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On the existence of bounded Palais-Smale sequences and application to a Landesman-Lazer-type problem set on 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 uH 1 ( N ) to the problem -Δu+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 a(0,+] as u+; and (ii) the mapping [0,+)uf(x,u)u -1 is non-decreasing, a.e. x N .

The paper gives a new and interesting perspective in the critical point theory and its applications to the study of variational problems.


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