Critical exponents of weighted Sobolev embeddings for radial functions

Abstract

We confirm in this paper that the number $q^{\ast }\left(\alpha \right)=\frac{p(N+\alpha )}{N-p}$ with $\alpha >0$ is exactly the critical exponent for the embedding from the weighted Sobolev space $W_r^{1,p}\left({\mathbb{R}}^N\right)$ of radial functions into $L^q({\mathbb{R}}^N,{\left|x\right|}^{\alpha })$ where $1<p<N$. We name $q^{\ast }\left(\alpha \right)$ as the upper Hénon–Sobolev critical exponent of the embedding.

Introduction

Let $D$ be a spherically symmetric open domain in ${\mathbb{R}}^N$ with $N⩾3$ and $B_R=\{x\in {\mathbb{R}}^N|\left|x\right|<R\}$. Let $C_0^{\infty }\left(D\right)$ denote the collection of smooth functions with compact support and $C_{0,r}^{\infty }\left(D\right)≔\{u\in C_0^{\infty }\left(D\right)|u\mathrm{is radial}\}$. For $1<p<N$, let $D_r^{1,p}\left({\mathbb{R}}^N\right)$ be the completion of $C_{0,r}^{\infty }\left({\mathbb{R}}^N\right)$ under the norm ${\Vert u\Vert }_{D_r^{1,p}\left({\mathbb{R}}^N\right)}={\left({\int }_{{\mathbb{R}}^N}{|\nabla u|}^{p}dx\right)}^{1∕p}$, and $W_{0,r}^{1,p}\left(B_R\right)$ be the completion of $C_{0,r}^{\infty }\left(B_R\right)$ with norm $\Vert u\Vert ={\left({\int }_{B_R}{|\nabla u|}^{p}dx\right)}^{1∕p}$. For $q⩾1,K\left(\left|x\right|\right)⩾0$, define

$$L^q(D;K\left(\left|x\right|\right))=\left\{u:D\to \mathbb{R}|u\mathrm{is measurable},{\int }_{D}K\left(\left|x\right|\right){\left|u\right|}^{q}dx<\infty \right\}.$$

Set $W_r^{1,p}({\mathbb{R}}^N;K\left(\left|x\right|\right))≔D_r^{1,p}\left({\mathbb{R}}^N\right)\cap L^p({\mathbb{R}}^N;K\left(\left|x\right|\right))$. Especially, for $K\left(\left|x\right|\right)\equiv 1$, we set $L^q\left(D\right)≔L^q(D;1)$, $W_r^{1,p}\left({\mathbb{R}}^N\right)≔W_r^{1,p}(R^N;1)$.

Let $V,Q$ be nonnegative, continuous in $(0,\infty )$ and satisfy

• there exist $a,a_0\in \mathbb{R}$ such that ${lim\; inf}_{r\to \infty }\frac{V\left(r\right)}{r^a}>0$, ${lim\; inf}_{r\to 0}\frac{V\left(r\right)}{r^{a_0}}>0$;

• there exist $b,b_0\in \mathbb{R}$ such that ${lim\; sup}_{r\to \infty }\frac{Q\left(r\right)}{r^b}<\infty$, ${lim\; sup}_{r\to 0}\frac{Q\left(r\right)}{r^{b_0}}<\infty ,Q\left(r\right)>0$.

Define

$$q_{\ast }=\left\{\begin{array}{cc}\frac{p^2(N-1+b)-ap}{p(N-1)+a(p-1)},\hfill & b⩾a>-p,\hfill \\ \frac{p(N+b)}{N-p},\hfill & b⩾-p,a⩽-p,\hfill \\ p,\hfill & b⩽max\{a,-p\},\hfill \end{array}\right.q^{\ast }=\left\{\begin{array}{cc}\frac{p(N+b_0)}{N-p},b_0⩾-p,\hfill & a_0⩾-p,\hfill \\ \frac{p^2(N-1+b_0)-a_{0}p}{p(N-1)+a_0(p-1)},\hfill & \begin{array}{c}-p⩾a_0⩾\frac{1-N}{p-1}p\hfill \\ \text{and}{b}_0⩾a_0,\hfill \end{array}\hfill \\ \infty ,\hfill & a_0⩽\frac{1-N}{p-1}p,b_0⩾a_0.\hfill \end{array}\right.$$

In [1], [2], Su, Wang and Willem proved the following embedding theorem.

Theorem A

Let $1<p<N$. Assume $\left(V\right)$ and $\left(Q\right)$, then we have embedding

$$W_r^{1,p}({\mathbb{R}}^N;V)\hookrightarrow L^q({\mathbb{R}}^N;Q)$$

for $q_{\ast }⩽q⩽q^{\ast }$ when $q^{\ast }<\infty$ and for $q_{\ast }⩽q<\infty$ when $q^{\ast }=\infty$. Furthermore, the embedding is compact for $q_{\ast }<q<q^{\ast }$. And if $b<max\{a,-p\}$ and $b_0>min\{-p,a_0\}$, the embedding is also compact for $q=p$.

We take $V\left(\left|x\right|\right)\equiv 1$ and $Q\left(\left|x\right|\right)={\left|x\right|}^{\alpha }$ with $\alpha ⩾-p$ in Theorem A and denote

$$q_{\ast }\left(\alpha \right)=\left\{\begin{array}{cc}p,\hfill & -p⩽\alpha ⩽0,\hfill \\ \frac{p(N+\alpha -1)}{N-1},\hfill & \alpha >0,\hfill \end{array}\right.q^{\ast }\left(\alpha \right)=\frac{p(N+\alpha )}{N-p}.$$

Then we can obtain the following embedding theorem.

Theorem B

Assume $1<p<N$ and $\alpha ⩾-p$. Then the embedding

$$W_r^{1,p}\left({\mathbb{R}}^N\right)\hookrightarrow L^q({\mathbb{R}}^N;{\left|x\right|}^{\alpha })$$

is continuous for $q_{\ast }\left(\alpha \right)⩽q⩽q^{\ast }\left(\alpha \right)$ and it is compact for $q_{\ast }\left(\alpha \right)<q<q^{\ast }\left(\alpha \right)$.

Restricted to the unit ball of ${\mathbb{R}}^N$, we have the following embedding theorem.

Theorem C

Assume $1<p<N,\alpha ⩾-p$. Then the embedding

$$W_{0,r}^{1,p}\left(B_1\right)\hookrightarrow L^q(B_1;{\left|x\right|}^{\alpha })$$

is continuous for $1⩽q⩽q^{\ast }\left(\alpha \right)$ and it is compact for $1⩽q<q^{\ast }\left(\alpha \right)$.

It is well-known in the literature (see [3], [4]) that for $\alpha =0$, $q^{\ast }\left(\alpha \right)=\frac{Np}{N-p}≕p^{\ast }$ ($q_{\ast }\left(\alpha \right)=p$, resp.) is exactly the upper (lower, resp.) critical exponent of the Sobolev embedding $W_r^{1,p}\left({\mathbb{R}}^N\right)\hookrightarrow L^q\left({\mathbb{R}}^N\right)$. This means that the embedding $W_r^{1,p}\left({\mathbb{R}}^N\right)\hookrightarrow L^q\left({\mathbb{R}}^N\right)$ is compact for all $q\in (p,p^{\ast })$ (see [5] for $p=2$) and it ceases to be true for $q=p$ and $q=p^{\ast }$. For $\alpha =-p$, $q^{\ast }\left(\alpha \right)=p$ is the Hardy critical exponent of the embeddings (1.1), (1.2). For $-p<\alpha <0$, $q^{\ast }\left(\alpha \right)$ was named as the upper Hardy–Sobolev critical exponents of the embeddings (1.1), (1.2), and it was proved in [6] that the embeddings were not compact as $q=q^{\ast }\left(\alpha \right)$ and $q=q_{\ast }\left(\alpha \right)$. However, there are no similar results in the literature for the case $\alpha >0$. In this paper we give a positive answer by proving that for $\alpha >0$, the number $q^{\ast }\left(\alpha \right)$ is exactly the upper critical exponent of embeddings (1.1), (1.2). This result may be of its own meaning. Precisely, we will prove the following theorems.

Theorem 1.1

Assume $\left(\mathrm{V}\right)$ and $\left(\mathrm{Q}\right)$ with $a_0⩾-p$, $b_0⩾-p$. There is no embedding from $W_r^{1,p}({\mathbb{R}}^N;V)$ into $L^q({\mathbb{R}}^N;Q)$ for any $q>q^{\ast }$. The embedding $W_r^{1,p}({\mathbb{R}}^N;V)\hookrightarrow L^{q^{\ast }}({\mathbb{R}}^N;Q)$ is not compact.

Following the arguments in [1], [2] we have the following embedding theorem on the unit ball of ${\mathbb{R}}^N$ with $\left(\mathrm{Q}\right)$ replaced by

• $Q:(0,1]\to (0,\infty )$ is continuous, and there is $b_0\in \mathbb{R}$ such that ${lim\; sup}_{r\to 0}\frac{Q\left(r\right)}{r^{b_0}}<\infty$.

Theorem 1.2

Assume $\left({\mathrm{Q}}^{\prime }\right)$ with $b_0⩾-p$. Then the embedding $W_{0,r}^{1,p}\left(B_1\right)\hookrightarrow L^q(B_1;Q)$ is continuous for $1⩽q⩽q^{\ast }$ and it is compact for $1⩽q<q^{\ast }$.

We note here that Theorem C is a corollary of Theorem 1.2. The case $p=2$ of Theorem C was first proved by Ni in [7].

Theorem 1.3

Assume $\left({\mathrm{Q}}^{\prime }\right)$ with $b_0⩾-p$. There is no embedding from $W_{0,r}^{1,p}\left(B_1\right)$ into $L^q(B_1;Q)$ for any $q>q^{\ast }$. The embedding $W_{0,r}^{1,p}\left(B_1\right)\hookrightarrow L^{q^{\ast }}(B_1;Q)$ is not compact.

By Theorem A and Theorem 1.1, Theorem 1.3 we see that $q^{\ast }\left(\alpha \right)$ for all $\alpha ⩾-p$ is critical exponent for the embeddings (1.1), (1.2). We will name $q^{\ast }\left(\alpha \right)$ with $\alpha >0$ the upper Hénon–Sobolev critical exponent for the embeddings due to the reason that Hénon [8] first raised the model of the semilinear elliptic equation with $\alpha >0$

$$\left\{\begin{array}{cc}-\Delta u={\left|x\right|}^{\alpha }{u}^{q-1},u>0\hfill & \mathrm{in}{B}_1,\hfill \\ u=0\hfill & \mathrm{on}\partial B_1\hfill \end{array}\right.$$

in studying the rotating stellar structures in 1973. From the above conclusions, we see that the upper critical exponents of the embeddings depend on behavior of weighted functions near the origin. For $-p⩽\alpha ⩽0$, $q_{\ast }\left(\alpha \right)=p$ is the lower critical exponent of the embedding (1.1), see [4] for the case $p=2,\alpha =0$. It is still an open problem whether or not $q_{\ast }\left(\alpha \right)$ with $\alpha >0$ is the lower critical exponent for the embedding (1.1).

The elliptic equations involving critical exponents are very important topics in the applications of variational methods since the appearance of the pioneering work of Brezis and Nirenberg [9]. Due to the above facts in the forthcoming papers [10], [11], [12], [13], [14] we apply the variational methods to deal with the existence of the solutions for elliptic equations with upper Hénon–Sobolev critical exponent.