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A variant of mountain-pass theorem. (English) Zbl 1299.35121
The authors establish a version of the mountain-pass theorem, in which the functional involved satisfies a cut-off Palais-Smale condition, weaker than the classical one (Theorem 2.1). They show that either the critical point is not a local minimum or the functional admits a sequence of distinct local minima. They obtain also a multiplicity result of critical points (Theorem 2.3) and give a precise localization of them.
Finally, they present an application to a two-point boundary value problem, and obtain a precise estimate of the \(L^{\infty }\) norm of the solutions.
35J60 Nonlinear elliptic equations
35J20 Variational methods for second-order elliptic equations
49J35 Existence of solutions for minimax problems
49J52 Nonsmooth analysis