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May  2020, 40(5): 2671-2685. doi: 10.3934/dcds.2020145

Multiple positive solutions for a Schrödinger logarithmic equation

Claudianor O. Alves 1, and  Chao Ji 2,,

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

* Corresponding author: Chao Ji

Received  May 2019 Revised  December 2019 Published  March 2020

Fund Project: C.O. Alves was partially supported by CNPq/Brazil 304804/2017-7 and C. Ji was partially supported by Shanghai Natural Science Foundation(18ZR1409100)

This article concerns with the existence of multiple positive solutions for the following logarithmic Schrödinger equation
$ \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. $
where
$ \epsilon >0 $
,
$ N \geq 1 $
 and
$ V $
 is a continuous function with a global minimum. Using variational method, we prove that for small enough
$ \epsilon>0 $
, the "shape" of the graph of the function
$ V $
 affects the number of nontrivial solutions.
Keywords: Variational method, logarithmic Schrödinger equation, positive solutions, multiple solutions.
Mathematics Subject Classification: Primary: 35A15, 35J10; Secondary: 35B09.
Citation: Claudianor O. Alves, Chao Ji. Multiple positive solutions for a Schrödinger logarithmic equation. Discrete & Continuous Dynamical Systems - A, 2020, 40 (5) : 2671-2685. doi: 10.3934/dcds.2020145
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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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