Liouville Results and Asymptotics of Solutions of a Quasilinear Elliptic Equation with Supercritical Source Gradient Term

Marie-Françoise Bidaut-Véron

doi:10.1515/ans-2020-2109

Publicly Available Published by De Gruyter October 16, 2020

Liouville Results and Asymptotics of Solutions of a Quasilinear Elliptic Equation with Supercritical Source Gradient Term

Marie-Françoise Bidaut-Véron

From the journal Advanced Nonlinear Studies

https://doi.org/10.1515/ans-2020-2109

Abstract

We consider the elliptic quasilinear equation -Δm⁢u=up⁢|∇⁡u|q in ℝN, q≥m and p>0, 1<m<N. Our main result is a Liouville-type property, namely, all the positive C1 solutions in ℝN are constant. We also give their asymptotic behaviour; all the solutions in an exterior domain ℝN∖Br0 are bounded. The solutions in Br0∖{0} can be extended as continuous functions in Br0. The solutions in ℝN∖{0} has a finite limit l≥0 as |x|→∞. Our main argument is a Bernstein estimate of the gradient of a power of the solution, combined with a precise Osserman-type estimate for the equation satisfied by the gradient.

Keywords: Liouville Property; Bernstein Method; Keller–Osserman Estimates

MSC 2010: 35J92

1 Introduction

In this paper, we study local and global properties of positive solutions of the equation

(1.1) - div ( | ∇ u | m - 2 ∇ u ) : = - Δ m u = u p | ∇ u | q

in ℝN (N≥1, 1<m<N and p>0) in the supercritical case q≥m. We are concerned with the Liouville property in ℝN, which is whether all the positive C1 solutions are constant. We also study the asymptotic behaviour of any solution of (1.1) near a singularity in the punctured ball Br0∖{0}, in ℝN∖{0} or in an exterior domain ℝN∖Br0.

In the case q=0, equation (1.1) reduces to the classical Lane–Emden–Fowler equation

- Δ m ⁢ u = u p ,

which has already been the subject of countless publications. One of the questions solved is that the Liouville property holds if and only if

p < p m ∗ : = N ⁢ ( m - 1 ) + m N - m .

Note that pm∗ is the Sobolev exponent. Since it is impossible to quote all the articles on the subject, we only mention here the pioneering works and references therein. Gidas and Spruck [21] first showed the nonexistence of positive solutions in ℝN for m=2 and p<p2∗. They combine the Bernstein technique applied in the equation satisfied by the gradient of a suitable power of u with delicate integral estimates ensuring the Harnack inequality; see also [11]. Then the complete behaviour up to the case p=p2∗ was obtained by moving plane methods by [13]; see also [15]. In the general case m>1, the nonexistence of nontrivial solutions for p<p2∗ was proved in a beautiful article of Serrin and Zou [33]; then the extension to the case p=p2∗ was done by [18] for m<2, then [36] for 1<m<2, and finally [30] for any m>1, for a special class of solutions.

When p=0, (1.1) reduces to the Hamilton–Jacobi equation

- Δ m ⁢ u = | ∇ ⁡ u | q .

The Liouville property was proved in [26] for m=2, and in [6] for any m>1, using the Bernstein technique. In that case, the nonexistence holds for any q>m-1, without any sign condition on the solution. Estimates of the gradient for more general problems can be found in [25].

For the general case of equation (1.1), consider the range of exponents p>0, p+q+1-m>0. As in the case q=0, there exists a “first subcritical case”, where

p < N ⁢ ( m - 1 ) N - m - ( N - 1 ) ⁢ q N - m ,

for which any supersolution in ℝN of equation (1.1) is constant, from [19]. Beyond this case, a second critical case appears when 0≤q<m-1; indeed, there exist radial positive nonconstant solutions of (1.2) whenever p≥pm,q∗, where

p m , q ∗ = N ⁢ ( m - 1 ) + m N - m - q ⁢ ( ( N - 1 ) ⁢ q - N ⁢ ( m - 1 ) + m ) ( N - m ) ⁢ ( m + 1 - q ) ;

see [14, 5].

When m=2<N and p>0, the equation

(1.2) - Δ ⁢ u = u p ⁢ | ∇ ⁡ u | q

was studied in [5] for 0<q≤2. The case q=2 could be solved explicitly by a change of the unknown function, showing that the Liouville property holds for any p>0. Using a direct Bernstein technique, we obtained a first range of values of (p,q) for which the Liouville property holds; in particular, it holds when p+q-1<4N-1, covering the first subcritical case. Using an integral Bernstein technique in the spirit of [21], we obtained a wider range of (p,q) ensuring the Liouville property, recovering Gidas and Spruck result p<N+2N-2 when q=0. However, some deep questions remained unsolved: Does the property hold for any p<p2,q∗ when q<1? Does it hold for any p>0 when 1≤q<2?

In a recent article, Filippucci, Pucci and Souplet [20, Theorem 1.1] considered the case m=2, q>2 of a superquadratic growth in the gradient, a case which was not covered by [5]. They proved the following.

Theorem ([20, Theorem 1.1]).

Any classical positive and bounded solution of equation (1.2) in RN with q≥2 and p>0 is constant.

In this article, we prove that the Liouville property holds true not only for (1.2) but for the quasilinear equation (1.1) without the assumption of boundedness on the solution. Our main result is the following.

Theorem 1.

Let u be any positive C1⁢(RN) solution of equation (1.1) with 1<m<N and q≥m, p≥0. Then u is constant.

We show that the case q=m can still be solved explicitly, giving the complete behaviour of the solutions of the equation; see Theorem 1. Next we assume q>m. We prove that all the solutions in an exterior domain are bounded, and we give the asymptotic behaviour (|x|→0 and |x|→∞) of the solutions in ℝN∖{0}.

Theorem 2.

Assume 1<m<N, q>m, p≥0. Then any positive C1 solution u of (1.1) in RN∖Br0 is bounded. If u is a nonconstant solution, then |∇⁡u| does not vanish for |x|>r0. Moreover, any positive solution u in RN∖{0} satisfies

lim | x | → ∞ ⁡ u ⁢ ( x ) = l ≥ 0 .

If l>0, there exist constants C1,C2>0 such that, for |x| large enough,

(1.3) C 1 ⁢ | x | N - m m - 1 ≤ | u ⁢ ( x ) - l | ≤ C 2 ⁢ | x | N - m m - 1 .

Concerning the solutions in Br0∖{0} and in particular in ℝN∖{0}, we proved an estimate of the gradient, showing that the solution is continuous up to 0 but the gradient is singular at 0.

Theorem 3.

Assume 1<m<N, q>m, p≥0. Any positive solution u in Br0∖{0} is bounded near 0; it can be extended as a continuous function in Br0 such that u⁢(0)>0, and for any x∈Br02∖{0},

(1.4) | ∇ ⁡ u ⁢ ( x ) | ≤ C ⁢ | x | - 1 q - m + 1 ,

where C=C⁢(N,p,q,m,u). Finally,

| u ⁢ ( x ) - u ⁢ ( 0 ) | ≤ C ⁢ | x | q - m q - m + 1

near 0, where C=C⁢(N,p,q,m,u⁢(0)). Moreover, if u is defined in RN∖{0}, then u⁢(x)≤u⁢(0) in RN∖{0}.

Note that the exponent involved in (1.4) is independent of p; actually, the solution behaves like a solution of the Hamilton–Jacobi equation

- Δ m ⁢ u = c ⁢ | ∇ ⁡ u | q with ⁢ c = u p ⁢ ( 0 ) .

Finally, we make an exhaustive study of the radial solutions for q>m, showing the sharpness of the nonradial results. We reduce the study to the one of an autonomous quadratic polynomial system of order 2, following the technique introduced in [7]. Compared to other classical techniques, it provides a complete description of all the positive solutions, in particular the global ones, without questions of regularity. We prove the following.

Theorem 4.

Assume 1<m<N, q>m, p≥0 and u is any positive nonconstant radial solution r↦u⁢(r) of (1.1) in an interval (a,b)⊆(0,∞).

If a = 0 , then u is bounded, decreasing and singular,

(1.5) lim r → 0 ⁡ u = u 0 > 0 , lim r → 0 ⁡ r ⁢ | u ′ | q - m + 1 = a m , q u 0 p , a m , q = ( N - 1 ) ⁢ q - N ⁢ ( m - 1 ) q + 1 - m .

And for given u 0 > 0 , there exist infinitely many such solutions.
If b = ∞ , then u admits a limit l ≥ 0 at infinity and

(1.6) lim r → ∞ ⁡ r N - m m - 1 ⁢ | u ⁢ ( r ) - l | = k > 0 .

Furthermore, for given l > 0 , c≠0, there exists a unique local solution near ∞ such that

(1.7) lim r → ∞ ⁡ r N - m m - 1 ⁢ ( u ⁢ ( r ) - l ) = c .
For any u 0 > 0 , there exist infinitely many solutions in ( 0 , ∞ ) , decreasing, such that lim r → 0 ⁡ u = u 0 , but a unique one, satisfying

(1.8) lim r → 0 ⁡ u = u 0 𝑎𝑛𝑑 lim r → ∞ ⁡ u = 0 .

There exist infinitely many solutions defined on an interval ( 0 , ρ ) such that lim r → ρ ⁡ u = 0 , and an infinity such that lim r → ρ ⁡ u ′ = - ∞ . Finally, there exist an infinity of solutions in ( ρ , ∞ ) such that lim r → ρ ⁡ u = 0 , and an infinity of solutions such that lim r → ρ ⁡ u ′ = ∞ .

Note that Theorems 2 and 3 lead to the following natural question: are all the solutions in ℝN∖{0} radially symmetric? This is still an open problem, even in the case p=0 of the Hamilton–Jacobi equation.

To conclude this paper, we improve another result of [20], where it was noticed that [20, Theorem 1.1] was still valid for p<0, q≥2. We prove here a much more general result covering the case p=0.

Theorem 5.

Assume 1<m<N, p≤0 and p+q+1-m>0. There exists a constant C=C⁢(N,p,q,m)>0 such that, for any positive C1 solution u of (1.1) in a bounded domain Ω,

| ∇ u ( x ) | ≤ C dist ( x , ∂ Ω ) - 1 q + 1 - m for all x ∈ Ω .

If Ω=RN, then u is constant.

Let us give a brief comment on the analogous equation with an absorption term

- Δ m ⁢ u + u p ⁢ | ∇ ⁡ u | q = 0 .

In the case m=2, 0<q<2, a complete classification of the solutions with isolated singularities was performed in [16]. A main contribution was recently given by the same authors in [17], where they obtained optimal estimates of the gradient for any 1<m≤N, p,q≥0, p+q-m+1>0, still by the Bernstein method.

Our paper is organised as follows. In Section 2, we first treat the case q=m. In Section 3, we give the main ideas of our proofs when q>m=2, and we introduce some tools for the general case q>m>1. Our main theorems are proved in Section 4, and Section 5 is devoted to the radial case. The extension to the case p≤0 is given in Section 6.

2 The Case q=m

If q=m, we can express explicitly the solutions of (1.1). We prove the following.

Theorem 1.

Let 1<m<N, p≥0, q=m. Then

any C 1 positive solution in ℝ N is constant;
any nonconstant positive solution in ℝ N ∖ B r 0 has a limit l at ∞ , and

lim | x | → ∞ ⁡ | x | m - N m - 1 ⁢ | u - l | = c > 0 ;
any positive solution in B r 0 ∖ { 0 } extends as a continuous function in B r 0 or satisfies

(2.1) lim x → 0 ⁡ u p + 1 | ln ⁡ | x | | = ( N - m ) ⁢ ( p + 1 ) m - 1 ;
any positive solution in ℝ N ∖ { 0 } is radial.

Proof.

We use a change of variable already considered in [23]; the equation takes the form

(2.2) - Δ m ⁢ u = β ⁢ ( u ) ⁢ | ∇ ⁡ u | m with ⁢ β ⁢ ( u ) = u p .

We set

γ ( τ ) = ∫ 0 τ β ( θ ) d θ = τ p + 1 p + 1 and U ( x ) = Ψ ( u ( x ) ) = ∫ 0 u ⁢ ( x ) e γ ⁢ ( θ ) m - 1 d θ : = ∫ 0 u ⁢ ( x ) e θ p + 1 ( p + 1 ) ⁢ ( m - 1 ) d θ .

A function u is a solution of (1.1) if and only if the above function U satisfies -Δm⁢U=0, and if u is nonnegative not identically 0, U is m-harmonic and positive. Conversely, u is derived from U by

(2.3) u ⁢ ( x ) = Ψ - 1 ⁢ ( U ⁢ ( x ) ) = ∫ 0 U ⁢ ( x ) d ⁢ s 1 + g ⁢ ( s ) , where g ⁢ ( s ) = ∫ 0 s β ⁢ ( Ψ ⁢ ( w ) ) ⁢ d ⁢ w = ∫ 0 s Ψ p ⁢ ( w ) ⁢ d ⁢ w .

(i) If u is a solution in ℝN of (2.2), it is constant. Indeed, any nonnegative m-harmonic function U defined in ℝN is constant, from the Harnack inequality; see [29, 31] and [33, Theorem II].

(ii) If u is defined in ℝN∖Br0, then U is bounded, it admits a limit L at ∞ and there holds

| U ⁢ ( x ) - L | ≤ C ⁢ | x | p - N p - 1 near ⁢ ∞ ;

see [2] for more general results. Clearly, the same properties hold for u (with another limit).

(iii) If u is defined in Br0∖{0}, it follows from [31] that either U extends as a continuous m-harmonic function in Br0, or it behaves like k⁢|x|p-Np-1 near 0, so (2.1) holds.

(iv) If u is a solution in ℝN∖{0}, it is proved in [24] that U is radial and endows the form

U ⁢ ( x ) = k ⁢ | x | m - N m - 1 + λ with ⁢ k , λ ≥ 0 .

Then u is radial, and using (2.3), it has the expression

u ⁢ ( x ) = ∫ 0 λ d ⁢ s 1 + g ⁢ ( s ) + ∫ 0 k ⁢ | x | m - N m - 1 d ⁢ s 1 + g ⁢ ( s - λ ) . ∎

3 Main Arguments of the Proofs

3.1 Ideas in the Case m=2

Before detailing the proof of Theorem 1, for q>m, we give an overview of it in the simple case of equation (1.2) with m=2, p>0,q>2. We set u=vb with b∈(0,1) and obtain

- Δ ⁢ v = ( b - 1 ) ⁢ | ∇ ⁡ v | m v + b q - 1 ⁢ v s ⁢ | ∇ ⁡ v | q

with s=1-q+b⁢(p+q-1). Next we make the equation satisfied by z=|∇⁡v|2 explicit. Taking into account the Böchner formula and the Cauchy–Schwarz inequality in ℝN,

- 1 2 ⁢ Δ ⁢ z + 1 N ⁢ ( Δ ⁢ v ) 2 + 〈 ∇ ⁡ ( Δ ⁢ v ) , ∇ ⁡ v 〉 ≤ - 1 2 ⁢ Δ ⁢ z + | Hess ⁡ v | 2 + 〈 ∇ ⁡ ( Δ ⁢ v ) , ∇ ⁡ v 〉 = 0 ,

we get an estimate of the form, with universal constants Ci>0,

- Δ ⁢ z + C 1 ⁢ v 2 ⁢ s ⁢ z q ≤ C 2 ⁢ z 2 v 2 + C 3 ⁢ 1 v ⁢ 〈 ∇ ⁡ z , ∇ ⁡ v 〉 + C 4 ⁢ v s ⁢ z q - 2 2 ⁢ 〈 ∇ ⁡ z , ∇ ⁡ v 〉 ;

then

(3.1) - Δ ⁢ z + C 5 ⁢ v 2 ⁢ s ⁢ z q ≤ C 6 ⁢ z 2 v 2 + C 7 ⁢ | ∇ ⁡ z | 2 z .

Using the Hölder inequality, we deduce

- Δ ⁢ z + C 8 ⁢ v 2 ⁢ s ⁢ z q ≤ C 9 ⁢ v - 2 ⁢ ( q + 2 ⁢ s ) q - 2 + C 7 ⁢ | ∇ ⁡ z | 2 z .

The crucial step is an estimate of Osserman-type in a ball Bρ valid for functions satisfying the inequality

- Δ ⁢ z + α ⁢ ( x ) ⁢ z k ≤ β ⁢ ( x ) + d ⁢ | ∇ ⁡ z | 2 z in ⁢ B ρ ,

where k>1. This is proved in Lemma 1 below, and it asserts that

z ⁢ ( x ) ≤ C ⁢ ( N , k , d ) ⁢ ( 1 ρ 2 ⁢ max B ¯ ρ ⁡ 1 α ) 1 k - 1 + ( max B ¯ ρ ⁡ β α ) 1 k in ⁢ B ρ 2 .

Then we take b=q-2p+q-1, in the same spirit as in [20], so that Bα is constant and α-1⁢(x)=v2⁢(x). We obtain an estimate

max B ¯ ρ 2 ⁡ | ∇ ⁡ v | ≤ C ⁢ ( ( max B ¯ ρ ⁡ v ρ ) 1 q - 1 + 1 ) .

But any solution in ℝN satisfies, for any ρ≥1,

(3.2) max B ρ ⁡ v ≤ v ⁢ ( 0 ) + C ⁢ ρ ⁢ max B ¯ ρ ⁡ | ∇ ⁡ v | ≤ C ⁢ ρ ⁢ ( 1 + max B ¯ ρ ⁡ | ∇ ⁡ v | ) ,

which yields

max B ¯ ρ 2 ⁡ | ∇ ⁡ v | ≤ C ⁢ ( ( max B ¯ ρ ⁡ | ∇ ⁡ v | ) 1 q - 1 + 1 ) .

Using the bootstrap method developed in [9, 6] based on the fact that 1q-1<1, we deduce that |∇⁡v|∈L∞⁢(ℝN). Note that the boundedness of |∇⁡v| had been obtained in [20] but under the extra assumption u∈L∞⁢(ℝN), an assumption that we get rid of. Returning to u=vb, it means that

- Δ ⁢ u = u p ⁢ | ∇ ⁡ u | q ≤ C ⁢ | ∇ ⁡ u | 2 u ,

and the same happens for u-l, where l=infℝN⁡u. It implies that wl=(u-l)σ is subharmonic for σ large enough. Then, from [6] (see also Lemma 3 below) and since u is superharmonic,

sup B R ⁡ w l ≤ C ⁢ ( 1 | B 2 ⁢ R | ⁢ ∫ B 2 ⁢ R w 1 σ ) σ = C ⁢ ( 1 | B 2 ⁢ R | ⁢ ∫ B 2 ⁢ R ( u - l ) ) σ ≤ C ′ ⁢ ( inf B R ⁡ u - l ) σ .

Since C′ is independent of R, it follows that supℝN⁡wl=0; thus u≡l.

Next we consider a solution in an exterior domain, and we replace (3.2) by a more precise comparison estimate between v⁢(x) and its infimum on a sphere of radius |x| and use the fact that this infimum is bounded as r→∞. Then we can show that u is still bounded and obtain the behaviour near ∞ by a careful study of u and wl. Finally, we study the behaviour in Br0∖{0} by the Bernstein technique, not relative to v but directly to u; that means, we take b=1 so that s=p. From (3.1), the function ξ=|∇⁡u|2 satisfies

- Δ ⁢ ξ + C 5 ⁢ u 2 ⁢ p ⁢ ξ q ≤ C 6 ⁢ ξ 2 u 2 + C 7 ⁢ | ∇ ⁡ z | 2 z ,

and k=infBr02∖{0}⁡u is positive by the strong maximum principle; thus

- Δ ⁢ ξ + C 8 ⁢ ξ q ≤ C 9 ⁢ ξ 2 + C 7 ⁢ | ∇ ⁡ z | 2 z ≤ C 8 2 ⁢ ξ q + C 11 + C 7 ⁢ | ∇ ⁡ z | 2 z ,

from which we deduce the estimates of ξ.

3.2 Some Tools

In the sequel, we use the Bernstein method. In the case p=0, it appeared that the square of the gradient is a subsolution of an elliptic equation with absorption, for which one can find estimates from above of Osserman-type. In the case of equation (1.1), the problem is more difficult, but such upper estimates were also a main step in study of [5] of equation (1.2) for q<2. Here also, they constitute a crucial step of our proofs below. The following lemma gives an Osserman-type property of such equations, extending [5, Lemma 2.2]; see also [4, Proposition 2.1].

Lemma 1.

Let Ω be a domain of RN, and z∈C⁢(Ω)∩C2⁢(G), where G={x∈Ω:z⁢(x)≠0}. Let

w ↦ 𝒜 ⁢ w = - ∑ i , j = 1 N a i ⁢ j ⁢ ∂ 2 ⁡ w ∂ ⁡ x i ⁢ ∂ ⁡ x j

be a uniformly elliptic operator in the open set G,

(3.3) θ ⁢ | ξ | 2 ≤ ∑ i , j = 1 N a i ⁢ j ⁢ ξ i ⁢ ξ j ≤ Θ ⁢ | ξ | 2 , θ > 0 .

Suppose that, for any x⊂G,

𝒜 ⁢ ( z ) + α ⁢ ( x ) ⁢ z k ≤ β ⁢ ( x ) + d ⁢ | ∇ ⁡ z | 2 z ,

with k>1, d=d⁢(N,p,q), and α,β are continuous in Ω and α is positive. Then there exists c=c⁢(N,p,q,k)>0 such that, for any ball B¯⁢(x0,ρ)⊂Ω, there holds

z ⁢ ( x 0 ) ≤ c ⁢ ( 1 ρ 2 ⁢ max B ρ ⁢ ( x 0 ) ⁡ 1 α ) 1 k - 1 + ( max B ρ ⁢ ( x 0 ) ⁡ β α ) 1 k .

Proof.

Let B¯⁢(x0,ρ)⊂Ω. We can assume that z⁢(x0)≠0. Let r=|x-x0|. Let w be the function defined in Bρ⁢(x0) by w⁢(x)=λ⁢(ρ2-r2)-2k-1+μ, where λ,μ>0. Let G1 be a connected component of {x∈Bρ⁢(x0);z⁢(x)>w⁢(x)}. Then G1⊂G and G1¯⊂B¯⁢(x0,ρ)⊂G. We define ℒ⁢w in Bρ⁢(x0) by

ℒ ⁢ ( w ) = 𝒜 ⁢ ( w ) + α ⁢ ( x ) ⁢ w k - β ⁢ ( x ) - d ⁢ | ∇ ⁡ w | 2 w .

Then

w x i = 4 ⁢ λ k - 1 ⁢ ( ρ 2 - r 2 ) - 2 k - 1 - 1 ⁢ x i ,

w x i ⁢ x j = 4 ⁢ λ k - 1 ⁢ ( ρ 2 - r 2 ) - 2 k - 1 - 1 ⁢ δ i ⁢ j + 4 ⁢ λ ⁢ ( k + 1 ) ( k - 1 ) 2 ⁢ ( ρ 2 - r 2 ) - 2 k - 1 - 2 ⁢ x i ⁢ x j ,

𝒜 ⁢ ( w ) = - ∑ i , j = 1 N a i ⁢ j ⁢ w x i ⁢ x j = 4 ⁢ λ k - 1 ⁢ ( ρ 2 - r 2 ) - 2 k - 1 - 1 ⁢ ( - ∑ i , j = 1 N a i ⁢ j ⁢ δ i ⁢ j ) + 4 ⁢ λ ⁢ ( k + 1 ) ( k - 1 ) 2 ⁢ ( ρ 2 - r 2 ) - 2 k - 1 - 2 ⁢ ( - ∑ i , j = 1 N a i ⁢ j ⁢ x i ⁢ x j ) ≥ - Θ ⁢ ( 4 ⁢ λ ⁢ N k - 1 ⁢ ( ρ 2 - r 2 ) - 2 k - 1 - 1 + 4 ⁢ λ ⁢ ( k + 1 ) ( k - 1 ) 2 ⁢ ( ρ 2 - r 2 ) - 2 k - 1 - 2 ⁢ r 2 ) = - Θ ⁢ 4 ⁢ λ ⁢ N k - 1 ⁢ ( ρ 2 - r 2 ) - 2 k - 1 - 2 ⁢ ( N ⁢ ( ρ 2 - r 2 ) + k + 1 k - 1 ⁢ r 2 ) = - Θ ⁢ 4 ⁢ λ k - 1 ⁢ ( ρ 2 - r 2 ) - 2 k - 1 - 2 ⁢ ( N ⁢ ρ 2 + ( k + 1 k - 1 - N ) ⁢ r 2 ) ,

| ∇ ⁡ w | 2 = 16 ⁢ λ 2 ( k - 1 ) 2 ⁢ ( ρ 2 - r 2 ) - 4 k - 1 - 2 ⁢ r 2 ⟹ | ∇ ⁡ w | 2 w ≤ 16 ⁢ λ ( k - 1 ) 2 ⁢ ( ρ 2 - r 2 ) - 2 k - 1 - 2 ⁢ r 2 ,

w k = ( λ ⁢ ( ρ 2 - r 2 ) - 2 k - 1 + μ ) k ≥ μ k + λ k ⁢ ( ρ 2 - r 2 ) - 2 ⁢ k k - 1 = μ k + λ k ⁢ ( ρ 2 - r 2 ) - 2 k - 1 - 2 .

We deduce from this series of inequalities

ℒ ⁢ ( w ) ≥ α ⁢ ( x ) ⁢ μ k - β ⁢ ( x ) + λ ⁢ ( ρ 2 - r 2 ) - 2 ⁢ k k - 1 ⁢ ( λ k - 1 ⁢ C ⁢ ( x ) - Θ ⁢ 4 k - 1 ⁢ ( N ⁢ ρ 2 + ( k + 1 k - 1 - N ) ⁢ r 2 ) ) - 16 ⁢ d ⁢ r 2 ( k - 1 ) 2 ≥ α ⁢ ( x ) ⁢ μ k - β ⁢ ( x ) + λ ⁢ ( ρ 2 - r 2 ) - 2 ⁢ k k - 1 ⁢ ( λ k - 1 ⁢ C ⁢ ( x ) - c ′ ⁢ ρ 2 ) ,

where c′=Θ⁢4k-1⁢(2⁢N+k+1k-1)+16⁢d(k-1)2=c′⁢(N,p,q,k). We deduce that ℒ⁢(w)≥0 if we impose

μ k ≥ max B ρ ⁢ ( x 0 ) ⁡ β α and λ k - 1 ≥ c ′ ⁢ ρ 2 ⁢ max B ρ ⁢ ( x 0 ) ⁡ 1 α .

If x1∈G1 is such that z⁢(x1)-w⁢(x1)=maxG1⁡(z-w)>0, then ∇⁡z⁢(x1)=∇⁡w⁢(x1), and 𝒜⁢(z-w)⁢(x1)≥0. Therefore,

0 ≥ ℒ ( z - w ) ( x 1 ) ) = 𝒜 ( z - w ) ( x 1 ) + α ( x ) ( z k - w k ) ( x 1 ) + d ( | ∇ ⁡ w | 2 w - | ∇ ⁡ z | 2 z ) .

Since the last term is positive, it is a contradiction. Then z≤w in Bρ⁢(x0). In particular, z⁢(x0)≤w⁢(x0). ∎

We also use a bootstrap argument, initially used in [9, Lemma 2.2] and then in [6] in more general form.

Lemma 2.

Let d,h∈R with d∈(0,1), and let y be a positive nondecreasing function on some interval (r1,∞). Assume that there exist K>0 and ε0>0 such that, for any ε∈(0,ε0] and r>r1,

y ⁢ ( r ) ≤ K ⁢ ε - h ⁢ y d ⁢ ( r ⁢ ( 1 + ε ) ) .

Then there exists C=C⁢(K,d,h,ε0) such that sup(r1,∞)⁡y≤C.

Proof.

Consider the sequence {εn}={ε0⁢2-n}n≥1. Since the series ∑εn is convergent, the sequence

{ P m } : = { ∏ i = 1 m ( 1 + ε i ) } m ≥ 1

is convergent too, with limit P>0. Then there holds, for any r>r1,

y ⁢ ( r ) ≤ K ⁢ ε 1 - h ⁢ y d ⁢ ( r ⁢ ( 1 + ε 1 ) ) = K ⁢ ε 1 - h ⁢ y d ⁢ ( r ⁢ P 1 ) .

We deduce by induction

y ⁢ ( r ) ≤ K 1 + d + ⋯ + d m ⁢ ( ε 1 - h ⁢ ε 2 - h ⁢ d ⁢ … ⁢ ε m - h ⁢ d m - 1 ) ⁢ y d m ⁢ ( P m ⁢ r ) = ( K ⁢ ε 0 - h ) 1 + d + ⋯ + d m ⁢ ( 2 h ⁢ ( 1 + 2 ⁢ d + ⋯ + m ⁢ d m - 1 ) ) ⁢ y d m ⁢ ( P m ⁢ r ) ,

and r⁢Pm→r⁢P, dm→0; thus (y⁢(Pm⁢r))dm→1. Therefore, we deduce that, for any r>r1,

y ⁢ ( r ) ≤ ( K ⁢ ε 0 - h ) ∑ m = 1 ∞ d m ⁢ 2 ∑ m = 1 ∞ m ⁢ d m - 1 = ( K ⁢ ε 0 - h ) 1 1 - d ⁢ 2 d ( 1 - d ) 2 . ∎

We also mention below a property of m-subharmonic functions given in [6, Lemma 2.1]. Its proof is also based upon a bootstrap method and is valid for more general quasilinear operators.

Lemma 3.

Let u∈Wloc1,m⁢(Ω) be nonnegative, m-subharmonic function in a domain Ω of RN. Then, for any τ>0, there exists a constant C=C⁢(N,m,τ)>0 such that, for any ball B2⁢ρ⁢(x0)⊂Ω and any ε∈(0,12],

sup B ρ ⁢ ( x 0 ) ⁡ u ≤ C ⁢ ε - N ⁢ m 2 τ 2 ⁢ ( 1 | B ( 1 + ε ) ⁢ ρ ⁢ ( x 0 ) | ⁢ ∫ B ( 1 + ε ) ⁢ ρ ⁢ ( x 0 ) u τ ) 1 τ .

Finally, we use some simple properties of mean value on spheres of m-superharmonic functions, in the same spirit as the ones given in [1, Lemmas 3.7, 3.8, 3.9] for mean values on annulus, and in [12] for m=2. For the sake of completeness, we recall their proofs.

Lemma 4.

Let u∈C1⁢(Ω) be nonnegative, m-superharmonic in Ω.

If Ω = ℝ N ∖ B r 0 , then r ↦ μ ⁢ ( r ) := inf | x | = r ⁡ u is bounded in ( r 0 , ∞ ) , and strictly monotone or constant for large r.
If Ω = B r 0 ∖ { 0 } , then r ↦ μ ⁢ ( r ) is nonincreasing in ( 0 , r 0 ) .

Proof.

(i) Let r>r0 be fixed. The function

f ⁢ ( x ) = μ ⁢ ( r ) ⁢ ( 1 - ( | x | r 0 ) m - N m - 1 )

is m-harmonic, and u≥f on ∂⁡Br∪∂⁡Br0; therefore, u≥f in Br¯∖Br0. Let k>0 be large enough such that 1-kp-Np-1≥12. If we take r>k⁢r0 and any x such that |x|=k⁢r0, we obtain

u ⁢ ( x ) ≥ μ ⁢ ( k ⁢ r 0 ) ≥ f ⁢ ( x ) = μ ⁢ ( r ) ⁢ ( 1 - k p - N p - 1 ) ≥ 1 2 ⁢ μ ⁢ ( r ) ,

so μ⁢(r) is bounded for r>k⁢r0. For any r2>r1>r0, φ(r1,r2):=infBr2¯∖B1u=min(μ(r1),μ(r2)) from the maximum principle. Then φ is nonincreasing in r2 and nondecreasing in r1. If μ has a strict local minimum at some point r, then, for 0<δ<δ0 small enough, μ⁢(r)<φ⁢(r-δ0,r+δ0)≤φ⁢(r-δ,r+δ), which yields a contradiction as δ→0. Then μ is monotone. If it is constant on two intervals (a,b) and (a′,b′) with b<a′ and nonconstant on (b,a′), it follows by Vazquez’s maximum principle [35] that u is constant on B¯b∖Ba and on B¯b′∖Ba′ but nonconstant on Ba′∖B¯b. This means, always by Vazquez’s maximum principle,

either min⁡{μ⁢(r):a<r<b′}=μ⁢(a) (if μ is nondecreasing) and the minimum of u in B¯b′∖Ba is achieved in any point in Bb∖B¯a, hence u is constant in B¯b′∖Ba,
or min⁡{μ⁢(r):a<r<b′}=μ⁢(a) (if μ is nonincreasing) and the minimum of u in B¯b′∖Ba is achieved in any point in Bb′∖B¯a′, hence u is constant in B¯b′∖Ba.

In both cases, we obtain a contradiction. Hence μ is either strictly monotone for r large enough, or it is constant, and so is u.

(ii) For given r1<r0 and δ>0, there exists εδ≤r1 such that, for 0<ε<εδ, we have δ⁢εm-Nm-1≥μ⁢(r1). Let h⁢(x)=μ⁢(r1)-δ⁢|x|m-Nm-1. Then u≥h on ∂⁡Br1∪∂⁡Bε, and then u≥h in Br1¯∖Bε. Making ε→0 and then δ→0, one gets u≥μ⁢(r1) in Br1∖{0}; thus μ⁢(r)≥μ⁢(r1) for r<r1. ∎

4 Proof of the Main Results

4.1 Proof of the Liouville Property for q>m

We first give a general Bernstein estimate for solutions of equation (1.1).

Lemma 1.

Let u be any C1 positive solution of (1.1) in a domain Ω, with m>1 and p,q arbitrary real numbers. Let G={x∈Ω:|∇⁡u⁢(x)|≠0}. Let u=vb with b∈R∖{0} and z=|∇⁡v|2. Then the operator

(4.1) w ↦ 𝒜 ⁢ ( w ) = - Δ ⁢ w - ( m - 2 ) ⁢ D 2 ⁢ w ⁢ ( ∇ ⁡ v , ∇ ⁡ v ) | ∇ ⁡ v | 2 = - ∑ i , j = 1 N a i ⁢ j ⁢ v x i ⁢ x j ,

with ai,j depending on ∇⁡v, is uniformly elliptic in G, and for any ε>0, there exists Cε=Cε⁢(N,m,p,q,b,ε) such that

(4.2) - 1 2 ⁢ 𝒜 ⁢ ( z ) + ( 1 - ε N ⁢ ( b - 1 ) 2 ⁢ ( m - 1 ) 2 - ( 1 - b ) ⁢ ( m - 1 ) ) ⁢ z 2 v 2 + 1 - 2 ⁢ ε N ⁢ | b | 2 ⁢ ( q - m + 1 ) ⁢ v 2 ⁢ s ⁢ z q + 2 - m + ( 1 N ⁢ 2 ⁢ ( b - 1 ) ⁢ ( m - 1 ) - s ) ⁢ | b | q - m ⁢ b ⁢ v s - 1 ⁢ z q + 4 - m 2 ≤ C ε ⁢ | ∇ ⁡ z | 2 z .

Proof.

The following identities hold if u=vb:

∇ ⁡ u = b ⁢ v b - 1 ⁢ ∇ ⁡ v ,

| ∇ ⁡ u | m - 2 ⁢ ∇ ⁡ u = | b | m - 2 ⁢ b ⁢ v ( b - 1 ) ⁢ ( m - 1 ) ⁢ | ∇ ⁡ v | m - 2 ⁢ ∇ ⁡ v ,

Δ m ⁢ u = | b | m - 2 ⁢ b ⁢ ( v ( b - 1 ) ⁢ ( m - 1 ) ⁢ Δ m ⁢ v + ( b - 1 ) ⁢ ( m - 1 ) ⁢ v ( b ( m - 1 ) - m ⁢ | ∇ ⁡ v | m ) ,

- v ( b - 1 ) ⁢ ( m - 1 ) ⁢ Δ m ⁢ v = ( b - 1 ) ⁢ ( m - 1 ) ⁢ v ( b ( m - 1 ) - m ⁢ | ∇ ⁡ v | m + | b | q ⁢ v b ⁢ p + ( b - 1 ) ⁢ q ⁢ | ∇ ⁡ v | q ,

and finally

(4.3) - Δ m ⁢ v = ( b - 1 ) ⁢ ( m - 1 ) ⁢ | ∇ ⁡ v | m v + | b | q - m ⁢ b ⁢ v s ⁢ | ∇ ⁡ v | q ,

with

(4.4) s = m - 1 - q + b ⁢ ( p + q - m + 1 ) .

We set z=|∇⁡v|2. Then, in G,

- Δ m ⁢ v = f ⇔ - Δ ⁢ v - ( m - 2 ) ⁢ D 2 ⁢ v ⁢ ( ∇ ⁡ v , ∇ ⁡ v ) | ∇ ⁡ v | 2 = f ⁢ | ∇ ⁡ v | 2 - m ,

from which identity we infer

- Δ ⁢ v = ( m - 2 ) ⁢ D 2 ⁢ v ⁢ ( ∇ ⁡ v , ∇ ⁡ v ) | ∇ ⁡ v | 2 + ( b - 1 ) ⁢ ( m - 1 ) ⁢ | ∇ ⁡ v | 2 v + | b | q - m ⁢ b ⁢ v s ⁢ | ∇ ⁡ v | q + 2 - m

where

〈 Hess ⁡ v ⁢ ( ∇ ⁡ v ) , ∇ ⁡ v 〉 = D 2 ⁢ v ⁢ ( ∇ ⁡ v , ∇ ⁡ v ) = 1 2 ⁢ 〈 ∇ ⁡ z , ∇ ⁡ v 〉 .

We recall the Böchner formula combined with Cauchy–Schwarz inequality

- 1 2 ⁢ Δ ⁢ z + 1 N ⁢ ( Δ ⁢ v ) 2 + 〈 ∇ ⁡ ( Δ ⁢ v ) , ∇ ⁡ v 〉 ≤ - 1 2 ⁢ Δ ⁢ z + | Hess ⁡ v | 2 + 〈 ∇ ⁡ ( Δ ⁢ v ) , ∇ ⁡ v 〉 = 0 .

Since

- Δ ⁢ v = m - 2 2 ⁢ 〈 ∇ ⁡ z , ∇ ⁡ v 〉 z + ( b - 1 ) ⁢ ( m - 1 ) ⁢ z v + | b | q - m ⁢ b ⁢ v s ⁢ z q + 2 - m 2 ,

we deduce

〈 ∇ ⁡ ( Δ ⁢ v ) , ∇ ⁡ v 〉 = - m - 2 2 ⁢ 〈 ∇ ⁡ 〈 ∇ ⁡ z , ∇ ⁡ v 〉 z , ∇ ⁡ v 〉 + ( 1 - b ) ⁢ ( m - 1 ) ⁢ 〈 ∇ ⁡ z v , ∇ ⁡ v 〉 - | b | q - m ⁢ b ⁢ ( s ⁢ v s - 1 ⁢ z q + 4 - m 2 + q + 2 - m 2 ⁢ v s ⁢ z q - m 2 ⁢ 〈 ∇ ⁡ z , ∇ ⁡ v 〉 ) .

We observe that

〈 ∇ ⁡ z v , ∇ ⁡ v 〉 = 〈 ∇ ⁡ z , ∇ ⁡ v 〉 v - z 2 v 2 and 〈 ∇ ⁡ z , ∇ ⁡ v 〉 2 z 2 ≤ | ∇ ⁡ z | 2 z ;

thus

- m - 2 2 ⁢ 〈 ∇ ⁡ 〈 ∇ ⁡ z , ∇ ⁡ v 〉 z , ∇ ⁡ v 〉 = - m - 2 2 ⁢ ( D 2 ⁢ z ⁢ ( ∇ ⁡ v , ∇ ⁡ v ) z + 1 2 ⁢ | ∇ ⁡ z | 2 z - 〈 ∇ ⁡ z , ∇ ⁡ v 〉 2 z 2 ) ≥ - m - 2 2 ⁢ D 2 ⁢ z ⁢ ( ∇ ⁡ v , ∇ ⁡ v ) z - | m - 2 | ⁢ | ∇ ⁡ z | 2 z .

We define the operator 𝒜 by (4.1); it satisfies (3.3) with θ=min⁡(1,m-1) and Θ=max⁡(1,m-1), so it is uniformly elliptic in G. Therefore,

(4.5) - 1 2 ⁢ 𝒜 ⁢ ( z ) + 1 N ⁢ ( Δ ⁢ v ) 2 - ( 1 - b ) ⁢ ( m - 1 ) ⁢ z 2 v 2 - | b | q - m ⁢ b ⁢ s ⁢ v s - 1 ⁢ z q + 4 - m 2 ≤ ( b - 1 ) ⁢ ( m - 1 ) ⁢ 〈 ∇ ⁡ z , ∇ ⁡ v 〉 v + ( q + 2 - m ) ⁢ | b | q - m ⁢ b 2 ⁢ v s ⁢ z q - m 2 ⁢ 〈 ∇ ⁡ z , ∇ ⁡ v 〉 + | m - 2 | ⁢ | ∇ ⁡ z | 2 z .

For ε>0, there holds, by Hölder’s inequality,

q + 2 - m 2 ⁢ v s ⁢ z q - m 2 ⁢ 〈 ∇ ⁡ z , ∇ ⁡ v 〉 ≤ ε N ⁢ | b | 2 ⁢ ( q - m + 1 ) ⁢ v 2 ⁢ s ⁢ z q + 2 - m + C ε ⁢ | ∇ ⁡ z | 2 z ,

( Δ ⁢ v ) 2 = ( m - 2 2 ⁢ 〈 ∇ ⁡ z , ∇ ⁡ v 〉 z + ( b - 1 ) ⁢ ( m - 1 ) ⁢ z v + | b | q - m ⁢ b ⁢ v s ⁢ z q + 2 - m 2 ) 2 ≥ ( b - 1 ) 2 ⁢ ( m - 1 ) 2 ⁢ z 2 v 2 + | b | 2 ⁢ ( q - m + 1 ) ⁢ v 2 ⁢ s ⁢ z q + 2 - m + 2 ⁢ ( b - 1 ) ⁢ ( m - 1 ) ⁢ | b | q - m ⁢ b ⁢ v s - 1 ⁢ z q + 4 - m 2 - | m - 2 | ⁢ | ∇ ⁡ z | z ⁢ ( | b - 1 | ⁢ ( m - 1 ) ⁢ z v + | b | q - m + 1 ⁢ v s ⁢ z q + 2 - m 2 ) ,

and for any ε>0,

| m - 2 | ⁢ | ∇ ⁡ z | z ⁢ | b - 1 | ⁢ ( m - 1 ) ⁢ z v ≤ ε N ⁢ ( b - 1 ) 2 ⁢ ( m - 1 ) 2 ⁢ z 2 v 2 + C ε ⁢ | ∇ ⁡ z | 2 z ,

| m - 2 | ⁢ | ∇ ⁡ z | z ⁢ | b | q - m + 1 ⁢ v s ⁢ z q + 2 - m 2 ≤ ε N ⁢ | b | 2 ⁢ ( q - m + 1 ) ⁢ v 2 ⁢ s ⁢ z q + 2 - m + C ε ⁢ | ∇ ⁡ z | 2 z ;

thus (4.2) follows. ∎

Proof of Theorem 1.

We use Lemma 1 with b∈(0,1), combined with the estimate

( 1 N ⁢ 2 ⁢ ( b - 1 ) ⁢ ( m - 1 ) - s ) ⁢ | b | q - m ⁢ b ⁢ v s - 1 ⁢ z q + 4 - m 2 ≤ ε N ⁢ | b | 2 ⁢ ( q - m + 1 ) ⁢ v 2 ⁢ s ⁢ z q + 2 - m + C ε ⁢ z 2 v 2 .

Then there exist constants Ci>0 depending only on m,b,N,p,q such that

1 2 ⁢ 𝒜 ⁢ ( z ) + C 1 ⁢ v 2 ⁢ s ⁢ z q + 2 - m ≤ C 2 ⁢ z 2 v 2 + C 3 ⁢ | ∇ ⁡ z | 2 z .

Next we choose s=-1 in (4.4); thus

b = q - m p + q - m + 1 ,

which is positive because q>m. Using the Hölder inequality, we deduce

𝒜 ⁢ ( z ) + C 4 ⁢ z q + 2 - m - C 5 v 2 ≤ 𝒜 ⁢ ( z ) + C 1 ⁢ z q + 2 - m - C 2 ⁢ z 2 v 2 ≤ C 3 ⁢ | ∇ ⁡ z | 2 z .

If we apply Lemma 1 with

α ⁢ ( x ) = C 4 v 2 ⁢ ( x ) , β ⁢ ( x ) = C 5 v 2 ⁢ ( x ) , k = q + 2 - m ,

we deduce that any solution in B¯ρ⁢(x0), ρ>0, satisfies

z ⁢ ( x 0 ) ≤ C 6 ⁢ ( 1 α ⁢ ρ 2 ) 1 k - 1 + ( C 5 C 4 ) 1 k ≤ C 7 ⁢ ( ( max B ρ ⁢ ( x 0 ) ⁡ v ρ ) 2 q + 1 - m + 1 ) ,

which yields

(4.6) | ∇ ⁡ v ⁢ ( x 0 ) | ≤ C 8 ⁢ ( ( max B ¯ ρ ⁢ ( x 0 ) ⁡ v ρ ) 1 q + 1 - m + 1 ) ,

where we observe that 1q+1-m<1 since q>m. Let ε∈(0,12]. As a consequence, for any solution in B2⁢R (or even B¯3⁢R2), considering any x0∈B¯R and taking ρ=R⁢ε, we get

(4.7) max B ¯ R ⁡ | ∇ ⁡ v | ≤ c ⁢ ( ( max B ¯ R ⁢ ( 1 + ε ) ⁡ v ε ⁢ R ) 1 q + 1 - m + 1 ) ≤ c ⁢ ε - 1 q + 1 - m ⁢ ( ( max B ¯ R ⁢ ( 1 + ε ) ⁡ v R ) 1 q + 1 - m + 1 ) ,

max B ¯ R ⁢ ( 1 + ε ) ⁡ v ≤ v ⁢ ( 0 ) + R ⁢ ( 1 + ε ) ⁢ max B ¯ R ⁢ ( 1 + ε ) ⁡ | ∇ ⁡ v | ,

max B ¯ R ⁢ ( 1 + ε ) ⁡ v R ≤ 1 + v ⁢ ( 0 ) R + ( 1 + ε ) ⁢ max B ¯ R ⁢ ( 1 + ε ) ⁡ | ∇ ⁡ v | ≤ c 0 ⁢ ( 1 R + max B ¯ R ⁢ ( 1 + ε ) ⁡ | ∇ ⁡ v | ) ,

where c0=2+v⁢(0) depends on v⁢(0). If R≥1,

( max B ¯ R ⁢ ( 1 + ε ) ⁡ v R ) 1 q + 1 - m + 1 ≤ c 0 1 q + 1 - m ( 1 + max B ¯ R ⁢ ( 1 + ε ) | ∇ v | ) 1 q + 1 - m ) + 1 ≤ c 1 ( 1 + max B ¯ R ⁢ ( 1 + ε ) | ∇ v | ) 1 q + 1 - m .

Then, from (4.7),

y ( R ) : = 1 + max B ¯ R | ∇ v | ≤ 1 + c 2 ε - 1 q + 1 - m ( 1 + max B ¯ R ⁢ ( 1 + ε ) | ∇ v | ) 1 q + 1 - m ≤ c 3 ε - 1 q + 1 - m ( 1 + max B ¯ R ⁢ ( 1 + ε ) | ∇ v | ) 1 q + 1 - m .

Using the definition of y, this is

y ⁢ ( R ) ≤ c ⁢ ε - 1 q + 1 - m ⁢ ( y ⁢ ( ( 1 + ε ) ⁢ R ) ) 1 q + 1 - m ,

where c depends on v⁢(0). Therefore, y⁢(R) is bounded as a consequence of Lemma 2. Thus |∇⁡v| is bounded, and using the definition of v with the value of b, up+1⁢|∇⁡u|q-m∈L∞⁢(ℝN). Next we consider any l≥0 such that u-l>0. The function ul=u-l satisfies

0 ≤ - Δ m ⁢ u l ≤ C ∞ ⁢ | ∇ ⁡ u | m u ≤ C ∞ ⁢ | ∇ ⁡ u l | m u l ,

with C∞=∥up+1⁢|∇⁡u|q-m∥L∞⁢(ℝN). Then the function wl=ulσ, with σ>1 to be specified below, satisfies

- Δ m ⁢ w l = σ m - 1 u l ( - Δ m u l + ( σ - 1 ) ( m - 1 ) | ∇ ⁡ u l | m u l ) ( σ - 1 ) ⁢ ( m - 1 ) ≤ σ m - 1 ⁢ ( ( σ - 1 ) ⁢ ( m - 1 ) - C ∞ ) ⁢ u l σ ⁢ ( m - 1 ) - m ⁢ | ∇ ⁡ u l | m .

Therefore, wl is m-subharmonic for σ large enough.

We first take l=0, so w=uσ. By Lemma 3, for any τ>0, there exists a constant Cτ=Cτ⁢(N,m,τ) such that

(4.8) sup B R ⁡ w ≤ C τ ⁢ ( 1 | B 2 ⁢ R | ⁢ ∫ B 2 ⁢ R w τ ) 1 τ = C τ ⁢ ( 1 | B 2 ⁢ R | ⁢ ∫ B 2 ⁢ R u τ ⁢ σ ) 1 τ ,

and since u is m-superharmonic, there holds, for any θ∈(0,N⁢(m-1)N-m) (see [34]),

(4.9) inf B R ⁡ u ≥ c θ ⁢ ( 1 | B 2 ⁢ R | ⁢ ∫ B 2 ⁢ R u θ ) 1 θ .

Taking τ=θσ, we deduce

(4.10) sup B R ⁡ u = ( sup B R ⁡ w ) 1 σ ≤ C τ 1 σ ⁢ ( 1 | B 2 ⁢ R | ⁢ ∫ B 2 ⁢ R u τ ⁢ σ ) 1 s ⁢ σ ≤ C τ 1 σ c θ ⁢ inf B R ⁡ u .

This means that u, and also w, satisfies the Harnack inequality in ℝN,

sup B R ⁡ w ≤ C τ c θ σ ⁢ inf B R ⁡ w .

But r↦μ⁢(r)=inf|x|=r⁡u=infBr⁡u from the maximum principle is nonincreasing, so it has a limit L≥0 as r→∞. This implies that u is bounded and l=infℝN⁡u≥0. If we replace u by ul and w by wl, then (4.8) holds with w and u replaced respectively by wl and ul since wl is m-subharmonic, but also (4.9) holds with u replaced by ul since ul is m-superharmonic. Thus supBR⁡wl≤C⁢(infBR⁡ul)σ. Therefore, supBR⁡wl tends to 0 as R→∞. Then wl≡0, and thus u≡l. ∎

4.2 Asymptotic Behaviour near ∞

In this section, we consider the behaviour of solutions defined in an exterior domain.

Proof of Theorem 2.

Consider a nonnegative solution u=vb (0<b<1) of (1.1) in ℝN∖Br0. From (4.6), the function v satisfies, in B¯ρ⁢(x0) (ρ>0),

(4.11) | ∇ ⁡ v ⁢ ( x 0 ) | ≤ C ⁢ ( ( max B ¯ ρ ⁢ ( x 0 ) ⁡ v ρ ) 1 q + 1 - m + 1 )

with C=C⁢(N,p,q,m). Here we denote by ci some positive constants depending on r0,N,p,q,m. Let R>4⁢r0 and 0<ε≤14. Applying (4.11) with ρ=ε⁢R, we get

| ∇ ⁡ v ⁢ ( x 0 ) | ≤ c 1 ⁢ ( ( max B ¯ ε ⁢ R ⁢ ( x 0 ) ⁡ v ε ⁢ R ) 1 q + 1 - m + 1 ) ≤ c 1 ⁢ ε - 1 q + 1 - m ⁢ ( ( max B ¯ ε ⁢ R ⁢ ( x 0 ) ⁡ v R ) 1 q + 1 - m + 1 ) ,

then

max | x | = R ⁡ | ∇ ⁡ v | ≤ c 2 ⁢ ( ( max R ⁢ ( 1 - ε 2 ) ≤ | x | ≤ R ⁢ ( 1 + ε 2 ) ⁡ v ε ⁢ R ) 1 q + 1 - m + 1 ) ≤ c 3 ⁢ ( ( max R 1 + ε ≤ | x | ≤ R ⁢ ( 1 + ε ) ⁡ v ε ⁢ R ) 1 q + 1 - m + 1 ) ,

max R 2 ≤ | x | ≤ 2 ⁢ R ⁡ | ∇ ⁡ v | ≤ c 4 ⁢ ( ( max R 2 ⁢ ( 1 + ε ) ≤ | x | ≤ 2 ⁢ R ⁢ ( 1 + ε ) ⁡ v ε ⁢ R ) 1 q + 1 - m + 1 ) ,

and finally,

1 + max R 2 ≤ | x | ≤ 2 ⁢ R ⁡ | ∇ ⁡ v | ≤ c 5 ⁢ ε - 1 q + 1 - m ⁢ ( ( max R 2 ⁢ ( 1 + ε ) ≤ | x | ≤ 2 ⁢ R ⁢ ( 1 + ε ) ⁡ v R ) 1 q + 1 - m + 1 ) .

From Lemma 4 (i), μ⁢(r)=inf|x|=r⁡u=(inf|x|=r⁡v)b is bounded; let M=maxr≥r0⁡μ⁢(r). Note that M depends on u. Now, for any x such that |x|=ρ, there exists at least one point xρ where v⁢(xρ)=inf|x|=ρ⁡v. We can join any point x∈Sρ to xρ by a connected chain of balls of radius ε⁢ρ with points xi∈Sρ, and this chain can be constructed so that it has at most πε elements. Considering one ball containing x and joining it to a ball containing xρ, we get

v ⁢ ( x ) ≤ v ⁢ ( x ρ ) + C N ⁢ ε - 1 ⁢ ρ ⁢ max ρ 1 + ε ≤ | x | ≤ ρ ⁢ ( 1 + ε ) ⁡ | ∇ ⁡ v | ≤ M 1 b + C N ⁢ ε - 1 ⁢ ρ ⁢ max ρ 1 + ε ≤ | x | ≤ ρ ⁢ ( 1 + ε ) ⁡ | ∇ ⁡ v | .

Then

max R 2 ⁢ ( 1 + ε ) ≤ | x | ≤ 2 ⁢ R ⁢ ( 1 + ε ) ⁡ v ≤ c M 1 ⁢ ( 1 + ε - 1 ⁢ R ⁢ max R 2 ⁢ ( 1 + 3 ⁢ ε ) ≤ | x | ≤ 2 ⁢ R ⁢ ( 1 + 3 ⁢ ε ) ⁡ | ∇ ⁡ v | ) ≤ c M 2 ⁢ ε - 1 ⁢ R ⁢ ( 1 + max R 2 ⁢ ( 1 + 3 ⁢ ε ) ≤ | x | ≤ 2 ⁢ R ⁢ ( 1 + 3 ⁢ ε ) ⁡ | ∇ ⁡ v | ) ,

1 ε ⁢ R ⁢ max R 2 ⁢ ( 1 + ε ) ≤ | x | ≤ 2 ⁢ R ⁢ ( 1 + ε ) ⁡ v ≤ c M 3 ⁢ ε - 2 ⁢ ( 1 + max R 2 ⁢ ( 1 + 3 ⁢ ε ) ≤ | x | ≤ 2 ⁢ R ⁢ ( 1 + 3 ⁢ ε ) ⁡ | ∇ ⁡ v | ) .

Using estimate (4.7), we obtain

(4.12) 1 + max R 2 ≤ | x | ≤ 2 ⁢ R ⁡ | ∇ ⁡ v | ≤ c M 4 ⁢ ε - 2 q + 1 - m ⁢ ( 1 + max R 2 ⁢ ( 1 + 3 ⁢ ε ) ≤ | x | ≤ 2 ⁢ R ⁢ ( 1 + 3 ⁢ ε ) ⁡ | ∇ ⁡ v | ) 1 q + 1 - m .

Let {εn}n≥1 be a positive decreasing sequence such that Pn:=∏j=1n(1+εj)→2 and Θn:=∏j=1nεj+1dj→Θ>0 when n→∞. It is easy to find such sequences such that εj∼2-j. For R2≤a<2⁢R≤b, we set

y ⁢ ( a , b ) = 1 + max a ≤ | x | ≤ b ⁡ | ∇ ⁡ v | .

Then (4.12) with (a,b)=(R2,2⁢R) and ε1=3⁢ε asserts that

y ⁢ ( R 2 , 2 ⁢ R ) ≤ c 5 ⁢ ε 1 - h ⁢ ( y ⁢ ( R 2 ⁢ ( 1 + ε 1 ) , 2 ⁢ R ⁢ ( 1 + ε 1 ) ) ) d with ⁢ h = 2 q + 1 - m ⁢ and ⁢ d = 1 q + 1 - m ∈ ( 0 , 1 ) .

Applying (4.12) with (a,b)=(R2⁢Pn,2⁢R⁢Pn), we obtain

y ⁢ ( R 2 ⁢ P n , 2 ⁢ R ⁢ P n ) ≤ c 5 ⁢ ε n + 1 - h ⁢ ( y ⁢ ( R 2 ⁢ P n + 1 , 2 ⁢ R ⁢ P n + 1 ) ) d .

By induction, we deduce

y ⁢ ( R 2 , 2 ⁢ R ) ≤ c 5 1 + d + d 2 + ⋯ + d n ⁢ Θ n - h ⁢ ( y ⁢ ( R 2 ⁢ P n + 1 , 2 ⁢ R ⁢ P n + 1 ) ) d n + 1 .

Since y⁢(R2⁢Pn+1,2⁢R⁢Pn+1)→y⁢(R4,4⁢R), we obtain

1 + max R 2 ≤ | x | ≤ 2 ⁢ R | ∇ v | ≤ c 5 d 1 - d Θ - h : = C ( M , N , p , q , m ) .

Then we conclude again that |∇⁡v| is bounded for |x|≥4⁢r0, then in ℝN∖Br0 since we have assumed that u∈C1⁢ℝN∖Br0. We consider again the function w=uσ, for σ depending on r0, large enough so that (σ-1)⁢(m-1)≥∥up+1⁢|∇⁡u|q-m∥L∞⁢(ℝN∖Br0). As in the proof of Theorem 1, we conclude that w is m-subharmonic in ℝN∖B¯r0. Hence u satisfies the Harnack inequality using estimate (4.10). Therefore, for any R>2⁢r0,

sup R 2 ≤ | x | ≤ 3 ⁢ R 2 ⁡ u ≤ C ⁢ inf R 2 ≤ | x | ≤ 3 ⁢ R 2 ⁡ u .

Since u is m-superharmonic, it follows by the strong maximum principle that it cannot have any local minimum in ℝN∖B¯r0. Since uσ is m-subharmonic, it cannot have any local maximum too, and u shares this property. As a consequence, |∇⁡u| does not vanish in ℝN∖B¯r0. The function r↦μ is bounded by Lemma 4; hence u is also bounded by the above Harnack inequality. Finally, μ⁢(r) is monotone for large r, so it admits a limit l≥0 when r→∞.

If μ⁢(r) is nonincreasing for r≥r1>r0, then u-l≥0, so we can consider the function wl instead of w. Then

max R ≤ | x | ≤ 2 ⁢ R ⁡ w l ≤ C ⁢ ( inf R ≤ | x | ≤ 2 ⁢ R ⁡ ( u - l ) ) σ .

Then wl tends to 0; thus u tends to l as |x|→∞. Since u-l is m-superharmonic in ℝN∖Br0, then there holds

u ⁢ ( x ) - l ≥ C ⁢ | x | m - N m - 1

with C=C⁢(r0,N,m,u); see for example [10, Proposition 2.6], [33, Lemma 2.3]. This is the case in particular when u is a solution in ℝN∖{0}. Note that the radial solutions such that μ is nonincreasing are precisely defined in (0,∞).

Now it follows from the upper estimate of y⁢(R) that the function u satisfies

- Δ m ⁢ u = u p ⁢ | ∇ ⁡ u | q ≤ C ⁢ | ∇ ⁡ u | m u

in ℝN∖Br0. Next suppose that l>0. Then -Δm⁢u≤C′⁢|∇⁡u|m. The function U (still used in case q=m), defined by U=(m-1)⁢(eu-lm-1-1), satisfies -Δm⁢U≤0, and U tends to 0 at ∞. Then there exists Rε>0 such that U⁢(x)≤ε for |x|≥Rε. For R>Rε, the function x↦ω(x):=ε+(sup|z|=r0U(z))(|x|r0)m-Nm-1 is m-harmonic in BR∖Br0; hence it is larger than U. Letting ε→0, we get U≤C⁢|x|m-Nm-1 near ∞, and U has the same behaviour as u-l, so we deduce the estimate from above,

u ⁢ ( x ) - l ≤ C ⁢ | x | m - N m - 1 .

Then we get estimate (1.3). ∎

Remark 2.

(i) In case u is defined in ℝN∖{0} and l=0, we obtain the estimates

C 1 ⁢ | x | m - N m - 1 ≤ u ⁢ ( x ) ≤ C 2 ⁢ | x | 1 σ ⁢ m - N m - 1 .

It would be interesting to improve the estimate from above.

(ii) If u is defined in ℝN∖Br0 and if μ is nonincreasing, we have proved that u has a limit l≥0 as |x|→∞. If μ is nondecreasing, we only obtain that μ⁢(r)=inf|x|=r⁡u⁢(x) has a limit l, and sup|x|=r⁡u⁢(x) has a limit λ≥l. Indeed, the function w is m-subharmonic positive and bounded, so the function r↦sup|x|=r⁡w=(sup|x|=r⁡u)σ is also monotone for large r and has a limit λσ. We have w=uσ≤λσ, so sup|x|=r⁡u is also nondecreasing. But we cannot prove that λ=l.

4.3 Behaviour near an Isolated Singularity

In this section, we study the behaviour of solutions with an isolated singularity at the origin.

Proof of Theorem 3.

Let u be a nonnegative solution u of (1.1) in Br0∖{0}. We apply directly the Bernstein method to u; we obtain by Lemma 1 with b=1, and then s=p. Setting ξ=|∇⁡u|2, we get

1 2 ⁢ 𝒜 ⁢ ( ξ ) + C 1 ⁢ u 2 ⁢ p ⁢ z q + 2 - m ≤ C 2 ⁢ ξ 2 u 2 + C 3 ⁢ | ∇ ⁡ ξ | 2 ξ .

By the strong maximum principle, there exists a constant a0>0 depending on r0 and N,p,q such that u≥a0 in Br02∖{0}. Therefore, there holds

1 2 ⁢ 𝒜 ⁢ ( ξ ) + C 1 2 ⁢ p ⁢ a 0 2 ⁢ p ⁢ z q + 2 - m ≤ C 2 ⁢ ξ 2 a 0 2 + C 3 ⁢ | ∇ ⁡ ξ | 2 ξ

in Br02∖{0}. Then, from Lemma 1, we deduce the inequality

z ⁢ ( x 0 ) ≤ c ⁢ ( ( 1 a 0 2 ⁢ p ⁢ ρ 2 ) 1 q + 1 - m + ( 1 a 0 2 ⁢ ( p + 1 ) ) 1 q + 2 - m ) ≤ c 0 2 ⁢ ( 1 ρ 2 q + 1 - m + 1 )

for any ball B¯ρ⁢(x0)⊂Br02∖{0}, with c=c⁢(N,p,q,m) and c02=c⁢(a0-pq+1-m+a0-p+1q+2-m). Hence, for any x∈B¯R∖{0}, with R≤min⁡(1,r08),

| ∇ ⁡ u ⁢ ( x ) | ≤ 2 ⁢ c 0 | x | 1 q + 1 - m ≤ 2 ⁢ c 0 | x | 1 q + 1 - m .

As a consequence, considering any xR such that x,x′∈B¯R2∖{0}, there holds

| u ⁢ ( x ′ ) - u ⁢ ( x ) | ≤ 2 ⁢ c 0 ⁢ R q - m q + 1 - m .

Since q>m, u is bounded near 0. Then, with constants C>0 depending on a0,

- Δ m ⁢ u = f ≤ C ⁢ | ∇ ⁡ u | q ≤ C ⁢ | x | - q q + 1 - m .

Then f∈LlocNm+ε⁢(BR4) since N-Nm⁢qq+1-m=N⁢(m-1)⁢(q-m)m⁢(q+1-m)>0. Thus, from [31], u can be extended as a continuous function, a solution of the equation in the sense of distributions. Then we deduce that, for any x∈B¯R2∖{0}, |u⁢(0)-u⁢(x)|≤c0⁢|x|q-mq+1-m. Moreover, replacing r0 by ρ>0 small enough such that u⁢(x)≥u⁢(0)2 in Bρ, then a0≥u⁢(0)2; hence c0≤C⁢(N,p,q,m,u⁢(0)), and for |x|≤min⁡(1,ρ8), we infer |u⁢(0)-u⁢(x)|≤C⁢|x|q-mq+1-m.

Next assume that u is defined in ℝN∖{0} and is not constant. Then u is bounded since it is bounded near 0 and ∞. Then r↦μ⁢(r)=inf|x|=r⁡u is nonincreasing; thus μ⁢(r)≤u⁢(0). Indeed, for all ε>0, we have μ⁢(|x|)≤u⁢(x)≤u⁢(0)+ε for any |x|≤rε; then, from the monotone decreasingness, μ⁢(r)≤u⁢(0)+ε for any r>0. From Theorem 2, lim|x|→∞⁡u=l=limr→∞⁡μ, and then necessarily l≤u⁢(0). Suppose that there exists x≠0 such that u⁢(x)>u⁢(0); then u has a maximum in ℝN∖{0}, but |∇⁡u| cannot vanish in ℝN∖{0} by Theorem 2, so we get a contradiction. ∎

5 Radial Case

If u is a positive radial solution of (1.1) and if we denote for simplicity u⁢(r)=u⁢(x) with r=|x|, then u satisfies the ordinary differential equation

(5.1) ( | u ′ | m - 2 ⁢ u ′ ) ′ + N - 1 r ⁢ | u ′ | m - 2 ⁢ u ′ + u p ⁢ | u ′ | q = r 1 - N ⁢ ( r N - 1 ⁢ | u ′ | m - 2 ⁢ u ′ ) ′ + u p ⁢ | u ′ | q = 0 .

We begin with a simple observation about the set of zeros of u′. We have shown above that any solution of the exterior problem is either constant, or its gradient does not vanish. In the radial case, the proof is elementary.

Proposition 1.

Assume q>m-1, p≥0. Then any nonnegative radial solution of (5.1) on a segment [r1,r2]⊂(0,∞) is constant or strictly monotone.

Proof.

By the strong maximum principle [35], we can assume that u>0 on (r1,r2). The function

r ↦ W ( r ) : = r N - 1 | u ′ ( r ) | m - 2 u ′ ( r )

is nonincreasing. Suppose that u′ has two zeros ρ1 and ρ2 in (r1,r2); then, by integrating W′ and using the equation, we deduce that u′≡0 on [ρ1,ρ2]; hence u is constant therein; therefore, we can assume that [ρ1,ρ2] is the maximal subinterval of [r1,r2] where u′ vanishes. If [r1,ρ1]≠[r1,r2], for example r1<ρ1, then u′>0 on (r1,ρ1), where u′⁢(r)=(r1-N⁢W⁢(r))1m-1. By (5.1),

(5.2) m - 1 m - 1 - q ⁢ ( W m - 1 - q m - 1 ) ′ = | W | - q m - 1 ⁢ W ′ = - r N - 1 - ( N - 1 ) ⁢ q m - 1 ⁢ u p

on (r1,ρ1), and limr→ρ1⁡u′⁢(r)=0. Since m-1-q<0, this implies limr→ρ1⁡Wm-1-qm-1⁢(r)=∞, a contradiction since u is bounded on [r1,ρ1]. We proceed similarly if r1=ρ1 but ρ2<r2, or if ρ1=ρ2. Hence either u is constant, or it is strictly monotone. ∎

Next we make a complete description of the radial solutions for p≥0, q>m.

5.1 The Case p=0

This case p=0 of the Hamilton–Jacobi equation is well known since equation (5.2) can be directly integrated, so the solutions are explicit, and are an Ariadne thread for studying the case p>0. We find different types of nonconstant solutions according to the sign of u′,

{ u ′ = r 1 - N m - 1 ⁢ ( C 1 - a m , q - 1 ⁢ r - a m , q ) - 1 q - m + 1 , u ′ = - r 1 - N m - 1 ⁢ ( C 2 + a m , q - 1 ⁢ r - a m , q ) - 1 q - m + 1 ,

where am,q=(N-1)⁢q-N⁢(m-1)m-1>0 since q>N⁢(m-1)N-1>m-1, and the value of u follows by integration, with the requirement that u>0. The solutions such that C1>0 satisfy

lim r → 0 ⁡ r 1 q - m + 1 ⁢ u ′ ⁢ ( r ) = - a m , q 1 q - m + 1 ;

then limr→0⁡u⁢(r)=u0>0 since q>m. The conclusions of Theorem 4 follow in that case.

5.2 The Case p>0

Equation (5.1) can be reduced to an autonomous system since it is invariant by the transformation u↦Tλ⁢u (λ>0) given by

(5.3) T λ ⁢ u ⁢ ( x ) = λ - q - m p + q + 1 - m ⁢ u ⁢ ( λ ⁢ x ) .

Here we perform a change of unknown, introduced in [7], which consists in a differentiation of the equation, as in the Bernstein technique. We set

X ⁢ ( t ) = - r ⁢ u ′ ⁢ ( r ) u ⁢ ( r ) , Z ⁢ ( t ) = - r ⁢ u p ⁢ | u ′ | q - m ⁢ u ′ , t = ln ⁡ r ,

and obtain the following quadratic system of Kolmogorov type, valid for any point t where u′⁢(t)≠0, and any reals m,p,q:

(5.4) { X t = X ⁢ ( X - N - m m - 1 + Z m - 1 ) , Z t = Z ⁢ ( N - N - 1 m - 1 ⁢ q - p ⁢ X + q + 1 - m m - 1 ⁢ Z )

in the region Q={(X,Z)∈ℝ2:X⁢Z>0}. Note that the trajectories X=0 and Z=0 are not admissible in our study. Since p+q≠m-1, we can recover u and u′ by

(5.5) u = ( r q - m ⁢ | Z | ⁢ | X | m - 1 - q ) 1 p + q - m + 1 , u ′ = - X ⁢ u r = ( r - ( p + 1 ) ⁢ | Z | ⁢ | X | p ) 1 p + q - m + 1 ⁢ sign ⁡ ( - X ) .

The fixed points of the system in Q¯ are

N 0 = ( 0 , a m , q ) = ( 0 , ( N - 1 ) ⁢ q - N ⁢ ( m - 1 ) q + 1 - m ) , O = ( 0 , 0 ) , A 0 = ( N - m m - 1 , 0 ) .

We begin by a local study of the different points and the corresponding results for the solutions of (5.1).

Lemma 2.

Let p>0 and q>m.

The point N 0 is a source with eigenvalues

0 < λ 1 = q - m q + 1 - m < λ 2 = ( N - 1 ) ⁢ q - N ⁢ ( m - 1 ) m - 1

and eigenvectors v 1 = ( 1 , c m , q ) with c m , q > 0 and v 2 = ( 0 , 1 ) . Then there exist infinitely many singular decreasing solutions u of ( 5.1 ) defined near 0 , satisfying ( 1.5 ).
The point O is a sink with eigenvalues

0 > ξ 1 = - N - m m - 1 > ξ 2 = N - N - 1 m - 1 ⁢ q

and eigenvectors u 1 = ( 1 , 0 ) and u 2 = ( 0 , 1 ) . Then there exist infinitely many solutions u of ( 1.1 ) defined for large r and either increasing or decreasing near ∞ , satisfying ( 1.6 ) and ( 1.7 )
The point A 0 is a saddle point with eigenvalues

μ 1 = - ( N - m ) ⁢ p + ( N - 1 ) ⁢ q - ( m - 1 ) ⁢ N m - 1 < 0 < μ 2 = N - m m - 1

and eigenvectors w 1 = ( 1 , - d m , q ) with d m , q > 0 and w 2 = ( 1 , 0 ) . Then, for any c > 0 , there exists a unique solution u of ( 5.1 ) defined at least for large r such that lim r → ∞ ⁡ r N - m m - 1 ⁢ u = c > 0 .

Proof.

(i) We perform the linearisation at N0; setting Z=am,q+Z¯, we get, with X>0 and Z>0,

{ X t = q - m q + 1 - m ⁢ X , Z ¯ t = a m , q ⁢ ( - p ⁢ X + q + 1 - m m - 1 ⁢ Z ¯ ) ,

which gives the eigenvalues λ1,λ2 and their respective eigenvectors, with the value

c m , q = p ( N - 1 ) q - N ) ( m - 1 ) ( N - 1 ) ⁢ q 2 - 2 ⁢ N ⁢ ( m - 1 ) ⁢ q + ( m - 1 ) ⁢ ( N ⁢ ( m - 1 ) + m ) .

So N0 is a source; the particular trajectory X=0 associated to λ2 is not admissible. There exists an infinity of trajectories starting from N0 as t→-∞, associated to the eigenvalue λ1; the solutions (X,Z) satisfy X>0, limt→-∞⁡e-q-mq+1-m⁢t⁢X=C0, where C0>0 is arbitrary and limt→-∞⁡Z=am,q; then, from (5.5) and the definition of Z, there exist infinitely many decreasing singular solutions u of (5.1) defined near 0, satisfying (1.5).

(ii) The linearisation at O gives the system

{ X t = - N - m m - 1 ⁢ X , Z t = ( N - N - 1 m - 1 ⁢ q ) ⁢ Z ,

which admits the eigenvalues ξ1,ξ2. So O is a sink, two particular trajectories are the axis X=0 and Z=0, which are not admissible. There is an infinity of trajectories converging to O as t→∞, tangent to the axis Z=0, associated to the eigenvalue ξ1, with either X,Z>0, or X,Z<0. They satisfy

(5.6) X ∼ t → ∞ C 1 ⁢ e - N - m m - 1 ⁢ t , Z ∼ t → ∞ C 2 ⁢ e ( N - N - 1 m - 1 ⁢ q ) ⁢ t with ⁢ C 1 , C 2 > 0 .

The corresponding solutions u of (5.1) are defined for large r, and either decreasing or increasing; from (5.5), we obtain limr→∞⁡u=(C1m-1-q⁢C2)1p+q-m+1=l>0 and limr→∞⁡rN-1m-1⁢u′=-l⁢C1; thus limr→∞⁡rN-mm-1⁢(u-l)=-l⁢C1. Thus (1.6) and (1.7) follow. The uniqueness property follows from the uniqueness of a trajectory satisfying (5.6) for given C1,C2; see also Remark 3 below.

(iii) Linearisation at A0: setting X=N-mm-1+X¯, we get

{ X ¯ t = N - m m - 1 ⁢ ( X ¯ + Z m - 1 ) , Z t = - ( N - m ) ⁢ p + ( N - 1 ) ⁢ q - ( m - 1 ) ⁢ N m - 1 ⁢ Z ,

which admits the eigenvalues μ1<0<μ2 and the eigenvectors with

d m , q = m - 1 N - m ⁢ ( N - m + | μ 1 | ⁢ ( m - 1 ) ) .

It is a saddle point. The trajectory X=0 associated to μ2 is not admissible. Then a unique trajectory 𝒯A0 converges to A0 as t→∞. By the scaling (5.3), we deduce the uniqueness property for u. ∎

Next we give a complete description of the local and global solutions in the phase plane leading to the conclusions of Theorem 4.

Proof of Theorem 4 when p>0.

We consider the sets

ℒ X = { ( X , Z ) ∈ Q : X t = 0 } = { ( X , Z ) ∈ Q : X - N - m m - 1 + Z m - 1 = 0 } ,

ℒ Z = { ( X , Z ) ∈ Q : Z t = 0 } = { ( X , X ) ∈ Q : N - N - 1 m - 1 ⁢ q - p ⁢ X + q + 1 - m m - 1 ⁢ Z } .

The straight line ℒX has an extremity at A0 with slope -(m-1), and the slope of 𝒯A0 is -dm,q<-(m-1), so 𝒯A0 is above ℒX near t=∞. The line ℒX has an extremity at N0 with slope p⁢(m-1)q+1-m and is located above ℒX for X>0. The trajectories issued from N0 have the slope cm,q, and we check that it is greater than p⁢(m-1)q+1-m because q>1-m, so they start above ℒZ.

(i) The trajectory 𝒯A0 stays in the region ℛ={0<X<N-mm-1;X-N-mm-1+Zm-1>0} which is negatively invariant. Then Xt stays positive; thus X is increasing, hence bounded. Either 𝒯A0 stays under ℒZ; then Zt<0 and Z is bounded; thus 𝒯A0 converges to N0. Or it crosses the line ℒZ at time t0, and for t<t0, there holds Zt>0 so that Z stays bounded, and 𝒯A0 still converges to N0. In fact, the second eventuality holds because of the slope of the eigenvector at N0. So the trajectory 𝒯A0 joins N0 to A0. By scaling, for any u0>0, there exists a unique solution u defined in (0,∞) satisfying (1.8).

(ii) All the trajectories with one point in the bounded invariant region ℛ′ delimited by the axis X=0, Z=0 and 𝒯A0 join N0 to O, and the corresponding solutions u are positive on (0,∞), decreasing and satisfy (1.5). The trajectories with one point in the region ℛ′′⊂Q above 𝒯A0 converge to N0 as t→-∞ and satisfy Xt>0 since 𝒯A0 is above ℒX and cannot be bounded since there is no fixed point in this region. They can be of two types.

Either they cross ℒZ; then, after crossing, Z is decreasing, necessarily to 0; then, from (5.5), u is defined in a maximal interval (0,ρ) with u⁢(ρ)=0. Such solutions exist because by any point on ℒZ passes a trajectory.
Or they stay above ℒZ; thus Z increases to ∞; in this case, from (5.5), u is defined in a maximal interval (0,ρ) with limr→ρ⁡u′=-∞.

Let us show the existence of such solutions. For given c>0, we define

ℒ c = { ( X , Z ) ∈ Q : X > 0 , Z = c ⁢ X + a m , q } .

We compute the field on this line and show that it is entering the region above ℒc for c large enough; indeed, we obtain

Z t X t - c = Z ⁢ ( q + 1 - m m - 1 ⁢ ( Z - a m , q ) - p ⁢ X ) X ⁢ ( X - N - m m - 1 + c ⁢ X + a m , q m - 1 ) - c = Z ⁢ ( c ⁢ q + 1 - m m - 1 - p ) X - N - m m - 1 + c ⁢ X + a m , q m - 1 - c > ( c ⁢ ( q + 1 - m ) - p ⁢ ( m - 1 ) ) ⁢ ( c ⁢ X + a m , q ) ( m - 1 + c ) ⁢ X + a m , q - ( N - m ) - c ,

and

( c ⁢ ( q + 1 - m ) - p ⁢ ( m - 1 ) ) ⁢ ( c ⁢ X + a m , q ) - c ⁢ ( ( m - 1 + c ) ⁢ X + a m , q - ( N - m ) ) = c ⁢ X ⁢ ( c ⁢ ( q - m ) - ( p + 1 ) ⁢ ( m - 1 ) - ( p + a m , q ⁢ ( m - 1 ) ) + c ⁢ ( q - m ) ⁢ a m , q + N - m ) - p ⁢ ( m - 1 ) ⁢ a m , q

is positive for large c since q-m>0. All the solutions with one point above ℒc stay in this region, so above ℒZ, which proves the existence.

(iii) All the trajectories with one point in {(X,Z)∈Q:X<0} satisfy Xt>0 from (5.4). Then X increases necessarily up to 0, and then Zt>0 for large t; thus (X,Z) converges to O, and u is defined for r large enough, increasing and limr→∞⁡u=l>0.

Either they cross ℒZ; then, before crossing, Z is decreasing, necessarily to 0; then, from (5.5), u is defined in a maximal interval (0,ρ) with u⁢(ρ)=0. Such solutions exist as above.
Or they stay under ℒZ; thus X and Z decrease to -∞; in this case, from (5.5), u is defined in a maximal interval (0,ρ) with limr→ρ⁡u′=-∞.

Let us show their existence. For given k>0, we compute the field on the line ℒk={(X,Z)∈Q:X<0,Z=k⁢X}. On this line, Xt>0 and

Z t - k ⁢ X t = Z ⁢ ( N - N - 1 m - 1 ⁢ q - p ⁢ X + q + 1 - m m - 1 ⁢ Z - X + N - m m - 1 - Z m - 1 ) = Z ⁢ ( ( q - m m - 1 - p + 1 k ) ⁢ Z - N - 1 m - 1 ⁢ ( q - m ) ) = Z 2 ⁢ ( q - m m - 1 - p + 1 k ) + N - 1 m - 1 ⁢ ( q - m ) ⁢ | Z |

is positive for large k. The region below ℒk is therefore negatively invariant; then the existence follows. This concludes the proof. ∎

Remark 3.

The change of variable u⁢(r)=u~⁢(s), s=rm-Nm-1, introduced in [22], and also used in [12] in case m=2<N, leads to the equation

(5.7) ( | u ~ s | s m - 2 ⁢ u ~ ) s + ( m - N m - 1 ) q - m ⁢ s N - 1 N - m ⁢ ( q - m ) ⁢ u ~ p ⁢ | u ~ s | q = 0 .

Hence, if u is not constant, u~s does not vanish, from Remark 1, and (5.7) is equivalent to

(5.8) ( m - 1 ) ⁢ u ~ s ⁢ s + ( m - N m - 1 ) q - m ⁢ s N - 1 N - m ⁢ ( q - m ) ⁢ u ~ p ⁢ | u ~ s | q - m + 2 = 0 .

In particular, we find again the existence and uniqueness of local solutions near ∞, satisfying (1.7) for given l≥0 and c≠0 (c>0 if l=0); indeed, the problem reduces to equation (5.8) with the initial conditions u~⁢(0)=l and u~s⁢(0)=c.

6 The Case p<0

Proof of Theorem 5.

We still consider u=vb with b>0; we recall that, from (4.3) and (4.4),

- Δ m ⁢ v = ( b - 1 ) ⁢ ( m - 1 ) ⁢ | ∇ ⁡ v | m v + b q - m + 1 ⁢ v s ⁢ | ∇ ⁡ v | q

with s=1-q+m+b⁢(p+q-m+1). Next we take

b = q + 1 - m p + q - m + 1 ;

thus here b≥1 and s=0, so

- Δ m ⁢ v = ( b - 1 ) ⁢ ( m - 1 ) ⁢ | ∇ ⁡ v | m v + b q - m + 1 ⁢ | ∇ ⁡ v | q ,

where the two terms have the same sign. Then z=|∇⁡v|2 satisfies

𝒜 ⁢ ( v ) = - Δ ⁢ v - m - 2 2 ⁢ 〈 ∇ ⁡ z , ∇ ⁡ v 〉 z = ( b - 1 ) ⁢ ( m - 1 ) ⁢ z v + b q - m + 1 ⁢ z q + 2 - m 2 .

From (4.5), we get

- 1 2 𝒜 ( z ) + 1 N ( Δ v ) 2 + ( b - 1 ) ( m - 1 ) z 2 v 2 ≤ ( b - 1 ) ( m - 1 ) ( 〈 ∇ ⁡ z , ∇ ⁡ v 〉 v + q + 2 - m 2 b q - m + 1 z q - m 2 〈 ∇ z , ∇ v 〉 ,

where now the term in z2v2 has a positive coefficient. Since b≥1, we get an estimate of the form

𝒜 ⁢ ( z ) + C 1 ⁢ z q + 2 - m ≤ C 3 ⁢ | ∇ ⁡ z | 2 z .

Since q+2-m>1, we deduce the estimate in any ball Bρ⁢(x0),

| ∇ ⁡ v ⁢ ( x 0 ) | ≤ C ⁢ ( 1 ρ ) 1 q + 1 - m ,

from Lemma 1, where C is a universal constant, which leads to the conclusions.∎

Communicated by Patrizia Pucci

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Received: 2020-08-28

Accepted: 2020-10-03

Published Online: 2020-10-16

Published in Print: 2021-02-01

Liouville Results and Asymptotics of Solutions of a Quasilinear Elliptic Equation with Supercritical Source Gradient Term

Abstract

1 Introduction

Theorem ([20, Theorem 1.1]).

Theorem 1.

Theorem 2.

Theorem 3.

Theorem 4.

Theorem 5.

2 The Case q=m

Theorem 1.

Proof.

3 Main Arguments of the Proofs

3.1 Ideas in the Case m=2

3.2 Some Tools

Lemma 1.

Proof.

Lemma 2.

Proof.

Lemma 3.

Lemma 4.

Proof.

4 Proof of the Main Results

4.1 Proof of the Liouville Property for q>m

Lemma 1.

Proof.

Proof of Theorem 1.

4.2 Asymptotic Behaviour near ∞

Proof of Theorem 2.

Remark 2.

4.3 Behaviour near an Isolated Singularity

Proof of Theorem 3.

5 Radial Case

Proposition 1.

Proof.

5.1 The Case p=0

5.2 The Case p>0

Lemma 2.

Proof.

Proof of Theorem 4 when p>0.

Remark 3.

6 The Case p<0

Proof of Theorem 5.

References

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