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Deng, Zhiying; Huang, Yisheng

Multiple symmetric results for quasilinear elliptic systems involving singular potentials and critical Sobolev exponents in \(\mathbb{R}^N\). (English) Zbl 1472.35183

Abstr. Appl. Anal. 2014, Article ID 430976, 14 p. (2014).
Summary: This paper deals with a class of quasilinear elliptic systems involving singular potentials and critical Sobolev exponents in \(\mathbb{R}^N\). By using the symmetric criticality principle of Palais and variational methods, we prove several existence and multiplicity results of \(G\)-symmetric solutions under certain appropriate hypotheses on the potentials and parameters.

Cited in 1 Document

MSC:

35J62 Quasilinear elliptic equations
35J47 Second-order elliptic systems
35A01 Existence problems for PDEs: global existence, local existence, non-existence
35A15 Variational methods applied to PDEs

Keywords:

quasilinear elliptic systems; singular potentials; critical exponents; existence; variational methods
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References:

[1] Dautray, R.; Lions, J. L., Mathematical analysis and numerical methods for science and technology, Physical Origins and Classical Methods (1990), Berlin, Germany: Springer, Berlin, Germany · Zbl 0683.35001
[2] García Azorero, J. P.; Peral Alonso, I., Hardy inequalities and some critical elliptic and parabolic problems, Journal of Differential Equations, 144, 2, 441-476 (1998) · Zbl 0918.35052 · doi:10.1006/jdeq.1997.3375
[3] Ghoussoub, N.; Yuan, C., Multiple solutions for quasi-linear PDEs involving the critical Sobolev and Hardy exponents, Transactions of the American Mathematical Society, 352, 12, 5703-5743 (2000) · Zbl 0956.35056 · doi:10.1090/S0002-9947-00-02560-5
[4] Kang, D., On the quasilinear elliptic problems with critical Sobolev-HARdy exponents and HARdy terms, Nonlinear Analysis: Theory, Methods & Applications, 68, 7, 1973-1985 (2008) · Zbl 1135.35042 · doi:10.1016/j.na.2007.01.024
[5] Chen, Z.; Zou, W., On an elliptic problem with critical exponent and Hardy potential, Journal of Differential Equations, 252, 2, 969-987 (2012) · Zbl 1233.35078 · doi:10.1016/j.jde.2011.09.042
[6] Ramaswamy, M.; Santra, S., Uniqueness and profile of positive solutions of a critical exponent problem with Hardy potential, Journal of Differential Equations, 254, 11, 4347-4372 (2013) · Zbl 1286.35093 · doi:10.1016/j.jde.2013.03.002
[7] Liu, H.; Zhao, L., Nontrivial solutions for a class of critical elliptic equations of Caffarelli-Kohn-Nirenberg type, Journal of Mathematical Analysis and Applications, 404, 2, 317-325 (2013) · Zbl 1307.35123 · doi:10.1016/j.jmaa.2013.03.016
[8] Carrião, P. C.; Demarque, R.; Miyagaki, O. H., Existence and non-existence of solutions for p-Laplacian equations with decaying cylindrical potentials, Journal of Differential Equations, 255, 10, 3412-3433 (2013) · Zbl 1321.35065 · doi:10.1016/j.jde.2013.07.037
[9] Deng, Y.; Jin, L., On symmetric solutions of a singular elliptic equation with critical Sobolev-Hardy exponent, Journal of Mathematical Analysis and Applications, 329, 1, 603-616 (2007) · Zbl 1241.35055 · doi:10.1016/j.jmaa.2006.06.070
[10] Waliullah, S., Minimizers and symmetric minimizers for problems with critical Sobolev exponent, Topological Methods in Nonlinear Analysis, 34, 2, 291-326 (2009) · Zbl 1200.35128
[11] Deng, Z.; Huang, Y., On \(G\)-symmetric solutions of a quasilinear elliptic equation involving critical Hardy-Sobolev exponent, Journal of Mathematical Analysis and Applications, 384, 2, 578-590 (2011) · Zbl 1229.35100 · doi:10.1016/j.jmaa.2011.06.012
[12] Bianchi, G.; Chabrowski, J.; Szulkin, A., On symmetric solutions of an elliptic equation with a nonlinearity involving critical Sobolev exponent, Nonlinear Analysis: Theory, Methods & Applications, 25, 1, 41-59 (1995) · Zbl 0823.35051 · doi:10.1016/0362-546X(94)E0070-W
[13] Bartsch, T.; Willem, M., Infinitely Many Non-Radial Solutions of an Euclidean Scalar Field Equation (1992), Heidelberg, Germany: Mathematisches Institut, Universitat Heidelberg, Heidelberg, Germany
[14] Chabrowski, J., On the existence of \(G\)-symmetric entire solutions for semilinear elliptic equations, Rendiconti del Circolo Matematico di Palermo, 41, 3, 413-440 (1992) · Zbl 0793.35027 · doi:10.1007/BF02848946
[15] Wu, T.-F., The Nehari manifold for a semilinear elliptic system involving sign-changing weight functions, Nonlinear Analysis: Theory, Methods & Applications, 68, 6, 1733-1745 (2008) · Zbl 1151.35342 · doi:10.1016/j.na.2007.01.004
[16] Nyamoradi, N., Existence and multiplicity of solutions to a singular elliptic system with critical Sobolev-Hardy exponents and concave-convex nonlinearities, Journal of Mathematical Analysis and Applications, 396, 1, 280-293 (2012) · Zbl 1255.35117 · doi:10.1016/j.jmaa.2012.06.026
[17] Lü, D.; Xiao, J., Multiple solutions for weighted nonlinear elliptic system involving critical exponents, Mathematical and Computer Modelling, 55, 3-4, 816-827 (2012) · Zbl 1255.35100 · doi:10.1016/j.mcm.2011.09.007
[18] Li, Y.; Gao, W., Existence of multiple solutions for singular quasilinear elliptic system with critical sobolev-hardy exponents and concave-convex terms, Acta Mathematica Scientia B, 33, 1, 107-121 (2013) · Zbl 1289.35116 · doi:10.1016/S0252-9602(12)60197-0
[19] Alves, C. O.; de Morais Filho, D. C.; Souto, M. A. S., On systems of elliptic equations involving subcritical or critical Sobolev exponents, Nonlinear Analysis, 42, 5, 771-787 (2000) · Zbl 0958.35037 · doi:10.1016/S0362-546X(99)00121-2
[20] Long, J.; Yang, J., Existence results for critical singular elliptic systems, Nonlinear Analysis: Theory, Methods & Applications, 69, 11, 4199-4214 (2008) · Zbl 1159.35338 · doi:10.1016/j.na.2007.10.048
[21] de Souza, M., On a singular class of elliptic systems involving critical growth in \(R^2\), Nonlinear Analysis: Real World Applications, 12, 2, 1072-1088 (2011) · Zbl 1206.35095 · doi:10.1016/j.nonrwa.2010.09.001
[22] Benmouloud, S.; Echarghaoui, R.; Sbaï, S. M., Multiplicity of positive solutions for a critical quasilinear elliptic system with concave and convex nonlinearities, Journal of Mathematical Analysis and Applications, 396, 1, 375-385 (2012) · Zbl 1255.35119 · doi:10.1016/j.jmaa.2012.05.078
[23] Sun, J.; Chen, H.; Nieto, J. J., On ground state solutions for some non-autonomous Schrödinger-Poisson systems, Journal of Differential Equations, 252, 5, 3365-3380 (2012) · Zbl 1241.35057 · doi:10.1016/j.jde.2011.12.007
[24] Deng, Z.; Huang, Y., Existence of symmetric solutions for singular semilinear elliptic systems with critical Hardy-Sobolev exponents, Nonlinear Analysis: Real World Applications, 14, 1, 613-625 (2013) · Zbl 1388.35040 · doi:10.1016/j.nonrwa.2012.07.021
[25] Huang, D.; Li, Y., Multiplicity of solutions for a noncooperative \(p\)-Laplacian elliptic system in \(R^N\), Journal of Differential Equations, 215, 1, 206-223 (2005) · Zbl 1181.35060 · doi:10.1016/j.jde.2004.09.001
[26] Palais, R. S., The principle of symmetric criticality, Communications in Mathematical Physics, 69, 1, 19-30 (1979) · Zbl 0417.58007 · doi:10.1007/BF01941322
[27] Lions, P. L., The concentration-compactness principle in the calculus of variations, Revista Matemática Iberoamericana, 1, 1, part 1, 145-201 (1985) · Zbl 0704.49005 · doi:10.4171/RMI/6
[28] Lions, P., The concentration-compactness principle in the calculus of variations. The limit case, part 2, Revista Matemática Iberoamericana, 1, 2, 45-121 (1985) · Zbl 0704.49006 · doi:10.4171/RMI/12
[29] Ambrosetti, A.; Rabinowitz, P. H., Dual variational methods in critical point theory and applications, Journal of Functional Analysis, 14, 349-381 (1973) · Zbl 0273.49063
[30] Brezis, H.; Nirenberg, L., Positive solutions of nonlinear elliptic equations involving critical Sobolev exponents, Communications on Pure and Applied Mathematics, 36, 4, 437-477 (1983) · Zbl 0541.35029 · doi:10.1002/cpa.3160360405
[31] Rabinowitz, H., Methods in Critical Point Theory with Applications to Differential Equations. Methods in Critical Point Theory with Applications to Differential Equations, CBMS Regional Conference Series in Mathematics (1986), American Mathematical Society · Zbl 0609.58002
[32] Brézis, H.; Lieb, E., A relation between pointwise convergence of functions and convergence of functionals, Proceedings of the American Mathematical Society, 88, 3, 486-490 (1983) · Zbl 0526.46037 · doi:10.2307/2044999
[33] Han, P., The effect of the domain topology on the number of positive solutions of an elliptic system involving critical Sobolev exponents, Houston Journal of Mathematics, 32, 4, 1241-1257 (2006) · Zbl 1200.35116
[34] Ekeland, I., Nonconvex minimization problems, Bulletin of the American Mathematical Society, 1, 3, 443-474 (1979) · Zbl 0441.49011 · doi:10.1090/S0273-0979-1979-14595-6
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