THE DIRICHLET PROBLEMS OF A CLASS OF SEMILINEAR EQUATIONS

In this paper we proved the exstence of solutions to the following Diriehlet problem: $\left\{\begin{array}{l}-\left(\frac{\Delta}{2}+\mu\right) u+f(u)=\nu, \quad \text { in } D, \\ \left.u\right|_{\partial D}=g,\end{array}\right.$ where D is a bounded regular domain of R^d,μ and v are signed measures belonging to the generalizead Kato class GK_d,f is a continuously differentiable function on R¹,g is a continuous function on $\partial D$.The main tools are Dirichlet forms and Markov processes.