Invariant Sets of Descending Flow in Critical Point Theory with Applications to Nonlinear Differential Equations

By Zhaoli Liu, Jingxian Sun. 2001

218 citations recorded by the source corpus.

Abstract Properties of invariant sets of descending flow defined by a pseudogradient vector field of a functional in a Banach space are studied. In this way, several critical points can be found by constructing different invariant sets on which the functional is bounded below. Under suitable conditions, the existence of at least four critical points of a functional is proved, each critical point being in a certain invariant set. The theoretical results are applied to nonlinear elliptic boundary value problems and nonlinear systems of ordinary differential equations. In variant cases, at least four solutions are obtained for these equations.

Subfields: Applied Mathematics, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/c3151bc2e85bf53d61534a15a03569f0ac66f5be

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Citations in the corpus

A note on additional properties of sign changing solutions to superlinear elliptic equations (2003, background): “Here F (x, t) := ∫ t 0 f(x, s) ds is a primitive of f .”
Nodal Solutions of a p‐Laplacian Equation (2005, unknown): “De:ne as in [32]: CXðOÞ :1⁄4 O [ fu 2 X0 : there exists t> 0 such that ’ðuÞ 2 Og: By @CXðOÞ we denote the boundary of CXðOÞ in X.”
Infinitely many sign-changing solutions for Kirchhoff type problems in R3 (2019, methodology): “method of invariant sets of descending flow has been used widely in dealing with sign-changing solutions of elliptic problem, see [2, 3, 19] and the references therein.”
Schrödinger equations with concave and convex nonlinearities (2005, background): “The main idea in [3, 17, 27, 28, 29 ] is to make use of the cones of positive functions and negative functions in C 1 0 (Ω).”
Sign Changing Solutions of Superlinear Schrödinger Equations (2003, background): “We may again quote Liu and Sun (2001) for the existence of an element w∗n ∈ Cn so that the omega limit set w∗n = ⋂ 0≤t≤T w∗n clos ( ⋃ t≤s T w∗n s w∗n ) contains a critical point un ∈ E\D .”
Infinitely many sign-changing solutions for the nonlinear Schrödinger–Poisson system (2014, methodology): “However, with the presence of the coupling term φu, the techniques of constructing invariant sets of descending flow in [6, 7, 8, 9, 10, 11, 24, 25] can not be directly applied to system (1.”