## 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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