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A bifurcation diagram of solutions to an elliptic equation with exponential nonlinearity in higher dimensions

Part of: Boundary value problems Elliptic equations and systems

Published online by Cambridge University Press:  17 October 2017

Hiroaki Kikuchi  and
Juncheng Wei
Hiroaki Kikuchi
Affiliation:
Department of Mathematics, Tsuda College, 2-1-1 Tsuda-machi, Kodaira-shi, Tokyo 187-8577, Japan ([email protected])
Juncheng Wei
Affiliation:
Department of Mathematics, University of British Columbia, Vancouver V6T 1Z2, Canada ([email protected])
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Abstract

We consider the following semilinear elliptic equation:

where B1 is the unit ball in ℝd, d ≥ 3, λ > 0 and p > 0. Firstly, following Merle and Peletier, we show that there exists an eigenvalue λp,∞ such that (*) has a solution (λp,∞,Wp) satisfying lim|x|→0Wp(x) = ∞. Secondly, we study a bifurcation diagram of regular solutions to (*). It follows from the result of Dancer that (*) has an unbounded bifurcation branch of regular solutions that emanates from (λ, u) = (0, 0). Here, using the singular solution, we show that the bifurcation branch has infinitely many turning points around λp,∞ when 3 ≤ d ≤ 9. We also investigate the Morse index of the singular solution in the d ≥ 11 case.

Keywords

exponential nonlinearitysingular solutionbifurcation diagramMorse index

MSC classification

Secondary: 34B18: Positive solutions of nonlinear boundary value problems 35J25: Boundary value problems for second-order elliptic equations 35J60: Nonlinear elliptic equations
Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 148 , Issue 1 , February 2018 , pp. 101 - 122
Copyright
Copyright © Royal Society of Edinburgh 2018 

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