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Let $d\geq 2$, $\delta \in (0,1)$ and let $\mathcal{L}_{d,\delta}$ be the second order differential operator defined by \begin{align*} \mathcal{L}_{d,\delta}(f)(x) = \Delta(f)(x)-\delta \|x\|^{\delta-2}\langle x;\nabla(f)(x)\rangle. \end{align*} Consider the weak formulation problem in $L^2(\mu_\delta)$: \begin{align*} \mathcal{L}_{d,\delta}(f) = g \end{align*} for some data $g \in L^2(\mu_\delta)$ such that \begin{align*} \int_{\mathbb{R}^d} g(x) \mu_\delta(dx) = \int_{\mathbb{R}^d} g(x) \exp\left(-\|x\|^{\delta}\right) dx = 0. \end{align*} What can be said about the solution $f$ (if it exists) of such a weak formulation problem? I am looking for any references regarding this type of PDE with singular coefficients. Note that you don't have Poincaré inequality.

Thanks in advance.

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  • $\begingroup$ you can use spherical harmonic and blow up analysis to get results in various weighted spaces. $\endgroup$
    – Math604
    Commented Feb 22, 2021 at 18:09
  • $\begingroup$ see this paper for the methods The Supercritical Lane–Emden–Fowler Equation in Exterior Domains Juan Dávila,Manuel del Pino & Monica Musso (this is where I learned these techniques from) $\endgroup$
    – Math604
    Commented Feb 22, 2021 at 20:09

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