Global bifurcation for asymptotically linear Schrödinger equations

Article

Abstract

We prove global asymptotic bifurcation for a very general class of asymptotically linear Schrödinger equations \({\left\{\begin{array}{lll}\Delta u + f(x, u)u = \lambda u \quad {\rm in} \; \mathbb{R}^N,\\ u \in H^1(\mathbb{R}^N) \backslash \{0\}, \quad N \; \geqslant \; 1.\qquad\qquad\qquad(1)\end{array}\right.}\) The method is topological, based on recent developments of degree theory. We use the inversion \({u\to v:= u/\Vert u\Vert_X^2}\) in an appropriate Sobolev space \({X=W^{2,p}(\mathbb{R}^{N}),}\) and we first obtain bifurcation from the line of trivial solutions for an auxiliary problem in the variables \({(\lambda,v) \in {\mathbb R}\times X.}\) This problem has a lack of compactness and of regularity, requiring a truncation procedure. Going back to the original problem, we obtain global branches of positive/negative solutions ‘bifurcating from infinity’. We believe that, for the values of λ covered by our bifurcation approach, the existence result we obtain for positive solutions of (1) is the most general so far.

Mathematics subject classification (2010)

Primary 35J61 Secondary 35B32 

Keywords

Asymptotically linear Schrödinger equations Semilinear elliptic eigenvalue problems Global bifurcation Unbounded domains 

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Copyright information

© Springer Basel AG 2012

Authors and Affiliations

  1. 1.Maxwell Institute for Mathematical Sciences, Department of MathematicsHeriot-Watt UniversityEdinburghScotland

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