Related papers: H\"older regularity of solutions for Schr\"odinger…

We derive H\"older regularity estimates for operators associated with a time independent Schr\"odinger operator of the form $-\Delta+V$. The results are obtained by checking a certain condition on the function $T1$. Our general method…

Let L be a Schr\"odinger operator of the form L=-\Delta+V, where the nonnegative potential V satisfies a reverse H\"older inequality. Using the method of L-harmonic extensions we study regularity estimates at the scale of adapted H\"older…

We study the Schr\"{o}dinger equation: \begin{eqnarray} - \Delta u+V(x)u+f(x,u)=0,\qquad u\in H^{1}(\mathbb{R}^{N}),\nonumber \end{eqnarray} where $V$ is periodic and $f$ is periodic in the $x$-variables, $0$ is in a gap of the spectrum of…

We study the following nonlinear Schr\"odinger equation $$-\Delta u + V(x) u = g(x,u),$$ where V and g are periodic in x. We assume that 0 is a right boundary point of the essential spectrum of $-\Delta+V$. The superlinear and subcritical…

In this paper we discuss the solvability of the Neumann and Regularity boundary value problem of elliptic Schr\"odinger-type equation $-\DIV(A(x)\nabla u(x,t))+V(x)u(x,t)=0$ with bounded measurable uniformly elliptic coefficinets $A(x)$…

In this work, we investigate the regularization mechanisms of the Schr\"odinger equation with a spatial potential $$ i\partial_t u+\Delta u+\eta u =0, $$ where $\eta$ denotes a given spatial potential. The regularity of solutions…

Let $V$ be a potential on $\RR^3$ that is smooth everywhere except at a discrete set $\maS$ of points, where it has singularities of the form $Z/\rho^2$, with $\rho(x) = |x - p|$ for $x$ close to $p$ and $Z$ continuous on $\RR^3$ with $Z(p)…

We consider the nonlinear Schr\"{o}dinger equation $-\Delta u + V(x) u = \Gamma(x) |u|^{p-1}u$ in $\R^n$ where the spectrum of $-\Delta+V(x)$ is positive. In the case $n\geq 3$ we use variational methods to prove that for all $p\in…

We study the Schr\"odinger equations $-\Delta u + V(x)u = f(x,u)$ in $\mathbb{R}^N$ and $-\Delta u - \lambda u = f(x,u)$ in a bounded domain $\Omega\subset\mathbb{R}^N$. We assume that $f$ is superlinear but of subcritical growth and…

We establish that the initial value problem for the quadratic non-linear Schr\"odinger equation $$ iu_t - \Delta u = u^2$$ where $u: \R^2 \times \R \to \C$, is locally well-posed in $H^s(\R^2)$ when $s > -1$. The critical exponent for this…

In this paper we consider the vector-valued Schr\"{o}dinger operator $-\Delta + V$, where the potential term $V$ is a matrix-valued function whose entries belong to $L^1_{\rm loc}(\mathbb{R}^d)$ and, for every $x\in\mathbb{R}^d$, $V(x)$ is…

We prove H\"older regularity estimates up to the boundary for weak solutions $u$ to nonlocal Schr\"odinger equations subject to exterior Dirichlet conditions in an open set $\Omega\subset \mathbb{R}^N$. The class of nonlocal operators…

We prove the existence results for the Schr\"odinger equation of the form $$ -\Delta u + V(x) u = g(x,u), \quad x \in \mathbb{R}^N, $$ where $g$ is superlinear and subcritical in some periodic set $K$ and linear in $\mathbb{R}^N \setminus…

In this paper, we provide the boundedness property of the Riesz transforms associated to the Schr\"odinger operator $\mathcal{L}=-\Delta + \mathbf{V}$ in a new weighted Morrey space which is the generalized version of many previous Morrey…

Let $(X,d,\mu)$ be a metric measure space satisfying a doubling property with the upper/lower dimension $Q\ge n>1$, and admitting an $L^2$-Poincar\'e inequality. In this article, we establish the H\"{o}lder continuity and a Liouville-type…

We study the Schr\"{o}dinger equation: \begin{equation} - \Delta u+V(x)u=f(x,u) ,\qquad u\in H^{1}(\mathbb{R}^{N}),\nonumber \end{equation} where $V$ is periodic and $f$ is periodic in the $x$-variables, 0 is in a gap of the spectrum of the…

For the Schr\"odinger equation, $ (i \partial_t + \Delta) u = 0 $ on a torus, an arbitrary non-empty open set $ \Omega $ provides control and observability of the solution: $ \| u |_{t = 0} \|_{L^2 (\T^2)} \leq K_T \| u \|_{L^2 ([0,T]…

In this paper we introduce a class of generalized Morrey spaces associated with Schr\"odinger operator $L=-\Delta+V$. Via a pointwise estimate, we obtain the boundedness of the operators $V^{\beta_{2}}(-\Delta+V)^{-\beta_{1}}$ and their…

For the stationary nonlinear Schr\"odinger equation $-\Delta u+ V(x)u- f(u) = \lambda u$ with periodic potential $V$ we study the existence and stability properties of multibump solutions with prescribed $L^2$-norm. To this end we introduce…

This paper investigates the localization properties of solutions to the semi-classical Schr\"odinger equation on closed Riemann surfaces. Unlike classical studies that assume a smooth potential, our work addresses the challenges arising…