*(English)*Zbl 0935.35044

The author applies a “monotonicity trick” introduced by Struwe in order to derive an existence result for a large class of functionals having a mountain-pass geometry. The abstract theorem establishes, essentially, the existence of a bounded Palais-Smale sequence at the mountain-pass level. This result is then applied to deduce the existence of a positive solution $u\in {H}^{1}\left({\mathbb{R}}^{N}\right)$ to the problem $-{\Delta}u+Ku=f(x,u)$, where $K$ is a positive constant, provided that the energy functional associated to the above problem has a mountain-pass geometry. The nonlinearity $f$ is assumed to satisfy the following conditions: (i) $f(x,u){u}^{-1}\to a\in (0,+\infty ]$ as $u\to +\infty $; and (ii) the mapping $[0,+\infty )\ni u\mapsto f(x,u){u}^{-1}$ is non-decreasing, a.e. $x\in {\mathbb{R}}^{N}$.

The paper gives a new and interesting perspective in the critical point theory and its applications to the study of variational problems.