Structural stability for variable exponent elliptic problems. I. The $p(x)$-laplacian kind problems. Article - Janvier 2010

We study the structural stability (i.e., the continuous dependence on coefficients) of solutions of the elliptic problems under the form $$b(u_n)-\div mathfraka_n(x,\Grad u_n)=f_n.$$ The equation is set in a bounded domain $\Om$ of $\R^N$ and supplied with the homogeneous Dirichlet boundary condition on $\ptl\Om$. Here $b$ is a non-decreasing function on $\R$, and $\Bigl(\mathfraka_n(x,\xi)\Bigr)_n$ is a family of applications which verifies the classical Leray-Lions hypotheses but with a variable summability exponent $p_n(x)$, $1