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.


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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[1] Ambrosetti, A.; Rabinowitz, P. H., Dual variational methods in critical point theory and applications, J. Funct. Anal., 14, 349-381 (1973) · Zbl 0273.49063
[2] Bartsch, T.; Clapp, M., Critical point theory for indefinite functionals with symmetries, J. Funct. Anal., 138, 107-136 (1996) · Zbl 0853.58027
[3] Brezis, H., On a characterization of flow-invariant sets, Comm. Pure Appl. Math., 23, 261-263 (1970) · Zbl 0191.38703
[4] Brezis, H.; Nirenberg, L., Remarks on finding critical points, Comm. Pure Appl. Math., 64, 939-963 (1991) · Zbl 0751.58006
[5] Chang, K. C., Infinite Dimensional Morse Theory and Multiple Solution Problems (1993), Birkhäuser: Birkhäuser Boston · Zbl 0779.58005
[6] Chang, K. C., A variant mountain pass lemma, Sci. Sinica (Ser. A), 26, 1241-1255 (1983) · Zbl 0544.35044
[7] Costa, D. G.; Magalhães, C. A., A variational approach to subquatratic perturbations of elliptic systems, J. Differential Equations, 111, 103-122 (1994) · Zbl 0803.35052
[8] Dancer, E. N.; Du, Y., A note on multiple solutions of some semilinear elliptic problems, J. Math. Anal. Appl., 211, 626-640 (1997) · Zbl 0880.35046
[9] Dancer, E. N.; Du, Y., Multiple solutions of some semilinear elliptic equations via the generalized Conley index, J. Math. Anal. Appl., 189, 848-871 (1995) · Zbl 0834.35049
[10] Deimling, K., Ordinary Differential Equations in Banach Spaces. Ordinary Differential Equations in Banach Spaces, Lecture Notes in Math., 596 (1977), Springer-Verlag: Springer-Verlag New York · Zbl 0361.34050
[11] Hofer, H., Variational and topological methods in patially ordered Hilbert spaces, Math. Ann., 261, 493-514 (1982) · Zbl 0488.47034
[12] Li, S. J., Periodic solutions of non-autonomous second order systems with superlinear terms, Differential Integral Equations, 5, 1419-1424 (1992) · Zbl 0757.34021
[13] Li, S. J.; Willem, M., Applications of local linking to critical point theory, J. Math. Anal. Appl., 189, 6-32 (1995) · Zbl 0820.58012
[14] Liu, Z., Multiple Solutions of Differential Equations (1992), Shandong University: Shandong University Jinan
[15] Liu, Z., A topological property of boundary of bounded open sets in plane, J. Shandong Univ., 29, 299-304 (1994) · Zbl 0831.54027
[16] Milnor, J., Morse Theory (1963), Princeton Univ. Press: Princeton Univ. Press Princeton · Zbl 0108.10401
[17] Palais, R. S., Critical point theory and the minimax principle, Proc. Sympos. Pure Math. (1970), Amer. Math. Soc: Amer. Math. Soc Providence, p. 185-202 · Zbl 0212.28902
[18] Palais, R. S., Lusternik-Schnirelmann theory on Banach manifolds, Topology, 5, 115-132 (1966) · Zbl 0143.35203
[19] Peral, I.; van der Vorst, R., On some quasilinear systems, Rocky Mountain J. Math., 27, 913-927 (1997) · Zbl 0901.35030
[20] Rabinowitz, P. H., Minimax Methods in Critical Point Theory with Applications to Differential Equations. Minimax Methods in Critical Point Theory with Applications to Differential Equations, CBMS, Regional Conf. Ser. in Math., 65 (1986), Amer. Math. Soc: Amer. Math. Soc Providence · Zbl 0609.58002
[21] Stone, A. H., Paracompactness and product spaces, Bull. Amer. Math. Soc., 54, 977-982 (1948) · Zbl 0032.31403
[22] Sun, J., On Some Problems about Nonlinear Operators (1984), Shandong University: Shandong University Jinan
[23] Sun, J., The Schauder condition in the critical point theory, Chinese Sci. Bull., 31, 1157-1162 (1986) · Zbl 0603.47045
[24] Sun, J.; Liu, Z., Calculus of variations and super- and sub-solutions in reversed order, Acta Math. Sinica, 37, 512-514 (1994) · Zbl 0810.47059
[25] Wang, Z. Q., On a superlinear elliptic equation, Anal. Nonlineaire, 8, 43-58 (1991) · Zbl 0733.35043
[26] Whyburn, G. T., Topological Analysis (1958), Princeton Univ. Press: Princeton Univ. Press Princeton · Zbl 0080.15903
[27] Zou, W., Solutions for resonant elliptic systems with nonodd or odd nonlinearities, J. Math. Anal. Appl., 223, 397-417 (1998) · Zbl 0921.35062
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