In this paper, we establish the existence and uniqueness theorem for solutions of the exterior Dirichlet problem for Hessian quotient equations with prescribed asymptotic behavior at infinity. This extends the previous related results on the Monge-Ampère equations and on the Hessian equations, and rearranges them in a systematic way. Based on the Perron's method, the main ingredient of this paper is to construct some appropriate subsolutions of the Hessian quotient equation, which is realized by introducing some new quantities about the elementary symmetric functions and using them to analyze the corresponding ordinary differential equation related to the generalized radially symmetric subsolutions of the original equation.
In this paper, we obtain the uniqueness and existence of viscosity solutions with prescribed asymptotic behavior at infinity to Hessian equations in exterior domains.
In this paper, we obtain the uniqueness and existence of viscosity solutions with prescribed asymptotic behavior at infinity to Hessian quotient equations in exterior domains.
Prostatic cancer (PCa) is a prevalent form of malignancy based on its high associated levels of mortality and morbidity across the world. MicroRNAs (miRNAs) are significant in the advancement of prostatic cancer. The current study is aimed at exploring the potential roles of miR-373 in PCa. In turn, the study conducted a qRT-PCR test to determine the levels of mRNA. A western blot test was also executed in determining the protein level. The processes of transwell assay and wound healing were integrated in the detection of the potential for PCa cells to invade and migrate. The integration of dual luciferase reporter assay is critical in determining the levels of luciferase activity among prostatic cancer cells. Then, the results showed a net decrease of miR-373 within prostatic cancer cells and tissues. Upregulated miR-373 reduced the invasion and migration potential of PCa cells. Moreover, overexpressed miR-373 increased the levels of E-cadherin and FSP1 as epithelial cell markers. Similarly, the overregulation of miR-373 brought about the upregulation of mesenchymal markers (N-cadherin, Snail, and vimentin). The study predicted runt-related transcription factor 2 (RUNX2) to be a target of miR-373. The luciferase activity of PCa cells was decreased after the cotransfection with miR-373 mimics and RUNX2 3′ untranslated region (3′UTR) wild type (WT). Moreover, RUNX2 became upregulated in PCa cells and tissues. The upregulation of miR-373 decreased the mRNA and protein level of RUNX2. However, overexpressed RUNX2 abated the roles of miR-373 in the intrusion and migration of PCa cells and in regulating the expression of epithelial cell markers and mesenchymal markers. In short, miR-373 may regulate the EMT of PCa cells via targeting RUNX2. The miR-373/RUNX2 axis provides a therapeutic target for PCa.
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