Nonhomogeneous Dirichlet Problems with Unbounded Coefficient in the Principal Part

The main result of the paper establishes the existence of a bounded weak solution for a nonlinear Dirichlet problem exhibiting full dependence on the solution u and its gradient ∇u in the reaction term, which is driven by a p-Laplacian-type operator with a coefficient G(u) that can be unbounded. Through a special Moser iteration procedure, it is shown that the solution set is uniformly bounded. A truncated problem is formulated that drops that G(u) be unbounded. The existence of a bounded weak solution to the truncated problem is proven via the theory of pseudomonotone operators. It is noted that the bound of the solution for the truncated problem coincides with the uniform bound of the original problem. This estimate allows us to deduce that for an appropriate choice of truncation, one actually resolves the original problem.