Solutions of nonlocal problem with critical exponent

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December 2020, 19(12): 5591-5608. doi: 10.3934/cpaa.2020253

Solutions of nonlocal problem with critical exponent

Qingfang Wang ^, and Hua Yang

School of Mathematics and computer science, Wuhan Polytechnic University, Wuhan 430023, China

^* Corresponding author

Received May 2020 Revised July 2020 Published December 2020 Early access October 2020

Fund Project: The first author is supported by NSF grant 11701439

This paper deals with the system with linearly coupled of nonlocal problem with critical exponent,

$ \begin{equation*} \begin{cases} (-\Delta)^\alpha u+\lambda_1u = |u|^{2_\alpha^*-2}u+\beta v , \quad x\in \Omega , \\ (-\Delta)^\alpha v+\lambda_2v = |v|^{2_\alpha^*-2}v+\beta u , \,\quad x\in \Omega , \\ u = v = 0,\ \ \qquad \qquad \qquad \quad \quad \qquad \,x\in \partial\Omega. \end{cases} \end{equation*} $

Here

$ \Omega $

is a smooth bounded domain in

$ {\mathbb{R}}^N(N>4\alpha) $

,

$ 0<\alpha<1 $

,

$ \lambda_1,\lambda_2>-\lambda_1(\Omega) $

are constants,

$ \lambda_1(\Omega) $

is the first eigenvalue of fractional Laplacian with Dirichlet boundary,

$ 2_\alpha^* = \frac{2N}{N-2\alpha} $

is the Sobolev critical exponent and

$ \beta\in {\mathbb{R}} $

is a coupling parameter. By variational method, we prove that this system has a positive ground state solution for some

$ \beta>0 $

. Via a perturbation argument, by doing some delicate estimates for the nonlocal term, we overcome some difficulties and find a positive higher energy solution when

$ |\beta| $

is small. Moreover, the asymptotic behaviors of the positive ground state and higher energy solutions as

$ \beta\rightarrow 0 $

are analyzed.

Keywords: Linearly coupled, critical exponent, asymptotic behavior, nonlocal problem.

Mathematics Subject Classification: Primary: 35R11; Secondary: 35B33, 35B65.

Citation: Qingfang Wang, Hua Yang. Solutions of nonlocal problem with critical exponent. Communications on Pure and Applied Analysis, 2020, 19 (12) : 5591-5608. doi: 10.3934/cpaa.2020253

References:

[1]	A. Ambrosetti and E. Colorado, Standing waves of some coupled nonlinear Schrödinger equations, J. London Math. Soc., 75 (2007), 67-82. doi: 10.1112/jlms/jdl020.
[2]	A. Ambrosetti, E. Colorado and D. Ruiz, Multi-bump solitons to linearly coupled systems of nonlinear Schrödinger equations, Calc. Var. Partial Differ. Equ., 30 (2007), 85-112. doi: 10.1007/s00526-006-0079-0.
[3]	D. Applebaum, Levy processes-from probability to finance and quantum groups, Notices Amer. Math. Soc., 51 (2004), 1336-1347.
[4]	T. Bartsch, N. Dancer and Z.Q. Wang, A Liouville theorem, a priori bounds, and bifurcating branches of positive solutions for a nonlinear elliptic system, Calc. Var. Partial Differ. Equ., 37 (2010), 345-361. doi: 10.1007/s00526-009-0265-y.
[5]	C. Br$\ddot{a}$ndle, E. Colorado, A. de Pablo and U. Sanches, A concave-convex elliptic problem involving the fractional Laplacian, Proc. Roy. Soc. Edinburgh Sect A, 143 (2013), 39-71. doi: 10.1017/S0308210511000175.
[6]	H. Brezis and E. Lieb, A relation between pointwise convergence of functions and convergence of functions, Proc. Am. Math. Soc., 88 (1983), 486-490. doi: 10.1007/978-3-642-55925-9_42.
[7]	X. Cabre and J. Tan, Positive solutions of nonlinear problems involving the square root of Laplacian, Adv. Math., 224 (2010), 2052-2093. doi: 10.1016/j.aim.2010.01.025.
[8]	L. Caffarelli and L. Silvestre, An extension problem related to the fractional Laplacian, Commun. Pure Differ. Equ., 32 (2007), 1245-1260. doi: 10.1080/03605300600987306.
[9]	X. Chang and Z.Q. Wang, Nodal and multiple solutions of nonlinear problems involving the fractional Laplacian, J. Differ. Equ., 256 (2014), 2965-2992. doi: 10.1016/j.jde.2014.01.027.
[10]	Z. Chen and W. Zou, On linearly coupled Schrödinger systems, Proc. Am. Math. Soc., 142 (2014), 323-333. doi: 10.1090/S0002-9939-2013-12000-9.
[11]	J. D$\acute{a}$vila, M. De Pino and J. Wei, Concentrating standing waves for the fractional nonlinear Schr$\ddot{o}$dinger equation, J. Differ. Equ., 256 (2014), 858-892. doi: 10.1016/j.jde.2013.10.006.
[12]	A. Garroni and S. M$\ddot{u}$ller, $\Gamma$-limit of a phase-field model of dislocations, SIAM J. Math. Anal., 36 (2005), 1943-1964. doi: 10.1137/s003614100343768x.
[13]	Q. Guo and X. He, Least energy solutions for a weakly coupled fractional Schrödinger system, Nonlinear Anal., 132 (2016), 141-159. doi: 10.1016/j.na.2015.11.005.
[14]	X. He, M. Squassina and W. Zou, The Nehari manifold for fractional systems involving critical nonlinearities, Commun. Pure Appl. Anal., 15 (2016), 1285-1308. doi: 10.3934/cpaa.2016.15.1285.
[15]	C. Lin and S. Peng, Segregated vector solutions for linearly coupled nonlinear Schrödinger systems, Indiana Univ. Math. J., 63 (2014), 939-967. doi: 10.1512/iumj.2014.63.5310.
[16]	W. Long and S. Peng, Positive vector solutions for a schrödinger system with external source terms, Nonlinear Differ. Equ. Appl., 27 (2020), 36pp. doi: 10.1007/s00030-019-0608-0.
[17]	W. Long and S. Peng, Segregated vector solutions for a class of Bose-Einstein systems, J. Differ. Equ., 257 (2014), 207-230. doi: 10.1016/j.jde.2014.03.019.
[18]	W. Long, Z. Tang and S. Yan, Many synchronized vector solutions for a Bose-Einstein system, preprint.
[19]	D. Lv and S. Peng, On the positive vector solutions for nonlinear fractional systems with linear coupling, Discrete contin. dyn. syst. Ser. A, 37 (2017), 3327-3352. doi: 10.1515/ans-2015-5024.
[20]	S Peng, W. Shuai and Q. Wang, Multiple positive solutions for linearly coupled nonlinear elliptic systems with critical exponent, J. Differ. Equ., 263 (2017), 709-731. doi: 10.1016/j.jde.2017.02.053.
[21]	J. Tan, The Br$\acute{e}$zis-Nirenberg type problem involving the square root of the Laplacian, Calc. Var. Partial Differ. Equ., 42 (2011), 21-41. doi: 10.1007/s00526-010-0378-3.
[22]	E. Valdinoci, From the long jump random walk to the fractional Laplacian, arXiv: 0901.3261.
[23]	S. Yan, J. Yang and X. Yu, Equations involving fractional Laplacian operator: compactness and application, J. Funct. Anal., 269 (2015), 47-79. doi: 10.1016/j.jfa.2015.04.012.

show all references

References:

[1]	A. Ambrosetti and E. Colorado, Standing waves of some coupled nonlinear Schrödinger equations, J. London Math. Soc., 75 (2007), 67-82. doi: 10.1112/jlms/jdl020.
[2]	A. Ambrosetti, E. Colorado and D. Ruiz, Multi-bump solitons to linearly coupled systems of nonlinear Schrödinger equations, Calc. Var. Partial Differ. Equ., 30 (2007), 85-112. doi: 10.1007/s00526-006-0079-0.
[3]	D. Applebaum, Levy processes-from probability to finance and quantum groups, Notices Amer. Math. Soc., 51 (2004), 1336-1347.
[4]	T. Bartsch, N. Dancer and Z.Q. Wang, A Liouville theorem, a priori bounds, and bifurcating branches of positive solutions for a nonlinear elliptic system, Calc. Var. Partial Differ. Equ., 37 (2010), 345-361. doi: 10.1007/s00526-009-0265-y.
[5]	C. Br$\ddot{a}$ndle, E. Colorado, A. de Pablo and U. Sanches, A concave-convex elliptic problem involving the fractional Laplacian, Proc. Roy. Soc. Edinburgh Sect A, 143 (2013), 39-71. doi: 10.1017/S0308210511000175.
[6]	H. Brezis and E. Lieb, A relation between pointwise convergence of functions and convergence of functions, Proc. Am. Math. Soc., 88 (1983), 486-490. doi: 10.1007/978-3-642-55925-9_42.
[7]	X. Cabre and J. Tan, Positive solutions of nonlinear problems involving the square root of Laplacian, Adv. Math., 224 (2010), 2052-2093. doi: 10.1016/j.aim.2010.01.025.
[8]	L. Caffarelli and L. Silvestre, An extension problem related to the fractional Laplacian, Commun. Pure Differ. Equ., 32 (2007), 1245-1260. doi: 10.1080/03605300600987306.
[9]	X. Chang and Z.Q. Wang, Nodal and multiple solutions of nonlinear problems involving the fractional Laplacian, J. Differ. Equ., 256 (2014), 2965-2992. doi: 10.1016/j.jde.2014.01.027.
[10]	Z. Chen and W. Zou, On linearly coupled Schrödinger systems, Proc. Am. Math. Soc., 142 (2014), 323-333. doi: 10.1090/S0002-9939-2013-12000-9.
[11]	J. D$\acute{a}$vila, M. De Pino and J. Wei, Concentrating standing waves for the fractional nonlinear Schr$\ddot{o}$dinger equation, J. Differ. Equ., 256 (2014), 858-892. doi: 10.1016/j.jde.2013.10.006.
[12]	A. Garroni and S. M$\ddot{u}$ller, $\Gamma$-limit of a phase-field model of dislocations, SIAM J. Math. Anal., 36 (2005), 1943-1964. doi: 10.1137/s003614100343768x.
[13]	Q. Guo and X. He, Least energy solutions for a weakly coupled fractional Schrödinger system, Nonlinear Anal., 132 (2016), 141-159. doi: 10.1016/j.na.2015.11.005.
[14]	X. He, M. Squassina and W. Zou, The Nehari manifold for fractional systems involving critical nonlinearities, Commun. Pure Appl. Anal., 15 (2016), 1285-1308. doi: 10.3934/cpaa.2016.15.1285.
[15]	C. Lin and S. Peng, Segregated vector solutions for linearly coupled nonlinear Schrödinger systems, Indiana Univ. Math. J., 63 (2014), 939-967. doi: 10.1512/iumj.2014.63.5310.
[16]	W. Long and S. Peng, Positive vector solutions for a schrödinger system with external source terms, Nonlinear Differ. Equ. Appl., 27 (2020), 36pp. doi: 10.1007/s00030-019-0608-0.
[17]	W. Long and S. Peng, Segregated vector solutions for a class of Bose-Einstein systems, J. Differ. Equ., 257 (2014), 207-230. doi: 10.1016/j.jde.2014.03.019.
[18]	W. Long, Z. Tang and S. Yan, Many synchronized vector solutions for a Bose-Einstein system, preprint.
[19]	D. Lv and S. Peng, On the positive vector solutions for nonlinear fractional systems with linear coupling, Discrete contin. dyn. syst. Ser. A, 37 (2017), 3327-3352. doi: 10.1515/ans-2015-5024.
[20]	S Peng, W. Shuai and Q. Wang, Multiple positive solutions for linearly coupled nonlinear elliptic systems with critical exponent, J. Differ. Equ., 263 (2017), 709-731. doi: 10.1016/j.jde.2017.02.053.
[21]	J. Tan, The Br$\acute{e}$zis-Nirenberg type problem involving the square root of the Laplacian, Calc. Var. Partial Differ. Equ., 42 (2011), 21-41. doi: 10.1007/s00526-010-0378-3.
[22]	E. Valdinoci, From the long jump random walk to the fractional Laplacian, arXiv: 0901.3261.
[23]	S. Yan, J. Yang and X. Yu, Equations involving fractional Laplacian operator: compactness and application, J. Funct. Anal., 269 (2015), 47-79. doi: 10.1016/j.jfa.2015.04.012.

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