Multiplicity of positive solutions for some quasilinear Dirichlet problems on bounded domains in ℝ n

Dimitrios A. Kandilakis; Athanasios N. Lyberopoulos

Commentationes Mathematicae Universitatis Carolinae (2003)

  • Volume: 44, Issue: 4, page 645-658
  • ISSN: 0010-2628

Abstract

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We show that, under appropriate structure conditions, the quasilinear Dirichlet problem - div ( | ∇ u | p - 2 ∇ u ) = f ( x , u ) , x ∈ Ω , u = 0 , x ∈ ∂ Ω , where Ω is a bounded domain in ℝ n , 1 < p < + ∞ , admits two positive solutions u 0 , u 1 in W 0 1 , p ( Ω ) such that 0 < u 0 ≤ u 1 in Ω , while u 0 is a local minimizer of the associated Euler-Lagrange functional.

How to cite

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Kandilakis, Dimitrios A., and Lyberopoulos, Athanasios N.. "Multiplicity of positive solutions for some quasilinear Dirichlet problems on bounded domains in $\mathbb {R}^n$." Commentationes Mathematicae Universitatis Carolinae 44.4 (2003): 645-658. <http://eudml.org/doc/249209>.

@article{Kandilakis2003,
abstract = {We show that, under appropriate structure conditions, the quasilinear Dirichlet problem \[ \left\lbrace \begin\{array\}\{ll\}-\operatorname\{div\}(|\nabla u|^\{p-2\}\nabla u) =f(x,u), \quad & x\in \Omega , \ u=0, & x\in \partial \Omega , \end\{array\}\right.\] where $\Omega $is a bounded domain in $\mathbb \{R\}^n$, $1<p<+\infty $, admits two positive solutions $u_\{0\}$, $u_\{1\}$ in $W_\{0\}^\{1,p\}(\Omega )$ such that $0<u_\{0\}\le u_\{1\}$ in $\Omega $, while $u_\{0\}$ is a local minimizer of the associated Euler-Lagrange functional.},
author = {Kandilakis, Dimitrios A., Lyberopoulos, Athanasios N.},
journal = {Commentationes Mathematicae Universitatis Carolinae},
keywords = {$p$-Laplacian; positive solutions; sub- and supersolutions; local minimizers; Palais-Smale condition; -Laplacian; positive solutions; local minimizers; Palais-Smale condition},
language = {eng},
number = {4},
pages = {645-658},
publisher = {Charles University in Prague, Faculty of Mathematics and Physics},
title = {Multiplicity of positive solutions for some quasilinear Dirichlet problems on bounded domains in $\mathbb \{R\}^n$},
url = {http://eudml.org/doc/249209},
volume = {44},
year = {2003},
}

TY - JOUR
AU - Kandilakis, Dimitrios A.
AU - Lyberopoulos, Athanasios N.
TI - Multiplicity of positive solutions for some quasilinear Dirichlet problems on bounded domains in $\mathbb {R}^n$
JO - Commentationes Mathematicae Universitatis Carolinae
PY - 2003
PB - Charles University in Prague, Faculty of Mathematics and Physics
VL - 44
IS - 4
SP - 645
EP - 658
AB - We show that, under appropriate structure conditions, the quasilinear Dirichlet problem \[ \left\lbrace \begin{array}{ll}-\operatorname{div}(|\nabla u|^{p-2}\nabla u) =f(x,u), \quad & x\in \Omega , \ u=0, & x\in \partial \Omega , \end{array}\right.\] where $\Omega $is a bounded domain in $\mathbb {R}^n$, $1<p<+\infty $, admits two positive solutions $u_{0}$, $u_{1}$ in $W_{0}^{1,p}(\Omega )$ such that $0<u_{0}\le u_{1}$ in $\Omega $, while $u_{0}$ is a local minimizer of the associated Euler-Lagrange functional.
LA - eng
KW - $p$-Laplacian; positive solutions; sub- and supersolutions; local minimizers; Palais-Smale condition; -Laplacian; positive solutions; local minimizers; Palais-Smale condition
UR - http://eudml.org/doc/249209
ER -

References

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