April  2022, 21(4): 1329-1342. doi: 10.3934/cpaa.2022020

On the critical Schrödinger-Poisson system with $ p $-Laplacian

School of Mathematical Sciences, Capital Normal University, Beijing 100048, China

*Corresponding author

Received  July 2021 Revised  December 2021 Published  April 2022 Early access  January 2022

Fund Project: Supported by KZ202010028048 and NSFC(11771302, 12171326)

In this paper we consider the critical quasilinear Schrödinger-Poisson system
$ \begin{eqnarray*} \left \{\begin{array}{ll} -\Delta_p u+|u|^{p-2}u+\mu\phi |u|^{p-2}u = \lambda|u|^{q-2}u+|u|^{p^*-2}u,&\mathrm{in} \ \mathbb{R}^3,\\ -\Delta \phi = |u|^p, &\mathrm{in}\ \mathbb{R}^3, \end{array} \right. \end{eqnarray*} $
where
$ \frac{3}{2}<p<3 $
,
$ \Delta_p u = \hbox{div}(|\nabla u|^{p-2}\nabla u) $
,
$ p<q<p^*: = \frac{3p}{3-p} $
and
$ \mu,\lambda>0 $
. Based upon the variational approach, the ground state solutions and the nontrivial solutions are obtained depending on the parameters
$ q $
,
$ \mu $
and
$ \lambda $
.
Citation: Yao Du, Jiabao Su, Cong Wang. On the critical Schrödinger-Poisson system with $ p $-Laplacian. Communications on Pure and Applied Analysis, 2022, 21 (4) : 1329-1342. doi: 10.3934/cpaa.2022020
References:
[1]

A. Ambrosetti, On Schrödinger-Poisson Systems, Milan J. Math., 76 (2008), 257-274.  doi: 10.1007/s00032-008-0094-z.

[2]

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.

[3]

A. Ambrosetti and R. Ruiz, Multiple bound states for the Schrödinger-Poisson problem, Commun. Contemp. Math., 10 (2008), 391-404.  doi: 10.1142/S021919970800282X.

[4]

A. Azzollini and A. Pomponio, Ground state solutions for the nonlinear Schrödinger-Maxwell equations, J. Math. Anal. Appl., 345 (2008), 90-108.  doi: 10.1016/j.jmaa.2008.03.057.

[5]

V. Benci and D. Fortunato, An eigenvalue problem for the Schrödinger-Maxwell equations, Topol. Methods Nonlinear Anal., 11 (1998), 283-293.  doi: 10.12775/TMNA.1998.019.

[6]

J. Benedikt, P. Girg, L. Kotrla and P. Takáč, Origin of the $p$-Laplacian and A. Missbach, Electron. J. Differ. Equ., (2018), 17 pp.

[7]

L. Boccardo and F. Murat, Almost everywhere convergence of the gradients of solutions to elliptic and parabolic equations, Nonlinear Anal., 19 (1992), 581-597.  doi: 10.1016/0362-546X(92)90023-8.

[8]

H. Brézis 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.

[9]

T. D'Aprile and D. Mugnai, Solitary waves for nonlinear Klein-Gordon-Maxwell and Schrödinger-Maxwell equations, Proc. Roy. Soc. Edinburgh Sect. A, 134 (2004), 893-906.  doi: 10.1017/S030821050000353X.

[10]

T. D'Aprile and D. Mugnai, Non-existence results for the coupled Klein-Gordon-Maxwell equations, Adv. Nonlinear Stud., 4 (2004), 307-322.  doi: 10.1515/ans-2004-0305.

[11]

Y. Du, J. Su and C. Wang, The Schr$\ddot{\mathrm{o}}$dinger-Poisson system with $p$-Laplacian, Appl. Math. Lett., 120 (2021), 107286, 7 pp. doi: 10.1016/j.aml.2021.107286.

[12]

Y. Du, J. Su and C. Wang, On a quasilinear Schr$\ddot{\mathrm{o}}$dinger-Poisson system, J. Math. Anal. Appl., 505 (2022), 125446, 14 pp. doi: 10.1016/j.jmaa.2021.125446.

[13]

L. Jeanjean, Existence of solutions with prescribed norm for semilinear elliptic equations, Nonlinear Anal., 28 (1997), 1633-1659.  doi: 10.1016/S0362-546X(96)00021-1.

[14]

L. Jeanjean and S. Le Coz, An existence and stability result for standing waves of nonlinear Schrödinger equations, Adv. Differential Equations, 11 (2006), 813-840. 

[15]

E.H. Lieb and M. Loss, Analysis, American Mathematical Society, 2001. doi: 10.1090/gsm/014.

[16]

Z. Liu and S. Guo, On ground state solutions for the Schrödinger-Poisson equations with critical growth, J. Math. Anal. Appl., 412 (2014), 435-448.  doi: 10.1016/j.jmaa.2013.10.066.

[17]

Z. LiuZ. Zhang and S. Huang, Existence and nonexistence of positive solutions for a static Schrödinger-Poisson-Slater equation, J. Differential Equations, 266 (2019), 5912-5941.  doi: 10.1016/j.jde.2018.10.048.

[18]

D. Ruiz, The Schrödinger-Poisson equation under the effect of a nonlinear local term, J. Funct. Anal., 237 (2006), 655-674.  doi: 10.1016/j.jfa.2006.04.005.

[19]

G. Talenti, Best constant in Sobolev inequality, Ann. Mat. Pura Appl., 110 (1976), 353-372.  doi: 10.1007/BF02418013.

[20]

C. Wang and J. Su, The ground states of Hénon equations for $p$-Laplacian in $\mathbb{R}^N$ involving upper weighted critical exponents, preprint.

[21]

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

[22]

M. Willem, Functional analysis. Fundamentals and applications, Birkhäuser/Springer, New York, 2013. doi: 10.1007/978-1-4614-7004-5.

[23]

L. Zhao and F. Zhao, Positive solutions for Schrödinger-Poisson equations with a critical exponent, Nonlinear Anal., 70 (2009), 2150-2164.  doi: 10.1016/j.na.2008.02.116.

[24]

X. J. Zhong and C. L. Tang, Ground state sign-changing solutions for a Schrödinger-Poisson system with a critical nonlinearity in $\mathbb{R}^3$, Nonlinear Anal. Real World Appl., 39 (2018), 166-184.  doi: 10.1016/j.nonrwa.2017.06.014.

show all references

References:
[1]

A. Ambrosetti, On Schrödinger-Poisson Systems, Milan J. Math., 76 (2008), 257-274.  doi: 10.1007/s00032-008-0094-z.

[2]

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.

[3]

A. Ambrosetti and R. Ruiz, Multiple bound states for the Schrödinger-Poisson problem, Commun. Contemp. Math., 10 (2008), 391-404.  doi: 10.1142/S021919970800282X.

[4]

A. Azzollini and A. Pomponio, Ground state solutions for the nonlinear Schrödinger-Maxwell equations, J. Math. Anal. Appl., 345 (2008), 90-108.  doi: 10.1016/j.jmaa.2008.03.057.

[5]

V. Benci and D. Fortunato, An eigenvalue problem for the Schrödinger-Maxwell equations, Topol. Methods Nonlinear Anal., 11 (1998), 283-293.  doi: 10.12775/TMNA.1998.019.

[6]

J. Benedikt, P. Girg, L. Kotrla and P. Takáč, Origin of the $p$-Laplacian and A. Missbach, Electron. J. Differ. Equ., (2018), 17 pp.

[7]

L. Boccardo and F. Murat, Almost everywhere convergence of the gradients of solutions to elliptic and parabolic equations, Nonlinear Anal., 19 (1992), 581-597.  doi: 10.1016/0362-546X(92)90023-8.

[8]

H. Brézis 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.

[9]

T. D'Aprile and D. Mugnai, Solitary waves for nonlinear Klein-Gordon-Maxwell and Schrödinger-Maxwell equations, Proc. Roy. Soc. Edinburgh Sect. A, 134 (2004), 893-906.  doi: 10.1017/S030821050000353X.

[10]

T. D'Aprile and D. Mugnai, Non-existence results for the coupled Klein-Gordon-Maxwell equations, Adv. Nonlinear Stud., 4 (2004), 307-322.  doi: 10.1515/ans-2004-0305.

[11]

Y. Du, J. Su and C. Wang, The Schr$\ddot{\mathrm{o}}$dinger-Poisson system with $p$-Laplacian, Appl. Math. Lett., 120 (2021), 107286, 7 pp. doi: 10.1016/j.aml.2021.107286.

[12]

Y. Du, J. Su and C. Wang, On a quasilinear Schr$\ddot{\mathrm{o}}$dinger-Poisson system, J. Math. Anal. Appl., 505 (2022), 125446, 14 pp. doi: 10.1016/j.jmaa.2021.125446.

[13]

L. Jeanjean, Existence of solutions with prescribed norm for semilinear elliptic equations, Nonlinear Anal., 28 (1997), 1633-1659.  doi: 10.1016/S0362-546X(96)00021-1.

[14]

L. Jeanjean and S. Le Coz, An existence and stability result for standing waves of nonlinear Schrödinger equations, Adv. Differential Equations, 11 (2006), 813-840. 

[15]

E.H. Lieb and M. Loss, Analysis, American Mathematical Society, 2001. doi: 10.1090/gsm/014.

[16]

Z. Liu and S. Guo, On ground state solutions for the Schrödinger-Poisson equations with critical growth, J. Math. Anal. Appl., 412 (2014), 435-448.  doi: 10.1016/j.jmaa.2013.10.066.

[17]

Z. LiuZ. Zhang and S. Huang, Existence and nonexistence of positive solutions for a static Schrödinger-Poisson-Slater equation, J. Differential Equations, 266 (2019), 5912-5941.  doi: 10.1016/j.jde.2018.10.048.

[18]

D. Ruiz, The Schrödinger-Poisson equation under the effect of a nonlinear local term, J. Funct. Anal., 237 (2006), 655-674.  doi: 10.1016/j.jfa.2006.04.005.

[19]

G. Talenti, Best constant in Sobolev inequality, Ann. Mat. Pura Appl., 110 (1976), 353-372.  doi: 10.1007/BF02418013.

[20]

C. Wang and J. Su, The ground states of Hénon equations for $p$-Laplacian in $\mathbb{R}^N$ involving upper weighted critical exponents, preprint.

[21]

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

[22]

M. Willem, Functional analysis. Fundamentals and applications, Birkhäuser/Springer, New York, 2013. doi: 10.1007/978-1-4614-7004-5.

[23]

L. Zhao and F. Zhao, Positive solutions for Schrödinger-Poisson equations with a critical exponent, Nonlinear Anal., 70 (2009), 2150-2164.  doi: 10.1016/j.na.2008.02.116.

[24]

X. J. Zhong and C. L. Tang, Ground state sign-changing solutions for a Schrödinger-Poisson system with a critical nonlinearity in $\mathbb{R}^3$, Nonlinear Anal. Real World Appl., 39 (2018), 166-184.  doi: 10.1016/j.nonrwa.2017.06.014.

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