A remark on the uniqueness of positive solutions to semilinear elliptic equations with double-power nonlinearities. (Q636954)

The paper studies the existence and uniqueness of positive solutions of the problem \[ \Delta u -\omega u +u^p-u^{2p-1} =0\quad\text{ in }\mathbb R^n,\quad u(x)\to 0\quad\text{as}\;| x| \to \infty. \] It is shown that the problem has exactly one positive solution for \[ p(7p-5)/[4(p+1)(2p-1)^2] \leq \omega <p/(p+1)^2. \]