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BY-NC-ND 3.0 license Open Access Published by De Gruyter April 19, 2017

Quasilinear elliptic equations with critical potentials

Lorenzo D’Ambrosio and Enzo Mitidieri

Abstract

We study Liouville theorems for problems of the form

divL(𝒜(x,u,Lu))+V(x)|u|p-2u=a(x)|u|q-1uon N

in the framework of Carnot groups. Here 𝒜 is a vector-valued function satisfying Carathéodory condition and L denotes an horizontal gradient, V is a given singular potential, a is a measurable scalar function and q>p-1. Particular emphasis is given to the case when V is a Hardy or Gagliardo–Nirenberg potential. The results are new even in the canonical Euclidean setting.

1 Introduction

In this paper we study Liouville theorems for a class of possibly singular quasilinear elliptic equations and inequalities of the form

(1.1)divL(𝒜(x,u,Lu))+V(x)|u|p-2u=a(x)|u|q-1uon N.

Here 𝒜 is a given vector-valued function satisfying Carathéodory conditions (see below for the precise assumptions), p>1, V0 is a singular potential function, a:N is a nonnegative measurable function and q>p-1.

In the last years Liouville theorems for a wide class of weakly elliptic quasilinear problems were studied among others by Farina and Serrin [9] and Pucci and Serrin [18], where sharp interesting results were proved.

Similar problems have been studied among others in the semilinear case in [1], where nonexistence of solutions of the Schrödinger equation

(1.2)Δu+λV(x)u=f(x,u)on N{0}

was proved by reducing the problem to an ODE inequality by applying the spherical mean operator to (1.2) and using some convexity argument. For our problem (1.1), a radial reduction is in general not possible even if the differential operator is linear. So we need to proceed differently.

In order to achieve our goal, the main technique that we use throughout this paper will be a combination of three ingredients: the quasilinear version of Kato inequalities [8] and a slight modification of the test functions method together with an idea introduced in [15].

Roughly speaking the proof or our main results will be organised in two steps. The first is to apply Kato’s inequality to (1.1) reducing the problem to the study of the nonnegative solutions of

divL(𝒜(x,u,Lu))+V(x)|u|p-2ua(x)uq,u0,on N.

The second step will be the application of some a priori estimates proved during the course. These estimates depend on two parameters α and R. By using an idea first introduced in [15, see proof of Theorem 4.1], we can choose α large enough and then by letting R+ we conclude.

We point out that when dealing with equations or inequalities other fine techniques based on Keller–Osserman ideas ([13] and [17], respectively) are available. However, the application of these later ideas need special strong assumptions on the differential operator and on the nonlinearity, see [7, 6].

Our results allow us to consider, as special case in the Euclidean setting the following:

(1.3)Δpu+λ1|x|p|u|p-2u=a(x)|u|q-1uon N.

We have:

Theorem 1.1

Let N>p>1, q>p-1 and let aLloc1(RN) be nonnegative functions such that

a(x)c1|x|θfor |x| large,

with p>θ.

  1. Let uWloc1,p(N)Lloc(N) be a weak solution of

    (1.4)Δpu+λ1|x|p|u|p-2ua(x)|u|q-1uon N.

    Let 0<λ(N-pp)p and let x0 be the unique solution of the equation

    (x-1+p)pλ=x(Q-p)p,x1.

    If

    q(N-θ)(p-1)+x0(p-θ)N-p,

    then u0 a.e. on N. In particular, if u is a solution of

    Δpu+λ1|x|p|u|p-2u=a(x)|u|q-1uon N,

    then u0 a.e. on N.

  2. Let a(x)=|x|-θ. If λ>(N-pp)p, then inequality (1.4) has a positive bounded ground state solution.

  3. Let a(x)=|x|-θ. If 0<λ(N-pp)p and q>qcr, then inequality (1.4) has a positive bounded ground state solution.

Throughout this paper we endow N with a group law such that it becomes a Carnot group. As usual, L stands for the horizontal gradient as described in Appendix A.

Considering the operator appearing in (1.1), we shall require that 𝒜 is 𝐖-p-𝐂 (see Definition 2.1) and

(1.5)a(x)cψk|x|Lθfor |x|L large,
(1.6)C1ψh|x|LνV(x)C2ψp|x|Lpfor |x|L large.

In addition we will assume that a Hardy inequality holds for the potential V, that is there exists λH>0 such that

(1.7)N|Lϕ|pλHNV|ϕ|pfor any ϕ𝒞01(N).

In what follows, for simplicity, we deal with locally bounded solutions in the setting of Carnot groups. We note that if the function 𝒜 is 𝐒-p-𝐂 (see Definition 2.1) and V belongs to LlocQ/p(N) or to the Morrey space MQ/(p-ϵ)(N), then the positive solutions of (1.1) belong to Lloc(N). This is due to the fact that for 𝐒-p-𝐂 operator a weak Harnack inequality holds. See [14] for the Euclidean case and [3] for the Carnot group setting. The validity of (1.7) with V=ψp/||Lp is established among other Hardy inequalities in [5], see Theorem A.3.

This paper is organised as follows. In Section 2 we fix some notations and point out some examples of differential operators for which our results apply when considering problems of type (1.1). In Section 3 we prove the main results of this paper for (1.1) when we assume that the potential function V is of Hardy type (see (1.7) below). This section contains also the main a-priori estimates that play a crucial role in the paper and a short discussion on the sharpness of the results proved in this work.

Finally, in Section 4 we consider some quasilinear problems related to a class of weighted Gagliardo–Nirenberg-type inequality. Namely, we consider problems whose prototype in Euclidean setting reads as

Δpu+λV(x)up-1a(x)uq+μW(x)up-1,u0on N,

where the functions V and W are related by a weighted Gagliardo–Nirenberg-type inequality.

In addition, for easy reference, we recall some basic facts that we use throughout the paper in the Appendix.

2 Notations and definitions

As pointed out in the preceding section, a setting in which our results apply is the framework of Carnot groups. For details see Appendix A. In this paper and || stand respectively for the usual gradient in N and the Euclidean norm. In the Carnot groups framework we denote by ||L a homogeneous norm and we set ψ:=|L||L|. For R>0 we define the ball BR:={xN:|x|L<R} and by AR we denote the annulus AR:=B2RBR.

Let ΩN be an open set and p>1. We define the space

WL,loc1,p(Ω):={uLlocp(Ω):|Lu|Llocp(Ω)}.

As a canonical particular setting, our framework contains the Euclidean space (N,||) with || the Euclidean norm. In this case L= is the isotropic gradient and divL is the divergence operator. Here, Q=N is the dimension of the space. In this case, ψ1 and BR is the Euclidean open ball of radius R centered at the origin. The space WL,loc1,p(Ω) is the usual Sobolev space Wloc1,p(Ω).

The results we state in this paper in this setting can be proved with slight changes for nonlinear problems associated to more degenerate elliptic operators. For instance for operators generated by the vector field L such that L is homogeneous of degree one with respect to an anisotropic dilation δR as specified in [8]. However, to simplify the exposition, we prefer to limit ourselves to study our problems in the Carnot groups settings.

In what follows we shall assume that 𝒜:N××ll is a Carathéodory function, that is for each t and ξl the function 𝒜(,t,ξ) is measurable; and for a.e. xN, 𝒜(x,,) is continuous.

We consider operators L “generated” by 𝒜, that is

L(u)(x)=divL(𝒜(x,u(x),Lu(x))).

Our model cases are the p-Laplacian operator, the mean curvature operator and some related generalizations. See Examples 2.3 below.

Definition 2.1

Let 𝒜:N××ll be a Carathéodory function. The function 𝒜 is called weakly elliptic if it generates a weakly elliptic operator L, i.e.

(WE)𝒜(x,t,ξ)ξ0for each xN,t,ξl,𝒜(x,0,ξ)=0or𝒜(x,t,0)=0.

Let p1. The function 𝒜 is called 𝐖-p-𝐂 (weakly-p-coercive) if 𝒜 is (WE) and it generates a weakly-p-coercive operator L, i.e. if there exists a constant k2>0 such that

(\mbox{${\mathbf{W}}$-${p}$-${\mathbf{C}}$})(𝒜(x,t,ξ)ξ)p-1k2|𝒜(x,t,ξ)|pfor each xN,t,ξl.

Let p>1. The function 𝒜 is called S-p-C (strongly-p-coercive) (see [16]) if there exist constants k1,k2>0 such that

(\mbox{${\mathbf{S}}$-${p}$-${\mathbf{C}}$})(𝒜(x,t,ξ)ξ)k1|ξ|pk2|𝒜(x,t,ξ)|pfor each xN,t,ξl.

Definition 2.2

Let ΩN be an open set and let f:Ω××l be a Carathéodory function. Let p1. We say that uWL,loc1,p(Ω) is a weak solution of

divL(𝒜(x,u,Lu))f(x,u,Lu)on Ω

if 𝒜(,u,u)Llocp(Ω), f(,u,Lu)Lloc1(Ω), and for any nonnegative φ𝒞01(Ω) we have

-Ω𝒜(x,u,Lu)LφΩf(x,u,Lu)φ.
Examples 2.3

Consider the following examples.

  1. Let p>1. The p-Laplacian operator defined on suitable functions u by

    Δp,Lu=divL(|Lu|p-2Lu)

    is an operator generated by

    𝒜(x,t,ξ):=|ξ|p-2ξ

    which is 𝐒-p-𝐂.

  2. The mean curvature operator in nonparametric form

    Tu:=divL(Lu1+|Lu|2)

    is generated by

    𝒜(x,t,ξ):=ξ1+|ξ|2.

    In this case 𝒜 is 𝐖-p-𝐂 with 1p2 and of mean curvature type but it is not S-2-C.

  3. Let p>1 and define

    Lu:=i=1Ni(|iu|p-2iu).

    The operator L is 𝐒-p-𝐂.

  4. The operator defined by

    div(|u|u|u|+|u|)

    is 𝐖-2-𝐂.

See [8] for further examples.

3 Quasilinear equations related to Hardy inequality

In this section we prove the main results of this paper for (1.1). Here we assume that the potential function V is of Hardy type (see (1.7) below). Our first main result is the following.

Theorem 3.1

Let Q>p>1. Let A be S-p-C and let a,VLloc1(RN) be nonnegative functions satisfying (1.5) and (1.6) with pν>θ and phk0. Assume that (1.7) holds and let λ be such that 0<λλHk1 where λH is the best constant in (1.7) and k1 is the constant structure appearing in the definition of S-p-C (see Definition 2.1). Let uWL,loc1,p(RN)Lloc(RN) be a weak solution of

(3.1)divL(𝒜(x,u,Lu))+λV(x)|u|p-2ua(x)|u|q-1uon N.

Then au0 a.e. on RN. Moreover, if λ<λHk1, then u0 a.e. on RN. In particular, if A is odd[1] and u solves the equation

(3.2)divL(𝒜(x,u,Lu))+λV(x)|u|p-2u=a(x)|u|q-1uon N,

then au=0 a.e. on RN. In addition if λ<λHk1, then u=0 a.e. on RN.

The proof is one of the consequences of the reduction principles stated in [8]. Indeed, in view of the reduction principles it follows that it suffices to study nonnegative solutions of the inequality related to (3.2). Notice that the case λ0 has been considered in [8]. Hence, in what follows we shall focus our attention to the case λ>0.

Theorem 3.2

Let Q>p>1. Let A be S-p-C and let a,VLloc1(RN) be nonnegative functions satisfying (1.5) and (1.6) with pν>θ and phk0. Assume that (1.7) holds and let λ be such that 0<λλHk1 where λH is the best constant in (1.7) and k1 is the constant structure appearing in the definition of S-p-C. Let uWL,loc1,p(RN)Lloc(RN) be a nonnegative weak solution of

(3.3)divL(𝒜(x,u,Lu))+λV(x)up-1a(x)uq,u0on N.

If

(3.4)p-1<q(Q-θ)(p-1)+x0(ν-θ)Q-ν,

where x01 is the unique solution of the equation

(x-1+p)pλ=xλHk1pp,x1,

then au0 a.e. on RN. Moreover, if a>0 or if λ<λHk1, then u0 a.e. on RN.

If L=Δp,G=divL(|L|p-2L) and V is the related Hardy potential, we obtain the following.

Corollary 3.3

Let Q>p>1 and let aLloc1(RN) be nonnegative functions satisfying (1.5) with p>θ, pk0 and with a homogeneous norm ||L such that

-Δp,G|x|Lp-Qp-1=cδ0

and ψ:=|L||L|.[2] Suppose that uWL,loc1,p(RN)Lloc(RN) is a weak solution of

Δp,Gu+λψp|x|Lp|u|p-2ua(x)|u|q-1uon N.

If 0<λ(Q-pp)p and x0 is the unique solution of the equation

(x-1+p)pλ=x(Q-p)p,x1,

then u0 a.e. on RN, provided

(3.5)p-1<q(Q-θ)(p-1)+x0(p-θ)Q-p.

In particular, if u is a solution of

(3.6)Δp,Gu+λψp|x|Lp|u|p-2u=a(x)|u|q-1uon N,

then u0 a.e. on RN.

Remark 3.4

(i) If p=2, we know an explicit formula of x0, namely

x0=Q-22λ[Q-2+(Q-2)2-4λ]-1.

(ii) If aconstant, then (3.4) becomes

p-1<q<qcr:=Q(p-1)+x0pQ-p

and for λ<λH, qcr>qs:=Q(p-1)+pQ-p. Notice that qs is the Sobolev exponent associated to the operator Δp,G. For the case of the standard Laplacian operator in the Euclidean space and the corresponding problem (3.6) see [1].

The above results can be generalised as follows.

Theorem 3.5

Let Q>p>1, let A be S-p-C, let q>p-1 and let a,VLloc1(RN) be nonnegative functions. Assume that there exist R0,M>0, α1, s1 such that

(3.7)R-pq+αq-p+1ARa-p+α-1q-p+1ψpq+αq-p+1<Mfor R>R0

and

(3.8)RQ(s-1)R0<|x|L<RVsq+αq-p+1a-sp+α-1q-p+1<Mfor R>R0.

Suppose that (1.7) holds with V satisfying

V(x)C2ψp|x|Lpfor |x|L large.

Let uWL,loc1,p(RN)Lloc(RN) be a weak solution of (3.3). If λk1λHα(pp+α-1)p, then

u(x)=0for a.e. |x|>R0  𝑎𝑛𝑑  a(x)u(x)=0for a.e. |x|<R0.

In particular, u0 a.e. on RN provided a is a.e. positive or α>1.

For the proof, we need the following lemma. Notice that we assume that 𝒜 satisfies a 𝐖-p-𝐂 condition only.

Lemma 3.6

Let A be W-p-C and let a,VLloc1(RN) be nonnegative functions. Assume that for q>p-1 there exist R0,M>0, α1, s1 such that (3.7) and (3.8) hold. If uWL,loc1,p(RN)Lloc(RN) is a weak solution of (3.3), then

(3.9)Nauq+α<+,
(3.10)N𝒜(x,uLu)Luuα-1<+,
(3.11)NVup+α-1<+

and

(3.12)Nauq+α+αN𝒜(x,u,Lu)Luuα-1λNVuα+p-1.

Proof.

Since uLloc(N), we can apply [8, Lemma 5.1 and Remark 5.6]. Using φ=uαϕ as test function[3] in Definition 2.2, we obtain

Nauq+αϕ+αN𝒜(x,u,Lu)Luuα-1ϕ-N𝒜(x,u,Lu)Lϕuα+λNVuα+p-1ϕ.

Since 𝒜 is 𝐖-p-𝐂, by the Hölder and Young inequality, it follows that

(3.13)Nauq+αϕ+c1αN𝒜(x,u,Lu)Luuα-1ϕc2α1-pNuα+p-1|Lϕ|pϕp-1+λNVuα+p-1ϕ,

where

c1:=1-ϵppk2>0,c2:=pppϵp

and ϵ>0 is sufficiently small. By Hölder’s inequality with exponent χ:=q+αα+p-1, from the right-hand side of (3.13) we obtain

Ωauq+αϕ+c1αΩ𝒜(x,u,Lu)Luuα-1ϕ
(3.14)c2α1-p(Sauq+αϕ)1/χ(S|Lϕ|pχϕpχ-1a-α+p-1q-p+1)1/χ+λΩVuα+p-1ϕ.

Thus by Young’s inequalities, we get

(3.15)c3Nauq+αϕ+c1αN𝒜Luuα-1ϕc4N|Lϕ|pχϕpχ-1a-χ/χ+λNVup+α-1ϕ,

where

c3:=1-ϵχc2α1-pχ,c4:=c2α1-pϵχχ

and ϵ>0.

Now we shall estimate the right-hand side of (3.15). We have

NVup+α-1ϕ=BR0Vup+α-1ϕ+|x|L>R0Vup+α-1ϕ
(3.16)C(α,u,V,R0)+(|x|L>R0auq+αϕ)1/χ(|x|L>R0Vχa-χ/χϕ)1/χ,

which by Young’s inequality yields

c5Nauq+αϕ+c1αN𝒜Luuα-1ϕc4AR|Lϕ|pχϕpχ-1a-χ/χ+λC(α,u,V,R0)+c6|x|L>R0Vχa-χ/χϕ,

where

c5:=c3-λϵχχ,c6:=λϵχχ

and ϵ>0.

Let ϕ0𝒞01() be a standard cut off function. Setting ϕ(x):=ϕ0(|δ1/Rx|L), for R>R0, by (3.7), we have

(3.17)AR|Lϕ|pχϕpχ-1a-χ/χc(ϕ0)R-pχARa-χ/χψpχ<M.

If s=1, then the hypothesis (3.8) assures that |x|L>R0Vχa-χ/χϕ is uniformly bounded by the constant M. If s>1, then an application of Hölder’s inequality with exponent s yields

|x|L>R0Vχa-χ/χϕR0<|x|L<2RVχa-χ/χ|B2R|1/s(R0<|x|L<2RVsχa-sχ/χ)1/s<Ms.

Therefore, the right-hand side of (3.15) is uniformly bounded with respect to R, that is

c3BRauq+α+c1αBR𝒜Luuα-1C.

Letting R+, we obtain auq+α,𝒜(x,u,Lu)Luuα-1L1(N). In other words (3.9) and (3.10) hold. Thus (3.16) implies (3.11).

Next, from (3.9) we have

ARauq+α0as R+,

which, in turn by (3.17) and (3.14), implies

Nauq+α+α(1-ϵppk2)N𝒜(x,u,Lu)Luuα-1λNVuα+p-1.

Letting ϵ0 in the above inequality, (3.12) follows. ∎

Lemma 3.7

Let the hypotheses of Lemma 3.6 be fulfilled. Set

WA,α1,p:={vWL,loc1,p(N):N𝒜(x,v,Lv)Lvvα-1<+,NVvp+α-1<+}.

Suppose that the following Hardy-type inequality holds:

(3.18)N𝒜(x,v,Lv)Lvvα-1λH,αNVvp+α-1for any vWA,α1,pLloc(N),v0.

Let uWL,loc1,p(RN)Lloc(RN) be a weak solution of (3.3). If λαλH,α, then

auq+α0a.e. on N.

Proof.

From (3.12) and (3.18) we have

Nauq+α+(α-λλH,α)N𝒜(x,u,Lu)Luuα-10.

Since each addendum in the above inequality is nonnegative the claim follows. ∎

In order to prove Theorem 3.5 we need the following generalization of the weak maximum principle. This results is essentially based on the validity of (1.7). Indeed, we are in a position to apply some of the results of [8] obtaining a weak maximum principle for (3.19). For sake of completeness we state it here.

Theorem 3.8

Theorem 3.8 (Generalized weak maximum principle)

The following statements hold.

  1. Assume that (1.7) holds and 𝒜 is S-p-C. Let uWL1,p(BR) be a weak solution of

    divL(𝒜(x,u,Lu))+λV(x)|u|p-2u0,u0on BR0

    with λ<λHk1. Then u0 a.e. on BR.

  2. Let Q>p>1 and let S be a homogeneous norm such that Δp,LSp-Qp-1=cδ0 on 𝔾. Let uWL1,p(BR) be a weak solution of

    Δp,Lu+λ|LS|pSp|u|p-2u0on BR,u0on BR0,

    with λ(Q-pp)p. Then u0 a.e. on BR.

Proof of Theorem 3.5.

First of all we observe that by density argument the Hardy inequality (1.7) holds for functions in DL1,p(N) (see Appendix A.2 for definition).

Set β:=1+α-1p. We claim that uβDL1,p(N). Indeed, from (3.10) we have

N|Luβ|p=βpN|Lu|p|u|α-1<+.

From (3.11) it follows that

NV|uβ|p=NV|u|p+α-1<+.

Therefore by Theorem A.1 we deduce that uβDL1,p(N).

Since 𝒜 is 𝐒-p-𝐂, from (1.7) we obtain

N𝒜(x,u,Lu)Luuα-1k1N|Lu|puα-1=k1βpN|Luβ|pk1βpλHNV|u|p+α-1.

By the above Lemma 3.7, we deduce that if

λαk1βpλH=k1λHα(pp+α-1)p,

then auq+α0.

Now if a is positive, it is clear that u0. Otherwise (3.7) implies a(x)0 for a.e. |x|L>R0. Therefore u0 on NBR0. Hence,

(3.19)divL(𝒜(x,u,Lu))+λV(x)up-10,u0on BR0,u=0on BR0.

Therefore, since α>1, we have

λk1λHα(pp+α-1)p<k1λH.

Indeed, it is simple to verify that the function

f(α):=α(pp+α-1)p

has a negative first derivative

f(α)=(pp+α-1)p(p-1)(1-α)p+α-1

for α>1 and f(1)=1. Hence λk1λHf(α)<k1λHf(1). Therefore, from i) of Theorem 3.8 it follows that u0 provided λ<k1λH. ∎

Proof of Theorem 3.2.

From (1.5) we have

R-pq+αq-p+1ARa-p+α-1q-p+1ψpq+αq-p+1R-pq+αq-p+1+θp+α-1q-p+1cARψpq+αq-p+1-kp+α-1q-p+1CRΘ1

with

Θ1:=Q+θ(α+p-1)-p(q+α)q-p+1.

Condition (3.7) is fulfilled provided Θ10, that is if

q(Q-θ)(p-1)+α(p-θ)Q-p.

On the other hand, if s>1 we have

RQ(s-1)R0<|x|L<RVsq+αq-p+1a-sp+α-1q-p+1CRQ(s-1)R0<|x|L<R|x|L-sνq+αq-p+1+sθp+α-1q-p+1ψshq+αq-p+1-skp+α-1q-p+1
CRQ(s-1)R0RρQ-1-sνq+αq-p+1+sθp+α-1q-p+1.

Condition (3.8) holds provided Qs-sνq+αq-p+1+sθp+α-1q-p+10, that is if

(3.20)q(Q-θ)(p-1)+α(ν-θ)Q-ν.

Therefore, taking into account that pν>θ, inequalities (3.7) and (3.8) follow provided (3.20) holds.

The conclusion of Theorem 3.5 holds if for some α1 we have αk1λ(α-1+p)pppλH, or equivalently,

f(α):=(α-1+p)pαppλHk1λ.

Notice that the function f for α1 is increasing, f(1)=pp and f(+)=+. Therefore f(α)ppλHk1λ provided α[1,x0], where x01 is the only solution of

f(α)=ppλHk1λ.

If λ<λHk1, then x0>1. Hence α>1. From Theorem 3.5 we complete the proof. ∎

Proof of Corollary 3.3.

By Kato inequality (or reduction principle [8]) it is enough to consider nonnegative solutions. In order to apply Theorem 3.2, we need to check that inequality (1.7) holds for V=ψp|x|Lp and that the best constant therein is given by

λH=(Q-pp)p.

This fact is proved in [5] (see Theorem A.3 below). Therefore, from Theorem 3.2, it follows that au0 a.e. on N. Further, if λ<λH, we are done. Otherwise assume λ=λH. Since (1.5) holds, we have that u(x)=0 for a.e. |x|L>R0 and u is a solution of

Δp,Gu+λHψp|x|Lpup-10,u0on BR0,u=0on BR0.

By (ii) of Theorem 3.8 we complete the proof. ∎

3.1 Sharpness of the results

In this subsection we point out that many of our assumptions (the bounds on q and λ, N>p, νp and θ<p) are necessary in order to have a Liouville-type result. Namely, we prove that if an assumption is not satisfied, then there are nontrivial solution, that is we show the sharpness of our results.

We claim that the exponents provided in Theorems 3.2, 3.1 Corollary 3.3 are sharp. To this end, we deal with the inequality

(3.21)Δpu+λup-1|x|puq|x|θon N.

We begin proving that:

Claim 1

If λ>λH, then (3.21) has a bounded positive solution for any q>p-1.

Claim 2

If λλH and q>qcr:=(N-θ)(p-1)+x0(p-θ)N-p, then (3.21) has a bounded positive solution.

Obviously, a positive function u solves inequality (3.21) if we show that

Δpu+λup-1|x|puq|x|θ1.

Indeed, let q>p-1, N>p>θ. We define

u(x):=c(1+|x|β)-p-θβ(q-p+1)

for β>0 and c>0 that we choose later. Clearly, uWloc1,p(N)𝒞(N), it is bounded and vanishes at infinity. Now, by computation, for r=|x|, we have

Δpu+λup-1r-puqr-θ=(p-θq-p+1)pcp-1-q(1+rβ)p-θβrp-θR(rβ1+rβ)
(p-θq-p+1)pcp-1-qR(rβ1+rβ),

where R(t):=a2tp+a1tp-1+a0, and

a0:=(q-p+1p-θ)pλ>0,
a1:=-(q-p+1)β(p-1)+N-pp-θ<0,
a2:=(p-1)β(q-p+1)+p-θp-θ>0.

Therefore setting m:=min[0,1]R(t), we have

Δpu+λup-1r-puqr-θ(p-θq-p+1)pcp-1-qm.

If m>0, then u is a solution of (3.21) for c>0 small enough. Therefore, our goal is to choose β>0 so that m=m(β)>0.

Now, observing that R(0)=a0>0 and R(+)=+, it follows that the function R has a minimum. Since R has only one critical point

t0=(q-p+1)(β(p-1)+N-p)p(β(q-p+1)+p-θ),

we have

m(β)=R(t0)=(q-p+1p)p[(pp-θ)pλ-1p-θ(β(p-1)+N-p)p(β(q-p+1)+p-θ)p-1],

Proof of Claim 1. Let λ>λH=(N-pp)p. It is easy to check that

m(0)=(q-p+1p)p[(pp-θ)pλ-(N-p)p(p-θ)p]=(q-p+1p-θ)p[λ-λH]>0.

Therefore, by continuity, for β>0 small enough we have that m(β)>0, and hence we obtain the claim.

Proof of Claim 2. For the sake of simplicity we set β:=γ(N-p). Let q>qcr. We write q as

q:=(N-θ)(p-1)+x0(p-θ)+ϵ(p-θ)(p-1)N-p>qcr.

By computation we have

m(β)=m(γ(N-p))=((q-p+1)(N-p)p(p-θ))p[(pN-p)pλ-(γ(p-1)+1)p(γ(x0+(ϵ+1)(p-1)+1p-1)]
=:((q-p+1)(N-p)p(p-θ))pg(γ).

We reach our goal by choosing γ>0 such that g(γ)>0. It is easy to see that the function g has a maximum at

γ0:=ϵ(p-1)+x0-1p+ϵ(p-1)+x0-1.

From the definition of x0, that is λ:=x0(N-p)p(x0-1+p)p, we have

g(γ0)=(px0-1+p)px0-(pp+ϵ(p-1)+x0-1)p(x0+ϵ(p-1)).

Therefore, g(γ0)>0 if and only if

(1+ϵ(p-1)p+x0-1)p>1+ϵ(p-1)x0.

Next by using Bernoulli’s inequality and the fact that x01, we obtain

(1+ϵ(p-1)p+x0-1)p>1+pϵ(p-1)p+x0-11+ϵ(p-1)x0.

This completes the proof.

Claim 3

If p>N, we can construct a positive solution of (3.1).

We first observe that in this case a Hardy inequality as (1.7) cannot hold. Moreover, the potentials V and a cannot be very singular, that is for x close to the origin, V(x) cannot behave as |x|p. Having this in mind, let us consider

Δpu+λV(x)up-1a(x)uq,u0on N.

We claim that the above inequality admits a positive solution for potentials V satisfying (1.6) with ν=p and any θ<p, λ>0, q>p-1. Let

V(x):=1(1+|x|β)p/β,a(x):=1(1+|x|β)θ/β

with β:=p-Np-1. Indeed, by a direct computation it is easy to see that the function

u(x):=c(1+|x|β)αwith α:=p-θ(q-p+1)β

satisfies

Δpu+λup-1Vauq=αpβpcp-1-q[λ(q-p+1p-θ)p+a2tp-pβ]
αpβpcp-1-qλ(q-p+1p-θ)p
=cp-1-qλ

where we have set

t:=|x|β1+|x|βanda2:=(p-1)(p+β(q-p+1))p-θ=(p-1)(1+1α).

By choosing a suitable c>0 we complete the proof.

As we can see we have not considered the case N=p. This particular situation will be studied elsewhere.

Claim 4

We observe that the assumption θ<p in Theorem 3.2 is necessary in order to prove a nonexistence result.

Indeed, if Va, inequality (3.3) has positive nonconstant solutions. To see this, consider the special case divL(𝒜(x,,L))=Δp. Let v be a nonnegative bounded solution of Δpv0 (see [8, Remark 11.8]). By choosing a suitable c>0 and ϵ>0, it follows that the function u:=ϵ+cv is a bounded positive solution of (3.3).

Claim 5

Assumption νp is also necessary in order to prove a nonexistence result. Dealing with a potential V(x)=1|x|ν with ν<p, our original inequality (3.3) has solutions.

Indeed, the inequality

Δu+λu1|x|uq,u0on 3,

possesses a positive 𝒞(3) ground state solution for any λ>0 and q>1. Indeed, defining

u(x):=ce-λ264|x|2,withcq-1=3λ232,

we have

Δu+λu1|x|=uqλ1024cq-1e(q-1)λ264|x|21024-96λ|x|+λ3|x|3|x|
uqλ1024cq-11024-96λ|x|+λ3|x|3|x|
uqλ1024cq-196λuq.

4 Quasilinear equations related to weighted Gagliardo–Nirenberg-type inequality

Let uWL,loc1,p(N)Lloc(N) be a weak solution of

(4.1)divL(𝒜(x,u,Lu))+λV(x)up-1a(x)uq+μW(x)up-1,u0on N.

In our main result of this section we require that the following weighted Gagliardo–Nirenberg inequality holds:

(4.2)CV,WNV(x)|u|p(N|Lu|p)1/γ(NW(x)|u|p)1/γ

for any function uDL1,p(N) such that NW(x)|u|p<+. Notice that if γ=1, then (4.2) coincides with (1.7).

A concrete example of (4.2) is the classical weighted Gagliardo–Nirenberg inequality, that is choosing V(x):=|x|-1 inequality (4.2) holds with W1 and γ=2. In this case the best constant is given by

CV,W=N-12.

For potentials V and W satisfying (4.2) we have the following result, which is the analogue of Theorem 3.5.

Theorem 4.1

Let Q>p>1, let A be S-p-C, q>p-1 and let a,V,WLloc1(RN) be nonnegative functions. Assume that there exist R0,M>0, α1, s1 such that (3.7) and (3.8) holds. Suppose that (4.2) holds with V satisfying

(4.3)V(x)C2ψp|x|Lpfor |x|L large.

Let uWL,loc1,p(RN)Lloc(RN) be a weak solution of (4.1). If λ>0 and

(4.4)μ(λγCV,W)γ(γ-1)((p+α-1)pppαk1)γ-1,

then

auq+α0a.e. on N.

Proof.

If u is a solution of (4.1), then u solves (3.3) as well. Therefore Lemma 3.6 applies and it is easy to check that under the same hypotheses we also have

NW(x)up+α-1<+,

and

(4.5)Nauq+α+μNWuα+p-1+αN𝒜(x,u,Lu)Luuα-1λNVuα+p-1.

Again we argue as in the proof of Theorem 3.5. Setting β:=1+α-1p, using (4.3) and (3.10) and arguing as in the proof of Theorem 3.5, we obtain uβDL1,p(N). Therefore, (4.2) applies for v:=uβ=u1p(α+p-1) and from (4.5) and the fact that 𝒜 is 𝐒-p-𝐂, we obtain

Nauq+α+μNWvp+αk1N|Lu|puα-1λCV,W(N|Lv|p)1/γ(NW(x)|v|p)1/γ.

Now by using Young inequality with ϵ>0,

(4.6)Nauq+α+(μ-λCV,Wγϵγ)NWvp+(αk1βp-λϵγCV,Wγ)N|Lv|p0.

Choosing ϵ so that ϵ-γ=λβpαk1CV,Wγ, from (4.6) and our assumption on μ, we have the thesis. ∎

The following result is the analogue of Theorem 3.2.

Theorem 4.2

Let Q>p>1. Let A be S-p-C and let a,V,WLloc1(RN) be nonnegative functions satisfying (1.5) and (1.6) with pν>θ and phk0. Assume that (4.2) holds. Let λ>0 and

μμ1:=(λCV,Wγk1)γ(γ-1)k1,

where CV,W is the best constant in (4.2) and k1 is the constant structure appearing in the definition of S-p-C (see Definition 2.1). Let uWL,loc1,p(RN)Lloc(RN) be a weak solution of (4.1). If

p-1<q(Q-θ)(p-1)+x0(ν-θ)Q-ν,

where x01 is the unique solution of the equation

(x-1+p)px=(μμ1)γ-1pp,x1,

then au0 a.e. on RN.

Proof.

In order to apply Theorem 4.1 we argue as in proof of Theorem 3.2. Hence, conditions (3.7) and (3.8) are fulfilled if (3.20) holds for some α1. On the other hand the hypotheses we made on μ, x0 and q assure that condition (4.4) holds. An application of Theorem 4.1 completes the proof. ∎


Dedicated to our dearest Professor Patrizia Pucci on the occasion of her birthday, with great esteem and friendship


Funding statement: The authors are supported by the MIUR Bando PRIN 2015 2015KB9WPT_001. Enzo Mitidieri acknowledges the support from FRA 2015: Equazioni differenziali: teoria qualitativa e computazionale, Università degli Studi di Trieste.

A Carnot groups

A.1 Basic facts

Here, we quote some facts on Carnot groups and refer the interested reader to [10, 11, 2, 4] for a more detailed information on this structures.

A Carnot group is a connected, simply connected, nilpotent Lie group 𝔾 of dimension N with graded Lie algebra 𝒢=V1Vr such that [V1,Vi]=Vi+1 for i=1,,r-1 and [V1,Vr]=0. Such an integer r is called the step of the group. We set l=n1=dimV1, n2=dimV2,,nr=dimVr. A Carnot group 𝔾 of dimension N can be identified, up to an isomorphism, with the structure of a homogeneous Carnot group(N,,δR) defined as follows: We identify 𝔾 with N endowed with a Lie group law . We consider N split in r subspaces N=n1×n2××nr with n1+n2++nr=N and ξ=(ξ(1),,ξ(r)) with ξ(i)ni. We shall assume that for any R>0 the dilation δR(ξ)=(Rξ(1),R2ξ(2),,Rrξ(r)) is a Lie group automorphism. The Lie algebra of left-invariant vector fields on (N,) is 𝒢. For i=1,,n1=l let Zi be the unique vector field in 𝒢 that coincides with /ξi(1) at the origin. We require that the Lie algebra generated by Z1,,Zl is the whole 𝒢.

We denote with the vector field :=(Z1,,Zl)T and we call it the canonical horizontal vector field in 𝔾. The canonical sub-Laplacian on 𝔾 is the second order differential operator defined by i=1lZi2.

Along the paper we choose X1,,Xl stands for a basis of span{Z1,,Zl}. We denote with L the vector field L:=(X1,,Xl)T and we call it the a horizontal vector field in 𝔾. Moreover, the vector fields X1,,Xl are homogeneous of degree 1 with respect to δR. In this case

Q=i=1rini=i=1ridimVi

is called the homogeneous dimension of 𝔾.

For i=1,,l, Xi* stands for the formal adjoint of Xi. Hence, we shall use the notation

divL(h)=-i=1lXi*hi

for any vector field h=(h1,,hl)T𝒞1(Ω,l).

A sub-Laplacian on 𝔾 is the second order differential operator defined by ΔG=i=1lXi2 and for p>1 the p-sub-Laplacian operator is given by

Δp,Gu:=i=1l-Xi*(|Lu|p-2Xiu).

Since X1,,Xl generate the whole graded Lie algebra 𝒢, the sub-Laplacian ΔG satisfies the Hörmander hypoellipticity condition.

A nonnegative continuous function S:N+ is called a homogeneous norm on 𝔾 if S(ξ-1)=S(ξ), S(ξ)=0 if and only if ξ=0, and it is homogeneous of degree 1 with respect to δR (i.e. S(δR(ξ))=RS(ξ)). A homogeneous norm S defines on 𝔾 a pseudo-distance defined as d(ξ,η):=S(ξ-1η), which in general is not a distance. If S and S~ are two homogeneous norms, then they are equivalent, that is there exists a constant C>0 such that C-1S(ξ)S~(ξ)CS(ξ). Let S be a homogeneous norm, then there exists a constant C>0 such that C-1|ξ|S(ξ)C|ξ|1/r, for S(ξ)1. Examples of homogeneous norms are Sδ() defined as

Sδ(x):=(i=1N(xim)dδi)1md,

where d:=δ1δ2δN and m is the lowest even integer such that mmax{δ1d,,δNd}, or as

S(ξ):=(i=1r|ξi|2r!i)12r!.

Notice that if S is a homogeneous norm differentiable a.e., then |LS| is homogeneous of degree 0 with respect to δR; hence |LS| is bounded.

Special examples of Carnot groups are the Euclidean spaces Q. Moreover, if Q3, then any Carnot group is the ordinary Euclidean space Q.

The simplest nontrivial example of a Carnot group is the Heisenberg group 1=3. For an integer n1, the Heisenberg group n is defined as follows: let ξ=(ξ(1),ξ(2)) with ξ(1):=(x1,,xn,y1,,yn) and ξ(2):=t. We endow 2n+1 with the group law ξ^ξ~:=(x^+x~,y^+y~,t^+t~+2i=1n(x~iy^i-x^iy~i)). We consider the vector fields

Xi:=xi+2yit,Yi:=yi-2xitfor i=1,,n,

and the associated Heisenberg gradient H:=(X1,,Xn,Y1,,Yn)T. The Kohn Laplacian ΔH is then the operator defined by

ΔH:=i=1nXi2+Yi2.

The family of dilations is given by δR(ξ):=(Rx,Ry,R2t) with homogeneous dimension Q=2n+2. In n a canonical homogeneous norm is defined as

|ξ|H:=((i=1nxi2+yi2)2+t2)14.

A.2 Mollifiers

On a Carnot Group there is a “good” notion of mollifier. Let 𝔾 be a homogeneous Carnot group on N and let S be a fixed homogeneous norm on 𝔾. For every x𝔾 and every r>0, the set

BS(x,r):={y𝔾:S(x-1y)<r}

is called the S-ball with center at x and radius r. For a fixed point x𝔾 and a set A𝔾, the number

distS(x,A):=inf{S(x-1a):aA}.

is called S-distance of x from A. Let Ω𝔾 and ϵ>0 we define

ΩS,ϵ:={xΩ:distS(x,Ω)>ϵ}.

In order to avoid cumbersome notations we shall omit the norm S in the above symbols.

Let m𝒞0(𝔾), m0, be given such that

supp(m)BS(0,1)andm=1.

For any η>0 we set mη:=η-Qm(δ1/η(x)). The family (mη)η will be called a family of mollifiers.

Let Ω𝔾 be an open set and let uLloc1(Ω). For any xΩη we define

uη:=(u𝔾mη)(x):=B(x,η)u(y)mη(xy-1)𝑑y=B(0,η)u(y-1x)mη(y)𝑑y

calling uη a mollified of u related to the homogeneous norm S.

It is easy that check that if uLloc1(Ω), then

uηuas η0in Lloc1(Ω).

See [2].

A.3 A characterization of the space DL1,p(N)

As in the Euclidean case, for 1<p<Q the space DL1,p(N) is defined as the closure of 𝒞01(N) with respect to the norm |Lu|p=(|Lu|p)1/p. Let uLloc1(N). It is clear that the assumption that the distribution |Lu| belongs to Lp(N) does not guarantee that uDL1,p(N).

We have the following.

Theorem A.1

Let 1<p<Q and let ||L be a homogeneous norm. Let uLloc1(RN) be such that |Lu|Lp(RN). If there exists R0>0 such that

(A.1)|x|L>R0ψp|x|Lp|u|p<,

then uDL1,p(RN).

Proof.

Let ϕ0: be a usual cut off function and let ϕR:=ϕ0(|δ1/Rx|L). We have

|L(ϕRu)-Lu|p=|uLϕR+ϕRLu-Lu|p|uLϕR|p+|ϕRLu-Lu|p.

Now by Lebesgue dominated convergence theorem it follows that

|ϕRLu-Lu|pp=N(1-ϕR)p|Lu|p0.

On the other hand,

|uLϕR|pp=R<|x|L<2R|u|p|ϕ0|p(|x|LR)ψpRpR<|x|L<2R|u|pC(ϕ0,p)ψp|x|Lp

which, by (A.1), implies

limR+|uLϕR|pp=0.

Therefore for any R>0, ϕRuL1(N) has compact support and ϕRuu in DL1,p(N).

Finally, if u is smooth we are done. Otherwise, by a standard mollification argument we complete the proof. ∎

A.4 The Kato inequality for quasilinear weakly elliptic operators

In this subsection we shall recall that suitable versions of the Kato inequality [12] proved in [8] hold for some quasilinear weakly elliptic operators.

Let Ω be an open set contained in N, let p1 and let uWloc1,p(Ω). As usual, we denote by sign, sign+ and u+ the functions defined as follows:

sign(t):={1if t>0,0if t=0,-1if t<0,sign+(t):={1if t>0,0if t0,u+:=sign+(u)u.
Theorem A.2

Theorem A.2 (Kato inequality: The quasilinear case)

Let A be (WE). Let fLloc1(Ω) and let uWL,loc1,p(Ω) be a weak solution of

divL(𝒜(x,u,Lu))fon Ω.

Then u+ is a weak solution of

divL(𝒜(x,u+,Lu+))sign+ufon Ω.

Moreover, if

divL(𝒜(x,u,Lu))=fon Ω,

and A is odd, i.e.

𝒜(x,-t,-ξ)=-𝒜(x,t,ξ),

then |u| satisfies

divL(𝒜(x,|u|,L|u|))signufon Ω.

A.5 Hardy inequality in the Carnot framework

The following result has been proved in [5].

Theorem A.3

Let p>1. Let d:ΩR be a nonnegative nonconstant measurable function and let αR, α0, such that

d-p|Ld|p,d(α-1)(p-1)|Ld|p-1Lloc1(Ω).

If -ΔL,p(dα)0 in the weak sense, then for every uC01(Ω) we have

(|α|(p-1)p)pΩ|u|pdp|Ld|p𝑑xΩ|Lu|p𝑑x.

In particular, if S is a homogeneous norm such that[4]-ΔL,pSp-Qp-1=cδ0 on G with Q>p>1, then[5]

(Q-pp)p𝔾|u|pSp|LS|p𝑑x𝔾|Lu|p𝑑x,uDL1,p(𝔾),

where the constant (Q-pp)p is sharp and it is not achieved.

References

[1] Benguria R. D., Lorca S. and Yarur C. S., Nonexistence results for solutions of semilinear elliptic equations, Duke Math. J. 74 (1994), 615–634. 10.1215/S0012-7094-94-07422-XSearch in Google Scholar

[2] Bonfiglioli A., Lanconelli E. and Uguzzoni F., Stratified Lie groups and Potential Theory for Their Sub-Laplacians, Springer Monogr. Math., Springer, Berlin, 2007. Search in Google Scholar

[3] Capogna L., Danielli D. and Garofalo N., An embedding theorem and the Harnack inequality for nonlinear subelliptic equations, Comm. Partial Differential Equations 18 (1993), 1765–1794. 10.1080/03605309308820992Search in Google Scholar

[4] Coulhon T., Holopainen I. and Saloff-Coste L., Harnack inequality and hyperbolicity for subelliptic p-Laplacians with applications to Picard type theorems, Geom. Funct. Anal. 11 (2001), 1139–1191. 10.1007/s00039-001-8227-3Search in Google Scholar

[5] D’Ambrosio L., Hardy-type inequalities related to degenerate elliptic differential operators, Ann. Sc. Norm. Super. Pisa Cl. Sci. (5) 4 (2005), 451–486. 10.2422/2036-2145.2005.3.04Search in Google Scholar

[6] D’Ambrosio L. and Mitidieri E., A priori estimates, positivity results, and nonexistence theorems for quasilinear degenerate elliptic inequalities, Adv. Math. 224 (2010), 967–1020. 10.1016/j.aim.2009.12.017Search in Google Scholar

[7] D’Ambrosio L. and Mitidieri E., Nonnegative solutions of some quasilinear elliptic inequalities and applications, Sb. Math. 201 (2010), 856–871. 10.1070/SM2010v201n06ABEH004094Search in Google Scholar

[8] D’Ambrosio L. and Mitidieri E., A priori estimates and reduction principles for quasilinear elliptic problems and applications, Adv. Differential Equations 17 (2012), no. 9–10, 935–1000. Search in Google Scholar

[9] Farina A. and Serrin J., Entire solutions of completely coercive quasilinear elliptic equations, J. Differential Equations 250 (2011), 4367–4408. 10.1016/j.jde.2011.02.007Search in Google Scholar

[10] Folland G. B., Subelliptic estimates and function spaces on nilpotent Lie groups, Ark. Mat. 13 (1975), 161–207. 10.1007/BF02386204Search in Google Scholar

[11] Folland G. B. and Stein E. M., Hardy Spaces on Homogeneous Groups, Math. Notes 28, Princeton University Press, Princeton, 1982. 10.1515/9780691222455Search in Google Scholar

[12] Kato T., Schrödinger operators with singular potentials, Israel J. Math. 13 (1972), 135–148. 10.1007/BF02760233Search in Google Scholar

[13] Keller J. B., On solutions of Δu=f(u), Comm. Pure Appl. Math. 10 (1957), 503–510. 10.1002/cpa.3160100402Search in Google Scholar

[14] Maly J. and Ziemer W. P., Fine Regularity of Solutions of Elliptic Differential Equations, Math. Surveys Monogr., American Mathematical Society, Providence, 1997. 10.1090/surv/051Search in Google Scholar

[15] Mitidieri E. and Pohozaev S. I., Non-Existence of Positive Solutions for Quasilinear Elliptic Problems on , Tr. Mat. Inst. Steklova 227 (1999), 192–222. Search in Google Scholar

[16] Mitidieri E. and Pohozaev S. I., A priori estimates and the absence of solutions of nonlinear partial differential equations and inequalities, Tr. Mat. Inst. Steklova 234 (2001), 1–384. Search in Google Scholar

[17] Osserman R., On the inequality Δuf(u), Pacific J. Math. 7 (1957), 1641–1647. 10.2140/pjm.1957.7.1641Search in Google Scholar

[18] Pucci P. and Serrin J., A remark on entire solutions of quasilinear elliptic equations, J. Differential Equations 250 (2011), 675–689. 10.1016/j.jde.2010.04.018Search in Google Scholar

Received: 2017-4-10
Accepted: 2017-4-10
Published Online: 2017-4-19
Published in Print: 2017-5-1

© 2017 by De Gruyter

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