Neumann problem for a quasilinear elliptic equation in a varying domain (Q2499647)

The author investigates the sequence of solutions \(u^{(s)}\) of the boundary value problem \[ \begin{aligned} Au^{(s)}=- \sum^n_{i=1} {\partial\over\partial x_i} \,\Biggl(a_i\Biggl(x, {\partial u^{(s)}\over\partial x}\Biggr)\Biggr)= f &\quad\text{in }\Omega^{(s)},\\ {\partial u^{(s)}\over\partial\nu_A}:= \sum^n_{i=1} a_i\Biggl(x,{\partial u^{(s)}\over\partial x}\Biggr)\cos(\nu, x_i)= 0 &\quad\text{on }\partial F^{(s)},\\ u^{(s)}= 0 &\quad\text{on }\partial\Omega, \end{aligned} \] where \(\Omega\) is a bounded domain in \(\mathbb{R}^N\), \(N\geq 3\), with a sufficiently smooth boundary \(\partial\Omega\), \(F^{(s)}\) is a closed set in \(\Omega\) depending on the parameter \(s\) running through the set of natural numbers, \(\nu\) is a normal vector to \(\partial F^{(s)}\), \(f\) is a function defined and compactly supported inside of \(\Omega\) (the support of \(f\) does not intersect a small neighborhood of \(F^{(s)}\)), \(A: W^1_p(\mathbb{R}^N)\to W^1_{p'}(\mathbb{R}^N)\) is a monotone operator satisfying appropriate conditions.