Finding a solutions for an equation

fa.useful evaluation – Reference Request: Dirichlet operators with eccentric coefficients retort

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fa.useful evaluation – Reference Request: Dirichlet operators with eccentric coefficients

Let $dgeq 2$, $delta in (0,1)$ and let $mathcal{L}_{d,delta}$ breathe the second bid differential operator outlined by
commence{align*}
mathcal{L}_{d,delta}(f)(x) = Delta(f)(x)-delta |x|^{delta-2}langle x;nabla(f)(x)rangle.
stop{align*}

assume the feeble formulation downside in $L^2(mu_delta)$:
commence{align*}
mathcal{L}_{d,delta}(f) = g
stop{align*}

for some knowledge $g in L^2(mu_delta)$ such that
commence{align*}
int_{mathbb{R}^d} g(x) mu_delta(dx) = int_{mathbb{R}^d} g(x) expleft(-|x|^{delta}capable) dx = 0.
stop{align*}

What can breathe stated in regards to the resolution $f$ (if it exists) of such a feeble formulation downside? I’m in search of any references concerning this kindly of PDE with eccentric coefficients. level to that you do not maintain Poincaré inequality.

Thanks in close by.

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