A smooth global branch of solutions for a semilinear elliptic equation on \({\mathbb{R}^N}\)

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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.

Mathematics Subject Classification (2000)

35J60 35B32 

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© Springer-Verlag 2009

Authors and Affiliations

  1. 1.OxPDE, Mathematical InstituteUniversity of OxfordOxfordUK

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