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Mat. Zametki, 1998, Volume 64, Issue 4, Pages 543–548 (Mi mz1428)  

This article is cited in 1 scientific paper (total in 1 paper)

On asymptotics of solutions to semilinear elliptic equations near the first eigenvalue of the nonperturbed problem

Ya. Sh. Il'yasov

Steklov Mathematical Institute, Russian Academy of Sciences

Abstract: The following elliptic equations with $p$-Laplacian
$$ -\Delta_pu=\lambda g(x)|u|^{p-2}u+f(x)|u|^{\gamma-2}u $$
are considered in the entire space $\mathbb R^N$ and in the bounded domain with the Dirichlet boundary conditions. By the fibering method for the basic positive solutions of these equations, we derive the following asymptotic formula
$$ u^\lambda=(\lambda_1-\lambda)^{1/(\gamma-p)}u_1 +o((\lambda_1-\lambda)^{1/(\gamma-p)}) $$
for $\lambda\uparrow\lambda_1$, where $\lambda_1$ is the first eigenvalue and $u_1$ is the corresponding eigenfunction of nonperturbed problem ($f=0$).

DOI: https://doi.org/10.4213/mzm1428

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English version:
Mathematical Notes, 1998, 64:4, 471–475

Bibliographic databases:

UDC: 517.956.2
Received: 25.06.1997

Citation: Ya. Sh. Il'yasov, “On asymptotics of solutions to semilinear elliptic equations near the first eigenvalue of the nonperturbed problem”, Mat. Zametki, 64:4 (1998), 543–548; Math. Notes, 64:4 (1998), 471–475

Citation in format AMSBIB
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\by Ya.~Sh.~Il'yasov
\paper On asymptotics of solutions to semilinear elliptic equations near the first eigenvalue of the nonperturbed problem
\jour Mat. Zametki
\yr 1998
\vol 64
\issue 4
\pages 543--548
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\crossref{https://doi.org/10.4213/mzm1428}
\mathscinet{http://www.ams.org/mathscinet-getitem?mr=1687228}
\zmath{https://zbmath.org/?q=an:0926.35108}
\transl
\jour Math. Notes
\yr 1998
\vol 64
\issue 4
\pages 471--475
\crossref{https://doi.org/10.1007/BF02314627}
\isi{http://gateway.isiknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=PARTNER_APP&SrcAuth=LinksAMR&DestLinkType=FullRecord&DestApp=ALL_WOS&KeyUT=000079258700026}


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    Citing articles on Google Scholar: Russian citations, English citations
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    This publication is cited in the following articles:
    1. Il'yasov Ya., Runst T., “Positive Solutions of Indefinite Equations with P-Laplacian and Supercritical Nonlinearity”, Complex Var. Elliptic Equ., 56:10-11, SI (2011), 945–954  crossref  mathscinet  zmath  isi  scopus  scopus
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