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Well-posedness and scattering for a system of quadratic derivative nonlinear Schrödinger equations with low regularity initial data
Two sequences of solutions for indefinite superlinear-sublinear elliptic equations with nonlinear boundary conditions
1. | Department of Mathematics, Faculty of Science and Engineering, Saga University, Saga, 840-8502, Japan |
2. | Department of Mathematics, Graduate School of Science, Osaka City University, 3-3-138 Sugimoto Sumiyoshi-ku, Osaka-shi, Osaka, 558-8585, Japan |
References:
[1] |
A. Ambrosetti and P. H. Rabinowitz, Dual variational methods in critical point theory and applications, J. Funct. Anal., 14 (1973), 349-381. |
[2] |
A. Ambrosetti, H. Brezis and G. Cerami, Combined effects of concave and convex nonlinearities in some elliptic problems, J. Funct. Anal., 122 (1994), 519-543.
doi: 10.1006/jfan.1994.1078. |
[3] |
T. Bartsch and M. Willem, On an elliptic equation with concave and convex nonlinearities, Proc. Amer. Math. Soc., 123 (1995), 3555-3561.
doi: 10.2307/2161107. |
[4] |
D. C. Clark, A variant of the Lusternik-Schnirelman theory, Indiana Univ. Math. J., 22 (1972), 65-74. |
[5] |
D. G. DeFigueiredo, J.-P. Gossez and P. Ubilla, Local superlinearity and sublinearity for indefinite semilinear elliptic problems, J. Funct. Anal., 199 (2003), 452-467.
doi: 10.1016/S0022-1236(02)00060-5. |
[6] |
D. G. DeFigueiredo, J.-P. Gossez and P. Ubilla, Multiplicity results for a family of semilinear elliptic problems under local superlinearity and sublinearity, J. Eur. Math. Soc., 8 (2006), 269-288.
doi: 10.4171/JEMS/52. |
[7] |
J. Garcia-Azorero, I. Peral and J. D. Rossi, A convex-concave problem with a nonlinear boundary condition, J. Differential Equations, 198 (2004), 91-128.
doi: 10.1016/S0022-0396(03)00068-8. |
[8] |
R. Kajikiya, A critical point theorem related to the symmetric mountain pass lemma and its applications to elliptic equations, J. Funct. Anal., 225 (2005), 352-370.
doi: 10.1016/j.jfa.2005.04.005. |
[9] |
R. Kajikiya, Superlinear elliptic equations with singular coefficients on the boundary, Nonlinear Analysis, T.M.A., 73 (2010), 2117-2131.
doi: 10.1016/j.na.2010.05.039. |
[10] |
O. A. Ladyzhenskaya and N. N. Ural'tseva, Linear and Quasilinear Elliptic Equations, translated by Scripta Technica, Inc. Mathematics in Science and Engineering, Vol. 46, Academic Press, 1968. |
[11] |
D. Naimen, Existence of infinitely many solutions for nonlinear Neumann problems with indefinite coefficients, Submitted for publications. |
[12] |
P. H. Rabinowitz, Minimax Methods in Critical Point Theory with Applications to Differential Equations, CBMS Regional Conference Series in Mathematics Vol. 65, American Mathematical Society, Providence, RI, 1986. |
[13] |
M. Struwe, Variational Methods, 2nd edition, Springer, Berlin, 1996. |
[14] |
Z.-Q. Wang, Nonlinear boundary value problems with concave nonlinearities near the origin, Nonlinear Differential Equations Appl., 8 (2001), 15-33.
doi: 10.1007/PL00001436. |
show all references
References:
[1] |
A. Ambrosetti and P. H. Rabinowitz, Dual variational methods in critical point theory and applications, J. Funct. Anal., 14 (1973), 349-381. |
[2] |
A. Ambrosetti, H. Brezis and G. Cerami, Combined effects of concave and convex nonlinearities in some elliptic problems, J. Funct. Anal., 122 (1994), 519-543.
doi: 10.1006/jfan.1994.1078. |
[3] |
T. Bartsch and M. Willem, On an elliptic equation with concave and convex nonlinearities, Proc. Amer. Math. Soc., 123 (1995), 3555-3561.
doi: 10.2307/2161107. |
[4] |
D. C. Clark, A variant of the Lusternik-Schnirelman theory, Indiana Univ. Math. J., 22 (1972), 65-74. |
[5] |
D. G. DeFigueiredo, J.-P. Gossez and P. Ubilla, Local superlinearity and sublinearity for indefinite semilinear elliptic problems, J. Funct. Anal., 199 (2003), 452-467.
doi: 10.1016/S0022-1236(02)00060-5. |
[6] |
D. G. DeFigueiredo, J.-P. Gossez and P. Ubilla, Multiplicity results for a family of semilinear elliptic problems under local superlinearity and sublinearity, J. Eur. Math. Soc., 8 (2006), 269-288.
doi: 10.4171/JEMS/52. |
[7] |
J. Garcia-Azorero, I. Peral and J. D. Rossi, A convex-concave problem with a nonlinear boundary condition, J. Differential Equations, 198 (2004), 91-128.
doi: 10.1016/S0022-0396(03)00068-8. |
[8] |
R. Kajikiya, A critical point theorem related to the symmetric mountain pass lemma and its applications to elliptic equations, J. Funct. Anal., 225 (2005), 352-370.
doi: 10.1016/j.jfa.2005.04.005. |
[9] |
R. Kajikiya, Superlinear elliptic equations with singular coefficients on the boundary, Nonlinear Analysis, T.M.A., 73 (2010), 2117-2131.
doi: 10.1016/j.na.2010.05.039. |
[10] |
O. A. Ladyzhenskaya and N. N. Ural'tseva, Linear and Quasilinear Elliptic Equations, translated by Scripta Technica, Inc. Mathematics in Science and Engineering, Vol. 46, Academic Press, 1968. |
[11] |
D. Naimen, Existence of infinitely many solutions for nonlinear Neumann problems with indefinite coefficients, Submitted for publications. |
[12] |
P. H. Rabinowitz, Minimax Methods in Critical Point Theory with Applications to Differential Equations, CBMS Regional Conference Series in Mathematics Vol. 65, American Mathematical Society, Providence, RI, 1986. |
[13] |
M. Struwe, Variational Methods, 2nd edition, Springer, Berlin, 1996. |
[14] |
Z.-Q. Wang, Nonlinear boundary value problems with concave nonlinearities near the origin, Nonlinear Differential Equations Appl., 8 (2001), 15-33.
doi: 10.1007/PL00001436. |
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