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Invariant sets of descending flow in critical point theory with applications to nonlinear differential equations. (English) Zbl 0995.58006

Let \(f:X\to\mathbb R\) be a \({\mathcal{C}}^1\)-functional on a Banach space \(X\) satisfying the Palais-Smale condition. Let \( D_1, D_2\subset X\) be positive invariant subsets with respect to the flow \(u(t,u_0)\) associated to a pseudogradient vector field of \(f\).
The author proves several variations of a result which yields the existence of four critical points, a local minimum \(u_1\in D_1\cap D_2\), two mountain pass points \(u_2\in D_1\setminus D_2\), \(u_3\in D_2\setminus D_1\), and \(u_4\in X\setminus (D_1\cup D_2)\) of Morse index \(2\) type. The solutions \(u_2, u_3, u_4\) are obtained via a dynamical systems argument, not by a standard min-max argument. They lie on the boundary of the set \(\{u_0\in X:u(t,u_0)\in D_1\cap D_2\) for some \(t\geq 0\}\), the domain of attraction of \(D_1\cap D_2\).
The abstract results are applied to the semilinear elliptic boundary value problem \(-\Delta u=f(u)\) on a bounded smooth domain in \(\mathbb R ^N\) with Dirichlet boundary conditions. Conditions on \(f\) yield a sub/supersolution pair \(\phi <\psi\), and \(D_1=[\phi,\infty)\), \(D_2=(-\infty,\psi]\) are order cones. A second application deals with periodic solutions of nonautonomous second order Hamiltonian systems.

MSC:

58E05 Abstract critical point theory (Morse theory, Lyusternik-Shnirel’man theory, etc.) in infinite-dimensional spaces
37J45 Periodic, homoclinic and heteroclinic orbits; variational methods, degree-theoretic methods (MSC2010)
35J65 Nonlinear boundary value problems for linear elliptic equations
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