Non trivial \(L^q\) solutions to the Ginzburg-Landau equation
The Ginzburg-Landau equation \(-\Delta u = u(1-| u| ^2)\) in \(\mathbb R^n\) is studied for \(n=3\) or \(4\). The existence of a nontrivial solution to this equation satisfying a decay condition at infinity is obtained. More precisely, it is shown that for \(f \in L^2(S^{n-1})\) small enough, there exists \(a > 0\) such that the Ginzburg-Landau equation has a unique solution \(u\) such that \(\| u\| _{L^4} < a\) and \[ \lim_{R\to\infty}\frac{1}{R}\int_{B_R} \left| \nabla u - iu\frac{x}{| x| } + 2i(2\pi i)^{\frac{n-1}{2}} \frac{x}{| x| }f(\frac{x}{| x| }) \frac{e^{-i| x| }}{| x| ^{\frac{n-1}{2}}}\right| ^2\,dx = 0. \] For \(n=4\), a solution in \(L^q\) can be obtained for \(q \in [4,30/7]\). The proof relies on the contraction principle and a priori estimates for the Helmholtz operator \(F \mapsto (\Delta + I)^{-1}F\), and the Fourier transform operator \[ \widehat{f\,d\sigma(x)} = \int_{S^{n-1}} e^{-ix\cdot s}f(s)\,ds, \] with \(\sigma\) a singular measure on \(S^{n-1}\).
scientific article; zbMATH DE number 2051439