Skip to main content Skip to main navigation menu Skip to site footer
  • Login
  • Language
    • English
    • Język Polski
  • Menu
  • Home
  • Current
  • Online First
  • Archives
  • About
    • About the Journal
    • Submissions
    • Editorial Team
    • Privacy Statement
    • Contact
  • Login
  • Language:
  • English
  • Język Polski

Topological Methods in Nonlinear Analysis

Corrigendum and addendum to "Non-autonomous quasilinear elliptic equations and Ważewski's principle"
  • Home
  • /
  • Corrigendum and addendum to "Non-autonomous quasilinear elliptic equations and Ważewski's principle"
  1. Home /
  2. Archives /
  3. Vol 56, No 1 (September 2020) /
  4. Articles

Corrigendum and addendum to "Non-autonomous quasilinear elliptic equations and Ważewski's principle"

Authors

  • Matteo Franca https://orcid.org/0000-0002-1219-4768

Keywords

$p$-laplace equations, invariant manifold, non-smooth systems, radial solutions, ground states, Fowler transformation, Ważewski's principle

Abstract

In this addendum we fill a gap in a proof and we correct some results appearing in \cite{F6}. In the original paper \cite{F6} we classified positive solutions for the following equation \begin{equation*} \Delta_{p}u+K(r) u^{\sigma-1}=0 \end{equation*} where $r=|x|$, $x \in \RR^n$, $n> p> 1$, $\sigma ={n p}/({n-p})$ and $K(r)$ is a function strictly positive and bounded. In fact \cite{F6} had two main purposes. First, to establish asymptotic conditions which are sufficient for the existence of ground states with fast decay and to classify regular and singular solutions: these results are correct but need some non-trivial further explanations. Second to establish some computable conditions on $K$ which are sufficient to obtain multiplicity of ground states with fast decay in a non-perturbation context. Also in this case the original argument contained a flaw: here we correct the assumptions of \cite{F6} by performing a new nontrivial construction. A third purpose of this addendum is to generalize results of \cite{F6} to a slightly more general equation \begin{equation*} \Delta_p u+ r^{\delta}K(r) u^{\sigma(\delta)-1}=0 \end{equation*} where $\delta> -p$, and $\sigma(\delta) ={p(n+\delta)}/({n-p})$.

References

F. Battelli and R. Johnson, On positive solutions of the scalar curvature equation when the curvature has variable sign, Nonlinear Anal. 47 (2001), 1029–1037.

G. Bianchi and H. Egnell, A variational approach to the equation ∆u + Ku(n+2)/(n−2) = 0 in Rn , Arch. Ration. Mech. Anal. 122 (1993), 159–182.

G. Bianchi, Non-existence and symmetry of solutions to the scalar curvature equation, Comm. Partial Differential Equations 21 (1996), 229–234.

M.F. Bidaut-Véron, Local and global behavior of solutions of quasilinear equations of Emden–Fowler type, Arch. Ration. Mech. Anal. 107 (1989), 293–324.

L. Caffarelli, B. Gidas and J. Spruck, Asymptotic symmetry and local behavior of semilinear elliptic equations with critical sobolev growth, Comm. Pur. Appl. Math. 42 (1989), no. 3, 271–297.

C.C. Chen and C.S. Lin, Blowing up with infinite energy of conformal metrics on S n , Comm. Partial Differential Equations 24 (1999), 785–799.

C.C. Chen and C.S. Lin, On the asymptotic symmetry of singular solutions of the scalar curvature equations, Math. Ann. 313 (1999), 229–245.

E. Coddington and N. Levinson, Theory of Ordinary Differential Equations, Mc Graw Hill, New York, 1955.

F. Dalbono and M. Franca, Nodal solutions for supercritical Laplace equations, Comm. Math. Phys. 347 (2016), 875–901.

W.Y. Ding and W.M. Ni, On the elliptic equation ∆u + Ku(n+2)/(n−2) = 0 and related topics, Duke Math. J. 52 (1985), 485–506.

I. Flores and M. Franca, Multiplicity results for the scalar curvature equation, J. Differential Equations 259 (2015), 4327–4355.

M. Franca, Non-autonomous quasilinear elliptic equations and Ważewski’s principle, Topol. Methods Nonlinear Anal. 23 (2004), 213–238.

M. Franca, Structure theorems for positive radial solutions of the generalized scalar curvature equation, Funkc. Ekv. 52 (2009), 343–369.

M. Franca and R. Johnson, Ground states and singular ground states for quasilinear partial differential equations with critical exponent in the perturbative case, Adv. Nonlinear Stud. 4 (2004), 93–120.

B. Gidas, W.M. Ni and L. Nirenberg, Symmetry and related properties via the maximum principle, Comm. Math. Phys. 68 (1979), no. 3, 209–243.

R. Johnson, X.B. Pan and Y.F. Yi, The Melnikov method and elliptic equations with critical exponent, Indiana Math. J. 43 (1994), 1045–1077.

R. Johnson, X.B. Pan and Y.F. Yi, Singular ground states of semilinear elliptic equations via invariant manifold theory, Nonlinear Anal. 20 (1993), 1279–1302.

N. Kawano, W.M. Ni and S. Yotsutani, A generalized Pohozaev identity and its applications, J. Math. Soc. Japan 42 (1990), 541–564.

N. Kawano, E. Yanagida and S. Yotsutani, Structure theorems for positive radial solutions to div(|Du|m−2 Du) + K(|x|)uq = 0 in Rn , J. Math. Soc. Japan 45 (1993), 719–742.

D. Papini and F. Zanolin, Periodic points and chaotic-like dynamics of planar maps associated to nonlinear Hill’s equations with indefinite weight, Georgian Math. J. 9 (2002), 339–366.

P. Pucci and R. Servadei, Existence, non-existence and regularity of radial ground states for p-Laplacain equations with singular weights, Ann. Inst. H. Poincaré Anal. Non Linéaire 25 (2008), 505–537.

B. Sciunzi, Classification of positive D1,p (Rn )-solutions to the critical p-Laplace equation in Rn , Adv. Math. 291 (2016), 12–23.

J. Wei and S. Yan, Infinitely many solutions for the prescribed scalar curvature problem on SN , J. Funct. Anal. 258 (2010), 3048–3081.

S. Yan, Concentration of solutions for the scalar curvature equation on RN , J. Differential Equations 163 (2000), 239–264.

E. Yanagida and S. Yotsutani, Global structure of positive solutions to equations of Matukuma type, Arch. Rational Mech. Anal. 134 (1996), 199–226.

Downloads

  • PREVIEW
  • FULL TEXT

Published

2020-09-05

How to Cite

1.
FRANCA, Matteo. Corrigendum and addendum to "Non-autonomous quasilinear elliptic equations and Ważewski’s principle". Topological Methods in Nonlinear Analysis. Online. 5 September 2020. Vol. 56, no. 1, pp. 1 - 30. [Accessed 4 July 2025].
  • ISO 690
  • ACM
  • ACS
  • APA
  • ABNT
  • Chicago
  • Harvard
  • IEEE
  • MLA
  • Turabian
  • Vancouver
Download Citation
  • Endnote/Zotero/Mendeley (RIS)
  • BibTeX

Issue

Vol 56, No 1 (September 2020)

Section

Articles

Stats

Number of views and downloads: 0
Number of citations: 0

Search

Search

Browse

  • Browse Author Index
  • Issue archive

User

User

Current Issue

  • Atom logo
  • RSS2 logo
  • RSS1 logo

Newsletter

Subscribe Unsubscribe
Up

Akademicka Platforma Czasopism

Najlepsze czasopisma naukowe i akademickie w jednym miejscu

apcz.umk.pl

Partners

  • Akademia Ignatianum w Krakowie
  • Akademickie Towarzystwo Andragogiczne
  • Fundacja Copernicus na rzecz Rozwoju Badań Naukowych
  • Instytut Historii im. Tadeusza Manteuffla Polskiej Akademii Nauk
  • Instytut Kultur Śródziemnomorskich i Orientalnych PAN
  • Instytut Tomistyczny
  • Karmelitański Instytut Duchowości w Krakowie
  • Ministerstwo Kultury i Dziedzictwa Narodowego
  • Państwowa Akademia Nauk Stosowanych w Krośnie
  • Państwowa Akademia Nauk Stosowanych we Włocławku
  • Państwowa Wyższa Szkoła Zawodowa im. Stanisława Pigonia w Krośnie
  • Polska Fundacja Przemysłu Kosmicznego
  • Polskie Towarzystwo Ekonomiczne
  • Polskie Towarzystwo Ludoznawcze
  • Towarzystwo Miłośników Torunia
  • Towarzystwo Naukowe w Toruniu
  • Uniwersytet im. Adama Mickiewicza w Poznaniu
  • Uniwersytet Komisji Edukacji Narodowej w Krakowie
  • Uniwersytet Mikołaja Kopernika
  • Uniwersytet w Białymstoku
  • Uniwersytet Warszawski
  • Wojewódzka Biblioteka Publiczna - Książnica Kopernikańska
  • Wyższe Seminarium Duchowne w Pelplinie / Wydawnictwo Diecezjalne „Bernardinum" w Pelplinie

© 2021- Nicolaus Copernicus University Accessibility statement Shop