Existence of solutions for singular elliptic equations with mixed boundary conditions

Abstract

In this article, we investigate generalized solutions for elliptic equations involving double-phase $(p(x),q(x))$-Laplacian operators with Hardy potential in variable exponent spaces. $(p(x),q(x))$-Laplacian operators include p-Laplacian, q-Laplacian, $p(x)$-Laplacian and $q(x)$-Laplacian operators. Additionally, $(p(x),q(x))$-Laplacian elliptic equations with singular coefficients under mixed boundary conditions are seldom mentioned in previous work. Some new theorems of the existence on the generalized solutions are reestablished for such equations via variational methods when the nonlinearity satisfies suitable hypotheses in variable exponent Lebesgue spaces.

Keywords:

• Double-phase Laplacian operator
• Hardy potential
• mixed boundary condition
• variable exponent space

2020 Mathematics Subject Classifications:

• 35J15
• 35J20
• 35J25

Disclosure statement

No potential conflict of interest was reported by the author(s).

Additional information

Funding

The authors extend their appreciation to the Deanship of Scientific Research at Northern Border University, Arar, KSA for funding this research work through the project number NBU-FFR-2024-1706-02. This work is supported by Natural Science Foundation of Shandong Province, China (ZR2021MA070). M.K. Hamdani was supported by the Tunisian Military Research Center for Science and Technology Laboratory LR19DN01.