August  2022, 21(8): 2561-2586. doi: 10.3934/cpaa.2022060

Singular quasilinear critical Schrödinger equations in $ \mathbb {R}^N $

1. 

Department of Mathematics – University of Firenze, Viale Morgagni 40-44 – 50134 Firenze, Italy

2. 

Department of Mathematics – University of Perugia, Via Vanvitelli 1 – 06123 Perugia, Italy

*Corresponding author

Received  November 2021 Revised  January 2022 Published  August 2022 Early access  March 2022

Fund Project: The authors are members of the Gruppo Nazionale per l'Analisi Matematica, la Probabilità e le loro Applicazioni (GNAMPA) of the Istituto Nazionale di Alta Matematica (INdAM). RF was partly supported by Fondo Ricerca di Base di Ateneo Esercizio 2017-19 of the University of Perugia, named Problemi con non linearità dipendenti dal gradiente

We prove multiplicity results for solutions, both with positive and negative energy, for a class of singular quasilinear Schrödinger equations in the entire $ \mathbb {R}^N $ involving a critical term, nontrivial weights and positive parameters $ \lambda $, $ \beta $, covering several physical models, coming from plasma physics as well as high-power ultra short laser in matter. Also the symmetric setting is investigated. Our proofs relay on variational tools, including concentration compactness principles because of the delicate situation of the double lack of compactness. In addition, a necessary reformulation of the original problem in a suitable variational setting, produces a singular function, delicate to be managed.

Citation: Laura Baldelli, Roberta Filippucci. Singular quasilinear critical Schrödinger equations in $ \mathbb {R}^N $. Communications on Pure and Applied Analysis, 2022, 21 (8) : 2561-2586. doi: 10.3934/cpaa.2022060
References:
[1]

S. AdachiM. Shibata and T. Watanabe, Blow-up phenomena and asymptotic profiles of ground states of quasilinear elliptic equations with $H^1$-supercritical nonlinearities, J. Differ. Equ., 256 (2014), 1492-1514.  doi: 10.1016/j.jde.2013.11.004.

[2]

S. Adachi and T. Watanabe, $G$-invariant positive solutions for a quasilinear Schrödinger equation, Adv. Differ. Equ., 16 (2011), 289-324. 

[3]

C. O. AlvesG. M. Figueiredo and U. B. Severo, Multiplicity of positive solutions for a class of quasilinear problems, Adv. Differ. Equ., 14 (2009), 911-942. 

[4]

L. Baldelli, Y. Brizi and R. Filippucci, Multiplicity results for $(p, q)$-Laplacian equations with critical exponent in $\mathbb {R}^N$ and negative energy, Calc. Var. Partial Differ. Equ., 60 (2021), 30 pp. doi: 10.1007/s00526-020-01867-6.

[5]

L. Baldelli, Y. Brizi and R. Filippucci, On symmetric solutions for ($p, q$)-Laplacian equations in $\mathbb {R}^N$ with critical terms, J. Geom. Anal., 32 (2022), 25 pp. doi: 10.1007/s12220-021-00846-3.

[6]

A.K. Ben-NaoumC. Troestler and M. Willem, Extrema problems with critical Sobolev exponents on unbounded domains, Nonlinear Anal., 26 (1996), 823-833.  doi: 10.1016/0362-546X(94)00324-B.

[7]

H. Brezis and E. Lieb, A relation between pointwise convergence of functions and convergence of functionals, Proc. Amer. Math. Soc., 88 (1983), 486-490.  doi: 10.2307/2044999.

[8]

J. Chabrowski, Concentration-compactness principle at infinity and semilinear elliptic equations involving critical and subcritical Sobolev exponents, Calc. Var. Partial Differ. Equ., 3 (1995), 493-512.  doi: 10.1007/BF01187898.

[9]

M. Colin and L. Jeanjean, Solutions for a quasilinear Schrödinger equations: a dual approach, Nonlinear Anal., 56 (2004), 213-226.  doi: 10.1016/j.na.2003.09.008.

[10]

Y. B. DengS. J. Peng and S. S. Yan, Critical exponents and solitary wave solutions for generalized quasilinear Schrödinger equations, J. Differ. Equ., 260 (2016), 1228-1262.  doi: 10.1016/j.jde.2015.09.021.

[11]

R. FilippucciP. Pucci and F. Robert, On a $p$-Laplace equation with multiple critical nonlinearities, J. Math. Pures Appl., 91 (2009), 156-177.  doi: 10.1016/j.matpur.2008.09.008.

[12]

J. Garcia Azorero and I. Peral Alonso, Multiplicity of solutions for elliptic problems with critical exponent or with a nonsymmetric term, Trans. Amer. Math. Soc., 323 (1991), 877-895.  doi: 10.2307/2001562.

[13]

M. Ghergu and G. Singh, On a class of mixed Choquard-Schrödinger-Poisson system, Discrete Contin. Dyn. Syst. S, 12 (2019), 297-309.  doi: 10.3934/dcdss.2019021.

[14]

Y. He, X. Luo and V. D. Rădulescu, Nodal multi-peak standing waves of fourth-order Schrödinger equations with mixed dispersion, J. Geom. Anal., 32 (2022), 36 pp. doi: 10.1007/s12220-021-00795-x.

[15]

M. Ishiwata and M. $\hat{O}$tani, Concentration compactness principle at infinity with partial symmetry and its application, Nonlinear Anal., 51 (2002), 391-407.  doi: 10.1016/S0362-546X(01)00836-7.

[16]

L. Jeanjean and K. Tanaka, A positive solution for a nonlinear Schrödinger equation on $\mathbb {R}^N$, Indiana Univ. Math. J., 54 (2005), 443-464.  doi: 10.1512/iumj.2005.54.2502.

[17]

M. A. Krasnosel'skii, Topological Methods in the Theory of Nonlinear Integral Equations, A Pergamon Press Book, The Macmillan Co., New York, 1964.

[18]

Z. Li and Y. Wang, Solutions to singular quasilinear elliptic equations on bounded domains, Electron. J. Differ. Equ., 2018 (2018), 1-12. 

[19]

P. L. Lions, The concentration-compacteness principle in the calculus of variations. The locally compact case. Ⅰ, Ann. Inst. H. Poincaré Anal. Non Linéaire, 1 (1984), 109-145. 

[20]

P. L. Lions, The concentration-compacteness principle in the calculus of variations. The limit case. Ⅰ, Rev. Mat. Iberoam., 1 (1985), 145-201.  doi: 10.4171/RMI/6.

[21]

J. Q. LiuY. Q. Wang and Z. Q. Wang, Soliton solutions for quasilinear Schrödinger equations Ⅱ, J. Differ. Equ., 187 (2003), 473-493.  doi: 10.1016/S0022-0396(02)00064-5.

[22]

R. S. Palais, The principle of symmetric criticality, Commun. Math. Phys., 69 (1979), 19-30. 

[23]

N. S. Papageorgiou, V. D. Rădulescu and D. D. Repovs, Nonlinear Analysis-Theory and Methods, Springer Monographs in Mathematics, Springer, Cham, 2019. doi: 10.1007/978-3-030-03430-6.

[24]

U. Severo, Existence of weak solutions for quasilinear elliptic equations involving the $p$-laplacian, Electron. J. Differ. Equ., 2008 (2008), 1-16. 

[25]

Y. T. Shen and Y. J. Wang, A class of generalized quasilinear Schrödinger equations, Commun. Pure Appl. Anal., 15 (2016), 853-870.  doi: 10.3934/cpaa.2016.15.853.

[26]

A. Elvesde B. Silva and Gilberto F. Vieira, Quasilinear asymptotically periodic Schrödinger equations with critical growth, Calc. Var. and Partial Differ. Equ., 39 (2010), 1-33.  doi: 10.1007/s00526-009-0299-1.

[27]

Y. Wang, Multiplicity of solutions for singular quasilinear Schrödinger equations with critical exponents, J. Math. Anal. Appl., 458 (2018), 1027-1043.  doi: 10.1016/j.jmaa.2017.10.015.

[28]

Y. WangZ. Li and A. A. Abdelgadir, On singular quasilinear Schrödinger equations with critical exponents, Math. Methods Appl. Sci., 40 (2017), 5095-5108.  doi: 10.1002/mma.4373.

[29]

Y. WangY. Zhang and Y. Shen, Multiple solutions for quasilinear Schrödinger equations involving critical exponent, Appl. Math. Comput., 216 (2010), 849-856.  doi: 10.1016/j.amc.2010.01.091.

[30]

Y. J. Wang and W. M. Zou, Bound states to critical quasilinear Schrödinger equations, Nonlinear Differ. Equ. Appl., 19 (2012), 19-47.  doi: 10.1007/s00030-011-0116-3.

[31]

L. Wen, S. Chen and V. D. Rădulescu, Axially symmetric solutions of the Schrödinger-Poisson system with zero mass potential in $\mathbb {R}^2$, Appl. Math. Lett., 104 (2020), 7 pp. doi: 10.1016/j.aml.2020.106244.

[32]

M. Willem, Minimax Theorems, Birkhäuser Boston, Inc., Boston, MA, 1996. doi: 10.1007/978-1-4612-4146-1.

[33]

J. YangY. J. Wang and A. A. Abdelgadir, Soliton solutions for quasilinear Schrödinger equations, J. Math. Phys., 54 (2013), 1-19.  doi: 10.1063/1.4811394.

show all references

References:
[1]

S. AdachiM. Shibata and T. Watanabe, Blow-up phenomena and asymptotic profiles of ground states of quasilinear elliptic equations with $H^1$-supercritical nonlinearities, J. Differ. Equ., 256 (2014), 1492-1514.  doi: 10.1016/j.jde.2013.11.004.

[2]

S. Adachi and T. Watanabe, $G$-invariant positive solutions for a quasilinear Schrödinger equation, Adv. Differ. Equ., 16 (2011), 289-324. 

[3]

C. O. AlvesG. M. Figueiredo and U. B. Severo, Multiplicity of positive solutions for a class of quasilinear problems, Adv. Differ. Equ., 14 (2009), 911-942. 

[4]

L. Baldelli, Y. Brizi and R. Filippucci, Multiplicity results for $(p, q)$-Laplacian equations with critical exponent in $\mathbb {R}^N$ and negative energy, Calc. Var. Partial Differ. Equ., 60 (2021), 30 pp. doi: 10.1007/s00526-020-01867-6.

[5]

L. Baldelli, Y. Brizi and R. Filippucci, On symmetric solutions for ($p, q$)-Laplacian equations in $\mathbb {R}^N$ with critical terms, J. Geom. Anal., 32 (2022), 25 pp. doi: 10.1007/s12220-021-00846-3.

[6]

A.K. Ben-NaoumC. Troestler and M. Willem, Extrema problems with critical Sobolev exponents on unbounded domains, Nonlinear Anal., 26 (1996), 823-833.  doi: 10.1016/0362-546X(94)00324-B.

[7]

H. Brezis and E. Lieb, A relation between pointwise convergence of functions and convergence of functionals, Proc. Amer. Math. Soc., 88 (1983), 486-490.  doi: 10.2307/2044999.

[8]

J. Chabrowski, Concentration-compactness principle at infinity and semilinear elliptic equations involving critical and subcritical Sobolev exponents, Calc. Var. Partial Differ. Equ., 3 (1995), 493-512.  doi: 10.1007/BF01187898.

[9]

M. Colin and L. Jeanjean, Solutions for a quasilinear Schrödinger equations: a dual approach, Nonlinear Anal., 56 (2004), 213-226.  doi: 10.1016/j.na.2003.09.008.

[10]

Y. B. DengS. J. Peng and S. S. Yan, Critical exponents and solitary wave solutions for generalized quasilinear Schrödinger equations, J. Differ. Equ., 260 (2016), 1228-1262.  doi: 10.1016/j.jde.2015.09.021.

[11]

R. FilippucciP. Pucci and F. Robert, On a $p$-Laplace equation with multiple critical nonlinearities, J. Math. Pures Appl., 91 (2009), 156-177.  doi: 10.1016/j.matpur.2008.09.008.

[12]

J. Garcia Azorero and I. Peral Alonso, Multiplicity of solutions for elliptic problems with critical exponent or with a nonsymmetric term, Trans. Amer. Math. Soc., 323 (1991), 877-895.  doi: 10.2307/2001562.

[13]

M. Ghergu and G. Singh, On a class of mixed Choquard-Schrödinger-Poisson system, Discrete Contin. Dyn. Syst. S, 12 (2019), 297-309.  doi: 10.3934/dcdss.2019021.

[14]

Y. He, X. Luo and V. D. Rădulescu, Nodal multi-peak standing waves of fourth-order Schrödinger equations with mixed dispersion, J. Geom. Anal., 32 (2022), 36 pp. doi: 10.1007/s12220-021-00795-x.

[15]

M. Ishiwata and M. $\hat{O}$tani, Concentration compactness principle at infinity with partial symmetry and its application, Nonlinear Anal., 51 (2002), 391-407.  doi: 10.1016/S0362-546X(01)00836-7.

[16]

L. Jeanjean and K. Tanaka, A positive solution for a nonlinear Schrödinger equation on $\mathbb {R}^N$, Indiana Univ. Math. J., 54 (2005), 443-464.  doi: 10.1512/iumj.2005.54.2502.

[17]

M. A. Krasnosel'skii, Topological Methods in the Theory of Nonlinear Integral Equations, A Pergamon Press Book, The Macmillan Co., New York, 1964.

[18]

Z. Li and Y. Wang, Solutions to singular quasilinear elliptic equations on bounded domains, Electron. J. Differ. Equ., 2018 (2018), 1-12. 

[19]

P. L. Lions, The concentration-compacteness principle in the calculus of variations. The locally compact case. Ⅰ, Ann. Inst. H. Poincaré Anal. Non Linéaire, 1 (1984), 109-145. 

[20]

P. L. Lions, The concentration-compacteness principle in the calculus of variations. The limit case. Ⅰ, Rev. Mat. Iberoam., 1 (1985), 145-201.  doi: 10.4171/RMI/6.

[21]

J. Q. LiuY. Q. Wang and Z. Q. Wang, Soliton solutions for quasilinear Schrödinger equations Ⅱ, J. Differ. Equ., 187 (2003), 473-493.  doi: 10.1016/S0022-0396(02)00064-5.

[22]

R. S. Palais, The principle of symmetric criticality, Commun. Math. Phys., 69 (1979), 19-30. 

[23]

N. S. Papageorgiou, V. D. Rădulescu and D. D. Repovs, Nonlinear Analysis-Theory and Methods, Springer Monographs in Mathematics, Springer, Cham, 2019. doi: 10.1007/978-3-030-03430-6.

[24]

U. Severo, Existence of weak solutions for quasilinear elliptic equations involving the $p$-laplacian, Electron. J. Differ. Equ., 2008 (2008), 1-16. 

[25]

Y. T. Shen and Y. J. Wang, A class of generalized quasilinear Schrödinger equations, Commun. Pure Appl. Anal., 15 (2016), 853-870.  doi: 10.3934/cpaa.2016.15.853.

[26]

A. Elvesde B. Silva and Gilberto F. Vieira, Quasilinear asymptotically periodic Schrödinger equations with critical growth, Calc. Var. and Partial Differ. Equ., 39 (2010), 1-33.  doi: 10.1007/s00526-009-0299-1.

[27]

Y. Wang, Multiplicity of solutions for singular quasilinear Schrödinger equations with critical exponents, J. Math. Anal. Appl., 458 (2018), 1027-1043.  doi: 10.1016/j.jmaa.2017.10.015.

[28]

Y. WangZ. Li and A. A. Abdelgadir, On singular quasilinear Schrödinger equations with critical exponents, Math. Methods Appl. Sci., 40 (2017), 5095-5108.  doi: 10.1002/mma.4373.

[29]

Y. WangY. Zhang and Y. Shen, Multiple solutions for quasilinear Schrödinger equations involving critical exponent, Appl. Math. Comput., 216 (2010), 849-856.  doi: 10.1016/j.amc.2010.01.091.

[30]

Y. J. Wang and W. M. Zou, Bound states to critical quasilinear Schrödinger equations, Nonlinear Differ. Equ. Appl., 19 (2012), 19-47.  doi: 10.1007/s00030-011-0116-3.

[31]

L. Wen, S. Chen and V. D. Rădulescu, Axially symmetric solutions of the Schrödinger-Poisson system with zero mass potential in $\mathbb {R}^2$, Appl. Math. Lett., 104 (2020), 7 pp. doi: 10.1016/j.aml.2020.106244.

[32]

M. Willem, Minimax Theorems, Birkhäuser Boston, Inc., Boston, MA, 1996. doi: 10.1007/978-1-4612-4146-1.

[33]

J. YangY. J. Wang and A. A. Abdelgadir, Soliton solutions for quasilinear Schrödinger equations, J. Math. Phys., 54 (2013), 1-19.  doi: 10.1063/1.4811394.

Figure 1.  $ h(t) $
Figure 2.  $ \overline{h}(t) $
Figure 3.  $ h(t) $
Figure 4.  $ \overline{h}(t) $
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