Abstract
In this paper, existence, localization and uniqueness results of solutions to elliptic Dirichlet boundary value problems are established. The approach is based on the nonlinear alternative of Leray-Schauder, the Brouwer fixed point theorem and the Galerkin method.MSC: 35J60, 47H10.
Highlights
In this paper, we consider the boundary value problem â§âšâdiv(|âu|p(x)â âu) = f (x, u) in, â©u = on â, ( . )where â RN is a nonempty bounded open set with smooth boundary â, p = p(x) â C+( ) with < pâ := min p(x) †p+ := max p(x) < +â and f : Ă R â R is a continuous function.The operator âdiv(|âu|p(x)â âu) is said to be the p(x)-Laplacian and becomes pLaplacian when p(x) ⥠p
One of the most studied models leading to a problem of this type is the model of motion of electro-rheological fluids, which are characterized by their ability to drastically change the mechanical properties under the influence of an exterior electro-magnetic field [, ]
Problems with variable exponent growth conditions appear in the mathematical modeling of stationary thermo-rheological viscous flows of non-Newtonian fluids and in the mathematical description of the processes filtration of an ideal baro-tropic gas through a porous medium [, ]
Summary
Many authors have studied the existence of solutions for problem A useful method for the investigation of solutions to semilinear problems is based on the Leray-Schauder continuation principle, or equivalently, on Schaefers fixed point theorem. The aim of this paper is to present new existence, localization and uniqueness results for solutions to problem
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