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 to the problem , where is a positive constant, provided that the energy functional associated to the above problem has a mountain-pass geometry. The nonlinearity is assumed to satisfy the following conditions: (i) as ; and (ii) the mapping is non-decreasing, a.e. .
The paper gives a new and interesting perspective in the critical point theory and its applications to the study of variational problems.