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Existence of a nontrival solution for Dirichlet problem involving p(x)-Laplacian

Sylwia Barnaś (2014)

Discussiones Mathematicae, Differential Inclusions, Control and Optimization

In this paper we study the nonlinear Dirichlet problem involving p(x)-Laplacian (hemivariational inequality) with nonsmooth potential. By using nonsmooth critical point theory for locally Lipschitz functionals due to Chang [6] and the properties of variational Sobolev spaces, we establish conditions which ensure the existence of solution for our problem.

Existence of entropy solutions for degenerate quasilinear elliptic equations in L 1

Albo Carlos Cavalheiro (2014)

Communications in Mathematics

In this article, we prove the existence of entropy solutions for the Dirichlet problem ( P ) - div [ ω ( x ) 𝒜 ( x , u , u ) ] = f ( x ) - div ( G ) , in Ω u ( x ) = 0 , on Ω where Ω is a bounded open set of N , N 2 , f L 1 ( Ω ) and G / ω [ L p ' ( Ω , ω ) ] N .

Existence of entropy solutions to nonlinear degenerate parabolic problems with variable exponent and L 1 -data

Abdelali Sabri, Ahmed Jamea, Hamad Talibi Alaoui (2020)

Communications in Mathematics

In the present paper, we prove existence results of entropy solutions to a class of nonlinear degenerate parabolic p ( · ) -Laplacian problem with Dirichlet-type boundary conditions and L 1 data. The main tool used here is the Rothe method combined with the theory of variable exponent Sobolev spaces.

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