Radial non-positive solutions for nonlinear equation \(\Delta u+| x| ^{\ell}| u| ^{p-1}u=0\) on the ball
The author studies the radial solutions of the nonlinear boundary value problem \[ (P)\quad \Delta u+| x|^{\ell}| u|^{p- 1}u=0\quad in\quad \Omega =\{x| \quad | x| <1\}\subset {\mathbb{R}}\quad n;\quad u=0\quad on\quad \partial \Omega, \] where \(n\geq 3\), \(\ell \geq 0\) and \(p>1\). The result proved is the following. If \(p\in (1,(n+2+2\ell)/(n-2)),\) for each positive integer k there exists a unique radial solution \(u=u(r)\) of (P), such that \(u(0)>0\) and u(r) has exactly k-1 zeros in(0,1). If \(p\in [(n+2+2\ell)/(n-2),+\infty),\) there exists no radial solution of (P) except the trivial one.
scientific article; zbMATH DE number 4065507