Discontinuity, Nonlinearity, and Complexity 14(2) (2025) 417--426 | DOI:10.5890/DNC.2025.06.013

Mykola Ivanovich Yaremenko

Department of Mathematics, ``Igor Sikorsky Kyiv Polytechnic Institute", National Technical University of Ukraine,

37, Prospect Beresteiskyi (former Peremohy), Kyiv, Ukraine, 03056, Ukraine

Abstract

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 $ \lambda$ 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.

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