Asymptotic behavior of positive solutions of a Dirichlet problem involving supercritical nonlinearities

Giovanni Anello; Giuseppe Rao

Commentationes Mathematicae Universitatis Carolinae (2013)

  • Volume: 54, Issue: 4, page 485-491
  • ISSN: 0010-2628

Abstract

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Let p > 1 , q > p , λ > 0 and s ∈ ] 1 , p [ . We study, for s → p - , the behavior of positive solutions of the problem - Δ p u = λ u s - 1 + u q - 1 in Ω , u ∣ ∂ Ω = 0 . In particular, we give a positive answer to an open question formulated in a recent paper of the first author.

How to cite

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Anello, Giovanni, and Rao, Giuseppe. "Asymptotic behavior of positive solutions of a Dirichlet problem involving supercritical nonlinearities." Commentationes Mathematicae Universitatis Carolinae 54.4 (2013): 485-491. <http://eudml.org/doc/260586>.

@article{Anello2013,
abstract = {Let $p>1$, $q>p$, $\lambda >0$ and $s\in ]1,p[$. We study, for $s\rightarrow p^-$, the behavior of positive solutions of the problem $-\Delta _p u = \lambda u^\{s-1\}+u^\{q-1\}$ in $\Omega $, $u_\{\mid \partial \Omega \}=0$. In particular, we give a positive answer to an open question formulated in a recent paper of the first author.},
author = {Anello, Giovanni, Rao, Giuseppe},
journal = {Commentationes Mathematicae Universitatis Carolinae},
keywords = {elliptic boundary value problems; positive solutions; variational methods; asymptotic behavior; combined nonlinearities; elliptic boundary value problems; positive solutions; variational methods; asymptotic behavior; combined nonlinearities},
language = {eng},
number = {4},
pages = {485-491},
publisher = {Charles University in Prague, Faculty of Mathematics and Physics},
title = {Asymptotic behavior of positive solutions of a Dirichlet problem involving supercritical nonlinearities},
url = {http://eudml.org/doc/260586},
volume = {54},
year = {2013},
}

TY - JOUR
AU - Anello, Giovanni
AU - Rao, Giuseppe
TI - Asymptotic behavior of positive solutions of a Dirichlet problem involving supercritical nonlinearities
JO - Commentationes Mathematicae Universitatis Carolinae
PY - 2013
PB - Charles University in Prague, Faculty of Mathematics and Physics
VL - 54
IS - 4
SP - 485
EP - 491
AB - Let $p>1$, $q>p$, $\lambda >0$ and $s\in ]1,p[$. We study, for $s\rightarrow p^-$, the behavior of positive solutions of the problem $-\Delta _p u = \lambda u^{s-1}+u^{q-1}$ in $\Omega $, $u_{\mid \partial \Omega }=0$. In particular, we give a positive answer to an open question formulated in a recent paper of the first author.
LA - eng
KW - elliptic boundary value problems; positive solutions; variational methods; asymptotic behavior; combined nonlinearities; elliptic boundary value problems; positive solutions; variational methods; asymptotic behavior; combined nonlinearities
UR - http://eudml.org/doc/260586
ER -

References

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  1. Anello G., On the Dirichlet problem involving the equation - Δ p u = λ u s - 1 , Nonlinear Anal. 70 (2009), 2060–2066. MR2492142
  2. Anello G., 10.1007/s00605-010-0189-9, Monatsh. Math. 162 (2011), 1–18. Zbl1206.35135MR2747340DOI10.1007/s00605-010-0189-9
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  5. Il'yasov Y., 10.1016/j.na.2004.10.022, Nonlinear Anal. 61 (2005), 211–236. Zbl1190.35112MR2122250DOI10.1016/j.na.2004.10.022
  6. Liebermann G.M., 10.1016/0362-546X(88)90053-3, Nonlinear Anal. 12 (1988), no. 11, 1203–1219. MR0969499DOI10.1016/0362-546X(88)90053-3
  7. Moser J., 10.1002/cpa.3160130308, Comm. Pure Appl. Math. 13 (1960), 457–478. Zbl0111.09301MR0170091DOI10.1002/cpa.3160130308

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