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Nontrivial solutions for Kirchhoff type equations via Morse theory
Schrödinger-like operators and the eikonal equation
1. | Departamento de Ciencias Básicas, UAM-A, Av. San Pablo 180, Col. Reynosa, Mèxico D. F. 02200, Mexico |
References:
[1] |
R. Adams, "Sobolev Spaces," Pure and Applied Mathematics, Vol. 65, Academic Press, 1975. |
[2] |
S. Agmon, "Lectures on Elliptic Boundary Value Problems," D. Van Nostrand Co. In., 1965. |
[3] |
S. Agmon, "Unicité et convexité dans les problèmes différentiels," Séminaire de Mathématiques Supérieures, No. 13 (Été, 1965) Les Presses de l'Université de Montréal, Montreal, Que., 1966. |
[4] |
S. Agmon, Lower bounds for solutions of Schrdinger equations, Journal D'analyse Mathématique, 23 (1970), 1-25, |
[5] |
S. Agmon, "Lectures on Exponential Decay of Solutions of Second-Order Elliptic Equations. Bounds on Eigenfunctions of N-Body Schrödinger Operators," Mathematical Notes 29, Princeton University Press, 1982. |
[6] |
S. Agmon, On the asymptotic behavior of solutions of Schröinger type equations in unbounded domains, Analyse mathématique et applications, 122, Gauthier-Villars, Montrouge, 1988. |
[7] |
S. Agmon, Representation theorems for solutions of the Helmholtz equation on $\mathbbR^n$, Differential Operators and Spectral Theory, 27-43, Amer. Math. Soc. Transl. Ser. 2, 189, Amer. Math. Soc., Providence, RI, 1999. |
[8] |
S. Agmon, J. Cruz-Sampedro and I. Herbst, Generalized Fourier transform for Schrödinger operators with potentials of order zero, Journal of Functional Analysis, 167 (1999), 345-369. |
[9] |
S. Agmon and L. Hörmander, Asymptotic properties of solutions of differential equations with simple characteristics, Journal D'Analyse Mathématique, 30 (1976). |
[10] |
S. Agmon and L. Nirenberg, Lower bounds and uniqueness theorems for solutions of differential equations in a Hilbert space, Comm. Pure Appl. Math., 20 (1967), 207-229. |
[11] |
G. Barles, On eikonal equations associated with Schrödinger operators with nonspherical radiation conditions, Commun. in Partial Differential Equations, 12 (1987), 263-283. |
[12] |
M. Ben-Artzi, Unitary equivalence and scattering theory for Stark-like Hamiltonians, J. Math. Phys., 25 (1984), 951-964. |
[13] |
P. Constantin, Scattering for Schröinger operators in a class of domains with noncompact boundaries, J. Funct. Anal., 44 (1981), 87-119. |
[14] |
J. Cruz-Sampedro, Exact asymptotic behavior at infinity of solutions to abstract second-order differential inequalities in Hilbert spaces, Math. Z., 237 (2001), 727-235. |
[15] |
J. Cruz-Sampedro, Boundary values of the resolvent of Schrödinger hamiltonians with potentials of order zero, Discrete Contin. Dyn. Syst., 33 (2013), 1061-1076. |
[16] |
A. Hassell, R. Melrose and A. Vasy, Spectral and scattering theory for symbolic potentials of order zero, Adv. Math., 181 (2004), 1-87. |
[17] |
A. Hassell, R. Melrose and A. Vasy, Microlocal propagation near radial points and scattering for symbolic potentials of order zero, Anal. PDE, 1 (2008), 127-196. |
[18] |
L. Hörmander, "The Analysis of Linear Partial Differential Operators III," Springer-Verlag, Berlin, 1985.
doi: 978-3-540-49938-1. |
[19] |
W. Jäger, Über das Dirichletsche Außenraumproblem für die Schwingungsgleichung, Math. Z., 95 (1967), 299-323. |
[20] |
W. Jäger, Zur Theorie der Schwingungsgleichung mit variablen Koeffizienten in Außengebieten, Math. Z., 102 (1967), 62-88. |
[21] |
W. Jäger, Das asymptotische Verhalten von Lsngen eines Typs von Differentialgleichungen, Math. Z., 112 (1969), 26-36. |
[22] |
W. Jäger and P. Rejto, Limiting absorption principle for some Schrödinger operators with exploding potentials. II, J. Math. Anal. Appl., 95 (1983), 169-194. |
[23] |
D. Jerison and C. Kenig, Unique continuation and absence of positive eigenvalues for Schrödinger operators, Ann. Math., 121 (1985), 463-494. |
[24] |
A. Jensen and P. Perry, Commutator methods and Besov space estimates for Schrödinger operators, J. Operator Theory, 14 (1985), 181-188. |
[25] |
P. Lions, "Generalized Solutions of Hamilton-Jacobi Equations," Pitman, London, 1982. |
[26] |
M. Reed and B. Simon, "Methods of Modern Mathematical Physics, II Fourier Analysis Self-Adjontness," New York, Academic Press, 1978. |
[27] |
M. Reed and B. Simon, "Methods of Modern Mathematical Physics, III Sacattering Theory," New York, Academic Press, 1979. |
[28] |
M. Reed and B. Simon, "Methods of Modern Mathematical Physics, IV Analysis of Operators," Academic Press, 1978. |
[29] |
Y. Saitō, "Spectral Representations for Schrödinger Operators with Long-range Potentials," Lecture Notes in Mathematics, 727. Springer, Berlin, 1979.
doi: 978-3-540-35132-0. |
[30] |
Y. Saitō, Schrödinger operators with a nonspherical radiation condition, Pacific J. Math., 126 (1987), 331-359. |
[31] |
I. Sigal, "Scattering Theory for Many-Body Quantum Mechanical Systems," Lecture Notes in Mathematics 1011, Springer Verlag, 1983.
doi: 978-3-540-38664-3. |
show all references
References:
[1] |
R. Adams, "Sobolev Spaces," Pure and Applied Mathematics, Vol. 65, Academic Press, 1975. |
[2] |
S. Agmon, "Lectures on Elliptic Boundary Value Problems," D. Van Nostrand Co. In., 1965. |
[3] |
S. Agmon, "Unicité et convexité dans les problèmes différentiels," Séminaire de Mathématiques Supérieures, No. 13 (Été, 1965) Les Presses de l'Université de Montréal, Montreal, Que., 1966. |
[4] |
S. Agmon, Lower bounds for solutions of Schrdinger equations, Journal D'analyse Mathématique, 23 (1970), 1-25, |
[5] |
S. Agmon, "Lectures on Exponential Decay of Solutions of Second-Order Elliptic Equations. Bounds on Eigenfunctions of N-Body Schrödinger Operators," Mathematical Notes 29, Princeton University Press, 1982. |
[6] |
S. Agmon, On the asymptotic behavior of solutions of Schröinger type equations in unbounded domains, Analyse mathématique et applications, 122, Gauthier-Villars, Montrouge, 1988. |
[7] |
S. Agmon, Representation theorems for solutions of the Helmholtz equation on $\mathbbR^n$, Differential Operators and Spectral Theory, 27-43, Amer. Math. Soc. Transl. Ser. 2, 189, Amer. Math. Soc., Providence, RI, 1999. |
[8] |
S. Agmon, J. Cruz-Sampedro and I. Herbst, Generalized Fourier transform for Schrödinger operators with potentials of order zero, Journal of Functional Analysis, 167 (1999), 345-369. |
[9] |
S. Agmon and L. Hörmander, Asymptotic properties of solutions of differential equations with simple characteristics, Journal D'Analyse Mathématique, 30 (1976). |
[10] |
S. Agmon and L. Nirenberg, Lower bounds and uniqueness theorems for solutions of differential equations in a Hilbert space, Comm. Pure Appl. Math., 20 (1967), 207-229. |
[11] |
G. Barles, On eikonal equations associated with Schrödinger operators with nonspherical radiation conditions, Commun. in Partial Differential Equations, 12 (1987), 263-283. |
[12] |
M. Ben-Artzi, Unitary equivalence and scattering theory for Stark-like Hamiltonians, J. Math. Phys., 25 (1984), 951-964. |
[13] |
P. Constantin, Scattering for Schröinger operators in a class of domains with noncompact boundaries, J. Funct. Anal., 44 (1981), 87-119. |
[14] |
J. Cruz-Sampedro, Exact asymptotic behavior at infinity of solutions to abstract second-order differential inequalities in Hilbert spaces, Math. Z., 237 (2001), 727-235. |
[15] |
J. Cruz-Sampedro, Boundary values of the resolvent of Schrödinger hamiltonians with potentials of order zero, Discrete Contin. Dyn. Syst., 33 (2013), 1061-1076. |
[16] |
A. Hassell, R. Melrose and A. Vasy, Spectral and scattering theory for symbolic potentials of order zero, Adv. Math., 181 (2004), 1-87. |
[17] |
A. Hassell, R. Melrose and A. Vasy, Microlocal propagation near radial points and scattering for symbolic potentials of order zero, Anal. PDE, 1 (2008), 127-196. |
[18] |
L. Hörmander, "The Analysis of Linear Partial Differential Operators III," Springer-Verlag, Berlin, 1985.
doi: 978-3-540-49938-1. |
[19] |
W. Jäger, Über das Dirichletsche Außenraumproblem für die Schwingungsgleichung, Math. Z., 95 (1967), 299-323. |
[20] |
W. Jäger, Zur Theorie der Schwingungsgleichung mit variablen Koeffizienten in Außengebieten, Math. Z., 102 (1967), 62-88. |
[21] |
W. Jäger, Das asymptotische Verhalten von Lsngen eines Typs von Differentialgleichungen, Math. Z., 112 (1969), 26-36. |
[22] |
W. Jäger and P. Rejto, Limiting absorption principle for some Schrödinger operators with exploding potentials. II, J. Math. Anal. Appl., 95 (1983), 169-194. |
[23] |
D. Jerison and C. Kenig, Unique continuation and absence of positive eigenvalues for Schrödinger operators, Ann. Math., 121 (1985), 463-494. |
[24] |
A. Jensen and P. Perry, Commutator methods and Besov space estimates for Schrödinger operators, J. Operator Theory, 14 (1985), 181-188. |
[25] |
P. Lions, "Generalized Solutions of Hamilton-Jacobi Equations," Pitman, London, 1982. |
[26] |
M. Reed and B. Simon, "Methods of Modern Mathematical Physics, II Fourier Analysis Self-Adjontness," New York, Academic Press, 1978. |
[27] |
M. Reed and B. Simon, "Methods of Modern Mathematical Physics, III Sacattering Theory," New York, Academic Press, 1979. |
[28] |
M. Reed and B. Simon, "Methods of Modern Mathematical Physics, IV Analysis of Operators," Academic Press, 1978. |
[29] |
Y. Saitō, "Spectral Representations for Schrödinger Operators with Long-range Potentials," Lecture Notes in Mathematics, 727. Springer, Berlin, 1979.
doi: 978-3-540-35132-0. |
[30] |
Y. Saitō, Schrödinger operators with a nonspherical radiation condition, Pacific J. Math., 126 (1987), 331-359. |
[31] |
I. Sigal, "Scattering Theory for Many-Body Quantum Mechanical Systems," Lecture Notes in Mathematics 1011, Springer Verlag, 1983.
doi: 978-3-540-38664-3. |
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