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Let $\Omega$ be a smooth bounded simply connected domain in $\mathbb{R}^2$.We investigate the existence of critical points of the energy $E_\varepsilon(u)=1/2\int_\Omega |\nabla u|^2+1/(4\varepsilon^2)\int_\Omega (1-|u|^2)^2$,where the complex map $u$ has modulus one and prescribed degree $d$ on theboundary. Under suitable nondegeneracy assumptions on $\Omega$, we proveexistence of critical points for small $\varepsilon$. More can be said when theprescribed degree equals one. First, we obtain existence of critical points indomains close to a disc. Next, we prove that critical points exist in "most" ofthe domains.

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