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On the existence of bounded Palais-Smale sequences and application to a Landesman-Lazer-type problem set on \(\mathbb{R}^N\). (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 \(u\in H^1({\mathbb R}^N)\) 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) \(f(x,u)u^{-1}\rightarrow a\in (0,+\infty ]\) as \(u\rightarrow +\infty\); 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.

MSC:
35J60 Nonlinear elliptic equations
35A15 Variational methods applied to PDEs
49J35 Existence of solutions for minimax problems
58E05 Abstract critical point theory (Morse theory, Lyusternik-Shnirel’man theory, etc.) in infinite-dimensional spaces
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