Existence and non-existence of solutions to the Ginzburg-Landau equations in a semi-infinite superconducting film

For the problem \begin{gather*} \frac {\psi ^{\prime \prime }}{\kappa ^2} = \psi ^{3} - \psi + A^{2}\psi , A^{\prime \prime } = \psi ^{2}A, \psi ^{\prime }(0)= \psi (\infty )=0, A^{\prime }(0)=A^{\prime }(\infty )=h , \end{gather*} it is proved for type II superconductors ($\kappa >1/\sqrt {2}$) that

1. No solutions can exist for $h \leq 1/\sqrt {2}$ other than the normal state $\psi \equiv 0$, $A=hx+C$;

2. Positive solutions ($\psi >0$) exist whenever $1/\sqrt {2}<h<h_{c_3}\approx 1.7\kappa$;

3. As $h \downarrow 1/\sqrt {2}$, the limit of any converging subsequence satisfies $A=0, \; \psi =1$ at infinity.