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On the Hardy–Sobolev equation

Part of: Elliptic equations and systems Partial differential equations

Published online by Cambridge University Press:  16 January 2017

E. N. Dancer ,
F. Gladiali  and
M. Grossi
E. N. Dancer
Affiliation:
School of Mathematics and Statistics, University of Sydney, NSW 2006 Australia (norman.dancer@sydney.edu.au) and Department of Mathematics, Talbot Building, Swansea University, Singleton Park, Swansea SA2 8PP, UK
F. Gladiali
Affiliation:
Dipartimento PolComIng – Matematica e Fisica, Università di Sassari, Via Piandanna 4, 07100 Sassari, Italy (fgladiali@uniss.it)
M. Grossi
Affiliation:
Dipartimento di Matematica, Università degli Studi di Roma ‘La Sapienza’, Piazzale Aldo Moro 2, 00185 Roma, Italy (massimo.grossi@uniroma1.it)
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In this paper we study the problem

where Ω = ℝ N or Ω = B 1, N ⩾ 3, p > 1 and . Using a suitable map we transform problem (1) into another one without the singularity 1/|x|2. Then we obtain some bifurcation results from the radial solutions corresponding to some explicit values of λ.

Keywords

semilinear elliptic equationspositive solutionsbifurcation theoryHardy inequalitySobolev inequality

MSC classification

Secondary: 35J15: Second-order elliptic equations 35J61: Semilinear elliptic equations 35B32: Bifurcation 35B33: Critical exponents 35B09: Positive solutions 35B06: Symmetries, invariants, etc.
Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 147 , Issue 2 , April 2017 , pp. 299 - 336
Copyright
Copyright © Royal Society of Edinburgh 2017