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Multiple positive solutions for a Schrödinger logarithmic equation
1. | Unidade Acadêmica de Matemática, Universidade Federal de Campina Grande, Campina Grande, PB, CEP:58429-900, Brazil |
2. | Department of Mathematics, East China University of Science and Technology, Shanghai 200237, China |
$ \left\{ \begin{array}{lc} -{\epsilon}^2\Delta u+ V(x)u = u \log u^2, & \mbox{in} \quad \mathbb{R}^{N}, \\ u \in H^1(\mathbb{R}^{N}), & \; \\ \end{array} \right. $ |
$ \epsilon >0 $ |
$ N \geq 1 $ |
$ V $ |
$ \epsilon>0 $ |
$ V $ |
References:
[1] |
C. O. Alves and D. C. de Morais Filho, Existence of concentration of positive solutions for a Schrödinger logarithmic equation, Z. Angew. Math. Phys., 69 (2018), Art. 144, 22 pp.
doi: 10.1007/s00033-018-1038-2. |
[2] |
D. M. Cao and E. S. Noussair,
Multiplicity of positive and nodal solutions for nonlinear elliptic problem in $\mathbb{R}^{N}$, Ann. Inst. Henri Poincaré, 13 (1996), 567-588.
doi: 10.1016/S0294-1449(16)30115-9. |
[3] |
P. d'Avenia, E. Montefusco and M. Squassina, On the logarithmic Schrödinger equation, Commun. Contemp. Math., 16 (2014), 1350032, 15 pp.
doi: 10.1142/S0219199713500326. |
[4] |
P. d'Avenia, M. Squassina and M. Zenari,
Fractional logarithmic Schrödinger equations, Math. Methods Appl. Sci., 38 (2015), 5207-5216.
doi: 10.1002/mma.3449. |
[5] |
M. Degiovanni and S. Zani,
Multiple solutions of semilinear elliptic equations with one-sided growth conditions. Nonlinear operator theory, Math. Comput. Model., 32 (2000), 1377-1393.
doi: 10.1016/S0895-7177(00)00211-9. |
[6] |
C. Ji and A. Szulkin,
A logarithmic Schrödinger equation with asymptotic conditions on the potential, J. Math. Anal. Appl., 437 (2016), 241-254.
doi: 10.1016/j.jmaa.2015.11.071. |
[7] |
E. H. Lieb and M. Loss, Analysis, 2nd Edition, Graduate Studies in Math. 14, American Mathematical Society, Providence, RI, 2001.
doi: 10.1090/gsm/014. |
[8] |
P.-L. Lions,
The concentration-compactness principle in the calculus of variations. The locally compact case. Ⅱ, Ann. Inst. H. Poincaré Anal. Non Linéaire, 1 (1984), 223-283.
doi: 10.1016/S0294-1449(16)30422-X. |
[9] |
M. Squassina and A. Szulkin,
Multiple solution to logarithmic Schrödinger equations with periodic potential, Cal. Var. Partial Differential Equations, 54 (2015), 585-597.
doi: 10.1007/s00526-014-0796-8. |
[10] |
M. Squassina and A. Szulkin, Multiple solution to logarithmic Schrödinger equations with periodic potential, Cal. Var. Partial Differential Equations, 54 (2015), 585–597, http://dx.doi.org/10.1007/s00526-017-1127-7.
doi: 10.1007/s00526-014-0796-8. |
[11] |
A. Szulkin,
Minimax principles for lower semicontinuous functions and applications to nonlinear boundary value problems, Ann. Inst. H. Poincaré Anal. Non Linéaire, 3 (1986), 77-109.
doi: 10.1016/S0294-1449(16)30389-4. |
[12] |
K. Tanaka and C. X. Zhang, Multi-bump solutions for logarithmic Schrödinger equations, Cal. Var. Partial Differential Equations, 56 (2017), Art. 33, 35 pp.
doi: 10.1007/s00526-017-1122-z. |
[13] |
M. Willem, Minimax Theorems, Progress in Nonlinear Differential Equations and their Applications, 24. Birkhäuser Boston, Inc., Boston, MA, 1996.
doi: 10.1007/978-1-4612-4146-1. |
[14] |
K. G. Zloshchastiev,
Logarithmic nonlinearity in the theories of quantum gravity: Origin of time and observational consequences, Grav. Cosmol., 16 (2010), 288-297.
doi: 10.1134/S0202289310040067. |
show all references
References:
[1] |
C. O. Alves and D. C. de Morais Filho, Existence of concentration of positive solutions for a Schrödinger logarithmic equation, Z. Angew. Math. Phys., 69 (2018), Art. 144, 22 pp.
doi: 10.1007/s00033-018-1038-2. |
[2] |
D. M. Cao and E. S. Noussair,
Multiplicity of positive and nodal solutions for nonlinear elliptic problem in $\mathbb{R}^{N}$, Ann. Inst. Henri Poincaré, 13 (1996), 567-588.
doi: 10.1016/S0294-1449(16)30115-9. |
[3] |
P. d'Avenia, E. Montefusco and M. Squassina, On the logarithmic Schrödinger equation, Commun. Contemp. Math., 16 (2014), 1350032, 15 pp.
doi: 10.1142/S0219199713500326. |
[4] |
P. d'Avenia, M. Squassina and M. Zenari,
Fractional logarithmic Schrödinger equations, Math. Methods Appl. Sci., 38 (2015), 5207-5216.
doi: 10.1002/mma.3449. |
[5] |
M. Degiovanni and S. Zani,
Multiple solutions of semilinear elliptic equations with one-sided growth conditions. Nonlinear operator theory, Math. Comput. Model., 32 (2000), 1377-1393.
doi: 10.1016/S0895-7177(00)00211-9. |
[6] |
C. Ji and A. Szulkin,
A logarithmic Schrödinger equation with asymptotic conditions on the potential, J. Math. Anal. Appl., 437 (2016), 241-254.
doi: 10.1016/j.jmaa.2015.11.071. |
[7] |
E. H. Lieb and M. Loss, Analysis, 2nd Edition, Graduate Studies in Math. 14, American Mathematical Society, Providence, RI, 2001.
doi: 10.1090/gsm/014. |
[8] |
P.-L. Lions,
The concentration-compactness principle in the calculus of variations. The locally compact case. Ⅱ, Ann. Inst. H. Poincaré Anal. Non Linéaire, 1 (1984), 223-283.
doi: 10.1016/S0294-1449(16)30422-X. |
[9] |
M. Squassina and A. Szulkin,
Multiple solution to logarithmic Schrödinger equations with periodic potential, Cal. Var. Partial Differential Equations, 54 (2015), 585-597.
doi: 10.1007/s00526-014-0796-8. |
[10] |
M. Squassina and A. Szulkin, Multiple solution to logarithmic Schrödinger equations with periodic potential, Cal. Var. Partial Differential Equations, 54 (2015), 585–597, http://dx.doi.org/10.1007/s00526-017-1127-7.
doi: 10.1007/s00526-014-0796-8. |
[11] |
A. Szulkin,
Minimax principles for lower semicontinuous functions and applications to nonlinear boundary value problems, Ann. Inst. H. Poincaré Anal. Non Linéaire, 3 (1986), 77-109.
doi: 10.1016/S0294-1449(16)30389-4. |
[12] |
K. Tanaka and C. X. Zhang, Multi-bump solutions for logarithmic Schrödinger equations, Cal. Var. Partial Differential Equations, 56 (2017), Art. 33, 35 pp.
doi: 10.1007/s00526-017-1122-z. |
[13] |
M. Willem, Minimax Theorems, Progress in Nonlinear Differential Equations and their Applications, 24. Birkhäuser Boston, Inc., Boston, MA, 1996.
doi: 10.1007/978-1-4612-4146-1. |
[14] |
K. G. Zloshchastiev,
Logarithmic nonlinearity in the theories of quantum gravity: Origin of time and observational consequences, Grav. Cosmol., 16 (2010), 288-297.
doi: 10.1134/S0202289310040067. |
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