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Existence, uniqueness, and nondegeneracy of positive solutions of semilinear elliptic equations. (English) Zbl 0582.35046
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 \(u(x)=u(| x|,p),\quad T'(p)\neq 0,\quad u'(T(p),p)\neq 0.\)
Reviewer: R.Schaaf

35J65 Nonlinear boundary value problems for linear elliptic equations
35B32 Bifurcations in context of PDEs
35B35 Stability in context of PDEs
35A05 General existence and uniqueness theorems (PDE) (MSC2000)
Full Text: DOI
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