Let F be a dense vector subspace of a Banach space E, and f be a functional defined on F, in which f is not expected to have any critical point. We extend the variational theory to seek generalized critical points of f in E (not in F). Using these results we find solutions in \(W_ 0^{1,2}(\Omega)\) of the following equation \[ (P)\quad -\Delta u+c=Ke^ u, \] where \(\Omega\) is a bounded domain in \(R^ 2\), and c and K are measurable functions on \(\Omega\). Our results generalize those by [\textit{J. L. Kazdan} and \textit{F. W. Warner} [Ann. Math., II. Ser. 99, 203- 219 (1974; Zbl 0278.53031)]. In order to solve (P), we have improved the Sobolev inequality and the Trudinger inequality. The applications to singular elliptic equations in \(R^ n\), \(n\geq 3\), are given.