arXiv subjects

Rainer Mandel

Publications and source records attributed to Rainer Mandel.

38 records · Page 3Linked to original sources

arXiv2013

Minimal energy solutions for repulsive nonlinear Schrödinger systems

In this paper we establish existence and nonexistence results concerning fully nontrivial minimal energy solutions of the nonlinear Schrödinger system \begin{align*} \begin{gathered} -Δu + \, u = |u|^{2q-2}u + b|u|^{q-2}u|v|^q \quad\text{in}\R^n, -Δv + ω^2 v = |v|^{2q-2}v + b|u|^q|v|^{q-2}v\quad\text{in}\R^n. \end{gathered} \end{align*} We consider the repulsive case $b<0$ and assume that the exponent $q$ satisfies $1<q<\frac{n}{n-2}$ in case $n\geq 3$ and $1<q<\infty$ in case $n=1$ or $n=2$. For space dimensions $n\geq 2$ and arbitrary $b<0$ we prove the existence of fully nontrivial nonnegative solutions which converge to a solution of some optimal partition problem as $b\to -\infty$. In case $n=1$ we prove that minimal energy solutions exist provided the coupling parameter $b$ has small absolute value whereas fully nontrivial solutions do not exist if $1<q\leq 2$ and $b$ has large absolute value.

math.AP↗

arXiv2011

Distributional solutions of the stationary nonlinear Schrödinger equation: singularities, regularity and exponential decay

We consider the nonlinear Schrödinger equation $-Δu + V(x) u = Γ(x) |u|^{p-1}u$ in $\R^n$ where the spectrum of $-Δ+V(x)$ is positive. In the case $n\geq 3$ we use variational methods to prove that for all $p\in (\frac{n}{n-2},\frac{n}{n-2}+\eps)$ there exist distributional solutions with a point singularity at the origin provided $\eps>0$ is sufficiently small and $V,Γ$ are bounded on $\R^n\setminus B_1(0)$ and satisfy suitable Hölder-type conditions at the origin. In the case $n=1,2$ or $n\geq 3,1<p<\frac{n}{n-2}$, however, we show that every distributional solution of the more general equation $-Δu + V(x) u = g(x,u)$ is a bounded strong solution if $V$ is bounded and $g$ satisfies certain growth conditions.

math.AP↗