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

Abstract

We are interested in finding prescribed $L^2$-norm solutions to inhomogeneous nonlinear Schrödinger (INLS) equations. For $N\ge 3$ we treat the equation with combined Hardy-Sobolev power-type nonlinearities $$ -Δu+λu=μ|x|^{-b}|u|^{q-2}u+|x|^{-d}|u|^{2^*_{d}-2}u \;\;\mbox{in}\;\; \mathbb{R}^N,\, N\ge 3 $$ where $\hat{I}»\in \mathbb{R}$, $\hat{I}¼>0$, $0<b,d<2$, $2+(4-2b)/N<q<2+(4-2b)/(N-2)$ and $2_d^{\ast }=2(N-d)/(N-2)$ is the Hardy-Sobolev critical exponent, while for $N=2$ we investigate the equation with critical exponential growth \begin{equation}\nonumber \begin{aligned} &-Δu+λu=|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 $0<b,d<2$.