Article Contents
Article Contents

# Infinitely many symmetric solutions for anisotropic problems driven by nonhomogeneous operators

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.

Mathematics Subject Classification: Primary: 35J60; Secondary: 35A15, 35B38, 47H14, 58E05.

 Citation:

•  A. Ambrosetti  and  P. Rabinowitz , Dual variational methods in critical point theory and applications, J. Functional Anal., 14 (1973) , 349-381. A. Azzollini , Minimum action solutions for a quasilinear equation, J. Lond. Math. Soc., 92 (2015) , 583-595.  doi: 10.1112/jlms/jdv050. 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. 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. 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. H. Brezis  and  F. Browder , Partial differential equations in the 20th century, Adv. Math., 135 (1998) , 76-144.  doi: 10.1006/aima.1997.1713. 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. 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. 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. 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. T. C. Halsey , Electrorheological fluids, Science, 258 (1992) , 761-766. 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. 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. 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. 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. R. Palais , The principle of symmetric criticality, Commun. Math. Phys., 69 (1979) , 19-30.  doi: 10.1007/BF01941322. 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. 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. 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. V. V. Zhikov , Lavrentiev phenomenon and homogenization for some variational problems, C. R. Acad. Sci. Paris, Sér. I, 316 (1993) , 435-439.