-
Previous Article
Classification of non-topological solutions of an elliptic equation arising from self-dual gauged Sigma model
- CPAA Home
- This Issue
-
Next Article
Orbitally symmetric systems with applications to planar centers
Ground state solution of critical Schrödinger equation with singular potential
School of Mathematics and Big Data, Anhui University of Science and Technology, Huainan, Anhui 232001, China |
$ \begin{equation*} \begin{aligned} -\Delta u + V(|x|)u = f(u),\ \, x\in \mathbb{R}^{N}, \end{aligned} \end{equation*} $ |
$ N\geqslant 3 $ |
$ V $ |
$ \alpha\in(0,2)\cup(2,\infty) $ |
$ f $ |
References:
[1] |
M. Badiale and S. Rolando,
A note on nonlinear elliptic problems with singular potentials, Atti Accad. Naz. Lincei Rend. Lincei Mat. Appl., 17 (2006), 1-13.
doi: 10.4171/RLM/450. |
[2] |
M. Badiale, V. Benci and S. Rolando,
A nonlinear elliptic equation with singular potential and applications to nonlinear field equations, J. Eur. Math. Soc., 9 (2007), 355-381.
doi: 10.4171/JEMS/83. |
[3] |
M. Badiale, M. Guida and S. Rolando, Elliptic equations with decaying cylindrical potentials and power-type nonlinearities, Adv. Differ. Equ., 12 (2007), 1321-1362.
doi: euclid.ade/1355867405. |
[4] |
M. Badiale, M. Guida and S. Rolando, A nonexistence result for a nonlinear elliptic equation with singular and decaying potential, Commun. Contemp. Math., 17 (2015), 21 pp.
doi: 10.1142/S0219199714500242. |
[5] |
M. Badiale, M. Guida and S. Rolando, Compactness and existence results in weighted Sobolev spaces of radial functions Part Ⅱ: Existence, Nonlinear Differ. Equ. Appl., 23 (2016), 34 pp.
doi: 10.1007/s00030-016-0411-0. |
[6] |
M. Badiale, M. Guida and S. Rolando,
Compactness and existence results for the p-Laplace equations, J. Math. Anal. Appl., 451 (2017), 345-370.
doi: 10.1016/j.jmaa.2017.02.011. |
[7] |
M. Badiale, L. Pisani and S. Rolando,
Sum of weighted Lebesgue spaces and nonlinear elliptic equations, Nonlinear Differ. Equ. Appl., 18 (2011), 369-405.
doi: 10.1007/s00030-011-0100-y. |
[8] |
M. Badiale, S. Greco and S. Rolando,
Radial solutions of a biharmonic equation with vanishing or singular radial potentials, Nonlinear Appl., 185 (2019), 97-122.
doi: 10.1016/j.na.2019.01.011. |
[9] |
V. Benci and D. Fortunato, Variational Methods in Nonlinear Field Equations, Springer, Cham, 2014. |
[10] |
H. Berestycki and P. L. Lions,
Nonlinear scalar field equations, I. Existence of a ground state, Arch. Rational Mech. Anal., 82 (1983), 313-345.
doi: 10.1007/BF00250555. |
[11] |
H. Brézis and E. H. Lieb,
A relation between pointwise convergence of functions and convergence of functionals, Proc. Amer. Math. Soc., 88 (1983), 486-490.
doi: 10.2307/2044999. |
[12] |
F. Catrina,
Nonexistence of positive radial solutions for a problem with singular potential, Adv. Nonlinear Anal., 3 (2014), 1-13.
doi: 10.1515/anona-2013-0023. |
[13] |
P. C. Carrião, R. Demarque and O. H. Miyagaki,
Nonlinear biharmonic problems with singular potentials, Commun. Pure Appl. Anal., 13 (2014), 2141-2154.
doi: 10.3934/cpaa.2014.13.2141. |
[14] |
M. Conti, S. Crotti and D. Pardo, On the existence of positive solutions for a class of singular elliptic equations, Adv. Differ. Equ., 3 (1998), 111-132.
doi: euclid.ade/1366399907. |
[15] |
R. Demarque and O. H. Miyagaki,
Radial solutions of inhomogeneous fourth order elliptic equations and weighted sobolev embeddings, Adv. Nonlinear Anal., 4 (2015), 135-151.
doi: 10.1515/anona-2014-0041. |
[16] |
P. C. Fife,
Asymptotic states for equations of reaction and diffusion, Bull. Amer. Math. Soc., 84 (1978), 693-726.
doi: 10.1090/S0002-9904-1978-14502-9. |
[17] |
R. Filippucci, P. Pucci and F. Robert,
On a $p$-Laplace equation with multiple critical nonlinearities, J. Math. Pures Appl., 50 (2014), 156-177.
doi: 10.1016/j.matpur.2008.09.008. |
[18] |
W. Ni,
A nonlinear Dirichlet problem on the unit ball and its applications, Indiana Univ. Math. J., 31 (1982), 801-807.
doi: 10.1512/iumj.1982.31.31056. |
[19] |
G. Palatucci and A. Pisante,
Improved Sobolev embeddings, profile decomposition, and concentration-compactness for fractional Sobolev spaces, Calc. Var. Partial Differ. Equ., 50 (2014), 799-829.
doi: 10.1007/s00526-013-0656-y. |
[20] |
S. Rolando,
Multiple nonradial solutions for a nonlinear elliptic problem with singular and decaying radial potential, Adv. Nonlinear Anal., 8 (2019), 885-901.
doi: 10.1515/anona-2017-0177. |
[21] |
J. Su and R. Tian,
Weighted Sobolev embeddings and radial solutions of inhomogeneous quasilinear elliptic equations, Commun. Pure Appl. Anal., 9 (2010), 885-904.
doi: 10.3934/cpaa.2010.9.885. |
[22] |
J. Su, Z. Wang and M. Willem,
Nonlinear Schrödinger equations with unbounded and decaying potentials, Commun. Contemp. Math., 9 (2007), 571-583.
doi: 10.1142/S021919970700254X. |
[23] |
J. Su, Z. Wang and M. Willem,
Weighted Sobolev embedding with unbounded and decaying radial potentials, J. Differ. Equ., 238 (2007), 201-219.
doi: 10.1016/j.jde.2007.03.018. |
[24] |
S. Terracini, On positive entire solutions to a class of equations with a singular coefficient and critical exponent, Adv. Differ. Equ., 1 (1996), 241-264.
doi: euclid.ade/1366896239. |
[25] |
P. Tolksdorf,
Regularity for a more general class of quasilinear elliptic equations, J. Differ. Equ., 51 (1984), 126-150.
doi: 10.1016/0022-0396(84)90105-0. |
[26] |
C. Vincent and S. Phatak,
Accurate momentum-space method for scattering by nuclear and Coulomb potentials, Phys. Rev., 10 (1974), 391-394.
|
[27] |
J. L. Vàzquez,
A strong maximum principle for some quasilinear elliptic equations, Appl. Math. Optim., 12 (1984), 191-202.
doi: 10.1007/BF01449041. |
[28] |
Y. Yang, Solitons in Field Theory and Nonlinear Analysis, Springer Monographs in Mathematics, Springer-Verlag, New York, 2001. |
show all references
References:
[1] |
M. Badiale and S. Rolando,
A note on nonlinear elliptic problems with singular potentials, Atti Accad. Naz. Lincei Rend. Lincei Mat. Appl., 17 (2006), 1-13.
doi: 10.4171/RLM/450. |
[2] |
M. Badiale, V. Benci and S. Rolando,
A nonlinear elliptic equation with singular potential and applications to nonlinear field equations, J. Eur. Math. Soc., 9 (2007), 355-381.
doi: 10.4171/JEMS/83. |
[3] |
M. Badiale, M. Guida and S. Rolando, Elliptic equations with decaying cylindrical potentials and power-type nonlinearities, Adv. Differ. Equ., 12 (2007), 1321-1362.
doi: euclid.ade/1355867405. |
[4] |
M. Badiale, M. Guida and S. Rolando, A nonexistence result for a nonlinear elliptic equation with singular and decaying potential, Commun. Contemp. Math., 17 (2015), 21 pp.
doi: 10.1142/S0219199714500242. |
[5] |
M. Badiale, M. Guida and S. Rolando, Compactness and existence results in weighted Sobolev spaces of radial functions Part Ⅱ: Existence, Nonlinear Differ. Equ. Appl., 23 (2016), 34 pp.
doi: 10.1007/s00030-016-0411-0. |
[6] |
M. Badiale, M. Guida and S. Rolando,
Compactness and existence results for the p-Laplace equations, J. Math. Anal. Appl., 451 (2017), 345-370.
doi: 10.1016/j.jmaa.2017.02.011. |
[7] |
M. Badiale, L. Pisani and S. Rolando,
Sum of weighted Lebesgue spaces and nonlinear elliptic equations, Nonlinear Differ. Equ. Appl., 18 (2011), 369-405.
doi: 10.1007/s00030-011-0100-y. |
[8] |
M. Badiale, S. Greco and S. Rolando,
Radial solutions of a biharmonic equation with vanishing or singular radial potentials, Nonlinear Appl., 185 (2019), 97-122.
doi: 10.1016/j.na.2019.01.011. |
[9] |
V. Benci and D. Fortunato, Variational Methods in Nonlinear Field Equations, Springer, Cham, 2014. |
[10] |
H. Berestycki and P. L. Lions,
Nonlinear scalar field equations, I. Existence of a ground state, Arch. Rational Mech. Anal., 82 (1983), 313-345.
doi: 10.1007/BF00250555. |
[11] |
H. Brézis and E. H. Lieb,
A relation between pointwise convergence of functions and convergence of functionals, Proc. Amer. Math. Soc., 88 (1983), 486-490.
doi: 10.2307/2044999. |
[12] |
F. Catrina,
Nonexistence of positive radial solutions for a problem with singular potential, Adv. Nonlinear Anal., 3 (2014), 1-13.
doi: 10.1515/anona-2013-0023. |
[13] |
P. C. Carrião, R. Demarque and O. H. Miyagaki,
Nonlinear biharmonic problems with singular potentials, Commun. Pure Appl. Anal., 13 (2014), 2141-2154.
doi: 10.3934/cpaa.2014.13.2141. |
[14] |
M. Conti, S. Crotti and D. Pardo, On the existence of positive solutions for a class of singular elliptic equations, Adv. Differ. Equ., 3 (1998), 111-132.
doi: euclid.ade/1366399907. |
[15] |
R. Demarque and O. H. Miyagaki,
Radial solutions of inhomogeneous fourth order elliptic equations and weighted sobolev embeddings, Adv. Nonlinear Anal., 4 (2015), 135-151.
doi: 10.1515/anona-2014-0041. |
[16] |
P. C. Fife,
Asymptotic states for equations of reaction and diffusion, Bull. Amer. Math. Soc., 84 (1978), 693-726.
doi: 10.1090/S0002-9904-1978-14502-9. |
[17] |
R. Filippucci, P. Pucci and F. Robert,
On a $p$-Laplace equation with multiple critical nonlinearities, J. Math. Pures Appl., 50 (2014), 156-177.
doi: 10.1016/j.matpur.2008.09.008. |
[18] |
W. Ni,
A nonlinear Dirichlet problem on the unit ball and its applications, Indiana Univ. Math. J., 31 (1982), 801-807.
doi: 10.1512/iumj.1982.31.31056. |
[19] |
G. Palatucci and A. Pisante,
Improved Sobolev embeddings, profile decomposition, and concentration-compactness for fractional Sobolev spaces, Calc. Var. Partial Differ. Equ., 50 (2014), 799-829.
doi: 10.1007/s00526-013-0656-y. |
[20] |
S. Rolando,
Multiple nonradial solutions for a nonlinear elliptic problem with singular and decaying radial potential, Adv. Nonlinear Anal., 8 (2019), 885-901.
doi: 10.1515/anona-2017-0177. |
[21] |
J. Su and R. Tian,
Weighted Sobolev embeddings and radial solutions of inhomogeneous quasilinear elliptic equations, Commun. Pure Appl. Anal., 9 (2010), 885-904.
doi: 10.3934/cpaa.2010.9.885. |
[22] |
J. Su, Z. Wang and M. Willem,
Nonlinear Schrödinger equations with unbounded and decaying potentials, Commun. Contemp. Math., 9 (2007), 571-583.
doi: 10.1142/S021919970700254X. |
[23] |
J. Su, Z. Wang and M. Willem,
Weighted Sobolev embedding with unbounded and decaying radial potentials, J. Differ. Equ., 238 (2007), 201-219.
doi: 10.1016/j.jde.2007.03.018. |
[24] |
S. Terracini, On positive entire solutions to a class of equations with a singular coefficient and critical exponent, Adv. Differ. Equ., 1 (1996), 241-264.
doi: euclid.ade/1366896239. |
[25] |
P. Tolksdorf,
Regularity for a more general class of quasilinear elliptic equations, J. Differ. Equ., 51 (1984), 126-150.
doi: 10.1016/0022-0396(84)90105-0. |
[26] |
C. Vincent and S. Phatak,
Accurate momentum-space method for scattering by nuclear and Coulomb potentials, Phys. Rev., 10 (1974), 391-394.
|
[27] |
J. L. Vàzquez,
A strong maximum principle for some quasilinear elliptic equations, Appl. Math. Optim., 12 (1984), 191-202.
doi: 10.1007/BF01449041. |
[28] |
Y. Yang, Solitons in Field Theory and Nonlinear Analysis, Springer Monographs in Mathematics, Springer-Verlag, New York, 2001. |
[1] |
Wentao Huang, Jianlin Xiang. Soliton solutions for a quasilinear Schrödinger equation with critical exponent. Communications on Pure and Applied Analysis, 2016, 15 (4) : 1309-1333. doi: 10.3934/cpaa.2016.15.1309 |
[2] |
David Gómez-Castro, Juan Luis Vázquez. The fractional Schrödinger equation with singular potential and measure data. Discrete and Continuous Dynamical Systems, 2019, 39 (12) : 7113-7139. doi: 10.3934/dcds.2019298 |
[3] |
Wenmin Gong, Guangcun Lu. On Dirac equation with a potential and critical Sobolev exponent. Communications on Pure and Applied Analysis, 2015, 14 (6) : 2231-2263. doi: 10.3934/cpaa.2015.14.2231 |
[4] |
Hiroshi Isozaki, Hisashi Morioka. A Rellich type theorem for discrete Schrödinger operators. Inverse Problems and Imaging, 2014, 8 (2) : 475-489. doi: 10.3934/ipi.2014.8.475 |
[5] |
Yuxia Guo, Zhongwei Tang. Multi-bump solutions for Schrödinger equation involving critical growth and potential wells. Discrete and Continuous Dynamical Systems, 2015, 35 (8) : 3393-3415. doi: 10.3934/dcds.2015.35.3393 |
[6] |
Jian Zhang, Shihui Zhu, Xiaoguang Li. Rate of $L^2$-concentration of the blow-up solution for critical nonlinear Schrödinger equation with potential. Mathematical Control and Related Fields, 2011, 1 (1) : 119-127. doi: 10.3934/mcrf.2011.1.119 |
[7] |
Xu Zhang, Shiwang Ma, Qilin Xie. Bound state solutions of Schrödinger-Poisson system with critical exponent. Discrete and Continuous Dynamical Systems, 2017, 37 (1) : 605-625. doi: 10.3934/dcds.2017025 |
[8] |
Kaimin Teng, Xiumei He. Ground state solutions for fractional Schrödinger equations with critical Sobolev exponent. Communications on Pure and Applied Analysis, 2016, 15 (3) : 991-1008. doi: 10.3934/cpaa.2016.15.991 |
[9] |
Mengyao Chen, Qi Li, Shuangjie Peng. Bound states for fractional Schrödinger-Poisson system with critical exponent. Discrete and Continuous Dynamical Systems - S, 2021, 14 (6) : 1819-1835. doi: 10.3934/dcdss.2021038 |
[10] |
Brahim Alouini. Finite dimensional global attractor for a damped fractional anisotropic Schrödinger type equation with harmonic potential. Communications on Pure and Applied Analysis, 2020, 19 (9) : 4545-4573. doi: 10.3934/cpaa.2020206 |
[11] |
Grégoire Allaire, M. Vanninathan. Homogenization of the Schrödinger equation with a time oscillating potential. Discrete and Continuous Dynamical Systems - B, 2006, 6 (1) : 1-16. doi: 10.3934/dcdsb.2006.6.1 |
[12] |
Younghun Hong. Scattering for a nonlinear Schrödinger equation with a potential. Communications on Pure and Applied Analysis, 2016, 15 (5) : 1571-1601. doi: 10.3934/cpaa.2016003 |
[13] |
Zhiyan Ding, Hichem Hajaiej. On a fractional Schrödinger equation in the presence of harmonic potential. Electronic Research Archive, 2021, 29 (5) : 3449-3469. doi: 10.3934/era.2021047 |
[14] |
Mouhamed Moustapha Fall, Veronica Felli. Unique continuation properties for relativistic Schrödinger operators with a singular potential. Discrete and Continuous Dynamical Systems, 2015, 35 (12) : 5827-5867. doi: 10.3934/dcds.2015.35.5827 |
[15] |
Pavel I. Naumkin, Isahi Sánchez-Suárez. On the critical nongauge invariant nonlinear Schrödinger equation. Discrete and Continuous Dynamical Systems, 2011, 30 (3) : 807-834. doi: 10.3934/dcds.2011.30.807 |
[16] |
Yu Su, Zhaosheng Feng. Ground state solutions for the fractional problems with dipole-type potential and critical exponent. Communications on Pure and Applied Analysis, 2022, 21 (6) : 1953-1968. doi: 10.3934/cpaa.2021111 |
[17] |
Stefano Pasquali. A Nekhoroshev type theorem for the nonlinear Klein-Gordon equation with potential. Discrete and Continuous Dynamical Systems - B, 2018, 23 (9) : 3573-3594. doi: 10.3934/dcdsb.2017215 |
[18] |
Maria-Magdalena Boureanu. Fourth-order problems with Leray-Lions type operators in variable exponent spaces. Discrete and Continuous Dynamical Systems - S, 2019, 12 (2) : 231-243. doi: 10.3934/dcdss.2019016 |
[19] |
Yu Su, Zhaosheng Feng. Ground state solutions and decay estimation of Choquard equation with critical exponent and Dipole potential. Discrete and Continuous Dynamical Systems - S, 2022 doi: 10.3934/dcdss.2022112 |
[20] |
Xia Sun, Kaimin Teng. Positive bound states for fractional Schrödinger-Poisson system with critical exponent. Communications on Pure and Applied Analysis, 2020, 19 (7) : 3735-3768. doi: 10.3934/cpaa.2020165 |
2021 Impact Factor: 1.273
Tools
Metrics
Other articles
by authors
[Back to Top]