| Title:
|
Positive solutions of the $p$-Laplace Emden-Fowler equation in hollow thin symmetric domains (English) |
| Author:
|
Kajikiya, Ryuji |
| Language:
|
English |
| Journal:
|
Mathematica Bohemica |
| ISSN:
|
0862-7959 (print) |
| ISSN:
|
2464-7136 (online) |
| Volume:
|
139 |
| Issue:
|
2 |
| Year:
|
2014 |
| Pages:
|
145-154 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
We study the existence of positive solutions for the $p$-Laplace Emden-Fowler equation. Let $H$ and $G$ be closed subgroups of the orthogonal group $O(N)$ such that $H \varsubsetneq G \subset O(N)$. We denote the orbit of $G$ through $x\in \mathbb {R}^N$ by $G(x)$, i.e., $G(x):=\{gx\colon g\in G \}$. We prove that if $H(x)\varsubsetneq G(x)$ for all $x\in \overline {\Omega }$ and the first eigenvalue of the $p$-Laplacian is large enough, then no $H$ invariant least energy solution is $G$ invariant. Here an $H$ invariant least energy solution means a solution which achieves the minimum of the Rayleigh quotient among all $H$ invariant functions. Therefore there exists an $H$ invariant $G$ non-invariant positive solution. (English) |
| Keyword:
|
Emden-Fowler equation |
| Keyword:
|
group invariant solution |
| Keyword:
|
least energy solution |
| Keyword:
|
positive solution |
| Keyword:
|
variational method |
| MSC:
|
35B09 |
| MSC:
|
35J20 |
| MSC:
|
35J25 |
| MSC:
|
35J92 |
| idZBL:
|
Zbl 06362249 |
| idMR:
|
MR3238830 |
| DOI:
|
10.21136/MB.2014.143845 |
| . |
| Date available:
|
2014-07-14T08:06:58Z |
| Last updated:
|
2020-07-29 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/143845 |
| . |
| Reference:
|
[1] Bhattacharya, T.: A proof of the Faber-Krahn inequality for the first eigenvalue of the $p$-Laplacian.Ann. Mat. Pura Appl., IV. Ser. 177 (1999), 225-240. Zbl 0966.35091, MR 1747632, 10.1007/BF02505910 |
| Reference:
|
[2] Borel, A.: Le plan projectif des octaves et les sphères comme espaces homogènes.French C. R. Acad. Sci., Paris 230 (1950), 1378-1380. Zbl 0041.52203, MR 0034768 |
| Reference:
|
[3] Kajikiya, R.: Least energy solutions of the Emden-Fowler equation in hollow thin symmetric domains.J. Math. Anal. Appl. 406 (2013), 277-286. MR 3062420, 10.1016/j.jmaa.2013.04.068 |
| Reference:
|
[4] Kajikiya, R.: Multiple positive solutions of the Emden-Fowler equation in hollow thin symmetric domains.Calc. Var. Partial Differ. Equ (to appear). |
| Reference:
|
[5] Kajikiya, R.: Partially symmetric solutions of the generalized Hénon equation in symmetric domains.Topol. Methods Nonlinear Anal (to appear). MR 1980135 |
| Reference:
|
[6] Kristály, A.: Asymptotically critical problems on higher-dimensional spheres.Discrete Contin. Dyn. Syst. 23 (2009), 919-935. Zbl 1154.35051, MR 2461832, 10.3934/dcds.2009.23.919 |
| Reference:
|
[7] Lindqvist, P.: On the equation $ div (|\nabla u|^{p-2}\nabla u) + \lambda |u|^{p-2}u =0$.Proc. Am. Math. Soc. 109 (1990), 157-164. MR 1007505 |
| Reference:
|
[8] Montgomery, D., Samelson, H.: Transformation groups of spheres.Ann. Math. (2) 44 (1943), 454-470. Zbl 0063.04077, MR 0008817, 10.2307/1968975 |
| Reference:
|
[9] Pontryagin, L. S.: Topological Groups.Transl. from the Russian, (3rd ed.), Classics of Soviet Mathematics Gordon and Breach, New York (1986). Zbl 0882.01025, MR 0898007 |
| . |
