Nonlocal planar Schr\"odinger-Poisson systems in the fractional Sobolev limiting case

We study the nonlinear Schr\"odinger equation for the $s-$fractional $p-$Laplacian strongly coupled with the Poisson equation in dimension two and with $p=\frac2s$, which is the limiting case for the embedding of the fractional Sobolev space $W^{s,p}(\mathbb{R}^2)$. We prove existence of solutions by means of a variational approximating procedure for an auxiliary Choquard equation in which the uniformly approximated sign-changing logarithmic kernel competes with the exponential nonlinearity. Qualitative properties of solutions such as symmetry and decay are also established by exploiting a suitable moving planes technique

Skull base metastases from a malignant solitary fibrous tumor of the liver. A case report and literature review

Solitary fibrous tumors (SFTs) of the liver are rarely described; only 38 cases have been reported in literature, most of which have shown benign clinical characteristics, and only 3 of these cases exhibited malignant variants. In this study, we present a 24-year-old woman with a 1-month history of a rapidly enlarging abdominal mass and a CT showing an exophytic heterogeneous liver mass with a firm parietal bone mass. The patient underwent a transcatheter arterial chemoembolization (TACE) before operation, and an extended right hepatectomy and craniectomy with a negative margin was performed under general anesthesia. The masses showed histological features of oval spindle cells haphazardly arranged in the classic short-storiform or so-called patternless pattern of solitary fibrous tumors. The tumor cells showed positive immunohistochemical reactions to CD34 and bcl-2. The tumor recurred in the residual liver 2 months after operation, metastatic osteoblastic lesions in the thoracic and lumbar vertebrae were identified 3 months after the operation, and lumbar vertebrae metastasis 7 months after operation paralyzed the patient. The patient underwent percutaneous ethanol injection therapy (PEI) and chemotherapy, but the patient died because of the uncontrolled tumor 16 months after the initial operation. To our knowledge, this is the first case of malignant solitary fibrous liver tumors with skeletal metastasis

Feasibility and Efficacy of S-Adenosyl-L-methionine in Patients with HBV-Related HCC with Different BCLC Stages

Aims. To understand the feasibility and efficacy of treatment with SAMe in patients with hepatitis B-related HCC with different Barcelona Clinic Liver Cancer (BCLC) stages. Methods. We retrospectively enrolled 697 patients with BCLC early-stage (stages 0-A) and advanced-stage (stages B-C) HCC who underwent SAMe therapy (354 cases) or no SAMe therapy (343 cases). The baseline characteristics, postoperative recoveries, and 24-month overall survival rates of the patients in the 2 groups were compared. Cox regression model analysis was performed to confirm the independent variables influencing the survival rate. Results. For patients in the early-stage (BCLC stages A1–A4) group, little benefit of SAMe therapy was observed. For advanced-stage (BCLC B-C) patients, SAMe therapy reduced alanine aminotransferase (ALT) and aspartate transaminase (AST) levels and effectively delayed the recurrence time and enhanced the 24-month survival rate. Cox regression model analysis in the advanced-stage group revealed that treatment with SAMe, preoperative viral load, and Child-Pugh grade were independent variables influencing survival time. Conclusion. SAMe therapy exhibited protective and therapeutic efficacy for BCLC advanced-stage HBV-related HCC patients. And the efficacy of SAMe therapy should be further explored in randomized prospective clinical trials

Multiplicity and concentration of positive solutions for the fractional Schrödinger–Poisson systems with critical growth

In this paper, we study the multiplicity and concentration of solutions for the following critical fractional Schrödinger–Poisson system: \begin{eqnarray*} \left\{ \begin{array}{ll} \epsilon^{2s}(-\triangle)^{s} {u}+ V(x)u+\phi u =f(u)+|u|^{2^*_{s}-2}u &\mbox{in}\,\,\R^3, \\[2.5mm] \epsilon^{2t}(-\triangle)^{t}{\phi}=u^2 &\mbox{in}\,\, \R^3, \end{array} \right. \end{eqnarray*

Existence of positive ground state solutions for Kirchhoff type equation with general critical growth

We study the existence of positive ground state solutions for the nonlinear Kirchhoff type equation \begin{cases} \displaystyle -\bigg(a+b\int_{\mathbb R^3}|\nabla u|^2\bigg)\Delta {u}+V(x)u =f(u) & \mbox{in }\mathbb R^3, \\ \noalign{\medskip} u\in H^1(\mathbb R^3), \quad u> 0 & \mbox{in } \mathbb R^3, \end{cases} % where a,b> 0 are constants, $f\in C(\mathbb R,\mathbb R)$ has general critical growth. We generalize a Berestycki-Lions theorem about the critical case of Schrödinger equation to Kirchhoff type equation via variational methods. Moreover, some subcritical works on Kirchhoff type equation are extended to the current critical case

Multiple semiclassical states for coupled Schrödinger-Poisson equations with critical exponential growth

In this paper, we study the multiplicity of positive solutions for a class of Schrödinger-Poisson systems. Working in a variational setting, we prove the existence and multiplicity of positive solutions for the system when the Plank\u27s constant is small and the potential satisfies some suitable conditions. We show that the number of positive solutions depends on the profile of the potential and that each solution concentrates around its corresponding global minimum point of the potential in the semi-classical limit. We also study the exponential decay