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Compactness results for linearly perturbed Yamabe problem on manifolds with boundary
Sign-changing solutions for a parameter-dependent quasilinear equation
1. | LMAM, School of Mathematical Science, Peking University, Beijing 100871, China |
2. | Department of Mathematics, Yunnan Normal University, Kunming 650500, China |
3. | Department of Mathematics and Statistics, Utah State University, Logan, UT 84322, USA |
$ \begin{equation*} \left\{ \begin{aligned} &\Delta u+\frac{1}{2}u\Delta u^2+\lambda |u|^{r-2}u = 0, \ \ \ \text{in}\,\,\Omega,\\ &u = 0\quad\text{on}\,\,\partial\Omega, \end{aligned} \right. \end{equation*} $ |
$ \Omega\subset\mathbb{R}^N(N\geq3) $ |
$ \lambda>0,\, r\in(2,4) $ |
$ \lambda $ |
References:
[1] |
A. Ambrosetti and P. H. Rabinowitz,
Dual variational methods in critical point theory and applications, J. Funct. Anal., 14 (1973), 349-381.
doi: 10.1016/0022-1236(73)90051-7. |
[2] |
T. Bartsch, K.-C. Chang and Z.-Q. Wang,
On the Morse indices of sign changing solutions of nonlinear elliptic problems, Math. Z., 233 (2000), 655-677.
doi: 10.1007/s002090050492. |
[3] |
T. Bartsch, Z. Liu and T. Weth,
Sign changing solutions of superlinear Schrödinger equations, Commun. Partial Differential Equations, 29 (2004), 25-42.
doi: 10.1081/PDE-120028842. |
[4] |
T. Bartsch and Z.-Q. Wang,
On the existence of sign changing solutions for semilinear Dirichlet problems, Topol. Methods Nonlinear Anal., 7 (1996), 115-131.
doi: 10.12775/TMNA.1996.005. |
[5] |
F. G. Bass and N. N. Nasonov,
Nonlinear electromagnetic-spin waves, Phys. Rep., 189 (1990), 165-223.
doi: 10.1016/0370-1573(90)90093-H. |
[6] |
H. Brezis, Functional Analysis, Sobolev Spaces and Partial Differential Equations, Universitext, Springer, New York, 2011. |
[7] |
D. C. Clark,
A variant of the Lusternik-Schnirelman theory, Indiana Univ. Math. J., 22 (1972/1973), 65-74.
doi: 10.1512/iumj.1973.22.22008. |
[8] |
M. Colin and L. Jeanjean,
Solutions for a quasilinear Schrödinger equation: A dual approach, Nonlinear Anal., 56 (2004), 213-226.
doi: 10.1016/j.na.2003.09.008. |
[9] |
R. W. Hasse,
A general method for the solution of nonlinear soliton and kink Schrödinger equations, Z. Phys. B, 37 (1980), 83-87.
doi: 10.1007/BF01325508. |
[10] |
Y. Jing, Z. Liu and Z.-Q. Wang, Multiple solutions of a parameter-dependent quasilinear elliptic equation, Calc. Var. Partial Differential Equations, 55 (2016), 150, 26 pp.
doi: 10.1007/s00526-016-1067-7. |
[11] |
Y. Jing, Z. Liu and Z.-Q. Wang,
Existence results for a singular quasilinear elliptic equation, J. Fixed Point Theory Appl., 19 (2017), 67-84.
doi: 10.1007/s11784-016-0341-9. |
[12] |
Y. Jing, Z. Liu and Z.-Q. Wang, Parameter-dependent multiplicity results of sign-changing solutions for quasilinear elliptic equations, preprint. |
[13] |
M. Kosevich, A. Ivanov and S. Kovalev,
Magnetic solutions, Phys. Rep., 194 (1990), 117-238.
|
[14] |
S. Kurihara,
Large-amplitude quasi-solitons in superfluid films, J. Phys. Soc. Jap., 50 (1981), 3801-3805.
doi: 10.1143/JPSJ.50.3801. |
[15] |
O. A. Ladyzhenskaya and N. N. Ural'tseva, Linear and Quasilinear Elliptic Equations, Academic Press, New York-London, 1968.
![]() ![]() |
[16] |
S. Li and Z.-Q. Wang,
Ljusternik-Schnirelman theory in partially ordered Hilbert spaces, Trans. Amer. Math. Soc., 354 (2002), 3207-3227.
doi: 10.1090/S0002-9947-02-03031-3. |
[17] |
G. M. Lieberman,
The natural generalizationj of the natural conditions of Ladyzhenskaya and Ural'tseva for elliptic equations, Commun. Partial Differential Equations, 16 (1991), 311-361.
doi: 10.1080/03605309108820761. |
[18] |
A. G. Litvak and A. M. Sergeev,
One dimensional collapse of plasma waves, JEPT Letters, 27 (1978), 517-520.
|
[19] |
J.-Q. Liu, X.-Q. Liu and Z.-Q. Wang,
Multiple sign-changing solutions for quasilinear elliptic equations via perturbation method, Commun. Partial Differential Equations, 39 (2014), 2216-2239.
doi: 10.1080/03605302.2014.942738. |
[20] |
J. Liu, X. Liu and Z.-Q. Wang,
Sign-changing solutions for coupled nonlinear Schrödinger equations with critical growth, J. Differential Equations, 261 (2016), 7194-7236.
doi: 10.1016/j.jde.2016.09.018. |
[21] |
J. Liu, X. Liu and Z.-Q. Wang,
Multiple mixed states of nodal solutions for nonlinear Schrödinger systems, Calc. Var. Partial Differential Equations, 52 (2015), 565-586.
doi: 10.1007/s00526-014-0724-y. |
[22] |
X.-Q. Liu, J.-Q. Liu and Z.-Q. Wang,
Quasilinear elliptic equations via perturbation method, Proc. Amer. Math. Soc., 141 (2013), 253-263.
doi: 10.1090/S0002-9939-2012-11293-6. |
[23] |
X.-Q. Liu, J.-Q. Liu and Z.-Q. Wang,
Quasilinear elliptic equations with critical growth via perturbation method, J. Differential Equations, 254 (2013), 102-124.
doi: 10.1016/j.jde.2012.09.006. |
[24] |
X. Liu, J. Liu and Z.-Q. Wang,
Localized nodal solutions for quasilinear Schrödinger equations, J. Differential Equations, 267 (2019), 7411-7461.
doi: 10.1016/j.jde.2019.08.003. |
[25] |
Z. Liu and J. Sun,
Invariant sets of descending flow in critical point theory with applications to nonlinear differential equations, J. Differential Equations, 172 (2001), 257-299.
doi: 10.1006/jdeq.2000.3867. |
[26] |
J. Liu and Z.-Q. Wang,
Soliton solutions for quasilinear Schrödinger equations, Proc. Amer. Math. Soc., 131 (2003), 441-448.
doi: 10.1090/S0002-9939-02-06783-7. |
[27] |
J.-Q. Liu, Y.-Q. Wang and Z.-Q. Wang,
Soliton solutions for quasilinear Schrödinger equation, II, J. Differential Equations, 187 (2003), 473-493.
doi: 10.1016/S0022-0396(02)00064-5. |
[28] |
J.-Q. Liu, Y.-Q. Wang and Z.-Q. Wang,
Solutions for quasilinear Schrödinger equations via the Nehari method, Commun. Partial Differential Equations, 29 (2004), 879-901.
doi: 10.1081/PDE-120037335. |
[29] |
X. Liu and J. Zhao,
$p$-Laplacian equation in $\mathbb{R}^N$ with finite potential via the truncation method, Adv. Nonlinear Stud., 17 (2017), 595-610.
doi: 10.1515/ans-2015-5059. |
[30] |
V. G. Makhan'kov and V. K. Fedyanin,
Nonlinear effects in quasi-one-dimensional models of condensed matter theory, Phys. Rep., 104 (1984), 1-86.
doi: 10.1016/0370-1573(84)90106-6. |
[31] |
M. Porkolab and M. V. Goldman,
Upper hybrid solitons and oscillating two-stream instabilities, Phys. Fluids, 19 (1976), 872-881.
doi: 10.1063/1.861553. |
[32] |
G. R. W. Quispel and H. W. Capel,
Equation of motion for the Heisenberg spin chain, Phys. A, 110 (1982), 41-80.
doi: 10.1016/0378-4371(82)90104-2. |
[33] |
P. H. Rabinowitz, Minimax Methods in Critical Point Theory with Applications to Differential Equations, CBMS Regional Conference, Series in Mathematics, American Mathematical Society, Vol. 65, 1986.
doi: 10.1090/cbms/065. |
[34] |
J. Zhao, X. Liu and J. Liu,
$p$-Laplacian equations in $\mathbb{R}^N$ with finite potential via truncation method, the critical case, J. Math. Anal. Appl., 455 (2017), 58-88.
doi: 10.1016/j.jmaa.2017.03.085. |
show all references
References:
[1] |
A. Ambrosetti and P. H. Rabinowitz,
Dual variational methods in critical point theory and applications, J. Funct. Anal., 14 (1973), 349-381.
doi: 10.1016/0022-1236(73)90051-7. |
[2] |
T. Bartsch, K.-C. Chang and Z.-Q. Wang,
On the Morse indices of sign changing solutions of nonlinear elliptic problems, Math. Z., 233 (2000), 655-677.
doi: 10.1007/s002090050492. |
[3] |
T. Bartsch, Z. Liu and T. Weth,
Sign changing solutions of superlinear Schrödinger equations, Commun. Partial Differential Equations, 29 (2004), 25-42.
doi: 10.1081/PDE-120028842. |
[4] |
T. Bartsch and Z.-Q. Wang,
On the existence of sign changing solutions for semilinear Dirichlet problems, Topol. Methods Nonlinear Anal., 7 (1996), 115-131.
doi: 10.12775/TMNA.1996.005. |
[5] |
F. G. Bass and N. N. Nasonov,
Nonlinear electromagnetic-spin waves, Phys. Rep., 189 (1990), 165-223.
doi: 10.1016/0370-1573(90)90093-H. |
[6] |
H. Brezis, Functional Analysis, Sobolev Spaces and Partial Differential Equations, Universitext, Springer, New York, 2011. |
[7] |
D. C. Clark,
A variant of the Lusternik-Schnirelman theory, Indiana Univ. Math. J., 22 (1972/1973), 65-74.
doi: 10.1512/iumj.1973.22.22008. |
[8] |
M. Colin and L. Jeanjean,
Solutions for a quasilinear Schrödinger equation: A dual approach, Nonlinear Anal., 56 (2004), 213-226.
doi: 10.1016/j.na.2003.09.008. |
[9] |
R. W. Hasse,
A general method for the solution of nonlinear soliton and kink Schrödinger equations, Z. Phys. B, 37 (1980), 83-87.
doi: 10.1007/BF01325508. |
[10] |
Y. Jing, Z. Liu and Z.-Q. Wang, Multiple solutions of a parameter-dependent quasilinear elliptic equation, Calc. Var. Partial Differential Equations, 55 (2016), 150, 26 pp.
doi: 10.1007/s00526-016-1067-7. |
[11] |
Y. Jing, Z. Liu and Z.-Q. Wang,
Existence results for a singular quasilinear elliptic equation, J. Fixed Point Theory Appl., 19 (2017), 67-84.
doi: 10.1007/s11784-016-0341-9. |
[12] |
Y. Jing, Z. Liu and Z.-Q. Wang, Parameter-dependent multiplicity results of sign-changing solutions for quasilinear elliptic equations, preprint. |
[13] |
M. Kosevich, A. Ivanov and S. Kovalev,
Magnetic solutions, Phys. Rep., 194 (1990), 117-238.
|
[14] |
S. Kurihara,
Large-amplitude quasi-solitons in superfluid films, J. Phys. Soc. Jap., 50 (1981), 3801-3805.
doi: 10.1143/JPSJ.50.3801. |
[15] |
O. A. Ladyzhenskaya and N. N. Ural'tseva, Linear and Quasilinear Elliptic Equations, Academic Press, New York-London, 1968.
![]() ![]() |
[16] |
S. Li and Z.-Q. Wang,
Ljusternik-Schnirelman theory in partially ordered Hilbert spaces, Trans. Amer. Math. Soc., 354 (2002), 3207-3227.
doi: 10.1090/S0002-9947-02-03031-3. |
[17] |
G. M. Lieberman,
The natural generalizationj of the natural conditions of Ladyzhenskaya and Ural'tseva for elliptic equations, Commun. Partial Differential Equations, 16 (1991), 311-361.
doi: 10.1080/03605309108820761. |
[18] |
A. G. Litvak and A. M. Sergeev,
One dimensional collapse of plasma waves, JEPT Letters, 27 (1978), 517-520.
|
[19] |
J.-Q. Liu, X.-Q. Liu and Z.-Q. Wang,
Multiple sign-changing solutions for quasilinear elliptic equations via perturbation method, Commun. Partial Differential Equations, 39 (2014), 2216-2239.
doi: 10.1080/03605302.2014.942738. |
[20] |
J. Liu, X. Liu and Z.-Q. Wang,
Sign-changing solutions for coupled nonlinear Schrödinger equations with critical growth, J. Differential Equations, 261 (2016), 7194-7236.
doi: 10.1016/j.jde.2016.09.018. |
[21] |
J. Liu, X. Liu and Z.-Q. Wang,
Multiple mixed states of nodal solutions for nonlinear Schrödinger systems, Calc. Var. Partial Differential Equations, 52 (2015), 565-586.
doi: 10.1007/s00526-014-0724-y. |
[22] |
X.-Q. Liu, J.-Q. Liu and Z.-Q. Wang,
Quasilinear elliptic equations via perturbation method, Proc. Amer. Math. Soc., 141 (2013), 253-263.
doi: 10.1090/S0002-9939-2012-11293-6. |
[23] |
X.-Q. Liu, J.-Q. Liu and Z.-Q. Wang,
Quasilinear elliptic equations with critical growth via perturbation method, J. Differential Equations, 254 (2013), 102-124.
doi: 10.1016/j.jde.2012.09.006. |
[24] |
X. Liu, J. Liu and Z.-Q. Wang,
Localized nodal solutions for quasilinear Schrödinger equations, J. Differential Equations, 267 (2019), 7411-7461.
doi: 10.1016/j.jde.2019.08.003. |
[25] |
Z. Liu and J. Sun,
Invariant sets of descending flow in critical point theory with applications to nonlinear differential equations, J. Differential Equations, 172 (2001), 257-299.
doi: 10.1006/jdeq.2000.3867. |
[26] |
J. Liu and Z.-Q. Wang,
Soliton solutions for quasilinear Schrödinger equations, Proc. Amer. Math. Soc., 131 (2003), 441-448.
doi: 10.1090/S0002-9939-02-06783-7. |
[27] |
J.-Q. Liu, Y.-Q. Wang and Z.-Q. Wang,
Soliton solutions for quasilinear Schrödinger equation, II, J. Differential Equations, 187 (2003), 473-493.
doi: 10.1016/S0022-0396(02)00064-5. |
[28] |
J.-Q. Liu, Y.-Q. Wang and Z.-Q. Wang,
Solutions for quasilinear Schrödinger equations via the Nehari method, Commun. Partial Differential Equations, 29 (2004), 879-901.
doi: 10.1081/PDE-120037335. |
[29] |
X. Liu and J. Zhao,
$p$-Laplacian equation in $\mathbb{R}^N$ with finite potential via the truncation method, Adv. Nonlinear Stud., 17 (2017), 595-610.
doi: 10.1515/ans-2015-5059. |
[30] |
V. G. Makhan'kov and V. K. Fedyanin,
Nonlinear effects in quasi-one-dimensional models of condensed matter theory, Phys. Rep., 104 (1984), 1-86.
doi: 10.1016/0370-1573(84)90106-6. |
[31] |
M. Porkolab and M. V. Goldman,
Upper hybrid solitons and oscillating two-stream instabilities, Phys. Fluids, 19 (1976), 872-881.
doi: 10.1063/1.861553. |
[32] |
G. R. W. Quispel and H. W. Capel,
Equation of motion for the Heisenberg spin chain, Phys. A, 110 (1982), 41-80.
doi: 10.1016/0378-4371(82)90104-2. |
[33] |
P. H. Rabinowitz, Minimax Methods in Critical Point Theory with Applications to Differential Equations, CBMS Regional Conference, Series in Mathematics, American Mathematical Society, Vol. 65, 1986.
doi: 10.1090/cbms/065. |
[34] |
J. Zhao, X. Liu and J. Liu,
$p$-Laplacian equations in $\mathbb{R}^N$ with finite potential via truncation method, the critical case, J. Math. Anal. Appl., 455 (2017), 58-88.
doi: 10.1016/j.jmaa.2017.03.085. |
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