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

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

2011-10-11

arxiv.org/abs/1110.2314v1

Mathematics

Analysis of PDEs

Scientific paper

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 $p\in (\frac{n}{n-2},\frac{n}{n-2}+\eps)$ there exist distributional solutions with a point singularity at the origin provided $\eps>0$ is sufficiently small and $V,\Gamma$ are bounded on $\R^n\setminus B_1(0)$ and satisfy suitable H\"{o}lder-type conditions at the origin. In the case $n=1,2$ or $n\geq 3,1

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