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On the existence of bounded Palais-Smale sequences and application to a Landesman-Lazer-type problem set on $\bbfR^N$. (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 $u\in H^1({\Bbb R}^N)$ 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. $x\in{\Bbb R}^N$. The paper gives a new and interesting perspective in the critical point theory and its applications to the study of variational problems.

35J60Nonlinear elliptic equations
35A15Variational methods (PDE)
49J35Minimax problems (existence)
58E05Abstract critical point theory
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