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29. Synchronous Urinary Bladder Urothelial Carcinoma and Transitional Cell Carcinoma of the Ovary: A Case Report

Author

Ivana Rizzuto, Laura Slade, Yang Yanling, and Martin Klaus Oehler

Subjects
transitional cell carcinoma, ovarian metastases, bladder cancer, Neoplasms. Tumors. Oncology. Including cancer and carcinogens, RC254-282
Abstract

Transitional cell carcinoma (TCC) of the ovary is a rare subtype of epithelial ovarian tumours defined as a tumour composed of epithelial elements, histologically resembling urothelium and its neoplasms. Ovarian metastases from primary urinary tract carcinomas are rare. The differential diagnosis of primary TCC of the ovary versus metastatic bladder TCC is challenging because of histological similarity. We present the case of a 49-year-old premenopausal woman who was initially diagnosed with non-invasive papillary urothelial carcinoma of bladder (NIPUC) and after 2 years with a synchronous TCC of the ovary while being investigated for suspected relapse. She underwent a radical cystectomy, total hysterectomy, bilateral salpingo-oopharectomy, and pelvic lymph node dissection. The final diagnosis of synchronous NIPUC of the bladder and TCC of the ovary was made by histopathology and immunohistochemical studies.

Published
2023
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30. Nodal solutions with a prescribed number of nodes for the Kirchhoff-type problem with an asymptotically cubic term

Author

Wang Tao, Yang Yanling, and Guo Hui

Subjects
kirchhoff equation, nodal solutions, miranda theorem, gluing method, 35j20, 35j60, 35q40, Analysis, QA299.6-433
Abstract

In this article, we study the following Kirchhoff equation: (0.1)−(a+b‖∇u‖L2(R3)2)Δu+V(∣x∣)u=f(u)inR3,-(a+b\Vert \nabla u{\Vert }_{{L}^{2}\left({{\mathbb{R}}}^{3})}^{2})\Delta u+V\left(| x| )u=f\left(u)\hspace{1.0em}\hspace{0.1em}\text{in}\hspace{0.1em}\hspace{0.33em}{{\mathbb{R}}}^{3}, where a,b>0a,b\gt 0, VV is a positive radial potential function, and f(u)f\left(u) is an asymptotically cubic term. The nonlocal term b‖∇u‖L2(R3)2Δub\Vert \nabla u{\Vert }_{{L}^{2}\left({{\mathbb{R}}}^{3})}^{2}\Delta u is 3-homogeneous in the sense that b‖∇tu‖L2(R3)2Δ(tu)=t3b‖∇u‖L2(R3)2Δub\Vert \nabla tu{\Vert }_{{L}^{2}\left({{\mathbb{R}}}^{3})}^{2}\Delta \left(tu)={t}^{3}b\Vert \nabla u{\Vert }_{{L}^{2}\left({{\mathbb{R}}}^{3})}^{2}\Delta u, so it competes complicatedly with the asymptotically cubic term f(u)f\left(u), which is totally different from the super-cubic case. By using the Miranda theorem and classifying the domain partitions, via the gluing method and variational method, we prove that for each positive integer kk, equation (0.1) has a radial nodal solution Uk,4b{U}_{k,4}^{b}, which has exactly k+1k+1 nodal domains. Moreover, we show that the energy of Uk,4b{U}_{k,4}^{b} is strictly increasing in kk, and for any sequence {bn}→0+,\left\{{b}_{n}\right\}\to {0}_{+}, up to a subsequence, Uk,4bn{U}_{k,4}^{{b}_{n}} converges strongly to Uk,40{U}_{k,4}^{0} in H1(R3){H}^{1}\left({{\mathbb{R}}}^{3}), where Uk,40{U}_{k,4}^{0} also has k+1k+1 nodal domains exactly and solves the classical Schrödinger equation: −aΔu+V(∣x∣)u=f(u)inR3.-a\Delta u+V\left(| x| )u=f\left(u)\hspace{1.0em}\hspace{0.1em}\text{in}\hspace{0.1em}\hspace{0.33em}{{\mathbb{R}}}^{3}. Our results extend the ones in Deng et al. from the super-cubic case to the asymptotically cubic case.

Published
2023
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