Sobolev versus Hölder local minimizers and existence of multiple solutions for a singular quasilinear equation

Jacques Giacomoni; Ian Schindler; Peter Takáč

Annali della Scuola Normale Superiore di Pisa - Classe di Scienze (2007)

  • Volume: 6, Issue: 1, page 117-158
  • ISSN: 0391-173X

Abstract

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We investigate the following quasilinear and singular problem, t o 2 . 7 c m - Δ p u = λ u δ + u q in Ω ; u | ∂ Ω = 0 , u > 0 in Ω , t o 2 . 7 c m (P) where Ω is an open bounded domain with smooth boundary, 1 < p < ∞ , p - 1 < q ≤ p * - 1 , λ > 0 , and 0 < δ < 1 . As usual, p * = N p N - p if 1 < p < N , p * ∈ ( p , ∞ ) is arbitrarily large if p = N , and p * = ∞ if p > N . We employ variational methods in order to show the existence of at least two distinct (positive) solutions of problem (P) in W 0 1 , p ( Ω ) . While following an approach due to Ambrosetti-Brezis-Cerami, we need to prove two new results of separate interest: a strong comparison principle and a regularity result for solutions to problem (P) in C 1 , β ( Ω ¯ ) with some β ∈ ( 0 , 1 ) . Furthermore, we show that δ < 1 is a reasonable sufficient (and likely optimal) condition to obtain solutions of problem (P) in C 1 ( Ω ¯ ) .

How to cite

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Giacomoni, Jacques, Schindler, Ian, and Takáč, Peter. "Sobolev versus Hölder local minimizers and existence of multiple solutions for a singular quasilinear equation." Annali della Scuola Normale Superiore di Pisa - Classe di Scienze 6.1 (2007): 117-158. <http://eudml.org/doc/272290>.

@article{Giacomoni2007,
abstract = {We investigate the following quasilinear and singular problem,\[ \hbox\{t\}o 2.7cm\{\}\left\lbrace \begin\{array\}\{ll\} - \Delta \_p u = \frac\{\lambda \}\{u^\delta \} + u^q \quad & \mbox\{ in \}\,\Omega ;\\ u\vert \_\{\partial \Omega \} = 0 ,\quad u &gt; 0\quad & \mbox\{ in \}\,\Omega , \end\{array\} \right.\hbox\{t\}o 2.7cm\{\}\hbox\{ \{\rm (P)\}\} \]where $\Omega $ is an open bounded domain with smooth boundary, $1 &lt; p &lt; \infty $, $p-1 &lt; q\le p^\{*\} - 1$, $\lambda &gt; 0$, and $0 &lt; \delta &lt; 1$. As usual, $p^\{*\} = \frac\{Np\}\{N-p\}$ if $1 &lt; p &lt; N$, $p^\{*\}\in (p,\infty )$ is arbitrarily large if $p = N$, and $p^\{*\} = \infty $ if $p &gt; N$. We employ variational methods in order to show the existence of at least two distinct (positive) solutions of problem (P) in $W_0^\{1,p\}(\Omega )$. While following an approach due to Ambrosetti-Brezis-Cerami, we need to prove two new results of separate interest: a strong comparison principle and a regularity result for solutions to problem (P) in $C^\{1,\beta \}(\overline\{\Omega \})$ with some $\beta \in (0,1)$. Furthermore, we show that $\delta &lt; 1$ is a reasonable sufficient (and likely optimal) condition to obtain solutions of problem (P) in $C^1(\overline\{\Omega \})$.},
author = {Giacomoni, Jacques, Schindler, Ian, Takáč, Peter},
journal = {Annali della Scuola Normale Superiore di Pisa - Classe di Scienze},
keywords = {quasilinear equation; singular equation; positive solutions; multiplicity; regularity; comparison principle; local minimizer; saddle point},
language = {eng},
number = {1},
pages = {117-158},
publisher = {Scuola Normale Superiore, Pisa},
title = {Sobolev versus Hölder local minimizers and existence of multiple solutions for a singular quasilinear equation},
url = {http://eudml.org/doc/272290},
volume = {6},
year = {2007},
}

TY - JOUR
AU - Giacomoni, Jacques
AU - Schindler, Ian
AU - Takáč, Peter
TI - Sobolev versus Hölder local minimizers and existence of multiple solutions for a singular quasilinear equation
JO - Annali della Scuola Normale Superiore di Pisa - Classe di Scienze
PY - 2007
PB - Scuola Normale Superiore, Pisa
VL - 6
IS - 1
SP - 117
EP - 158
AB - We investigate the following quasilinear and singular problem,\[ \hbox{t}o 2.7cm{}\left\lbrace \begin{array}{ll} - \Delta _p u = \frac{\lambda }{u^\delta } + u^q \quad & \mbox{ in }\,\Omega ;\\ u\vert _{\partial \Omega } = 0 ,\quad u &gt; 0\quad & \mbox{ in }\,\Omega , \end{array} \right.\hbox{t}o 2.7cm{}\hbox{ {\rm (P)}} \]where $\Omega $ is an open bounded domain with smooth boundary, $1 &lt; p &lt; \infty $, $p-1 &lt; q\le p^{*} - 1$, $\lambda &gt; 0$, and $0 &lt; \delta &lt; 1$. As usual, $p^{*} = \frac{Np}{N-p}$ if $1 &lt; p &lt; N$, $p^{*}\in (p,\infty )$ is arbitrarily large if $p = N$, and $p^{*} = \infty $ if $p &gt; N$. We employ variational methods in order to show the existence of at least two distinct (positive) solutions of problem (P) in $W_0^{1,p}(\Omega )$. While following an approach due to Ambrosetti-Brezis-Cerami, we need to prove two new results of separate interest: a strong comparison principle and a regularity result for solutions to problem (P) in $C^{1,\beta }(\overline{\Omega })$ with some $\beta \in (0,1)$. Furthermore, we show that $\delta &lt; 1$ is a reasonable sufficient (and likely optimal) condition to obtain solutions of problem (P) in $C^1(\overline{\Omega })$.
LA - eng
KW - quasilinear equation; singular equation; positive solutions; multiplicity; regularity; comparison principle; local minimizer; saddle point
UR - http://eudml.org/doc/272290
ER -

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