Existence of entropy solutions for degenerate elliptic unilateral problems with variable exponents

Abstract

In this article, we study the following degenerate unilateral problems: $$ -\mbox{ div} (a(x,\nabla u))+H(x,u,\nabla u)=f,$$ which is subject to the Weighted Sobolev spaces with variable exponent $W^{1,p(x)}_{0}(\Omega,\omega)$, where $\omega$ is a weight function on $\Omega$, ($\omega$ is a measurable, a.e. strictly positive function on $\Omega$ and satisfying some integrability conditions). The function $H(x,s,\xi)$ is a nonlinear term satisfying some growth condition but no sign condition and the right hand side $f\in L^1(\Omega)$.

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

“…Bénilan, L. Boccardo, T. Gallouet, R. Gariepy, M. Pierre, J.L. Vazquez in [6], this notion was then adapted by many authors to study some nonlinear elliptic and parabolic problems with a constant or variable exponent and Dirichlet or Neumann boundary conditions (see for example [1], [4], [14], [16], [19], and [20]).…”