therein (in fact, for a more general nonlinear term H(x, u, ∇u) instead of g(u)|∇u|2). They consider in the lower order term a continuous gin Rwhich does not satisfy any growth restriction and the sign condition g(s)s≥0 for every s∈Ris assumed. Thanks to the presence of the lower order term the Dirichlet problem associated to the equation is allowed to have finite energy weak solutions. 1 2 J. CARMONA, P. J. MART´ INEZ-APARICIO, AND A. SU´ AREZ In [15] and [7] some of the above results were extended to the case of systems. Specifically, in [7] the authors study systems of elliptic equations with quadratic gradient. They consider a general system (1.3) −∆ui+Hi(x, u, ∇u) = ai(x) in Ω, u= 0 on ∂Ω, i= 1, . . . , n where u= (u1, . . . , un), ai∈H−1(Ω) and the quadratic terms Hi(x, u, ∇u) satisfy a more general one-side condition than the sign condition, but in the case Hi(x, u, ∇u) = gi(u)|∇u|2this one-side hypothesis is equivalent to the sign condition. In their case giis continuous in Rnand they prove the existence of solution in the Sobolev space. In the last years, equation (1.2) has attracted much attention by the presence of singular terms in front of the gradient, see [1, 2, 8] and references therein. In [11] we prove that a sub-supersolution method works for equations of the form −∆u+|∇u|2 uα=f(λ, u) and we apply it to different models. In this paper we focus our attention in systems with quadratic gradient and singular terms as (1.1). Let us mention that the sub-supersolution method is valid for semilinear systems, see for instance [13] and [20]. In this case, when g1≡g2≡0, the natural extension of the scalar definition of sub-supersolution depends on the monotonicity of the functions f1and f2with respect to vand u, respectively. A general definition was given in [13] and [20] where a pair of functions (u, v), (u, v), u, u, v, v ∈H1(Ω) ∩ L∞(Ω) is called a sub-supersolution if u≤u, v ≤vin Ω, u≤0≤u, v ≤0≤von ∂Ω, and −∆u≤f1(x, u, v),−∆u≥f1(x, u, v),∀v∈[v, v], −∆v≤f2(x, u, v),−∆v≥f2(x, u, v),∀u∈[u, u], where, given two ordered functions z≤w, we have denoted [z, w] := {q∈L∞(Ω) : z(x)≤q(x)≤w(x)} (see also [19] where it is proved the validity of the method for singular semilinear systems). Assuming the existence of a sub-supersolution, (u, v), (u, v), there exists a solution (u, v)∈I≡[u, u]×[v,v] of the semilinear system (i.e. (1.1) with g1≡g2≡0). When the reaction terms depend on the gradient, i.e., −∆u=f1(x, u, v, ∇u, ∇v),−∆v=f2(x, u, v, ∇u, ∇v) and the functions f1and f2are regular verifying some hypotheses, the definition is (see [21]) −∆u≤f1(x, u, v, ∇u, ∇v),−∆u≥f1(x, u, v, ∇u, ∇v),∀v∈[v, v], −∆v≤f2(x, u, v, ∇u, ∇v),−∆v≥f2(x, u, v, ∇u, ∇v),∀u∈[u, u]. Assuming again the existence of a sub-supersolution, the existence of a solution (u, v)∈Ifollows. NONLINEAR ELLIPTIC SINGULAR SYSTEMS WITH NATURAL GROWTH 3 In this paper, we use the above definition of sub-supersolution, and taking advantage of the form of equation, we overcome the singularities difficulty of the system (1.1) with respect to uand v. Hence, for our system (1.1) we define a couple of sub-super solution as follows −∆u+g1(v)|∇u|2 uα≤f1(x, u, v),−∆u+g1(v)|∇u|2 uα≥f1(x, u, v),∀v∈[v, v], −∆v+g2(u)|∇v|2 vγ≤f2(x, u, v),−∆v+g2(u)|∇v|2 vγ≥f2(x, u, v),∀u∈[u, u]. Moreover, we apply this method to prove existence of positive solution for some systems, including the classical Lotka-Volterra models confronted with the Laplacian operator perturbed by a singular gradient term, that is, the following systems (1.4)          −∆u+g1(v)|∇u|2 uα=u(λ−u−bv) in Ω, −∆v+g2(u)|∇v|2 vγ=v(µ−v−cu) in Ω, u=v= 0 on ∂Ω, where λ, µ ∈IR and b·c > 0. Here, u(x) and v(x) denote the densities of two species, λand µrepresent the growth rates of the species, band cmeasure the interaction rates between both species; if b, c > 0 they are competing and if b, c < 0 cooperating. Moreover, in (1.4) a nonlinear convective term is included with a singular term. This term is accompanied by a nonlinear function depending on the other species. We give conditions on λand µthat assure the existence of a coexistence state of (1.4), that is, a solution with both components positive. The structure of the article is: in Section 2 we study an auxiliary scalar equation that we use in Section 3 to prove the validity of the sub-supersolution method for (1.1). Section 4 is devoted to applications of the method. 2. An auxiliary scalar equation Frizzing one of the unknown in each equation of (1.1) we are led to consider the scalar boundary value problem (2.1)    −∆w+m(x)|∇w|2 wθ=f(x, w) in Ω, w= 0 on ∂Ω, for convenient functions mand fand a parameter 0 < θ < 1. In order to use monotone methods for (2.1), as was pointed out in [11] for m(x) constant, it is useful to consider a positive function g∈C(0,+∞) such that, denoting G(u) = Ru 1g(s)ds, the function e−G(s)belongs to L1(0,1) and we define also Ψ by Ψ(s) := Zs 0 e−G(t)dt, s > 0. Observe that if there exists M≥0 such that f(x, s) + Ms is nondecreasing for a.e. x∈Ω then f(x, s)+MΨ(s)eG(s)is also nondecreasing for a.e. x∈Ω. Thus, adding the term MΨ(w)eG(w)in (2.1) it becomes    −∆w+m(x)|∇w|2 wθ+MΨ(w)eG(w)=f(x, w) + MΨ(w)eG(w)in Ω, w= 0 on ∂Ω. 4 J. CARMONA, P. J. MART´ INEZ-APARICIO, AND A. SU´ AREZ Therefore, in order to study (2.1) using sub-supersolution, we need to establish a comparison principle for the problem (2.2)    −∆w+m(x)|∇w|2 wθ+MΨ(w)eG(w)=f0(x) in Ω, w= 0 on ∂Ω, where 0 < m(x)∈L∞(Ω), 0 < θ < 1 and 0 ≤f0(x)∈L2N/(N+2)(Ω), f06≡ 0. Observe that this problem is similar to that studied in [11] but here the function g, from which are defined Gand Ψ, is arbitrary and non necessary related with the gradiend lower order term. When M= 0, that is,    −∆w+m(x)|∇w|2 wθ=f0(x) in Ω, u= 0 on ∂Ω, this problem has solution (see [8]) and it is unique (see [3]). The concept of sub and super-solution for the problem (2.2) is the following: Definition 2.1. A sub-solution of (2.2) is a function u∈H1 0(Ω) such that 0 < u a.e. in Ω, |∇u|2 uγ, Ψ(u)eG(u)∈L1(Ω) and for every φ∈H1 0(Ω) ∩L∞(Ω), φ ≥0, ZΩ ∇u· ∇φ+ZΩ m(x)|∇u|2 uθφ+MZΩ Ψ(u)eG(u)φ≤ZΩ f0(x)φ. Similarly u∈H1(Ω) such that 0 < u a.e. in Ω, |∇u|2 uθ,Ψ(u)eG(u)∈L1(Ω) and for every φ∈H1 0(Ω) ∩L∞(Ω), φ ≥0, ZΩ ∇u· ∇φ+ZΩ m(x)|∇u|2 uθφ+MZΩ Ψ(u)eG(u)φ≥ZΩ f0(x)φ, is called a super-solution of (2.2). We say that u∈H1 0(Ω) is a solution of (2.2) if it is a sub and super-solution of (2.2). We recall some classical results about the regularity of the equation (2.2) with f0(x) = f(x, w(x)), that is the non-linear equation (2.3)    −∆w+m(x)|∇w|2 wθ+MΨ(w)eG(w)=f(x, w) in Ω, w= 0 on ∂Ω. Concretely, it can be deduced from [22] the following two lemmas, summarizing some known L∞(Ω)-estimates for sub-solutions of (2.3). The first one deals with a subcritical function f, here the L∞(Ω)-estimate follows from a standard bootstrap argument. Lemma 2.2. Assume that there exists C > 0such that |f(x, s)| ≤ C(1 + |s|q) (q < (N+ 2)/(N−2)) for every s≥0, a.e. x∈Ω, and that uis a sub-solution of (2.3), then u∈L∞(Ω). Remark 2.3. Once we have proved that it is bounded, under conditions of the previous lemma, we have that any solution uis continuous in Ω arguing as in [14] (see Remark 2.6 in [1] for a detailed proof). Moreover, since ∂Ω is smooth, u∈C0,α(Ω) for some α∈(0,1). NONLINEAR ELLIPTIC SINGULAR SYSTEMS WITH NATURAL GROWTH 5 Lemma 2.4. Assume that there exists s0such that f(x, s)≤0a.e. x∈Ωand for every s > s0. Assume also that uis a sub-solution of (2.3), then u∈L∞(Ω) and kuk∞≤s0. We can prove that the problem (2.2) has solution arguing as in Lemma 3.3 of [11]. We include here a sketch of the proof in order to show how to deal with the term m(x)∈L∞(Ω). Lemma 2.5. Assume that g∈L1(0,1). Then there exists a solution for (2.2). Proof. We use an approximative scheme, namely (2.4) −∆un+Bn(x, un,∇un) = min{f0(x), n}in Ω, un= 0 on ∂Ω, where the function Bn(x, s, p) is given, for every (x, s, p)∈Ω×R×RNand n∈N, by Bn(x, s, p) = m(x)s+|p|2 (1 n+s+)θ+1(1 + 1 n|p|2)+ +MeRs+ 1g(t+1 n)dt Rs+ 0e−Rt 1g(σ+1 n)dσdt 1 + 1 neRs+ 1g(t+1 n)dt Rs+ 0e−Rt 1g(σ+1 n)dσdt . Since Bn(x, s, p)s≥0 and Bn(x, s, p)≤ kmkL∞(Ω)n(nθ+M), the existence of solution un∈H1 0(Ω) of (2.4) is deduced from [16]. Moreover, since Bn(x, s, p)≥0 then −∆un≤f0(x) and, using [22], un∈L∞(Ω) and the sequence unis bounded in L∞(Ω), that is, there exists R > 0 such that kunkL∞(Ω) ≤R. Moreover, taking u− nas test function we obtain that un≥0. Similarly, taking un as test function and using the positivity of the lower order term we get that un is bounded in H1 0(Ω). Even more, taking min{un, ε}/ε as test function and using Fatou Lemma as ε→0 yields that ZΩ Bn(x, un,∇un)≤ kf0k1. Therefore unweakly converges to u∈H1 0(Ω), ∇un→ ∇ua.e. (see [9, Theorem 2.1]) and using Fatou Lemma as n→ ∞, m(x)|∇u|2 uθχ{u>0}∈L1(Ω) and Ψ(u)eG(u)χ{u>0}∈L1(Ω). In particular, since gis integrable at zero, we have that Ψ(u)eG(u)is bounded at zero and thus, Ψ(u)eG(u)∈L1(Ω). In order to pass to the limit and to prove that uis the solution of (2.2) it is essential to prove that u > 0. In order to do that we follow the ideas in [8]. Given ˜m≥ kmkL∞(Ω) we take e−˜mRun 1 1 tθdtφ, with 0 ≤φ∈C∞ 0(Ω), as test function in (2.4) and we obtain ZΩ e−˜mRun 1 1 tθdt∇un· ∇φ+ZΩBn(x, un,∇un)−˜m uθ ne−˜mRun 1 1 tθdtφ= =ZΩ min{f0(x), n}e−˜mRun 1 1 tθdtφ≥ZΩ min{f0(x),1}e−˜mRun 1 1 tθdtφ.(2.5) 6 J. CARMONA, P. J. MART´ INEZ-APARICIO, AND A. SU´ AREZ We can use now that for 0 < s < R Bn(x, s, p)−˜m sθe−Rs 1 1 tθdt ≤m(x)−˜m sθ|p|2e−Rs 1 1 tθdt+(2.6) +MeRs 1(g(t+1 n)−˜m tθ)dt Zs 0 e−Rt 1g(σ+1 n)dσdt ≤ ≤MeRs 1(g(t+1 n)−˜m tθ)dt Zs 0 e−Rt 1g(σ+1 n)dσdt ≤ ≤CZs 0 e−˜mRt 1 1 σθdσdt. The last inequality is due to the fact that eRs 1(g(t+1 n)−˜m tθ)dt =eRs+1 n 1+ 1 n g(σ)dσ−Rs 1 ˜m tθdt ≤eRR+1 1g(σ)dσ+R1 0 ˜m tθdt and Zs 0 e−Rt 1g(σ+1 n)dσdt =Zs 0 eRt 1(˜m σθ−g(σ+1 n))dσe−˜mRt 1 1 σθdσdt ≤ ≤eRR+1 1 ˜m σθdσ+R1 0g(σ+1 n)dσ Zs 0 e−˜mRt 1 1 σθdσdt ≤ ≤eRR+1 1 ˜m σθdσ+R2 0g(σ)dσ Zs 0 e−˜mRt 1 1 σθdσdt. Thus, we can take C=MeRR+1 1g(σ)dσ+R1 0 ˜m tθdteRR+1 1 ˜m σθdσ+R2 0g(σ)dσ, using (2.6) in (2.5) and denoting ˜ Ψ(s) = Rs 0e−˜mRt 1 1 σθdσdt we get ZΩ ∇˜ Ψ(un)· ∇φ+˜ CZΩ ˜ Ψ(un)φ≥ZΩ min{f0(x),1}e−˜mRun 1 1 tθdtφ. From now on the proof deals exactly as in [11]. Passing to the limit in the previous inequality it follows that ZΩ ∇˜ Ψ(u)· ∇φ+˜ CZΩ ˜ Ψ(u)φ≥ZΩ min{f0(x),1}e−˜mRu 1 1 tθdtφ. Thus, the strong maximum principle allows us to assure that 0 <˜ Ψ(u)≤ eR1 0 ˜m σθdσu, in particular u > 0, and we can to pass to the limit in the approximated problem to deduce that u∈H1 0(Ω) is a solution of (2.2) arguing as in [8].  Respect to the uniqueness we prove below a comparison principle for this equation that assures that this solution is unique. This comparison principle is one of the keystones of the proof of our method. In the case M= 0 it correspond to the comparison principle in [3, Corollary 3.5]. We include here, for the convenience of the reader, the proof of that result with the new term MΨ(u)eG(u)at the left-hand side of the equation, is that to say, we prove a comparison principle for the problem (2.2) although the proof follows with no significant change that of Theorem 1.1 in [3]. Proposition 2.6. Assume that 0< θ < 1and 0< m(x)∈L∞(Ω). Let u, u ∈ C(Ω) be, respectively, a sub and a super-solution of (2.2). Suppose also that g∈ NONLINEAR ELLIPTIC SINGULAR SYSTEMS WITH NATURAL GROWTH 7 C1(0,+∞),e−G(t)∈L1(0,1),g≥0and there exists τ≥0such that for a.e. x∈Ω and for every 0< s < max{kukL∞(Ω),kukL∞(Ω)}we have (2.7) τ−g0(s)−m(x)θ sθ+1 +−g(s) + m(x) sθg(s)≥−g(s) + m(x) sθ2 . Then u≤u. Proof. We will use the usual function Gε(s) = (s−ε)+for every s∈R. We also define w= Ψ(u)−Ψ(u) and observe that Gε(w) is bounded and has compact support in Ω. In particular, e−G(u), e−G(u), g(u), g(u) are bounded in the support of Gε(w). Thus, for nequal to the integer part of τ+ 1, we can take e−G(u)Gε(w)n as test function in the inequality satisfied by uand e−G(u)Gε(w)nin the inequality satisfied by u. Subtracting we have 0≥ZΩ−g(u) + m(x) uθe−G(u)|∇u|2Gε(w)n− −ZΩ−g(u) + m(x) uθe−G(u)|∇u|2Gε(w)n+ +nZΩ Gε(w)n−1(e−G(u)∇u−e−G(u)∇u)· ∇w, where we have used that ZΩ (Ψ(u)−Ψ(u))Gε(w)n≥0. We denote s= Ψ−1(tΨ(u) + (1 −t)Ψ(u)) and ξ=t∇Ψ(u) + (1 −t)∇Ψ(u), this means that 0≥Z{w>ε} Gε(w)nZ1 0 d dt −g(s) + m(x) sθeG(s)|ξ|2dt+ +nZ{w>ε} Gε(w)n−1|∇w|2. Now we perform the derivative and we get 0≥Z{w>ε} wGε(w)nZ1 0−g0(s)−m(x)θ sθ+1 e2G(s)|ξ|2dt+ +Z{w>ε} wGε(w)nZ1 0−g(s) + m(x) sθg(s)e2G(s)|ξ|2dt+ +Z{w>ε} Gε(w)nZ1 0−g(s) + m(x) sθeG(s)2ξ· ∇wdt+ +nZ{w>ε} Gε(w)n−1|∇w|2. Multiplying by τ nand taking into account that, by Young’s inequality, τ n Gε(w)n−g(s) + m(x) sθeG(s)2ξ· ∇w ≤ ≤τ2 nGε(w)n−1|∇w|2+Gε(w)n+1 n−g(s) + m(x) sθ2 e2G(s)|ξ|2, 8 J. CARMONA, P. J. MART´ INEZ-APARICIO, AND A. SU´ AREZ it follows that 0≥τ1−τ nZ{w>ε} Gε(w)n−1|∇w|2+ +Z{w>ε}Z1 0 wGε(w)nτe2G(s) nh−g0(s)−m(x)θ sθ+1 |ξ|2+ +−g(s) + m(x) sθg(s)|ξ|2−Gε(w) τw −g(s) + m(x) sθ2 |ξ|2idt ≥0. The last inequality due to the fact that Gε(w)/w ≤1, M≥0 and (2.7). We deduce that the integrands are zero, which implies that Gε(w) = 0 for every ε > 0, i.e., w+≡0, concluding the proof.  The following technical result plays an essential role in the further work, and it was proved in [3, Corollary 3.5] (see condition (3.6) of that paper). Lemma 2.7. Fix m∈L∞(Ω),m > 0in Ωand ν > 0. Then, there exist g∈ C1(0,∞)∩L1(0,1)1and τsuch that if uand uare a sub and a supersolution of (2.2) such that max{kukL∞(Ω),kukL∞(Ω)} ≤ ν,gsatisfies condition (2.7), and as consequence u≤u. In fact, gdepends on kmkL∞(Ω) and θ, but neither νnor τ. Specifically, fixed d, C, m1with 0<θ<d<1,C > 0and m1≤min (dC, C d−θ 1−θ1−θ), for any m∈L∞(Ω) with kmk∞≤m1we can choose g(s)≡gθ,d,C (s)given by gθ,d,C(s) =        dC sθ, s < θ C1 1−θ, dθ θs +θ C1 1−θ(1 −θ) , s ≥θ C1 1−θ, for every s > 0. Moreover, τis such that τ > max        dC +m1 C(1 −d),2dm2 1ν2(1−θ)+θ2 (1 −d)θ2, 2m2 11−θ d−θ2(1−θ)+ 2d2C2 d(1 −d)C21−m1 C1−θ d−θ1−θ       . 3. The sub-supersolution method Now, we are ready to state the method of sub and super-solutions in order to get existence of solution of (1.1). In view of the results of the previous section the concept of sub and super-solution for (1.1) is the following. Definition 3.1. A pair (u, u),(v, v) is a sub-supersolution of (1.1) if u, u, v, v ∈ H1(Ω) ∩C(Ω), u, v ∈H1 0(Ω) such that (1) 0 < u ≤u, 0 < v ≤valmost everywhere in Ω, (2) |∇u|2 uα,|∇u|2 uα,|∇v|2 vγ,|∇v|2 vγ∈L1(Ω), 1in particular e−G(t)∈L1(0,1) NONLINEAR ELLIPTIC SINGULAR SYSTEMS WITH NATURAL GROWTH 9 (3) for every φ∈H1 0(Ω) ∩L∞(Ω), φ > 0, (3.1) ZΩ ∇u· ∇φ+ZΩ g1(v)|∇u|2 uαφ−ZΩ f1(x, u, v)φ≤0≤ ≤ZΩ ∇u· ∇φ+ZΩ g1(v)|∇u|2 uαφ−ZΩ f1(x, u, v)φ∀v∈[v, v], (4) for every φ∈H1 0(Ω) ∩L∞(Ω), φ > 0, (3.2) ZΩ ∇v· ∇φ+ZΩ g2(u)|∇v|2 vγφ−ZΩ f2(x, u, v)φ≤0≤ ≤ZΩ ∇v· ∇φ+ZΩ g2(u)|∇v|2 vγφ−ZΩ f2(x, u, v)φ∀u∈[u, u]. Remark 3.2. Observe that any of the four inequalities in (3.1) and (3.2) is verified if it is satisfied in the classical sense, for instance, the first inequality in (3.1) is satisfied if uis twice differentiable and −∆u+g1(v)|∇u|2 uα−f1(x, u, v)≤0 a.e. x∈Ω,∀v∈[v, v]. Theorem 3.3. Assume that (u, u),(v, v)is pair of sub-supersolution of (1.1) and denote I:= [u, u]×[v, v]⊂C(Ω) ×C(Ω). Assume also the following conditions on f1, f2, g1and g2: (F) There exists a constant M≥0such that the maps s7→ f1(x, s, v)+Ms and r7→ f2(x, u, r) + Mr are positive and increasing for (s, r)∈[0,supΩu]× [0,supΩv]and for all (u, v)∈I. (G) g1,g2are non-negative functions and gi(s)=0if and only if s= 0 for i= 1,2. Then, there exists a solution (u, v)of (1.1) such that (u, v)∈I. Proof. Taking into account (G), there exits a positive number 0 < m1such that 0< g1(z(x)), g2(w(x)) ≤m1,∀(w, z)∈I. Then, taking ν= max{kvkL∞(Ω),kukL∞(Ω)}, by Lemma 2.7 there exist h1, h2∈ C1(0,+∞)∩L1(0,1) and τ1, τ2≥0 such that for a.e. x∈Ω, for every 0 < s < ν and for every (w, z)∈Iwe have τ1−h0 1(s)−g1(z(x))α sα+1 +−h1(s) + g1(z(x)) sαh1(s)≥ ≥−h1(s) + g1(z(x)) sα2 and τ2−h0 2(s)−g2(w(x))γ sγ+1 +−h2(s) + g2(w(x)) sγh2(s)≥ ≥−h2(s) + g2(w(x)) sγ2 . We would like to remark again that neither hinor τidepend on (w, z), see Lemma 2.7. 16 J. CARMONA, P. J. MART´ INEZ-APARICIO, AND A. SU´ AREZ satisfied. In order to verify the second inequality in (3.1) and in (3.2) it is enough to take K1−p−q 1≥λkekp+q ∞and K2(−∆E) + g2(u)K2−γ 2 |∇E|2 Eγ≥µKm+n 2emEn,∀u∈[u, u], for which it suffices that g2(u)K2−γ−m−n 2|σ|2≥µkekm ∞kEkn+γ ∞,∀u∈[u, u]. If g2is increasing, then g2(u)≥g2(0) and so we need that 2 −γ > m +nand g2(0) >0. Thus, in this case, K1and K2depend only on λand µ, respectively. However, if g2is decreasing then g2(u)≥g2(K1kek∞)>0 and, using that 2−γ > m +nwe can choose K2depending on K1(which depends on λ) and µ. On the other hand, the first inequality in (3.1) is satisfied if εis small enough and (3.2) is satisfied if aϕa(1−(m+n))−2 1|∇ϕ1|2h(1 −a) + g2(u)aϕa(1−α) 1i+ϕ1−(m+n) 1≤µ, ∀u∈[u, u]. If g2is increasing, then g2(u)≤g2(K1(λ)e)≤K3(λ), and so the above inequality is true for µ > K(λ) for some constant K(λ)>0. If g2is decreasing, then g2(u)≤g2(0) and so the above inequality is true for µ large and independent of λ. This completes the proof.  Remark 4.5. (1) Similar result for the case 2 −α > p +q≥1 and m+n < 1. (2) We could obtain results for any positive function g2imposing more restrictive conditions in λand µ. Using similar arguments of the proofs of the above results, we can show the following result. Theorem 4.6. Assume that 1≤p+q < 2−αand 1≤m+n < 2−γ. (1) Assume that g1and g2are increasing and g1(0), g2(0) >0. Then, there exist K1(µ)and K2(λ)such that if λ>K1(µ)and µ>K2(λ)system (4.6) possesses at least a positive solution. (2) Assume that g1is increasing, g2decreasing and g1(0) >0. Then, there exist K1(µ)and K2such that if λ > K1(µ)and µ > K2, system (4.6) possesses at least a positive solution. (3) Assume that g1and g2are decreasing. Then, there exist K1, K2>0such that if λ > K1and µ > K2, system (4.6) possesses at least a positive solution. 4.2. Example 2: competition Lotka-Volterra system. We consider the system (4.7)          −∆u+g1(v)|∇u|2 uα=u(λ−u−bv) in Ω, −∆v+g2(u)|∇v|2 vγ=v(µ−v−cu) in Ω, u=v= 0 on ∂Ω, where λ, µ ∈R,b, c ≥0, and g1and g2verify (G). When g1≡g2≡0 system (4.7) is the classical competition Lotka-Volterra model, studied extensively in the last years, see for instance [10]. NONLINEAR ELLIPTIC SINGULAR SYSTEMS WITH NATURAL GROWTH 17 First, it is clear that if λ≤λ1or µ≤λ1then (4.7) does not have positive solution. So, assume that λ, µ > λ1. Again, it is clear that f1(x, u, v) = u(λ−u−bv) and f2(x, u, v) = v(µ−v−cu) verify (F) for any pair of sub-supersolution of (4.7). Theorem 4.7. Assume that one of the following conditions holds: (1) g1and g2are increasing and (λ, µ)satisfies (4.8) λ>λ1(bΘ[µ,γ,g2(0)])and µ > λ1(cΘ[λ,α,g1(0)]); (2) g1and g2are decreasing and (λ, µ)satisfies (4.9) λ>λ1(bΘ[µ,γ,g2(λ)])and µ>λ1(cΘ[λ,α,g1(µ)]); (3) g1is increasing, g2is decreasing and (λ, µ)satisfies (4.10) λ>λ1(bΘ[µ,γ,g2(0)])and µ>λ1(cΘ[λ,α,g1(µ)]). Then (4.7) possesses a least a positive solution. Remark 4.8. Observe that conditions (4.8), (4.9) and (4.10) define regions in the plane (λ, µ) which could eventually be empty. For the semilinear case, that is g1≡g2≡0, it can be shown, see for example [18] and [17], that these regions are not empty, imposing some conditions (bor csmall). The study of these regions are out of the scope of this paper, but let us remark some aspects. Observe that the map λ∈[λ1,∞)7→ λ1(cΘ[λ,α,g1(0)]) is increasing. Hence, for example, the region defined by (4.8) in not empty if bor cis small. Proof. (1) Assume that g1and g2are increasing. Then, take (u, v) = (Θ[λ,α,g1(0)],Θ[µ,γ,g2(0)]), (u, v) = (Θ[λ−bΘ[µ,γ,g2(0)],α,R],Θ[µ−cΘ[λ,α,g1(0)],γ,S]), for some positive constants Rand Sto be chosen. Since u, v, u and vare solutions of logistic equations as (4.3), then items (1) and (2) of Definition 3.1 are satisfied. Using the equation of u, it can be shown that usatisfies the second inequality in (3.1) if u(λ−u)−g1(0)|∇u|2 |u|α≥u(λ−u−bv)−g1(v)|∇u|2 |u|α, or equivalently, buv + (g1(v)−g1(0))|∇u|2 |u|α≥0, which is true because g1in increasing and v > 0. For u, we need that |∇u|2 |u|α(g1(v)−R)≤0,∀v∈[v, v]. Take R≥g1(v). Observe that by the increase of the map λ+m7→ Θ[λ+m,α,k], it follows that u≤Θ[λ,α,R]≤Θ[λ,α,g1(0)] =u, this last inequality because R≥g1(0). 18 J. CARMONA, P. J. MART´ INEZ-APARICIO, AND A. SU´ AREZ (2) Assume that g1and g2are decreasing. Then, take (u, v) = (Θ[λ,α,g1(µ)],Θ[µ,γ,g2(λ)]), (u, v) = (Θ[λ−bΘ[µ,γ,g2(λ)],α,g1(0)],Θ[µ−cΘ[λ,α,g1(µ)],γ,g2(0)]). Indeed, observe that, with a similar argument to the used in the first paragraph, u satisfies the second inequality in (3.1) if u(λ−u)−g1(µ)|∇u|2 |u|α≥u(λ−u−bv)−g1(v)|∇u|2 |u|α, for what it is sufficient that g1(v)≥g1(µ). But, from (4.2) we have that v≤θµ≤µ, and since g1is decreasing, it follows that g1(v)≥g1(µ). With respect to u, it can be proved that usatisfies the first inequality in (3.1) because g1(v)≤g1(0). Again, it can shown that u≤u. (3) Assume that g1is increasing and g2is decreasing. Then, take in this case (u, v) = (Θ[λ,α,g1(0)],Θ[µ,γ,g2(λ)]), (u, v) = (Θ[λ−bΘ[µ,γ,g2(λ)],α,R],Θ[µ−cΘ[λ,α,g1(0)],γ,g2(0)]).  4.3. Example 3: symbiotic Lotka-Volterra system. We consider the system (4.11)          −∆u+g1(v)|∇u|2 uα=u(λ−u+bv) in Ω, −∆v+g2(u)|∇v|2 vγ=v(µ−v+cu) in Ω, u=v= 0 on ∂Ω, where λ, µ ∈R,b, c > 0, g1and g2verify (G). Theorem 4.9. Assume that bc < 1and (λ, µ)satisfies (4.12) λ>λ1(−bΘ[µ,γ,g2])and µ > λ1(−cΘ[λ,α,g1]), where gi=gi(0) when giis decreasing and g1=g1µ+cλ 1−bc when g1is increasing, and g2=g2λ+bµ 1−bc when g2is increasing. Then (4.11) possesses a least a positive solution. Proof. First, recall that Θ[µ,γ,g2]≤µ, and then if λand µverify (4.12), we have that λ>λ1(−bΘ[µ,γ,g2])≥λ1(−bµ) = λ1−bµ, and so λ+bµ > 0. Analogously, µ+cλ > 0. Now, take (u, v)=(R, S) NONLINEAR ELLIPTIC SINGULAR SYSTEMS WITH NATURAL GROWTH 19 where Rand Sare large positive constants and (u, v) = (Θ[λ+bΘ[µ,γ,g2],α,g1],Θ[µ+cΘ[λ,α,g1],γ,g2]). Indeed, Rand Smust verify λ−R+bS ≤0 and µ−S+cR ≤0. Since bc < 1, we can take R=λ+bµ 1−bc , S =µ+cλ 1−bc . On the other hand, uis subsolution provided of g1(v)≤g1,∀v∈[v, v]. Then, if g1is decreasing (respectively increasing) we can take g1=g1(0) (respectively g1=g1(v) = g1(µ+cλ 1−bc )). Finally, observe that u= Θ[λ+bΘ[µ,γ,g2],α,g1]≤λ+b(Θ[µ,γ,g2])M≤λ+bµ ≤λ+bµ 1−bc =u.  Remark 4.10. Observe again that condition (4.12) could define an empty region in the plane (λ, µ). As in Remark 4.8 we point out that the maps λ∈[λ1,∞)7→ λ1(−cΘ[λ,α,g1(0)]) and µ∈[λ1,∞)7→ λ1(−bΘ[µ,γ,g2(0)]) are decreasing, and so the region defined by (4.12) is non empty when g1and g2 are decreasing, see also [12] for the semilinear case g1≡g2≡0. Acknowledgements. Research supported by MICINN Ministerio de Ciencia e Innovaci´on, Spain under grants MTM2012-31799 (JC and PJMA) and MTM2012-31304 (AS) and Junta de Andaluc´ıa FQM-116 (PJMA), FQM-194 (JC) and FQM-131 (AS). References [1] D. Arcoya, J. Carmona, T. Leonori, P. J. Mart´ınez-Aparicio, L. Orsina and F. Petitta, Existence and nonexistence of solutions for singular quadratic quasilinear equations, J. Differential Equations 246 (2009), no. 10, 4006-4042. [2] D. Arcoya, J. Carmona and P. J. Mart´ınez-Aparicio, Bifurcation for quasilinear elliptic singular BVP, Comm. Partial Differential Equations 36 (2011), no. 4, 670-692. [3] D. Arcoya, J. Carmona and P. J. Mart´ınez-Aparicio, Comparison principle for elliptic equations in divergence with singular lower order terms having natural growth. Preprint 2014. [4] A. Bensoussan, L. Boccard and F. Murat, On a nonlinear partial differential equation having natural growth terms and unbounded solution. Ann. Inst. H. Poincar´e Anal. Non Lin´eaire 5 (1988), no. 4, 347–364. [5] L. Boccardo, F. Murat and J. P. Puel, Existence de solutions non born´ees pour certaines ´equations quasi-lin´eaires. Portugal. Math. 41 (1982), 507–534. [6] L. Boccardo, F. Murat and J. P. Puel, Existence de solutions faibles pour des ´equations elliptiques quasi-lin´eaires `a croissance quadratique. In Nonlinear partial differential equations and their applications. Coll`ege de France Seminar, Vol. IV (Paris, 1981/1982), 19–73. Res. Notes in Math. 84. Pitman, Boston, Mass.-London, 1983. 20 J. CARMONA, P. J. MART´ INEZ-APARICIO, AND A. SU´ AREZ [7] A. Bensoussan and L. Boccardo, Nonlinear Systems of Elliptic Equations with Natural Growth Conditions and Sign Conditions. Applied Mathematics and Optimization 46 (2002), 143–66. [8] L. Boccardo, Dirichlet problems with singular and quadratic gradient lower order terms, ESAIM: Control, Optimization and the Calculus of Variations, 14 (2008) 411-426. [9] L. Boccardo and F. Murat, Almost everywhere convergence of the gradients of solutions to elliptic and parabolic equations,Nonlinear Anal. 19 (1992) 581–597. [10] R. S. Cantrell and C. Cosner, Spatial ecology via reaction-diffusion equations, Wiley Series in Mathematical and Computational Biology. John Wiley & Sons, Ltd., Chichester, (2003). [11] J. Carmona, P. J. Mart´ınez-Aparicio and A. Su´arez, Existence and non-existence of positive solutions for nonlinear elliptic singular equations with natural growth. Nonlinear Analysis 89 (2013), 157-169. [12] M. Delgado, J. L´opez-G´omez and A. Su´arez, On the symbiotic Lotka-Volterra model with diffusion and transport effects, J. Differential Equations 160 (2000), no. 1, 175-262. [13] J. Hern´andez, Qualitative methods for nonlinear diffusion equations, Nonlinear diffusion problems (Montecatini Terme, 1985), 47118, Lecture Notes in Math., 1224, Springer, Berlin, 1986. [14] O. Ladyzenskaya and N. Uralt’seva, Linear and quasilinear elliptic equations; Translated by Scripta Technica. - New York, Academic Press, 1968. [15] R. Landes, On the existence of weak solutions of perturbated systems with critical growth. J. Reine Angew. Math. 393 (1989), 21–38. [16] J. Leray and J. L. Lions, Quelques r´esultats de Visik sur les probl`emes elliptiques non lin´eaires par les m´ethodes de Minty-Browder, Bull. Soc. Math. France, 93 (1965), 97–107. [17] J. L´opezG´omez and R. Pardo, Coexistence regions in Lotka-Volterra models with diffusion, Nonl. Anal. T.M.A. 19, 11-28 (1992). [18] J. L´opezG´omez and J. C. Sabina, Coexistence states and global attractivity for some convective diffusive competing species models, Trans. A.M.S 347, 3797-3833 (1995). [19] M. Montenegro and A. Su´arez, Existence of a positive solution for a singular system, Proc. Roy. Soc. Edinburgh Sect. A 140 (2010), 435-447 [20] C. V. Pao, “Nonlinear Parabolic and Elliptic Equations”, Plenum Press, New York, 1992. [21] W. H. Ruan, One-parameter family of invariant set for nonweakly coupled nonlinear parabolic systems, J. Math. Anal. Appl., 189 (1995), 763-780. [22] G. Stampacchia, Le probl`eme de Dirichlet pour les ´equations elliptiques du second ordre `a coefficients discontinus, Ann. Inst. Fourier (Grenoble), 15 (1965), 189–258. (JC) Departamento de Matem´ aticas, Universidad de Almer´ ıa, Ctra. Sacramento s/n, La Ca˜ nada de San Urbano, 04120 - Almer´ ıa, Spain. [email protected] (PJMA) Departamento de Matem´ atica Aplicada y Estad´ ıstica, Universidad Polit´ ecnica de Cartagena, 30202 - Murcia, Spain. pedroj.mar[email protected] (AS) Departamento de Ecuaciones Diferenciales y An´ alisis Num´ erico, Facultad de Matem´ aticas, Calle Tarfia s/n, 41012-Sevilla, Spain. [email protected]