Existence of critical points with semi-stiff boundary conditions for singular perturbation problems in simply connected planar domains

Let Ω be a smooth bounded simply connected domain in R2. We investigate the existence of critical points of the energy Eε(u) = 1/2 ́ Ω |∇u|2 + 1/(4ε2) ́ Ω(1−|u|), where the complex map u has modulus one and prescribed degree d on the boundary. Under suitable nondegeneracy assumptions on Ω, we prove existence of critical points for small ε. More can be said when the prescribed degree equals one. First, we obtain existence of critical points in domains close to a disc. Next, we prove that critical points exist in “most” of the domains.