On boundary value problems for degenerate quasilinear elliptic equations and inequalities
The boundary-value problem for a quasilinear elliptic equation \[ \text{div }A(x, \nabla u)= f,\quad x\in\Omega,\quad u= 0,\quad x\in\partial\Omega \] is considered. Solutions are sought for in weighted Sobolev spaces as extremum points of the variational inequality \[ L(u,\nu- u)\geq (f,\nu- u),\quad \forall\nu\in K,\quad u\in K. \] Using the fixed point theory and similar techniques, the authors study a nonlinear perturbed problem, where \(f\) depends not only on \(x\) but also on \(u\), the growth of \(f\) with respect to \(u\) being restricted. The existence and continuity of the solutions is proved under the restriction \[ | F(x,u)|\leq a(x)+ b| u|^\sigma,\quad f(x,u)= \int_\Omega F(x,u(x))\nu(x) dx. \] For the quasilinear degenerate case it is also proved that the existence of subsolutions and supersolutions follows from the existence of extremal solutions of the variational inequality.
scientific article; zbMATH DE number 1206648
scientific article