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Morse theory methods for a class of quasi-linear elliptic systems of higher order. (English) Zbl 1427.58004

In this paper, the author suggests a local Morse theory for a class of functionals on Hilbert spaces, which are not twice continuously differentiable. To describe this class, let \(H\) be a Hilbert space, \(X\) a dense linear subspace, and \(\mathcal{L}\in C^1(V,\mathbb{R})\), where \(V\) is an open neighborhood of \(0\in H\), such that \(\mathcal{L}'(0)=0\). It is required that the gradient \(\nabla\mathcal{L}\) has a Gâteaux derivative \(B(u)\), a symmetric bounded operator on \(H\) at every \(u\in V\cap X\), and that \(B: V\cap X \to L_s(H)\) has a decomposition \(B=P+Q\) with each \(P(u)\) positive definite and \(Q(u)\) compact. Additionally required are also
(1) All eigenfunctions of the operator \(B(0)\) that correspond to non-positive eigenvalues belong to \(X\).
(2) For any sequence \((x_k)\) in \(V\cap X\) with \(\|x_k\|\to 0\), \(\|P(x_k)u-P(0)u\|\to 0\) for any \(u\in H\).
(3) The map \(Q: V\cap X \to L(H)\) is continuous at \(0\) with respect to the topology on \(H\).
(4) For any sequence \((x_k)\) in \(V\cap X\) with \(\|x_k\|\to 0\), there exist constants \(C_0>0\) and \(k_0\in \mathbb{N}\) such that \((P(x_k)u,u) \geq C_0\|u\|^2\) for all \(u\in H\) and for all \(k\geq k_0\).
For functions satisfying these conditions, the author then derives a Morse-Palais Lemma, and under the assumption that \(X=H\), a generalization of the Gromoll-Meyer splitting theorem. Comparisons with other hypotheses for local Morse theories are given, as well as new applications of his hypothesis.

MSC:

58E05 Abstract critical point theory (Morse theory, Lyusternik-Shnirel’man theory, etc.) in infinite-dimensional spaces
35J62 Quasilinear elliptic equations
49J50 Fréchet and Gateaux differentiability in optimization
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[1] Aubin, J.P., Ekeland, I.: Applied Nonlinear Analysis. Wiley, New York (1984) · Zbl 0641.47066
[2] Bartsch, T.; Szulkin, A.; Willem, M.; Krupka, D. (ed.); Saunders, D. (ed.), Morse theory and nonlinear differential equations, 41-73 (2008), Amsterdam · Zbl 1236.58023 · doi:10.1016/B978-044452833-9.50003-6
[3] Berger, M.: Nonlinearity and Functional Analysis. Academic Press, New York (1977) · Zbl 0368.47001
[4] Bobylev, N.A., Burman, Y.M.: Morse lemmas for multi-dimensional variational problems. Nonlinear Anal. 18, 595-604 (1992) · Zbl 0766.58013 · doi:10.1016/0362-546X(92)90213-X
[5] Bott, R.: Nondegenerate critical manifold. Ann. Math. 60, 248-261 (1954) · Zbl 0058.09101 · doi:10.2307/1969631
[6] Brezis, H.: Functional Analysis, Sobolev Spaces and Partial Differential Equations. Universitext. Springer, New York (2011) · Zbl 1220.46002
[7] Browder, F.E.: Nonlinear elliptic boundary value problems and the generalized topological degree. Bull. Am. Math. Soc. 76, 999-1005 (1970) · Zbl 0201.18401 · doi:10.1090/S0002-9904-1970-12530-7
[8] Browder, F.E.: Fixed point theory and nonlinear problem. Bull. Am. Math. Soc. (New Ser.) 9, 1-39 (1983) · Zbl 0533.47053 · doi:10.1090/S0273-0979-1983-15153-4
[9] Caklovic, L., Li, S.J., Willem, M.: A note on Palais-Smale condition and coercivity. Differ. Integral Equ. 3, 799-800 (1990) · Zbl 0782.58019
[10] Carmona, J., Cingolani, S., Martínez-Aparicio, P.-J., Vannella, G.: Regularity and Morse index of the solutions to critical quasilinear elliptic systems. Commun. Partial Differ. Equ. 38(10), 1675-1711 (2013) · Zbl 1304.35269 · doi:10.1080/03605302.2013.816856
[11] Chang, K.C.: Morse theory on Banach space and its applications to partial differential equations. Chin. Ann. Math. Ser. B 4, 381-399 (1983) · Zbl 0534.58020
[12] Chang, K.C.: Infinite Dimensional Morse Theory and Multiple Solution Problem. Birkhäuser, Basel (1993) · doi:10.1007/978-1-4612-0385-8
[13] Chang, K.C.: Methods in Nonlinear Analysis. Springer Monogaphs in Mathematics. Springer, Berlin (2005) · Zbl 1081.47001
[14] Chang, K.C., Ghoussoub, H.: The Conley index and the critical groups via an extension of Gromoll-Meyer theory. Topol. Methods Nonlinear Anal. 7, 77-93 (1996) · Zbl 0898.58006 · doi:10.12775/TMNA.1996.003
[15] Chen, C.Y., Kristensen, J.: On coercive variational integrals. Nonlinear Anal. Theory Methods Appl. 153, 213-229 (2017) · Zbl 1361.49007 · doi:10.1016/j.na.2016.09.011
[16] Cingolani, S., Degiovanni, M.: On the Poincaré-Hopf theorem for functionals defined on Banach spaces. Adv. Nonlinear Stud. 9, 679-699 (2009) · Zbl 1187.58016 · doi:10.1515/ans-2009-0406
[17] Cingolani, S., Degiovanni, M., Vannella, G.: Critical group estimates for nonregular critical points of functionals associated with quasilinear elliptic equations. J. Elliptic Parabol. Equ. 1, 75-87 (2015) · Zbl 1386.35108 · doi:10.1007/BF03377369
[18] Cingolani, S., Degiovanni, M., Vannella, G.: Amann-Zehnder type results for \[p\] p-Laplace problems. Ann. Mat. Pura Appl. (4) 197(2), 605-640 (2018) · Zbl 1391.35147 · doi:10.1007/s10231-017-0694-8
[19] Cingolani, S., Vannella, G.: Marino-Prodi perturbation type results and Morse indices of minimax critical points for a class of functionals in Banach spaces. Ann. Mat. 186, 155-183 (2007) · Zbl 1232.58006 · doi:10.1007/s10231-005-0176-2
[20] Dalbono, F., Portaluri, A.: Morse-Smale index theorems for elliptic boundary deformation problems. J. Differ. Equ. 253(2), 463-480 (2012) · Zbl 1245.35020 · doi:10.1016/j.jde.2012.04.008
[21] Degiovanni, M.: On topological and metric critical point theory. J. Fixed Point Theory Appl. 7(1), 85-102 (2010) · Zbl 1205.58007 · doi:10.1007/s11784-009-0001-4
[22] Duc, D.M., Hung, T.V., Khai, N.T.: Morse-Palais lemma for nonsmooth functionals on normed spaces. Proc. Am. Math. Soc. 135, 921-927 (2007) · Zbl 1139.58006 · doi:10.1090/S0002-9939-06-08662-X
[23] Duc, D.M., Hung, T.V., Khai, N.T.: Critical points of non-\[C^2\] C2 functionals. Topol. Methods Nonlinear Anal. 29, 35-68 (2007) · Zbl 1134.58005
[24] Ekeland, I.: An inverse function theorem in Frechet spaces. Ann. Inst. Henri Poincaré Anal. Non Linéaire 28(1), 91-105 (2011) · Zbl 1256.47037 · doi:10.1016/j.anihpc.2010.11.001
[25] Feckan, M.: An inverse function theorem for continuous mappings. J. Math. Anal. Appl. 185(1), 118-128 (1994) · Zbl 0829.46027 · doi:10.1006/jmaa.1994.1236
[26] Ghoussoub, N.: Duality and Perturbation Methods in Critical Point Theory. Cambridge University Press, Cambridge (2008) · Zbl 1143.58300
[27] Giaquinta, M.: Multiple Integrals in the Calculus of Variations and Nonlinear Elliptic Systems. Annals of Mathematics Studies. Princeton University Press, Princeton (1983) · Zbl 0516.49003
[28] Gromoll, D., Meyer, W.: On differentiable functions with isolated critical points. Topology 8, 361-369 (1969) · Zbl 0212.28903 · doi:10.1016/0040-9383(69)90022-6
[29] Jiang, M.: A generalization of Morse lemma and its applications. Nonlinear Anal. 36, 943-960 (1999) · Zbl 0927.58005 · doi:10.1016/S0362-546X(97)00701-3
[30] Krasnosel’skii, M.A.: Topological Methods in the Theory of Nonlinear Integral Equations. McMillan, New York (1964) · Zbl 0111.30303
[31] Lazer, A., Solimini, S.: Nontrivial solutions of operator equations and Morse indices of critical points of min-max type. Nonlinear Anal. TMA 12, 761-775 (1988) · Zbl 0667.47036 · doi:10.1016/0362-546X(88)90037-5
[32] Lu, G.: Corrigendum to “The Conley conjecture for Hamiltonian systems on the cotangent bundle and its analogue for Lagrangian systems” [J. Funct. Anal. 256(9):2967-3034 (2009)]. J. Funct. Anal. 261, 542-589 (2011) · Zbl 1245.53059 · doi:10.1016/j.jfa.2011.02.027
[33] Lu, G.: The splitting lemmas for nonsmooth functionals on Hilbert spaces I. Discret. Contin. Dyn. Syst. 33(7), 2939-2990 (2013) · Zbl 1279.58004 · doi:10.3934/dcds.2013.33.2939
[34] Lu, G.: The splitting lemmas for nonsmooth functionals on Hilbert spaces II. Topol. Methods Nonlinear Anal. 44, 277-335 (2014) · Zbl 1361.58006 · doi:10.12775/TMNA.2014.048
[35] Lu, G.: The splitting lemmas for nonsmooth functionals on Hilbert spaces. arxiv:1102.2062 · Zbl 1279.58004
[36] Lu, G.: Splitting lemmas for the Finsler energy functional on the space of \[H^1\] H1-curves. Proc. Lond. Math. Soc. 113(3), 24-76 (2016) · Zbl 1406.58007 · doi:10.1112/plms/pdw022
[37] Lu, G.: Nonsmooth generalization of some critical point theorems for \[C^2\] C2 functionals. Sci. Sin. Math. 46, 615-638 (2016). https://doi.org/10.1360/N012015-00375. (in Chinese) · Zbl 1499.58011 · doi:10.1360/N012015-00375
[38] Lu, G.: Morse theory methods for quasi-linear elliptic systems of higher order. arXiv:1702.06667 · Zbl 1427.58004
[39] Lu, G.: Parameterized splitting theorems and bifurcations for potential operators. arXiv:1712.03479 · Zbl 1528.58007
[40] Lu, G.: Variational methods for Lagrangian systems of higher order, A book in progress
[41] Marino, A., Prodi, G.: Metodi perturbativi nella teoria di Morse. Boll. Un. Mat. Ital. 11, 1-32 (1975) · Zbl 0311.58006
[42] Mawhin, J., Willem, M.: Critical Point Theory and Hamiltonian Systems. Applied Mathematical Sciences, vol. 74. Springer, New York (1989) · Zbl 0676.58017 · doi:10.1007/978-1-4757-2061-7
[43] Milnor, J.: Morse Theory. Annals of Mathematical Studies, vol. 51. Princeton University Press, Princeton, NJ (1963) · Zbl 0108.10401
[44] Morrey Jr., C.B.: Multiple Integrals in the Calculus of Variations. Reprint of the 1966 Classics in Mathematics. Springer, Berlin (2008) · Zbl 1213.49002 · doi:10.1007/978-3-540-69952-1
[45] Morse, M.: The Calculus of Variations in the Large, vol. 18. American Mathematical Society, Colloquium Publications, Ann Arbor (1934) · JFM 60.0450.01
[46] Moser, J.: Minimal solutions of variational problems on a torus. Ann. Inst. Henri Poincaré 3, 229-272 (1986) · Zbl 0609.49029 · doi:10.1016/S0294-1449(16)30387-0
[47] Motreanu, D., Motreanu, V., Papageorgiou, N.: Topological and Variational Methods with Applications to Nonlinear Boundary Value Problems. Springer, New York (2014) · Zbl 1292.47001 · doi:10.1007/978-1-4614-9323-5
[48] Palais, R.: Morse theory on Hilbert manifolds. Topology 2, 299-340 (1963) · Zbl 0122.10702 · doi:10.1016/0040-9383(63)90013-2
[49] Palais, R.: Foundations of Global Non-linear Analysis, vol. 44. W. A. Benjamin, New York (1968) · Zbl 0164.11102
[50] Palais, R.S., Smale, S.: A generalized Morse theory. Bull. Am. Math. Soc. 70, 165-172 (1964) · Zbl 0119.09201 · doi:10.1090/S0002-9904-1964-11062-4
[51] Perera, K., Agarwal, R.P., O’Regan, D.: Morse Theoretic Aspects of \[p\] p-Laplacian Type Operators. Mathematical Surveys and Monographs, vol. 161. American Mathematical Society, Providence (2010) · doi:10.1090/surv/161
[52] Skrypnik, I.V.: Nonlinear Elliptic Equations of a Higher Order. Naukova Dumka, Kiev (1973). (in Russian) · Zbl 0296.35032
[53] Skrypnik, I.V.: Solvability and properties of solutions of nonlinear elliptic equations. J. Sov. Math. 12, 555-629 (1979) · doi:10.1007/BF01089138
[54] Skrypnik, I.V.: Methods for Analysis of Nonlinear Elliptic Boundary Value Problems. Translations of Mathematical Monographs, vol. 139. American Mathematical Society, Providence (1994) · doi:10.1090/mmono/139
[55] Smale, S.: Morse theory and a non-linear generalization of the Dirichlet problem. Ann. Math. 80, 382-396 (1964) · Zbl 0131.32305 · doi:10.2307/1970398
[56] Smale, S.: On the Morse index theorem. J. Math. Mech. 14, 1049-1056 (1965) · Zbl 0166.36102
[57] Ströhmer, G.: About the morse theory for certain vartional problems. Math. Ann. 270, 275-284 (1985) · Zbl 0537.35006 · doi:10.1007/BF01456186
[58] Tromba, A.J.: A general approach to Morse theory. J. Differ. Geom. 12, 47-85 (1977) · Zbl 0344.58012 · doi:10.4310/jdg/1214433845
[59] Uhlenbeck, K.: Morse theory on Banach manifolds. J. Funct. Anal. 10, 430-445 (1972) · Zbl 0241.58002 · doi:10.1016/0022-1236(72)90039-0
[60] Uhlenbeck, K.: The Morse index theorem in Hilbert space. J. Differ. Geom. 8, 555-564 (1973) · Zbl 0277.58002 · doi:10.4310/jdg/1214431958
[61] Vakhrameev, S.A.: Critical point theory for smooth functions on Hilbert manifolds with singularities and its application to some optimal control problems. J. Sov. Math. 67, 2713-2811 (1993) · doi:10.1007/BF01455151
[62] Vannella, G.: Morse theory applied to a \[T^2\] T2-equivriant problem. Topol. Methods Nonlinear Anal. 17, 41-53 (2001) · Zbl 0992.35035 · doi:10.12775/TMNA.2001.003
[63] Viterbo, C.: Indice de Morse des points critiques obtenus par minimax. Ann. Inst. Henri Poincaré 5, 221-225 (1988) · Zbl 0695.58007 · doi:10.1016/S0294-1449(16)30345-6
[64] Wang, Z.Q.: Equivariant Morse theory for isolated critical orbits and its applications to nonlinear problems. In: Chern, S.S (ed.) Partial Differential Equations, Proceedings of the Seventh Symposium on Differential Geometry and Differential Equations held in Tianjin, June 23-July 5, 1986. Lecture Notes in Mathematics, vol. 1306, pp. 202-221. Springer, Berlin (1988)
[65] Wasserman, G.: Equivariant differential topology. Topology 8, 127-150 (1969) · Zbl 0215.24702 · doi:10.1016/0040-9383(69)90005-6
[66] Wendl, C.: Lectures on Holomorphic Curves in Symplectic and Contact Geometry, math.SG. arXiv:1011.1690 · Zbl 1490.53002
[67] Zou, W.M., Schechter, M.: Critical Point Theory and Its Applications. Springer, New York (2006) · Zbl 1125.58004
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