Remarks on a class of integro-differential problems

A class of variational elliptic problems involving integro-differential diffusion and Dirichlet boundary condition is studied. The absence of a maximum principle is observed and sign-changing solutions are obtained when the nonlinearity is a sublinear or superlinear power of u+, depending on the size of a positive parameter multiplying the nonlocal term. More general integro-differential operators, for which the problem is no longer variational, are also considered.