On the construction of solutions of the Landau-Lifshitz equation (Q5933741)

Let \(\Omega\) be a smooth bounded domain in \(\mathbb{R}^2\), \(H\in \mathbb{R}^3\) and \(\gamma\in C^\infty (\Omega , S^2)\). The paper deals with the study of the stationary Landau-Lifshitz system with external magnetic field \(H\): \(\Delta u+|\nabla u|^2u-(H,u)u+H=0\) in \(\Omega\), subject to the boundary condition \(u=0\) on \(\partial\Omega\). The authors study the existence and the behaviour of small as well as large solutions to this problem. The main result of the paper may be described as follows: for any local nondegenerate maximum point \(a\) of an appropriate ``energy'' functional, there exists a solution of the above Landau-Lifshitz system which concentrates at \(a\) as \(H\) tends to zero and \(\gamma\equiv \text{const}\).