American Institute of Mathematical Sciences

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

Infinitely many symmetric solutions for anisotropic problems driven by nonhomogeneous operators

Dušan D. Repovš

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.

Keywords: Anisotropic elliptic problem, nonhomogeneous differential operator, variable exponent, symmetric mountain pass theorem.
Mathematics Subject Classification: Primary: 35J60; Secondary: 35A15, 35B38, 47H14, 58E05.
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

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.   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

[1]

Claudianor O. Alves, Giovany M. Figueiredo, Marcelo F. Furtado. Multiplicity of solutions for elliptic systems via local Mountain Pass method. Communications on Pure & Applied Analysis, 2009, 8 (6) : 1745-1758. doi: 10.3934/cpaa.2009.8.1745
[2]

Liping Wang, Dong Ye. Concentrating solutions for an anisotropic elliptic problem with large exponent. Discrete & Continuous Dynamical Systems, 2015, 35 (8) : 3771-3797. doi: 10.3934/dcds.2015.35.3771
[3]

Dorota Bors. Application of Mountain Pass Theorem to superlinear equations with fractional Laplacian controlled by distributed parameters and boundary data. Discrete & Continuous Dynamical Systems - B, 2018, 23 (1) : 29-43. doi: 10.3934/dcdsb.2018003
[4]

Ian Schindler, Kyril Tintarev. Mountain pass solutions to semilinear problems with critical nonlinearity. Conference Publications, 2007, 2007 (Special) : 912-919. doi: 10.3934/proc.2007.2007.912
[5]

Shouchuan Hu, Nikolaos S. Papageorgiou. Nonlinear Neumann equations driven by a nonhomogeneous differential operator. Communications on Pure & Applied Analysis, 2011, 10 (4) : 1055-1078. doi: 10.3934/cpaa.2011.10.1055
[6]

Li Yin, Jinghua Yao, Qihu Zhang, Chunshan Zhao. Multiple solutions with constant sign of a Dirichlet problem for a class of elliptic systems with variable exponent growth. Discrete & Continuous Dynamical Systems, 2017, 37 (4) : 2207-2226. doi: 10.3934/dcds.2017095
[7]

Yinbin Deng, Shuangjie Peng, Li Wang. Existence of multiple solutions for a nonhomogeneous semilinear elliptic equation involving critical exponent. Discrete & Continuous Dynamical Systems, 2012, 32 (3) : 795-826. doi: 10.3934/dcds.2012.32.795
[8]

Prashanta Garain, Tuhina Mukherjee. Quasilinear nonlocal elliptic problems with variable singular exponent. Communications on Pure & Applied Analysis, 2020, 19 (11) : 5059-5075. doi: 10.3934/cpaa.2020226
[9]

SYLWIA DUDEK, IWONA SKRZYPCZAK. Liouville theorems for elliptic problems in variable exponent spaces. Communications on Pure & Applied Analysis, 2017, 16 (2) : 513-532. doi: 10.3934/cpaa.2017026
[10]

Xudong Shang, Jihui Zhang, Yang Yang. Positive solutions of nonhomogeneous fractional Laplacian problem with critical exponent. Communications on Pure & Applied Analysis, 2014, 13 (2) : 567-584. doi: 10.3934/cpaa.2014.13.567
[11]

Dmitry Glotov, P. J. McKenna. Numerical mountain pass solutions of Ginzburg-Landau type equations. Communications on Pure & Applied Analysis, 2008, 7 (6) : 1345-1359. doi: 10.3934/cpaa.2008.7.1345
[12]

Christopher Grumiau, Marco Squassina, Christophe Troestler. On the Mountain-Pass algorithm for the quasi-linear Schrödinger equation. Discrete & Continuous Dynamical Systems - B, 2013, 18 (5) : 1345-1360. doi: 10.3934/dcdsb.2013.18.1345
[13]

Yuan Li. Extremal solution and Liouville theorem for anisotropic elliptic equations. Communications on Pure & Applied Analysis, 2021, 20 (12) : 4063-4082. doi: 10.3934/cpaa.2021144
[14]

Carla Baroncini, Julián Fernández Bonder. An extension of a Theorem of V. Šverák to variable exponent spaces. Communications on Pure & Applied Analysis, 2015, 14 (5) : 1987-2007. doi: 10.3934/cpaa.2015.14.1987
[15]

V. V. Motreanu. Uniqueness results for a Dirichlet problem with variable exponent. Communications on Pure & Applied Analysis, 2010, 9 (5) : 1399-1410. doi: 10.3934/cpaa.2010.9.1399
[16]

Mónica Clapp, Marco Squassina. Nonhomogeneous polyharmonic elliptic problems at critical growth with symmetric data. Communications on Pure & Applied Analysis, 2003, 2 (2) : 171-186. doi: 10.3934/cpaa.2003.2.171
[17]

Mostafa Bendahmane, Kenneth Hvistendahl Karlsen, Mazen Saad. Nonlinear anisotropic elliptic and parabolic equations with variable exponents and $L^1$ data. Communications on Pure & Applied Analysis, 2013, 12 (3) : 1201-1220. doi: 10.3934/cpaa.2013.12.1201
[18]

Huiling Li, Xiaoliu Wang, Xueyan Lu. A nonlinear Stefan problem with variable exponent and different moving parameters. Discrete & Continuous Dynamical Systems - B, 2020, 25 (5) : 1671-1698. doi: 10.3934/dcdsb.2019246
[19]

V. V. Motreanu. Multiplicity of solutions for variable exponent Dirichlet problem with concave term. Discrete & Continuous Dynamical Systems - S, 2012, 5 (4) : 845-855. doi: 10.3934/dcdss.2012.5.845
[20]

Marco Donatelli, Luca Vilasi. Existence of multiple solutions for a fourth-order problem with variable exponent. Discrete & Continuous Dynamical Systems - B, 2021  doi: 10.3934/dcdsb.2021141

2020 Impact Factor: 2.425

Tools

Metrics

  • PDF downloads (130)
  • HTML views (102)
  • Cited by (1)

Other articles
by authors

[Back to Top]

Copyright © 2021 American Institute of Mathematical Sciences | Privacy Policy