1 Introduction
In this paper, we prove a priori estimates for the solutions of elliptic systems involving quasilinear operators in divergence form in an open set Ω⊆ℝN. The simplest problem that we have in mind is the classical model
(1.1){-Δu=f(u,v)inΩ⊆ℝN,-Δv=g(u,v)inΩ⊆ℝN,
where f,g:[0,∞)×[0,∞)→[0,∞) are given nonnegative continuous functions.
More generally, we prove a priori estimates for the solutions of elliptic systems in an open set Ω⊆ℝN involving two quasilinear operators in divergence form. Specifically, we shall study problems of the type
(P){-div(𝒜p(x,u,∇u))≥f(x,u,v)inΩ,-div(𝒜q(x,v,∇v))≥g(x,u,v)inΩ,u≥0,v≥0inΩ,
where 𝒜p,𝒜q:Ω×ℝ×ℝN→ℝN are weakly p-coercive and weakly q-coercive respectively, that is, p>1, q>1, and there exist a,b>0 such that
(𝒜p(x,t,w)⋅w)≥a|𝒜p(x,t,w)|p′ for all(x,t,w)∈Ω×ℝ×ℝN,
(𝒜q(x,t,w)⋅w)≥b|𝒜q(x,t,w)|q′ for all(x,t,w)∈Ω×ℝ×ℝN,
f,g:Ω×[0,∞)×[0,∞)→[0,∞) are Carathédory functions, and for u and v, a weak Harnack inequality holds (for further details and definitions, see Section 2).
In this setting, we prove some a priori bounds for weak solutions of system (P). We shall use some of the ideas developed in [4], where the case of scalar problems was considered.
Our main result is Theorem 3, in which we give a sufficient condition for the nonexistence of nontrivial solutions of (P) in the case Ω=ℝN, and the following local assumptions on f(x,u,v)=f(u,v) and g(x,u,v)=g(u,v), concerning their behavior near zero, hold. We note that this is the first attempt to study nonexistence of positive solutions for quasilinear elliptic systems in this generality. As it is well known, besides their intrinsic interest, these nonexistence theorems can be used to prove existence results for related Dirichlet problems in bounded domains via the so called blow-up technique and suitable index theorems. See, for instance, [12] and the references therein. In addition, we point out that our approach can be used to study similar quasilinear systems in the framework of Carnot groups in the same spirit as [4, 5]. For sake of brevity and in order to avoid cumbersome notations, we restrict our attention to the standard euclidean case.
Assumption 1 (Assumptions on the nonlinearities).
The functions f,g:[0,∞)×[0,∞)→[0,∞) are continuous and satisfy the following conditions:
There exist p1≥0 and q1>0 such that
(f0)lim inft+τ→0f(t,τ)tp1τq1>0 (possibly infinity).
There exist p2>0 and q2≥0 such that
(g0)lim inft+τ→0g(t,τ)tp2τq2>0 (possibly infinity).
On the possible solution (u,v) of the system, we do not require any kind of behavior at infinity. Indeed, we only assume that it belongs to a local Sobolev function space for which the integrals of the relevant quantities make sense. Under these hypotheses, a special case of our main nonexistence theorem applied to (1.1) reads as follows:
Theorem 1.
Suppose the functions f,g:[0,∞)×[0,∞)→[0,∞) are continuous and satisfy (f0) and (g0). Let (u,v) be a weak solution of (1.1) such that
essinfℝNu=essinfℝNv=0.
If 0≤p1<1, 0≤q2<1 and
(1.2)N[1-(1-p1)(1-q2)p2q1]≤max{2+2q1+2p1(1-q2)p2q1,2+2p2+2q2(1-p1)p2q1},
then u=0 or v=0 a.e. in RN. This result is sharp.
In [1, Theorem 5.3], a less general sufficient condition for the nonexistence has been proved in the case f(x,u,v)=|x|αup1vq1 and g(x,u,v)=|x|βup2vq2. While, the same sufficient condition (1.2) has been considered in [3, Theorem V.3] for radial solutions of (P) and the differential system involves the (Δp,Δq) operators, in the special case f(u,v)=up1vq1 and g(u,v)=up2vq2. In Remark 5, we prove also that condition (1.2) is sharp, in the sense that when it does not hold, we are able to construct an explicit nontrivial solution of (P) in the special case when the system involves the same p-Laplacian operator.
We emphasize that conditions (f0) and (g0) allow to study problems with singular nonlinearities. For instance, dealing with f(t,τ)=τ-1, it is easy to construct a function f~(t,τ) such that f(t,τ)≥f~(t,τ) and it satisfies (f0) with p1=0 and any q1>0.
We also prove a nonexistence result for a nonautonomous system of inequalities, in which
f(x,u,v)=a(x)f(u,v) and g(x,u,v)=b(x)g(u,v),
where a,b are positive measurable functions, and f(u,v), g(u,v) satisfy conditions (f0) and (g0), respectively.
As a final remark, we note that, among others, Bourgain [2] studied a stationary Schrödinger system with critical exponents for the Bose–Einstein condensate
{-Δu=upvqinℝN,-Δv=vpuqinℝN.
For earlier results concerning nonexistence of radial positive solutions of the more general model,
{-Δu=up1vq1inℝℕ,-Δv=up2vq2inℝℕ,
where p1,q1,p2,q2>0, see [6].
The paper is organized as follows: In Section 2, we give some useful definitions and preliminary results, focusing on the weak Harnack inequality and its consequences. Section 3 is totally devoted to the general a priori estimates for weak solutions of problem (P), while in Section 4, we prove our main results concerning the nonexistence of nontrivial solutions of (P) when f(x,u,v)=f(u,v) and g(x,u,v)=g(u,v). In Section 5, we prove a nonexistence theorem for the nonautonomous system (P) with f(x,u,v)=a(x)f(u,v) and g(x,u,v)=b(x)g(u,v).
2 Preliminaries
Let 𝒜:ℝN×ℝ×ℝN→ℝN be a Carathéodory function, that is, for each t∈ℝ and w∈ℝN, 𝒜(⋅,t,w) is measurable and for a.e. x∈ℝN, 𝒜(x,⋅,⋅) is continuous. We consider operators Lgenerated by𝒜, that is,
L(u)(x)=div(𝒜(x,u(x),∇u(x))).
Our model cases are the p-Laplace operator, the mean curvature operator and some related generalizations.
Let Ω⊆ℝN be an open set. Let p>1, and let 𝒜p:Ω×ℝ×ℝN→ℝN be a Carathéodory function. The function 𝒜p is called W-p-C, weakly p-coercive, if there exists a constant a>0 such that
(W-p-C)(𝒜p(x,t,w)⋅w)≥a|𝒜p(x,t,w)|p′ for all(x,t,w)∈Ω×ℝ×ℝN.
The function 𝒜p is called S-p-C, strongly p-coercive, if there exist two constants a,a~>0 such that
(S-p-C)(𝒜p(x,t,w)⋅w)≥a~|w|p≥a|𝒜p(x,t,w)|p′ for all(x,t,w)∈Ω×ℝ×ℝN,
see [1, 8, 10] for details.
Example 1.
Clearly, if 𝒜p is S-p-C, then 𝒜p is W-p-C.
Let p>1. The p-Laplace operator Δp(⋅)=div(|∇(⋅)|p-2∇(⋅)) is generated by 𝒜p(x,t,w)=|w|p-2w, which is S-p-C. In particular, when p=2, the Laplace operator Δ(⋅) is S-2-C.
The mean curvature operator
div(∇(⋅)1+|∇(⋅)|2),generated by𝒜p(x,t,w)=w1+|w|2,
is W-2-C, but not S-2-C.
For further details and comments, we refer to [4, Section 1].
In what follows, we denote by 𝒜p a weakly p-coercive operator. Furthermore, BR stands for the ball of radius R>0, that is, BR={x:|x|<R}, and AR is the annulus B2R∖BR¯. Therefore, we have
|BR|=∫BRdx=RN∫|x|<1dx=wNRN and |AR|=wN(2N-1)RN,
where wN is the measure of the unit ball B1 in ℝN.
Consider the system of inequalities
(2.1){-div(𝒜p(x,u,∇u))≥f(x,u,v)inΩ,-div(𝒜q(x,v,∇v))≥g(x,u,v)inΩ,u≥0,v≥0inΩ,
where Ω⊆ℝN is an open set, 𝒜p,𝒜q:Ω×ℝ×ℝN→ℝN are W-p-C and W-q-C, respectively, and
f,g:Ω×[0,∞)×[0,∞)→[0,∞)
are Carathédory functions.
Let p≥1. Throughout the paper, we shall denote
Wloc1,p(Ω):-{u∈Llocp(Ω):|∇u|∈Llocp(Ω)}.
Definition 1.
A pair of functions (u,v)∈Wloc1,p(Ω)×Wloc1,q(Ω) is a weak solution of (2.1) if
f(⋅,u,v),g(⋅,u,v)∈Lloc1(Ω),|𝒜p(⋅,u,∇u)|∈Llocp′(Ω),|𝒜q(⋅,v,∇v)|∈Llocq′(Ω),
and the following inequalities hold
for all nonnegative functions ϕ1, ϕ2∈C01(Ω):
(2.2)∫Ω(𝒜p(x,u,∇u)⋅∇ϕ1)≥∫Ωf(x,u,v)ϕ1,
(2.3)∫Ω(𝒜q(x,v,∇v)⋅∇ϕ2)≥∫Ωg(x,u,v)ϕ2.
Moreover, we say that a weak solution (u,v) is trivial if u=0 or v=0 a.e. in ℝN.
Lemma 1 (Weak Harnack inequality [10, 11]).
If u∈Wloc1,p(RN) is a weak solution of
{-div(𝒜p(x,u,∇u))≥0𝑖𝑛ℝN,u≥0𝑖𝑛ℝN,
𝒜p is S-p-C and N>p>1, then for any σ∈(0,N(p-1)N-p), there exists a constant cH>0 independent of u such that, for all R>0,
(1|BR|∫BRuσ)1σ≤cHessinfBR/2u.
As in [4], we introduce the following definition:
Definition 2.
Let u be a weak solution of
{-div(𝒜p(x,u,∇u))≥0inΩ,u≥0inΩ,
where Ω⊆ℝN is an open set. We say that the weak Harnack inequality holds for u with exponent σ>0 if there exists a constant cH>0 independent of u such that, for any R>0 for which B2R⊂Ω, we have
(WH)(1|BR|∫BRuσ)1σ≤cHessinfBR/2u.
Remark 1.
Inequality (WH) implies immediately that u∈Llocσ(Ω) and that either u≡0 or u>0 in Ω. Moreover, we point out that, by Hölder’s inequality, if (WH) holds with exponent σ, it also holds with any exponent σ0∈(0,σ).
The following is a direct consequence of (WH).
Proposition 1.
If (WH) holds for two nonnegative functions u and v, then (WH) also holds for u+v. Furthermore, there exists a positive constant C independent of u and v for which
essinfBR(u+v)≤C(essinfBR/2u+essinfBR/2v)
for all R>0 such that B2R⊂Ω.
Proof.
Let σ,δ>0 be the exponents for which (WH) holds for u and v, respectively. Suppose that σ≤δ, then (WH) holds with exponent σ for both u and v. Now, for all R>0 such that B2R⊂Ω, we get
(2.4)(∫BR(u+v)σ)1σ≤c[(∫BRuσ)1σ+(∫BRvσ)1σ]
with c:-max{1,2(1-σ)/σ}. Indeed, if σ≥1, inequality (2.4) is the subadditivity of the Lσ(BR)-norm, while if σ<1, (2.4) follows immediately from the fact that (u+v)σ≤uσ+vσ and by the convexity of the power (⋅)1/σ. Hence, by (2.4) and (WH), on u and v, we have
(2.5)(1|BR|∫BR(u+v)σ)1σ≤c[(1|BR|∫BRuσ)1σ+(1|BR|∫BRvσ)1σ]≤CH(essinfBR/2u+essinfBR/2v)≤CHessinfBR/2(u+v),
where CH:-c⋅cH. That is, (WH) holds for u+v.
On the other hand,
(1|BR|∫BR(u+v)σ)1σ≥essinfBR(u+v),
thus, by (2.5),
essinfBR(u+v)≤CH(essinfBR/2u+essinfBR/2v).
∎
Remark 2.
Obviously, the same conclusion of Proposition 1 holds for any finite number of nonnegative functions verifying (WH).
3 A priori estimates
In this section, we prove some integral a priori bounds of the solutions of the system of inequalities (2.1) in which we recall that Ω⊆ℝN is an open set, 𝒜p,𝒜q:Ω×ℝ×ℝN→ℝN are W-p-C and W-q-C, respectively, that is, p>1, q>1, and there exist a,b>0 such that
(𝒜p(x,t,w)⋅w)≥a|𝒜p(x,t,w)|p′for all(x,t,w)∈Ω×ℝ×ℝN,
(𝒜q(x,t,w)⋅w)≥b|𝒜q(x,t,w)|q′for all(x,t,w)∈Ω×ℝ×ℝN,
and f,g:Ω×[0,∞)×[0,∞)→[0,∞) are Carathédory functions.
Theorem 2.
Let (u,v) be a weak solution of (2.1). Then, for all test functions ϕ1,ϕ2, every ℓ≥0 and every α,β<0, we get
(3.1)∫Ωf(x,u,v)uℓαϕ1+c1∫Ω(𝒜p(x,u,∇u)⋅∇u)uℓα-1ϕ1≤c2∫Ωuℓα-1+p|∇ϕ1|pϕ1p-1,
∫Ωg(x,u,v)vℓβϕ2+c~1∫Ω(𝒜q(x,v,∇v)⋅∇v)vℓβ-1ϕ2≤c~2∫Ωvℓβ-1+q|∇ϕ2|qϕ2q-1,
where
uℓ:-u+ℓ,
vℓ:-v+ℓ,
c1:-|α|-ηp′/ap′,
c2:-η-p/p, η>0,
c~1:-|β|-μq′/bq′,
c~2:-μ-q/q and
μ>0.
If η,μ are so small that c1,c~1>0, then, for all α,β<0 and ℓ≥0,
(3.2)∫Ωf(x,u,v)ϕ1≤c3(∫Ωuℓα-1+p|∇ϕ1|pϕ1p-1)1p′(∫Ωuℓ(1-α)(p-1)|∇ϕ1|pϕ1p-1)1p,
∫Ωg(x,u,v)ϕ2≤c~3(∫Ωvℓβ-1+q|∇ϕ2|qϕ2q-1)1q′(∫Ωvℓ(1-β)(q-1)|∇ϕ2|qϕ2q-1)1q,
where c3:-(c2/ac1)1/p′ and c~3:-(c~2/bc~1)1/q′.
If uα-1+p,u(1-α)(p-1)∈Lloc1(AR), vβ-1+q,v(1-β)(q-1)∈Lloc1(AR) with R>0 such that B2R⋐Ω, then, for all α,β<0, there exist c4,c~4>0 for which
(3.3)1|BR|∫BRf(x,u,v)≤c4R-p(1|AR|∫ARuα-1+p)1p′(1|AR|∫ARu(1-α)(p-1))1p,
1|BR|∫BRg(x,u,v)≤c~4R-q(1|AR|∫ARvβ-1+q)1q′(1|AR|∫ARv(1-β)(q-1))1q.
If there exist σ>p-1, δ>q-1 such that uσ,vδ∈Lloc1(Ω), then
(3.4)1|BR|∫BRf(x,u,v)≤c4R-p(1|AR|∫ARuσ)p-1σ,
1|BR|∫BRg(x,u,v)≤c~4R-q(1|AR|∫ARvδ)q-1δ.
In particular, if (WH) holds with exponent σ>p-1 for u and with exponent δ>q-1 for v, then the following inequalities hold for some appropriate constants c5,c~5>0:
(3.5)1|BR|∫BRf(x,u,v)≤c5R-p(essinfBRu)p-1,
1|BR|∫BRg(x,u,v)≤c~5R-q(essinfBRv)q-1.
Proof.
We follow essentially the proof of [4, Theorem 2.1].
Fix a test function ϕ1, and set
r:-dist(supp(ϕ1),∂Ω),
Ωr:-{y∈Ω:dist(y,∂Ω)>r}.
For ε∈(0,r) and ℓ>0, we define
wε(x):-{ℓ+∫ΩrDε(x-y)u(y)dyifx∈Ωr,0ifx∈Ω∖Ωr,
where (Dε)ε is a family of mollifiers. Thus, we can choose wεαϕ1 as test function in (2.2). We have
∫Ωf(x,u,v)wεαϕ1+|α|∫Ω(𝒜p(x,u,∇u)⋅∇wε)wεα-1ϕ1≤∫Ω|𝒜p(x,u,∇u)|⋅|∇ϕ1|wεα.
Since wε→uℓ, ∇wε→∇u in Llocp(Ωr) as ε→0, by Lebesgue’s dominated convergence theorem and by duality, we get
∫Ωf(x,u,v)uℓαϕ1+|α|∫Ω(𝒜p(x,u,∇u)⋅∇u)uℓα-1ϕ1≤∫Ω|𝒜p(x,u,∇u)|⋅|∇ϕ1|uℓα=∫Ω|𝒜p(x,u,∇u)|uℓ(α-1)/p′ϕ11/p′⋅uℓ(α-1+p)/p|∇ϕ1|ϕ1-1/p′≤ηp′p′∫Ω|𝒜p(x,u,∇u)|p′uℓα-1ϕ1+1ηpp∫Ωuℓα-1+p|∇ϕ1|pϕ11-p≤ηp′ap′∫Ω(𝒜p(x,u,∇u)⋅∇u)uℓα-1ϕ1+1ηpp∫Ωuℓα-1+p|∇ϕ1|pϕ11-p,
where, in the last steps, we used Hölder’s and Young’s inequalities and the (W-p-C) condition for 𝒜p. This completes the proof of the first inequality in (3.1) when ℓ>0.
Analogously, it is possible to prove the second one. Indeed, fix a test function ϕ2, and set
r:-dist(supp(ϕ2),∂Ω),Ωr:-{y∈Ω:dist(y,∂Ω)>r}.
For ε∈(0,r) and ℓ>0, define
w~ε(x):-{ℓ+∫ΩrDε(x-y)v(y)dyifx∈Ωr,0ifx∈Ω∖Ωr,
use w~εβϕ2 as test function in (2.3), and proceed as above. The case ℓ=0 follows immediately from the case ℓ>0 by an application of Beppo–Levi’s theorem and letting ℓ→0.
From now on, we only prove the inequalities concerning f, as an argument to obtain the other estimates in exactly the same way.
In order to prove (3.2), use (2.2), and consider ℓ>0. Thus, the weak p-coercivity of 𝒜p, Hölder’s inequality and (3.1) imply
∫Ωf(x,u,v)ϕ1≤∫Ω|𝒜p(x,u,∇u)||∇ϕ1|≤(∫Ω1a(𝒜p(x,u,∇u)⋅∇u)uℓ(α-1)ϕ1)1p′(∫Ωuℓ(1-α)(p-1)|∇ϕ1|pϕ11-p)1p≤c3(∫Ωuℓα-1+p|∇ϕ1|pϕ11-p)1p′(∫Ωuℓ(1-α)(p-1)|∇ϕ1|pϕ11-p)1p.
Also here, it is enough to apply Beppo–Levi’s monotone convergence theorem
and/or Lebesgue’s dominated convergence theorem
to prove the remaining case ℓ=0.
Let ϕ0∈C01(ℝ) be such that 0≤ϕ0≤1, cϕ0:-∥|ϕ0′|p/ϕ0p-1∥∞<∞ and
ϕ0(t)={1,if|t|<1,0,if|t|>2.
Define ϕ1(x):-ϕ0(|x/R|) so that
|∇ϕ1(x)|pϕ1(x)p-1=|ϕ0′(|x/R|)|pϕ0p-1(|x/R|)R-p≤cϕ0R-p.
Hence, using ϕ1 as test function in (3.2) with ℓ=0, we get
∫Ωf(x,u,v)ϕ1≤c3(∫ARuα-1+pcϕ0R-p)1p′(∫ARu(1-α)(p-1)cϕ0R-p)1p,
and so, since |AR|=wN(2N-1)RN=(2N-1)|BR|, we have
1|BR|∫BRf(x,u,v)≤c3(2N-1)cϕ0R-p(1|AR|∫ARuα-1+p)1p′(1|AR|∫ARu(1-α)(p-1))1p,
which gives (3.3) with c4:-c3(2N-1)cϕ0.
Estimates (3.4) follow easily from (3.3) by applying Hölder’s inequality. Finally, if (WH) holds, by (3.4), we obtain
1|BR|∫BRf(x,u,v)≤c4(1-12N)1-pσR-p(1|B2R|∫B2Ruσ)p-1σ≤c5R-p(essinfBRu)p-1
with c5:-c4(1-12N)(1-p)/σcHp-1.
∎
4 Some Liouville-type theorems
In this section, we shall prove the main results of this paper.
Consider the problem
(4.1){-div(𝒜p(x,u,∇u))≥f(u,v)inℝN,-div(𝒜q(x,v,∇v))≥g(u,v)inℝN,u≥0,v≥0inℝN.
Throughout this section, without further mentioning, we shall assume the following:
Assumption 2.
The functions Ap,Aq:RN×R×RN→RN are W-p-C and W-q-C, respectively, N>max{p,q}, (WH) holds for u with exponent σ>p-1 and for v with exponent δ>q-1, and Assumption 1 holds.
Example 2.
Besides all the functions f such that f(t,τ)≥ctp1τq1 for every (t,τ)∈[0,∞)×[0,∞), an example of a function satisfying condition (f0) is given by f(t,τ)=sin2tsin2τ in [0,∞)×[0,∞). Clearly, in this case, f satisfies (f0) with p1=q1=2.
Lemma 2 (cf. [4, Lemma 3.1]).
Let u:RN→[0,∞) be a function such that essinfRNu=0. Assume that (WH) holds with exponent σ>0. Then, for all ε>0,
limR→∞|AR/2∩Tεu||AR/2|=1,limR→∞|BR∩Tεu||BR|=1,
where Tεu={x∈RN:u(x)<ε} and AR=B2R∖BR¯.
Lemma 3.
Let (u,v) be a weak solution of (4.1) such that essinfRNu=essinfRNv=0. If f(u(x),v(x))=0 for a.a. x∈RN, then u=0 or v=0 a.e. in RN. Similarly, if g(u(x),v(x))=0 for a.a. x∈RN, then u=0 or v=0 a.e. in RN.
Proof.
Suppose that f(u(x),v(x))=0 for a.a. x∈ℝN. Thanks to Proposition 1, we can apply Lemma 2 to the function u+v. Hence, by (f0), we get
(essinfBRu)p1(essinfBRv)q1≤1|AR/2∩Tε|∫AR/2∩Tεup1vq1≤c1|AR/2∩Tε|∫AR/2f(u,v)=0
for R sufficiently large and ε>0, where Tε={x∈ℝN:u(x)+v(x)<ε}. Using (WH) on u and v, we conclude that u=0 or v=0 a.e. in ℝN. If g(u(x),v(x))=0 for a.a. x∈ℝN, the proof is similar.
∎
Let us introduce the matrix
ℋ=(p1-p+1q1p2q2-q+1),
(4.2)D:--detℋ=p2q1-(p-1-p1)(q-1-q2).
Lemma 4.
Let (u,v) be a nontrivial weak solution of (4.1) such that essinfRNu=essinfRNv=0. Then there exists a constant c>0 such that, for all ε>0 and R>0 sufficiently large, the following estimates hold:
(4.3)(essinfAR/2∩Tεu)p1-p+1(essinfAR/2∩Tεv)q1≤cR-p,
(essinfAR/2∩Tεu)p2(essinfAR/2∩Tεv)q2-q+1≤cR-q,
where Tε={x∈RN:u(x)+v(x)<ε},
(4.4)∫BRf(u,v)≤cR-p|AR/2|(essinfBRu)p-1≤cRN[1-p-1p2+q2(p-1)p2(q-1)]-p-qq2(p-1)p2(q-1)(∫AR/2g(u,v))p-1p2(∫BRg(u,v))q2(p-1)p2(q-1),
(4.5)∫BRg(u,v)≤cR-q|AR/2|(essinfBRv)q-1≤cRN[1-q-1q1+p1(q-1)q1(p-1)]-q-pp1(q-1)q1(p-1)(∫AR/2f(u,v))q-1q1(∫BRf(u,v))p1(q-1)q1(p-1).
In particular, if q2≤q-1, then, for R sufficiently large,
(4.6)(essinfAR/2∩Tεu)1-(p-1-p1)(q-1-q2)p2q1≤cR-p(q-1-q2)+qq1p2q1,
(4.7)∫BRf(u,v)≤cRNDp2q1-p-qq1(p-1)+pp1(q-1-q2)p2q1(∫Sf(u,v))(p-1-p1)(q-1-q2)p2q1
with S=AR/2 or S=BR. If p1≤p-1, then, for R sufficiently large,
(4.8)(essinfAR/2∩Tεv)1-(p-1-p1)(q-1-q2)p2q1≤cR-q(p-1-p1)+pp2p2q1,
(4.9)∫BRg(u,v)≤cRNDp2q1-q-pp2(q-1)+qq2(p-1-p1)p2q1(∫Sg(u,v))(p-1-p1)(q-1-q2)p2q1
with S=AR/2 or S=BR.
Proof.
Fix ε>0. By the first inequality of (3.5),
we get
∫BRf(u,v)≤cR-p|AR/2|(essinfBRu)p-1≤cR-p|AR/2|(essinfAR/2∩Tεu)p-1.
On the other hand, using (f0), we have
∫BRf(u,v)≥∫AR/2∩Tεf(u,v)≥c∫AR/2∩Tεup1vq1,
hence,
∫AR/2∩Tεup1vq1≤cR-p|AR/2|(essinfAR/2∩Tεu)p-1.
Therefore,
(essinfAR/2∩Tεu)p1(essinfAR/2∩Tεv)q1≤cR-p|AR/2||AR/2∩Tε|(essinfAR/2∩Tεu)p-1,
and so, by Proposition 1 and by Lemma 2 applied to the function u+v, we obtain
(essinfAR/2∩Tεu)p1-p+1(essinfAR/2∩Tεv)q1≤cR-p
for R sufficiently large.
Similarly, from the second inequality of the system, we prove the second inequality of (4.3).
By (3.5) and (g0), for R sufficiently large, it follows that
∫BRf(u,v)≤cR-p|AR/2|(essinfBRu)p-1≤cR-p|AR/2|(essinfAR/2∩Tεu)p-1≤cR-p|AR/2|(essinfAR/2∩Tεv)-q2(p-1)p2(1|AR/2∩Tε|∫AR/2∩Tεup2vq2)p-1p2≤cR-p|AR/2||AR/2∩Tε|p-1p2(essinfBRv)-q2(p-1)p2(∫AR/2∩Tεg(u,v))p-1p2≤cR-p|AR/2|1-p-1p2(R-q|BR|∫BRg(u,v))q2(p-1)p2(q-1)(∫AR/2g(u,v))p-1p2,
where, in the last step, we have applied Lemma 2 to the function u+v, which, thanks to Proposition 1, satisfies all the required assumptions. Similarly, working on the second inequality of (4.1), we obtain (4.5).
Combining the two inequalities in (4.3) and using the assumption q2≤q-1, we immediately get (4.6),
(essinfAR/2∩Tεu)1-(q-1-q2)(p-1-p1)p2q1≤cR-p(q-1-q2)+qq1p2q1
for R sufficiently large.
From (4.4) and (4.5), we obtain
∫BRf(u,v)≤cRN[1-p-1p2+q2(p-1)p2(q-1)]-p-qq2(p-1)p2(q-1)(∫Sg(u,v))p-1p2(1-q2q-1),
∫BRg(u,v)≤cRN[1-q-1q1+p1(q-1)q1(p-1)]-q-pp1(q-1)q1(p-1)(∫Sf(u,v))q-1q1(1-p1p-1)
with S=AR/2 or S=BR, being f and g nonnegative and AR/2⊂BR. Since q2≤q-1, these two inequalities imply
∫BRf(u,v)≤cR-p-qp-1p2+pp1(q2-q+1)q1p2|AR/2|1-(p1-p+1)(q2-q+1)p2q1(∫Sf(u,v))(q-1)(p-1)q1p2(1-p1p-1)(1-q2q-1).
Similarly, under the assumption p1≤p-1, we can prove (4.8) and (4.9).
∎
Theorem 3.
Let p1<p-1, q2<q-1 and
(4.10)N≥min{(N-pp-1p2+N-qq-1q2)q1q1-q2+q-1+(N-pp-1p1+N-qq-1q1)q-1-q2q1-q2+q-1,(N-pp-1p1+N-qq-1q1)p2p2-p1+p-1+(N-pp-1p2+N-qq-1q2)p-1-p1p2-p1+p-1}.
If (u,v) is a weak solution of (4.1) such that essinfRNu=essinfRNv=0, then either u=0 or v=0 a.e. in RN.
We note that (4.10) is equivalent to
(4.11)N[1-(p-1-p1)(q-1-q2)p2q1]≤max{p+qq1(p-1)+pp1(q-1-q2)p2q1,q+pp2(q-1)+qq2(p-1-p1)p2q1}
when p1<p-1 and q2<q-1. Indeed, starting from (4.10), when the minimum is the first quantity in the brackets, we get
N≥N-pp-1⋅p2q1+p1(q-1-q2)q1-q2+q-1+Nq1q1-q2+q-1-qq1q1-q2+q-1=q-1-q2q1-q2+q-1{N-pp-1(p2q1q-1-q2+p1)-qq1q-1-q2}+Nq1q1-q2+q-1,
that is,
Nq-1-q2q1-q2+q-1≥q-1-q2q1-q2+q-1{N-pp-1(p2q1q-1-q2+p1)-qq1q-1-q2}.
Now, since q2<q-1, we get
N≥{N-pp-1(p2q1q-1-q2+p1)-qq1q-1-q2},
Multiplying both sides by (p-1)(q-1-q2)p2q1, we have
(N-p)[1+p1(q-1-q2)p2q1]-qq1(p-1)p2q1≤N(p-1)(q-1-q2)p2q1,
namely,
N[1-(p-1-p1)(q-1-q2)p2q1]≤p+qq1(p-1)+pp1(q-1-q2)p2q1.
Similarly, we can easily prove the second part of the equivalence.
Proof of Theorem 3.
We shall distinguish two cases depending on whether the constant D defined in (4.2), as well as the left side of (4.11), is positive or nonpositive.
Case D>0.
Suppose that
p+qq1(p-1)+pp1(q-1-q2)p2q1≥q+pp2(q-1)+qq2(p-1-p1)p2q1,
the remaining case being analogous. Without loss of generality, we prove the theorem only when
N[1-(p-1-p1)(q-1-q2)p2q1]=p+qq1(p-1)+pp1(q-1-q2)p2q1.
Suppose, by contradiction, that both u>0 and v>0 in ℝN. By (4.7), we have
(4.12)∫BRf(u,v)≤c(∫AR/2f(u,v))(p-1-p1)(q-1-q2)p2q1≤c(∫BRf(u,v))(p-1-p1)(q-1-q2)p2q1,
hence f(u,v)∈L1(ℝN). Thus, by the first inequality of (4.12), letting R→∞, we get f(u,v)=0 a.e. in ℝN. By Lemma 3, we conclude that either u=0 or v=0 a.e. in ℝN. This contradiction proves the claim.
Case D≤0.
Note that, in this case, condition (4.11) is trivially satisfied. Suppose, by contradiction, that both u>0 and v>0. Clearly, p(q2-q+1)-qq1<0 and q(p1-p+1)-pp2<0, since p1<p-1 and q2<q-1.
Hence, if D<0, by (4.6) and (4.8), R large and ε>0, we get
essinfAR/2∩Tεu≥cR-p(q-1-q2)-qq1p2q1-(p-1-p1)(q-1-q2),essinfAR/2∩Tεv≥cR-q(p-1-p1)-pp2p2q1-(p-1-p1)(q-1-q2).
Therefore,
limR→∞essinfAR/2∩Tεu≥∞,limR→∞essinfAR/2∩Tεv≥∞,
which is impossible.
Next, if D=0, then, by (4.6) and R large, it follows that
1≤cR-p(q-1-q2)-qq1.
Clearly, by letting R→∞, we reach a contradiction.
∎
Remark 3.
In Theorem 3, as well as in all the nonexistence theorems of this paper, we require that the solutions of the system have an essential infimum on ℝN equal to zero. If, for instance, f(u,v)≥cup1vq1 in all of ℝN, the assumption on the essential infimum of u and v is quite natural. Indeed, if essinfℝNu>0 and essinfℝNv>0, then every solution (u,v) of (4.1) is also a solution of
(4.13){-div(𝒜p(x,u,∇u))≥const.>0inℝN,-div(𝒜q(x,v,∇v))≥g(u,v)inℝN,u≥0,v≥0inℝN.
The first inequality of (4.13) does not have any weak solutions (see e.g. [4, Corollary 2.4]), therefore also system (4.1) has no weak solutions.
Furthermore, if essinfℝNv=0 and 𝒜p does not depend explicitly on u, we have the following result.
Corollary 1.
Let (u,v) be a weak solution of the problem
(4.14){-div(𝒜p(x,∇u))≥up1vq1𝑖𝑛ℝN,-div(𝒜q(x,v,∇v))≥up2vq2𝑖𝑛ℝN,u≥0,v≥0𝑖𝑛ℝN
with q2<q-1. If essinfRNv=0 and essinfRNu>0, then v=0 a.e. in RN.
Proof.
Put u0:-essinfℝNu>0 and u~:-u-u0. Then (u~,v) solves the problem
(4.15){-div(𝒜p(x,∇u~))≥(u~+u0)p1vq1inℝN,-div(𝒜q(x,v,∇v))≥(u~+u0)p2vq2inℝN,u~≥0,v≥0inℝN.
Consider the functions f,g:[0,∞)×[0,∞)→[0,∞) defined by
f(t,τ)=(t+u0)p1τq1 and g(t,τ)=(t+u0)p2τq2
for all (t,τ)∈[0,∞)×[0,∞). It follows that
lim inft+τ→0f(t,τ)tp~1τq1=lim inft+τ→0(t+u0)p1τq1tp~1τq1=+∞>0 for allp~1>0,
lim inft+τ→0g(t,τ)tp~2τq2=lim inft+τ→0(t+u0)p2τq2tp~2τq2=+∞>0 for allp~2>0,
that is, f and g satisfy (f0) and (g0) with exponents p~1,q1,p~2,q2. Next, by choosing p~1 and p~2 so small so that p~1<p-1 and p~2q1<(p-1-p~1)(q-1-q2), we see that we can apply Theorem 3 to problem (4.15). Consequently, u-u0=0 or v=0 a.e. in ℝN. If v=0 a.e. in ℝN, we are done. On the other hand, if u=u0 a.e. in ℝN, then, by the first inequality of (4.14), it follows that v=0 a.e. in ℝN.
∎
Obviously, an analogous result as above can be obtained when essinfℝNu=0, essinfℝNv>0, p1<p-1, and 𝒜q does not depend explicitly on v.
Remark 4.
In the case p=q, p1<p-1, q2<p-1 and D>0, condition (4.11) is sharp also for systems of equations. Indeed, if
N[1-(p-1-p1)(p-1-q2)p2q1]>max{p+pq1(p-1)+pp1(p-1-q2)p2q1,p+pp2(p-1)+pq2(p-1-p1)p2q1},
then we can construct an explicit nontrivial solution of the problem
(4.16){-div(|∇u|p-2∇u)=f(u,v)inℝN,-div(|∇v|p-2∇v)=g(u,v)inℝN,u≥0,v≥0inℝN,
where f,g:[0,∞)×[0,∞)→[0,∞) are continuous and such that, for all (t,τ)∈[0,1]×[0,1],
f=f(t,τ)=(αpp-1)p-1τ(α+1)(p-1)/β{N-(α+1)p+(α+1)pt1/α},
g=g(t,τ)=(βpp-1)p-1t(β+1)(p-1)/α{N-(β+1)p+(β+1)pτ1/β},
where
α:-(p-1)(q1+p-1)p2q1-(p-1)2,β:-(p-1)(p2+p-1)p2q1-(p-1)2.
Hence f and g satisfy (f0) and (g0) with exponents p1=0, q1=(α+1)(p-1)/β, p2=(β+1)(p-1)/α, q2=0. By straightforward calculation, it follows that the functions defined by
u(x)=1(1+|x|p/(p-1))α,v(x)=1(1+|x|p/(p-1))β
are weak solutions of (4.16).
Remark 5.
In the case p1<p-1, q2<q-1 and D>0, condition (4.11) is sharp for systems of inequalities. Indeed, if
(4.17)N[1-(p-1-p1)(q-1-q2)p2q1]>max{p+qq1(p-1)+pp1(q-1-q2)p2q1,q+pp2(q-1)+qq2(p-1-p1)p2q1},
then (4.1) has a nontrivial solution. Indeed, if (4.17) holds, then we can construct an explicit solution of the problem
(4.18){-div(|∇u|p-2∇u)≥f(u,v)inℝN,-div(|∇v|q-2∇v)≥g(u,v)inℝN,u≥0,v≥0inℝN,
where f and g satisfy (f0) and (g0), respectively. Consider the functions defined by
u(x)=1(1+|x|p/(p-1))α,α:-p-1p⋅qq1+p(q-1-q2)p2q1-(p-1-p1)(q-1-q2),v(x)=1(1+|x|q/(q-1))β,β:-q-1q⋅pp2+q(p-1-p1)p2q1-(p-1-p1)(q-1-q2).
Denoting ϱ:-|x|, an easy computation shows that
-Δpuup1vq1=(αpp-1)p-1(1+ϱp/(p-1))αp1-(α+1)(p-1)-1(1+ϱq/(q-1))βq1{[N-(α+1)p]ϱp/(p-1)+N},
-Δqvup2vq2=(βqq-1)q-1(1+ϱq/(q-1))βq2-(β+1)(q-1)-1(1+ϱp/(p-1))αp2{[N-(β+1)q]ϱq/(q-1)+N}.
By (4.17) and our assumptions p1<p-1 and q2<q-1, it follows that N>(α+1)p and N>(β+1)q.
Hence, if we denote
h1(ϱ):-(αpp-1)p-1(1+ϱp/(p-1))αp1-(α+1)(p-1)-1(1+ϱq/(q-1))βq1{[N-(α+1)p]ϱp/(p-1)+N},
h2(ϱ):-(βqq-1)q-1(1+ϱq/(q-1))βq2-(β+1)(p-1)-1(1+ϱp/(p-1))αp2{[N-(β+1)q]ϱq/(q-1)+N},
it follows that h1(ϱ)>0 and h2(ϱ)>0 for all ϱ≥0. Moreover, by the definitions of α and β, we get
pp-1[αp1-(α+1)(p-1)]+qq-1βq1=0,
pp-1αp2+qq-1[βq2-(β+1)(q-1)]=0,
hence,
limϱ→∞h1(ϱ)=(αpp-1)p-1⋅[N-(α+1)p]>0,
and similarly,
limϱ→∞h2(ϱ)=(βqq-1)q-1⋅[N-(β+1)q]>0.
Therefore, since h1 and h2 are continuous functions, there are two positive constants C1 and C2 such that h1(ϱ)≥C1 and h2(ϱ)≥C2 for all ϱ≥0. Thus, we have, for all x∈ℝN,
-Δpuup1vq1≥C1>0 and -Δqvup2vq2≥C2>0,
that is, (u,v) is a nontrivial solution of (4.18) with f(u,v)=C1up1vq1 and g(u,v)=C2up2vq2.
By our construction, it follows that 0<u(x)≤1 and 0<v(x)≤1 for all x∈ℝN. Hence this counterexample works also for all continuous functions f,g:[0,∞)×[0,∞)→[0,∞) such that f(t,τ)=C1tp1τq1 and g(t,τ)=C2tp2τq2 for all (t,τ)∈[0,1]×[0,1] and nonnegative elsewhere.
Corollary 2.
Let (u,v) be a weak solution of the system
(4.19){-div(𝒜p(x,u,∇u))≥f(v)𝑖𝑛ℝN,-div(𝒜q(x,v,∇v))≥g(u)𝑖𝑛ℝN,u≥0,v≥0𝑖𝑛ℝN,
where f,g:[0,∞)→[0,∞) are continuous functions satisfying the following conditions:
There exists q1>0 such that
f0´lim inft→0+f(t)tq1>0 (possibly infinity).
There exists p2>0 such that
g0´lim inft→0+g(t)tp2>0 (possibly infinity).
If
(4.20)N[1-(p-1)(q-1)p2q1]≤max{p+q(p-1)p2,q+p(q-1)q1}
and essinfRNu=essinfRNv=0, then u≡v≡0 a.e. in RN.
Proof.
By Theorem 3, with p1=q2=0, we have that u=0 or v=0 a.e. in ℝN. If v≡0, by W-q-C, for 𝒜q, we have 𝒜q(⋅,v,∇v)=0 a.e. in ℝN, and in turn, g(u)=0 a.e. in ℝN.
Thus, by (WH) on the first inequality of system (4.19), g0´ and Lemma 2, we obtain, for R large,
(1|B2R|∫B2Ruσ)1σ≤cHessinfBRu≤cR-N/p2(∫AR/2g(u))1p2=0,
that is, u=0 a.e. in ℝN.
∎
Remark 6.
Note that condition (4.20) is equivalent to
(4.21)max{qq1+p(q-1)p2q1-(p-1)(q-1)-N-pp-1,pp2+q(p-1)p2q1-(p-1)(q-1)-N-qq-1}≥0.
This is the assumption required in [7, Theorem 2.1], when f(v)=vq1 and g(u)=up2. In [7, Section 3], the authors prove also that the nonexistence result is sharp, in the sense that if (4.21) is not valid, they are able to construct a solution (u,v)≠(0,0) of (4.19). Corollary 2 in a more general setting has been studied in [5].
Remark 7.
Consider the problem
(4.22){-div(|∇u|p-2∇u)≥up1vq1inℝN,-div(|∇v|q-2∇v)≥up2vq2inℝN,u≥0,v≥0inℝN
with p,q>1, p1,q2≥0 and p2,q1>0. As pointed out in [1, Remark 5.1], it is possible to obtain a nonoptimal sufficient condition of nonexistence for (4.22), as a consequence of Corollary 2. Since -Δp and -Δq are S-p-C and S-q-C, respectively, inequality (WH) holds for both u and v. Hence, by Remark 1, either u≡0 or u>0 in ℝN, and analogously, either v≡0 or v>0 in ℝN.
Therefore, with a change of variables, we can obtain, from problem (4.22), a system of the type (4.19). More precisely, let θ,τ∈(0,1). Set w:-uθ, z:-vτ. Then
{-Δpw≥Cw[p1-(1-θ)(p-1)]/θzq1/τinℝN,-Δqz≥Cwp2/θz[q2-(1-τ)(q-1)]/τinℝN,
where C>0. When p1<p-1 and q2<q-1, we can choose θ=1-p1p-1, τ=1-q2q-1 and find
{-Δpw≥Czq1/τinℝN,-Δqz≥Cwp2/θinℝN.
Hence, if q1τ>q-1, p2θ>p-1 (i.e. q1>q-1-q2 and p2>p-1-p1), and if we require condition (4.20) with q1τ in place of q1 and p2θ in place of p2, that is,
(4.23)N[1-(p-1-p1)(q-1-q2)p2q1]≤max{p+q(p-1-p1)p2,q+p(q-1-q2)q1},
then problem (4.22) has no nontrivial solutions by Corollary 2. Nevertheless, condition (4.23) is not sharp, as Theorem 3 proves.
Theorem 4.
Assume that p1≤p-1 and q2≤q-1. If (u,v) is a weak solution of (4.1) such that essinfRNu=essinfRNv=0 and
(4.24)N[1-(p-1-p1)(q-1-q2)p2q1]<max{p+qq1(p-1)+pp1(q-1-q2)p2q1,q+pp2(q-1)+qq2(p-1-p1)p2q1},
then u=0 or v=0 a.e. in RN.
Proof.
From Theorem 3, if p1<p-1 and q2<q-1, we already know a stronger result. Therefore, we prove this result only when p1=p-1 and q2≤q-1, and we omit the similar proof in the case p1≤p-1 and q2=q-1. Suppose, by contradiction, that problem (4.1) admits a nontrivial solution (u,v). By (4.7) and (4.9), we have, for R sufficiently large,
∫BRf(u,v)≤cRN-p-p1qq1+p(q-1-q2)p2q1,∫BRg(u,v)≤cRN-q-p(q-1)q1.
By hypothesis (4.24) and letting R→∞, we get f(u,v)=0 or g(u,v)=0 a.e. in ℝN. We complete the proof by using Lemma 3.
∎
Lemma 5.
Let (u,v) be a weak solution of (4.1) such that essinfRNu=essinfRNv=0. If there exists z∈[0,1] such that
(4.25){p2p-1z+(p1p-1-1)(1-z)≥0,(q2q-1-1)z+q1q-1(1-z)≥0,N>(N-pp-1p1+N-qq-1q1)(1-z)+(N-pp-1p2+N-qq-1q2)z,
then u=0 or v=0 a.e. in RN.
Proof.
By contradiction, if u>0 and v>0, from (4.4) and (4.5), we have, for R large and for all z∈[0,1],
(∫BRf(u,v))p2p-1z(∫BRg(u,v))(q2q-1-1)z≤cRN(p2p-1+q2q-1-1)z-pp2p-1z-qq2q-1z,
(∫BRf(u,v))(p1p-1-1)(1-z)(∫BRg(u,v))q1q-1(1-z)≤cRN(q1q-1+p1p-1-1)(1-z)-qq1q-1(1-z)-pp1p-1(1-z),
and so
(4.26)(∫BRf(u,v))α(∫BRg(u,v))β≤cR-γ,
where
α:-p2p-1z+(p1p-1-1)(1-z),β:-(q2q-1-1)z+q1q-1(1-z),γ:-N-(N-pp-1p2+N-qq-1q2)z-(N-pp-1p1+N-qq-1q1)(1-z).
By (4.25),
α≥0, β≥0 and γ>0,
hence, by (4.26), f(u,v)=0 or g(u,v)=0 a.e. in ℝN, and so, by Lemma 3, we have that u=0 or v=0 a.e. in ℝN. This completes the proof.
∎
Theorem 5.
Let (u,v) be a weak solution of (4.1) such that essinfRNu=essinfRNv=0.
If p1≥p-1,
q2≤q-1 and
(4.27)N>min{N-pp-1p1+N-qq-1q1,(N-pp-1p2+N-qq-1q2)q1q1-q2+q-1+(N-pp-1p1+N-qq-1q1)q-1-q2q1-q2+q-1},
then u=0 or v=0 a.e. in ℝN.
If p1≤p-1,
q2≥q-1 and
(4.28)N>min{N-pp-1p2+N-qq-1q2,(N-pp-1p1+N-qq-1q1)p2p2-p1+p-1+(N-pp-1p2+N-qq-1q2)p-1-p1p2-p1+p-1},
then u=0 or v=0 a.e. in ℝN.
If p1≥p-1,
q2≥q-1 and
(4.29)N>min{N-pp-1p1+N-qq-1q1,N-pp-1p2+N-qq-1q2},
then u=0 or v=0 a.e. in ℝN.
Proof.
(i) Let (u,v) be a weak solution of (4.1). By Lemma 5, for all z∈[0,1] satisfying (4.25), we have that (u,v) is trivial. Now, system (4.25) is equivalent to
(4.30){z≤q1q1+q-1-q2,N>(N-pp-1p1+N-qq-1q1)(1-z)+(N-pp-1p2+N-qq-1q2)z,
since p1≥p-1 and q2≤q-1. Put
φ(z):-(N-pp-1p1+N-qq-1q1)(1-z)+(N-pp-1p2+N-qq-1q2)z.
If
N-pp-1p1+N-qq-1q1≤N-pp-1p2+N-qq-1q2,
then φ is nondecreasing, and we obtain the best condition taking z=0 in the second inequality of (4.30), namely,
N>N-pp-1p1+N-qq-1q1.
While, if
N-pp-1p1+N-qq-1q1>N-pp-1p2+N-qq-1q2,
then we have the best condition taking z=q1q1+q-1-q2 in second inequality of (4.30), that is
N>(N-pp-1p2+N-qq-1q2)q1q1-q2+q-1+(N-pp-1p1+N-qq-1q1)q-1-q2q1-q2+q-1.
Finally, by an easy calculation, we see that
(N-pp-1p2+N-qq-1q2)q1q1-q2+q-1+(N-pp-1p1+N-qq-1q1)q-1-q2q1-q2+q-1<N-pp-1p1+N-qq-1q1
if and only if
N-pp-1p2+N-qq-1q2<N-pp-1p1+N-qq-1q1.
This completes the proof of the first part of the theorem.
(ii) The proof is similar to the proof of (i), and it is omitted.
(iii) Let (u,v) be as in the statement. By Lemma 5, for all z∈[0,1] satisfying (4.25), we have that (u,v) is trivial. Now, system (4.25) is equivalent to
(4.31)N>(N-pp-1p1+N-qq-1q1)(1-z)+(N-pp-1p2+N-qq-1q2)z,
since p1≥p-1 and q2≥q-1. Now, if
N-pp-1p1+N-qq-1q1≤N-pp-1p2+N-qq-1q2,
then the function φ defined in part (i) is nondecreasing, and we obtain the best condition taking z=0 in (4.31), namely,
N>N-pp-1p1+N-qq-1q1.
While, if
N-pp-1p1+N-qq-1q1>N-pp-1p2+N-qq-1q2,
then we have the best condition taking z=1 in (4.31), that is,
N>N-pp-1p2+N-qq-1q2.∎
Remark 8.
If p1=p-1 and q2=q-1, then (4.27) jointly with (4.28) give the same condition as (4.29). Moreover, in this case, this curve is equivalent to condition (4.24).
Theorem 6.
Let (u,v) be a weak solution of (4.1) with essinfRNu=essinfRNv=0.
If q2<q-1,
D>0 and
(4.32)N≥(N-pp-1p2+N-qq-1q2)q1q1-q2+q-1+(N-pp-1p1+N-qq-1q1)q-1-q2q1-q2+q-1,
then u=0 or v=0 a.e. in ℝN. In particular, if q2<q-1,
p1≥p-1 and (4.32) holds, then (u,v) is trivial.
If p1<p-1,
D>0 and
(4.33)N≥(N-pp-1p1+N-qq-1q1)p2p2-p1+p-1+(N-pp-1p2+N-qq-1q2)p-1-p1p2-p1+p-1,
then u=0 or v=0 a.e. in ℝN. In particular, if p1<p-1,
q2≥q-1 and (4.33) holds, then (u,v) is trivial.
Proof.
(i) By contradiction, let (u,v) be a nontrivial weak solution of (4.1). By (4.4) and (4.5), for R sufficiently large, we have
(4.34)(∫BRf(u,v))p2p-1(∫BRg(u,v))q2q-1≤cRN(p2p-1-1+q2q-1)-pp2p-1-qq2q-1∫AR/2g(u,v)
and
(4.35)(∫BRg(u,v))q1q-1(∫BRf(u,v))p1p-1≤cRN(q1q-1-1+p1p-1)-qq1q-1-pp1p-1∫AR/2f(u,v).
By (4.34),
∫BRg(u,v)≥c(Rpp2p-1+qq2q-1-N(p2p-1-1+q2q-1))q-1q-1-q2(∫BRf(u,v))p2(q-1)(p-1)(q-1-q2)
since q2<q-1.
Combining this last inequality with (4.35), we get
(4.36)∫AR/2f(u,v)≥cRpp1p-1+qq1q-1-N(p1p-1+q1q-1-1)+[pp2p-1+qq2q-1-N(p2p-1+q2q-1-1)]q1q-1-q2(∫BRf(u,v))p1p-1+p2q1(p-1)(q-1-q2),
and so
(4.37)(∫BRf(u,v))D(p-1)(q-1-q2)≤cR-γ
for R large, where
γ:-pp1p-1+qq1q-1-N(p1p-1+q1q-1-1)+[pp2p-1+qq2q-1-N(p2p-1+q2q-1-1)]q1q-1-q2.
By hypothesis (i), γ≥0. Now, by (4.37), ∫ℝNf(u,v)<∞, since D(p-1)(q-1-q2)>0, so that by (4.36), f(u,v)=0 a.e. in ℝN. The contradiction follows by Lemma 3. For the second part of the statement, it is enough to note that if q2<q-1 and p1≥p-1, then D>0.
(ii) The proof is analogous to the proof of (i).
∎
Remark 9.
Note that, in the case p1=p-1 and q2=q-1, condition (4.25) is equivalent to (4.24). Moreover, when
(4.38)q2<q-1,p1≥p-1,N-pp-1p1+N-qq-1q1≤N-pp-1p2+N-qq-1q2,
then condition (4.32) is stronger than (4.27). Similarly, if
(4.39)p1<p-1,q2≥q-1,N-pp-1p2+N-qq-1q2≤N-pp-1p1+N-qq-1q1,
then (4.33) is stronger than (4.28).
Now we prove that conditions (4.32) and (4.33) are sharp at least when (4.38) and (4.39) hold, respectively.
For simplicity, we show a counterexample for (4.32) when p=q.
Let q2<p-1≤p1, p1+q1≤p2+q2 and
(4.40)N(p-1)N-p<(p2+q2)q1q1-q2+q-1+(p1+q1)q-1-q2q1-q2+q-1.
We prove that, under these assumptions, the system
{-div(|∇u|p-2∇u)≥f(u,v)inℝN,-div(|∇v|p-2∇v)≥g(u,v)inℝN,u≥0,v≥0inℝN,
where f and g satisfy (f0) and (g0), admits a nontrivial solution. Consider the functions
u(x)=1(1+|x|p/(p-1))α,α:-(p-1)(q1+p-1-q2)p2q1-(p-1-p1)(p-1-q2),
v(x)=1(1+|x|p/(p-1))β,β:-(p-1)(p2+p-1-p1)p2q1-(p-1-p1)(p-1-q2),
and denote ϱ:-|x|.
By straightforward computation, we know that
-Δpuup1vq1=h1(ϱ),-Δqvup2vq2=h2(ϱ),
where
h1(ϱ):-(αpp-1)p-1(1+ϱp/(p-1))αp1+βq1-(α+1)(p-1)-1{[N-(α+1)p]ϱp/(p-1)+N},
h2(ϱ):-(βqq-1)q-1(1+ϱp/(p-1))αp2+βq2-(β+1)(p-1)-1{[N-(β+1)p]ϱp/(p-1)+N}.
The exponents α and β are such that
{αp1+βq1-(α+1)(p-1)-1=0,αp2+βq2-(β+1)(p-1)-1=0,
hence, the expressions of h1 and h2 become simply
h1(ϱ):-(αpp-1)p-1{[N-(α+1)p]ϱp/(p-1)+N},
h2(ϱ):-(βqq-1)q-1{[N-(β+1)p]ϱp/(p-1)+N}.
By (4.40) and our assumptions q2<p-1≤p1 and p1+q1≤p2+q2, it follows that N>(α+1)p and N>(β+1)q.
Hence, h1(ϱ)>0 and h2(ϱ)>0 for all ϱ≥0.
For simplicity, we summarize the results obtained in Theorems 3, 4 and 5 for p=q.
Corollary 3.
Consider system (4.1) with p=q. Let (u,v) be a weak solution of this problem such that
essinfℝNu=essinfℝNv=0.
Denote A1:-p1+q1, A2:-p2+q2, α=p2p2-p1+p-1, β=q1q1-q2+p-1 and
(4.41)N(p-1)N-p≥min{A1α+A2(1-α),A2β+A1(1-β)},
(4.42)N(p-1)N-p>min{A1α+A2(1-α),A2β+A1(1-β)},
(4.43)N(p-1)N-p>min{A1,A2β+A1(1-β)},
(4.44)N(p-1)N-p>min{A2,A1α+A2(1-α)},
(4.45)N(p-1)N-p>min{A1,A2},
(4.46)N(p-1)N-p≥A2β+A1(1-β),
(4.47)N(p-1)N-p≥A1α+A2(1-α).
Then, under the assumptions described in Table 1, it follows that either u=0 or v=0 a.e. in RN.
In problem (4.1), we have excluded the cases q1=0 or p2=0. In this final part of the section, we would like to show that these cases can be treated essentially with the tools used in [4] for the inequalities. For simplicity, we consider now problem (4.1) with p=q. Moreover, we require that the functions f,g:[0,∞)×[0,∞)→[0,∞) are continuous and satisfy conditions (f0) and (g0), introduced in Section 1, with q1=0 or p2=0.
Theorem 7.
Consider system (4.1). Let q1=0 or p2=0, and suppose that p1>p-1, q2>p-1 and
(4.48)N(p-1)N-p≥min{p1+q1,p2+q2}.
If (u,v) is a weak solution of (4.1) such that essinfRNu=essinfRNv=0, then u=0 or v=0 a.e. in RN.
Proof.
Suppose that q1=0. If
p1≤p2+q2,
we can proceed as in the proofs of [4, Theorems 3.3 and 3.4]. Indeed, for R large and for all ε>0, we have, by (3.5) and (f0),
∫BRf(u,v)≤cR-p|AR/2|(essinfBRu)p-1≤cR-p|AR/2|(1|AR/2∩Tε|∫AR/2∩Tεup1)p-1p1≤cR-p|AR/2|1-p-1p1(∫AR/2f(u,v))p-1p1≤cRN-p-N(p-1)p1(∫AR/2f~(u,v))p-1p1.
Hence, f(u,v)=0 a.e. in ℝN by (4.48), since p1>p-1. Therefore, (u,v) is trivial.
If p1>p2+q2, we distinguish two cases. If u=0 a.e. in ℝN, then we are done. If u>0 a.e. in ℝN, then, by (3.5), (g0) and Lemma 2, we obtain, for R large enough and for all ε>0,
(4.49)∫BRg(u,v)≤cR-p|AR/2|(essinfAR/2∩Tεu)p2(p-1)q2(1|AR/2∩Tε|∫AR/2∩Tεup2vq2)p-1q2≤cR-p+N(1-p-1q2)+(N-p)p2q2(∫AR/2g(u,v))p-1q2(∫BRf(u,v))p2q2,
that is,
(∫BRf(u,v))p2q2∫BRg(u,v)≤cR-p+N(1-p-1q2)+(N-p)p2q2(∫AR/2g(u,v))p-1q2≤cR-p+N(1-p-1q2)+(N-p)p2q2(∫BRg(u,v))p-1q2.
Now, by (4.48), we have that the exponent of R in the right side is nonpositive. Without loss of generality, we consider only the case for which the exponent of R is equal to 0, i.e., when (4.48) holds with the equality sign.
Thus, since q2>p-1, we can have three cases. Either ∫BRg(u,v)→0 and ∫BRf(u,v)→∞ as R→∞ or vice versa. In both cases, we conclude that g(u,v)=0 or f(u,v)=0 a.e. in ℝN, and we are done. In the third case, ∫BRg(u,v)→const. and ∫BRf(u,v)→const. as R→∞. Hence, in particular, g(u,v)∈L1(ℝN), and so ∫AR/2g(u,v)→0 as R→∞. Hence, we conclude that g(u,v)=0 a.e. in ℝN by (4.49). In the case p2=0, the proof is analogous.
∎
