A general multiplicity theorem for certain nonlinear equations in Hilbert spaces

Abstract:

In this paper, we prove the following general result. Let $X$ be a real Hilbert space and $J:X\to \textbf {R}$ a continuously Gâteaux differentiable, nonconstant functional, with compact derivative, such that \[ \limsup _{\|x\|\to +\infty }{{J(x)}\over {\|x\|^2}}\leq 0\ .\] Then, for each $r\in \ ]\inf _{X}J,\sup _{X}J[$ for which the set $J^{-1}([r,+\infty [)$ is not convex and for each convex set $S\subseteq X$ dense in $X$, there exist $x_0\in S\cap J^{-1}(]-\infty ,r[)$ and $\lambda >0$ such that the equation \[ x=\lambda J’(x)+x_0\] has at least three solutions.