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A Variation of the Mountain Pass Lemma and Applications
Journal of the London Mathematical Society, 1991This paper studies functionals \(f\in C^ 1(H,\mathbb{R})\) \((H\)-Hilbert space) satisfying the conditions of the mountain pass lemma with the exception of the PS condition. The author was able to find \(c\in\mathbb{R}\) such that for any rapidly decreasing function \(\psi:\mathbb{R}_ +\to\mathbb{R}_ +\) there is a sequence \((u_ j)\subset H ...
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Ekeland's variational principle and the mountain pass lemma
Acta Mathematica Sinica, English Series, 1985The mountain-pass lemma of Abrosetti and Rabinowitz gives conditions under which a smooth mapping \(F: X\to {\mathbb{R}}\), where X is a Banach space, has a critical point. The hypotheses are simply that F display mountain-pass structure relative to some points \(x_ 0\) and \(y_ 0\), and that the Palais-Smale compactness condition be verified.
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Mountain Pass Solutions for a Double-Well Energy
Journal of Differential Equations, 2002 We establish the existence of a mountain pass solution for a variational integral involving a quasiconvex function with a double-well structure in the geometrically linear elasticity setting.
Kewei Zhang
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Studies in Applied Mathematics, 1996
Assuming that a function g solves the classical Plateau problem for a disc‐type surface, we give conditions on a non zero function H in ℝ3 with respect to g to obtain multiple weak solutions to the corresponding problem with mean curvature H.
Lami Dozo, E., Mariani, M. C.
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Assuming that a function g solves the classical Plateau problem for a disc‐type surface, we give conditions on a non zero function H in ℝ3 with respect to g to obtain multiple weak solutions to the corresponding problem with mean curvature H.
Lami Dozo, E., Mariani, M. C.
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A Mountain Pass Lemma and its implications regarding the uniqueness of constrained minimizers
Optimization, 2010We present a version of the classical Mountain Pass Lemma and explain how to combine it with constraint qualifications to prove that nonlinear programming problems have a unique local minimizer.
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Mountain Passes and Saddle Points [PDF]
SIAM Review, 2015 Variational methods find solutions of equations by considering a solution as a critical point of an appropriately chosen function. Local minima and maxima are well-known types of critical points.
Bisgard, James
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Archive for Rational Mechanics and Analysis, 1992
This paper is concerned with the existence of a weak solution for the quasilinear elliptic Dirichlet boundary-value problem on \(\Omega \subset \mathbb{R}^ n\), \(n \geq 3\), \[ -\nabla \cdot \bigl( | \nabla u |^{p- 2}\nabla u \bigr)+| u |^{p-2}u=f(u) \] with \(2 \leq ...
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This paper is concerned with the existence of a weak solution for the quasilinear elliptic Dirichlet boundary-value problem on \(\Omega \subset \mathbb{R}^ n\), \(n \geq 3\), \[ -\nabla \cdot \bigl( | \nabla u |^{p- 2}\nabla u \bigr)+| u |^{p-2}u=f(u) \] with \(2 \leq ...
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2009
Summary: Let \(M\) be a complete \(C^1\)-Finsler manifold without boundary and \(f: M\to \mathbb{R}\) be a locally Lipschitz function. The classical proof of the well-known deformation lemma can not be extended to this case because integral lines may not exist.
Ribarska, Nadezhda +2 more
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Summary: Let \(M\) be a complete \(C^1\)-Finsler manifold without boundary and \(f: M\to \mathbb{R}\) be a locally Lipschitz function. The classical proof of the well-known deformation lemma can not be extended to this case because integral lines may not exist.
Ribarska, Nadezhda +2 more
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