Abstract
We show that a realization of the operator \({L=|x|^\alpha\Delta +c|x|^{\alpha-1} \frac{x}{|x|} \cdot\nabla -b|x|^{\alpha-2}}\) generates a semigroup in \({L^p(\mathbb{R}^N)}\) if and only if \({D_c=b+(N-2+c)^2/4 > 0}\) and \({s_1+\min\{0,2-\alpha\} < N/p < s_2+\max\{0,2-\alpha\}}\), where \({s_i}\) are the roots of the equation \({b+s(N-2+c-s)=0}\), or \({D_c=0}\) and \({s_0+\min\{0,2-\alpha\} < N/p < s_0+\max\{0,2-\alpha\}}\), where \({s_0}\) is the unique root of the above equation. The domain of the generator is also characterized.
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Metafune, G., Okazawa, N., Sobajima, M. et al. Scale invariant elliptic operators with singular coefficients. J. Evol. Equ. 16, 391–439 (2016). https://doi.org/10.1007/s00028-015-0307-1
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DOI: https://doi.org/10.1007/s00028-015-0307-1