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Ufimsk. Mat. Zh., 2013, Volume 5, Issue 2, Pages 18–30 (Mi ufa195)  

This article is cited in 2 scientific papers (total in 2 papers)

On existence of nodal solution to elliptic equations with convex-concave nonlinearities

V. E. Bobkov

Institute of Mathematics CS USC RAS, Chernyshevskii str., 112, 450008, Ufa, Russia

Abstract: In a bounded connected domain $\Omega \subset \mathbb{R}^N$, $N \geqslant 1$, with a smooth boundary, we consider the Dirichlet boundary value problem for elliptic equation with a convex-concave nonlinearity
\begin{equation*} \begin{cases} -\Delta u = \lambda |u|^{q-2} u + |u|^{\gamma-2} u, \quad x \in \Omega u|_{\partial \Omega} = 0, \end{cases} \end{equation*}
where $1< q< 2< \gamma < 2^*$. As a main result, we prove the existence of a nodal solution to this equation on the nonlocal interval $\lambda \in (-\infty, \lambda_0^*)$, where $\lambda_0^*$ is determined by the variational principle of nonlinear spectral analysis via fibering method.

Keywords: nodal solution, convex-concave nonlinearity, fibering method.

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English version:
Ufa Mathematical Journal, 2013, 5:2, 18–30 (PDF, 201 kB); https://doi.org/10.13108/2013-5-2-18

Bibliographic databases:

UDC: 517.9
MSC: 35D30, 35J25, 35J20, 35J60
Received: 05.03.2012

Citation: V. E. Bobkov, “On existence of nodal solution to elliptic equations with convex-concave nonlinearities”, Ufimsk. Mat. Zh., 5:2 (2013), 18–30; Ufa Math. J., 5:2 (2013), 18–30

Citation in format AMSBIB
\Bibitem{Bob13}
\by V.~E.~Bobkov
\paper On existence of nodal solution to elliptic equations with convex-concave nonlinearities
\jour Ufimsk. Mat. Zh.
\yr 2013
\vol 5
\issue 2
\pages 18--30
\mathnet{http://mi.mathnet.ru/ufa195}
\mathscinet{http://www.ams.org/mathscinet-getitem?mr=3430773}
\elib{http://elibrary.ru/item.asp?id=19063033}
\transl
\jour Ufa Math. J.
\yr 2013
\vol 5
\issue 2
\pages 18--30
\crossref{https://doi.org/10.13108/2013-5-2-18}


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    Citing articles on Google Scholar: Russian citations, English citations
    Related articles on Google Scholar: Russian articles, English articles

    This publication is cited in the following articles:
    1. Bobkov V., “Least Energy Nodal Solutions For Elliptic Equations With Indefinite Nonlinearity”, Electron. J. Qual. Theory Differ., 2014, no. 56, 1–15  mathscinet  isi  elib
    2. Bobkov V., Kolonitskii S., “on a Property of the Nodal Set of Least Energy Sign-Changing Solutions For Quasilinear Elliptic Equations”, Proc. R. Soc. Edinb. Sect. A-Math., 149:5 (2019), PII S0308210518000884, 1163–1173  crossref  mathscinet  zmath  isi  scopus
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