Boundedness of the extremal solution of some image-Laplacian problems Original Research Article

In this article we consider the pp-Laplace equation View the MathML source−Δpu=λf(u) on a smooth bounded domain of RNRN with zero Dirichlet boundary conditions. Under adequate assumptions on ff we prove that the extremal solution of this problem is in the energy class View the MathML sourceW01,p(Ω) independently of the domain. We also obtain LqLq and W1,qW1,q estimates for such a solution. Moreover, we prove its boundedness for some range of dimensions depending on the nonlinearity ff.