Remarks on a semilinear elliptic equation on \({\mathbb{R}}^ n\)
We consider the following semilinear elliptic equation \[ (1.1)\quad \Delta u-u+Q| u|^{p-1}u=0\quad in\quad {\mathbb{R}}^ n;\quad u\geq 0,\quad u\not\equiv 0\quad in\quad {\mathbb{R}}^ n,\quad and\quad u\to 0\quad at\quad \infty, \] where p is a constant satisfying \(1<p<(n+2)/(n- 2).\) (For \(n=2\), the number \((n+2)/(n-2)\) is considered to be \(\infty.)\) We study the existence of solutions of equation (1.1) with some restrictions on Q. The symmetry properties and nonexistence of solutions are investigated, we find a class of radially symmetric potentials Q for which positive nonradial solutions of (1.1) exist, and a nonexistence theorem is proved for certain radial potentials.
scientific article; zbMATH DE number 4082085
scientific article