In this paper, we study the quasilinear Schrödinger-Poisson system with critical Sobolev exponent $$\begin{aligned} \begin{cases} -\Delta_{p} u+|u|^{p-2}u+l(x)\phi |u|^{p-2}u=|u|^{p^{*}-2}u+\mu h(x)|u|^{q-2}u & \ \ \ \mathrm{in}\ \mathbb{R}^{3},\\ -\Delta \phi=l(x)|u|^{p} &\ \ \ \mathrm{in}\ \mathbb{R}^{3}, \end{cases} \end{aligned}$$ where $\mu >0$, $\frac{3}{2}<p<3$, $p\leqslant q<p^{*}=\frac{3p}{3-p}$ and $\Delta_{p} u= \hbox{div}(|\nabla u|^{p-2}\nabla u)$. Under certain assumptions on the functions $l$ and $h$, we employ the mountain pass theorem to establish the existence of positive solutions for this system.
Lanxin HUANG
,
Jiabao SU
. A POSITIVE SOLUTION FOR QUASILINEAR SCHRÖODINGER-POISSON SYSTEM WITH CRITICAL EXPONENT[J]. Acta mathematica scientia, Series B, 2025
, 45(4)
: 1247
-1264
.
DOI: 10.1007/s10473-025-0402-5
[1] Ambrosetti A. On Schrödinger-Poisson systems. Milan J Math, 2008, 76(1): 257-274
[2] Azzollini A. Concentration and compactness in nonlinear Schrödinger-Poisson system with a general nonlinearity. J Differential Equations, 2010, 249(7): 1746-1763
[3] Azzollini A, d'Avenia P, Pomponio A. On the Schrödinger-Maxwell equations under the effect of a general nonlinear term. Ann Inst H Poincaré Anal Non Linéaire, 2010, 27(2): 779-791
[4] Azzollini A, Pomponio A. Ground state solutions for the nonlinear Schrödinger-Maxwell equations. J Math Anal Appl, 2008, 345(1): 90-108
[5] Benci V, Fortunato D. An eigenvalue problem for the Schrödinger-Maxwell equations. Topol Methods Nonlinear Anal, 1998, 11(2): 283-293
[6] Benci V, Fortunato D. Solitary waves of the nonlinear Klein-Gordon equation coupled with the Maxwell equations. Rev Math Phys, 2002, 14(4): 409-420
[7] Boccardo L, Murat F. Almost everywhere convergence of the gradients of solutions to elliptic and parabolic equations. Nonlinear Anal, 1992, 19(6): 581-597
[8] Brézis H, Nirenberg L. Positive solutions of nonlinear elliptic equations involving critical Sobolev exponents. Comm Pure Appl Math, 1983, 36(4): 437-477
[9] Cerami G, Vaira G. Positive solutions for some non-autonomous Schrödinger-Poisson systems. J Differential Equations, 2010, 248(3): 521-543
[10] D'Aprile T, Mugnai D. Non-existence results for the coupled Klein-Gordon-Maxwell equations. Adv Nonlinear Stud, 2004, 4(3): 307-322
[11] D'Aprile T, Mugnai D. Solitary waves for nonlinear Klein-Gordon-Maxwell and Schrödinger-Maxwell equations. Proc Roy Soc Edinburgh Sect A, 2004, 134(5): 893-906
[12] d'Avenia P. Non-radially symmetric solutions of nonlinear Schrödinger equation coupled with Maxwell equations. Adv Nonlinear Stud, 2002, 2(2): 177-192
[13] d'Avenia P, Pomponio A, Vaira G. Infinitely many positive solutions for a Schrödinger-Poisson system. Nonlinear Anal, 2011, 74(16): 5705-5721
[14] Deng Y B, Shuai W, Yang X L. Sign-changing solutions for the nonlinear Schrödinger-Poisson system with critical growth. Acta Math Sci, 2023, 43B(5): 2291-2308
[15] Du Y, Su J, Wang C. On the critical Schrödinger-Poisson system with $p$-Laplacian. Commun Pure Appl Anal, 2022, 21(4): 1329-1342
[16] Du Y, Tang C L. Ground state solutions for a Schrödinger-Poisson system with unconventional potential. Acta Math Sci, 2020, 40(4): 934-944
[17] Furtado M F, Wang Y, Zhang Z H. Positive and nodal ground state solutions for a critical Schrödinger-Poisson system with indefinite potentials. J Math Anal Appl, 2023, 526(2): Art 127252
[18] Gilbarg D, Trudinger N.Elliptic Partial Differential Equations of Second Order. Berlin: Springer, 1998
[19] Huang L R, Rocha E M. A positive solution of a Schrödinger-Poisson system with critical exponent. Commun Math Anal, 2013, 15(1): 29-43
[20] Huang L R, Rocha E M, Chen J Q. Positive and sign-changing solutions of a Schrödinger-Poisson system involving a critical nonlinearity. J Math Anal Appl, 2013, 408(1): 55-69
[21] Kang J C, Liu X Q, Tang C L. Ground state sign-changing solutions for critical Schrödinger-Poisson system with steep potential well. J Geom Anal, 2023, 33(2): Art 59
[22] Lieb E H, Loss M. Analysis. Providence, RI: American Mathematical Society, 2001
[23] Liu Z S, Guo S J. On ground state solutions for the Schrödinger-Poisson equations with critical growth. J Math Anal Appl, 2014, 412(1): 435-448
[24] Qian A X, Liu J M, Mao A M. Ground state and nodal solutions for a Schrödinger-Poisson equation with critical growth. J Math Phys, 2018, 59: Art 121509
[25] Ruiz D. The Schrödinger-Poisson equation under the effect of a nonlinear local term. J Funct Anal, 2006, 237(2): 655-674
[26] Su J, Wang Z-Q. Sobolev type embedding and quasilinear elliptic equations with radial potentials. J Differential Equations, 2011, 250(1): 223-242
[27] Sun J T, Chen H B, Nieto J J. On ground state solutions for some non-autonomous Schrödinger-Poisson systems. J Differential Equations, 2012, 252(5): 3365-3380
[28] Talenti G. Best constant in Sobolev inequality. Ann Mat Pura Appl, 1976, 110: 353-372
[29] Wang Z P, Zhou H S. Sign-changing solutions for the nonlinear Schrödinger-Poisson system in $\mathbb{R}^3$. Calc Var Partial Differential Equations, 2015, 52: 927-943
[30] Willem M.Functional Analysis: Fundamentals and Applications. New York: Birkhäuser, 2013
[31] Willem M. Minimax Theorems. Boston: Birkhäuser, 1996
[32] Zhang J. On ground state and nodal solutions of Schrödinger-Poisson equations with critical growth. J Math Anal Appl, 2015, 428(1): 387-404
[33] Zhang J. On the Schrödinger-Poisson equations with a general nonlinearity in the critical growth. Nonlinear Anal, 2012, 75(18): 6391-6401
[34] Zhao L G, Zhao F K. Positive solutions for Schrödinger-Poisson equations with a critical exponent. Nonlinear Anal, 2009, 70(6): 2150-2164
[35] Zhong X J, Tang C L. Ground state sign-changing solutions for a Schrödinger-Poisson system with a critical nonlinearity in $\mathbb{R}^3$. Nonlinear Anal Real World Appl, 2018, 39: 166-184
