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September  2017, 16(5): 1587-1602. doi: 10.3934/cpaa.2017076

Multiple positive solutions for Schrödinger-Poisson system in $\mathbb{R}^{3}$ involving concave-convex nonlinearities with critical exponent

Miao-Miao Li and  Chun-Lei Tang ,

School of Mathematics and Statistics, Southwest University, Chongqing, 400715, China

* Corresponding author: Chun-Lei Tang

Received  April 2016 Revised  January 2017 Published  May 2017

Fund Project: The second author is supported by National Natural Science Foundation of China(No. 11471267).

In this paper, we study the existence of multiple positive solutions of the following Schrödinger-Poisson system with critical exponent
$\begin{equation*}\begin{cases}-Δ u-l(x)φ u=λ h(x)|u|^{q-2}u+|u|^{4}u,\ \text{in}\ \mathbb{R}^{3}, \\-Δφ=l(x)u^{2},\ \text{in}\ \mathbb{R}^{3},\end{cases}\end{equation*}$
where
$1 < q < 2 $
 and
$λ>0 $
. Under some appropriate conditions on
$ l$
 and
$h $
, we show that there exists
$λ^{*}>0 $
 such that the above problem has at least two positive solutions for each
$λ∈(0,λ^{*}) $
 by using the Mountain Pass Theorem and Ekeland's Variational Principle.
Keywords: Schrödinger-Poisson system, Mountain Pass Theorem, Ekeland's variational principle, critical exponent, concentration compactness principle.
Mathematics Subject Classification: Primary: 35J20; Secondary: 35J60.
Citation: Miao-Miao Li, Chun-Lei Tang. Multiple positive solutions for Schrödinger-Poisson system in $\mathbb{R}^{3}$ involving concave-convex nonlinearities with critical exponent. Communications on Pure &amp; Applied Analysis, 2017, 16 (5) : 1587-1602. doi: 10.3934/cpaa.2017076
References:
[1]

C. O. Alves, F. J. S. A. Corra and G. M. Figueiredo, On a class of nonlocal elliptic problems with critical growth, Differential Equation and Applications, 23 (2010), 409-417.  doi: 10.7153/dea-02-25.  Google Scholar

[2]

C. O. Alves, J. V. Goncalves and O. H. Miyagaki, Multiple positive solutions for semilinear elliptic equations in $\mathbb{R}^{N}$ involving critical exponents, Nonlinear Anal., 34 (1998), 593-615.  doi: 10.1016/S0362-546X(97)00555-5.  Google Scholar

[3]

A. Ambrosetti, J. G. Azorero and I. Peral, Elliptic variational problems in $\mathbb{R}^{N}$ with critical growth, J. Differential Equations, 168 (2000), 10-32.  doi: 10.1006/jdeq.2000.3875.  Google Scholar

[4]

A. Ambrosetti, H. Brezis and G. Cerami, Combined effects of concave and convex nonlinear in some elliptic problems, J. Funct. Anal., 122 (1994), 519-543.  doi: 10.1006/jfan.1994.1078.  Google Scholar

[5]

A. Ambrosetti and P. Rabinowitz, Dual variational methods in critical point theory and applications, J. Funct. Anal., 41 (1973), 349-381.   Google Scholar

[6]

A. Ambrosetti and D. Ruiz, Multiple bound states for the Schr$\ddot{\mathrm{o}}$dinger-Poisson problem, Commun. Contemp. Math., 10 (2008), 391-404.  doi: 10.1142/S021919970800282X.  Google Scholar

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

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V. Benci and D. Fortunato, An eigenvalue problem for the Schrödinger-Maxwell equations, Topol. Methods Nonlinear Anal., 11 (1998), 282-293.   Google Scholar

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V. Benci and D. Fortunato, Solitary waves of the nonlinear Klein-Gordon equation coupled with Maxwell equations, Rev. Math. Phys., 14 (2002), 409-420.  doi: 10.1142/S0129055X02001168.  Google Scholar

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R. Benguria, H. Brézis and E. H. Lieb, The Thomas-Ferim-von Weizsäcker theory of atoms and moleculars, Comm. Math. Phys., 79 (1981), 167-180.   Google Scholar

[11]

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

[12]

H. Brézis and L. Nirenberg, Positive solutions of nonlinear elliptic equations involving critical Sobolev exponents, Comm. Pure. Appl. Math., 36 (1983), 437-477.   Google Scholar

[13]

I. Catto and P. L. Lions, Binding of atoms and stability of molecules in Hartree and Thomas-Fermi type theories. PART 1: A necessary and sufficient condition for the stability of generalmolecular system, Comm. Partial Differential Equations, 17 (1992), 1051-1110.  doi: 10.1080/03605309208820878.  Google Scholar

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G. Cerimi and G. Vaira, Positive solutions for some non-autonomous Schrödinger-Poisson systems, J. Differential Equations, 248 (2010), 521-543.  doi: 10.1016/j.jde.2009.06.017.  Google Scholar

[15]

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

[16]

G. M. Coclite, A Multiplicity result for the nonlinear Schrödinger-Maxwell equations, Commun. Appl. Anal., 7 (2003), 417-423.   Google Scholar

[17]

L. Huang and E. M. Rocha, A positive solution of a Schrödinger-Poisson system with critical exponent, Communications in Mathematical Analysis, 15 (2013), 29-43.   Google Scholar

[18]

L. Huang, E. M. Rocha and J. Chen, Two positive solutions of a class of Schrödinger-Poisson system with indefinite nonlinearity, J. Differential Equations, 255 (2013), 2463-2483.  doi: 10.1016/j.jde.2013.06.022.  Google Scholar

[19]

L. Huang, E. M. Rocha and J. Chen, Positive and sign-changing solutions of a Schrödinger-Poisson system involving a critical nonlinearity, J. Math. Anal. Appl., 408 (2013), 55-69.  doi: 10.1016/j.jmaa.2013.05.071.  Google Scholar

[20]

E. H. Lieb, Thomas-Fermi and related theories and molecules, Rev. Modern Phys., 53 (1981), 603-641.  doi: 10.1103/RevModPhys.53.603.  Google Scholar

[21]

P. L. Lions, The concentration-compactness principle in the calculus of variations. The limit case, part 1, Rev. Mat. Iberoamericana, 1 (1985), 145-201.  doi: 10.4171/RMI/6.  Google Scholar

[22]

P. L. Lions, The concentration-compactness principle in the calculus of variations. The limit case, part 2, Rev. Mat. Iberoamericana, 1 (1985), 45-121.  doi: 10.4171/RMI/12.  Google Scholar

[23]

J. J. Nie and X. Wu, Exsistence and muitilicity of non-trivial solutions for Schrödinger-Kirchhoff-type equations with radial potential, Nonlinear Anal., 75 (2012), 3470-3479.  doi: 10.1016/j.na.2012.01.004.  Google Scholar

[24]

M. Reed and B. Simon, Methods of Modern Mathematical Physics Vols. Elsevier (Singapore) Pte Ltd, 2003. Google Scholar

[25]

G. Talenti, Best constant in Sobolev inequality, Ann. Math., 110 (1976), 353-372.  doi: 10.1007/BF02418013.  Google Scholar

[26]

G. Vaira, Ground states for Schrödinger-Poisson type systems, Ricerche mat., 60 (2011), 263-297.  doi: 10.1007/s11587-011-0109-x.  Google Scholar

[27]

G. Vaira, Existence of bounded states for Schrödinger-Poisson type systems, S. I. S. S. A., 251 (2012), 112-146.   Google Scholar

[28]

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

[29]

Y. -P. Gao, S. -L. Yu and C. -L. Tang, On positive ground state solution to the Schrödinger-Poisson system with the negative non-local term, Electron. J. Differential Equations 118 (2015), 11 pp.  Google Scholar

[30]

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.  Google Scholar

[31]

V. I. Bogachev, Measure Theory Springer, Berlin, 2007. doi: 10.1007/978-3-540-34514-5.  Google Scholar

[32]

Stationary solutions for a Schrodinger-Poisson system in R3, in Proceedings of the 2002 Fez Conference on Partial Differential Equations, Electron. J. Differ. Equ. Conf. , 9 (2002), 65-76. Southwest Texas State Univ. , San Marcos, TX.  Google Scholar

show all references

References:
[1]

C. O. Alves, F. J. S. A. Corra and G. M. Figueiredo, On a class of nonlocal elliptic problems with critical growth, Differential Equation and Applications, 23 (2010), 409-417.  doi: 10.7153/dea-02-25.  Google Scholar

[2]

C. O. Alves, J. V. Goncalves and O. H. Miyagaki, Multiple positive solutions for semilinear elliptic equations in $\mathbb{R}^{N}$ involving critical exponents, Nonlinear Anal., 34 (1998), 593-615.  doi: 10.1016/S0362-546X(97)00555-5.  Google Scholar

[3]

A. Ambrosetti, J. G. Azorero and I. Peral, Elliptic variational problems in $\mathbb{R}^{N}$ with critical growth, J. Differential Equations, 168 (2000), 10-32.  doi: 10.1006/jdeq.2000.3875.  Google Scholar

[4]

A. Ambrosetti, H. Brezis and G. Cerami, Combined effects of concave and convex nonlinear in some elliptic problems, J. Funct. Anal., 122 (1994), 519-543.  doi: 10.1006/jfan.1994.1078.  Google Scholar

[5]

A. Ambrosetti and P. Rabinowitz, Dual variational methods in critical point theory and applications, J. Funct. Anal., 41 (1973), 349-381.   Google Scholar

[6]

A. Ambrosetti and D. Ruiz, Multiple bound states for the Schr$\ddot{\mathrm{o}}$dinger-Poisson problem, Commun. Contemp. Math., 10 (2008), 391-404.  doi: 10.1142/S021919970800282X.  Google Scholar

[7]

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

[8]

V. Benci and D. Fortunato, An eigenvalue problem for the Schrödinger-Maxwell equations, Topol. Methods Nonlinear Anal., 11 (1998), 282-293.   Google Scholar

[9]

V. Benci and D. Fortunato, Solitary waves of the nonlinear Klein-Gordon equation coupled with Maxwell equations, Rev. Math. Phys., 14 (2002), 409-420.  doi: 10.1142/S0129055X02001168.  Google Scholar

[10]

R. Benguria, H. Brézis and E. H. Lieb, The Thomas-Ferim-von Weizsäcker theory of atoms and moleculars, Comm. Math. Phys., 79 (1981), 167-180.   Google Scholar

[11]

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

[12]

H. Brézis and L. Nirenberg, Positive solutions of nonlinear elliptic equations involving critical Sobolev exponents, Comm. Pure. Appl. Math., 36 (1983), 437-477.   Google Scholar

[13]

I. Catto and P. L. Lions, Binding of atoms and stability of molecules in Hartree and Thomas-Fermi type theories. PART 1: A necessary and sufficient condition for the stability of generalmolecular system, Comm. Partial Differential Equations, 17 (1992), 1051-1110.  doi: 10.1080/03605309208820878.  Google Scholar

[14]

G. Cerimi and G. Vaira, Positive solutions for some non-autonomous Schrödinger-Poisson systems, J. Differential Equations, 248 (2010), 521-543.  doi: 10.1016/j.jde.2009.06.017.  Google Scholar

[15]

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

[16]

G. M. Coclite, A Multiplicity result for the nonlinear Schrödinger-Maxwell equations, Commun. Appl. Anal., 7 (2003), 417-423.   Google Scholar

[17]

L. Huang and E. M. Rocha, A positive solution of a Schrödinger-Poisson system with critical exponent, Communications in Mathematical Analysis, 15 (2013), 29-43.   Google Scholar

[18]

L. Huang, E. M. Rocha and J. Chen, Two positive solutions of a class of Schrödinger-Poisson system with indefinite nonlinearity, J. Differential Equations, 255 (2013), 2463-2483.  doi: 10.1016/j.jde.2013.06.022.  Google Scholar

[19]

L. Huang, E. M. Rocha and J. Chen, Positive and sign-changing solutions of a Schrödinger-Poisson system involving a critical nonlinearity, J. Math. Anal. Appl., 408 (2013), 55-69.  doi: 10.1016/j.jmaa.2013.05.071.  Google Scholar

[20]

E. H. Lieb, Thomas-Fermi and related theories and molecules, Rev. Modern Phys., 53 (1981), 603-641.  doi: 10.1103/RevModPhys.53.603.  Google Scholar

[21]

P. L. Lions, The concentration-compactness principle in the calculus of variations. The limit case, part 1, Rev. Mat. Iberoamericana, 1 (1985), 145-201.  doi: 10.4171/RMI/6.  Google Scholar

[22]

P. L. Lions, The concentration-compactness principle in the calculus of variations. The limit case, part 2, Rev. Mat. Iberoamericana, 1 (1985), 45-121.  doi: 10.4171/RMI/12.  Google Scholar

[23]

J. J. Nie and X. Wu, Exsistence and muitilicity of non-trivial solutions for Schrödinger-Kirchhoff-type equations with radial potential, Nonlinear Anal., 75 (2012), 3470-3479.  doi: 10.1016/j.na.2012.01.004.  Google Scholar

[24]

M. Reed and B. Simon, Methods of Modern Mathematical Physics Vols. Elsevier (Singapore) Pte Ltd, 2003. Google Scholar

[25]

G. Talenti, Best constant in Sobolev inequality, Ann. Math., 110 (1976), 353-372.  doi: 10.1007/BF02418013.  Google Scholar

[26]

G. Vaira, Ground states for Schrödinger-Poisson type systems, Ricerche mat., 60 (2011), 263-297.  doi: 10.1007/s11587-011-0109-x.  Google Scholar

[27]

G. Vaira, Existence of bounded states for Schrödinger-Poisson type systems, S. I. S. S. A., 251 (2012), 112-146.   Google Scholar

[28]

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

[29]

Y. -P. Gao, S. -L. Yu and C. -L. Tang, On positive ground state solution to the Schrödinger-Poisson system with the negative non-local term, Electron. J. Differential Equations 118 (2015), 11 pp.  Google Scholar

[30]

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.  Google Scholar

[31]

V. I. Bogachev, Measure Theory Springer, Berlin, 2007. doi: 10.1007/978-3-540-34514-5.  Google Scholar

[32]

Stationary solutions for a Schrodinger-Poisson system in R3, in Proceedings of the 2002 Fez Conference on Partial Differential Equations, Electron. J. Differ. Equ. Conf. , 9 (2002), 65-76. Southwest Texas State Univ. , San Marcos, TX.  Google Scholar

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