Related papers: Bound states of two-dimensional Schr\"{o}dinger-Ne…
We investigate the ground states for the focusing, subcritical nonlinear Schr\"odinger equation with a point defect in dimension two, defined as the minimizers of the energy functional at fixed mass. We prove that ground states exist for…
We prove the existence of a class of orbitally stable bound state solutions to nonlinear Schr\"odinger equations with super-quadratic confinement in two and three spatial dimensions. These solutions are given by time-dependent rotations of…
The excited states of a charged particle interacting with the quantized electromagnetic field and an external potential all decay, but such a particle should have a true ground state--one that minimizes the energy and satisfies the…
We prove the existence of ground state solutions for a class of nonlinear elliptic equations, arising in the production of standing wave solutions to an associated family of nonlinear Schr\"odinger equations. We examine two constrained…
We study the ground states for the Schr\"odinger equation with a focusing nonlinearity and a point interaction in dimension three. We establish that ground states exist for every value of the mass; moreover they are positive, radially…
We study the existence, the nonexistence, and the shape of the ground states of a Nonlinear Schr\"odinger Equation on a manifold called hybrid plane, that consists of a half-line whose origin is connected to a plane. The nonlinearity is of…
We prove the existence of bound and ground states for a system of coupled nonlinear Schr\"odinger-Korteweg-de Vries equations, depending on the size of the coupling coefficient.
This paper is devoted to studying the following nonlinear biharmonic Schr\"odinger equation with combined power-type nonlinearities \begin{equation*} \begin{aligned} \Delta^{2}u-\lambda u=\mu|u|^{q-2}u+|u|^{4^*-2}u\quad\mathrm{in}\…
We investigate the existence and the singular limit of normalized ground states for focusing doubly nonlinear Schr\"odinger equations with both standard and concentrated nonlinearities on two-dimensional square grids. First, we provide…
We establish the uniqueness of ground states of some coupled nonlinear Schrodinger systems in the whole space. We firstly use Schwartz symmetrization to obtain the existence of ground states for a more general case. To prove the uniqueness…
We study existence and properties of ground states for the nonlinear Schr\"odinger equation with combined power nonlinearities \[ -\Delta u= \lambda u + \mu |u|^{q-2} u + |u|^{p-2} u \qquad \text{in $\mathbb{R}^N$, $N \ge 1$,} \] having…
We prove the existence of normalized ground state solutions for the biharmonic Schr\"odinger equation with combined nonlinearities and show that all ground states correspond to the local minima of the associated energy functional restricted…
We prove the existence of a ground state of the Maxwell--Schr\"odinger equations in one spatial dimension, describing a specified amount of free charge under the influence of a fixed charge. For one case (equal free and fixed charge, i.e.,…
We study the existence of ground states for the coupled Schr\"odinger system \begin{equation} \label{ellipticabstract} \left\{ \begin{array}{llll} -\Delta u+u&=&|u|^{2q-2}u+b|v|^q|u|^{q-2}u\\ -\Delta…
A class of nonlinear Schroedinger equations with critical power-nonlinearities and potentials exhibiting multiple anisotropic inverse square singularities is investigated. Conditions on strength, location, and orientation of singularities…
We prove existence of positive ground state solutions to the pseudo-relativistic Schr\"{o}dinger equation \begin{equation*} \left\{ \begin{array}{l} \sqrt{-\Delta +m^2} u +Vu = \left( W * |u|^{\theta} \right)|u|^{\theta -2} u \quad\text{in…
We prove the existence of ground state solutions for the nonlinear Schrodinger-Maxwell equations.
We prove the uniqueness of positive radial solutions for a class of quasi-linear elliptic problems containing, in particular, the quasi-linear Schrodinger equation.
Exactly solvable models play an extremely important role in many fields of quantum physics. In this study, the Schr\"{o}dinger equation is applied for a solution of a two--dimensional (2D) problem for two particles interacting via Kratzer,…
We establish uniqueness and radial symmetry of ground states for higher-order nonlinear Schr\"odinger and Hartree equations whose higher-order differentials have small coefficients. As an application, we obtain error estimates for…