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Lifespan of solutions to a fourth order parabolic PDE involving the Hessian modeling epitaxial growth
Solutions of nonlocal problem with critical exponent
School of Mathematics and computer science, Wuhan Polytechnic University, Wuhan 430023, China |
$ \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*} $ |
$ \Omega $ |
$ {\mathbb{R}}^N(N>4\alpha) $ |
$ 0<\alpha<1 $ |
$ \lambda_1,\lambda_2>-\lambda_1(\Omega) $ |
$ \lambda_1(\Omega) $ |
$ 2_\alpha^* = \frac{2N}{N-2\alpha} $ |
$ \beta\in {\mathbb{R}} $ |
$ \beta>0 $ |
$ |\beta| $ |
$ \beta\rightarrow 0 $ |
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. Google Scholar |
[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. Google Scholar |
[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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