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

Article type: Research Article

Authors: Pistoia, Angela

Affiliations: Dipartimento di Matematica Applicata “U. Dini”, Università di Pisa, 56127 Pisa, Italy E‐mail: pistoia@mail.dm.unipi.it

Abstract: 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.

Keywords: Semilinear Neumann problem, semilinear Dirichlet problem, singular perturbation, symmetry of solutions

Journal: Asymptotic Analysis, vol. 22, no. 2, pp. 149-161, 2000