Opuscula Mathematica
Opuscula Math. 46, no. 1 (2026), 101-119

Galerkin-type minimizers to a competing problem for \((\vec{p},\vec{q})\)-Laplacian with variable exponents

Zhenfeng Zhang
Mina Ghasemi
Calogero Vetro
Abstract. This study focuses on a sequence of approximate minimizers for the functional \[J(u)=\int\limits_{\Omega}\sum\limits_{i=1}^{N}\frac{1}{p_{i}(x)}\bigg|\frac{\partial u}{\partial x_{i}}\bigg|^{p_{i}(x)}dx-\mu\int\limits_{\Omega}\sum\limits_{i=1}^{N}\frac{1}{q_{i}(x)}\bigg|\frac{\partial u}{\partial x_{i}}\bigg|^{q_{i}(x)}dx-\int\limits_{\Omega} F(u(x))dx,\] where \(\Omega\subset\mathbb{R}^N\) (\(N\geq 3\)) is a bounded domain, and \(p_i,q_i\in C(\overline{\Omega})\) with \(1\lt p_i,q_i\lt +\infty\) for all \(i \in \{1,\ldots,N\}\). We establish the convergence result to the infimum of \(J(u)\) when \(F:\mathbb{R}\to\mathbb{R}\) is a locally Lipschitz function of controlled growth, following the Galerkin method. As an application, we establish the existence of solutions to a class of Dirichlet inclusions associated to the functional.
Keywords: anisotropic Sobolev space, Clarke generalized gradient, Dirichlet problem, Galerkin basis, \((\vec{p},\vec{q})\)-Laplacian with variable exponents.
Mathematics Subject Classification: 46E30, 47J22.
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References ▼
  1. [1] G. Bonanno, G. D'Aguí, A. Sciammetta, Multiple solutions for a class of anisotropic \(\overrightarrow{p}\)-Laplacian problems, Bound. Value Probl. 2023 (2023), 89. https://doi.org/10.1186/s13661-023-01774-7
  2. [2] M.-M. Boureanu, Infinitely many solutions for a class of degenerate anisotropic elliptic problems with variable exponent, Taiwanese J. Math. 15 (2011), 2291-2310. https://doi.org/10.11650/twjm/1500406435
  3. [3] K.C. Chang, Variational methods for non-differentiable functionals and their applications to partial differential equations, J. Math. Anal. Appl. 80 (1981), 102-129. https://doi.org/10.1016/0022-247x(81)90095-0
  4. [4] N. Chems Eddine, M.A. Ragusa, D.D. Repovš, On the concentration-compactness principle for anisotropic variable exponent Sobolev spaces and its applications, Fract. Calc. Appl. Anal. 27 (2024), 725-756. https://doi.org/10.1007/s13540-024-00246-8
  5. [5] F.H. Clarke, Optimization and Nonsmooth Analysis, John Wiley & Sons Inc., New York, NY, USA, 1983. https://doi.org/10.1137/1.9781611971309
  6. [6] J. Diblík, M. Galewski, I. Kossowski, D. Motreanu, On competing \((p,q)\)-Laplacian Dirichlet problem with unbounded weight, Differ. lntegral Equ. 38 (2025), 23-42. https://doi.org/10.57262/die038-0102-23
  7. [7] L. Diening, P. Harjulehto, P. Hästö, M. Rŭzĭcka, Lebesgue and Sobolev Spaces with Variable Exponents, Lecture Notes in Mathematics. Springer, Berlin/Heidelberg, Germany, 2011. https://doi.org/10.1007/978-3-642-18363-8
  8. [8] Y. El Yazidi, A. Charkaoui, S. Zeng, Finite element solutions for variable exponents double phase problems, Numer. Algor. (2025). https://doi.org/10.1007/s11075-025-02194-7
  9. [9] X. Fan, Anisotropic variable exponent Sobolev spaces and \(\overrightarrow{p}(x)\)-Laplacian equations, Complex Var. Elliptic Equ. 56 (2011), 623-642. https://doi.org/10.1080/17476931003728412
  10. [10] X. Fan, D. Zhao, On the spaces \(L^{p(x)}(\Omega)\) and \(W^{m,p(x)}(\Omega)\), J. Math. Anal. Appl. 263 (2001), 424-446. https://doi.org/10.1006/jmaa.2000.7617
  11. [11] I. Fragalà, F. Gazzola, B. Kawohl, Existence and nonexistence results for anisotropic quasilinear elliptic equations, Ann. I. H. Poincaré 21 (2004), 715-734. https://doi.org/10.1016/j.anihpc.2003.12.001
  12. [12] M. Galewski, D. Motreanu, On variational competing \((p, q)\)-Laplacian Dirichlet problem with gradient depending weight, Appl. Math. Lett. 148, (2024), 108881. https://doi.org/10.1016/j.aml.2023.108881
  13. [13] M. Galewski, D. Motreanu, Competing Operators and their Applications to Boundary Value Problems, SpringerBriefs in Mathematics. Springer Cham, Switzerland, 2026.
  14. [14] M. Ghasemi, C. Vetro, Z. Zhang, Dirichlet \(\mu\)-parametric differential problem with multivalued reaction term, Mathematics 13 (2025), 1295. https://doi.org/10.3390/math13081295
  15. [15] Z. Liu, R. Livrea, D. Motreanu, S. Zeng, Variational differential inclusions without ellipticity condition, Electron. J. Qual. Theory Differ. Equ. 2020 (2020), 43. https://doi.org/10.14232/ejqtde.2020.1.43
  16. [16] M. Mihăilescu, P. Pucci, V. Rădulescu, Eigenvalue problems for anisotropic quasilinear elliptic equations with variable exponent, J. Math. Anal. Appl. 340 (2008), 687-698. https://doi.org/10.1016/j.jmaa.2007.09.015
  17. [17] D. Motreanu, Hemivariational inequalities with competing operators, Commun. Nonlinear Sci. Numer. Simul. 130 (2024), 107741. https://doi.org/10.1016/j.cnsns.2023.107741
  18. [18] D. Motreanu, V.V. Motreanu, N.S. Papageorgiou, Topological and Variational Methods with Applications to Nonlinear Boundary Value Problems, Springer, New York, 2014. https://doi.org/10.1007/978-1-4614-9323-5
  19. [19] N.S. Papageorgiou, F. Vetro, P. Winkert, Sequences of nodal solutions for critical double phase problems with variable exponents, Z. Angew. Math. Phys. 75 (2024), 95. https://doi.org/10.1007/s00033-024-02226-7
  20. [20] V.D. Rădulescu, D.D. Repovš, Partial Differential Equations with Variable Exponents: Variational Methods and Qualitative Analysis, Monographs and Research Notes in Mathematics, CRC Press, Boca Raton, FL, USA, 2015. https://doi.org/10.1201/b18601
  21. [21] J. Rákosník, Some remarks to anisotropic Sobolev spaces I, Beiträge Anal. 13 (1979), 55-68.
  22. [22] J. Rákosník, Some remarks to anisotropic Sobolev spaces II, Beiträge Anal. 15 (1981), 127-140.
  23. [23] A. Razani, G.M. Figueiredo, A positive solution for an anisotropic \((p,q)\)-Laplacian, Discrete Contin. Dyn. Syst. Ser. S. 16 (2023), 1629-1643. https://doi.org/10.3934/dcdss.2022147
  24. [24] A. Razani, L. Tavares, J. Vanterler da C. Sousa, Existence and multiplicity results for a system involving an anisotropic \((\overrightarrow{p},\overrightarrow{q})\)-Laplacian type operator, J. Fixed Point Theory Appl. 27 (2025), 66. https://doi.org/10.1007/s11784-025-01218-y
  25. [25] L. Tavares, Multiplicity of solutions for an anisotropic variable exponent problem, Bound. Value Probl. 2022 (2022), 10. https://doi.org/10.1186/s13661-022-01591-4
  26. [26] F. Vetro, R. Efendiev, Systems of differential inclusions with competing operators and variable exponents, Opuscula Math. 45 (2025), 665-684. https://doi.org/10.7494/opmath.2025.45.5.665
  27. [27] S. Zeng, Y. Lu, V.D. Rădulescu, Anisotropic double phase elliptic inclusion systems with logarithmic perturbation and multivalued convections, Appl. Math. Optim. 92 (2025), 6. https://doi.org/10.1007/s00245-025-10278-y
About the authors ▼
Zhenfeng Zhang
https://orcid.org/0009-0007-5285-6731
Hohai University, School of Mathematics, Nanjing, 210098, P.R. China
Mina Ghasemi
https://orcid.org/0000-0001-8526-4176
Universities of Messina, Catania and Palermo, Doctoral School on Mathematics and Computational Sciences, Viale Ferdinando Stagno d’Alcontres, 98166 Messina, Italy
Calogero Vetro (corresponding author)
https://orcid.org/0000-0001-5836-6847
University of Palermo, Department of Mathematics and Computer Science, Via Archirafi 34, 90123, Palermo, Italy
About the article ▼
Communicated by Marek Galewski.
Received: 2025-09-23.
Revised: 2025-11-06.
Accepted: 2025-11-23.
Published online: 2026-01-27.
Cite this article as ▼
Zhenfeng Zhang, Mina Ghasemi, Calogero Vetro, Galerkin-type minimizers to a competing problem for \((\vec{p},\vec{q})\)-Laplacian with variable exponents, Opuscula Math. 46, no. 1 (2026), 101-119, https://doi.org/10.7494/OpMath.202511231.
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