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Multiple solutions for a critical quasilinear equation with Hardy potential
Department of Mathematical Science, Tsinghua University, Beijing, China |
$\begin{array}{l}\left\{ {\begin{array}{*{20}{c}}{ - \sum\limits_{i,j = 1}^N {{D_j}} ({a_{ij}}(u){D_i}u) + \frac{1}{2}\sum\limits_{i,j = 1}^N {{{a'}_{ij}}} (u){D_i}u{D_j}u - \frac{\mu }{{|x{|^2}}}u = au + |u{|^{{2^ * } - 2}}u}&{{\rm{in}}\;\Omega ,}\\\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;{u = 0}&{{\rm{on}}\;\partial \Omega ,}\end{array}} \right.\\\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\left( {\rm{P}} \right)\end{array}$ |
$ 2^ * = \frac{2N}{N-2} $ |
$ H_0^1(\Omega) $ |
$ L^p(\Omega) $ |
$ a>0 $ |
$ \Omega\subset \mathbb{R}^N $ |
$ a_{ij} $ |
$ N\geq 7 $ |
$ \mu\in[0,\mu^*) $ |
$ \mu^* $ |
References:
[1] |
A. Ambrosetti and P. 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] |
A. Ambrosetti and Z. Q. Wang,
Positive solutions to a class of quasilinear elliptic equations on $\mathbb R$, Discrete Contin. Dyn. Syst., 9 (2003), 55-68.
doi: 10.3934/dcds.2003.9.55. |
[3] |
H. Berestycki and M. Esteban,
Existence and bifurcation of solutions for an elliptic degenerate problem, J. Differential Equations, 134 (1997), 1-25.
doi: 10.1006/jdeq.1996.3165. |
[4] |
H. Brezis and L. Nirenberg,
Positive solutions of nonlinear elliptic equations involving critical Sobolev exponents, Commun. Pure Appl. Math., 36 (1983), 437-477.
doi: 10.1002/cpa.3160360405. |
[5] |
D. Cao and P. Han,
Solutions to critical elliptic equations with multi-singular inverse sequare potentials, J. Differential Equations, 224 (2006), 332-372.
doi: 10.1016/j.jde.2005.07.010. |
[6] |
D. Cao, S. Peng and S. Yan,
Multiplicity of solutions for the plasma problem in two dimensions, Adv. Math., 225 (2010), 2741-2785.
doi: 10.1016/j.aim.2010.05.012. |
[7] |
D. Cao, S. Peng and S. Yan,
Infinitely many solutions for $p-$Laplacian equation involving critical Sobolev growth, J. Funct. Anal., 262 (2012), 2861-2902.
doi: 10.1016/j.jfa.2012.01.006. |
[8] |
D. Cao and S. Yan,
Infinitely many solutions for an elliptic problem involving critical Sobolev growth and Hardy potential, Calc. Var. Part. Diff. Equ., 38 (2010), 471-501.
doi: 10.1007/s00526-009-0295-5. |
[9] |
A. Capozzi, D. Fortunato and G. Palmieri,
An existence result for nonlinear elliptic problems involving critical Sobolev exponents, Ann. Inst. H. Poincare and Non Lineaire, 2 (1985), 463-470.
doi: 10.1016/S0294-1449(16)30395-X. |
[10] |
G. Cerami, S. Solimini and M. Struwe,
Some existence results for superlinear elliptic problem involving critical exponents, J. Funct. Anal., 69 (1986), 289-306.
doi: 10.1016/0022-1236(86)90094-7. |
[11] |
M. Colin and L. Jeanjean, Solutions for a quasilinear Schrödinger equation: A dual approach, Nonlinear. Anal. Theor. Meth. App., 56 (2004), 213-226.
doi: 10.1016/j.na.2003.09.008. |
[12] |
J. M. Coron,
Topologie et cas limite des injections de Sobolev (Topology and limit case of Sobolev embeddings), C. R. Acad. Sci. Paris Ser. I Math., 199 (1984), 209-212.
|
[13] |
A. de Bouard, N. Hayashi and J. C. Saut,
Global existence of small solutions to a relativistic nonlinear Schrödinger equation, Comm. Math. Phys., 189 (1997), 73-105.
doi: 10.1007/s002200050191. |
[14] |
Y. Deng, Y. Guo and J. Liu, Existence of solutions for quasilinear elliptic equations with Hardy potential, J. Math. Phys., 57 (2016), 031503, 15 pp.
doi: 10.1063/1.4944455. |
[15] |
G. Divillanova and S. Solimini,
Concentration estimates and multiple solutions to elliptic problems at critical growth, Adv. Differential Equations, 7 (2002), 1257-1280.
|
[16] |
J. P. García Azorero and I. Peral Alonso,
Hardy inequalities and some critical elliptic and parabolic problems, J. Differential Equations, 144 (1998), 441-476.
doi: 10.1006/jdeq.1997.3375. |
[17] |
Y. Guo, J. Liu and Z. Wang,
On a Brezis- Nirenburg type quasilinear problem, J. Fixed Point Theory Appl., 19 (2017), 719-753.
doi: 10.1007/s11784-016-0371-3. |
[18] |
T. Kilpeläinen and J. Malý,
The Winer test and potential estimates for quasilinear elliptic equations, Acta Math., 172 (1994), 137-161.
doi: 10.1007/BF02392793. |
[19] |
A. M. Kosevich, B. A. Ivanov and A. S. Kovalev,
Magnetic solitons in superfluid films, J. Phys. Soc. Japan., 50 (1981), 3262-3267.
|
[20] |
L. Leblond and J. Marc,
Electron capture by polar molecules, Phys. Rev., 153 (1967), 1-4.
|
[21] |
J. Liu, X. Liu and Z.-Q. Wang,
Multiple sign-changing solutions for quasilinear elliptic equations via perturbation method, Comm. PDE, 39 (2014), 2216-2239.
doi: 10.1080/03605302.2014.942738. |
[22] |
J. Liu and Z.-Q. Wang,
Soliton solutions for quasilinear Schr$\ddot{o}$dinger equation Ⅰ, Proc. Amer. Math. Soc., 131 (2003), 441-448.
doi: 10.1090/S0002-9939-02-06783-7. |
[23] |
J. Liu, Y. Wang and Z.-Q. Wang,
Soliton solutions for quasilinear Schr$\ddot{o}$dinger equation Ⅱ, J. Differential Equations, 187 (2003), 473-493.
doi: 10.1016/S0022-0396(02)00064-5. |
[24] |
J. Liu, Y. Wang and Z.-Q. Wang,
Solutions for quasilinear Schr$\ddot{o}$dinger equation via the Nehari method, Comm. PDE, 29 (2004), 879-901.
doi: 10.1081/PDE-120037335. |
[25] |
X. Liu, J. 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. |
[26] |
X. Liu, J. 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. |
[27] |
X. Liu, J. Liu and Z. Wang,
Quasilinear equations via elliptic regularization method, Adv. Non. Stu., 13 (2013), 517-531.
doi: 10.1515/ans-2013-0215. |
[28] |
M. Maris, Profile decomposition for sequences of Borel measures, https://arXiv.org/abs/1410.6125. |
[29] |
M. Poppenberg, K. Schmitt and Z.-Q. Wang,
On the existence of soliton solutions to quasilinear Schrödinger equations, Calc. Var. Part. Diff. Equ., 14 (2002), 329-344.
doi: 10.1007/s005260100105. |
[30] |
M. Struwe,
A global compactness result for elliptic boundary value problems involving limiting nonlinearities, M. Math Z., 187 (1984), 511-517.
doi: 10.1007/BF01174186. |
[31] |
C. Tintarev, Concentration analysis and cocompactness, Concentration Analysis and Applications to PDE, 117-141, Trends Math., Birkh$\ddot{a}$user/Springer, Basel, 2013.
doi: 10.1007/978-3-0348-0373-1_7. |
[32] |
C. Tintarev and K. H. Fineseler, Concentration Compactness, Functional Analytic Grounds and Applications, Imperial College Press, London, 2007.
doi: 10.1142/p456.![]() ![]() ![]() |
show all references
References:
[1] |
A. Ambrosetti and P. 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] |
A. Ambrosetti and Z. Q. Wang,
Positive solutions to a class of quasilinear elliptic equations on $\mathbb R$, Discrete Contin. Dyn. Syst., 9 (2003), 55-68.
doi: 10.3934/dcds.2003.9.55. |
[3] |
H. Berestycki and M. Esteban,
Existence and bifurcation of solutions for an elliptic degenerate problem, J. Differential Equations, 134 (1997), 1-25.
doi: 10.1006/jdeq.1996.3165. |
[4] |
H. Brezis and L. Nirenberg,
Positive solutions of nonlinear elliptic equations involving critical Sobolev exponents, Commun. Pure Appl. Math., 36 (1983), 437-477.
doi: 10.1002/cpa.3160360405. |
[5] |
D. Cao and P. Han,
Solutions to critical elliptic equations with multi-singular inverse sequare potentials, J. Differential Equations, 224 (2006), 332-372.
doi: 10.1016/j.jde.2005.07.010. |
[6] |
D. Cao, S. Peng and S. Yan,
Multiplicity of solutions for the plasma problem in two dimensions, Adv. Math., 225 (2010), 2741-2785.
doi: 10.1016/j.aim.2010.05.012. |
[7] |
D. Cao, S. Peng and S. Yan,
Infinitely many solutions for $p-$Laplacian equation involving critical Sobolev growth, J. Funct. Anal., 262 (2012), 2861-2902.
doi: 10.1016/j.jfa.2012.01.006. |
[8] |
D. Cao and S. Yan,
Infinitely many solutions for an elliptic problem involving critical Sobolev growth and Hardy potential, Calc. Var. Part. Diff. Equ., 38 (2010), 471-501.
doi: 10.1007/s00526-009-0295-5. |
[9] |
A. Capozzi, D. Fortunato and G. Palmieri,
An existence result for nonlinear elliptic problems involving critical Sobolev exponents, Ann. Inst. H. Poincare and Non Lineaire, 2 (1985), 463-470.
doi: 10.1016/S0294-1449(16)30395-X. |
[10] |
G. Cerami, S. Solimini and M. Struwe,
Some existence results for superlinear elliptic problem involving critical exponents, J. Funct. Anal., 69 (1986), 289-306.
doi: 10.1016/0022-1236(86)90094-7. |
[11] |
M. Colin and L. Jeanjean, Solutions for a quasilinear Schrödinger equation: A dual approach, Nonlinear. Anal. Theor. Meth. App., 56 (2004), 213-226.
doi: 10.1016/j.na.2003.09.008. |
[12] |
J. M. Coron,
Topologie et cas limite des injections de Sobolev (Topology and limit case of Sobolev embeddings), C. R. Acad. Sci. Paris Ser. I Math., 199 (1984), 209-212.
|
[13] |
A. de Bouard, N. Hayashi and J. C. Saut,
Global existence of small solutions to a relativistic nonlinear Schrödinger equation, Comm. Math. Phys., 189 (1997), 73-105.
doi: 10.1007/s002200050191. |
[14] |
Y. Deng, Y. Guo and J. Liu, Existence of solutions for quasilinear elliptic equations with Hardy potential, J. Math. Phys., 57 (2016), 031503, 15 pp.
doi: 10.1063/1.4944455. |
[15] |
G. Divillanova and S. Solimini,
Concentration estimates and multiple solutions to elliptic problems at critical growth, Adv. Differential Equations, 7 (2002), 1257-1280.
|
[16] |
J. P. García Azorero and I. Peral Alonso,
Hardy inequalities and some critical elliptic and parabolic problems, J. Differential Equations, 144 (1998), 441-476.
doi: 10.1006/jdeq.1997.3375. |
[17] |
Y. Guo, J. Liu and Z. Wang,
On a Brezis- Nirenburg type quasilinear problem, J. Fixed Point Theory Appl., 19 (2017), 719-753.
doi: 10.1007/s11784-016-0371-3. |
[18] |
T. Kilpeläinen and J. Malý,
The Winer test and potential estimates for quasilinear elliptic equations, Acta Math., 172 (1994), 137-161.
doi: 10.1007/BF02392793. |
[19] |
A. M. Kosevich, B. A. Ivanov and A. S. Kovalev,
Magnetic solitons in superfluid films, J. Phys. Soc. Japan., 50 (1981), 3262-3267.
|
[20] |
L. Leblond and J. Marc,
Electron capture by polar molecules, Phys. Rev., 153 (1967), 1-4.
|
[21] |
J. Liu, X. Liu and Z.-Q. Wang,
Multiple sign-changing solutions for quasilinear elliptic equations via perturbation method, Comm. PDE, 39 (2014), 2216-2239.
doi: 10.1080/03605302.2014.942738. |
[22] |
J. Liu and Z.-Q. Wang,
Soliton solutions for quasilinear Schr$\ddot{o}$dinger equation Ⅰ, Proc. Amer. Math. Soc., 131 (2003), 441-448.
doi: 10.1090/S0002-9939-02-06783-7. |
[23] |
J. Liu, Y. Wang and Z.-Q. Wang,
Soliton solutions for quasilinear Schr$\ddot{o}$dinger equation Ⅱ, J. Differential Equations, 187 (2003), 473-493.
doi: 10.1016/S0022-0396(02)00064-5. |
[24] |
J. Liu, Y. Wang and Z.-Q. Wang,
Solutions for quasilinear Schr$\ddot{o}$dinger equation via the Nehari method, Comm. PDE, 29 (2004), 879-901.
doi: 10.1081/PDE-120037335. |
[25] |
X. Liu, J. 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. |
[26] |
X. Liu, J. 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. |
[27] |
X. Liu, J. Liu and Z. Wang,
Quasilinear equations via elliptic regularization method, Adv. Non. Stu., 13 (2013), 517-531.
doi: 10.1515/ans-2013-0215. |
[28] |
M. Maris, Profile decomposition for sequences of Borel measures, https://arXiv.org/abs/1410.6125. |
[29] |
M. Poppenberg, K. Schmitt and Z.-Q. Wang,
On the existence of soliton solutions to quasilinear Schrödinger equations, Calc. Var. Part. Diff. Equ., 14 (2002), 329-344.
doi: 10.1007/s005260100105. |
[30] |
M. Struwe,
A global compactness result for elliptic boundary value problems involving limiting nonlinearities, M. Math Z., 187 (1984), 511-517.
doi: 10.1007/BF01174186. |
[31] |
C. Tintarev, Concentration analysis and cocompactness, Concentration Analysis and Applications to PDE, 117-141, Trends Math., Birkh$\ddot{a}$user/Springer, Basel, 2013.
doi: 10.1007/978-3-0348-0373-1_7. |
[32] |
C. Tintarev and K. H. Fineseler, Concentration Compactness, Functional Analytic Grounds and Applications, Imperial College Press, London, 2007.
doi: 10.1142/p456.![]() ![]() ![]() |
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