Singular points of non-monotone potential operators

In this paper, we establish some results about the singular points of certain non-monotone potential operators. Here is a sample: If $X$ is an infinite-dimensional reflexive real Banach space and if $T:X\to X^*$ is a non-monotone, closed, continuous potential operator such that the functional $x\to \int_0^1T(sx)(x)ds$ is sequentially weakly lower semicontinuous and $\lim_{\|x\|\to +\infty}(\int_0^1T(sx)(x)ds+\varphi(x))=+\infty$ for all $\varphi\in X^*$, then the set of all singular points of $T$ is not $\sigma$-compact.