# well-posedness of elliptic pdes with gaussian weights

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.