Continuous dependence of solutions of elliptic BVPs on parameters

Abstract

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 $\{u_k\}_{k\in N}$) converges weakly to $x_{0}$ (corresponding to $u_{0}$) in $W^{1,q}_0(\Omega,R)$, provided that $\{u_k\}_{k\in N}$ 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.