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Goal of this paper is to study the following doubly nonlocal equation $(- \Delta)^s u + \mu u = (I_\alpha*F(u))F'(u) \quad {\rm{in}}\;{\mathbb{R}^N}\qquad\qquad\qquad\qquad ({\rm{P}}) $ in the case of general nonlinearities $ F \in C^1(\mathbb{R}) $ of Berestycki-Lions type, when $ N \geq 2 $ and $ \mu > 0 $ is fixed. Here $ (-\Delta)^s $, $ s \in (0, 1) $, denotes the fractional Laplacian, while the Hartree-type term is given by convolution with the Riesz potential $ I_{\alpha} $, $ \alpha \in (0, N) $. We prove existence of ground states of (P). Furthermore we obtain regularity and asymptotic decay of general solutions, extending some results contained in [23,61].

Background/aims: There have been few reports of primary carcinoma of the cystic duct (CCD) included in advanced cases. The aim of this study was to elucidate the clinical features of resected CCD., Methodology: Six cases of CCD were diagnosed in which the main carcinomatous component arose from the cystic bile duct, even if these carcinomas were accompanied by invasion beyond the cystic duct. Histopathologic findings (i.e., H.E. staining and cell proliferating potency assessed by ki-67 staining) were compared between the main lesion and invasive lesion of the CCD., Results: Abdominal ultrasonography revealed swelling of the gallbladder in 3 of the 6 patients, but not in the remaining 3, who were later diagnosed as having adenomyomatosis of the gallbladder, dystelectasis due to the carcinomatous infiltration, and atrophic gallbladder, respectively. On computed tomography, 4 of the 6 cases with nodular-type lesions fulfilling Farrar's criteria, the tumors showed contrast enhancement. Direct cholangiography demonstrated unilateral obstruction of the common bile duct in 4 out of the 6 cases. Intraductal ultrasonography revealed CCD in only 1 of the 6 cases. Advanced CCD shows 2 patterns of invasion; the hepatic hilum pattern and the confluence invasive pattern. The hepatic hilum pattern of invasion tends to be associated with a poorer prognosis. Histopathological study revealed papillary and/or well differentiated adenocarcinoma in the cases where the lesion predominantly involved the cystic duct, whereas those lesions which extended beyond the cystic duct were composed of moderate and/or poorly differentiated tubular adenocarcinoma. The latter was associated with a high cellular proliferative activity as assessed by immunocytochemical examination for ki-67. Invasion of the perineural space was often observed in the cases with advanced CCD., Conclusions: CCD showed the hepatic hilum and/or confluence pattern of invasion when the tumor extended beyond the cystic duct. CCD extending beyond the cystic duct was associated with more aggressive characteristics of the tumors, with perineural infiltration and histopathologic features resembling those of pancreatic cancer. It is concluded that CCDs extending beyond the cystic duct are more aggressive and associated with a poorer prognosis.