Nonlinear eigenvalue problems admitting eigenfunctions with known geometric properties

Abstract

We consider nonlinear eigenvalue problems of the form $$ A_0 y + B(y) y = \lambda y \tag{$*$} $$ in a real Hilbert space $\mathcal H$, where $A_0$ is a semi-bounded self-adjoint operator and, for every $y$ from a certain dense subspace $X$ of $\mathcal H, B(y)$ is a bounded symmetric linear operator. The left hand side is assumed to be the gradient of a functional $\psi \in C^1(x)$, and the associated linear problems $$ A_0 v + B(y) v = \mu v \tag{$**$} $$ are supposed to have discrete spectrum $(y \in X)$. We present a new topological method which permits, under appropriate assumptions, to construct solutions of ($*$) on a sphere $S_R := \{ y \in X \mid \|y\|_{\mathcal H} = R\}$ whose $\psi$-value is the $n$th Ljusternik-Schnirelman level of $\psi |_{S_R}$ and whose corresponding eigenvalue is the $n$th eigenvalue of the associated linear problem ($**$), where $R > 0$ and $n \in \mathbb N$ are given. In applications, the eigenfunctions thus found share any geometric property enjoyed by an $n$-th eigenfunction of a linear problem of the form ($**$). We discuss applications to nonlinear Sturm-Liouville problems, to the nonlinear Hill's equation, to periodic solutions of second-order systems, and to elliptic partial differential equations with radial symmetry.