Differential inclusion problems with convolution and discontinuous nonlinearities
The authors study the quasilinear inclusion problem \[-\Delta_pu-\mu(x)\Delta_qu\in f(x,\rho*u,\nabla(\rho*u))+[\underline{g}(u),\overline{g}(u)] \ \text{in}\ \Omega,\quad u=0\text{ on }\ \partial\Omega,\] on a bounded domain \(\Omega\subset \mathbb{R}^N\), \(N\ge 2\), with \(p\)- and \(q\)-Laplacians \[\Delta_p:W_0^{1,p}(\Omega)\to W^{-1,p'}(\Omega), \ \Delta_q:W_0^{1,q}(\Omega)\to W^{-1,q'}(\Omega),\ 1<q<p<+\infty.\] The problem reduces to a quasilinear Dirichlet boundary value problem if \(g\) is continuous. The main results are existence and uniqueness theorems and the dependence of the solutions on the parameters of the inclusion.
quasilinear Dirichlet boundary value problem
scientific article; zbMATH DE number 7293862