Existence and nonexistence of entire positive radial solutions for a class of Schrödinger elliptic systems involving a nonlinear operator

Zedong Yang; Guotao Wang; Ravi P. Agarwal; Haiyong Xu

doi:10.3934/dcdss.2020436

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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

Zedong Yang ^a, , Guotao Wang ^a,c, , Ravi P. Agarwal ^b,c,, and Haiyong Xu ^d,

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.

Keywords: Schrödinger elliptic system, nonlinear operator, radial solution, blow up, monotone iterative method.

Mathematics Subject Classification: Primary: 35A24, 35B09; Secondary: 35B44.

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 & 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. Google Scholar
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[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. Google Scholar
[6]	A. B. Dkhil, Positive solutions for nonlinear elliptic systems, Electron. J. Differential Equations, 2012 (239) (2012), 1-10. Google Scholar
[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. Google Scholar
[8]	J. B. Keller, On solutions of $\triangle z = \psi(z)$, Comm. Pure Appl. Math., 10 (1957), 503-510. doi: 10.1002/cpa.3160100402. Google Scholar
[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. Google Scholar
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[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[14]	R. Osserman, On the inequality $\triangle z\geq \psi(z)$, Pacific J. Math., 7 (1957), 1641-1647. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[27]	Z. Zhang, Existence of entire positive solutions for a class of semilinear elliptic systems, Electron. J. Differential Equations, 2010 (2010), 1-5. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar

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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[6]	A. B. Dkhil, Positive solutions for nonlinear elliptic systems, Electron. J. Differential Equations, 2012 (239) (2012), 1-10. Google Scholar
[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. Google Scholar
[8]	J. B. Keller, On solutions of $\triangle z = \psi(z)$, Comm. Pure Appl. Math., 10 (1957), 503-510. doi: 10.1002/cpa.3160100402. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[14]	R. Osserman, On the inequality $\triangle z\geq \psi(z)$, Pacific J. Math., 7 (1957), 1641-1647. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[27]	Z. Zhang, Existence of entire positive solutions for a class of semilinear elliptic systems, Electron. J. Differential Equations, 2010 (2010), 1-5. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar

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American Institute of Mathematical Sciences

Existence and nonexistence of entire positive radial solutions for a class of Schrödinger elliptic systems involving a nonlinear operator

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Existence and nonexistence of entire positive radial solutions for a class of Schrödinger elliptic systems involving a nonlinear operator

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