 0945 Fatima Aboud, Didier Robert
 Asymptotic expansion for non linear eigenvalue problems
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Mar 5, 09

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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
$L(\lambda)=\triangle +(P(x)\lambda)^2$ in $L^2(\R^d)$ where $P$ is a positive elliptic polynomial in $\R^d$
of degree $m\geq 2$.
It is known that for $d$ even, or $d=1$, or $d=3$ and $m\geq 6$, there exist $\lambda\in\C$ and $u\in L^2(\R^d)$,
$u\neq 0$, 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}.
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