# American Institute of Mathematical Sciences

April  2019, 12(2): 401-411. doi: 10.3934/dcdss.2019026

## 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

Dedicated to Professor Vicenţiu Rădulescu with deep esteem and admiration

Received  June 2017 Revised  December 2017 Published  August 2018

Fund Project: This work was supported by the Slovenian Research Agency grants P1-0292, J1-8131 and J1-7025.

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.

Citation: Dušan D. Repovš. Infinitely many symmetric solutions for anisotropic problems driven by nonhomogeneous operators. Discrete & Continuous Dynamical Systems - S, 2019, 12 (2) : 401-411. doi: 10.3934/dcdss.2019026
##### References:
 [1] A. Ambrosetti and P. Rabinowitz, Dual variational methods in critical point theory and applications, J. Functional Anal., 14 (1973), 349-381.   Google Scholar [2] A. Azzollini, Minimum action solutions for a quasilinear equation, J. Lond. Math. Soc., 92 (2015), 583-595.  doi: 10.1112/jlms/jdv050.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.   Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [11] T. C. Halsey, Electrorheological fluids, Science, 258 (1992), 761-766.   Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [16] R. Palais, The principle of symmetric criticality, Commun. Math. Phys., 69 (1979), 19-30.  doi: 10.1007/BF01941322.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [20] V. V. Zhikov, Lavrentiev phenomenon and homogenization for some variational problems, C. R. Acad. Sci. Paris, Sér. I, 316 (1993), 435-439.   Google Scholar

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##### References:
 [1] A. Ambrosetti and P. Rabinowitz, Dual variational methods in critical point theory and applications, J. Functional Anal., 14 (1973), 349-381.   Google Scholar [2] A. Azzollini, Minimum action solutions for a quasilinear equation, J. Lond. Math. Soc., 92 (2015), 583-595.  doi: 10.1112/jlms/jdv050.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.   Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [11] T. C. Halsey, Electrorheological fluids, Science, 258 (1992), 761-766.   Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [16] R. Palais, The principle of symmetric criticality, Commun. Math. Phys., 69 (1979), 19-30.  doi: 10.1007/BF01941322.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [20] V. V. Zhikov, Lavrentiev phenomenon and homogenization for some variational problems, C. R. Acad. Sci. Paris, Sér. I, 316 (1993), 435-439.   Google Scholar
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