# American Institute of Mathematical Sciences

October  2021, 14(10): 3821-3836. doi: 10.3934/dcdss.2020436

## 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

* Corresponding author: Ravi P. Agarwal

All authors equally contributed this manuscript

Received  April 2020 Revised  June 2020 Published  October 2021 Early access  November 2020

Fund Project: This work is supported by NSFC (No.11501342), NSF of Shanxi, China (No.201701D221007), Science and Technology Innovation Project of Shanxi Normal University (No.2019XSY027), the Graduate Innovation Program of Shanxi, China (No.2020SY337) and STIP (Nos.201802068 and 201802069)

In this paper, we study the positive solutions of the Schrödinger elliptic system
 $\begin{equation*} \begin{split} \left\{\begin{array}{ll}{\operatorname{div}(\mathcal{G}(|\nabla y|^{p-2})\nabla y) = b_{1}(|x|) \psi(y)+h_{1}(|x|) \varphi(z),}& {x \in \mathbb{R}^{n}(n \geq 3)}, \\ {\operatorname{div}(\mathcal{G}(|\nabla z|^{p-2})\nabla z) = b_{2}(|x|) \psi(z)+h_{2}(|x|) \varphi(y),} & {x \in \mathbb{R}^{n}},\end{array}\right. \end{split} \end{equation*}$
where
 $\mathcal{G}$
is a nonlinear operator. By using the monotone iterative technique and Arzela-Ascoli theorem, we prove that the system has the positive entire bounded radial solutions. Then, we establish the results for the existence and nonexistence of the positive entire blow-up radial solutions for the nonlinear Schrödinger elliptic system involving a nonlinear operator. Finally, three examples are given to illustrate our results.
Citation: Zedong Yang, Guotao Wang, Ravi P. Agarwal, Haiyong Xu. Existence and nonexistence of entire positive radial solutions for a class of Schrödinger elliptic systems involving a nonlinear operator. Discrete and Continuous Dynamical Systems - S, 2021, 14 (10) : 3821-3836. doi: 10.3934/dcdss.2020436
##### 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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##### 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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