## Radial regular and rupture solutions for a PDE problem with gradient term and two parameters

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- by Marius Ghergu and Yasuhito Miyamoto PDF
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**150**(2022), 1697-1709 Request permission

## Abstract:

We investigate radial solutions for the problem \[ \begin {cases} \displaystyle -\Delta U=\frac {\lambda +\delta |\nabla U|^2}{1-U},\; U>0 & \text {in}\ B,\\ U=0 & \text {on}\ \partial B, \end {cases} \] where $B\subset \mathbb {R}^N$ $(N\geq 2)$ denotes the open unit ball and $\lambda , \delta >0$ are real numbers. Two classes of solutions are considered in this work: (i)*regular solutions*, which satisfy $0<U<1$ in $B$, and (ii)

*rupture solutions*, which satisfy $U(0)=1$, and thus make the equation singular at the origin. Bifurcation with respect to parameter $\lambda >0$ is also discussed.

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## Additional Information

**Marius Ghergu**- Affiliation: School of Mathematics and Statistics, University College Dublin, Belfield, Dublin 4, Ireland; and Institute of Mathematics Simion Stoilow of the Romanian Academy, 21 Calea Grivitei St., 010702 Bucharest, Romania
- MR Author ID: 700524
- ORCID: 0000-0001-9104-5295
- Email: marius.ghergu@ucd.ie
**Yasuhito Miyamoto**- Affiliation: Graduate School of Mathematical Sciences, The University of Tokyo, 3-8-1 Komaba, Meguro-ku, Tokyo 153-8914, Japan
- MR Author ID: 752875
- ORCID: 0000-0002-7766-1849
- Email: miyamoto@ms.u-tokyo.ac.jp
- Received by editor(s): June 30, 2020
- Received by editor(s) in revised form: July 27, 2021
- Published electronically: January 20, 2022
- Additional Notes: The second author was supported by JSPS KAKENHI Grant Numbers 19H01797 and 19H05599.
- Communicated by: Ryan Hynd
- © Copyright 2022 American Mathematical Society
- Journal: Proc. Amer. Math. Soc.
**150**(2022), 1697-1709 - MSC (2020): Primary 34A12, 35B32; Secondary 35B40, 35J62
- DOI: https://doi.org/10.1090/proc/15861
- MathSciNet review: 4375756