A variant of the mountain pass theorem and variational gluing
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.
scientific article; zbMATH DE number 7350887
scientific article