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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

##### MSC:
 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)
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