Potential estimates in Orlicz spaces (Q1262408)

Using the capacity method, the author establishes inequalities useful in solving quasilinear Dirichlet problems of the form \[ (1-\Delta)^ mu=f(x,u)\quad in\quad \Omega \subset {\mathbb{R}}^ n,\quad u(x)=0\quad on\quad \partial \Omega, \] by means of topological methods. More precisely, he considers continuous even functions \(M_ 0\), M: \({\mathbb{R}}\to {\mathbb{R}}\) for which there are constants \(C_ 1\), \(C_ 2\) such that (a) \(M_ 0\) is strictly increasing in \(| t|\) and M is convex with \(M(2t)\leq C_ 1M(t),\) \(t>0;\) (b) \(M_ 0^{-1}(\int^{\infty}_{0}M_ 0(g(t))dM_ 0(t))\leq C_ 2M^{-1}(\int^{\infty}_{0}M(g(t))dM(t))\) for all decreasing functions g(t)\(\geq 0;\) (c) \(M_ 0(t)\to \infty\), M(t)\(\to \infty\) as \(t\to \infty\) and \(M(0)=0.\) There are interesting examples for which these conditions hold. Defining \(\rho_{\nu}(u,M)=\int M(u)d\nu,\) the author determines sufficient conditions on \(M_ 0\), M, \(\nu\) and \(\mu\) under which \[ M_ 0^{- 1}(\rho_{\nu}(u,M_ 0))\leq CM^{-1}(\rho_{\nu}((1-\Delta)^ mu,M)) \] holds for all \(u\in C^{\infty}({\mathbb{R}}^ n)\) and \[ M_ 0^{- 1}(\rho_{\mu}(u,M_ 0))\leq CM^{-1}(\rho_{\nu}(\Delta^ mu,M)) \] holds for all \(u\in C^{\infty}_ 0({\mathbb{R}}^ n).\) The author also offers applications and comments.