## 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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