Singular solutions for some semilinear elliptic equations
This paper studies solutions \(u\in C\) \(2(B_ R\setminus 0)\) of the equation \(-\Delta u+u\) \(p=0\), \(u\geq 0\) on \(B_ R\setminus 0\), the dimension of the underlying space being N. When \(1<p<N/(N-2)\) there are solutions with a singularity at 0. The presented paper shows that if non- negative smooth Dirichlet boundary data are prescribed there is a unique solution which is \(C^ 2\) on the closure of the domain. Moreover a unique solution can be specified satisfying the boundary conditions by giving an asymptotic form for the singularity at 0. The uniqueness statement as a consequence of the imposition of boundary data extends work of Veron. In addition the present paper removes the restriction to a ball and obtains the analogous results for any smooth domain.
scientific article; zbMATH DE number 4034217
scientific article