A variant of the mountain pass theorem and variational gluing
type: Item
instance of: scholarly article
title: A variant of the mountain pass theorem and variational gluing
zbMATH Open document ID: 1471.35124
author: Piero Montecchiari
author: Paul H. Rabinowitz
published in: Milan Journal of Mathematics
publication date: 2021-05-25T00:00:00Z
review text: The authors discuss a new generalization of the Mountain Pass Theorem obtained in their previous work [Calc. Var. Partial Differ. Equ. 57, No. 5, Paper No. 114, 31 p. (2018; Zbl 1405.37072)], and they demonstrate its application to two classes of problems: PDEs of the form \(-\Delta u + u = F_u(x,u)\) in \(\mathbb{R}^n\), and the second order Hamiltonian systems with a double well potential \(-\ddot{q} + V_q(t,q) = 0\), \(t \in \mathbb{R}\), \(q \in \mathbb{R}^m\), under certain assumption on \(F\) and \(V\) including periodicity in \(x\) and \(t\), respectively. The distinguishing feature of these classes of problems is the presence of the mountain pass geometry of the corresponding energy functionals with the simultaneous violation of an appropriate compactness condition (e.g., the Palais--Smale condition). The authors overview previous results on these classes of problems and provide sufficient conditions for the applicability of the generalized Mountain Pass Theorem in order to prove the existence of nontrivial solutions, including multibump and multitransition solutions. Moreover, the system of PDEs of the form \(-\Delta u + G_u(x,u)=0\) in \(\mathbb{R}^n\) is also discussed.
reviewed by: Vladimir Bobkov
Mathematics Subject Classification ID: 35J50
Mathematics Subject Classification ID: 35J47
Mathematics Subject Classification ID: 35J57
Mathematics Subject Classification ID: 34C37
zbMATH DE Number: 7350887
zbMATH Keywords: variational methods
zbMATH Keywords: mountain pass theorem
zbMATH Keywords: variational gluing
zbMATH Keywords: nondegeneracy condition
zbMATH Keywords: heteroclinic solutions
zbMATH Keywords: homoclinic solutions
zbMATH Keywords: double well potential
zbMATH Keywords: multitransition solutions
zbMATH Keywords: Hamiltonian systems
MaRDI profile type: Publication
full work available at URL: s00032-020-00318-3
OpenAlex ID: W3082423381
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DOI: 10.1007/S00032-020-00318-3
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description: scientific article; zbMATH DE number 7350887