A general multiplicity theorem for certain nonlinear equations in Hilbert spaces

In this paper, we prove the following general result: Let $X$ be a real Hilbert space and $J:X\to {\bf R}$ a continuously G\^ateaux 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.\par