Consider the weighted $L^p_\omega(\mathbb{R}^d)$ space on $\mathbb{R}^d$ be the set of Lebesgue measurable functions such that $$\|f\|_{L^p_\omega}=\int_{\mathbb{R}^d}|f|^p\omega_\mu(x)\,dx< \infty,$$ where $\omega_\mu(x)=\exp(-\mu |x|^2)$ or $\omega_\mu(x)=\exp(-\mu\sqrt{1+|x|^2})$ for some $\mu>0$. Consider the weighted sobolev space $W^{1,p}_\omega(\mathbb{R}^d)$ such that $$W^{1,p}_\omega(\mathbb{R}^d)=\{u\in L^p_\omega\mid \partial_{x_i}u\in L^p_\omega,\, i=1,\ldots, d \},$$ where $\partial_{x_i}$ denotes the weak derivative in the distribution sense. Similarly we define the high-order sobolev space $W^{2,p}_\omega(\mathbb{R}^d)$ such that $\partial_{x_ix_j}u\in L^p_\omega$ for all $i,j$.
I was wondering whether the Gagliardo–Nirenberg–Sobolev inequality for the classical Sobolev space still holds? In particular, whether for $1\le p<n$, we have $$ \|u\|_{L^{p^*}_\omega(\mathbf{R}^n)}\leq C \|Du\|_{L^{p}_\omega(\mathbf{R}^n)}.$$
with $1/p* = 1/p - 1/n$? Do you have a reference for this?
The main motivation is that I would like to find some weighted spaces on $\mathbb{R}^n$, such that $L^p_\omega(\mathbb{R}^n)\subset L^q_\omega(\mathbb{R}^n)$ if $q\ge p>1$, and then study the elliptic equation on $\mathbb{R}^n$, say $$ -\partial_i a^{ij}(x)\partial_j u+ru=f(x), \quad x\in \mathbb{R}^d, $$ where $a^{ij}$ is strictly elliptic. That is why I introduce this decaying weight, such that $\omega (x)dx$ is a finite measure on $\mathbb{R}^n$. The exponential weight is introduced to involve bounded functions or functions with polynomial growth at infinity.