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 Izv. RAN. Ser. Mat., 2002, Volume 66, Issue 6, Pages 19–48 (Mi izv408)

Non-local investigation of bifurcations of solutions of non-linear elliptic equations

Ya. Sh. Il'yasov

Abstract: We justify the projective fibration procedure for functionals defined on Banach spaces. Using this procedure and a dynamical approach to the study with respect to parameters, we prove that there are branches of positive solutions of non-linear elliptic equations with indefinite non-linearities. We investigate the asymptotic behaviour of these branches at bifurcation points. In the general case of equations with $p$-Laplacian we prove that there are upper bounds of branches of positive solutions with respect to the parameter.

DOI: https://doi.org/10.4213/im408

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English version:
Izvestiya: Mathematics, 2002, 66:6, 1103–1130

Bibliographic databases:

UDC: 517.95
MSC: 47J25, 58E05
Revised: 15.09.2000

Citation: Ya. Sh. Il'yasov, “Non-local investigation of bifurcations of solutions of non-linear elliptic equations”, Izv. RAN. Ser. Mat., 66:6 (2002), 19–48; Izv. Math., 66:6 (2002), 1103–1130

Citation in format AMSBIB
\Bibitem{Ily02} \by Ya.~Sh.~Il'yasov \paper Non-local investigation of bifurcations of solutions of non-linear elliptic equations \jour Izv. RAN. Ser. Mat. \yr 2002 \vol 66 \issue 6 \pages 19--48 \mathnet{http://mi.mathnet.ru/izv408} \crossref{https://doi.org/10.4213/im408} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=1970351} \zmath{https://zbmath.org/?q=an:1112.35311} \transl \jour Izv. Math. \yr 2002 \vol 66 \issue 6 \pages 1103--1130 \crossref{https://doi.org/10.1070/IM2002v066n06ABEH000408} 

• http://mi.mathnet.ru/eng/izv408
• https://doi.org/10.4213/im408
• http://mi.mathnet.ru/eng/izv/v66/i6/p19

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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. Il'yasov I. Sh., “A nonlocal study of families of elliptic equations with convex-concave nonlinearities”, Dokl. Math., 68:2 (2003), 258–260
2. Il'yasov Ya.Sh., “On global positive solutions of parabolic equations with a sign-indefinite nonlinearity”, Differ. Equ., 41:4 (2005), 548–556
3. Egorov Yu.V., Il'yasov Ya.Sh., “Multiple solutions to the Yamabe problem”, Dokl. Math., 74:1 (2006), 484–486
4. Ya. Sh. Il'yasov, “Bifurcation Calculus by the Extended Functional Method”, Funct. Anal. Appl., 41:1 (2007), 18–30
5. Il'yasov Ya., Egorov Y., “Hopf boundary maximum principle violation for semilinear elliptic equations”, Nonlinear Analysis-Theory Methods & Applications, 72:7–8 (2010), 3346–3355
6. V. E. Bobkov, “On existence of nodal solution to elliptic equations with convex-concave nonlinearities”, Ufa Math. J., 5:2 (2013), 18–30
7. Bobkov V., Il'Yasov Ya., “Asymptotic Behaviour of Branches for Ground States of Elliptic Systems”, Electron. J. Differ. Equ., 2013, 212
8. Bobkov V., “Least Energy Nodal Solutions For Elliptic Equations With Indefinite Nonlinearity”, Electron. J. Qual. Theory Differ., 2014, no. 56, 1–15
9. Jesús.I.ldefonso Díaz, Jesús Hernández, Yavdat Il’yasov, “On the existence of positive solutions and solutions with compact support for a spectral nonlinear elliptic problem with strong absorption”, Nonlinear Analysis: Theory, Methods & Applications, 2015
10. Ildefonso Diaz J., Hernandez J., Il'yasov Ya., “Flat solutions of some non-Lipschitz autonomous semilinear equations may be stable for N 3”, Chin. Ann. Math. Ser. B, 38:1 (2017), 345–378
11. Y. Sh. Il'yasov, E. E. Kholodnov, “On global instability of solutions to hyperbolic equations with non-Lipschitz nonlinearity”, Ufa Math. J., 9:4 (2017), 44–53
12. El Aidi M., “On a Weak Solution For a Doubly Critical Fourth-Order Semilinear Elliptic Equation in a Compact Manifold”, J. Math. Anal. Appl., 472:1 (2019), 864–878
13. Bobkov V., Drabek P., Ilyasov Ya., “Estimates on the Spectral Interval of Validity of the Anti-Maximum Principle”, J. Differ. Equ., 269:4 (2020), 2956–2976
14. de Albuquerque J.C., Silva K., “On the Extreme Value of the Nehari Manifold Method For a Class of Schrodinger Equations With Indefinite Weight Functions”, J. Differ. Equ., 269:7 (2020), 5680–5700
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