Asymptotic expansion for nonlinear eigenvalue problems

Abstract

In this paper we consider generalized eigenvalue problems for a family of operators with a quadratic dependence on a complex parameter. Our model is $L(\lambda )=-△+(P(x)-\lambda {)}^2$ in $L^2({\mathbb{R}}^d)$ where $P$ is a positive elliptic polynomial in ${\mathbb{R}}^d$ of degree $m\ge 2$. It is known that for $d$ even, or $d=1$, or $d=3$ and $m\ge 6$, there exist $λ\in\C$ and $u\in L^2({\mathbb{R}}^d)$, $u\ne 0$, such that $L(\lambda )u=0$. In this paper, we give a method to prove existence of non trivial solutions for the equation $L(\lambda )u=0$, valid in every dimension. This is a partial answer to a conjecture in \cite{herowa}.