Continuous dependence of solutions of elliptic BVPs on parameters

The continuous dependence of solutions for a certain class of elliptic PDE on functional parameters is studied in this paper. The main result is as follow: the sequence $\{x_k\}_{k\in N}$ of solutions of the Dirichlet problem discussed here (corresponding to parameters $\{x_k\}_{k\in N}$) converges weakly to $x_0$ (corresponding to $u_0$) in $W^{1,q}_0(\Omega,R)$, provided that $\{x_k\}_{k\in N}$ tends to $u_0$ a.e. in $\Omega$. Our investigation covers both sub and superlinear cases. We apply this result to some optimal control problems.