Non-radial sign-changing solutions for the Schroedinger-Poisson problem in the semiclassical limit

We study the existence of nonradial sign-changing solutions to the Schroedinger-Poisson system in dimension N>=3. We construct nonradial sign-changing multi-peak solutions whose peaks are displaced in suitable symmetric configurations and collapse to the same point. The proof is based on the Lyapunov-Schmidt reduction

Low Energy Solutions for the Semiclassical Limit of Schrodinger–Maxwell Systems

We show that the number of positive solutions of Schrodinger– Maxwell system on a smooth bounded domain depends on the topological properties of the domain. In particular we consider the Lusternik– Schnirelmann category and the Poincaré polynomial of the domain

Nonlinear Klein-Gordon-Maxwell systems with Neumann boundary conditions on a Riemannian manifold with boundary

Let (M,g) be a smooth compact, n dimensional Riemannian manifold, n=3,4 with smooth n-1 dimensional boundary. We search the positive solutions of the singularly perturbed Klein Gordon Maxwell Proca system with homogeneous Neumann boundary conditions or for the singularly perturbed Klein Gordon Maxwell system with mixed Dirichlet Neumann homogeneous boundary conditions. We prove that stable critical points of the mean curvature of the boundary generates solutions when the perturbation parameter is sufficiently small.Comment: arXiv admin note: text overlap with arXiv:1410.884

Relativistic point dynamics and Einstein formula as a property of localized solutions of a nonlinear Klein-Gordon equation

Einstein's relation E=Mc^2 between the energy E and the mass M is the cornerstone of the relativity theory. This relation is often derived in a context of the relativistic theory for closed systems which do not accelerate. By contrast, Newtonian approach to the mass is based on an accelerated motion. We study here a particular neoclassical field model of a particle governed by a nonlinear Klein-Gordon (KG) field equation. We prove that if a solution to the nonlinear KG equation and its energy density concentrate at a trajectory, then this trajectory and the energy must satisfy the relativistic version of Newton's law with the mass satisfying Einstein's relation. Therefore the internal energy of a localized wave affects its acceleration in an external field as the inertial mass does in Newtonian mechanics. We demonstrate that the "concentration" assumptions hold for a wide class of rectilinear accelerating motions

Stable standing waves for a class of nonlinear Schroedinger-Poisson equations

We prove the existence of orbitally stable standing waves with prescribed $L^2$-norm for the following Schr\"odinger-Poisson type equation \label{intro} %{%{ll} i\psi_{t}+ \Delta \psi - (|x|^{-1}*|\psi|^{2}) \psi+|\psi|^{p-2}\psi=0 \text{in} \R^{3}, %-\Delta\phi= |\psi|^{2}& \text{in} \R^{3},%. when $p\in \{8/3\}\cup (3,10/3)$. In the case $3<p<10/3$ we prove the existence and stability only for sufficiently large $L^2$-norm. In case $p=8/3$ our approach recovers the result of Sanchez and Soler \cite{SS} %concerning the existence and stability for sufficiently small charges. The main point is the analysis of the compactness of minimizing sequences for the related constrained minimization problem. In a final section a further application to the Schr\"odinger equation involving the biharmonic operator is given

Tamoxifen in treatment of hepatocellular carcinoma: a randomised controlled trial

Background Results from small randomised trials on tamoxifen in the treatment of hepatocellular carcinoma (HCC) are conflicting, We studied whether the addition of tamoxifen to best supportive care prolongs survival of patients with HCC. Methods Patients with any stage of HCC were eligible, irrespective of locoregional treatment. Randomisation was centralised, with a minimisation procedure accounting for centre, evidence of disease, and time from diagnosis. Patients were randomly allocated best supportive care alone or in addition to tamoxifen, Tamoxifen was given orally, 40 mg per day, from randomisation until death. Results 496 patients from 30 institutions were randomly allocated treatment from January, 1995, to January, 1997. Information was available for 477 patients. By Sept 15, 1997, 119 (50%) of 240 and 130 (55%) of 237 patients had died in the control and tamoxifen arms, respectively. Median survival was 16 months and 15 months (p=0.54), respectively, No differences were found within subgroups defined by prognostic variables. Relative hazard of death for patients receiving tamoxifen was 1.07 (95% CI 0.83-1.39). Interpretation Our findings show that tamoxifen is not effective in prolonging survival of patients with HCC

Non-radial sign-changing solutions for the Schrödinger–Poisson problem in the semiclassical limit

We study the Schroedinger– Poisson problem in R^N and construct non-radial sign-changing multi-peak solutions in the semiclassical limit. The peaks are displaced in suitable symmetric configurations and collapse to the same point as the parameter ε → 0. The proof is based on the Lyapunov–Schmidt reduction