Existence of Solution to Elliptic Equations with Generalized p(· )-Laplacian Operator in the Sobolev Spaces with Variable Exponents - Yaremenko, Mykola Ivanovich (Discontinuity, Nonlinearity, and Complexity, 2026)
In this article, we establish fairly general conditions 1) - 4) under which the Dirichlet problem for the parametrized elliptic partial differential equations involving p()-Laplacian has a weak solution in Sobolev spaces with variable exponents. The research employs variational methods and a mountain pass theorem in the variable exponent spaces. The existence of weak solutions to the Dirichlet boundary problem for the elliptic partial differential equation with a positive parameter λ is established in the variable exponent Sobolev space. The variable exponent Laplace equations play a prominent role in the modeling of diffusion processes with changing temperature and in fractional quantum mechanics. These results can be applied to image restoration problems.