On the existence of symmetric solutions to some singularly perturbed semilinear elliptic problems

We prove the existence of symmetric solutions to the singularly perturbed semilinear Dirichlet or Neumann problem $ \cases{-\varepsilon^2\Delta w+w=f(w) \quad \mbox{in}\ \varOmega,\cr w>0\quad \mbox{in}\ \varOmega,\cr \curpartial w/\curpartial\nu=0\quad \mbox{or}\quad w=0\quad \mbox{on}\ \curpartial\varOmega} $ under certain assumptions on the nonlinearity $f$ .