A smooth global branch of solutions for a semilinear elliptic equation on {\mathbb{R}^N} |
| |
Authors: | François Genoud |
| |
Institution: | 1. OxPDE, Mathematical Institute, University of Oxford, 24-29 St Giles, Oxford, OX1 3LB, UK
|
| |
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. |
| |
Keywords: | |
本文献已被 SpringerLink 等数据库收录! |
|