Symmetry results for critical anisotropic p-Laplacian equations in convex cones

Abstract

Given $n \ge 2$ and $1<p<n$, we consider the critical p-Laplacian equation $\Delta _p u + u^{p^*-1}=0$, which corresponds to critical points of the Sobolev inequality. Exploiting the moving planes method, it has been recently shown that positive solutions in the whole space are classified. Since the moving plane method strongly relies on the symmetries of the equation and the domain, in this paper we provide a new approach to this Liouville-type problem that allows us to give a complete classification of solutions in an anisotropic setting. More precisely, we characterize solutions to the critical p-Laplacian equation induced by a smooth norm inside any convex cone. In addition, using optimal transport, we prove a general class of (weighted) anisotropic Sobolev inequalities inside arbitrary convex cones.

Appendix A. Sharp anisotropic Sobolev inequalities with weight in convex cones

In this appendix we prove a sharp version of the anisotropic Sobolev inequality in cones by suitably adapting the optimal transportation proof of the Sobolev inequality in [CNV04, Theorem 2]. As we shall see, the proof not only applies to the case of arbitrary norms, but it also allows us to cover a large class of weights. In particular, our result extends the weighted isoperimetric inequalities from [CRS16, Theorem 1.3] to the full Sobolev range $p \in (1,n)$ (note that the case $p=1$ can be recovered letting $p\rightarrow 1^+$).

Theorem A.1

Let $p\in (1,n)$. Let $\Sigma$ be a convex cone and H a norm in ${\mathbb {R}}^n$. Let $w \in C^0({\overline{\Sigma }})$ be positive in $\Sigma$, homogeneous of degree $a \ge 0$, and such that $w^{1/a}$ is concave in case $a>0$. Then for any $f\in \mathcal {D}^{1,p}(\Sigma )$ we have

$$\begin{aligned} \biggl (\int _{\Sigma }|f(x)|^{\beta }w(x)\, dx\biggr )^{p/\beta } \le C_\Sigma (n,p,a,H,w) \int _{\Sigma }H^p(\nabla f(x))\, w (x) \,dx \end{aligned}$$

(A.1)

where

$$\begin{aligned} \beta =\frac{p(n+a)}{n+a-p} \,. \end{aligned}$$

(A.2)

Moreover, inequality (A.1) is sharp and the equality is attained if and only if $f=U_{\lambda ,x_0}^{H,a}$, where

$$\begin{aligned} U_{\lambda ,x_0}^{H,a} (x) := \left( \frac{\lambda ^{\frac{1}{p-1}}c(n,p,a,H,w) }{\lambda ^\frac{p}{p-1} + {\hat{H}}_0(x-x_0)^\frac{p}{p-1} } \right) ^{\frac{n+a-p}{p}} \end{aligned}$$

(A.3)

with $\lambda >0$ and ${\hat{H}}_0$ given by (1.7).

Furthermore, writing $\Sigma ={\mathbb {R}}^k\times \mathcal {C}$ with $k\in \{0,\dots ,n\}$ and with ${\mathcal {C}} \subset {\mathbb {R}}^{n-k}$ a convex cone that does not contain a line, then:

(i):

if $k=n$ then $\Sigma = {\mathbb {R}}^n$ and $x_0$ may be a generic point in ${\mathbb {R}}^n$;

(ii):

if $k\in \{1,\dots ,n-1\}$ then $x_0\in {\mathbb {R}}^k\times \mathcal {\{{\mathcal {O}}\}}$;

(iii):

if $k=0$ then $x_0=\mathcal {O}$.

Proof

We aim at proving that for any nonnegative $f,g \in L^{\beta }( \Sigma )$ with $\Vert f\Vert _{L^{\beta }(\Sigma )}=\Vert g\Vert _{L^{\beta }(\Sigma )}$ and such that $\nabla f \in L^p(\Sigma )$, we have that

$$\begin{aligned} \int _{\Sigma }g^\gamma w\,dx\le \dfrac{\gamma }{n+a}\left( \int _{\Sigma } H^p(\nabla f)\,w\,dx \right) ^{1/p}\left( \int _{\Sigma }H_0^{p'}g^\beta w\,dx\right) ^{1/p'}\, , \end{aligned}$$

(A.4)

with equality if $f=g=U_{\lambda ,x_0}^{H,a}$. The value of $\gamma$ will be specified later. As shown in [CNV04], inequality (A.4) implies the Sobolev inequality (A.1).

Let F and G be probability densities on $\Sigma$ and let $T:\Sigma \rightarrow \Sigma$ be the optimal transport map (see e.g. [Vil03]).³ It is well known that, by the transport condition $T_\# F=G$, one has

$$\begin{aligned} |\det (DT)|=\dfrac{F}{G\circ T} \end{aligned}$$

(see for instance [DF14, Section 3]). Then, if we choose

$$\begin{aligned} F=f^\beta w \quad \text {and} \quad G=g^\beta w \,, \end{aligned}$$

the Jacobian equation for T becomes

$$\begin{aligned} |\det (DT)|\,\dfrac{w\circ T}{w}=\dfrac{f^\beta }{g^\beta \circ T}\, . \end{aligned}$$

We observe that, since

$$\begin{aligned} T_{\#}(f^\beta w)=g^\beta w\, , \end{aligned}$$

then for any $0<\gamma <\beta$ we have

$$\begin{aligned} \int _{\Sigma }g^\gamma w\,dx= \int _{\Sigma }(g^{\gamma -\beta }\circ T ) f^\beta w\,dx = \int _{\Sigma } \left[ |\det (DT)|\,\dfrac{w\circ T}{w}\right] ^{\frac{\beta -\gamma }{\beta }}f^\gamma w\,dx\, . \end{aligned}$$

(A.5)

We choose $\gamma$ such that

$$\begin{aligned} \dfrac{\beta -\gamma }{\beta }=\dfrac{1}{n+a}\, \quad \text {i.e.} \quad \gamma =\frac{p(n+a-1)}{n+a-p} \,. \end{aligned}$$

Since $T=\nabla \varphi$ for some convex function $\varphi$, then DT is symmetric and nonnegative definite. In particular $\det (DT)\ge 0$, and it follows from Young and the arithmetic-geometric inequalities that

$$\begin{aligned} \begin{aligned} \left[ |\det (DT)|\,\dfrac{w\circ T}{w}\right] ^{\frac{1}{n+a}}&\le \dfrac{n}{n+a}\det (DT)^{1/n}+\dfrac{a}{n+a}\left( \dfrac{w\circ T}{w}\right) ^{1/a} \\&\le \dfrac{1}{n+a}\left[ \mathrm{div \,}(T)+a\left( \dfrac{w\circ T}{w}\right) ^{1/a}\right] \, . \end{aligned} \end{aligned}$$

Also, from the concavity of $w^{1/a}$ we have that

$$\begin{aligned} a\left( \dfrac{w\circ T}{w}\right) ^{1/a} \le \frac{\nabla w \cdot T}{w} \end{aligned}$$

(see [CRS16, Lemma 5.1]), hence

$$\begin{aligned} \left[ |\det (DT)|\,\dfrac{w\circ T}{w}\right] ^{\frac{1}{n+a}} \le \dfrac{1}{n+a}\left( \mathrm{div \,}(T)+ \frac{\nabla w \cdot T}{w} \right) \, . \end{aligned}$$

(A.6)

(If $a=0$ then w is just constant and (A.6) corresponds to the arithmetic-geometric inequality.) Noticing that

$$\begin{aligned} \mathrm{div \,}(T)+ \frac{\nabla w \cdot T}{w} = \frac{1}{w} \mathrm{div \,}(T w)\, \end{aligned}$$

combining (A.5) and (A.6) we have

$$\begin{aligned} \begin{aligned} \int _{\Sigma }g^\gamma w\,dx&\le \dfrac{1}{n+a}\int _{\Sigma }\mathrm{div \,}(T w) f^{\gamma }\,dx \\&=-\dfrac{\gamma }{n+a}\int _{\Sigma }w f^{\gamma -1}T\cdot \nabla f \,dx+ \dfrac{1}{n+a}\int _{\partial \Sigma }w f^\gamma T\cdot \nu \,d\sigma \, . \end{aligned} \end{aligned}$$

Here we notice that, since $T(x)\in {\overline{\Sigma }}$ for any $x \in {\overline{\Sigma }}$, the convexity of $\Sigma$ implies that $T \cdot \nu \le 0$ on $\partial \Sigma$. Thus we obtain

$$\begin{aligned} \int _{\Sigma }g^\gamma w \,dx\le -\dfrac{\gamma }{n+a}\int _{\Sigma } f^{\gamma -1}T\cdot \nabla f\,w \,dx \le \dfrac{\gamma }{n+a}\int _{\Sigma } f^{\gamma -1}{\hat{H}}_0(T)H(\nabla f)\,w\,dx\, , \end{aligned}$$

where the last inequality follows from the definition of the dual norm $H_0$ of H, and since ${\hat{H}}_0(x)=H_0(-x)$. Finally, setting $p'=\frac{p}{p-1}$, it follows by Holder’s inequality that

$$\begin{aligned} \begin{aligned} \int _{\Sigma } f^{\gamma -1}{\hat{H}}_0(T)H(\nabla f) \,w\,dx&\le \left( \int _{\Sigma } f^{p(\gamma -1)-\frac{p\beta }{p'}}H^p(\nabla f)\,w\,dx\right) ^{1/p}\left( \int _{\Sigma } {\hat{H}}_0^{p'}(T)\,f^{\beta }w\,dx \right) ^{1/p'}\\&=\left( \int _{\Sigma } H^p(\nabla f)\,w\,dx\right) ^{1/p}\left( \int _{\Sigma } {\hat{H}}_0^{p'}g^{\beta }w\,dx\right) ^{1/p'} \, , \end{aligned} \end{aligned}$$

where we used the transport condition $T_\#(f^\beta w)=g^\beta w$ and the identity

$$\begin{aligned} \gamma -1-\frac{\beta }{p'}=0\, . \end{aligned}$$

Hence, by this chain of inequalities we get (A.4).

In order to prove the sharpness of our Sobolev inequality we choose $f=g=U_{1,{\mathcal {O}}}^{H,a}$. In this particular case the transport map reduces to the identity map $T(x)=\nabla \varphi (x) = x$ and $\det (DT)=1$. Also the homogeneity of w implies that $\nabla w\cdot x=a\,w$. This implies that all the inequalities in the previous computations become equalities and we obtain (A.1).

Finally, to prove the characterization of the minimizers one can argue as in [FMP10, Appendix A] and [CNV04, Section 4]. More precisely, choose $g=U_{1,{\mathcal {O}}}^{H,a}$ and let f be a minimizer. As noticed in the proof of [CNV04, Theorem 5], one can assume that $f \ge 0$.

First one shows that the support of f is indecomposable (this is a measure-theoretic notion of the concept that $\{f>0\}$ is connected, see [FMP10, Appendix A] for a definition and more details). Indeed, otherwise one could write $f=f_1+f_2$ where $f_1$ and $f_2$ have disjoint supports. Then

$$\begin{aligned} \int _{\Sigma }H^p(\nabla f) w (x) dx=\int _{\Sigma }H^p(\nabla f_1) w (x) dx+\int _{\Sigma }H^p(\nabla f_2) w (x) dx \end{aligned}$$

and then by applying (A.1) and the fact that f is a minimizer, we would get

$$\begin{aligned} \biggl (\int _{\Sigma }f^{\beta }w(x) dx\biggr )^{p/\beta }\ge \biggl (\int _{\Sigma }f_1^{\beta }w(x) dx\biggr )^{p/\beta }+\biggl (\int _{\Sigma }f_2^{\beta }w(x) dx\biggr )^{p/\beta }. \end{aligned}$$

Since

$$\begin{aligned} \int _{\Sigma }f^{\beta }w(x) dx=\int _{\Sigma }f_1^{\beta }w(x) dx+\int _{\Sigma }f_2^{\beta }w(x) dx \end{aligned}$$

(because $f_1$ and $f_2$ have disjoint support), by concavity of the function $t\mapsto t^{p/\beta }$ we conclude that either $f_1$ or $f_2$ vanishes.

Once this is proved, one can then argue as in the proof of [CNV04, Proposition 6] to deduce (from the fact that all the inequalities in the proof given above must be equalities) that T must be of the form $T(x)=\lambda (x-x_0)$ for some $\lambda >0$ and $x_0 \in \Sigma$, from which the result follows easily. Finally, properties (i)-(ii)-(iii) on the location of $x_0$ follow for instance from the fact that T has to map $\Sigma$ onto $\Sigma$. $\square$