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Some recent results in critical point theory. (English) Zbl 1152.58303

The author deals with the problem \[ -\Delta u= f(x,u)\quad\text{in}\;\Omega, \quad u=0\quad\text{on}\;\partial\Omega, \tag{1} \] where \(\omega\subset\mathbb R^n\) is a bounded domain whose boundary is a smooth manifold, and \(f(x,t)\) is a continuous function on \(\overline\Omega\times\mathbb R^n\). He discusses recent results in abstract critical point theory concerning weak notions of linking, the so-called “monotonicity trick” on the existence of Palais-Smale sequences, and Morse index estimates. He also gives a few applications, concerning resonance problems and jumping nonlinearities.

MSC:

58E05 Abstract critical point theory (Morse theory, Lyusternik-Shnirel’man theory, etc.) in infinite-dimensional spaces
35J20 Variational methods for second-order elliptic equations
35J61 Semilinear elliptic equations
35J65 Nonlinear boundary value problems for linear elliptic equations
47J30 Variational methods involving nonlinear operators
49J35 Existence of solutions for minimax problems
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