Normalized solutions for INLS equation with critical Hardy-Sobolev type nonlinearities
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| Format: | Preprint |
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2024
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| author | Cardoso, Mykael de Oliveira, José Francisco Miyagaki, Olímpio |
| author_facet | Cardoso, Mykael de Oliveira, José Francisco Miyagaki, Olímpio |
| contents | 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 $λ\in\mathbb{R}$, $μ>0$, $0<b,d<2$, $2+(4-2b)/N<q<2+(4-2b)/(N-2)$ 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}
&-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 $0 < b,d < 2$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2407_09737 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Normalized solutions for INLS equation with critical Hardy-Sobolev type nonlinearities Cardoso, Mykael de Oliveira, José Francisco Miyagaki, Olímpio Analysis of PDEs 35B08, 35J75, 35J92, 35B33 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 $λ\in\mathbb{R}$, $μ>0$, $0<b,d<2$, $2+(4-2b)/N<q<2+(4-2b)/(N-2)$ 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} &-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 $0 < b,d < 2$. |
| title | Normalized solutions for INLS equation with critical Hardy-Sobolev type nonlinearities |
| topic | Analysis of PDEs 35B08, 35J75, 35J92, 35B33 |
| url | https://arxiv.org/abs/2407.09737 |