Ground states of \(-\Delta u=f(u)\) and the Emden-Fowler equation
Radially symmetric solutions of nonlinear Poisson equations in \({\mathbb{R}}^ n\) (n\(\geq 2):\) \[ (I)\quad -\Delta u=f(u),\quad u>0\quad in\quad {\mathbb{R}}^ n;\quad u(x)\to 0\quad as\quad x\to \infty, \] are considered. The search of radial solutions of Problem I leads to the ODE problem: \[ (II)\quad u''+((n-1)/r)u'+f(u)=0,\quad 0<r<\infty,\quad u(r)>0,\quad 0<r<\infty,\quad u'(0)=0,\quad u(r)\to 0\quad as\quad r\to 0. \] The paper completes the ODE approach to (I) by placing the problem in the context of the rather more general Emden-Fowler equation of the type: \[ (III)\quad y''+a(t)f(y)=0,\quad y>0,\quad -\infty \leq A<t<\infty,\quad y(A)=0,\quad \lim_{t\to \infty}y(t)\quad exists. \] Conditions on a(t) and f(y) are formulated which guarantee the existence of solutions of Problem III. This yields in particular results about Problem II but also about related problems.
scientific article; zbMATH DE number 3979600
scientific article