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Zbl 0935.35044
Jeanjean, Louis
On the existence of bounded Palais-Smale sequences and application to a Landesman-Lazer-type problem set on $\bbfR^N$.
(English)
[J] Proc. R. Soc. Edinb., Sect. A, Math. 129, No.4, 787-809 (1999). ISSN 0308-2105; ISSN 1473-7124/e

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. $x\in{\Bbb R}^N$. \par The paper gives a new and interesting perspective in the critical point theory and its applications to the study of variational problems.
[Vicentiu D.Rădulescu (Craiova)]
MSC 2000:
*35J60 Nonlinear elliptic equations
35A15 Variational methods (PDE)
49J35 Minimax problems (existence)
58E05 Abstract critical point theory

Keywords: critical point theory; Palais-Smale condition; mountain-pass geometry

Cited in: Zbl pre06120543 Zbl 1119.35085

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