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Let $\gamma_n: \mathbb{R}^n\to\mathbb{R}$ be the Gaussian distribution function defined by $$ \gamma_n(x):=(2 \pi)^{-\frac{n}{2}} e^{-\frac{|x|^2}{2}}. $$ Let $d\gamma_n$ denote the following measure (weight) $\gamma_n(x) dx$ and consequently $L^2(\mathbb{R}^n,d\gamma_n)$ and $H^k(\mathbb{R}^n,d\gamma_n)$, $k=1,2$, be the weighted versions of the Sobolev spaces with respect to the measure $d\gamma_n$.

We consider the following elliptic PDE on $\mathbb{R}^n$: $$ -a^{ij}(x)\partial_{ij}u+b^i\partial_iu+c(x)u=f,\tag{1} $$ where $(a^{i,j})$ satisfies the uniform elliptic condition on $\mathbb{R}^n$, and all coefficients $a^{i,j}$, $b^i$, $c$ are bounded and sufficiently smooth.

Then is there any reference saying whether the following well-posedness result holds or not: for all $f\in L^2(\mathbb{R}^n,d\gamma_n)$, (1) admits a unique solution $u\in H^2(\mathbb{R}^n,d\gamma_n)$?


I found a reference stating (without proof) that for $\gamma=\exp(-\mu\sqrt{1+|x|^2})$ with any $\mu>0$, the elliptic pde is well-posed in the corresponding weighted spaces.

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