Existence of weak solutions to boundary problem for elliptic equations with variable exponents under $p\left(\cdot \right)$-uniformly convexity condition

Articles in Press

Document Type : Research Paper

Author
Mykola I Yaremenko
National Technical University of Ukraine, ”Igor Sikorsky Kyiv Polytechnic Institute” Kyiv, Ukraine, 37, Prospect Beresteiskyi (former Peremohy), Kyiv, Ukraine, 03056
10.22075/ijnaa.2025.39735.5578
Abstract
The main goal of this work is to establish the existence of a weak solution in {{$W_{1,0}^{p\left(\cdot \right)} \left(\Omega \right)$}} to the boundary problem
\[\begin{array}{l} {-div\left(a\left(x,\; \nabla u\right)\right)+d\left(x\right)u\left|u\right|^{p\left(x\right)-2} =f\left(x,\; u\right),} \\ {\left. u\right|_{\partial \Omega } =0,} \end{array}\]
where $\Omega $ is a bounded domain in \textit{$R^{n} ,\; n\ge 3$}. The main term $a\left(x,\; \xi \right)=\frac{\partial A\left(x,\; \xi \right)}{\partial \xi } $ satisfies: $p\left(\cdot \right)$-uniformly convexity and $p\left(\cdot \right)$- sub-homogeneousness conditions; $d\in L^{\infty } \left(\Omega \right)$ and $d_{m} >0$; and $f$ is a real Caratheodory function on $\Omega \times R$, which satisfies the following non-standard growth condition {{$\left|f\left(x,\; s\right)\right|\le A_{1} \left(1+\left|s\right|^{\gamma \left(x\right)-1} \right)$}}, where {{$p_{S} <\gamma _{m} \le \gamma _{S} <p_{m}^{*} $}}, {{$c_{1} $}} is a positive constant, and $p^{*} \left(x\right)=\frac{np\left(x\right)}{n-p\left(x\right)} $ for $p\left(x\right)<n$.
Keywords
elliptic equation
Dirichlet problem
variable exponent space
variational method
variable Lebesgue space
uniformly convex condition
variable subhomogeneous
weak solution
[1] A. Ahmed and M.E. Vall, Multiplicity analysis of positive weak solutions in a quasi‑linear Dirichlet problem inspired by Kirchhoff‑type phenomena, Int. J. Nonlinear Anal. Appl. 16 (2024), no. 1, 359-369.
[2] J.X. Cen, A.A. Khan, D. Motreanu, and S.D. Zeng, Inverse problems for generalized quasi‑variational inequalities with application to elliptic mixed boundary value systems, Inverse Probl. 38 (2022), 065006.
[3] J. Chabrowski and D.G. Costa, A positive solution for a weighted p‑Laplace equation with Hardy‑Sobolev's critical exponent, Bull. Malays. Math. Sci. Soc. 47 (2024), 57.
[4] Y. Chen and S. Antontsev, Solutions to a (p(x);q(x))(p(x);q(x))‑biharmonic elliptic problem on a bounded domain, Bound. Value Probl. 2023 (2023), 41.
[5] S.G. Deng and Q.M. Zhou, Three weak solutions for a class of p(x)p(x)‑Kirchhoff type biharmonic problems, Eurasian Math. J. 14 (2023), 45-58.
[6] L. Diening, P. Harjulehto, P. Hasto, and M. R, Lebesgue and Sobolev spaces with variable exponents, Lecture Notes in Mathematics, vol. 2017, Springer‑Verlag, Heidelberg, 2011.
[7] L. Diening and M. Bulicek, Competing Kohn‑Spencer Laplacian systems with convection in non‑isotropic Folland‑Stein space, Complex Var. Elliptic Equ. 69 (2024), 1-25.
[8] P. Drabek and A.I. Benedetti, Nonstandard competing anisotropic (p;q)(p;q)‑Laplacians with convolution, Bound. Value Probl. 2022 (2022), 69.
[9] X.L. Fan and D. Zhao, On the generalized Orlicz‑Sobolev Space Wk,p(x)Wk,p(x), J. Gansu Educ. College 1 (1998), no. 1, 1-6.
[10] R. Filippucci and P. Pucci, Existence of positive radial solutions for a problem involving weighted Heisenberg p(⋅)p(⋅)‑Laplacian operator, AIMS Math. 8 (2023), 1-19.
[11] A. Fiscella and P.P. Agarwal, A study on a class of weighted elliptic problems with indefinite nonlinearities, Appl. Anal. 103 (2024), 1-20.
[12] H.H. Ha, K. Ho, and I. Sim, Infinitely many solutions for a generalized p(⋅)p(⋅)‑Laplace equation involving Leray‑Lions type operators, Math. Methods Appl. Sci. 45 (2022), 8450-8473.
[13] P. Harjulehto, P. Hasto, and O. Toivanen, Holder regularity of quasiminimizers under generalized growth conditions, Calc. Var. Partial Differential Equations 56 (2017), 22, 26pp.
[14] P. Hasto and J. Habermann, Non‑variational solutions for anisotropic (p;q)(p;q)‑Laplacian systems with nonhomogeneous growth, Mediterr. J. Math. 22 (2025), 107.
[15] K. Ho and Y.-H. Kim, The concentration‑compactness principles for Ws,p(⋅)Ws,p(⋅)​ (RN) and application, Adv. Nonlinear Anal. 10 (2021), 816-848.
[16] K. Ho, Y.-H. Kim, and J. Lee, Schrodinger p(⋅)p(⋅)‑Laplace equations in RN involving indefinite weights and critical growth, J. Math. Phys. 62 (2021), 111506.
[17] K. Ho and I. Sim, On degenerate p(x)p(x)‑Laplace equations involving critical growth with two parameters, Nonlinear Anal. 132 (2016), 95-114.
[18] Z. Li and R. Liu, Existence and concentration behavior of solutions to 1‑Laplace equations on RN, J. Differ. Equ. 272 (2021), 399-432.
[19] M. Mihailescu and D. Repovs, Existence of radial solutions for a p(x)p(x)‑Laplacian Dirichlet problem, Adv. Differ. Equ. 2021 (2021), 233.
[20] X. Qian, Multiplicity of positive solutions for a class of nonlocal problems involving critical exponent, Electron. J. Qual. Theory Differ. Equ. 57 (2021), 1-14.
[21] X. Qiu, X. Ding, Y. Tao, J. Qu, J. Li, and Y. Liu, Research on the flow around a circular cylinder near a wall for shear‑thinning power‑law fluids, J. Non‑Newton. Fluid Mech. 333 (2024), 105323.
[22] V.D. Radulescu and D.D. Repovs, A (p(⋅);q(⋅))(p(⋅);q(⋅))‑Laplacian problem with the Steklov boundary conditions, Eurasian Math. J. 13 (2022), 125-138.
[23] A. Razani, Z. Musbah, F. Safari, and E.S. Sevim, Some existence results for a class of Dirichlet problems with variable exponents, Bound. Value Probl. 2024 (2024), 19.
[24] M. Ruzicka and D. Breit, A solution of a nonstandard Dirichlet Finsler (p; q)‑Laplacian, Filomat 38 (2024), 2313-2325.
[25] S. Song and Y. Cui, Existence of solutions for integral boundary value problems of mixed fractional differential equations under resonance, Bound. Value Probl. 2020 (2020), 23.
[26] J. Wang, Y. Yuan, and G. Li, Multifuture contrast enhancement algorithm for digital media images based on the diffusion equation, Adv. Math. Phys. 2022 (2022).
[27] Z.Q. Wang and J. Byeon, Standing waves for nonlinear Schrödinger equations with singular potentials, Nonlinear Anal. 26 (2009), 943-958.
[28] M. Willem, Minimax Theorem, Birkhauser, Boston, 1996.
[29] M.I. Yaremenko, Variable exponent de Giorgi classes and regularity of minimizers of variational integrals with nonstandard growth conditions, Math. Montisnigri 63 (2025), 53-68.
[30] S. Zhang and B. Sun, Nonlinear differential equations involving mixed fractional derivatives with functional boundary data, Math. Methods Appl. Sci. 45 (2022), no. 10, 5930-5944.
[31] S.D. Zeng, S. Migorski, and A.A. Khan, Nonlinear quasi‑hemivariational inequalities: Existence and optimal control, SIAM J. Control Optim. 59 (2021), 1246-1274.
[32] S.D. Zeng, A.A. Khan, and S. Migorski, A new class of generalized quasi‑variational inequalities with applications to Oseen problems under non‑smooth boundary conditions, Sci. China Math. 67 (2024), 315-338.
International Journal of Nonlinear Analysis and Applications

Articles in Press, Corrected Proof
Available Online from 25 August 2026

  • Receive Date 16 November 2025
  • Revise Date 08 December 2025
  • Accept Date 11 December 2025