On second order elliptic equations and variational inequalities with anisotropic principal operators (Q519386)

The following boundary value problem is under consideration: \[ \mathrm{div}[\nabla\Phi(\nabla u)] = f(x, u), \quad u|_{\partial \Omega} = 0, \] where \(\Omega\) is a bounded domain in \({\mathbb R}^{n}\) with sufficiently smooth boundary, the convex function \(\Phi(x): {\mathbb R}^{n}\to [0,\infty]\) satisfies the following conditions: \(\Phi(0)=0\), \(\lim_{x\to \infty} \Phi(x)=\infty\), \(\Phi(x)=\Phi(-x)\) for all \(x\), the set \(\{x:\Phi(x)=\infty\}\) is separated from 0, and \(\Phi\) is lower semicontinuous. The problem is studied in an anisotropic Orlicz-Sobolev space associated with \(\Phi\). Under some additional conditions on the functions \(\Phi,f\), existence of solutions to this problem is proven and the corresponding variational inequality is studied.