Minimal mass blow-up solutions for nonlinear Schrödinger equations with a singular potential

Abstract

We consider the following nonlinear Schrödinger equation with an inverse potential: \[ i\frac{\partial u}{\partial t}+Δu+|u|^{\frac{4}{N}}u\pm\frac{1}{|x|^{2σ}}\log|x|u=0 \] in ${\mathbb{R}}^N$. From the classical argument, the solution with subcritical mass ($\Vert u{\Vert }_2<\Vert Q{\Vert }_2$) is global and bounded in $H^1({\mathbb{R}}^N)$. Here, $Q$ is the ground state of the mass-critical problem. Therefore, we are interested in the existence and behaviour of blow-up solutions for the threshold (${\Vert u_0\Vert }_2={\Vert Q\Vert }_2$).

arXiv commentarXiv admin note: text overlap with arXiv:2012.13887, arXiv:2109.08840, arXiv:2108.06205, arXiv:2007.15968

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• Bounds on the Lambert function and their application to the outage analysis of user cooperation
• Minimal blow-up solutions to the mass-critical inhomogeneous NLS equation
• Minimal mass blow-up solutions for nonlinear Schrödinger equations with an inverse potential
• Remarks on minimal mass blow up solutions for a double power nonlinear Schrödinger equation
• Minimal mass blow-up solutions for nonlinear Schrödinger equations with a potential