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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 EMAIL logo

Abstract

We consider the elliptic quasilinear equation -Δmu=up|u|q in N, qm and p>0, 1<m<N. Our main result is a Liouville-type property, namely, all the positive C1 solutions in N are constant. We also give their asymptotic behaviour; all the solutions in an exterior domain NBr0 are bounded. The solutions in Br0{0} can be extended as continuous functions in Br0. The solutions in N{0} has a finite limit l0 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.

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 N (N1, 1<m<N and p>0) in the supercritical case qm. We are concerned with the Liouville property in N, 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 N{0} or in an exterior domain NBr0.

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 N 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 N of equation (1.1) is constant, from [19]. Beyond this case, a second critical case appears when 0q<m-1; indeed, there exist radial positive nonconstant solutions of (1.2) whenever ppm,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<q2. 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 1q<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 q2 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 qm, p0. 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 N{0}.

Theorem 2.

Assume 1<m<N, q>m, p0. Then any positive C1 solution u of (1.1) in RNBr0 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 N{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, p0. 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 xBr02{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, p0 and u is any positive nonconstant radial solution ru(r) of (1.1) in an interval (a,b)(0,).

  1. 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.

  2. 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 , c0, there exists a unique local solution near such that

    (1.7) lim r r N - m m - 1 ( u ( r ) - l ) = c .

  3. 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 N{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, q2. We prove here a much more general result covering the case p=0.

Theorem 5.

Assume 1<m<N, p0 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<mN, p,q0, 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 p0 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, p0, q=m. Then

  1. any C 1 positive solution in N is constant;

  2. any nonconstant positive solution in N B r 0 has a limit l at , and

    lim | x | | x | m - N m - 1 | u - l | = c > 0 ;

  3. 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 ;

  4. any positive solution in 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 -ΔmU=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 N of (2.2), it is constant. Indeed, any nonnegative m-harmonic function U defined in N is constant, from the Harnack inequality; see [29, 31] and [33, Theorem II].

(ii) If u is defined in NBr0, 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 N{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 N,

- 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 N satisfies, for any ρ1,

(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(N). Note that the boundedness of |v| had been obtained in [20] but under the extra assumption uL(N), 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=infNu. 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 supNwl=0; thus ul.

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 zC(Ω)C2(G), where G={xΩ: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 xG,

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 {xBρ(x0);z(x)>w(x)}. Then G1G and G1¯B¯(x0,ρ)G. We define w in Bρ(x0) by

( 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

( 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=Θ4k-1(2N+k+1k-1)+16d(k-1)2=c(N,p,q,k). We deduce that (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 x1G1 is such that z(x1)-w(x1)=maxG1(z-w)>0, then z(x1)=w(x1), and 𝒜(z-w)(x1)0. Therefore,

0 ( 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 zw 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,hR 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,)yC.

Proof.

Consider the sequence {εn}={ε02-n}n1. 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 rPmrP, dm0; thus (y(Pmr))dm1. 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 uWloc1,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 B2ρ(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 uC1(Ω) be nonnegative, m-superharmonic in Ω.

  1. If Ω = 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.

  2. 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 uf on BrBr0; therefore, uf in Br¯Br0. Let k>0 be large enough such that 1-kp-Np-112. If we take r>kr0 and any x such that |x|=kr0, 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>kr0. 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 δ0. 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¯bBa and on B¯bBa but nonconstant on BaB¯b. This means, always by Vazquez’s maximum principle,

  1. either min{μ(r):a<r<b}=μ(a) (if μ is nondecreasing) and the minimum of u in B¯bBa is achieved in any point in BbB¯a, hence u is constant in B¯bBa,

  2. or min{μ(r):a<r<b}=μ(a) (if μ is nonincreasing) and the minimum of u in B¯bBa is achieved in any point in BbB¯a, hence u is constant in B¯bBa.

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 εδr1 such that, for 0<ε<εδ, we have δεm-Nm-1μ(r1). Let h(x)=μ(r1)-δ|x|m-Nm-1. Then uh on Br1Bε, and then uh in Br1¯Bε. Making ε0 and then δ0, one gets uμ(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={xΩ:|u(x)|0}. Let u=vb with bR{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 B2R (or even B¯3R2), considering any x0B¯R and taking ρ=Rε, 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 R1,

( 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-mL(N). Next we consider any l0 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-mL(N). 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 N,

sup B R w C τ c θ σ inf B R w .

But rμ(r)=inf|x|=ru=infBru from the maximum principle is nonincreasing, so it has a limit L0 as r. This implies that u is bounded and l=infNu0. 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 supBRwlC(infBRul)σ. Therefore, supBRwl tends to 0 as R. Then wl0, and thus ul. ∎

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 NBr0. 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>4r0 and 0<ε14. Applying (4.11) with ρ=εR, 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|=ru=(inf|x|=rv)b is bounded; let M=maxrr0μ(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|=ρv. We can join any point xSρ to xρ by a connected chain of balls of radius ερ with points xiSρ, 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}n1 be a positive decreasing sequence such that Pn:=j=1n(1+εj)2 and Θn:=j=1nεj+1djΘ>0 when n. It is easy to find such sequences such that εj2-j. For R2a<2Rb, we set

y ( a , b ) = 1 + max a | x | b | v | .

Then (4.12) with (a,b)=(R2,2R) and ε1=3ε 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)=(R2Pn,2RPn), 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(R2Pn+1,2RPn+1)y(R4,4R), 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|4r0, then in NBr0 since we have assumed that uC1NBr0. We consider again the function w=uσ, for σ depending on r0, large enough so that (σ-1)(m-1)up+1|u|q-mL(NBr0). As in the proof of Theorem 1, we conclude that w is m-subharmonic in NB¯r0. Hence u satisfies the Harnack inequality using estimate (4.10). Therefore, for any R>2r0,

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 NB¯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 NB¯r0. The function rμ 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 l0 when r.

If μ(r) is nonincreasing for rr1>r0, then u-l0, 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 NBr0, 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 N{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 NBr0. Next suppose that l>0. Then -ΔmuC|u|m. The function U (still used in case q=m), defined by U=(m-1)(eu-lm-1-1), satisfies -ΔmU0, and U tends to 0 at . Then there exists Rε>0 such that U(x)ε for |x|Rε. For R>Rε, the function xω(x):=ε+(sup|z|=r0U(z))(|x|r0)m-Nm-1 is m-harmonic in BRBr0; hence it is larger than U. Letting ε0, we get UC|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 N{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 NBr0 and if μ is nonincreasing, we have proved that u has a limit l0 as |x|. If μ is nondecreasing, we only obtain that μ(r)=inf|x|=ru(x) has a limit l, and sup|x|=ru(x) has a limit λl. Indeed, the function w is m-subharmonic positive and bounded, so the function rsup|x|=rw=(sup|x|=ru)σ is also monotone for large r and has a limit λσ. We have w=uσλσ, so sup|x|=ru 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 ua0 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