Elliptic partial differential equations involving variable exponential operators with supercritical growth in variable exponential Lebesgue spaces

Yaremenko, M.

doi:10.47974/JIM-2236

Open Access Research Article

Elliptic partial differential equations involving variable exponential operators with supercritical growth in variable exponential Lebesgue spaces

^*M. YaremenkoCorresponding authormath.kiev@gmail.comDepartment of Differential Equations Faculty of Physics and Mathematics National Technical University of UkraineKyiv, 03056, Ukraine0000-0002-9209-6059View full profile →

* Corresponding author · click or hover a name for details

pp. 1–17Online FirstMay 2026DOI: 10.47974/JIM-2236XML

Received:: 02 Sep 2024
Published Online:: 09 May 2026
Article type:: Research Article
Language:: EN
Article no.:: JIM-2236
Pages:: 1–17

Abstract

The article is dedicated to the investigation of existence of weak solutions to a quasilinear elliptic equation involving terms with variable exponents –d/dxi ai (x, u, ∇u) – λb (x) |u|p(x) − 2 u = f (x, u) in the variable exponential Sobolev space. Applying the Galerkin method for pseudo-monotone operators, we establish conditions under which there exists at least one weak solution to the boundary problem for such equations. Also, we prove the uniqueness of the weak solution under some additional monotony conditions on operator A0 and function f in functional Sobolev space.

Keywords

Elliptic equation Pseudo-monotone operator Galerkin method Critical growth Variable exponential Lebesgue space Variable Laplacian

Subject Classifications

35J6035J6535J6635J7537L9947F10

References

[1] E. Acerbi and G. Mingione, “Regularity results for stationary electro-rheological fluids,” Archive for Rational Mechanics and Analysis, vol. 164, no. 3, pp. 213–259 (2002).
[2] L. Boccardo, D. Giachetti, J. I. Díaz, and F. Murat, “Existence and regularity of renormalized solutions for some elliptic problems involving derivations of nonlinear terms,” Journal of Differential Equations, vol. 106, no. 2, pp. 215–237 (1993).
[3] Y. M. Chen, S. Levine, and M. Rao, “Variable exponent, linear growth functionals in image restoration,” SIAM Journal on Applied Mathematics, vol. 66, no. 4, pp. 1383–1406 (2006).
[4] G. Dal Maso, F. Murat, L. Orsina, and A. Prignet, “Renormalized solutions of elliptic equations with general measure data,” Annali della Scuola Normale Superiore di Pisa – Classe di Scienze (Series IV), vol. 28, no. 4, pp. 741–808 (1999).
[5] M. Ding, C. Zhang, and S. Zhou, “On optimal C1,α estimates for p(x)-Laplace type equations,” Nonlinear Analysis, vol. 200, Art. no. 112030 (2020).
[6] X. L. Fan and Q. H. Zhang, “Existence of solutions for p(x)-Laplacian Dirichlet problems,” Nonlinear Analysis, vol. 52, no. 8, pp. 1843–1852 (2003).
[7] X. L. Fan and D. Zhao, “On the spaces Lp(x) and Wm,p(x),” Journal of Mathematical Analysis and Applications, vol. 263, no. 2, pp. 424–446 (2001).
[8] X. L. Fan, Q. H. Zhang, and D. Zhao, “Eigenvalues of p(x)-Laplacian Dirichlet problem,” Journal of Mathematical Analysis and Applications, vol. 302, no. 2, pp. 306–317 (2005).
[9] J. Heinonen, T. Kilpeläinen, and O. Martio, Nonlinear Potential Theory of Degenerate Elliptic Equations. Oxford, U.K.: Oxford Univ. Press (1993).
[10] Y. C. Kim, “Nonlocal Harnack inequalities for nonlocal heat equations,” Journal of Differential Equations, vol. 267, no. 11, pp. 6691–6757 (2019), doi: 10.1016/j.jde.2019.07.006.
[11] G. Li, G. Motreanu, D. Wu, and H. Zhang, “Multiple solutions with constant sign for a (p,q)-elliptic system Dirichlet problem with product nonlinear term,” Boundary Value Problems, vol. 2018, Art. no. 67 (2018).
[12] D. Liu, B. Wang, and P. Zhao, “On the trace regularity results of Musielak–Orlicz–Sobolev spaces in a bounded domain,” Communications on Pure and Applied Analysis, vol. 15, no. 4, pp. 1643–1659 (2016).
[13] D. Liu and P. Zhao, “Solutions for a quasilinear elliptic equation in Musielak–Sobolev spaces,” Nonlinear Analysis: Real World Applications, vol. 26, pp. 315–329 (2016).
[14] A. Matsoukas and N. Yannakakis, “The double phase Dirichlet problem when the lowest exponent is equal to 1,” Journal of Mathematical Analysis and Applications, vol. 526, no. 1, Art. no. 127270 (2023).
[15] J. X. Yin, J. K. Li, and Y. Y. Ke, “Existence of positive solutions for the p(x)-Laplacian equation,” Rocky Mountain Journal of Mathematics, vol. 42, no. 5, pp. 1675–1758 (2012).
[16] X. J. Wang, “Neumann problems of semilinear elliptic equations involving critical Sobolev exponents,” Journal of Differential Equations, vol. 93, no. 2, pp. 283–310 (1991).
[17] Q. H. Zhang, “Existence of solutions for p(x)-Laplacian equations with singular coefficients in RNRN,” Journal of Mathematical Analysis and Applications, vol. 348, no. 1, pp. 38–50 (2008).
[18] D. Z. Zhang, K. H. Shi, Z. C. Guo, and B. Y. Wu, “A class of elliptic systems with discontinuous variable exponents and data for image denoising,” Nonlinear Analysis: Real World Applications, vol. 50, pp. 448–468 (2019).
[19] C. Zhang and S. Zhou, “Entropy and renormalized solutions for the p(x)-Laplacian equation with measure data,” Bulletin of the Australian Mathematical Society, vol. 82, no. 3, pp. 459–479 (2010).
[20] C. Zhang and X. Zhang, “Renormalized solutions for the fractional p(x)-Laplacian equation with L1 data,” Nonlinear Analysis, vol. 190, Art. no. 111610 (2020).
[21] X. L. Fan, Q. H. Zhang, and D. Zhao, “Eigenvalues of p(x)-Laplacian Dirichlet problem,” Journal of Mathematical Analysis and Applications, vol. 302, no. 2, pp. 306–317 (2005).
[22] Q. Zhang, “Sampling theorem on shift invariant subspaces in mixed Lebesgue spaces,” Journal of Interdisciplinary Mathematics, vol. 22, no. 2, pp. 121–128 (2019).

Views: 14Downloads: 5Citations: 0