Nonlinear homogeneous eigenvalue problem in R N : nonstandard variational approach

Pavel Drábek; Zakaria Moudan; Abdelfettah Touzani

Commentationes Mathematicae Universitatis Carolinae (1997)

  • Volume: 38, Issue: 3, page 421-431
  • ISSN: 0010-2628

Abstract

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The nonlinear eigenvalue problem for p-Laplacian - div ( a ( x ) | ∇ u | p - 2 ∇ u ) = λ g ( x ) | u | p - 2 u in ℝ N , u > 0 in ℝ N , lim | x | → ∞ u ( x ) = 0 , is considered. We assume that 1 < p < N and that g is indefinite weight function. The existence and C 1 , α -regularity of the weak solution is proved.

How to cite

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Drábek, Pavel, Moudan, Zakaria, and Touzani, Abdelfettah. "Nonlinear homogeneous eigenvalue problem in $R^N$: nonstandard variational approach." Commentationes Mathematicae Universitatis Carolinae 38.3 (1997): 421-431. <http://eudml.org/doc/248097>.

@article{Drábek1997,
abstract = {The nonlinear eigenvalue problem for p-Laplacian \[ \left\lbrace \begin\{array\}\{ll\}- \operatorname\{div\} (a(x) |\nabla u|^\{p-2\} \nabla u) = \lambda g (x) |u|^\{p-2\} u \text\{ in \} \mathbb \{R\}^N, \ u >0 \text\{ in \} \mathbb \{R\}^N, \mathop \{\lim \}\limits \_\{|x|\rightarrow \infty \} u(x) = 0, \end\{array\}\right.\] is considered. We assume that $1 < p < N$ and that $g$ is indefinite weight function. The existence and $C^\{1, \alpha \}$-regularity of the weak solution is proved.},
author = {Drábek, Pavel, Moudan, Zakaria, Touzani, Abdelfettah},
journal = {Commentationes Mathematicae Universitatis Carolinae},
keywords = {eigenvalue; the p-Laplacian; indefinite weight; regularity; -Laplacian; eigenvalue; indefinite weight; regularity},
language = {eng},
number = {3},
pages = {421-431},
publisher = {Charles University in Prague, Faculty of Mathematics and Physics},
title = {Nonlinear homogeneous eigenvalue problem in $R^N$: nonstandard variational approach},
url = {http://eudml.org/doc/248097},
volume = {38},
year = {1997},
}

TY - JOUR
AU - Drábek, Pavel
AU - Moudan, Zakaria
AU - Touzani, Abdelfettah
TI - Nonlinear homogeneous eigenvalue problem in $R^N$: nonstandard variational approach
JO - Commentationes Mathematicae Universitatis Carolinae
PY - 1997
PB - Charles University in Prague, Faculty of Mathematics and Physics
VL - 38
IS - 3
SP - 421
EP - 431
AB - The nonlinear eigenvalue problem for p-Laplacian \[ \left\lbrace \begin{array}{ll}- \operatorname{div} (a(x) |\nabla u|^{p-2} \nabla u) = \lambda g (x) |u|^{p-2} u \text{ in } \mathbb {R}^N, \ u >0 \text{ in } \mathbb {R}^N, \mathop {\lim }\limits _{|x|\rightarrow \infty } u(x) = 0, \end{array}\right.\] is considered. We assume that $1 < p < N$ and that $g$ is indefinite weight function. The existence and $C^{1, \alpha }$-regularity of the weak solution is proved.
LA - eng
KW - eigenvalue; the p-Laplacian; indefinite weight; regularity; -Laplacian; eigenvalue; indefinite weight; regularity
UR - http://eudml.org/doc/248097
ER -

References

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  2. Allegretto W., Huang Y.X., Eigenvalues of the indefinite weight p-Laplacian in weighted spaces, Funkc. Ekvac. 38 (1995), 233-242. (1995) Zbl0922.35114MR1356326
  3. Drábek P., Nonlinear eigenvalue for p-Laplacian in ℝ N , Math. Nach 173 (1995), 131-139. (1995) MR1336957
  4. Drábek, Huang Y.X., Bifurcation problems for the p-Laplacian in ℝ N , to appear in Trans. of AMS. 
  5. Fleckinger J., Manasevich R.F., Stavrakakis N.M., de Thelin F., Principal eigenvalues for some quasilinear elliptic equations on ℝ N , preprint. 
  6. Huang Y.X., Eigenvalues of the p-Laplacian in ℝ N with indefinite weight, Comment. Math. Univ. Carolinae 36 (1995), 519-527. (1995) MR1364493
  7. Lindqvist P., On the equation d i v ( | ∇ u | p - 2 ∇ u ) + λ | u | p - 2 u = 0 , Proc. Amer. Math. Society 109 (1990), 157-164. (1990) Zbl0714.35029MR1007505
  8. Serin J., Local behavior of solutions of quasilinear equations, Acta Math. 111 (1964), 247-302. (1964) MR0170096
  9. Tolkdorf P., Regularity for a more general class of quasilinear elliptic equations, J. Diff. Equations 51 (1984), 126-150. (1984) MR0727034
  10. Trudinger N.S., On Harnack type inequalities and their applications to quasilinear elliptic equations, Comm. Pure. Appl. Math. 20 (1967), 721-747. (1967) MR0226198

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