Multiplicity of weak solutions for mixed variable exponent triharmonic problems with hardy potentials
Abstract
This paper is devoted to the study of a nonlinear boundary value problem driven by a combination of mixed variable-exponent triharmonic operators and a Hardy-type singular potential. Specifically, we consider −a(x) Δp(x)3u − b(x) Δq(x)3u + m(x) (|u|s(x)−2u)/|x|3s(x) = μf(x,u), & in Ω, u = Δu = Δ2u = 0, & on ∂Ω, where Ω⊂ ℝN is a smooth bounded domain with N > 3. The operators Δp(x)3 and Δq(x)3 represent triharmonic operators with variable exponents, creating a framework of nonstandard growth. Under suitable assumptions on the variable exponents $p(x), q(x), s(x)$, the weight functions $a(x), b(x), m(x)$, and the nonlinearity f(x,u), we establish the existence of at least three distinct weak solutions. Our approach is variational, relying on critical point theory for functionals defined on variable-exponent Sobolev spaces. By combining Ricceri's and Bonanno-Marano's abstract three critical points theorems with new Hardy-Rellich inequalities adapted to the variable-exponent setting, we prove our main multiplicity result for small values of the parameter μ. To our knowledge, this work presents the first systematic treatment of sixth-order problems featuring simultaneously mixed variable exponents and Hardy-type singular potentials.
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BibTeX
@misc{soualhine2026multiplicity,
title = {{Multiplicity of weak solutions for mixed variable exponent triharmonic problems with hardy potentials}},
author = {Khalid Soualhine and Adnane Belakhdar and Mohamed karim Hamdani},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.33576},
eprint = {2610.33576},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.33576},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Multiplicity of weak solutions for mixed variable exponent triharmonic problems with hardy potentials
AU - Khalid Soualhine
AU - Adnane Belakhdar
AU - Mohamed karim Hamdani
PY - 2026
DA - 2026/10/24
PB - arXiv
DO - 10.48550/arxiv.2610.33576
UR - https://arxiv.org/abs/2610.33576
AB - This paper is devoted to the study of a nonlinear boundary value problem driven by a combination of mixed variable-exponent triharmonic operators and a Hardy-type singular potential. Specifically, we consider −a(x) Δ_p(x)^3u − b(x) Δ_q(x)^3u + m(x) (|u|^s(x)−2u)/|x|^3s(x) = μf(x,u), & in Ω, u = Δu = Δ^2u = 0, & on ∂Ω, where Ω⊂ ℝ^N is a smooth bounded domain with N > 3. The operators Δ_p(x)^3 and Δ_q(x)^3 represent triharmonic operators with variable exponents, creating a framework of nonstandard growth. Under suitable assumptions on the variable exponents $p(x), q(x), s(x)$, the weight functions $a(x), b(x), m(x)$, and the nonlinearity f(x,u), we establish the existence of at least three distinct weak solutions. Our approach is variational, relying on critical point theory for functionals defined on variable-exponent Sobolev spaces. By combining Ricceri's and Bonanno-Marano's abstract three critical points theorems with new Hardy-Rellich inequalities adapted to the variable-exponent setting, we prove our main multiplicity result for small values of the parameter μ. To our knowledge, this work presents the first systematic treatment of sixth-order problems featuring simultaneously mixed variable exponents and Hardy-type singular potentials.
ER -