On the evolution of self-interacting clusters and applications to semilinear equations with exponential nonlinearity
The elliptic boundary value problem of the form \[ \Delta u = \lambda \exp \bigl( -(u + v) \bigr) \quad \text{in } G, \qquad \int \partial u/ \partial n = M \quad \text{on } \partial G, \] is considered, where \(G \subset \mathbb{R}^ n\) is a bounded domain, \(V \in L^ \infty(G)\), \(M>0\), \(\lambda>0\) and \(u\) satisfies an additional compatible boundary condition. A free energy functional \(F\) is introduced with a constraint and its behaviour is studied. Using the method of Kazdan-Warner, a solution is obtained as a global minimizer of \(F\). Radial symmetry of solutions is also considered.
scientific article; zbMATH DE number 640327
scientific article