Multiple solutions for a critical quasilinear equation with Hardy potential

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November 2019, 12(7): 1977-2003. doi: 10.3934/dcdss.2019128

Multiple solutions for a critical quasilinear equation with Hardy potential

Fengshuang Gao ^1, and Yuxia Guo ^1,,

Department of Mathematical Science, Tsinghua University, Beijing, China

* Corresponding author

Received December 2017 Revised April 2018 Published December 2018

Fund Project: The authors are supported by NSFC grant 11771235, 11331010, 11571040.

In this paper, we investigate the following quasilinear equation involving a Hardy potential:

$\begin{array}{l}\left\{ {\begin{array}{*{20}{c}}{ - \sum\limits_{i,j = 1}^N {{D_j}} ({a_{ij}}(u){D_i}u) + \frac{1}{2}\sum\limits_{i,j = 1}^N {{{a'}_{ij}}} (u){D_i}u{D_j}u - \frac{\mu }{{|x{|^2}}}u = au + |u{|^{{2^ * } - 2}}u}&{{\rm{in}}\;\Omega ,}\\\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;{u = 0}&{{\rm{on}}\;\partial \Omega ,}\end{array}} \right.\\\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\left( {\rm{P}} \right)\end{array}$

where

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

is the Sobolev critical exponent for the embedding of

$ H_0^1(\Omega) $

into

$ L^p(\Omega) $

,

$ a>0 $

is a constant and

$ \Omega\subset \mathbb{R}^N $

is an open bounded domain which contains the origin. We will prove that under some suitable assumptions on

$ a_{ij} $

, when

$ N\geq 7 $

and

$ \mu\in[0,\mu^*) $

for some constant

$ \mu^* $

, problem (P) admits an unbounded sequence of solutions. To achieve this goal, we perform the subcritical approximation and the regularization perturbation.

Keywords: Quasilinear equations, critical exponents, Hardy potential, multiple bound state solutions.

Mathematics Subject Classification: 35B45, 35J25.

Citation: Fengshuang Gao, Yuxia Guo. Multiple solutions for a critical quasilinear equation with Hardy potential. Discrete and Continuous Dynamical Systems - S, 2019, 12 (7) : 1977-2003. doi: 10.3934/dcdss.2019128

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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