Normalized solutions for INLS equation with critical Hardy-Sobolev type nonlinearities

TL;DR

This paper establishes the existence of prescribed $L^2$-norm solutions for inhomogeneous nonlinear Schrödinger equations with critical Hardy-Sobolev nonlinearities in higher dimensions and exponential growth in two dimensions, extending previous results.

Contribution

It extends existence results for INLS equations with critical nonlinearities to cases with Hardy-Sobolev weights, covering a broader range of parameters.

Findings

Existence of solutions for $N eq 2$ with Hardy-Sobolev nonlinearities.

Existence of solutions for $N=2$ with exponential nonlinearities.

Extension of previous results to cases with $0 < b,d < 2$.

Abstract

We are interested in finding prescribed $L^2$-norm solutions to inhomogeneous nonlinear Schr\"{o}dinger (INLS) equations. For $N\ge 3$ we treat the equation with combined Hardy-Sobolev power-type nonlinearities $$ -\Delta u+\lambda u=\mu|x|^{-b}|u|^{q-2}u+|x|^{-d}|u|^{2^*_{d}-2}u \;\;\mbox{in}\;\; \mathbb{R}^N,\, N\ge 3 $$ where $\lambda\in\mathbb{R}$, $\mu>0$, $0<b,d<2$, $2+(4-2b)/N<q<2+(4-2b)/(N-2)$ and $2^*_{d}= 2(N-d)/(N-2)$ is the Hardy-Sobolev critical exponent, while for $N=2$ we investigate the equation with critical exponential growth \begin{equation}\nonumber \begin{aligned} &-\Delta u+\lambda u=|x|^{-b}f(u) \;\;\mbox{in}\;\; \mathbb{R}^2 \end{aligned} \end{equation} where the nonlinearity $f(s)$ behaves like $\exp(s^2)$ as $s\to\infty$. We extend the existence results due to Alves-Ji-Miyagaki (Calc. Var. 61, 2022) from $b =d= 0$ to the case $0 < b,d < 2$.