NDLI: A smooth global branch of solutions for a semilinear elliptic equation on $${\mathbb{R}^N}$$

Content Provider	Springer Nature Link
Author	Geud, François
Copyright Year	2009
Abstract	The existence of a global branch of positive spherically symmetric solutions $${\{(\lambda,u(\lambda)):\lambda\in(0,\infty)\}}$$ of the semilinear elliptic equation $$\Delta u - \lambda u + V(x)\|u\|^{p-1}u = 0 \quad \text{in}\,\mathbb{R}^N\,\text{with}\,N\geq3$$ is proved for $${1 < p < 1+\frac{4-2b}{N-2}}$$ , where $${b\in(0,2)}$$ is such that the radial function V vanishes at infinity like \|x\|−b . V is allowed to be singular at the origin but not worse than \|x\|−b . The mapping $${\lambda\mapsto u(\lambda)}$$ is of class $${C^r((0,\infty),H^1(\mathbb{R}^N))}$$ if $${V\in C^r(\mathbb{R}^N\setminus\{0\},\mathbb{R})}$$ , for r = 0, 1. Further properties of regularity and decay at infinity of solutions are also established. This work is a natural continuation of previous results by Stuart and the author, concerning the existence of a local branch of solutions of the same equation for values of the bifurcation parameter λ in a right neighbourhood of λ = 0. The variational structure of the equation is deeply exploited and the global continuation is obtained via an implicit function theorem.
Ending Page	232
Page Count	26
Starting Page	207
File Format	PDF
ISSN	09442669
e-ISSN	14320835
Journal	Calculus of Variations and Partial Differential Equations
Issue Number	1-2
Volume Number	38
Language	English
Publisher	Springer-Verlag
Publisher Date	2009-10-28
Publisher Place	Berlin, Heidelberg
Access Restriction	One Nation One Subscription (ONOS)
Subject Keyword	Calculus of Variations and Optimal Control; Optimization Systems Theory, Control Theoretical, Mathematical and Computational Physics Analysis Bifurcation Nonlinear elliptic equations
Content Type	Text
Resource Type	Article
Subject	Applied Mathematics Analysis

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