A Liouville-type theorem for an elliptic equation with superquadratic growth in the gradient

TL;DR

This paper proves that all positive bounded solutions to a certain elliptic PDE with superquadratic gradient growth are constant, using spherical averages and gradient bounds, extending previous results to the case p ≥ 2.

Contribution

It establishes a Liouville-type theorem for elliptic equations with superquadratic gradient growth, filling a gap for the case p ≥ 2.

Findings

Positive bounded solutions are constant.

The proof uses spherical averages and Bernstein gradient bounds.

Extends previous results to new parameter range p ≥ 2.

Abstract

We consider the elliptic equation $-\Delta u = u^q|\nabla u|^p$ in $\mathbb R^n$ for any $p\ge 2$ and $q>0$. We prove a Liouville-type theorem, which asserts that any positive bounded solution is constant. The proof technique is based on monotonicity properties for the spherical averages of sub- and super-harmonic functions, combined with a gradient bound obtained by a local Bernstein argument. This solves, in the case of bounded solutions, a problem left open in~\cite{BVGHV}, where the authors consider the case $0<p<2$. Some extensions to elliptic systems are also given.

Equations

$$-\Delta u=u^q|\nabla u{|}^p\text{ in }{\mathbb{R}}^n,$$

$$u(x)=\biggl{(}\frac{c\alpha}{\alpha^{2}+|x|^{2}}\biggr{)}^{(n-2)/2},\quad\alpha>0,\quad c=\sqrt{n(n-2)}.$$

$$-\Delta u\ge u^q|\nabla u{|}^p\text{ in }{\mathbb{R}}^n$$

$$(n-2)q+(n-1)p>n.$$

$$(n-2)q+(n-1)p\le nandp>1,$$

$$\{\begin{array}{ll}-\Delta u=f(x,u,v,\nabla u,\nabla v)& \text{mbox}in{\mathbb{R}}^n,\text{vspace}1mm\\ -\Delta v=g(x,u,v,\nabla u,\nabla v)& \text{mbox}in{\mathbb{R}}^n,\end{array}$$

$$(u,v,\xi ,\zeta )\in {\Gamma }_M:=[0,M{]}^2\times {\mathbb{R}}^n\times {\mathbb{R}}^n$$

$$uf(x,u,v,\xi ,\zeta )\le K(M)|\xi {|}^{2}andvg(x,u,v,\xi ,\zeta )\le C(M)|\zeta {|}^2.$$

$$\{\begin{array}{ll}-\Delta u=u^{q_1}{v}^{r_1}|\nabla u{|}^{p_1}& \text{mbox}in{\mathbb{R}}^n,\text{vspace}1mm\\ -\Delta v=v^{q_2}{u}^{r_2}|\nabla v{|}^{p_2}& \text{mbox}in{\mathbb{R}}^n,\end{array}$$

$$\bar{w}(R)=|S_R{|}^{-1}{\int }_{S_R}wd\sigma ,R>0,$$

$$w(x)\ge \underset{R\to \infty }{\lim }\bar{w}(R)\in [0,\infty ),for\; allx\in {\mathbb{R}}^n.$$

$$u=v^m$$

$$\nabla u=m{v}^{m-1}\nabla v,\Delta u=m{v}^{m-1}\Delta v+m(m-1)v^{m-2}|\nabla v{|}^2.$$

$$-m{v}^{m-1}\Delta v-m(m-1)v^{m-2}|\nabla v{|}^2=m^p{v}^{mq+(m-1)p}|\nabla v{|}^p.$$

$$-\Delta v=(m-1)\frac{|\nabla v{|}^2}{v}+m^{p-1}{v}^{mq+(m-1)(p-1)}|\nabla v{|}^p.$$

$$|\nabla \eta |\le C{\eta }^{\alpha },|\Delta \eta |+\frac{|\nabla \eta {|}^2}{\eta }\le C{\eta }^{\alpha }\text{mbox}in{B}_2,$$

$$w=|\nabla v{|}^2,z:=\eta w,$$

$$\Delta z=\mbox{div}\bigl{(}|\nabla v|^{2}\nabla\eta+\eta\nabla(|\nabla v|^{2})\bigr{)}=|\nabla v|^{2}\Delta\eta+2\langle\nabla(|\nabla v|^{2}),\nabla\eta\rangle+\eta\Delta(|\nabla v|^{2}).$$

$$⟨\nabla (\Delta v),\nabla v⟩$$

$$\displaystyle=\biggl{[}-\frac{m-1}{v}-\frac{p}{2}m^{p-1}v^{mq+(m-1)(p-1)}w^{(p-2)/2}\biggr{]}\langle\nabla w,\nabla v\rangle$$

$$-m^{p-1}[mq+(m-1)(p-1)]v^{m(q+p-1)-p}{w}^{(p+2)/2}+\frac{m-1}{v^2}{w}^2.$$

$$-\Delta w-⟨b,\nabla w⟩+2|D^{2}v{|}^2=2a{w}^2,$$

$$a=m^{p-1}[mq+(m-1)(p-1)]v^{m(q+p-1)-p}{w}^{(p-2)/2}-\frac{m-1}{v^2},$$

$$\displaystyle b=\biggl{[}2\frac{m-1}{v}+pm^{p-1}v^{mq+(m-1)(p-1)}w^{(p-2)/2}\biggr{]}\nabla v,$$

$$-\Delta z+2\eta |D^{2}v{|}^2=-w\Delta \eta -2⟨\nabla w,\nabla \eta ⟩+\eta ⟨b,\nabla w⟩+2\eta a{w}^2.$$

$$-\Delta z-⟨b,\nabla z⟩+2\eta |D^{2}v{|}^2=-w[\Delta \eta +⟨b,\nabla \eta ⟩]-2⟨\nabla w,\nabla \eta ⟩+2\eta a{w}^2.$$

$$|\nabla w|=|\nabla (|\nabla v{|}^2)|=2|(D^{2}v)\nabla v|\le 2|D^{2}v||\nabla v|.$$

$$|⟨\nabla w,\nabla \eta ⟩|\le 2|D^{2}v||\nabla v||\nabla \eta |\le 2\frac{|\nabla \eta {|}^2|\nabla v{|}^2}{\eta }+\frac{1}{2}\eta |D^{2}v{|}^2=2\frac{|\nabla \eta {|}^2}{\eta }w+\frac{1}{2}\eta |D^{2}v{|}^2.$$

$$\mathscr{L}z+\eta|D^{2}v|^{2}\leq\biggl{[}\mathcal{L}\eta+4\frac{|\nabla\eta|^{2}}{\eta}\biggr{]}w+2\eta aw^{2}.$$

$$mq+(m-1)(p-1)=-1⟺m(q+p-1)=p-2⟺m=\frac{p-2}{q+p-1}\in (0,1).$$

$$a=\Bigl{[}-m^{p-1}w^{(p-2)/2}+(1-m)\Bigr{]}v^{-2}$$

$$b=\Bigl{[}pm^{p-1}w^{(p-2)/2}-2(1-m)\Bigr{]}v^{-1}\nabla v.$$

$$2\eta a{w}^2$$

Full text

1 Dipartimento di Matematica e Informatica, Universitá degli Studi di Perugia Via Vanvitelli 1, I-06123 Perugia, Italy. E-mails: roberta.filippucci@unipg.it; patrizia.pucci@unipg.it

2 Université Paris 13, Sorbonne Paris Cité, CNRS UMR 7539, Laboratoire Analyse, Géométrie et Applications, 93430 Villetaneuse, France. E-mail: souplet@math.univ-paris13.fr

1. Introduction and main results

In this paper we are interested in proving a Liouville-type result for positive solutions of the elliptic equation

[TABLE]

where $p\geq 2$, $q>0$ and $n\geq 1$. Equation (1.1) when $p=0$ reduces to the celebrated Lane-Emden equation and in this case the well-known and deep result of Gidas and Spruck in [10] asserts that if $q<q_{S}$, where $q_{S}=(n+2)/(n-2)_{+}$ is the critical Sobolev exponent, then no positive solutions can exist. The result is sharp since it fails for $q\geq q_{S}$. In particular, for $q=q_{S}$ with $n\geq 3$, equation (1.1) admits the positive bounded Lane-Emden solutions

[TABLE]

If we now consider supersolutions of the Lane-Emden equation, the Liouville result remains true in the smaller range $q\leq q_{*}$, where $q_{*}=n/(n-2)_{+}$ is the so-called Serrin critical exponent (cf. e.g. [16]), and this condition is optimal for supersolutions; see [1] for a detailed description of some results in this direction. Of course, due to the large number of papers dealing with this topic and its generalizations to problems involving quasilinear elliptic operators, it is not possible to produce here an exhaustive bibliography.

The subcase of (1.1) when $q=0$ reduces to the well-known diffusive Hamilton-Jacobi equation. It was studied in [12] and it was proved there that any classical solution has to be constant if $p>1$. Thus, in that case, nonexistence holds without sign condition.

Our main result is the following.

Theorem 1.1**.**

Let $u$ be a positive bounded classical solution of (1.1), with $p\geq 2$ and $q>0$. Then $u$ is constant.

Remark 1.1**.**

(a) The result of Theorem 1.1 is delicate since the conclusion fails for supersolutions. Namely, for $p,q\geq 0$, there exists a positive, nonconstant bounded classical solution of

[TABLE]

whenever $n\geq 3$ and

[TABLE]

Such supersolution can be found under the form $c(1+|x|^{2})^{-\beta}$ for suitable $\beta,c>0$. Condition (1.3) is essentially optimal, at least in the superlinear range. Indeed, if

[TABLE]

then any positive solution of (1.2) must be constant; see [4, Theorem 7.1] and [14, Theorem 15.1] (and this remains of course true when $n\leq 2$ since any positive superharmonic function is then constant). We also refer to [14], [6] for extensions of this result to quasilinear problems.

(b) Although the case of equation (1.1) with negative $q$ does not seem to have been much studied in the literature, it is worth pointing out that Theorem 1.1 remains true for all $q>1-p$ (with $p\geq 2$), as can be checked by inspection of the proof (see also Remark 2.1).

(c) We point out that, in the case $q=0$, Theorem 1.1, treated in [12] and already discussed above, holds without assuming that $u$ is bounded.

Equation (1.1) with $q\in(0,2)$, $p+q>1$, was studied in detail in [2], cf. in particular Corollary B-1, and various regions for nonexistence were determined. We also refer to the earlier paper [3] where the case $q\in(0,N/(N-1))$ was considered, cf. [3, Corollary 2]. The case $q\geq 2$ was left essentially open, and we completely answer it here in the case of bounded solutions. Note that Theorem 1.1 is true without boundedness assumption in the radial case, as shown in [2]. We do not know if the conclusion of Theorem 1.1 remains true without the boundedness assumption on $u$ in the nonradial case for $n\geq 3$.

Let us briefly consider the coercive analogue of (1.1), namely $\Delta u=u^{q}|\nabla u|^{p}$. This problem is quite different since, in the case $p=0$ (the so-called Keller-Osserman problem) the threshold value of $q$ for nonexistence does not depend on the dimension. Early results for $p>0$ appeared in [13], while more recent ones can be found in [9], [5] and the references therein. See also [11] for results on radial solutions of related coercive systems. **

The basic idea of our proof is to use the monotone nonincreasing property of the spherical averages of the superharmonic function $u$ (cf. Lemma 2.2). A key observation is then that a suitably large power of the function $v:=u-\inf u$ is on the contrary subharmonic. The latter property is a consequence of a suitable gradient bound (cf. Lemma 2.1), obtained by a local Bernstein argument. The combination of these two opposite monotonicity properties eventually forces $u$ to be constant.

We stress that this proof is quite different from those in [12] and [2]. Indeed, the celebrated result by Lions for $q=0$, cf. Corollary IV of [12], is a direct consequence of a local Bernstein-type estimate of the gradient of generic $C^{2}$ solutions of $-\Delta u=|\nabla u|^{p}$ in a ball $B_{R}$, an estimate which forces the vanishing of the gradient upon letting $R\to\infty$. On the other hand, the Liouville Theorem B in [2] in the range $0<p<2$ (plus additional assumptions on $p,q$) is also proved by Bernstein-type arguments (significantly more delicate and technical than in the case $q=0$.)

Our methods can be used for more general problems, including systems. As one of the possible extensions, let us consider the following class of systems:

[TABLE]

where $f,g:\mathbb{R}^{n}\times[0,\infty)^{2}\times\mathbb{R}^{n}\times\mathbb{R}^{n}\to\mathbb{R}$. We have the following result.

Theorem 1.2**.**

Let $f,g$ be nonnegative.

(i) Assume that, for each $M>0$, there exists a constant $K=K(M)>0$ such that,

[TABLE]

Then any nonnegative, bounded, classical solution $(u,v)$ of (1.4) must be constant.

(ii) Assume that property (1.5) holds with $\Gamma_{M}$ replaced by $[0,M]^{2}\times B_{M}\times B_{M}$. Let $(u,v)$ be a nonnegative classical solution of (1.4) such that $u,v,|\nabla u|,|\nabla v|$ are bounded. Then $(u,v)$ must be constant.

Remark 1.2**.**

(a) Theorem 1.2(i) applies in particular for the model system

[TABLE]

with $p_{1}=p_{2}=2$ and any $q_{1},q_{2},r_{1},r_{2}\geq 0$, whereas Theorem 1.2(ii) applies for any $p_{1},p_{2}\geq 2$ and $q_{1},q_{2},r_{1},r_{2}\geq 0$. We refer to e.g. [7], [8] for related results on systems of inequalities corresponding to (1.6), under stronger restrictions on the exponents.

(b) We note that, in the scalar case (cf. Theorem 1.1), no boundedness assumption on the gradients was necessary. This is due to the possibility of proving a suitable gradient estimate by a Bernstein-type argument (cf. Lemma 2.1), a property which does not seem available in general for systems.

(c) As can be seen from the proof, Theorem 1.2 extends in a straightforward manner to systems of more than two equations. **

2. Proofs

In view of the proof of Theorem 1.1 we prepare two lemmas. Our first lemma is a gradient bound, that will be obtained by a local Bernstein argument.

Lemma 2.1**.**

Let $u$ be a positive bounded classical solution of (1.1), with $p>2$ and $q>0$. Then $u^{q+1}|\nabla u|^{p-2}$ is bounded.

Next, for any function $w$, we denote by

[TABLE]

the spherical average of $w$. We shall use the next result (see e.g. [15, Lemma 3.2]).

Lemma 2.2**.**

If $w\geq 0$ is superharmonic, then $\bar{w}$ is nonincreasing in $\mathbb{R}^{+}$ and

[TABLE]

Proof of Lemma 2.1.

Set

[TABLE]

for $m>0$ to be chosen later. We compute

[TABLE]

Substituting in (1.1), we obtain

[TABLE]

Hence, dividing by $mv^{m-1}$, we get

[TABLE]

For given $\alpha\in(0,1)$ any $x_{0}\in\mathbb{R}^{n}$, let $\eta\in C^{2}(\mathbb{R}^{n})$ be a cut-off function such that $0\leq\eta\leq 1$ in $\mathbb{R}^{n}$, $\eta=1$ in $B_{1}(x_{0})$ and $\eta=0$ in $B_{2}^{C}(x_{0})$ and such that

[TABLE]

with $C=C(\alpha,n)>0$ independent of $x_{0}$. Put

[TABLE]

so that

[TABLE]

Recalling the Bochner formula $\Delta w=2\langle\nabla(\Delta v),\nabla v\rangle+2|D^{2}v|^{2}$, where $|D^{2}v|^{2}=\sum_{ij}(v_{ij})^{2}$, and using (2.1), we compute

[TABLE]

Therefore, $w$ is a solution of

[TABLE]

while $z$ is a solution of

[TABLE]

Using $\eta\nabla w=\nabla z-w\nabla\eta$, we obtain

[TABLE]

As a consequence of the Cauchy-Schwarz inequality, we have

[TABLE]

Thus,

[TABLE]

Putting $\mathscr{L}z=-\Delta z-\langle b,\nabla z\rangle$, we get

[TABLE]

We now make the choice

[TABLE]

This yields

[TABLE]

and

[TABLE]

Hence, owing to $p>2$,

[TABLE]

Here and in the rest of the proof, $C$ denotes a generic positive constant depending only on $m,n,p,q$. For any $\varepsilon>0$, using

[TABLE]

we get

[TABLE]

where we used (2.2). Taking $\alpha=(p+1)/(p+2)\in(1/2,1)$, using Young’s inequality and the fact that $v$ is bounded, we have

[TABLE]

where $C_{\varepsilon}$ depends only on $m,n,p,q,\varepsilon$. Combining this with (2.4) and (2.6), we obtain

[TABLE]

Now choosing $\varepsilon=\frac{1}{2}m^{p-1}$, we get

[TABLE]

Since $z\geq 0$ has compact support, it attains its maximum at some point $\xi=\xi(x_{0})$, and at this point we have

[TABLE]

Hence

[TABLE]

Since $C$ is independent of $x_{0}$, we deduce that $|\nabla v|$ is bounded in $\mathbb{R}^{n}$, with

[TABLE]

From

[TABLE]

the conclusion of the lemma follows at once. ∎

Remark 2.1**.**

Lemma 2.1 remains valid for all $p>2$ and $q>1-p$. Note that in that case we have $m>0$ instead of $m\in(0,1)$ in (2.5), but the proof works without changes.**

Proof of Theorem 1.1.

Since $u$ is superharmonic, Lemma 2.2 gives

[TABLE]

Now, for $s\geq 2$ to be fixed later, set $z:=(u-\ell)^{s}\geq 0$. By direct computation we have

[TABLE]

Now choose $s\geq 1+\|u^{q+1}|\nabla u|^{p-2}\|_{\infty}$, noting that the latter quantity is finite thanks to Lemma 2.1 (this lemma is actually not required if $p=2$). It then follows that $\Delta z\geq 0$.

Thus, $z$ is subharmonic and bounded. Applying Lemma 2.2 to the superharmonic function $\|z\|_{\infty}-z\geq 0$, it follows that $\bar{z}$ is nondecreasing and that

[TABLE]

But

[TABLE]

We conclude that $z\equiv 0$, i.e. $u\equiv\ell$. ∎

Proof of Theorem 1.2.

We shall only prove assertion (i) since the proof of assertion (ii) is completely similar. Let $(u,v)$ be a classical solution of (1.1) such that $0\leq u,v\leq M$. Since $u$ is superharmonic, Lemma 2.2 gives

[TABLE]

Now, for $s\geq K(M)+1$ with $K(M)$ from (1.5), we set $z:=(u-\ell)^{s}\geq 0$. Again, direct computation shows that

[TABLE]

where we used assumption (1.5). Thus, $z$ is subharmonic and bounded. Applying Lemma 2.2 to the superharmonic function $\|z\|_{\infty}-z\geq 0$, it follows that $\bar{z}$ is nondecreasing and that

[TABLE]

But

[TABLE]

It follows that $z\equiv 0$, i.e. $u\equiv\ell$. By exchanging the roles of $u,v$ and of $f,g$, we deduce that $v$ is also constant. This completes the proof. ∎

Acknowledgements. Part of this work was done during a visit of PhS at the Dipartimento di Matematica e Informatica of the Università degli Studi di Perugia within the auspices of the INdAM – GNAMPA Projects 2018. He wishes to thank this institution for the kind hospitality. PhS is partly supported by the Labex MME-DII (ANR11-LBX-0023-01).

RF and PP were partly supported by the Italian MIUR project Variational methods, with applications to problems in mathematical physics and geometry (2015KB9WPT_009) and are members of the Gruppo Nazionale per l’Analisi Matematica, la Probabilità e le loro Applicazioni (GNAMPA) of the Istituto Nazionale di Alta Matematica (INdAM). The manuscript was realized within the auspices of the INdAM – GNAMPA Projects 2018 Problemi non lineari alle derivate parziali (Prot_U-UFMBAZ-2018-000384).