Global bifurcation for asymptotically linear Schr\"odinger equations

TL;DR

This paper establishes a comprehensive global bifurcation theory for asymptotically linear Schrödinger equations using topological degree methods, revealing extensive solution branches bifurcating from infinity.

Contribution

It introduces a novel topological approach with a truncation technique to analyze bifurcation in asymptotically linear Schrödinger equations, extending previous results.

Findings

Global bifurcation from infinity for solutions

Existence of positive and negative solution branches

Most general existence results for these equations to date

Abstract

We prove global asymptotic bifurcation for a very general class of asymptotically linear Schr\"odinger equations \begin{equation}\label{1} \{{array}{lr} \D u + f(x,u)u = \lam u \quad \text{in} \ {\mathbb R}^N, u \in H^1({\mathbb R}^N)\setmimus\{0\}, \quad N \ge 1. {array}. \end{equation} 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 $(\lambda,v) \in {\mathbb R} \x X$. 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 $\lambda$ covered by our bifurcation approach, the existence result we obtain for positive solutions of \eqref{1} is the most general so far