Three weak solutions for degenerate weighted quasilinear elliptic equations with indefinite weights and variable exponents

Khaled Kefi; Nasser S. Albalawi; Khaled Kefi; Nasser S. Albalawi

doi:10.3934/math.2025207

AIMS Mathematics

2025, Volume 10, Issue 2: 4492-4503. doi: 10.3934/math.2025207

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Research article

Three weak solutions for degenerate weighted quasilinear elliptic equations with indefinite weights and variable exponents

Khaled Kefi ^{1
,
,},
Nasser S. Albalawi ²

1.
Center for Scientific Research and Entrepreneurship, Northern Border University, Arar 73213, Saudi Arabia
2.
Department of Computer Sciences, Faculty of Computing and Information Technology, Northern Border University, Rafha, Saudi Arabia

Received: 30 December 2024 Revised: 19 February 2025 Accepted: 25 February 2025 Published: 28 February 2025
MSC : 35J15, 35J20, 35J25

This paper explores the multiplicity of weak solutions to a class of weighted elliptic problems with variable exponents, incorporating a Hardy term and a nonlinear indefinite source term. Using critical point theory applied to the associated energy functional, we establish the existence of at least three weak solutions under general assumptions on the weight function and the nonlinearity. This result has wide applicability, extending existing theories on quasilinear elliptic equations.
- variational methods,
- Hardy inequality,
- degenerate $ p(x) $-Laplacian operators
Citation: Khaled Kefi, Nasser S. Albalawi. Three weak solutions for degenerate weighted quasilinear elliptic equations with indefinite weights and variable exponents[J]. AIMS Mathematics, 2025, 10(2): 4492-4503. doi: 10.3934/math.2025207

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Abstract

This paper explores the multiplicity of weak solutions to a class of weighted elliptic problems with variable exponents, incorporating a Hardy term and a nonlinear indefinite source term. Using critical point theory applied to the associated energy functional, we establish the existence of at least three weak solutions under general assumptions on the weight function and the nonlinearity. This result has wide applicability, extending existing theories on quasilinear elliptic equations.

References

[1]	A. Aberqi, O. Benslimane, A. Ouaziz, On some weighted fractional $ p(., .) $-Laplacian problems, Palest. J. Math., 12 (2023), 56–66.
[2]	S. Antontsev, S. Shmarev, Evolution PDEs with nonstandard growth conditions. Existence, uniqueness, localization, blow-up, Paris: Atlantis Press, 2015.
[3]	L. Baldelli, R. Filippucci, Multiplicity results for generalized quasilinear critical Schrödinger equations in $ \mathbb{R}^N $, Nonlinear Differ. Equ. Appl., 31 (2024), 8. https://doi.org/10.1007/s00030-023-00897-1 doi: 10.1007/s00030-023-00897-1
[4]	G. Bonanno, S. Marano, On the structure of the critical set of non-differentiable functions with a weak compactness condition, Appl. Anal., 89 (2010), 1–10.
[5]	Z. Chaouai, M. Tamaazousti, A priori bounds and multiplicity results for slightly superlinear and sublinear elliptic $ p $-Laplacian equations, Nonlinear Anal., 237 (2023), 113388.
[6]	P. Drábek, A. Kufner, F. Nicolosi, Quasilinear elliptic equations with degenerations and singularities, Berlin-New York: de Gruyter, 1997.
[7]	D. Edmunds, J. Rakosnik, Sobolev embeddings with variable exponent, Stud. Math., 143 (2000), 267–293.
[8]	X. Fan, J. S. Shen, D. Zhao, Sobolev embedding theorems for spaces $ W^{k, p(x)} $, J. Math. Anal. Appl., 262 (2001), 749–760.
[9]	X. Fan, D. Zhao, On the spaces $ L^{p(x)} $ and $ W^{m, p(x)} $, J. Math. Anal. Appl., 263 (2001), 424–446.
[10]	X. Fan, Solutions for $ p(x) $-Laplacian Dirichlet problems with singular coefficients, J. Math. Anal. Appl., 312 (2005), 464–477.
[11]	X. Fan, X. Han, Existence and multiplicity of solutions for $ p(x) $-Laplacian equations in $ \mathbb{R}^{N} $, Nonlinear Anal., 59 (2004), 173–188.
[12]	P. J. García Azorero, I. Peral Alonso, Hardy inequalities and some critical elliptic and parabolic problems, J. Differ. Equ., 144 (1998), 441–476.
[13]	J. Simon, Régularité de la solution d'une équation non linéaire dans $ \mathbb{R}^{N} $, In: Journées d'Analyse Non Linéaire, Berlin: Springer, 1978,205–227.
[14]	K. Kefi, N. Chung, W. Abdelfattah, Triple weak solution for $ p(x) $-Laplacian like problem involving no flux boundary condition, Georgian Math. J., 2024. https://doi.org/10.1515/gmj-2024-2043
[15]	Y. H. Kim, L. Wang, C. Zhang, Global bifurcation of a class of degenerate elliptic equations with variable exponents, J. Math. Anal. Appl., 371 (2010), 624–637.
[16]	J. Liu, Q. An, Analysis of degenerate $ p $-Laplacian elliptic equations involving Hardy terms: Existence and numbers of solutions, Appl. Math. Lett., 160 (2025), 109330.
[17]	J. Liu, Z. Zhao, Existence of triple solutions for elliptic equations driven by $ p $-Laplacian-like operators with Hardy potential under Dirichlet-Neumann boundary conditions, Bound. Value Probl., 3 (2023), 3. https://doi.org/10.1186/s13661-023-01692-8 doi: 10.1186/s13661-023-01692-8
[18]	J. Liu, Z. Zhao, Leray-Lions type $ p(x) $-biharmonic equations involving Hardy potentials, Appl. Math. Lett., 149 (2024), 108907.
[19]	K. Ho, I. Sim, Existence and some properties of solutions for degenerate elliptic equations with exponent variable, Nonlinear Anal., 98 (2014), 146–164.
[20]	K. Ho, I. Sim, Existence results for degenerate $ p(x) $-Laplace equations with Leray-Lions type operators, Sci. China Math., 60 (2017), 133–146.

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