On the strong unique continuation of eigenfunctions with Laplacian operator in variable Lebesgue spaces
Authors
G. L. Basendwah
- Department of Mathematics, King Abdulaziz University, Jeddah 21589, Saudi Arabia.
N. M. Aloraini
- Department of Mathematics, College of Science, Qassim University, Buraydah 51452, Saudi Arabia.
B. Sultan
- Department of Mathematics, Quaid-I-Azam University, Islamabad 45320, Pakistan.
I.-L. Popa
- Department of Computing, Mathematics and Electronics, “1 Decembrie 1918” University of Alba Iulia, 510009 Alba Iulia, Romania.
- Faculty of Mathematics and Computer Science, Transilvania University of Brasov, Iuliu Maniu Street 50, 500091 Brasov, Romania.
Abstract
This note discusses a strong unique continuation property for the eigenfunctions of the \(q(\cdot)\)-Laplacian operator, where the potential \(\mathcal{V}\in L^{q(\cdot)}(\Gamma)\). Firstly we define some important notations and results for variable exponents. Our goal is to prove the strong unique continuation properties in variable Lebesgue space setting. We also discuss the problem \(-\mbox{div}(|\nabla \mu |^{q(\cdot)-2}\nabla \mu )+V|\mu|^{q(\cdot)-2}\mu=0\) in \(\Gamma\), where the potential \(\mathcal{V}\) is not zero and belongs to \(L^{q(\cdot)}(\Gamma)\) and \(\Gamma\) is a bounded domain in \(\mathbb{R}^n\). We will prove some important inequalities for variable exponent to prove our main results.
Share and Cite
ISRP Style
G. L. Basendwah, N. M. Aloraini, B. Sultan, I.-L. Popa, On the strong unique continuation of eigenfunctions with Laplacian operator in variable Lebesgue spaces, Journal of Mathematics and Computer Science, 41 (2026), no. 4, 564--570
AMA Style
Basendwah G. L., Aloraini N. M., Sultan B., Popa I.-L., On the strong unique continuation of eigenfunctions with Laplacian operator in variable Lebesgue spaces. J Math Comput SCI-JM. (2026); 41(4):564--570
Chicago/Turabian Style
Basendwah, G. L., Aloraini, N. M., Sultan, B., Popa, I.-L.. "On the strong unique continuation of eigenfunctions with Laplacian operator in variable Lebesgue spaces." Journal of Mathematics and Computer Science, 41, no. 4 (2026): 564--570
Keywords
- \(q(\cdot)\)-Laplacian operator
- eigenfunctions
- mathematical operators
MSC
References
-
[1]
A. Abdalmonem, O. Abdalrhman, H. E. Mohammed, Commutators of fractional integral with variable kernel on variable exponent Herz-Morrey spaces, Open J. Math. Anal., 3 (2019), 19–29
-
[2]
A. Abdalmonem, O. Abdalrhman, S. Tao, Boundedness of fractional integral with variable kernel and their commutators on variable exponent Herz spaces, Appl. Math., 7 (2016), 1165–1182
-
[3]
A. Abdalmonem, A. Scapellato, Intrinsic square functions and commutators on Morrey-Herz spaces with variable exponents, Math. Methods Appl. Sci., 44 (2021), 12408–12425
-
[4]
E. Acerbi, G. Mingione, Regularity results for a class of functionals with non-standard growth, Arch. Ration. Mech. Anal., 156 (2001), 121–140
-
[5]
R. E. Castillo, A note on unique continuation of eigenfunctions for p-Laplacian operator in a bounded domain Ω in Rn with a potential V in Lp(Ω), Complex Var. Elliptic Equ., 69 (2024), 1763–1769
-
[6]
F. Chiarenza, M. Frasca, A remark on a paper by C. Fefferman, Proc. Amer. Math. Soc., 108 (1990), 407–409
-
[7]
K. Kaushik, A. Kumar, A. Khan, T. Abdeljawad, Existence of solutions by fixed point theorem of general delay fractional differential equation with p-Laplacian operator, AIMS Math., 8 (2023), 10160–10176
-
[8]
H. Khan, F. Jarad, T. Abdeljawad, A. Khan, A singular ABC-fractional differential equation with p-Laplacian operator, Chaos Solitons Fractals, 129 (2019), 56–61
-
[9]
R. A. Khan, A. Khan, A. Samad, H. Khan, On existence of solutions for fractional differential equation with p-Laplacian operator, J. Fract. Calc. Appl., 5 (2014), 28–37
-
[10]
H. Khan, Y. Li, W. Chen, D. Baleanu, A. Khan, Existence theorems and Hyers-Ulam stability for a coupled system of fractional differential equations with p-Laplacian operator, Bound Value Probl., 2017 (2017), 16 pages
-
[11]
H. Khan, C. Tunc, W. Chen, A. Khan, Existence theorems and Hyers-Ulam stability for a class of hybrid fractional differential equations with p-Laplacian operator, J. Appl. Anal. Comput., 8 (2018), 1211–1226
-
[12]
J. Musielak, Orlicz Spaces and Modular Spaces, Springer-Verlag, Berlin (1983)
-
[13]
J. Musielak, W. Orlicz, On modular spaces, Studia Math., 18 (1959), 49–65
-
[14]
H. Nakano , Modulared Semi-Ordered Linear Spaces, Maruzen Co. Ltd., Tokyo (1950)
-
[15]
W. Orlicz, Über konjugierte Exponentenfolgen, Studia Math., 3 (1931), 200–211
-
[16]
M. R˚ užiˇcka, Electrorheological fluids: modeling and mathematical theory, Springer-Verlag, Berlin (2000)
-
[17]
I. Šarapudinov, On the topology of the space Lt(t)([0; 1]), Matem. Zametki., 26 (1979), 613–632
-
[18]
B. Sultan, M. Sultan, Boundedness of commutators of rough Hardy operators on grand variable Herz spaces, Forum Math., 36 (2024), 717–733
-
[19]
B. Sultan, M. Sultan, Sobolev-type theorem for commutators of Hardy operators in grand Herz spaces, Ukrainian Math. J., 76 (2024), 1196–1213
-
[20]
B. Sultan, M. Sultan, Boundedness of higher order commutators of Hardy operators on grand Herz-Morrey spaces, Bull. Sci. Math., 190 (2024), 19 pages
-
[21]
M. Sultan, B. Sultan, A note on the boundedness of Marcinkiewicz integral operator on continual Herz-Morrey spaces, Filomat, 39 (2025), 2017–2027
-
[22]
M. Sultan, B. Sultan, Boundedness of sublinear operators on grand central Orlicz-Morrey spaces, Bull. Sci. Math., 205 (2025), 11 pages
-
[23]
M. Sultan, B. Sultan, λ-central Musielak-Orlicz-Morrey spaces, Arab. J. Math. (Springer), 14 (2025), 357–363
-
[24]
M. Sultan, B. Sultan, A. Aloqaily, N. Mlaiki, Boundedness of some operators on grand Herz spaces with variable exponent, AIMS Math., 8 (2023), 12964–12985
-
[25]
M. Sultan, B. Sultan, R. E. Castillo, Weighted composition operator on Gamma spaces with variable exponent, J. Pseudo- Differ. Oper. Appl., 15 (2024), 38 pages
-
[26]
B. Sultan, M. Sultan, I. Khan, On Sobolev theorem for higher commutators of fractional integrals in grand variable Herz spaces, Commun. Nonlinear Sci. Numer. Simul., 126 (2023), 11 pages
-
[27]
B. Sultan, M. Sultan, A. Khan, T. Abdeljawad, Boundedness of an intrinsic square function on grand p-adic Herz-Morrey spaces, AIMS Math., 8 (2023), 26484–26497
-
[28]
M. Sultan, B. Sultan, A. Khan, T. Abdeljawad, Boundedness of Marcinkiewicz integral operator of variable order in grand Herz-Morrey spaces, AIMS Math., 8 (2023), 22338–22353
-
[29]
B. Sultan, M. Sultan, A. Khan, T. Abdeljawad, Boundedness of commutators of variable Marcinkiewicz fractional integral operator in grand variable Herz spaces, J. Inequal. Appl., 2024 (2024), 16 pages
-
[30]
I. Tsenov, Generalization of the problem of best approximation of a function in the space Ls, Uch. Zap. Dagest. Gos. Univ., 7 (1961), 25–37
-
[31]
W. M. Winslow, Induced fibration of suspensions, J. Appl. Phys., 20 (1949), 1137–1140
-
[32]
P. Zamboni, Unique continuation for non-negative solutions of quasilinear elliptic equations, Bull. Austral. Math. Soc., 64 (2001), 149–156
-
[33]
V. Zhikov, Averaging of functionals in the calculus of variations and elasticity, Math. USSR Izv., 29 (1987), 33–66
-
[34]
V. Zhikov, On passing to the limit in nonlinear variational problem, Sb. Math., 183 (1992), 427–459
