Boundedness of the extremal solution of some \(p\)-Laplacian problems (Q880298)

The author deals with the following Dirichlet problem for the \(p\)-Laplacian operator \(-\Delta_p u:= -\text{div}(|\nabla u|^{p-2}\nabla u)\), \[ -\Delta_p u=\lambda f(u)\quad\text{in }\Omega,\tag{1} \] \[ u= 0\quad\text{on }\partial\Omega,\tag{2} \] where \(\Omega\) is a smooth domain of \(\mathbb{R}^N\), \(p> 1\) and \(\lambda\) is a positive parameter. Under appropriate conditions on \(f\), the author proves existence of an extremal solution of (1)--(2). Moreover, he gives \(L^q\) and \(W^{1,q}_0\) estimates for the extremal solution.