Results 141 to 150 of about 781 (177)
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The Mountain Pass Theorem in Terms of Tangencies
Journal of Mathematical Analysis and ApplicationsSi Tiep Dinh, Tiến-Sơn Pham
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An Application of a Mountain Pass Theorem
Acta Mathematica Sinica, English Series, 2002The present paper is devoted to study the following Dirichlet problem: \[ -\Delta u=f(x,u), \quad x\in\Omega,\;u\in H^1_0(\Omega),\tag{1} \] where \(\Omega\) is a bounded smooth domain in \(\mathbb{R}^N\), with \(f(x,t)\) asymptotically linear in \(t\) at infinity.
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Minimization and Mountain-Pass Theorems
2001In this introductory chapter, we consider the concept on differentiability of mappings in Banach spaces, Frechet and Gâteaux derivatives, secondorder derivatives and general minimization theorems. Variational principles of Ekeland [Ek1] and Borwein & Preiss [BP] are proved and relations to the minimization problem are given. Deformation lemmata, Palais—
Maria do Rosário Grossinho +1 more
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2003
This 2003 book presents min-max methods through a study of the different faces of the celebrated Mountain Pass Theorem (MPT) of Ambrosetti and Rabinowitz. The reader is led from the most accessible results to the forefront of the theory, and at each step in this walk between the hills, the author presents the extensions and variants of the MPT in a ...
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This 2003 book presents min-max methods through a study of the different faces of the celebrated Mountain Pass Theorem (MPT) of Ambrosetti and Rabinowitz. The reader is led from the most accessible results to the forefront of the theory, and at each step in this walk between the hills, the author presents the extensions and variants of the MPT in a ...
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An alternative proof of the mountain pass theorem for a class of functionals
Journal of Global Optimization, 2012The authors present an alternative proof of the well known mountain pass theorem by the use of the Ekeland variational principle for a class of \(C^1\)-functionals.
Marcelo Montenegro, Adilson E. Presoto
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2007
Roughly speaking, the basic idea behind the so-called minimax method is the following: Find a critical value of a functional ϕ ∈ C1 (X, ℝ) as a minimax (or maximin) value c ∈ ℝ of ϕ over a suitable class A of subsets of X: $$ c = \mathop {\inf }\limits_{A \in \mathcal{A}} \mathop {\sup }\limits_{u \in A} \phi \left( u \right). $$
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Roughly speaking, the basic idea behind the so-called minimax method is the following: Find a critical value of a functional ϕ ∈ C1 (X, ℝ) as a minimax (or maximin) value c ∈ ℝ of ϕ over a suitable class A of subsets of X: $$ c = \mathop {\inf }\limits_{A \in \mathcal{A}} \mathop {\sup }\limits_{u \in A} \phi \left( u \right). $$
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A VARIANT OF THE MOUNTAIN-PASS THEOREM
2013An existence result for a critical point of mountain-pass type, where the classical Palais-Smale condition is not required, is presented. A multiple-critical-point result is then obtained As an application, the existence of two positive classical solutions for two-point boundary-value problems, without assuming any asymptotic condition on the ...
BONANNO, Gabriele, D'AGUI', GIUSEPPINA
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A Generalized Mountain Pass Theorem
20202010 Mathematics Subject Classification: 58E05, 58E30.
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A Mountain Pass Theorem and Moduli Space of Minimal Immersions
The Journal of Geometric AnalysisGiven two Banach spaces \(X\) and \(Y\), the author considers functionals \(\mathcal{A}\colon X\times Y\to \mathbb{R}\) of class \(C^1\) which satisfy certain conditions, including that partial maps \(\mathcal{A}(u,\cdot)\) admit a unique minimum for each \(u\in X\), and a weak Palais-Smale condition.
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Mountain Pass Theorems, Deformation Theorems, and Palais-Smale Conditions
2001Let E be a Banach space, X ⊂ E be an open subset, f ∈ C 1 (X, R) be a functional and $$\begin{array}{*{20}{c}} {K = \left\{ {x \in X:f'\left( x \right) = 0} \right\},} \\ {{K_c} = \left\{ {x \in X:f\left( x \right) = c,f'\left( x \right) = 0} \right\}} \end{array}$$ are the sets of critical points of f.
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