On the regularizing effect of strongly increasing lower order terms

We show an existence result of bounded weak solutions for some semilinear Dirichlet problems, even if the right hand side belongs only to L1(Ω). The model example is $$ \left\{ \begin{gathered} - \Delta u + h(u) = f(x) in \Omega , \hfill \\ u = 0 on \partial \Omega , \hfill \\ \end{gathered} \right. $$ where Ω is a bounded open set in ℝ N , h(s) is a continuous and increasing function such that $$ \mathop{{\lim }}\limits_{{s \to \sigma }} h(s) = + \infty $$ , for some δ>0 We also show a nonexistence result for some measures as data as in the model example $$ \left\{ \begin{gathered} - \Delta u + h(u) = {\delta _{{x0}}} in \Omega , \hfill \\ u = 0 on \partial \Omega , \hfill \\ \end{gathered} \right. $$ where $$ {\delta _{{{x_{0}}}}} $$ is the Dirac mass in x 0(x 0∈Ω).