Some cases of weak continuity in nonlinear Dirichlet problems (Q2324737)

The main goal of this paper is to show how to pass to the limit in some nonlinear elliptic problems, despite the weak convergence of the right-hand-sides. More especifically, the main result of this paper concerns the stability of weak solutions of the nonlinear Dirichlet problem \[ w\in W^{1,p}_0(\Omega), \ \ \ \ -\text{div}(a(x,\nabla w))=y, \] with respect to the weak convergence of the right-hand-side \(y\). Here \(\Omega\subset\mathbb{R}^N\) with \(N \ge 2\) is a bounded, open set, and \(p > 1\). It is assumed that \(a:\Omega\times\mathbb{R}^N\rightarrow\mathbb{R}^N\) satisfies, for almost every \(x\in\Omega\) and every \(\xi,\eta\in\mathbb{R}^N\) with \(\xi\neq \eta\) \[ a(x,\xi)\cdot\xi\ge \alpha|\xi|^p, \ \ |a(x,\xi)|\le\beta|\xi|^{p-1}, \] for some \(\alpha,\beta\) positive constants, and \[(a(x,\xi)-a(x,\eta))\cdot (\xi-\eta)>0. \] The author considers the nonlinear boundary value problems \[ -\text{div}(a(x,\nabla u_n))=f_n(x), \ \text{ in } \Omega\] \[u_n=0 \ \text{ on } \partial\Omega, \] and \[ -\text{div}(a(x,\nabla u_{\infty}))=f_{\infty}(x), \ \text{ in } \Omega \] \[ u_{\infty}=0 \ \text{ on } \partial\Omega. \] Under the assumption that \((f_n)_n\) converges weakly to \(f_{\infty}\) in \(L^{(p^*)'}(\Omega)\), the author proves that \((u_n)_n\) converges weakly, and only weakly, to \(u_{\infty}\) in \(W^{1,p}_0(\Omega)\). The Minty approach is used to overcome the difficulty of the passage to the limit. Furthermore, the author studies other nonlinear Dirichlet problems and minimization problems for integral functionals, where it is possible to pass to the limit, despite the weak convergence on the right-hand-sides.