Existence, uniqueness, and nondegeneracy of positive solutions of semilinear elliptic equations
This article studies positive solutions of the Dirichlet problem \[ (1)\quad \Delta u(x)+f(u(x))=0,\quad x\in D^ n,\quad u(x)=0,\quad x\in \partial D^ n \] where \(D^ n\) is an n-ball. Since positive solutions of (1) are radially symmetric the authors consider a general ODE-problem \[ (2)\quad u''+g(u,u',t)=0,\quad u'(0)=0,\quad u(R)=0. \] For (2) the so called time-map T(p) is defined on the set \(A=\{p\in R^+:\quad u(t,p)=0\) for some \(t>0\}\) via \(T(p)=\min \{t>0:\quad u(t,p)=0\}.\) (u(t,p) is the solution to (2) with \(u(0,p)=p.)\) u is a positive solution of (2) iff \(u=u(\cdot,p)\), \(p\in A\) and \(R=T(p)\). Uniqueness follows from A being connected and \(T'(p)>0\) or \(<0\). For \(g(u,u',t)=(n-1)/t\cdot u'+f(u)\) the authors give \((3)\quad (f(u)/u)'<0\) as a condition for \(T'>0\), and \((4)\quad (f(u)/u)'>0,\quad f''(u)<0\) for \(T'<0.\) For existence, conditions on the growth of f at \(+\infty\) are given for which \(A\neq \emptyset\), like \(f(u)/u^ k\to 1\) for \(u\to \infty\) and \(2/(n-2)>k.\) Proofs use comparison and phase plane arguments. As the authors point out the usual method of finding sub- and supersolutions does not apply if e.g. (4) holds for f, since then all positive solutions are unstable. Nondegeneracy of a solution u (i.e. 0 is not in the spectrum of the linearization of (1) about u) is shown to be equivalent to \(u(x)=u(| x|,p),\quad T'(p)\neq 0,\quad u'(T(p),p)\neq 0.\)
scientific article; zbMATH DE number 3931550