Positive solutions of the p -Laplace Emden-Fowler equation in hollow thin symmetric domains

Ryuji Kajikiya

Mathematica Bohemica (2014)

  • Volume: 139, Issue: 2, page 145-154
  • ISSN: 0862-7959

Abstract

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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 G ⊂ O ( N ) . We denote the orbit of G through x ∈ ℝ N by G ( x ) , i.e., G ( x ) : = { g x : g ∈ G } . We prove that if H ( x ) G ( x ) for all x ∈ Ω ¯ 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.

How to cite

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Kajikiya, Ryuji. "Positive solutions of the $p$-Laplace Emden-Fowler equation in hollow thin symmetric domains." Mathematica Bohemica 139.2 (2014): 145-154. <http://eudml.org/doc/261930>.

@article{Kajikiya2014,
abstract = {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 $H \UnimplementedOperator G \subset O(N)$. We denote the orbit of $G$ through $x\in \mathbb \{R\}^N$ by $G(x)$, i.e., $G(x):=\lbrace gx\colon g\in G \rbrace $. We prove that if $H(x)\UnimplementedOperator G(x)$ 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.},
author = {Kajikiya, Ryuji},
journal = {Mathematica Bohemica},
keywords = {Emden-Fowler equation; group invariant solution; least energy solution; positive solution; variational method; Emden-Fowler equation; -Laplacian; group invariant solution; least energy solution; positive solution; variational method},
language = {eng},
number = {2},
pages = {145-154},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {Positive solutions of the $p$-Laplace Emden-Fowler equation in hollow thin symmetric domains},
url = {http://eudml.org/doc/261930},
volume = {139},
year = {2014},
}

TY - JOUR
AU - Kajikiya, Ryuji
TI - Positive solutions of the $p$-Laplace Emden-Fowler equation in hollow thin symmetric domains
JO - Mathematica Bohemica
PY - 2014
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 139
IS - 2
SP - 145
EP - 154
AB - 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 $H \UnimplementedOperator G \subset O(N)$. We denote the orbit of $G$ through $x\in \mathbb {R}^N$ by $G(x)$, i.e., $G(x):=\lbrace gx\colon g\in G \rbrace $. We prove that if $H(x)\UnimplementedOperator G(x)$ 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.
LA - eng
KW - Emden-Fowler equation; group invariant solution; least energy solution; positive solution; variational method; Emden-Fowler equation; -Laplacian; group invariant solution; least energy solution; positive solution; variational method
UR - http://eudml.org/doc/261930
ER -

References

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  8. Montgomery, D., Samelson, H., 10.2307/1968975, Ann. Math. (2) 44 (1943), 454-470. (1943) Zbl0063.04077MR0008817DOI10.2307/1968975
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