On a class of nonlinear Schrödinger equations

This paper concerns the existence of standing wave solutions of nonlinear Schrödinger equations. Making a standing wave ansatz reduces the problem to that of studying the semilinear elliptic equation:

Bound states for semilinear Schrödinger equations with sign-changing potential

We study the existence and the number of decaying solutions for the semilinear Schrödinger equations \({-\varepsilon^{2}\Delta u + V(x)u = g(x,u)}\), \({\varepsilon > 0}\) small, and \({-\Delta u + \lambda V(x)u = g(x,u)}\), \({\lambda > 0}\) large. The potential V may change sign and g is either asymptotically linear or superlinear (but subcritical) in u as \({|u| \to \infty}\) .     

On a class of nonlinear Schrödinger equations with nonnegative potentials in two space dimensions

Mathematical Notes - This paper discusses a class of critical nonlinear Schrödinger equations which are closely related to several applications, in particular to Bose-Einstein condensates with...     

On superlinear Schr?dinger equations with periodic potential

We obtain ground state solutions for a wide class of superlinear Schr?dinger equations with periodic potential. The result improves the recent result of Szulkin and Weth (J Funct Anal 257:3802–3822, 2009). The main ingredient is the observation that even in the strongly indefinite case, all Cerami sequences for the energy functional are bounded.     

Asymptotically linear fractional Schrödinger equations

By exploiting a variational technique based upon projecting over the Poho?aev manifold, we prove existence of positive solutions for a class of nonlinear fractional Schrödinger equations having a nonhomogenous nonautonomous asymptotically linear nonlinearity.     

Instability of standing waves for a class of inhomogeneous Schrödinger equations with harmonic potential

Ricerche di Matematica - This notes studies the inhomogeneous non-linear Schrödinger equations with a harmonic potential $$\begin{aligned} i\partial _tu +\Delta u-|x|^2u+|x|^{b}|u|^{p-1}u=0....     

On fractional Schrödinger equation in α-dimensional fractional space

The Schrödinger equation is solved in $\alpha$-dimensional fractional space with a Coulomb potential proportional to $\frac{1}{r^{\beta ?2}}$, $2\le \beta \le 4$. The wave functions are studied in terms of spatial dimensionality $\alpha$ and $\beta$ and the results for $\beta =3$ are compared with those obtained in the literature.     

On the bands of the Schrödinger operator with a matrix potential

In this article, we consider the one-dimensional Schrödinger operator with a Hermitian periodic matrix potential Q. We investigate the bands and gaps of the spectrum and prove that most of the positive real axis is overlapped by m bands. Moreover, we find a condition on the potential Q for which the number of gaps in the spectrum of is finite.