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A direct method of moving planes for fully nonlinear nonlocal operators and applications
Existence and nonexistence of entire positive radial solutions for a class of Schrödinger elliptic systems involving a nonlinear operator
a. | School of Mathematics and Computer Science, Shanxi Normal University, Linfen, Shanxi 041004, China |
b. | Department of Mathematics, Texas A & M University, Kingsville, TX 78363-8202, USA |
c. | Nonlinear Analysis and Applied Mathematics (NAAM) Research Group, Department of Mathematics, Faculty of Science, King Abdulaziz University, Jeddah 21589, Saudi Arabia |
d. | College of Science & Technology, Ningbo University, Ningbo, Zhejiang 315211, China |
$ \begin{equation*} \begin{split} \left\{\begin{array}{ll}{\operatorname{div}(\mathcal{G}(|\nabla y|^{p-2})\nabla y) = b_{1}(|x|) \psi(y)+h_{1}(|x|) \varphi(z),}& {x \in \mathbb{R}^{n}(n \geq 3)}, \\ {\operatorname{div}(\mathcal{G}(|\nabla z|^{p-2})\nabla z) = b_{2}(|x|) \psi(z)+h_{2}(|x|) \varphi(y),} & {x \in \mathbb{R}^{n}},\end{array}\right. \end{split} \end{equation*} $ |
$ \mathcal{G} $ |
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D. Baleanu, S. Rezapour and H. Mohammadi, Some existence results on nonlinear fractional differential equations, Phil. Trans. R. Soc. A, 371 (2013), 20120144.
doi: 10.1098/rsta.2012.0144. |
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D. Baleanu, R. P. Agarwal, H. Mohammadi and S. Rezapour, Some existence results for a nonlinear fractional differential equation on partially ordered Banach spaces, Bound. Value Probl., 2013 (2013), 112.
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D.-P. Covei,
Symmetric solutions for an elliptic partial differential equation that arises in stochastic production planning with production constraints, Appl. Math. Comput., 350 (2019), 190-197.
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D.-P. Covei,
Large and entire large solution for a quasilinear problem, Nonlinear Anal., 70 (2009), 1738-1745.
doi: 10.1016/j.na.2008.02.057. |
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D.-P. Covei,
Radial and nonradial solutions for a semilinear elliptic system of Schrödinger type, Funkcial. Ekvac., 54 (2011), 439-449.
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A. B. Dkhil, Positive solutions for nonlinear elliptic systems, Electron. J. Differential Equations, 2012 (239) (2012), 1-10. |
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X. Dong and Y. Wei,
Existence of radial solutions for nonlinear elliptic equations with gradient terms in annular domains, Nonlinear Anal., 187 (2019), 93-109.
doi: 10.1016/j.na.2019.03.024. |
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J. B. Keller,
On solutions of $\triangle z = \psi(z)$, Comm. Pure Appl. Math., 10 (1957), 503-510.
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A. V. Lair,
Large solution of sublinear/superlinear elliptic equations, J. Math. Anal. Appl., 346 (2008), 99-106.
doi: 10.1016/j.jmaa.2008.05.047. |
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A. V. Lair,
A necessary and sufficient condition for the existence of large solutions to sublinear elliptic systems, J. Math. Anal. Appl., 365 (2010), 103-108.
doi: 10.1016/j.jmaa.2009.10.026. |
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A. V. Lair,
Entire large solutions to semilinear elliptic systems, J. Math. Anal. Appl., 382 (2011), 324-333.
doi: 10.1016/j.jmaa.2011.04.051. |
[12] |
A. V. Lair and A. W. Wood,
Existence of entire large positive solutions of semilinear elliptic systems, J. Differential Euqations, 164 (2000), 380-394.
doi: 10.1006/jdeq.2000.3768. |
[13] |
H. Li, P. Zhang and Z. Zhang,
A remark on the existence of entire positve solutions for a class of semilinear elliptic system, J. Math. Anal. Appl., 365 (2010), 338-341.
doi: 10.1016/j.jmaa.2009.10.036. |
[14] |
R. Osserman,
On the inequality $\triangle z\geq \psi(z)$, Pacific J. Math., 7 (1957), 1641-1647.
|
[15] |
K. Pei, G. Wang and Y. Sun,
Successive iterations and positive extremal solutions for a Hadamard type fractional integro-differential equations on infinite domain, Appl. Math. Comput., 312 (2017), 158-168.
doi: 10.1016/j.amc.2017.05.056. |
[16] |
J. Qin, G. Wang, L. Zhang and B. Ahmad, Monotone iterative method for a p-Laplacian boundary value problem with fractional conformable derivatives, Bound. Value Probl., 2019 (2019), 145.
doi: 10.1186/s13661-019-1254-5. |
[17] |
Y. Sun, L. Liu and Y. Wu,
The existence and uniqueness of positive monotone solutions for a class of nonlinear Schrödinger equations on infinite domains, J. Comput. Appl. Math., 321 (2017), 478-486.
doi: 10.1016/j.cam.2017.02.036. |
[18] |
G. Wang and X. Ren, Radial symmetry of standing waves for nonlinear fractional Laplacian Hardy-Schrödinger systems, Appl. Math. Lett., 110 (2020), 106560.
doi: 10.1016/j.aml.2020.106560. |
[19] |
G. Wang, X. Ren, Z. Bai and W. Hou,
Radial symmetry of standing waves for nonlinear fractional Hardy-Schrödinger equation, Appl. Math. Lett., 96 (2019), 131-137.
doi: 10.1016/j.aml.2019.04.024. |
[20] |
G. Wang,
Twin iterative positive solutions of fractional q-difference Schrödinger equations, Appl. Math. Lett., 76 (2018), 103-109.
doi: 10.1016/j.aml.2017.08.008. |
[21] |
G. Wang,
Explicit iteration and unbounded solutions for fractional integral boundary value problem on an infinite interval, Appl. Math. Lett., 47 (2015), 1-7.
doi: 10.1016/j.aml.2015.03.003. |
[22] |
G. Wang, K. Pei, R. P. Agarwal, L. Zhang and B. Ahmad,
Nonlocal Hadamard fractional boundary value problem with Hadamard integral and discrete boundary conditions on a half-line, J. Comput. Appl. Math., 343 (2018), 230-239.
doi: 10.1016/j.cam.2018.04.062. |
[23] |
G. Wang, J. Qin, L. Zhang and D. Baleanu, Explicit iteration to a nonlinear fractional Langevin equation with non-separated integro-differential strip-multi-point boundary conditions, Chaos Solitons Fractals, 131 (2020), 109476.
doi: 10.1016/j.chaos.2019.109476. |
[24] |
G. Wang, Z. Bai and L. Zhang,
Successive iterations for unique positive solution of a nonlinear fractional q-integral boundary value problem, J. Appl. Anal. Comput., 9 (2019), 1204-1215.
doi: 10.11948/2156-907X.20180193. |
[25] |
G. Wang, Z. Yang, L. Zhang and D. Baleanu, Radial solutions of a nonlinear $k$-Hessian system involving a nonlinear operator, Commun. Nonlinear Sci. Numer. Simulat., 91 (2020), 105396.
doi: 10.1016/j.cnsns.2020.105396. |
[26] |
D. Ye and F. Zhou,
Invariant criteria for existence of bounded positive solutions, Discrete Contin. Dyn. Syst., 12 (2005), 413-424.
doi: 10.3934/dcds.2005.12.413. |
[27] |
Z. Zhang,
Existence of entire positive solutions for a class of semilinear elliptic systems, Electron. J. Differential Equations, 2010 (2010), 1-5.
|
[28] |
X. Zhang, Y. Wu and Y. Cui,
Existence and nonexistence of blow-up solutions for a Schrödinger equation involving a nonlinear operator, Appl. Math. Lett., 82 (2018), 85-91.
doi: 10.1016/j.aml.2018.02.019. |
[29] |
X. Zhang, C. Mao, L. Liu and Y. Wu,
Exact iterative solution for an abstract fractional dynamic system model for Bioprocess, Qual. Theory Dyn. Syst., 16 (2017), 205-222.
doi: 10.1007/s12346-015-0162-z. |
[30] |
X. Zhang, L. Liu, Y. Wu and L. Caccetta,
Entire large solutions for a class of Schrödinger systems with a nonlinear random operator, J. Math. Anal. Appl., 423 (2015), 1650-1659.
doi: 10.1016/j.jmaa.2014.10.068. |
[31] |
X. Zhang, L. Liu, Y. Wu and Y. Cui,
The existence and nonexistence of entire large solutions for a quasilinear Schrödinger elliptic system by dual approach, J. Math. Anal. Appl., 464 (2018), 1089-1106.
doi: 10.1016/j.jmaa.2018.04.040. |
[32] |
L. Zhang and W. Hou, Standing waves of nonlinear fractional p-Laplacian Schrödinger equation involving logarithmic nonlinearity, Appl. Math. Lett., 102 (2020), 106149.
doi: 10.1016/j.aml.2019.106149. |
[33] |
L. Zhang, B. Ahmad and G. Wang,
Explicit iterations and extremal solutions for fractional differential equations with nonlinear integral boundary conditions, Appl. Math. Comput., 268 (2015), 388-392.
doi: 10.1016/j.amc.2015.06.049. |
[34] |
L. Zhang, B. Ahmad and G. Wang,
The existence of an extremal solution to a nonlinear system with the right-handed Riemann-Liouville fractional derivative, Appl. Math. Lett., 31 (2014), 1-6.
doi: 10.1016/j.aml.2013.12.014. |
show all references
References:
[1] |
D. Baleanu, S. Rezapour and H. Mohammadi, Some existence results on nonlinear fractional differential equations, Phil. Trans. R. Soc. A, 371 (2013), 20120144.
doi: 10.1098/rsta.2012.0144. |
[2] |
D. Baleanu, R. P. Agarwal, H. Mohammadi and S. Rezapour, Some existence results for a nonlinear fractional differential equation on partially ordered Banach spaces, Bound. Value Probl., 2013 (2013), 112.
doi: 10.1186/1687-2770-2013-112. |
[3] |
D.-P. Covei,
Symmetric solutions for an elliptic partial differential equation that arises in stochastic production planning with production constraints, Appl. Math. Comput., 350 (2019), 190-197.
doi: 10.1016/j.amc.2019.01.015. |
[4] |
D.-P. Covei,
Large and entire large solution for a quasilinear problem, Nonlinear Anal., 70 (2009), 1738-1745.
doi: 10.1016/j.na.2008.02.057. |
[5] |
D.-P. Covei,
Radial and nonradial solutions for a semilinear elliptic system of Schrödinger type, Funkcial. Ekvac., 54 (2011), 439-449.
doi: 10.1619/fesi.54.439. |
[6] |
A. B. Dkhil, Positive solutions for nonlinear elliptic systems, Electron. J. Differential Equations, 2012 (239) (2012), 1-10. |
[7] |
X. Dong and Y. Wei,
Existence of radial solutions for nonlinear elliptic equations with gradient terms in annular domains, Nonlinear Anal., 187 (2019), 93-109.
doi: 10.1016/j.na.2019.03.024. |
[8] |
J. B. Keller,
On solutions of $\triangle z = \psi(z)$, Comm. Pure Appl. Math., 10 (1957), 503-510.
doi: 10.1002/cpa.3160100402. |
[9] |
A. V. Lair,
Large solution of sublinear/superlinear elliptic equations, J. Math. Anal. Appl., 346 (2008), 99-106.
doi: 10.1016/j.jmaa.2008.05.047. |
[10] |
A. V. Lair,
A necessary and sufficient condition for the existence of large solutions to sublinear elliptic systems, J. Math. Anal. Appl., 365 (2010), 103-108.
doi: 10.1016/j.jmaa.2009.10.026. |
[11] |
A. V. Lair,
Entire large solutions to semilinear elliptic systems, J. Math. Anal. Appl., 382 (2011), 324-333.
doi: 10.1016/j.jmaa.2011.04.051. |
[12] |
A. V. Lair and A. W. Wood,
Existence of entire large positive solutions of semilinear elliptic systems, J. Differential Euqations, 164 (2000), 380-394.
doi: 10.1006/jdeq.2000.3768. |
[13] |
H. Li, P. Zhang and Z. Zhang,
A remark on the existence of entire positve solutions for a class of semilinear elliptic system, J. Math. Anal. Appl., 365 (2010), 338-341.
doi: 10.1016/j.jmaa.2009.10.036. |
[14] |
R. Osserman,
On the inequality $\triangle z\geq \psi(z)$, Pacific J. Math., 7 (1957), 1641-1647.
|
[15] |
K. Pei, G. Wang and Y. Sun,
Successive iterations and positive extremal solutions for a Hadamard type fractional integro-differential equations on infinite domain, Appl. Math. Comput., 312 (2017), 158-168.
doi: 10.1016/j.amc.2017.05.056. |
[16] |
J. Qin, G. Wang, L. Zhang and B. Ahmad, Monotone iterative method for a p-Laplacian boundary value problem with fractional conformable derivatives, Bound. Value Probl., 2019 (2019), 145.
doi: 10.1186/s13661-019-1254-5. |
[17] |
Y. Sun, L. Liu and Y. Wu,
The existence and uniqueness of positive monotone solutions for a class of nonlinear Schrödinger equations on infinite domains, J. Comput. Appl. Math., 321 (2017), 478-486.
doi: 10.1016/j.cam.2017.02.036. |
[18] |
G. Wang and X. Ren, Radial symmetry of standing waves for nonlinear fractional Laplacian Hardy-Schrödinger systems, Appl. Math. Lett., 110 (2020), 106560.
doi: 10.1016/j.aml.2020.106560. |
[19] |
G. Wang, X. Ren, Z. Bai and W. Hou,
Radial symmetry of standing waves for nonlinear fractional Hardy-Schrödinger equation, Appl. Math. Lett., 96 (2019), 131-137.
doi: 10.1016/j.aml.2019.04.024. |
[20] |
G. Wang,
Twin iterative positive solutions of fractional q-difference Schrödinger equations, Appl. Math. Lett., 76 (2018), 103-109.
doi: 10.1016/j.aml.2017.08.008. |
[21] |
G. Wang,
Explicit iteration and unbounded solutions for fractional integral boundary value problem on an infinite interval, Appl. Math. Lett., 47 (2015), 1-7.
doi: 10.1016/j.aml.2015.03.003. |
[22] |
G. Wang, K. Pei, R. P. Agarwal, L. Zhang and B. Ahmad,
Nonlocal Hadamard fractional boundary value problem with Hadamard integral and discrete boundary conditions on a half-line, J. Comput. Appl. Math., 343 (2018), 230-239.
doi: 10.1016/j.cam.2018.04.062. |
[23] |
G. Wang, J. Qin, L. Zhang and D. Baleanu, Explicit iteration to a nonlinear fractional Langevin equation with non-separated integro-differential strip-multi-point boundary conditions, Chaos Solitons Fractals, 131 (2020), 109476.
doi: 10.1016/j.chaos.2019.109476. |
[24] |
G. Wang, Z. Bai and L. Zhang,
Successive iterations for unique positive solution of a nonlinear fractional q-integral boundary value problem, J. Appl. Anal. Comput., 9 (2019), 1204-1215.
doi: 10.11948/2156-907X.20180193. |
[25] |
G. Wang, Z. Yang, L. Zhang and D. Baleanu, Radial solutions of a nonlinear $k$-Hessian system involving a nonlinear operator, Commun. Nonlinear Sci. Numer. Simulat., 91 (2020), 105396.
doi: 10.1016/j.cnsns.2020.105396. |
[26] |
D. Ye and F. Zhou,
Invariant criteria for existence of bounded positive solutions, Discrete Contin. Dyn. Syst., 12 (2005), 413-424.
doi: 10.3934/dcds.2005.12.413. |
[27] |
Z. Zhang,
Existence of entire positive solutions for a class of semilinear elliptic systems, Electron. J. Differential Equations, 2010 (2010), 1-5.
|
[28] |
X. Zhang, Y. Wu and Y. Cui,
Existence and nonexistence of blow-up solutions for a Schrödinger equation involving a nonlinear operator, Appl. Math. Lett., 82 (2018), 85-91.
doi: 10.1016/j.aml.2018.02.019. |
[29] |
X. Zhang, C. Mao, L. Liu and Y. Wu,
Exact iterative solution for an abstract fractional dynamic system model for Bioprocess, Qual. Theory Dyn. Syst., 16 (2017), 205-222.
doi: 10.1007/s12346-015-0162-z. |
[30] |
X. Zhang, L. Liu, Y. Wu and L. Caccetta,
Entire large solutions for a class of Schrödinger systems with a nonlinear random operator, J. Math. Anal. Appl., 423 (2015), 1650-1659.
doi: 10.1016/j.jmaa.2014.10.068. |
[31] |
X. Zhang, L. Liu, Y. Wu and Y. Cui,
The existence and nonexistence of entire large solutions for a quasilinear Schrödinger elliptic system by dual approach, J. Math. Anal. Appl., 464 (2018), 1089-1106.
doi: 10.1016/j.jmaa.2018.04.040. |
[32] |
L. Zhang and W. Hou, Standing waves of nonlinear fractional p-Laplacian Schrödinger equation involving logarithmic nonlinearity, Appl. Math. Lett., 102 (2020), 106149.
doi: 10.1016/j.aml.2019.106149. |
[33] |
L. Zhang, B. Ahmad and G. Wang,
Explicit iterations and extremal solutions for fractional differential equations with nonlinear integral boundary conditions, Appl. Math. Comput., 268 (2015), 388-392.
doi: 10.1016/j.amc.2015.06.049. |
[34] |
L. Zhang, B. Ahmad and G. Wang,
The existence of an extremal solution to a nonlinear system with the right-handed Riemann-Liouville fractional derivative, Appl. Math. Lett., 31 (2014), 1-6.
doi: 10.1016/j.aml.2013.12.014. |
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