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Critical Schrödinger-Hardy systems in the Heisenberg group
Infinitely many symmetric solutions for anisotropic problems driven by nonhomogeneous operators
Faculty of Education and Faculty of Mathematics and Physics, University of Ljubljana, SI-1000 Ljubljana, Slovenia |
We are concerned with the existence of infinitely many radial symmetric solutions for a nonlinear stationary problem driven by a new class of nonhomogeneous differential operators. The proof relies on the symmetric version of the mountain pass theorem.
References:
[1] |
A. Ambrosetti and P. Rabinowitz,
Dual variational methods in critical point theory and applications, J. Functional Anal., 14 (1973), 349-381.
|
[2] |
A. Azzollini,
Minimum action solutions for a quasilinear equation, J. Lond. Math. Soc., 92 (2015), 583-595.
doi: 10.1112/jlms/jdv050. |
[3] |
A. Azzollini, P. d'Avenia and A. Pomponio,
Quasilinear elliptic equations in $\mathbb {R}^N$ via variational methods and Orlicz-Sobolev embeddings, Calc. Var. Partial Differential Equations, 49 (2014), 197-213.
doi: 10.1007/s00526-012-0578-0. |
[4] |
S. Baraket, S. Chebbi, N. Chorfi and V. Rădulescu,
Non-autonomous eigenvalue problems with variable (p1, p2)-growth, Advanced Nonlinear Studies, 17 (2017), 781-792.
doi: 10.1515/ans-2016-6020. |
[5] |
P. Baroni, M. Colombo and G. Mingione,
Non-autonomous functionals, borderline cases and related function classes, St. Petersburg Mathematical Journal, 27 (2016), 347-379.
doi: 10.1090/spmj/1392. |
[6] |
H. Brezis and F. Browder,
Partial differential equations in the 20th century, Adv. Math., 135 (1998), 76-144.
doi: 10.1006/aima.1997.1713. |
[7] |
N. Chorfi and V. Rădulescu,
Standing wave solutions of a quasilinear degenerate Schroedinger equation with unbounded potential, Electronic Journal of the Qualitative Theory of Differential Equations, 37 (2016), 1-12.
|
[8] |
N. Chorfi and V. Rădulescu,
Small perturbations of elliptic problems with variable growth, Applied Mathematics Letters, 74 (2017), 167-173.
doi: 10.1016/j.aml.2017.05.007. |
[9] |
M. Colombo and G. Mingione,
Bounded minimisers of double phase variational integrals, Archive for Rational Mechanics and Analysis, 218 (2015), 219-273.
doi: 10.1007/s00205-015-0859-9. |
[10] |
R. Filippucci, P. Pucci and V. Rădulescu,
Existence and non-existence results for quasilinear elliptic exterior problems with nonlinear boundary conditions, Comm. Partial Differential Equations, 33 (2008), 706-717.
doi: 10.1080/03605300701518208. |
[11] |
T. C. Halsey,
Electrorheological fluids, Science, 258 (1992), 761-766.
|
[12] |
Y. Jabri, The Mountain Pass Theorem. Variants, Generalizations and Some Applications, Encyclopedia of Mathematics and its Applications, vol. 95, Cambridge University Press, Cambridge, 2003.
doi: 10.1017/CBO9780511546655. |
[13] |
I. H. Kim and Y. H. Kim,
Mountain pass type solutions and positivity of the infimum eigenvalue for quasilinear elliptic equations with variable exponents, Manuscripta Math., 147 (2015), 169-191.
doi: 10.1007/s00229-014-0718-2. |
[14] |
A. Kristaly, V. Rădulescu and C. Varga, Variational Principles in Mathematical Physics, Geometry, and Economics: Qualitative Analysis of Nonlinear Equations and Unilateral Problems, Encyclopedia of Mathematics and its Applications, No. 136, Cambridge University Press, Cambridge, 384, 2010.
doi: 10.1017/CBO9780511760631. |
[15] |
P. Marcellini,
Regularity and existence of solutions of elliptic equations with (p, q)-growth conditions, J. Differential Equations, 90 (1991), 1-30.
doi: 10.1016/0022-0396(91)90158-6. |
[16] |
R. Palais,
The principle of symmetric criticality, Commun. Math. Phys., 69 (1979), 19-30.
doi: 10.1007/BF01941322. |
[17] |
P. Pucci and V. Rădulescu, The impact of the mountain pass theory in nonlinear analysis: A mathematical survey, Boll. Unione Mat. Ital., Series IX, 3 (2010), 543–582. |
[18] |
P. Rabinowitz, Minimax Methods in Critical Point Theory with Applications to Differential Equations, CBMS Regional Conference Series in Mathematics, vol. 65, American Mathematical Society, Providence, RI, 1986.
doi: 10.1090/cbms/065. |
[19] |
V. Rădulescu and D. Repovš, Partial Differential Equations with Variable Exponents: Variational Methods and Qualitative Analysis, CRC Press, Taylor & Francis Group, Boca Raton FL, 2015.
doi: 10.1201/b18601. |
[20] |
V. V. Zhikov,
Lavrentiev phenomenon and homogenization for some variational problems, C. R. Acad. Sci. Paris, Sér. I, 316 (1993), 435-439.
|
show all references
Dedicated to Professor Vicenţiu Rădulescu with deep esteem and admiration
References:
[1] |
A. Ambrosetti and P. Rabinowitz,
Dual variational methods in critical point theory and applications, J. Functional Anal., 14 (1973), 349-381.
|
[2] |
A. Azzollini,
Minimum action solutions for a quasilinear equation, J. Lond. Math. Soc., 92 (2015), 583-595.
doi: 10.1112/jlms/jdv050. |
[3] |
A. Azzollini, P. d'Avenia and A. Pomponio,
Quasilinear elliptic equations in $\mathbb {R}^N$ via variational methods and Orlicz-Sobolev embeddings, Calc. Var. Partial Differential Equations, 49 (2014), 197-213.
doi: 10.1007/s00526-012-0578-0. |
[4] |
S. Baraket, S. Chebbi, N. Chorfi and V. Rădulescu,
Non-autonomous eigenvalue problems with variable (p1, p2)-growth, Advanced Nonlinear Studies, 17 (2017), 781-792.
doi: 10.1515/ans-2016-6020. |
[5] |
P. Baroni, M. Colombo and G. Mingione,
Non-autonomous functionals, borderline cases and related function classes, St. Petersburg Mathematical Journal, 27 (2016), 347-379.
doi: 10.1090/spmj/1392. |
[6] |
H. Brezis and F. Browder,
Partial differential equations in the 20th century, Adv. Math., 135 (1998), 76-144.
doi: 10.1006/aima.1997.1713. |
[7] |
N. Chorfi and V. Rădulescu,
Standing wave solutions of a quasilinear degenerate Schroedinger equation with unbounded potential, Electronic Journal of the Qualitative Theory of Differential Equations, 37 (2016), 1-12.
|
[8] |
N. Chorfi and V. Rădulescu,
Small perturbations of elliptic problems with variable growth, Applied Mathematics Letters, 74 (2017), 167-173.
doi: 10.1016/j.aml.2017.05.007. |
[9] |
M. Colombo and G. Mingione,
Bounded minimisers of double phase variational integrals, Archive for Rational Mechanics and Analysis, 218 (2015), 219-273.
doi: 10.1007/s00205-015-0859-9. |
[10] |
R. Filippucci, P. Pucci and V. Rădulescu,
Existence and non-existence results for quasilinear elliptic exterior problems with nonlinear boundary conditions, Comm. Partial Differential Equations, 33 (2008), 706-717.
doi: 10.1080/03605300701518208. |
[11] |
T. C. Halsey,
Electrorheological fluids, Science, 258 (1992), 761-766.
|
[12] |
Y. Jabri, The Mountain Pass Theorem. Variants, Generalizations and Some Applications, Encyclopedia of Mathematics and its Applications, vol. 95, Cambridge University Press, Cambridge, 2003.
doi: 10.1017/CBO9780511546655. |
[13] |
I. H. Kim and Y. H. Kim,
Mountain pass type solutions and positivity of the infimum eigenvalue for quasilinear elliptic equations with variable exponents, Manuscripta Math., 147 (2015), 169-191.
doi: 10.1007/s00229-014-0718-2. |
[14] |
A. Kristaly, V. Rădulescu and C. Varga, Variational Principles in Mathematical Physics, Geometry, and Economics: Qualitative Analysis of Nonlinear Equations and Unilateral Problems, Encyclopedia of Mathematics and its Applications, No. 136, Cambridge University Press, Cambridge, 384, 2010.
doi: 10.1017/CBO9780511760631. |
[15] |
P. Marcellini,
Regularity and existence of solutions of elliptic equations with (p, q)-growth conditions, J. Differential Equations, 90 (1991), 1-30.
doi: 10.1016/0022-0396(91)90158-6. |
[16] |
R. Palais,
The principle of symmetric criticality, Commun. Math. Phys., 69 (1979), 19-30.
doi: 10.1007/BF01941322. |
[17] |
P. Pucci and V. Rădulescu, The impact of the mountain pass theory in nonlinear analysis: A mathematical survey, Boll. Unione Mat. Ital., Series IX, 3 (2010), 543–582. |
[18] |
P. Rabinowitz, Minimax Methods in Critical Point Theory with Applications to Differential Equations, CBMS Regional Conference Series in Mathematics, vol. 65, American Mathematical Society, Providence, RI, 1986.
doi: 10.1090/cbms/065. |
[19] |
V. Rădulescu and D. Repovš, Partial Differential Equations with Variable Exponents: Variational Methods and Qualitative Analysis, CRC Press, Taylor & Francis Group, Boca Raton FL, 2015.
doi: 10.1201/b18601. |
[20] |
V. V. Zhikov,
Lavrentiev phenomenon and homogenization for some variational problems, C. R. Acad. Sci. Paris, Sér. I, 316 (1993), 435-439.
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